SPHERIC – Smoothed Particle Hydrodynamics

European Research Interest Community

SECOND INTERNATIONAL

WORKSHOP

Escuela Técnica Superior de Ingenieros Navales,

Universidad Politécnica de Madrid. Madrid, May 23rd-25th, 2007,

DEDICATED TO THE MEMORY OF

DR. LARRY LIBERSKY

Edited by Alejandro J. C. Crespo

Moncho Gómez-Gesteira Antonio Souto-Iglesias

Louis Delorme José María Grassa

Title page designed by Pedro Cid Samamed © aica ediciones Parque Tecnológico de Galicia 32009 San Ciprián de Viñas Ourense I.S.B.N.: XX-XXX-XXXX-X Depósito Legal: OU-XX/2007 Digital Printing: Oficode S.L.

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SPHERIC 2nd INTERNATIONAL WORKSHOP Universidad Politécnica de Madrid, Spain

May 23- 25, 2007

Organizing Committee: • M. Gómez- Gesteira (University of Vigo, Spain)

• A. Souto Iglesias (UPM, Spain)

• J.M. Grassa (CEDEX, Spain)

Scientific Committee:

• D. Violeau (EDF R&D, France)

• N. Quinlan (Nat. Univ. Ireland, Galway)

• B. Rogers (University Manchester, UK)

• E. Parkinson (Andritz VA TECH Hydro, Switzerland)

• D. Graham (University of Plymouth, Uk)

• P. Groenenboom (ESI BV, Netherlands)

• D. Le Touzé (Ecole Centrale de Nantes, France)

• M. Gómez Gesteira (University of Vigo, Spain)

• A. Panizzo (University of Rome, La Sapienza, Italy)

• R. Klessen (University of Heidelberg, Germany)

• A. Colagrossi (INSEAN, Italy)

• J. Favre (CSCS, Switzerland)

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Local Organizing Committee:

• L. Delorme (UPM, Spain)

• L. Pérez Rojas (UPM, Spain)

• N. Carette (UPM, Spain)

• A. J. C. Crespo (University of Vigo, Spain)

• M. deCastro (University of Vigo, Spain)

• I. Álvarez (University of Vigo, Spain)

• N. Lorenzo (University of Vigo, Spain)

• B. Diaz (University of Vigo, Spain)

Sponsors:

• Universidade de Vigo

• Universidad Politécnica de Madrid

• Escuela Técnica Superior de Ingenieros Navales

• Colegio Oficial de Ingenieros Navales y Oceánicos de España

• ERCOFTAC

• CEDEX

V

Dr. Larry Libersky

Austin 1946 – Los Alamos 2007 (picture taken during

first SPHERIC Workshop, Rome, May 2006)

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VII

A Professional Remembrance of Larry D. Libersky

Larry completed his PhD at New Mexico Institute of Mining and Technology in Physics, 1979. His thesis, “The Nature of Turbulence in Cumuli”, was written under the guidance of Prof. Al Petschek. As part of this work he studied improved Eulerian advection schemes, applying these to the behavior of turbulent clouds in the atmosphere.

After graduating, Larry worked as computational team leader with Pat Buckley and Per-Anders Persson at NM Tech's centers for high explosives research, including TERA, CETR and EMRTC. While there he used his experience in Eulerian hydrodynamics to model weapons effects physics, high explosive reaction physics and high rate solid mechanics. Larry also served as a consultant to the Air Force Weapons Laboratory where he worked with Ray Bell on fireball dynamics and cloud rise.

It was during Larry’s time at NM Tech that a key moment in his professional life occurred – Larry attended a lecture on a new numerical technique, Smooth Particle Hydrodynamics or SPH, which had recently been developed to simulate problems in astrophysics. This introduction was to lead, over time, to a major focus of Larry's research – the application of particle methods to high-energy physics, fracture, fragmentation, and material damage.

Starting in the early 90s, Larry began working with long-time collaborators Phil Randles and Ted Carney, who along with Al Petschek, worked to develop the Smooth Particle Hydrodynamics for material fracture and damage. The publication of “Smooth particle hydrodynamics with strength of materials’ was one of the earliest examples of the use of SPH for solid mechanics. This work allowed for simulation of fracture and fragmentation problems which continue to prove challenging to competing techniques. During this time, limitations of traditional SPH formulations in strength modeling became apparent, leading Larry and collaborators to actively pursue improved numerical formulations based on normalized SPH kernel estimates.

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In 1998, Larry moved from EMRTC to work at Los Alamos National Laboratory. During Larry's time at Los Alamos, deficiencies in traditional SPH led to the co-development with Phil Randles of DPD – Dual Particle Dynamics. Based on a key suggestion by Carl Dyka, the introduction of a second set of particles, “stress points”, provides a non-collocated particle method. Following the experience of finite elements for solid mechanics, this formulation greatly reduces the presence of spurious solution modes. DPD has allowed for greatly improved simulation of material failure under extreme deformation within a purely Lagrangian framework.

While actively engaged in the development of DPD, Larry remained open to improving traditional particle techniques. He continued to study the use of Moving Least Squares interpolation for collocated methods, and was particularly interested in the use of Hugoniot solvers applied with shock particle simulations.

Among Larry's many other achievements, he was a talented musician as guitarist and jazz trumpeter, a scratch golfer and a fiercely competitive basketball player. Larry’s personal beliefs lead him to write many articles on the relationship between science and religion (www.mountainofgod.com). Larry is survived by his wife Dee, his mother Evelyn, and his children Jason Libersky, Seth Brown and Rachel Ortiz.

Sadly, any summary of this kind will be incomplete, failing to mention the countless collaborators and colleagues who have benefited from knowing Larry. I am grateful for the time I have spent working with Larry over the last few years, and will miss him as both a mentor and as a friend.

Andrew Brydon, Los Alamos, March 2007.

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A Remembrance of Larry D. Libersky

Dr. Libersky has pioneered the development of Smoothed Particle Hydrodynamics (SPH) towards application to the dynamic response of solids making possible some of the first computer simulations of fracture and fragmentation within a continuum modeling framework. The three-dimensional SPH code, MAGI, written by Libersky and coworkers has been used to predict, for the first time and from first principles, the size distributions of material fractured under high strain rate loading. These highly successful simulations constitute a new computational approach which holds promise to explain phenomena important to a broad range of applications currently beyond the scope of numerical simulation using traditional methods. Dr. Libersky has co-authored ten papers in refereed journals since 1993, including two invited papers for special issues on “Meshless Methods” in Computer Methods in Applied Mechanics and Engineering.

I met Larry for the fist time in Freiburg at the HVIS in 1996. His work on the development of SPH and his enthusiasm for meshless methods initiated my interest in SPH. Since the first meeting he became a close friend and colleague who encouraged and helped with the development our SPH research, including a Los Alamos Cranfield University collaboration agreement. As a part of this arrangement Larry was a frequent and welcome visitor at Cranfield. In 2002 he spent six months as a visiting scientist at Cranfield. The invigorating discussions during long walks in the mountains around his beloved Magdalena and Los Alamos will always stay with me. In addition to being a valued professional colleague, Larry and his family became good friends and he will be missed.

Rade Vignjevic, Cranfield, April 2007.

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About SPHERIC

Smoothed Particle Hydrodynamics (SPH) was developed to study non-axisymmetric phenomena in astrophysics. Yet, a number of developments based on this approach have been launched by various research teams in association with industry. Fields like free-surface flows, where Eulerian methods can be difficult to apply, represent a very high potential of applications (waves, impact on dams, off-shore...) as the meshfree technique facilitates the simulation of highly distorted fluids/bodies. Furthermore, with the ever increasing size and cost reduction of computer clusters, parallel simulations allow large-scale simulations that were previously limited to mainframes.

Following the impulse generated by a collection of local initiatives, a need of fostering and clustering efforts and developments has been identified. The goal of SPHERIC is indeed to foster the spread of this simulation method within Europe (& abroad). It will form a framework for closer co-operation between research groups working on the subject and serve as a platform for the information exchange from science to industry. One of the most important goals is the assessment of this method for all its possible applications and its development. With an emphasis on validation, it will then be possible to consider transfer of information, and thereby technology, to companies not currently engaged in the technology.

These goals can best be achieved through regular workshops and collaborations. The workshops will be organized to serve the following goals

• To develop the basic scientific concepts, including parallelism and post-processing methods.

• To communicate the experience in the application of the technology.

• To foster communication between industry and academia.

• To discuss currently available as well as new concepts.

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• To give an overview of existing software and methods.

• To define and run benchmark test cases.

SIG workshops will not be limited to CFD applications, as one of the goals of the organization is to benefit from the experience already gained in other areas such as structural mechanics as fluid-structure is one of the natural applications of Smoothed Particle Hydrodynamics methods.

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The following laboratories are part of SPHERIC

Research and Development

• Ecole Centrale de Lyon (France) • Ecole Centrale de Nantes (France) • Universidad Politécnica de Madrid (Spain) • Universidade de Vigo (Spain) • Universidad Nacional de Educacion a Distancia, Madrid

(Spain) • Novosibirsk State University (Russia) • National Univeristy of Ireland, Galway • University of Manchester (U.K.) • University of Plymouth (U.K.) • Swiss National Supercomputing Centre - CSCS

(Switzerland) • Ecole Polytechnique Fédérale de Lausanne - EPFL

(Switzerland) • Swiss Federal Institute of Technology, ETH Zurich,

(Switzerland) • INSEAN (Italy) • L'Aquila University (Italy) • The Johns Hopkins University (U.S.A.) • University of Nottingham (U.K.) • University of Sydney (Australia) • North China Electric Power Station (P.R.C.) • University of Lancaster (U.K.) • ESI BV (The Netherlands) • University of Heidelberg • Université de Montpellier (France) • Groupe-Conseil LaSalle, Québec (Canada) • CSIRO Mathematical and Information Sciences (Australia) • Shanghai Jiao Tong University (P.R.C.) • Université du Havre (France) • Université de Rennes 1 (France) • Université de Savoie (France) • CIMNE Barcelona (Spain) • University of Palermo (Italy) • University of Genova (Italy)

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• CEDEX (Spain) • University of Pavia (Italy) • Dublin Institute of Technology (Ireland) • Imperial College London (U.K.) • Institute for Plasma Research (India) • University of Umea (Sweden) • Institute Francais du Pétrole (France)

Industry

• Electricité de France (EDF) • VA TECH HYDRO ANDRITZ (Switzerland) • CESI Ricerca S.p.A. • BAE Systems, U.K.

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CONTENTS

SPH2D Simulation: Validation and Accuracy to Experiments Using Different Code Compiling Options (Benchmark Test Case 5). Crespo, A. J. C., Gómez-Gesteira, M. and Dalrymple, R. A.

1

Special, hardware accelerated, parallel SPH code for galaxy evolution. Berczik, P., Nakasato, N., Berentzen, I., Spurzem, R., Marcus, G., Lienhart, G., Kugel, A., Maenner, R., Burkert, A., Wetzstein, M., Naab, T., Vasquez, H. and Vinogradov, S. B.

5

SPH simulation of sediment scour in reservoir sedimentation problems. Falappi, S., Gallati, M. and Maffio, A

9

High-velocity impact simulation by a hybrid SPH-FE method in PAM-SHOCK. Groenenboom, P.

13

Wave interactions with coastal structures: quantitative predictions using SPH. Issa, R., Lee, E-S., Violeau, D., Gariah, A., Stansby, P. and Laurence, D.

15

Hybridation of generation propagation models and SPH model: application to realistic dimensions. Crespo, A. J. C., Gómez-Gesteira, M. and Dalrymple, R. A.

19

On the use of an alternative water state equation in SPH. Molteni, D., Colagrossi, A. and Colicchio, G.

23

Applications of Generalised Smoothed Particle Hydrodynamics to Benchmark CFD Problems. Ha, Joseph

27

Numerical calculations of flow through an orifice conduit. Klapp, J., Sigalotti, L. Di G., Galindo, S. and Duarte, R.

32

SPHERIC Test Case 6: 2-D Incompressible flow around a moving square inside a rectangular box. Lee, E-S., Violeau, D., Laurence, D., Stansby, P. and Moulinec, C.

37

Smoothed Particle Hydrodynamics with radiative transfer in the flux-limited diffusion approximation. Whitehouse, S.

42

SPHERIC benchmark test case number 5: sensitivity analysis to numerical and physical parameters. Violeau, D., and Issa, R.

47

Interactive Visualization and Exploration of SPH Data. Biddiscombe, J., Graham, D., Maruzewski, P.

51

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Pressure measurement in 2D sloshing simulations with SPH. Delorme, L., Celigueta, M.A., Oñate, E. and Souto-Iglesias, A.

55

Enforcing boundary conditions in SPH applications involving bodies with right angles. Colagrossi, A., Colicchio, G. and Le Touzé, D.

59

An FPGA-based hardware coprocessor for SPH computations. Marus, G., Lienhart, G., Kugel, A., Mäenner, R. , Berczik, P., Spurzem, R. Wetzstein, M., Naab, T., Burkert, A.

63

Challenges related to particle regularization in SPH. Børve, S., Speith, R., Omang, M. and Trulsen, J.

67

SPH with Improved Ghost Particle Boundary Treatment. Yildiz, M.and Suleman, A.

71

Coupling between roll motion and 2D sloshing. Delorme, L., López-Pavón, C., Botia, E. and Zamora-Rodríguez, R.

75

A New Stable and Consistent Version of the SPH Method in Lagrangian Coordinates. Ferrari, A., Dumbser, M., Toro, E. F. and Armanini, A.

79

Degradation and Instability of Incompressible SPH Computations of Simple Viscous Flows. Hughes, J. P. and Graham, D. I.

83

SPH for Cold and Low Viscous Shear Flow. Imaeda, Y., Tsuribe, T. and Inutsuka, S.

87

SPH simulation of moderate Reynolds number flows. Lobovsky, L. and Vimmr, J.

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Extension of the Finite Volume Particle Method to Higher Order Accuracy and Viscous Flow. Nestor, R. M. and Quinlan, N. J.

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A new parallelized 3D SPH model: resolution of water entry problems and scalability study. Oger, G., Le Touzé, D., Alessandrini, B. and Maruzewski, P.

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Symmetry assumptions in SPH. Omang, M., Børve, S., Christensen, S. O. and Trulsen, J.

103

DNS SPH simulation of 2D wall-bounded turbulence. Robinson, M., Monaghan, J. and Mansour, J.

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Development of a Parallel SPH code for free-surface wave hydrodynamics. Rogers, B. D., Dalrymple, R. A., Stansby, P. K. and Laurence, D. L.

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SPH analysis of a planing surface. Savio, L., Brizzolara, S. and Viviani, M.

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Water wave propagation using SPH models. Guilcher, P. M., Ducrozet, G., Alessandrini, B. and Ferrant, P.

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Exact Kernel Integration in SPH. Basa, M., Quinlan, N. J. and Lastiwka, M.

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Some Industrial SPH Applications Undertaken at the BAE Systems Advanced Technology Centre. Banim, Robert

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Modeling Star Formation with SPH. Klessen, R. S. and Clark, P. C.

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Smoothed particle hydrodynamics model for multiphase flow in porous media. Tartakovsky, A. M., Meakin, P. and Ward, A.

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Reactive transport and biomass growth in porous media. Tartakovsky, A. M., Meakin, P. and Scheibe, T. D.

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SPH Modeling of Forced Water Waves. Narayanaswamy, M., Frandsen, J. and Dalrymple, R. A.

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An SPH model of wave breaking: quantitative comparisons to laboratory observations. Ely, A.C. and Swan, C.

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Dynamic Boundary Particles in SPH. Crespo, A. J. C., Gómez-Gesteira, M. and Dalrymple, R. A.

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Accuracy and stability of numerical schemes in SPH. Capone, T., Panizzo, A., Cecioni, C. and Dalrymple, R. A.

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Application of SPH-NSWE to simulate landslide generated waves. Del Guzzo, A. and Panizzo, A.

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A new treatment of solid boundaries for the SPH method. Marongiu, J.C., Leboeuf, F. and Parkinson E.

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SPH simulation of local scour processes. Sibilla, S. 169

An incompressible multi-phase SPH method. Hu, X. Y. and Adams, N. A.

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Multi-phase and Multi-material Flow Modelling using Smoothed Particle. Ha, J., Cleary, P. W., Prakash, M. and Sinnott, M.

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SPHERIC

1

SPH2D Simulation: Validation and Accuracy to Experiments Using Different Code Compiling Options

(Benchmark Test Case 5)

A. J. C. Crespo1, M. Gómez-Gesteira1 and R.A. Dalrymple2

1 Grupo de Física de la Atmósfera y del Océano, Universidad de Vigo, Ourense, Spain;

alexbexe@uvigo.es; mggesteira@uvigo.es 2 Department of Civil Engineering, Johns Hopkins University, Baltimore, USA; rad@jhu.edu

Abstract

Dam break evolution over dry and wet beds is analyzed in the framework of SPH (Smoothed Particle Hydrodynamics) model. The model showed to fit accurately both the experimental dam- break profiles and the measured velocities. In addition, the model allows using different compiling options, so several tests will be analyzed to look for the best validation.

1. Introduction Models based on Smoothed Particle

Hydrodynamics (SPH) are an option to address dam break evolution. SPH is a purely Lagrangian method developed during seventies (Lucy, 1977; Gingold and Monaghan, 1977; Monaghan, 2005) in astrophysics to study the collision of galaxies and the impacts of bolides on planets. The numerical method has been shown to be robust and applicable to a wide variety of other fields. Recently, SPH has been used for wave impact studies on offshore structures (Dalrymple et al., 2002; Gómez-Gesteira and Dalrymple, 2004; Gómez-Gesteira et al., 2005; Crespo et al., 2007).

The aim of this paper is the study of the dam break experiment and the effect of wet bottom in his evolution by means of the SPH model. Different compiling options of the code will be used to reproduce the experiment in order to find which options

improve the accuracy of the simulations compared to experiments.

2. The Experiment Here we use the experiments by

Janosi et al. (2004) to validate the SPH model of dam break evolution over a wet bed. The schematic arrangement of their experimental tank, which has two parts, is shown in Fig. 1. The channel, beginning at x=38cm, is 955 cm long and 15 cm wide. The bottom and side walls of the channel were constructed with glass, the second part, comprising the lock and lock gate, is 38 cm long and made from Plexiglas. The initial fill height of the lock (d0) for our comparisons is taken as 0.15m.

Figure 1: Schematic arrangement and

geometric dimensions of the dam-break experiments.

The position of the water front as a

function of time was determined from digitized pictures. The gate separating the lock from the rest of the tank was removed from above at an approximate constant velocity (Vgate = 1.5 ms-1). The movement of this gate will be shown to play a key role

2

when fitting numerical results to experimental ones, since the gate velocity is on the same order of magnitude as the wavefront celerity.

3. SPH Methodology The main features of the base case

(TEST 1 from now on) will be described in this section: - The cubic spline kernel developed by Monaghan and Lattanzio (1985) was used in our simulations. - The momentum equation given by Monaghan (1992) was used. - The artificial viscosity term given by Monaghan (1992) was calculated using α= 0.08 and β= 0. - Changes in the fluid density were calculated by means of the differential equation given by Monaghan (1992). - The relationship between pressure and density was assumed to follow the equation of state according to Batchelor (1974). - Particles were moved using the XSPH variant due to Monaghan (1989). - Fluid particles were initially placed on a staggered grid with zero initial velocity. - Dynamic boundaries (treated as quasi fluid particle, see Gómez-Gesteira et al., 2005), were used to mimic the boundaries. - The Verlet algorithm (Verlet, 1967), was used in our numerical simulations. - A variable time step ∆t was calculated according to Monaghan (1992)

4. Model Validation

4.1. Wave Profiles

Experimental wave profiles (Janosi et al. 2004) were digitized in order to be compared with SPH profiles. The dimensions of the digitized snapshots are 0.38 m ≤ X ≤ 1.04 m and 0.0 m ≤ Z ≤ 0.13 m. Distances were measured from the left-lower corner of the tank.

In Figure 2, d= 0.038 m was considered to compare numerical results and experiments. Experimental values are represented by dots and SPH values by a line. The model is observed to reproduce the experimental profile. In the first snapshot, the water initially placed behind the gate pushes the still water; second and third snapshots show the wave propagation and breaking.

Figure 2: Comparison between experimental and numerical profiles of dam-break evolution over a wet bed

(d=0.038m).

Apart from this visual comparison, the observed difference between numerical and experimental results can be quantified considering the RMS error. 4.2. Wave Velocity

The experimental and numerical velocities were averaged along the first 3 meters of the tank (Figure 3). Numerically, the position of the leading edge was calculated every 0.06 s and velocity was obtained by linear fitting. Both velocities and distances are depicted in a dimensionless form. Velocity and water depth (Fig. 1) are normalized considering d/d0 and VN=<v>/c, where

0gdc = .

3

0dd

0.6

1

1.4

1.8

-0.1 0.1 0.3 0.5 0.7

NV

0dd

The normalized velocity is observed to decrease with d. The agreement between experimental measurements (light dots) and numerical results (dark squares) is excellent in most of the cases. Note that SPH velocity for dry bed is higher than observed in experiments, since experiments were not performed on a real dry bed, due to the impossibility of drying completely the tank.

Figure 3: Comparison between experimental and numerical dam-break

velocity.

5. Comparison with experimental results

Different tests have been simulated to

compare the results obtained using different compiling options in the SPH code. The features described in section 3 correspond to TEST 1. Only one of the options has been changed in the other four cases as shown in the next table:

TEST 1 TEST 2

ALGORITHM Verlet Predictor-Corrector

KERNEL Cubic Spline

Cubic Spline

VISCOSITY Artificial Artificial BOUNDARY CONDITION

Dynamic Boundaries

Dynamic Boundaries

TEST 3 TEST 4 TEST 5 Verlet Verlet Verlet

Gaussian Cubic Spline

Cubic Spline

Artificial Laminar Artificial Dynamic

Boundaries Dynamic

Boundaries Repulsive

Forces

Table 1: Different test cases to simulate Janosi experiment.

Different parameters are analyzed to

find the best accuracy to Janosi experiment (considering the case d=0.018m) using the five tests. In the figure 4, the time step (a), the RMS error (b) and VN, ratio between the SPH velocity and experimental one, (c) are represented.

1 2 3 4 5

0.5

1

1.5x 10-4

∆t (

s)

1 2 3 4 50.0

0.005

0.01

0.015

RM

S

1 2 3 4 50.8

0.9

1

1.1

test

V N

(a)

(b)

(c)

Figure 4: Test results.

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TEST 1 and TEST 2 take the longest time step, so the run time used is lower than in the other tests. The RMS error is calculated comparing numerical wave profiles with experimental ones. The minimum errors are obtained using TEST 1 and TEST 4. The dam-break velocity in Fig. 4c is normalized with the velocity measured in the experiment d=0.018m (v=1.494 m/s). Once again, note that TEST 1 and TEST 4 show the best results.

6. Conclusions The 2D version of the SPH model has

proven to be a suitable tool to reproduce a dam break evolution over dry and wet beds. Experimental profiles and horizontal velocities are properly reproduced by the model. In conclusion, but only for this particular case (dam-break evolution over a wet bed), the experiment is properly simulated (with the longest time step) by SPH model using a Verlet algorithm, a cubic spline kernel, artificial viscosity (α=0.08) and dynamic boundaries.

7. Acknowledgments This work was partially supported by

Xunta de Galicia under proyect PGIDIT06PXIB383285PR.

8. References BATCHELOR, G. K. (1974). Introduction to fluid dynamics. Cambridge University Press. U.K. DALRYMPLE, R.A., KNIO, O, COX, D.T., GÓMEZ-GESTEIRA, M., and ZOU, S. (2002). Using a Lagrangian particle method for deck overtopping. Proceedings of Waves 2001, ASCE. 1082-1091. CRESPO, A. J. C., GÓMEZ-GESTEIRA, M. and DALRYMPLE, R. A. (2007). 3D SPH simulation of large waves mitigation with a dike. Journal of Hydraulic Research. In press.

GINGOLD, R. A. and MONAGHAN, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non- spherical stars. Mon. Not. R. Astr. Soc., 181: 375- 389. GÓMEZ-GESTEIRA, M. and DALRYMPLE R. A. (2004). Using a 3D SPH Method for Wave Impact on a Tall Structure. J. Wtrwy. Port, Coastal and Ocean Engrg. 130(2): 63-69. GÓMEZ-GESTEIRA, M., CERQUEIRO, D., CRESPO, C. and DALRYMPLE, R. A. (2005). Green water overtopping analyzed with a SPH model, Ocean Engineering. 32: 223-238. JANOSI, I. M., JAN, D., SZABO, K. G. and TEL, TAMAS. (2004). Turbulent drag reduction in dam-break flows. Experiments in Fluids, 37: 219-229. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. MONAGHAN, J. J. and LATANZIO, J.C. (1985). A refined method for astrophysical problems. Astron. Astrophys, 149: 135- 143. MONAGHAN, J. J. (1989). On the Problem of Penetration in Particle Methods. Journal Computational Physics, 82: 1-15. MONAGHAN, J. J. (1992). Smoothed particle hydrodynamics. Annual Rev. Astron. Appl., 30: 543- 574. MONAGHAN, J. J. (2005). Smoothed particle hydrodynamics, Reports on Progress in Physics, 68: 1703-1759. VERLET, L. (1967). Computer Experiments on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Phys. Rev. 159: 98-103.

SPHERIC

5

Special, hardware accelerated, parallel SPH code for galaxy evolution.

Peter BERCZIK 1,5 , Naohito NAKASATO 2, Ingo BERENTZEN 1 , Rainer SPURZEM 1,

Guillermo Marcus MARTINEZ 3 , Gerhard LIENHART 3, Andreas Kugel 3 , Reinhard Maenner 3, Andreas BURKERT 4 , Markus WETZSTEIN 4, Thorsten NAAB 4 , Hipolito VASQUEZ 4,

Stanislav B. VINOGRADOV 5

1 Astronomisches Rechen-Institut, Zentrum fur Astronomie Univ. Heidelberg, Monchhofstrasse 12-14, 69120 Heidelberg, Germany

2 Department of Computer Software, University of Aizu, Aizu-Wakamatsu, 965-8580, Japan 3 Department for Computer Science, Univ. Mannheim, Quadrat B6, 26, D-68131 Mannheim Germany

4 University-Observatory, Ludwig-Maximilian University, Scheinerstr. 1, D-81679 Munich, Germany 5 Main Astronomical Observatory, National Academy of Sciences of Ukraine, Zabolotnoho Str., 27,

Kiev, Ukraine, 03680

Abstract We present our first results from the recently developed parallel 3D SPH dynamical code for galaxy evolution. It follows the evolution of all basic components of a galaxy such as dark matter, stars, diffuse interstellar matter (ISM). Dark matter and stars are treated as collision less N-body systems. The ISM is numerically described by a smoothed particle hydrodynamics (SPH) approach. We perform our simulations on the recently built 32 node GRACE cluster at the Astronomisches Rechen-Institut. This system is a new type of supercomputer based on a standard PC's with GRAPE and a new kind of programmable special hardware (FPGA) cards calls MPRACE: http://www.ari.uni-heidelberg.de/grace/. The gravitational forces calculated using the combined parallel TREE-GRAPE algorithms which give us the expected speed ~20 GFlops per node. Pipelines and pipeline tools for MPRACE have been developed for SPH forces and as planned

are performing with the expected speed of ~5 GFlops per board.

1. Introduction In present day astrophysical

community the SPH technique is one of the most popular tools for 3D hydro-dynamical calculations. We present here our first attempt to integrate in one astrophysical SPH code the two new hardware techniques: GRAPE and FPGA.

The main idea in our code is to use simultaneously the GRAPE cards for the self-gravity calculation of the gas particles and the FPGA (own developed MPRACE) cards for the gaseous forces calculations between them. This basic concept, which we call, GRACE (GRAPE and MPRACE) already described in few papers: Spurzem et al. 1999; Spurzem & Kugel 1999; Spurzem et al. 2002.

In our present code we use the GRAPE6a hardware, which are a single PCI 32 bit/33 MHz card (4 chip) version of the GRAPE6 gravity calculation accelerator (Fukushige et al. 2005). Such hardware is specially designed as a very suitable tool for parallel usage of GRAPE cards in a many node PC cluster systems.

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Basic working concept of our code we can show in a next figure:

Figure 1: Common work of CPU + GRAPE + FPGA processing units in our SPH code.

Our own developed MPRACE PCI card based on 4 Xilinx Virtex II FPGA chip with the memory which can store ~250K particles. The library of the floating point arithmetic for our card has a 16 or 24 bit mantissa. We use for the standard SPH pipeline coding also the own developed high level programming language.

Each of the 32 host computers in our cluster has a double Intel Xeon 3.2 GHz CPU.

2. Code description We develop simultaneously two

version of the code. One serial and one parallel (under MPI library) both in ANSI-C languages.

For the self-gravity calculation of gas we use the very popular self coded TREE-GRAPE scheme (Makino 2004; Makino 1991; Fukushige et al. 2005). Such a scheme allows us to have a very fast self gravity calculation routine up to few million particles per single node.

The present version of our TREE-GRAPE code is shows a relatively

accurate force calculation for our SPH particles with the quite conservative opening angle θ=0.5 with the standard Plummer gravitational softening parameter ε=0.01.

After some “fine tuning” of our code we find such an optimal TREE construction parameters (see discussion in paper Fukushige et al. 2005) Nleaf=10 and Ncrit=3000. In all our further test runs we use these parameters.

As one of the standard test for our SPH code we use the so call Evrard tests. During this test we follow the adiabatic collapse of the initially “cold” gaseous sphere with radius R and mass M, with the initial density profile:

The total energy of the sphere self gravity we can easy derive from the expression:

The internal energy of the gaseous

sphere we initially set to the similar value for all SPH gas particles:

The initial velocity inside the sphere

we also set to zero. In the code we use the natural normalization units where we set:

In a next few figures we present the

speed and accuracy results of such a test for different number of particles.

rRM 1

2 2 ⋅⋅⋅

=π

ρ

1=== RMG

RMGEG

2

32 ⋅

⋅−=

RMGu ⋅

⋅= 05.0

7

Figure 2: CPU times spend for SPH calculation of one time step. The red line

shows the data if we use the CPU for SPH forces calculation. The green line shows the similar data if we use our MPRACE-1

board for SPH forces. The blue line shows the “overall” SPH speedup of the code

(~factor of 10).

Figure 3: Distribution along the radius of the absolute differences of acceleration (for 100k particles) if we use for SPH the CPU

or the MPRACE-1 board in the code.

Figure 4: The time evolution of energies during our adiabatic collapse test runs with different particle numbers (10K, 50K and 100K). The absolute error in total energy conservation during the whole period of

integration was less when ~0.1%.

Figure 5: The parallel code CPU time of one time step for different values of N and CPU numbers. The scaling of the overall

code is almost linear with NCPU for the selected range of N.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

Energy

t

EKIN

ETHE

EPOT

ETOT

10k 50k100k

0.001

0.01

0.1

1

10

100

1 4 16 64 256 1024

∆T (one shared timestep) [sec]

N [in K]

SPH on CPUSPH on MPRACE-1

ratio: CPU/MPRACE-1

10-8

10-7

10-6

10-5

10-4

10-3

0 0.2 0.4 0.6 0.8 1

|∆a

i| (CPU vs. MPRACE)

ri

0.1

1

10

100

1000

1 2 4 8 16

One timestep:

∆TCPU (sec)

Number of CPU: NCPU

N = 100k 200k 400k 800k 1000k∼NCPU

-1.15

8

3. First astrophysical results

As a quick and relatively simple astrophysical test case for our code we calculate the evolution and fragmentation of the isothermal molecular cloud – cloud collisions with different impact parameters. We repeat and significantly extend for a larger particle numbers (in some cases up to 1M SPH particles) our previous runs (see for more astrophysical detail the Vinogardov & Berczik 2006 paper).

Figure 6: The overall view of the clouds

collision and fragmentation.

Figure 7: The time evolution of the slope of the fragmented clumps cumulative mass

function: α−mmN ~)( .

4. Acknowledgments This work was supported by the Volkswagen Foundation under Grant No. I80 041-043, the Ministry of Science, Education and Arts of the state of Baden-Wurttemberg, Germany. P.B. acknowledges support from the German Science Foundation under SFB 439 (sub-project B11) “Galaxies in the Young Universe” at the University of Heidelberg.

5. References FUKUSHIGE, T. et al. (2005). GRAPE-6a: A single-card GRAPE-6 for parallel PC-GRAPE cluster system. PASJ, 57: 1009 – 1021. MAKINO, J. (1991). Treecode with a special-purpose processor. PASJ, 43: 621 – 638. MAKINO, J. (2004). A fast parallel treecode with GRAPE. PASJ, 56: 521 – 531. SPURZEM, R. et al. (1999). AHA-GRAPE: Adaptive Hydrodynamic Architecture – GRAvity PipE. arXiv:astro-ph/9906153. SPURZEM, R., and KUGEL, A. (1999). Towards the million body problem on the computer – no news since the tree-body-problem? arXiv:astro-ph/9906155. SPURZEM, R. et al. (2002). Collisional stellar dynamics, gas dynamics and special purpose computing. arXiv:astro-ph/0204326. VINOGRADOV, S.B., and BERCZIK, P. (2006). The study of colliding molecular clumps evolution. A&ApTr, 25: No.4, 299 – 316. arXiv:astro-ph/0701377.

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

1 2 3 4 5 6

Frag. slope:

α

T [in Myr]

β=0.2

N=2x4000 2x8000 2x16000 2x32000 2x64000

SPHERIC

9

SPH simulation of sediment scour in reservoir sedimentation problems

Stefano Falappi1, Mario Gallati2, Andrea Maffio3

1 PhD, Hydraulics and Environmental Dept., Pavia University via Ferrata 1, 27100, Pavia, Italy, falappi@unipv.it

2 Full Prof., Hydraulics and Environmental Dept., Pavia University via Ferrata 1, 27100, Pavia, Italy, gallati@unipv.it

3 Dr. Eng., Environment and Sustainable Dev. Dept. / CESI RICERCA, via Rubattino 54, 20134, Milano, Italy, andrea.maffio@cesiricerca.it

Abstract

This paper refers on SPH simulations of water and non-cohesive sediment scour flow in a laboratory flume schematizing a two-dimensional vertical cross-section of a reservoir, in the presence of deposited sediment bed, during dam bottom outlet opening maneuvers . In order to capture the scour phenomenon a model suitable for the treatment of granular material is implemented. Experimental data of sediment bed profile are then compared to results of SPH simulation, and discussed to focus on capabilities and limits of the proposed model.

1. Introduction Reservoir sedimentation may give rise

to safety hazards for population and infrastructures where the scheme is located, related to equipment malfunctioning (due to clogging of dam bottom outlets) and increase in dam static load. Numerical modeling may provide an useful tool to make predictive simulations of hydrodynamics and sediment transport problems associated with the above issues, to assess sediment scour and transport induced by outlet opening maneuvers for different operational strategies (e.g. reservoir elevation, opening duration, etc.). Hence, a research

activity is in progress to develop a SPH software tool to simulate local sediment scour and hydrodynamics in reservoir sedimentation problems. Reservoir sediment bed is included in the computational domain and modeled as a granular pseudo-fluid with the same set of equation used to model the water dynamics, provided a relationship is used to define and apparent viscosity (considering Mohr-Coulomb criteria) as a function of consistency and friction angle. To evaluate SPH modeling response, numerical simulations are carried out for a two-dimensional test case, based on a laboratory flume schematizing a simplified vertical section of a dam with a bottom outlet and a sediment bed. By opening the bottom outlet, flume drawdown and local scour (with sediment flushed downstream) are induced. Simulation results are compared to flume experimental measures. As research is aimed at developing tools to be used in engineering applications, emphasis is placed on evaluating SPH from the point of view of both accuracy and computational cost.

2. Numerical model The SPH model adopted here is bi-

dimensional in the vertical plane. Water is modeled as a slightly compressible Newtonian fluid in isothermal conditions, thus for each particle i the pressure p is

10

related to the density ρ through the sound speed c with the linearized state equation:

)( 02 ρρ −= ii cp (1)

The SPH numerical counterparts of the Navier-Stokes equations adopted in this work are (Morris et al. 1997):

( ) ijjij

ji Wm

DtD

∇⋅−=∑ vvρ (2)

( )g

v

v

+∂

∂++

+∇+

−=

∑

∑

rW

rm

Wpp

mDtD

ij

ij

ij

j ji

jij

ijji

jj

jj

i

ρρµµ

ρρ (3)

Deposited non-cohesive sediment bed is schematized as made up of granular material. Following the approach proposed by Gutfraind & Savage (1997), the granular material is simulated as a single phase fluid with equivalent density ρeq having a proper rheological law that includes a Mohr-Coulomb yield criterion. This approach enables the description of the granular material dynamics using (1),(2) and (3) providing the definition of an effective viscosity for each i particle as below:

S

seff I

pc

24sincos ϕϕ

µ+

= (4)

where p is the pressure of the particle, cs is the cohesion coefficient of the granular material, φ the material repose angle and I2S is the second invariant of the rate of deformation tensor as proposed by Rodriguez-Paz & Bonet (2003). The kernel function adopted in this work is the classical 3rd order spline (Monaghan, 1994). The time integration is reached by an explicit 1st order scheme in which the evaluation of the velocity is half time step staggered from the computational time of

particle position, density and pressure (Gallati & Braschi, 2003).

3. Model validation

3.1. Experiment

Experimental data provided by CESI RICERCA refer to a test case carried out in the laboratory flume shown in Fig.1. The experimental set up (0.2 m wide) is built up to model a simplified two-dimensional vertical cross-section of a reservoir, in the presence of a deposited non-cohesive (sand) sediment bed, with a downstream opening (representing a dam bottom outlet) located in the downstream right side of the flume. Sediments are composed by uniform sand with d50=0.1 mm and density ρs=2650 kg/m3; measured bed porosity is n=53%. In the initial condition the sediment bed thickness is 0.165 m and the water depth is 0.8 m. By extracting a constant discharge (7.9 l/s) from the bottom outlet (no water enters the flume upstream), scour of the deposited sediment bed is induced while flume water level is lowering. The test case represents an idealized flushing maneuver, by opening dam outlet works, inducing sediment bed scour and reservoir drawdown.

0.8

1.86 1.0

0.165 0.06

0.5

0.3150.835 0.45

Z

X

0.040.30.3

Figure 1: Flume geometry.

In Fig.2 the sediment bed profiles,

respectively, at the beginning and end of the test (t=48 s) are shown. The area between the two profiles is proportional to the sediment volume flushed downstream.

water

11

Experimetal profile of sediments

0.00

0.05

0.10

0.15

0.20

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15x [m]

h [m] Initail Profile

Profile at t=48s

Figure 2: Sediment bed profiles.

3.2. SPH simulation

The initial field is represented by a set of particles placed according to a regular grid with a step equal to 0.01 m. The total number of particles is close to 24000. The kernel scale h=0.0125 m is adopted. Sediment bed, included in the computational domain, is modeled as a fluid with an equivalent density of approximately ρeq=1780 kg/m3 (based on water and sediment densities and bed porosity). Material natural repose angle is assumed φ=45°. As stable bed slope with larger angle is evident from the experimental bed profile a certain amount of cohesion is considered (cs=1000 Pa).

To evaluate the time step, sound speed c=30 m/s and viscosity value bounded by a maximum value (µmax=500 Pa s) are assumed to ensure convergence and a reasonable time step according to Shao & Lo (2003). Boundary conditions for the solid walls are simulated using two different strategies: classical mirror particle technique for the horizontal flume surface, and fixed particles method for other solid boundaries (Gallati et al., 2006). To assign constant discharge at the downstream outlet, particle horizontal and vertical velocity components are set, respectively, equal to 0.658 m/s and 0 m/s.

As first approximation, no turbulence model is considered for the water flow (however, turbulence is not expected to play a major role, relative to flow convection, close to the flume bottom outlet where sediment scour is induced).

In Fig. 3 sequence of sediment scour and water level lowering is shown by comparing images of the experimental setup and SPH results at different times. Fair qualitative agreement is apparent. A quantitative comparison can be got by considering, respectively in Fig.4 and Fig.5, water level variation up to t=48 s and sediment bed profiles at t=48 s. The channel emptying sequence in Fig.4 is fairly reproduced. Flushed sediment volume (proportional to the area bounded by initial profiles and t=48 s profiles in Fig.5) is also reasonably captured, though differences can be identified by comparing experimental and SPH bed profiles. The SPH simulation performed with a standard PC Intel 3.0 GHz requires about 40h.

t=20s

t=45s

Figure 3: Comparison between experiment and simulation.

12

Water level

0

0.2

0.4

0.6

0.8

0 10 20 30 40 50t [s]

h [m]ExperimentSimulation

Figure 4: Water level in the channel.

Sediment profile at t=48s

0.00

0.05

0.10

0.15

0.20

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15x [m]

h [m]Initial

Experiment

Simulation

Figure 5: Comparison of sediment

profiles.

4. Conclusions SPH results are in reasonable agreement with the measured laboratory data, the model seems to be capable to reproduce the more relevant engineering quantities (e.g. flushed sediment volume). In the simulation the sediment bed is represented by a set of particles and the sediment-water interface is a simulation result, this avoids the implementation of algorithms and hydro-sedimentological relationships to define a moving boundary problem as necessary with Eulerian methods. On the other hand, this simulation strategy needs more particles with increase in the computational burden. SPH modeling extension to more realistic application in progress, possibly three-dimensional, would bring about a huge number of particles and computer time. Hence, development of a parallel version of the SPH code is also in progress. Implementation of more sophisticated SPH models (e.g. turbulence, granular material, etc.) should be a trade-off between expected accuracy improvement and application available data (in particular for actual reservoirs).

5. Acknowledgements

This work has been financed by the Ministry of Economic Development with the Research Fund for the Italian Electrical System under the Contract Agreement established with the Ministry Decree of March 23, 2006.

6. References GALLATI, M. and BRASCHI, G. (2003), Numerical simulation of the jump formation over a sill via SPH method. Proc. Int. Conf. on Fluid Flow Technologies (CMFF’03), Budapest. GALLATI, M., BRASCHI, G., FALAPPI, S. (2006), Esperienze di validazione sull’impatto di una massa fluida in un bacino, Atti del XXX Convegno di Idraulica e Costruzioni Idrauliche, Roma. GUTFRAIND, R. and SAVAGE, S.B. (1997), Smoothed particle hydrodynamics for the simulation of broken-ice fields. J. Comp. Phys., 134: 203-215. MONAGHAN, J.J. (1994), Simulating free surface flow with SPH. Journ. of Comp. Physics. 110: 399–406. MORRIS, J.P., FOX, P.J. and ZHU, Y. (1997). Modelling low Reynolds number incompressible flows using SPH, J. of Comp. Physics, 136: 214–226. RODRIGUEZ-PAZ, M.X. and BONET, J. (2003). A corrected smoothed particle hydrodynamics method for the simulation of debris flow. Numer. Methods Partial Differential Eq., 20: 140-163. SHAO, S. and LO, E.Y.M. (2003). Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources, 26: 787-800.

SPHERIC

13

High-velocity impact simulation by a hybrid SPH-FE method in PAM-SHOCK

Paul Groenenboom 1

1 PhD, ESI Group, ,Rotterdamse weg 183C, 2629HD, Delft, The Netherlands

Abstract This contribution discusses the method and the advantages of having a combination of both SPH and Finite Elements in a single simulation environment. A few examples of high-velocity impact are presented.

1. Introduction SPH is a suitable method to simulate

the dynamics of various impact phenomena without any limitations to the deformation of the materials involved. The method is less suitable for thin-walled or layered structures for which the finite element method is more appropriate. It may also be difficult to apply the correct boundary conditions to structures made up by smoothed particles. FE methods are, however, usually not able to handle impacts for which very large deformations occur, except by deleting severely distorted elements. Hence, a combination of SPH and FE within a single simulation , as available in the explicit PAM-SHOCK code, allows to use the best tools for each part. A new option within PAM-SHOCK allows that heavily distorted elements may be replaced by particles during the simulation.

2. Results and discussion

The explicit finite element code PAM-SHOCK provides material models, contact algorithms and other features to allow simulation of high-velocity impact events. The most relevant material models are also

available for the constitutive behaviour for SPH. Particles may interact with adjacent finite element structures through appropriate penalty-based contact algorithms (Campbell et al., 2000): ‘particles’ may have a sliding interface contact with, or be tied to, exterior faces of FE structures, allowing for an optimised hybrid solution. An example in 2D of this is shown in figure 1, representing several stages of the impact of a copper cylinder on a semi-infinite copper target at 2000 m/s. The projectile (blue) and the innermost part of the target is represented by particles, whereas the outer target consists of finite elements of the same material. The results are in excellent agreement with those of a pure element simulation. Figure 1: Distribution of the impactor and target (SPH/FE) at various stages.

14

An application involving large deformation in combination with thin-walled structures is bird-strike on an aircraft wing. The bird is represented by particles whereas shell elements are used to model aluminium or composite wing parts. The deformation of such a wing after being hit by a bird is shown in figure 2. Figure 2: Deformation of a wing leading edge structure after bird strike (SPH not shown).

One of the fields in which the hybrid SPH-FE method offers great advantages is for simulation of hyper-velocity impact on Wipple shields. At typical impact speeds of 10 km/s material shatters into a debris cloud and it may change phase. Hence, the projectile and the shield areas subjected to a direct impact are modelled with particles, whereas the outer regions may well be represented by finite elements. Such an approach allows simulation of such complex events even in three dimensions (Groenenboom, 1994). In some cases it may be difficult to decide where large deformations in the model are to be expected and for which particles might best be defined. An attractive alternative is to start with finite elements and when the deformation become quite large, not to remove these elements as in the erosion algorithm, but replace them by particles. This has now become available within a test version of PAM-SHOCK and has been demonstrated for penetration of a long tungsten rod into a thick plate of steel,

as shown in figure 3. Once selected element properties exceed user-defined limits, the element is replaced by a particle at the centre of gravity with the same mass and momentum. Each of these particles interacts with the remaining finite elements, and with each other. In the rod penetration example, the force between particles and finite elements exceeds the direct contact force between projectile and target, once a number of particles has been created. Although it may be possible to obtain a correct penetration depth with the erosion algorithm, an improper choice of elimination parameters may cause an unrealistic perforation. This is avoided when the elements are replaced by particles. Figure 3: Distribution of elements and particles in a quarter model section of a high-velocity penetration of a long rod.

3. Acknowledgments The bird strike analysis was performed

with the EC CRAHVI project.

4. References CAMPBELL, J., VIGNJEVIC, R. and LIBERSKY L., A contact algorithm for smoothed particle hydrodynamics, CMAME, 184: 49-65, 2000. GROENENBOOM, P.H.L., Numerical simulation of 2D and 3D hypervelocity impact using the SPH option in PAM-SHOCK, International Journal of Impact Engineering, 20 (1-5): 309-323, 1994.

SPHERIC

15

Wave interactions with coastal structures: quantitative predictions using SPH

Réza ISSA1, Eun-Sug LEE2, Damien VIOLEAU1, Asven GARIAH3, Peter STANSBY4, Dominique

LAURENCE4

1 Research Engineer, LNHE / EDF R&D, 6 quai Watier 78400 Chatou, France 2 PhD student, SMACE / The University of Manchester,

PO Box 88, Manchester, M60 1QD, UK 3 MSc student, LNHE / EDF R&D, 6 quai Watier 78400 Chatou, France

4 Professor, SMACE / The University of Manchester, PO Box 88, Manchester, M60 1QD, UK

Abstract Breaking waves can run up at the shoreline, inundating coastal regions and causing large property damage and loss of life, as regularly pointed out by extreme events occurring all around the World such as storm surges or tsunamis. In order to proceed to the design of sea defense structures, interactions with waves must be well understood and predicted. However, due to the mathematical difficulties caused by the complexities of the fluid motion such as wave runup, reflection, overtopping, etc., a fully theoretical approach is not possible. Thus most of the investigations regarding breaking waves are experimental and numerical. The SPH code Spartacus-2D, developed at EDF R&D, is herein used to model several phenomena related to waves. Simulations are compared with experiments, showing satisfactory agree-ment in all cases.

1. Introduction Improving the design of sea defenses and coastal protections mainly relies on a precise understanding and modelling of wave-structure interactions such as wave runup, overtopping and forces due to wave action on structure’s elements. The

complexity of processes involved in these phenomena, as well as the wide variety of shapes allowed to dykes or breakwaters (e.g. multiple berms, varying slopes, crown walls, etc.), most often lead coastal engineers to investigate these problems through experiments and/or numerical simulations. The Lagrangian numerical method SPH is well suited to model complex free-surface turbulent flows, due to its meshless feature. The SPH code Spartacus-2D is herein used to simulate four test cases related to wave structure interaction:

-Solitary wave propagation and breaking on a gentle slope;

- Wave shoaling on a beach; - Wave setup on a coral reef; - Wave overtopping over a dyke.

2. The Spartacus-2D code The Spartacus-2D code, developed at EDF R&D, is based on the classical SPH model proposed by Monaghan (1994) for weakly compressible free surface flows. It resolves conservative form of Navier-Stokes equations and includes the modelling of turbulent effect through several models presented in details and validated by Violeau and Issa (2007). Pressure is estimated from a state equation based on a numerical speed of sound. Traditional

16

spline-based kernels of order 3, 4 or 5 are implemented and solid boundaries are discretized with wall and fictitious particles whose motion is prescribed by the user in case of moving walls.

3. Applications

3.1. Solitary wave

Modeling of solitary wave generation and propagation is at first considered. An horizontal wave flume of L=15 m length initially contains water at depth d=0.5 m. The motion of the wave board (see figure 1) is set up according to the theory described in Hughes (1998) in order to generate a H=0.2 m height wave.

Figure 1: System geometry relative to

solitary wave generation. Classical SPH equations are discretized using approximately 310 000 particles. Since wave generation and propagation are mainly advection and pressure dominated, no turbulence model is here considered. Wave water volume (V_inf) and a theoretical wave length (L_95) are compared to theoretical values at t=3.42 s, after the wave board stopped, ant at t=5.40 s, before the wave impacts the right boundary. Table 1 reveals that Spartacus-2D is very accurate while generating a solitary wave even if numerical diffusion appears during propagation stage. Indeed, figure 2, representing wave surface elevation numerically and analytically obtained at several times, shows that slight outphasing increases with time between Spartacus-2D and the theory, as well as a slight amplitude decreasing is observed. However the accuracy is satisfactory enough to attest the ability of Spartacus-2D to generate a solitary wave and represent

its propagation.

Figure 2: Free surface elevation computed by Spartacus-2D and compared to theory

at several times.

t=3.42 s t=5,40 s Relative error on V_inf 0.0 % 1.9 % Relative error on L_95 0.96 % 2.81 % Table 1: Free surface elevation computed

by Spartacus-2D and theory.

3.2. Wave shoaling

When approaching the coastline, the shape of water waves is modified by the change in bottom level. This phenomenon, referred to as “wave shoaling”, is repro-duced here following the experimental setup carried out at the University of Manchester by Stansby and Feng (2004) in a wave flume of 13 m length, 0.5 m height and 0.3 m width, with a beach slope of 1:20 and a trapezoidal dyke (Figure 3). The flume is equipped with a piston-type wave paddle generating regular waves with period T=2.39 s and height H=0.10 m at the toe of the beach. Figure 4 presents some comparisons between Spartacus-2D, simulations obtained by solving the Boussinesq equations (see Lee et al. 2006 for more details) and experiments. Generally speaking, the surface elevation is correctly predicted and Spartacus-2D performs slightly better than considered Boussinesq model.

0

0,05

0,1

0,15

0,2

0 2 4 6 8 10 12 14 16x (m)

free

surfa

ce e

leva

tion

(m)

Theory t=3.42 sSpartacus-2D t=3.42 sTheory t=4.50 sSpartacus-2D t=4.50 sTheory t=5.40 sSpartacus-2D t=5.40 s

17

Measurement points P1 P2 P3 P4 P5 P6 P8 P10 P9 P11

H

wave paddle

Slope 1:20 Dyke

Measurement points P1 P2 P3 P4 P5 P6 P8 P10 P9 P11

H

wave paddle

Slope 1:20 DykeH

wave paddle

Slope 1:20 Dyke

Figure 3: Model configuration for the

modelling of wave shoaling.

Figure 4: Wave shoaling: time series of

surface elevation at 3 measurement points from figure 2. Solid line: experiments;

empty circles: Spartacus-2D; dotted line: Boussinesq model.

3.3. Wave setup

When propagating over a shallow water area, waves break and lose energy. This results in an increase of the mean water level named “wave setup”. This process is simulated here, based on the Gourlay’s (1996) experiments on waves propagating over a schematised coral reef in a wave flume of length 17 m with maximum water depth of 0.50 m. The

height of the reef was 0.40 m with 8 m wide of ‘ocean’ between the reef-face and wave generator and 5 m wide of ‘lagoon’ behind it. The ‘ocean’ bottom had a slope of 1:88; therefore, the effective reef height was 0.32 m (see figure 5). The results presented on figure 5 show that Spartacus-2D performs well on this case; in particular our SPH results are slightly better than Massel and Gourlay’s (2000) numerical predictions as well as a Boussinesq-based model.

Figure 5: Model configuration for the modelling of wave setup (distances are in

meters).

Figure 6: Wave setup: longitudinal distribution of mean water level.

3.4. Wave overtopping

Estimating wave overtopping over a dyke or a breakwater is a difficult task due to the complexity of the involved physical mechanisms. Empirical formulae were provided by Hebsgaard et al. (1998) and others, giving the flow rate per unit length of the dyke (in m3/s/m). We used here the model setup presented on section 3.2 to study the prediction of wave overtopping rate over different types of dykes with

18

varying slope, height, berm length, etc. and wave parameters (height and period). Figure 6 shows that our predictions are quantitatively in good agreement with the available experimental formulae.

1E-05

1E-04

1E-03

1E-02

1E-011E-04 1E-03 1E-02 1E-01

Q_SPH (m^3/s/m)

Figure 7: Wave overtopping rate over a dyke: empirical formulae from literature (symbols) versus Spartacus-2D results.

4. Conclusions Interactions between waves and coastal structures have been extensively simulated with Spartacus-2D code. Various phenomena have been studied (namely generation, propagation, shoaling, setup and overtopping), all showing satisfactory results when compared with experiments. These tests will allow the use of Spartacus-2D in the near future in order to improve the design of coastal waterworks.

Future work will consider 3D applications based on a parallel algorithm developed by the Ecole Centrale de Lyon, as well as the use of an incompressible algorithm developed by the University of Manchester (Lee et al. 2007).

5. References GOURLAY, M.R. (1996). Wave set-up on coral reefs. 1. Set-up and wave-generated flow on an idealised two dimensional horizontal reef. Coastal Eng. 27: 161-193.

HEBSGAARD M., SLOTH. P. and JUHL J. (1998). Wave overtopping of rubble-mound breakwaters. Proc. 26th Int. Conf. Coastal Eng., Copenhagen, Denmark. HUGHES S. A. (1998). Physical models and laboratory techniques in coastal engineering. World scientific. LI Y. and RAICHLEN F. (2003). Energy balance model for breaking solitary wave runup. J. Waterway, Port, Coastal & Ocean Eng. 129-2: 47-59. LEE E.S., VIOLEAU D., BENOIT M., ISSA R., LAURENCE D. and STANSBY P. (2006). Prediction of wave overtopping on coastal structures by using extended Boussinesq and SPH models. Proc. 30th Int. Conf. Coastal Eng., San Diego, USA. LEE E.S., MOULINEC C., VIOLEAU D., LAURENCE D. and Stansby P. (2007). Comparisons of weakly compressible and truly incompressible SPH algorithms for 2D flows. Submitted to J. Comput. Phys. MASSEL S.R. and GOURLAY M.R. (2000). On the modelling of wave breaking and set-up on coral reefs. Coastal Eng. 39: 1-27. MONAGHAN J.J. (1994). Simulating free surface flows with SPH. J. Comput. Physics, 110: 399-406. STANSBY P.K. and FENG. T. (2004). Surf zone wave overtopping a trapezoidal structure: 1-D modelling and PIV com-parison. Coastal Eng. 51: 483-500. VIOLEAU D. and ISSA R. (2007). Numerical modelling of complex turbulent free surface flows with the SPH Lagrangian method: an overview. Int. J. Num. Meth. Fluids 53(2): 277-304.

SPHERIC

19

Hybridation of generation propagation models and SPH model: application to realistic dimensions.

A. J. C. Crespo1, M. Gómez-Gesteira1 and R.A. Dalrymple2

1 Grupo de Física de la Atmósfera y del Océano, Facultad de Ciencias, Universidad de Vigo, 32004

Ourense, Spain, alexbexe@uvigo.es; mggesteira@uvigo.es 2 Department of Civil Engineering, Johns Hopkins University, Baltimore, USA rad@jhu.edu

Abstract

Standard generation- propagation

models (WAM, WAVEWATCH, SWAN) are commonly used for operational oceanography purposes with satisfactory results both in the open ocean and in near shore areas but they do not provide information about the possibility of flooding in coastal areas. A 2D HV version of SPH (Smoothed Particle Hydrodynamics) model is considered to propagate the signal provided by generation- propagation models to the coast and to analyze the nature of coastal structures overtopping as a tool to create maps of risk. SPH is applied using realistic dimensions and is shown to provide valuable information about overtopping (maximum water height and water velocity) under the stormy conditions suffered by the Galician coast on February 17, 2006.

1. Introduction Lagrangian methods constitute an

alternative approach to the study of wave collision with coastal structures. In particular, the so called SPH (Smoothed Particle Hydrodynamic) models (Gingold and Monaghan, 1977) have been successfully applied to different coastal processes (Gomez-Gesteira and Dalrymple, 2004; Gómez-Gesteira et al., 2005; Dalrymple and Rogers, 2006; Crespo et al., 2007).

In the particular case of the Iberian Peninsula, the periodic passage of storms gives rise to extreme wave events in the near shore area during winter time. However, no studies have been carried out in this area to analyze the impact of waves on coastal structures and determine their potential risk on human activities. Due to the high computational resources needed to simulate the wave behavior near and on coast, this kind of study cannot be carried out in real time, under the threat of a severe stormy event. Thus, it is necessary to create a map with zones of risk, which can be potentially flooded by extreme events. The map can be created by means of the following protocol: (1) Standard generation-propagation models (G-P models from now on) are run from the open ocean till the vicinity of the coast under the weather conditions corresponding to a particular extreme event; (2) The significant height (Hs) and peak period (Tp) associated to that event are recorded at some point of interest near shore; (3) Different realizations of SPH are performed near shore. Waves with height (H) and period (T) on the same order of magnitude as the ones provide by G-P models are generated by a wavemaker; (4) The wave impact on the coast generated by the different realizations of the model allows determining the potential risk of the coastal area to be flooded. SPH models have been mainly used in wavetanks with oversimplified geometries and unrealistic dimensions. Here the model

20

will be applied to a geometry that mimics a dock. This case, although is still simplified compared to extremely complex coastal structures, was designed with dimensions close to the real ones (bed and dock are similar to the typical ones in the Galician region) proving the capability of SPH to deal with more realistic problems.

2. Transferring information between models

Phase-averaged models (WAM

(Wamdi, 1988); SWAM (Booij et al., 1999); and WAVEWATCH (Tolman, 1991) are used at present for operational oceanography purposes with satisfactory results, both in terms of accuracy and computational, but these models work in completely different time and space scales. G-P models can be used in operative forecasting systems because they can simulate large computational domains (thousand of miles) for days is just a few hours, although with a coarse grid. On the other hand, the SPH model allows analyzing small areas with a fine spatial resolution (on the order of centimetres). Thus the model is well suited to describe coastal areas, especially flooding and wave collision with structures. Unfortunately, the computational resources needed by SPH only permit running seconds in hours with the present technology. Due to the huge difference in scales both models cannot be nested directly, thus the information provided by SWAN at hundreds of meters from coast (significant wave height and peak period) is passed to the SPH wavemaker. The piston-like movement of the wavemaker has been previously calibrated, in such a way that it generates a wave-train with the same properties as the wave pattern provided by SWAN. This wave-train propagates landward interacting with the coastal structures. The SPH model can be run in its 2D or 3D version depending on

the nature of the problem under scope. In addition, the limit of validity of SWAN model is determined by the presence of obstacles (bathymetric or man made) which give rise to the appearance of diffraction phenomena. Thus, intermediate models as REF/DIF (Kirby and Dalrymple, 1983) or Boussinesq (Chen et al., 2000), should be used in those cases where the transition from shelf to coast is not very abrupt and the incoming wave train interacts with different bathymetric obstacles before approaching the coast. A hybrid model to be used under these conditions is described in (Narayanaswamy and Dalrymple, 2005).

3. Case study: Feb 17, 2006

The case study under scope

corresponds to the storm occurred on February 17, 2006. The waves measured at A Coruña buoy showed significant wave heights ranging from 3 to 7 m and peak periods ranging from 10 to 15 s.

SPH model was implemented as described in Gómez- Gesteira et al. (2005). Fluid particles were initially placed on a staggered grid with zero initial velocity as described (dx=dz=0.5m). The particles are assigned an initial density ρ0, which is modified at each position depending on the water height column. Boundary fixed particles are treated as quasi fluid particles (Gómez-Gesteira and Dalrymple, 2004; Gómez-Gesteira et al., 2005, Crespo et al., 2007). Fixed particles (including bottom and fixed walls) are placed in two rows forming a staggered grid (dx=dz=0.5m) with zero initial velocity. Their positions and velocities remained unchanged during the numerical experiment. Wavemaker particles are initially placed in two rows with the same inter-particle spacing. Their velocities and positions are externally imposed to mimic the wavemaker following Gómez-Gesteira et al., 2005.

21

The numerical tank can be seen in Fig. 1.

Figure 1: Numerical tank.

The bed shows a constant slope

(3.73º) till the base of the coastal structure, this chosen bottom slope was estimated by the admiralty carts of the Galician coast. It corresponds to areas over sandy beds, where the depth decreases progressively landward. The dimensions and slope of the dock were taken from real coastal structures at the Galician coast. As we mention above, a SPH model can be used to provide risk maps in a certain area. Thus, different realizations of the model can be carried out around the reference values (Hs and Tp) provided by G-P models. The most extreme event considered in this study corresponds to a peak period of 16.0 s (equal to Tp provided by SWAN) and to a wave height of 8.85 m, which exceeds in about 18% the Hs provided by SWAN. One instant of the water overtopping can be observed in Figure 2. At t= 46.3 s water overtops the dock, the first frame corresponds to the position of the particles and the second one to the velocities.

280 290 300 310 320

-10

0

10

20Time= 46.3 s

280 290 300 310 320

-10

0

10

20Time= 46.3 s

3 m/s

Figure 2: Interaction between an extreme wave and a dock

Velocities were calculated at different

instants during the overtopping at the front of the deck (X = 302 m) and at different heights. Maximum velocities close to 10 ms-1 were observed at the first level (Z=11.6 m) at t=46.0 s.

Parameters describing overtopping for different wave conditions are represented in Fig. 3. All wave-trains are characterized by the same period (T= 16.0 s) and different wave heights (H). The maximum water height above the dock (Hmax in meters) is represented by squares and the maximum velocity attained by the overtopping jet at the dock tip (Vmax in ms-1) is represented by circles.

0

4

8

12

6 7 8 9H (m)

Figure 3: Water overtopping for different wave conditions.

Both Hmax and Vmax are observed to

increase with H, reaching the maximum

22

values for the extreme event described in Fig. 2 (Hmax=4.7 m and Vmax= 10.7 ms-1).

4. Summary

SPH models can be coupled with

standard G-P models to elaborate risk maps in coastal areas. A 2DHV version of SPH model allows propagating a wave train with T and H values close to Tp and Hs, from the near shore area to the coast. The model is used to characterize the overtopping of coastal structures, determining the maximum velocity and height attained over the structure.

Finally, SPH has been mainly used to model laboratory tests with reduced geometries. In the present application the model has proved to provide satisfactory results in cases with realistic dimensions.

5. Acknowledgments This work was partially supported by

Xunta de Galicia under proyect PGIDIT06PXIB383285PR.

6. References BOOIJ, N., RIS, R.C. and. HOLTHUIJSEN, L.H. (1999). A Third-Generation Wave Model for Coastal Regions, Part I, Model Description and Validation. J. Geophys. Res.C4, 104: 7649-7666. CHEN, Q., KIRBY, J.T., DALRYMPLE, R.A., KENNEDY, A.B., and CHAWLA, A. (2000). Boussinesq modeling of wave transformation, breaking and runup. II: 2D. J, Waterway, Port, Coastal, and Ocean Engineering, 126, 1: 48-56. CRESPO, A.J.C., GÓMEZ-GESTEIRA, M. and DALRYMPLE, R.A. (2007), 3D SPH Simulation of Large Waves Mitigation With A Dike. Journal of Hydraulic Research IAHR (in press).

DALRYMPLE, R.A. and ROGERS, B. (2006). Numerical modeling of water waves with the SPH method, Coastal Engineering, 53/2-3, 141-147. GINGOLD, A. and MONAGHAN, J. J. (1977). Smoothed Particle Hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astr. Soc., 181: 375-389. GÓMEZ-GESTEIRA,M. and DALRYMPLE, R.A. (2004). Using a 3D SPH Method for Wave Impact on a Tall Structure. Journal of Waterway, Port, Coastal and Ocean Engineering 130(2):63- 69. GÓMEZ- GESTEIRA, M., CERQUEIRO, D., CRESPO, C. and DALRYMPLE, R.A. (2005). Green water overtopping analyzed with a SPH model. Ocean Engineering 32:223- 238. KIRBY J.T. and DALRYMPLE, R.A.(1983). A parabolic equation for the combined refracion-diffraction of Stokes waves by mildly varying topography. J. Fluid Mech., 136:543-566.

NARAYANASWAMY, M. and DALRYMPLE, R.A. (2005). A hybrid finite element and SPH model for forced oscillations in basins. In Waves 2005. Ocean Waves Measurements and Analysis. TOLMAN, H.L., (1991). A third-generation model for wind on slowly varying unsteady and inhomogeneous depths and currents. J. Phys. Oceanogr., 21:782-797. WAMDI Group (Hasselmann, S., Hasselmann, K., Bauer, E., Janssen, P.A.E.M., Komen, G.J., Bertotti, L., Lionello, P., Guillaume, A., Cardone, V.C., Greenwood, J.A., Reistad, M., Zambresky, L. y Ewing, J.A.), (1988), The WAM model – a third generation ocean wave prediction model. J. Phys. Oceanogr., 18: 1775-1810.

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23

On the use of an alternative water state equation in SPH

Diego MOLTENI 1 , Andrea COLAGROSSI 2,3 and Giuseppina COLICCHIO 2,3

1 Dipartimento di Fisica e Tecnologie Relative, Viale delle Scienze 18, Palermo, Italy 2 INSEAN, Italian Ship Model Basin, via di Vallerano,139 , Roma, Italy

3 CESOS, Centre of Excellence for Ship and Ocean Structures, NTNU, Trondheim, Norway

Abstract In this work new equations of state for water are formulated, they improve both the consistency and the accuracy of the Smoothed Particles Hydrodynamics (SPH) method to treat incompressible fluid dynamics. When, both physical and artificial viscous terms are introduced, entropy increases. In liquids, the influence of this entropy variation is negligible, but common SPH formulations treats the water as a weakly compressible gas. Therefore, for violent fluid motions, this increase cannot be neglected anymore and the energy equation has to be introduced. This equation has been time integrated, and connected to alternative water state equations which have been studied heuristically. These equations contain explicitly an internal energy term which increases largely especially during impact phenomena. The new numerical scheme adopted presents free parameters that are to be set. A plausible value of them has been evaluated in the case of the dam-break problem with violent fluid/structure and fluid/fluid interactions.

1. Introduction It is well known that the SPH method

treats incompressible fluid dynamics essentially in two ways: by the Tait equation of state that allows small changes of the fluid density or by Poisson equation to compute the pressure field. The Poisson

approach prescribes exactly the free divergent velocity constrain, while the use of an equation of state permits to write an hyperbolic system of equations which can be integrated in a simple explicit way. In this second scenario it is possible to introduce also the thermal energy content of the fluid to improve the convergence and the accuracy of the numerical scheme. The equation of state frequently used for SPH is the Tait equation

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1

00

γ

ρρPP

(1)

For it, the speed of sound is: ( ) 12

0 0 0/ /c P γγ ρ ρ ρ −=

as for a perfect gas in adiabatic isentropic transformations. In the SPH simulations of water flows the speed of sound for the fluid at rest c0 (i.e. for ρ=ρ0) is fixed at the beginning of the simulation so that

( )uc tmax100 ≥ (2)

where maxt |u| is the maximum speed in modulo of the fluid, obtainable along the whole simulation. In these conditions the Mach number M=u/c0 is always less than 0.1 and it is possible to consider the fluid as weakly compressible. The velocity constrain (2) allows a large range of values for c0 which is usually chosen two orders of magnitude lower than the real value. Since the maximum fractional density variation 0/ρ ρ∆ is inversely proportional

to M2 (see e.g. [1]), it is possible to use the linearized version of the Tait equation which reads as:

24

( )20 0P c ρ ρ= − (3)

The Tait equation is based an a polytropic law and therefore the entropy level should be preserved. When, both physical and artificial viscous terms are introduced, entropy increases. In liquids the influence of entropy is negligible, but common SPH formulations treats the water as a weakly compressible gas. Therefore, for violent fluid motions, this increase cannot be neglected anymore and the energy equation

( )ePdtde v ∇∇+∇

Π+−=

→

σρ

(4)

has to be introduced. where e is the internal energy per unit mass, Π is the artificial viscosity pressure, the last term is a diffusive term added to the energy equation and σ is the thermal diffusion coefficient. The presence of a diffusive term allows to redistribute the jump of e generated during violent impacts. Consequently the pressure oscillations correlated to the shock are dumped out. We rewrote σ equal to khc where h is the smoothing length adopted and k is a free parameter used to tune the numerical thermal conductivity. When a free surface is present, it exchanges heat with environment very quickly, so it can be considered as isothermal. This constitutes a boundary condition for equation (4). We started to study heuristically new equations of state containing explicitly an internal energy term, for example

( ) ( )( )0002

0 ρρρρ −−Γ+−= eecP (5)

For this EoS (Equation of State) the sound speed becomes

( )02

02 eecc −Γ+= (6)

Here the leading term comes from the linearization of Tait equation and the second term depends explicitly on the specific internal energy. The importance of this term is weighted by the parameter Γ . Because e increases largely during impact phenomena, from equation (5) it is well

visible that the local speed of sound c becomes large. This sets a lower limit on the time step, increasing the time resolution only during these violent phases. Γ and k are free parameters that are to be set. This is achieved studying a classical dam break problem with the flow impacting against a vertical wall. This is a problem with violent fluid/structure and fluid/fluid interactions and it is the simplest way to represent the water on deck event on ships. It is assumed that a mass of water suddenly appears on the deck and flows along it. The deck structure can be invested by the flow and the knowledge of the local loads on them is important for the structural safety ( see e.g [2]). In this study an elementary evaluation of Γ, based on the gas like equation, furnishes a values of order 103 while for the diffusion coefficient k a value of order 10-2 has been adopted.

2. The new state equations An obvious approach would be to

adopt a physical equation of state of water, explicitly containing the entropy terms; it can be written in several forms and can be quit complex. A much simpler approach is to write co as a Taylor expansion in e:

( ) ( ) ( )2

2 20 0 0 0

0

. .cc e c e e e H Oe

⎛ ⎞∂= + − +⎜ ⎟∂⎝ ⎠

Substituting it in equation (3), the new EoS (5) can be obtained, where

0

2 /e e

c e=

⎡ ⎤Γ = ∂ ∂⎣ ⎦

This implies that the pressure and the sound speed increase (or decrease) also according to the internal energy. The new EoS has been introduced in the standard SPH formulation [1,3], where the time integration is performed with a forth order Runge-Kutta scheme and the density is reinitialized according to Colagrossi [3].

25

3. Results and discussion Aiming to model the behaviour of un-

viscous flow in presence of impact effects, we studied the break of a dam with a subsequent impact of the flow onto a vertical rigid wall. This case is largely studied in literature (see e.g. [2,3,4]). The details of the geometry of the problem are depicted in figure 1 where: L=120 cm, H=60 cm, Lw-L=202 cm, the pressure probe P1 is at a distance of 16 cm from the bottom and has a diameter of 18 cm. We performed three simulations with different spatial resolutions, respectively H/h=18.5, 37, 74, where h is the SPH smoothing length set equal to 1.33 times the particle size.

Figure 2 shows the time evolution of the total thermal energy

W

E e dV= ∫

for three discretizations of a classical SPH formulation [3]. A large amount of thermal energy is produced during the impact stages by the numerical procedure. This numerical energy increment suggests that the dependence of pressure form e can be hardly neglected. In fact for the coarsest discretization at t=9(H/g)1/2 half of the total mechanical energy has been dissipated in the numerical thermal energy. The particles configuration at time t=6.4(H/g)1/2 is shown in Fig. 3 for the Tait equation and in Fig. 4 for the new EoS [4]. The colours of the particles show the internal energy variation. Obviously the results obtained with the new equation show a more homogeneous distribution of the thermal energy. Even though the free surface does not show any significant deformation after the application of the EoS, a beneficiary effect is visible in the time history of pressure at the probe P1. Fig. 5 shows a comparison of the experimental time pressure evolution versus the numerical ones obtained with the best resolution SPH simulations

adopting the two different equations of state. The new results remain closer to the experimental data. This effect is obtained with no numerical damping of the mechanical energy. Practically the new scheme smoothes in space the pressure shocks generated at the impacts. The parameters chosen for this case are

gHc 200 = , Γ=5000, k=0.02.

Figure 1: Initial configuration of the particles.

Figure 2: Thermal energy versus time for the

three spatial resolutions.

26

Figure 3: Particles distribution at t=6.4 with Tait

state equation.

Figure 4: Particles distribution at t=6.4 with new

equation of state (see eq. 4).

Figure 5: Experimental pressure at the detection

point and simulation results.

4. Conclusions For weakly compressible fluid simulations with SPH, the introduction of an EoS taking into account the internal energy has shown a benefit in terms of milder pressure oscillations in the case of violent impacts. Further investigation of more meaningful EoS is ongoing.

Acknowledgments. The present research activity is supported partially by CESOS: Centre of excellence for Ship and Ocean Structures, NTNU, Trondheim, Norway, within the project, "Sloshing flows" and partially by Ministero dei Trasporti within the "Programma Sicurezza 2006-2008".

5. References MONAGHAN J.J. (1994). Simulating free surface flows with SPH. Journal of Computational Physics,110: 399–406. GRECO M. (2001) Two-dimensional Study of Green-Water Loading PhD Thesis University of Trondheim, Norway. COLAGROSSI A., LANDRINI M.(2003). Numerical simulation of interfacial flows by smoothed particle hydrodynamics. Journal of Computational Physics,191: 448–475. LEE T., ZHOU Z., CAO Y. (2002). Numerical simulations of hydraulic jumps in water sloshing and water impacting. Journal of Fluid Engineering, 124: 215-226.

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27

Applications of Generalised Smoothed Particle Hydrodynamics to Benchmark CFD Problems

Joseph Ha

PhD, CSIRO Mathematical and Information Sciences, Private Bag 33, Clayton South, Victoria 3169, Australia

Abstract This study examines the effectiveness of generalised smoothed particle hydrodynamics as a numerical method for solving partial differential equations. Numerical results of several benchmark CFD problems obtained by the method compare very well with those obtained from grid based methods.

1. Introduction The kernel approximations used in the

original SPH proposed by Lucy (1977) and Gingold and Monaghan (1977) suffer from certain inconsistencies. Various approaches to remedy these inconsistencies have been reported in the literature. It has been shown that the kernel approximations can be corrected so that they reproduce constant and linear functions exactly (see, for examples, Belytschko et al., 1996 and Dilts, 1999). Essentially, these methods replace the standard SPH approximant with more sophisticated interpolant that was constructed by imposing certain consistency conditions. Liu et al. (1995) showed that the reproducing kernel method provides boundary correction as well as removing the tensile instability. Chen and Beraun (2000), on the other hand, developed a generalised SPH method (GSPH) by applying the kernel estimate into the Taylor series expansion. Their formulation extends not only the ability of standard SPH to model partial differential equations with higher order

derivatives but to enforce boundary conditions directly as well. In addition, GSPH satisfies certain consistent and reproducing conditions.

Most applications of standard SPH are to simulate compressible fluids. The aim of this paper is to study the application of GSPH to some benchmark incompressible fluid problems for testing CFD codes. All the numerical examples are obtained from substituting each term of the governing equations by their corresponding GSPH derivative approximations directly. GSPH produces accurate results to the problem considered. It is less affected by particle disorder than standard SPH. Also, its implementation is similar to standard SPH. There is no dimensional difference between 1D, 2D and 3D as far as computer coding for its implementation is concerned.

2. Generalised SPH Applying the kernel approximation to

the Taylor series expansion for f(x) in the neighbourhood of x, Chen and Beraun (2000) derived results that improve the approximation accuracy of SPH. In 1D, the GSPH approximation of a function f(x) and its first derivative are given in equations (1) – (2).

∫∫

−

−=

')'(

'),'()'()(

dxxxW

dxhxxWxfxf (1)

28

∫∫

−

−=

')'(

'))'()(()(

dxxx

dxxfxf

dxxdf

dxdW

dxdW

(2)

The same procedure can be followed to derive approximations in higher dimensions. However, the derivative estimates for higher dimensions involve matrix inversion. It is thus computationally more expensive to use than conventional SPH. It is well appreciated that SPH is closely related to the finite element method. The main difference between the two methods is that the SPH kernel approximation of a function does not satisfy the Kronecker delta property. It is thus not possible to impose essential boundary conditions in conventional SPH. The inclusion of f(x) and df/dx in the first and second derivative estimates enable the direct insertion of Dirichlet and Neumann boundary conditions in the GSPH method.

3. Results and discussion

3.1. Heat conduction

First, to test that the GSPH method can impose boundary conditions directly, the following heat conduction problem is solved in the domain 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, initial condition T(x,y,0) = -1, Neumann boundary condition ∂T(x,1,t)/∂y = 0 at y = 1 and Dirichlet condition at the other boundaries T(0,y,t) = T(1,y,t) = T(x,0,t) = 1.

2

2

2

2

yT

xT

tT

∂∂

+∂∂

=∂∂

(3) where T denotes temperature and t time. The result shown in Figure 1 appears identical to the result obtained by Jeong et al. (2003) who implement the boundary conditions to the conventional SPH in a different way.

0.80.60.40.20.0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 1: Temperature profile at t = 0.08 s.

3.2. Burger equation

Here, the following 3D Burger’s equation is solved using GSPH and the numerical results are compared with its analytical solutions.

02 =∇−∇⋅+∂∂ vvvv ν

t (4)

Figure 2 shows the computed results for ν = 0.05 at various times compare well with the analytical solutions. The number of particles used is 41×41×41.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

U(x

,y,z

)

X=Y=Z

Viscosity = 0.05 (41)

Figure 2: GSPH solution (+) of 3D Burger’s equation compared to analytical

29

solution (solid line) along the line x = y = z at various times for viscosity of 0.05.

3.3. Incompressible Navier-Stokes equation

The GSPH method is next applied to three standard CFD test problems – 2D Poiseuille flow, 2D lid-driven cavity and natural convection in a square cavity. For the 2D Poiseuille flow and lid-driven cavity examples, the following Navier Stokes equation in 2D is solved

Fvpvvtv

+∇+∇1

−=∇⋅+∂∂ 2ν

ρ (5)

For the Poiseuille flow, the boundary conditions are v = (0,0) on y = 0 and y = L, where L = 0.001 and F = 1.25x10-6/Re. For the lid-driven cavity problem, the boundary conditions are v = (-1,0) on y=1 and v = (0,0) on the other three sides of the unit square. For an incompressible fluid, the Navier Stokes equation is complemented by the incompressibility constraint 0=⋅∇ v . In general, velocity

1+nv at time 1+nt obtained by solving Equation (5) does not satisfy the incompressibility constraint. This constraint on velocity must be satisfied at all times. In this paper, the following steps are iterated until 0≈⋅∇ v is reached.

1. 1+⋅∇−=∆ nkkp vγ

2. )(1 nk

nk pt ∆∇∆=∆ +v

Here, k is the iteration counter and kkk fff −=∆ 1+ . Upon convergence, the above procedure gives the new

pressure 1+np and divergent free velocity 1+nv for time 1+nt . The parameter γ

controls the rate of convergence and must

satisfy the stability

requirements tx ∆∆≤≤ 4/)(0 2γ . The iteration is equivalent to solving a Poisson equation for the pressure.

Figure 3 shows the GSPH solutions of 2D Poiseuille flow using 51 × 51 particles for Re = 0.0125 and 100 compare well with the analytical solutions. Sigalotti, et al. (2003) reported that their SPH solution for Re = 5 eventually becomes unstable after about 280 s. For the two cases considered here, the GSPH solutions do not exhibit any instability up to twice the time when the steady state solution is reached.

0.0e+00

2.0e-06

4.0e-06

6.0e-06

8.0e-06

1.0e-05

1.2e-05

1.4e-05

0 0.0002 0.0004 0.0006 0.0008 0.001

U (

m/s

)

Y (m)

Re = 0.0125

0.02 s

0.04 s

0.06 s

0.10 s

0.20 s

0.30 s

1.00 s

0.0e+00

2.0e-06

4.0e-06

6.0e-06

8.0e-06

1.0e-05

1.2e-05

1.4e-05

0 0.0002 0.0004 0.0006 0.0008 0.001

U (

m/s

)

Y (m)

Re = 100

100 s

200 s

300 s

500 s

700 s

1000 s

1300 s

1700 s

5000 s

Figure 3: GSPH results (+) of Poiseuille flow compared to series solutions (solid lines) for Re = 0.0125 and 100.

Figure 4 shows that the GSPH solutions for Re = 1000 using 129×129 particles and 3 different kernels compare well with the benchmark solutions 1, 2 and 3 of Ghia et al. (1982), Botella and Peyret (1998) and Erturk et al. (2005) respectively. In the figure, W3 denotes the cubic spline kernel, W4 the quartic spline kernel and W5 the quintic spline kernel. For this problem, W4 gives the best result with W3 and W5 giving similar results.

For incompressible fluid flow in a differentially heated square cavity of side L, the following equations are solved.

30

gvvvv )( rTTpt

−=∇−∇1

+∇⋅+∂∂ 2 βν

ρ

(6)

TTtT 2∇=∇⋅+

∂∂ αv (7)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.2 0.4 0.6 0.8 1

U

Y

Re 1000

[1][2][3]

W3W4W5

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.2 0.4 0.6 0.8 1

V

X

Re 1000

[1][2][3]

W3W4W5

Figure 4: Comparison between published results and GSPH solutions for Re = 1000 of the lid-driven cavity problem using different kernels.

The initial conditions are v(x, y, 0) = (0, 0) and rTyxT =)0,,( . The boundary conditions are v = (0, 0) on cavity boundary, hTtyT =),,0( , cTtyLT =),,( , ∂T(x,0,t)/∂y = ∂T(x,L,t)/∂y = 0. Here, rT , hT and cT denote the reference, hot and cold wall temperatures

respectively. Table 1 show that the GSPH results compare well with the benchmark solutions for Prandtl number 0.71 and Rayleigh numbers 104 - 106. In the table, the numbers enclosed by [] and () are the results of Leal et al. (1999) and de Vahl Davis (1983) respectively. Using a remeshed SPH approach, Chaniotis, et al. (2002) obtained the values of umax = 17.31 at y = 0.823 and vmax = 20.05 at x = 0.112 for Ra = 104 which are not as close to the results of Leal et al. and de Vahl Davis as the GSPH results.

Table 1: Comparison of natural convection results.

4. Conclusions This paper examines the effectiveness

of GSPH as a numerical method to solve partial differential equations and the incompressible Navier-Stokes equation in particular.

The numerical examples presented in the last section demonstrated that GSPH gives accurate results to the problems considered. Unlike conventional SPH, it has the advantage of being able to impose boundary conditions directly. Also, it is just as easy to implement as the conventional SPH. There is no dimensional difference between 1D, 2D and 3D as far as computer coding for their implementation is concerned. Apart from correcting the boundary deficiency problem, GSPH is

umax y vmax x [16.18] [0.823] [19.63] [0.119]

Ra = 410 16.18

0.822

19.63

0.119

(16.178) (0.823) (19.617) (0.119) [34.74] [0.855] [68.62] [0.066]

Ra = 510 34.76

0.853

68.64

0.0656

(34.73) (0.855) (68.59) (0.066) [64.83] [0.850] [220.6] [0.0379]

Ra = 610 64.91

0.847

220.72

0.0375

(64.63) (0.850) (219.36) (0.0379)

31

less affected by particle disorder than conventional SPH because of the normalisation term in the denominator (refer to Equations 1-2).

5. References BELYTSCHKO, T., KRONGAUZ, Y., ORGAN, D., LEMING, M. and KRYSL, P. (1996). Meshless methods: An overview and recent developments. Comput. Meth. Appl. Mech. Engng., 139: 3-47. BOTELLA, O. AND PEYRET, R. (1998). Benchmark spectral results on the lid-driven cavity flow. Computers & Fluids, 27: 421-433. CHANIOTIS, A. K., POULIKAKOS, D. AND KOUMOUTSAKOS, P. (2002) Remeshed Smoothed Particle Hydrodynamics for the Simulation of Viscous and Heat Conducting Flows. J. Computational Physics, 182: 67-90. CHEN, J.K. and BERAUN, J.E. (2000). A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems. Comput. Methods Appl. Mech. Engrg., 190: 225-239. DE VAHL DAVIS, G. (1983). Natural convection of air in a square cavity: a benchmark numerical solution. Int. J. Numer. Methods Fluids, 3: 243-264. DILTS, G.A. (1999). Moving-least-squares-particle hydrodynamics - I. Consistency and stability. International Journal for Numerical Methods in Engineering, 44: 1115-1155. ERTURK, E., CORKE, T. C. AND GÖKÇÖL, C. (2005). Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers. International Journal for Numerical Methods in Fluids, 48: 747-774.

GHIA, U., GHIA, K.N. AND SHIN, C.T. (1982). High-Re solutions of incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48: 387-411. GINGOLD, R. and MONAGHAN, J. (1977). Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. Roy. Astron. Soc., 181: 375-389. JEONG, J.H., JHON, M.S., HALOW, J.S. AND VAN OSDOL, J. (2003). Smoothed particle hydrodynamics: Applications to heat conduction. Computer Physics Communications, 153: 71-84. LEAL, M.A., PEREZ-GUERRERO, J.S. AND COTTA, R.M. (1999). Natural convection inside two-dimensional cavities: the integral transform method. Communications in Numerical Methods in Engineering, 15: 113-125. LIU, W.K., JUN, S., ADEE, J. and BELYTSCHKO, T. (1995). Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Engng., 38: 1655-1679. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. SIGALOTTI, L.D.G., KLAPP, J., SIRA, E., MELEÁN, Y. AND HASMY, A. (2003). SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers. J. of Computational Physics, 191: 622-638.

SPHERIC

32

Numerical calculations of flow through an orifice conduit

Jaime KLAPP1, Leonardo Di G. SIGALOTTI2, Salvador GALINDO1 and Ricardo DUARTE3

1 Ph. D., Instituto Nacional de Investigaciones Nucleares, ININ, Km. 36.5 Carretera México-Toluca, Ocoyoacac 52750, Estado de México, México.

2 Ph. D., Instituto Venezolano de Investigaciones Científicas, IVIC, Apartado Postal 21827, Caracas 1020A, Venezuela.

3 M. I., Instituto Nacional de Investigaciones Nucleares, ININ, Km. 36.5 Carretera México-Toluca, Ocoyoacac 52750, Estado de México, México.

Abstract

In this paper we present numerical calculations of the flow of a compressible fluid at high Reynolds number (Re~106) between two parallel plates with a sudden constriction, using the method of Smoothed Particle Hydrodynamics (SPH). The constriction is modelled as a thin orifice conduit in the direction of the main flow. A fully-developed laminar flow with a time increasing velocity is employed far upstream of the orifice as the inlet boundary condition. The detailed flow structure along the channel and the pressure drop through the orifice conduit are calculated up to the point where the flow velocity becomes critical (or choked). We find that critical flow occurs at the onset of sonic velocity when the absolute pressure downstream of the orifice attains a limit of about 0.54 times the upstream value, in good agreement with the conditions expected for choked gas flow in a pipe orifice.

1. Introduction The orifice meter is perhaps the oldest

and cheapest known device for measuring and regulating the flow of fluids. Flow evaluation in pipes and ducts through an orifice conduit (or choke) is important for many industrial applications involving single- and multi-phase flows. In particular, some of these applications include the

design of throttling valves for refrigeration systems, bypass systems for steam power plants and emergency relief in chemical and nuclear plants.

When a flowing fluid is accelerated by restricting the cross-sectional area of the flow stream, the fluid pressure drops because part of the pressure energy is converted into kinetic energy. As a result, the flow velocity through the orifice conduit increases. Also, the mass flow rate and the flow velocity increase monotonically with decreasing ratio of the downstream to the upstream pressure (p2/p1) and the flow is said to be subcritical. If the pressure drop across the orifice becomes sufficiently large, the flow velocity reaches a maximum value, which is equal to the local sound speed. Under these conditions, the flow becomes critical (or choked). It is well-known that for gas flow, sonic velocity occurs at a critical pressure ratio of 0.528. The purpose of this paper is to describe the numerical results obtained for the flow of a compressible fluid at Re~106 between two parallel plates with a sudden orifice constriction for both the subcritical and critical regimes, using the SPH method. In contrast to previous SPH calculations in this line (KLAPP, SIGALOTTI, GALINDO and SIRA 2005), here we use increasing spatial resolution. The results are compared with the expected trends for gas flow in a pipe orifice.

2. Numerical model

33

Here we solve the SPH continuity, x- and y-momentum equations for a viscous fluid and assume that there is no significant heat exchange between different parts of the fluid. Therefore, an isothermal pressure-density relation of the form p=c2ρ is used to close the set of fluid equations, where c is the local sound speed taken to be 2.0x104cm s-1. For the problem at hand, we consider a fluid density ρ=1.0 g cm-3 and kinematic viscosity ν=5.0x10-4cm2s-1 flowing in the positive x-direction between two parallel plates with a sudden constriction (see Fig. 1). The length of the plates in the z-direction is assumed to be very large compared to their separation in the y-direction (inlet/outlet opening) so that all locations in the z-axis appear essentially identical to one another (i.e., ∂/∂z=0) and so the flow need only be represented in the (x,y)-plane. The constriction is modelled as an orifice conduit of length l(~7.78cm) in the x-direction and opening width d(~1.33cm) in the y-direction, as shown in Fig. 1. Upstream of the orifice, the plates are separated by a distance D1~8.89cm and downstream of it by a distance D2~7.78cm. The total length of the channel, including the orifice conduit, is L=44.45cm. No-slip boundary conditions are imposed to mimic the sticking of the fluid to the walls through the use of image particles (TAKEDA, MIYAMA and SEKIYA 1994). Inlet boundary conditions were obtained by injecting particles at the entrance of the channel with a plane Poiseuille velocity profile given by

,41 21

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛=

Dytvv

τ (1)

where t is time, τ =0.2 s and v1=c(d/D1). In this way, at t=0 the inlet particles are at rest and their velocity increases linearly in time during the evolution.

Figure 1: Schematic design of the channel

with an orifice conduit.

3. Results The flow behaviour through an orifice

conduit is described here for the case in which the inlet flow velocity far upstream of the orifice increases linearly with time, thus allowing the pressure drop through the orifice conduit to increase in the course of the evolution. The mean pressure and velocity profiles along the channel are depicted in Fig. 2 at three distinct times during the evolution. The vertical lines in the middle of each panel mark the exact position of the entrance and exit of the orifice conduit. By 0.191s when p2/p1~0.61, the flow velocity at the exit of the orifice is close to the sound speed. At about t=0.2s when the velocity becomes sonic through the orifice conduit, the maximum inlet velocity is c(d/D)~0.15c [see Eq. (2)]. The bottom panels of Fig. 2 show the profiles at t=0.23s when the maximum mean flow velocity at the orifice is v0~1.04c and the pressure ratio is 0.546. The further evolution is characterized by rather steady conditions in which the profiles change only slightly and the overall pressure drop oscillates about 0.54. This value is close to the critical pressure ratio of 0.528 expected for gas flow in a pipe orifice when sonic conditions are achieved (MILLS 1968; OBERT and CAGGIOLI 1960). Note that after about 0.2s, the mean flow becomes supersonic within the jet at the vena contracta, that is, in the downstream section close to the orifice exits. At t=0.024s, when the calculation is terminated, the maximum velocity there is vch=1.54c.

34

Figure 2: Mean pressure in terms of the initial pressure p0 (left panels) and flow velocity in terms of the sound speed (right panels) through the channels at t=0.091, 0.191 and 0.230 s.

Figure 3: Dependence of the flow velocity on pressure ratio. The curve with dots is the maximum velocity in the orifice conduit and that with diamonds represents the maximum velocity along the entire channel.

Figure 4: Flow velocity contours at t=0.16s. Numbers on the box sides are in cm. The color scale denotes the velocity distribution and the numbers on the right side give the velocity in units of the sound speed.

Figure 5: Flow velocity contours at t=0.24s. Numbers on the box sides are in cm. The color scale denotes the velocity distribution and the numbers on the right side give the velocity in units of the sound speed.

35

Figure 3 shows the variation of the maximum flow velocity with the pressure ratio. The curve with dots gives the maximum velocity within the orifice conduit and the one with diamonds refers to the maximum velocity across the entire channel. The former curve shows that the flow within the orifice becomes choked at p2/p1~0.54 when the flow velocity reaches sonic conditions. Note that for p2/p1<0.60 (t>0.19s), the maximum velocity just downstream of the orifice (within the jet at the vena contracta; see Figs. 4 and 5) grows faster and eventually becomes supersonic.

Figures 4 and 5 display the flow structure along the entire channel at t=0.16 and 0.24s, respectively. In each figure, the top and bottom panels depicts velocity contour plots for the upstream and downstream sections of the channel, respectively, while the middle panel is a magnification of the flow structure through the orifice conduit. The blue regions correspond to zones where the flow velocity is lower. In the upstream section, the flow is accelerated in front of the orifice opening because of the restricted cross-sectional area (top panels of Figs. 4 and 5). As the pressure energy is converted into kinetic energy, the flow velocity increases through the orifice, reaching the highest values towards the outlet (middle panels of Figs. 4 and 5). Note that the flow velocity decays rapidly in the proximity of the solid walls where the fluid sticks due to the viscous effects. In the downstream section, a straight jet forms which then extends along the full length of the channel, as shown in the bottom panel of Fig. 4 at t=0.16s. The cross-sectional area of the jet at the vena contracta, close to the orifice exit plane where the streamlines are almost parallel to one another (see Fig. 6), is constrained by the orifice width. The maximum velocity (~0.78c at t=0.16s) within the jet always occurs in this region.

Figure 6 depicts the velocity vectors in the downstream section of the channel at

the same time of Fig. 4. Outside the jet, the flow is clearly turbulent. This is evidenced by the small eddies surrounding the jet, which keep recirculating. These small eddies are in turn embedded in a large one, which extends over the full length of the downstream section. When the maximum velocity at the vena contracta becomes supersonic by about 0.62s, the jet deforms downstream as shown in the bottom panel of Fig. 5 at t=0.24s. Since the flow is kept isothermal, there is no dissipation of the kinetic energy into heat within the smallest eddies. As a result, they grow in size and move around; as more and more rotational kinetic energy is transferred to them from the larger eddies. The interaction of this turbulent motion with the jet affects primarily the region downstream of the vena contracta, making it to wind and eventually expand back again to the full cross-section of the channel. This kind of instability has been observed experimentally (WILKES 1999).

Figure 6: Velocity vectors in the downstream section of the channel at t=0.16s. The maximum velocity is ~0.78c.

36

Figure 7: Discharge coefficient as a function of the Reynolds number through the orifice conduit.

Finally, Fig. 7 shows the discharge coefficient Cd, defined according to

,1

)(2

4

121

22

22

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

Dd

ppvc

md ρ

ρ (2)

as a function of the Reynolds number at the orifice (Reo-vmd/ν), where ρm and vm refer to the mean values of the density and velocity through the orifice conduit. We can see that the trends seen in Fig. 7 are in good agreement with MILLS (1968) experimental data for small openings (d/D1<2).

4. Conclusions We have presented numerical results

for the flow of a compressible (Re~106) fluid between two parallel plates with a sudden orifice constriction, using the method of Smoothed Particle Hydrodynamics.

We find that at the onset of sonic velocity through the orifice conduit, the absolute pressure downstream of the orifice attains a limit of ~0.54 times the

upstream value, in good agreement with the expected value of 0.528 for gas flow through an orifice pipe. Under these choked conditions, the flow in the downstream section of the channel becomes supersonic within the jet at the vena contracta, while on both sides of it the flow is turbulent. Interaction of this turbulent motion with the jet makes it to wind and become unstable over the full cross-section of the channel. When the velocity across the orifice becomes sonic, the flow reaches almost steady conditions and the discharge coefficient attains a limiting value of ~0.68 when Re~1.82x106 at the orifice, in good agreement with the experimental data.

5. Acknowledgments This work was partially supported by CONACYT under contract number U43534-R.

6. References MILLS, R.D. (1968) Numerical solutions of viscous flow through a pipe orifice at low Reynolds numbers. J Mech Eng Sci 10, 133-140. OBERT, E. and CAGGIOLI, R. (1960) Thermodynamics. New York: McGraw-Hill. TAKEDA, H., MIYAMA, S.M. and SEKIYA, M. (1994). Numerical simulation of viscous flow with smoothed particle hydrodynamics. Prog Theoret Phys 92, 939-960. WILKES, J.O. (1999). Fluid mechanics for chemical engineers. New Jersey: Prentice-Hall. KLAPP, J., SIGALOTTI, L. D., GALINDO, S. and SIRA, E. (2005). Two-Dimensional TREESPH Simulations of Choked Flow Systems. Rev. Mex. Fís. 51, 563-573.

SPHERIC

37

SPHERIC Test Case 6: 2-D Incompressible flow around a moving square inside a rectangular box

E.-S Lee1, D. Violeau2, D. Laurence3, P. Stansby4 and C. Moulinec5

1 PhD student, MACE / The University of Manchester, UK, e.lee-2@postgrad.manchester.ac.uk

2 Research Engineer, LNHE / EDF R&D, 6 quai Watier 78400 Chatou, France, damien.violeau@edf.fr

3 Professor, MACE / The University of Manchester, UK, dominique.laurence@manchester.ac.uk

4 Professor, MACE / The University of Manchester, UK, p.k.stansby@manchester.ac.uk 5 Research Engineer, LNHE / EDF R&D, 6 quai Watier 78400 Chatou,France,

chmoulinec@yahoo.com

Abstract The present work deals with a comparison between a truly incompressible SPH approach and a Finite Difference one to compute an incompressible flow around a moving square set in a rectangular box. All comparisons are made in terms of velocity magnitude and pressure at Re=50, 100, and 150. The traditional weakly compressible SPH approach is used for comparison at Re=100.

1. Introduction Truly incompressible SPH (ISPH) has

been developed to circumvent some shortcomings of the traditional weakly compressible SPH (WCSPH) method (Monaghan 1992) such as large artificial pressure fluctuations and small time-steps. The main difference between the two methods lies in the pressure estimation, ISPH requiring to solve a pressure Poisson equation (e.g. Cummins et al. 1999, Shao et al. 2003) while with WCSPH a pressure estimation is obtained by a fluid state equation. WCSPH generates up to 1% of density fluctuation depending on the type of flow (Monaghan 1994), which is

tolerable, but in some cases such as separated flows sensitivity to the artificial speed of sound has been observed (Issa et al. 2005).

The classical projection method (Chorin 1968) is used to solve the velocity-pressure coupling between time step n and n+1; the prediction step generates an auxiliary velocity field u* usually not divergence free. Assuming that un+1 field is divergence free with constant density ρ leads to the resolution of a Poisson equation which reads:

2 1 *unpt

ρ+∇ = ∇ ⋅∆

(1)

where p is the pressure, t the time. The pressure linear solver is the Bi-CGSTAB method (Van Der Vorst 1992). un+1 is finally obtained by adding the pressure gradient to field u* (see Lee et al. (2007)). This paper focuses on a test case consisting of a 2-D incompressible flow generated by a square solid moving in a rectangular box (see SPHERIC website for more details).

2. Results and discussion

38

The moving square box dimensions are 1x1 and the rectangular one 10x5. ISPH and FD simulations are run at three values of Reynolds number of 50, 100 and 150 (Re is based on the length of moving square box and maximum velocity of the box) and WCSPH only at Re=100. SPH simulations are performed with 600×300 particles in order to have identical grid resolution as the 600×300 cells in the FD solution. Time steps for ISPH and WCSPH compared to FD are indicated in Table 1.

Table I: Time steps for ISPH and WCSPH (GT is the time step in FD

(=0.9009E-02s)).

ISPH WCSPH

Re 50 100 150 100 Time step GT/4 GT/2 GT/10

Two physical instants are presented,

the first one at t=5s, where the square cylinder motion is not influenced by the end wall of the box and the second one at t=8s, before its impact on the box right wall.

2.1. SPHERIC test case 6 at Re=100

Comparisons of WCSPH, ISPH and FD are shown in Fig. 1 for velocity magnitude. All three patters look similar but the maximum backflow behind the cylinder is one level lower in WCSPH. Also, voids appear just behind the front corners of the moving box, a phenomenon which is cured in ISPH (see Issa et al. (2005) for more details).

Pressure fields are shown in Fig. 2. WCSPH shows extreme levels of fluctuations which cannot be interpreted as a physical pressure field, mainly in the wake of the square box. ISPH pressure pattern compare realistically with FD, even if the peak in front and depletion at the rear appear more smeared.

At t =5s (from the top WCSPH, ISPH and FD)

At t =8s (from the top WCSPH, ISPH and FD)

Figure 1: Velocity magnitude for Re=100.

39

At t =5s (from the top WCSPH, ISPH and FD)

At t =8s (from the top WCSPH, ISPH and FD)

Fig. 2: Pressure for Re=100.

2.2. SPHERIC test case 6 at Re=50 and 150

Comparisons between ISPH and FD only are now presented at Re=50 (Figs. 3 and 4) and 150 (Figs. 5. and 6) at t=5s and t=8s. The same fair agreement as previously is observed for the wake and acceleration around the corners. However

the sharp angle of the iso-lines in front at the symmetry line is in contrast with the rounded FD solution and is seen to be exacerbated at the lower Re number (Fig 3. t=5s). Similarly, the ISPH pressure fields are acceptable for Re=100 and 150 but for Re=50 the stagnation point pressure build-up propagates too far upstream.

At t =5s (top: ISPH, bottom: FD)

At t =8s (top: ISPH, bottom: FD)

Figure 3: Velocity magnitude for Re=50.

At t =5s (top: ISPH, bottom: FD)

40

At t =8s (top: ISPH, bottom: FD)

Figure 4: Pressure for Re=50.

At t =5s (top: ISPH, bottom: FD)

At t =8s (top: ISPH, bottom: FD)

Figure 5: Velocity magnitude for Re=150.

At t = 5s (top: ISPH, bottom: FD)

At t = 8s (top: ISPH, bottom: FD)

Figure 6: Pressure for Re=150.

In other words, it is as if the particles

were experiencing higher resistance to pressure induced acceleration than is realistic at this Re=50. Indeed, in Fig. 7 which show the history of pressure drag coefficient on the square we can observe that this one is systematically overestimated for Re=50 whereas for the higher Re the ISPH predictions are scattered around the FD solution. While a major improvement has been obtained by switching to a truly incompressible solver, it would be interesting to investigate alternative formulation of the viscous term (e.g. Morris et al. 1997) on this test case. Moreover, the new Laplacian equation solver can also be adapted to test the effect of solving these viscous terms implicitly.

41

Figure 7: Time histories of pressure drag

coefficients.

3. Conclusions Comparisons of ISPH and FD in the

case of a moving square box set in a rectangular box (SPHERIC test 6) are performed at Re=50, 100 and 150. Generally speaking, the results from ISPH are close to FD’s. ISPH shows a major improvement compared to WCSPH in computing flows involving bluff bodies, as no voids and no pressure fluctuations appear with ISPH. The void might be cured by increasing the numerical speed of sound, but the pressure fluctuation still remains. ISPH pressure pattern is smooth and looks fairly similar to FD’s. For the lower Re number case the pressure drag seemed to be overestimated and this

needs to be investigated further for different resolutions or discretisations of the viscous forces.

4. References CHORIN, A.J. (1968) Numerical solution of the Navier-Stokes equations. Math. Comp., 22: 745-762. CUMMINS, S. J. and RUDMAN, M. (1999). An SPH projection method. Journal of Computational Physics, 152: 584-607. ISSA, R., LEE, E.-S., VIOLEAU, D. and LAURENCE, D.R. (2005). Incompressible separated flows simulations with the Smoothed Particle Hydrodynamics gridless method. Int. J. Numer. Metho. Fluids, 47: 1101-1106. LEE, E.-S., MOULINEC, C., VIOLEAU, D., LAURENCE, D. and STANSBY, P. (2007) Simulations of 2-D laminar flow past a bluff body in a closed channel with a truly incompressible SPH. 32nd Congress of IAHR in Venice, Italy. MONAGHAN, J.J. (1994) Simulating free surface flows with SPH. Journal of Computational Physics, 110: 399-406. MORRIS, J.P., FOX, P.J. and ZHU, Y. (1997) Modelling low Reynolds number incompressible flows using SPH. Journal of Computational Physics, 136: 214-226. SHAO, S. and LO. E.Y.M. (2003). Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources, 26: 787-800. VAN DER VORST, H.A. (1992) Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmmetric linear systems, Siam J. Sci. Stat., 13: 631-644.

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42

Smoothed Particle Hydrodynamics with radiative transfer in the flux-limited diffusion approximation

Dr Stuart C. Whitehouse 1, 2

1 School of Physics, University of Exeter, Exeter, EX4 4QL, U.K. 2 Currently at Institut für Physik, Universität Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

Abstract I describe an implicit algorithm for implementing radiative transfer using the flux-limited diffusion approximation within the smoothed particle hydrodynamics (SPH) formalism (Whitehouse et al. 2005). This algorithm uses a flux-limiter to restrict the speed of radiation progagation to the speed of light in optically thin regions. The algorithm uses implicit timesteps to enable the hydrodynamic timestep to be used. This algorithm was implemented in a three-dimensional SPH code, and calculations studying the collapse of molecular cloud cores were performed (Whitehouse & Bate, 2006). It was found that although the specific heat capacities and opacities of the gas were identical for all calculations, the initial density- and velocity-distributions had a significant effect on the temperature evolution, even in spherically-symmetric cases. Previous simulations which used a barotropic equation of state are shown to underestimate the gas temperature. In general, the three-dimensional temperature distribution has a significantly different structure than that given by a barotropic equation of state.

1. Introduction and Previous Work

SPH is a powerful tool for studying hydrodynamical problems in astrophysics. However, a number of astrophysical phenomena require the inclusion of other

processes to accurately simulate their behaviour. One of these processes is the transfer of energy by radiation – radiative transfer.

There have been a few prior attempts to include radiative transfer in the SPH formalism, beginning with that of Lucy (1977), who used the diffusion approximation to examine the fission of protostars. Lorrimer (1983) followed on from Lucy's work, greatly increasing the number of particles used. This work showed that the particle distribution has a large effect on the accuracy of second derivatives in SPH.

Brookshaw (1985, 1986) also worked with the diffusion approximation, introducing the technique of using a Taylor expansion to reduce the second derivative to a first-order derivative. This technique was further improved on by Cleary & Monaghan (1999) who developed a similar technique for conduction, and it is this technique upon which this work is based.

Viau (2001) and Bastien et al. (2004, 2006), present an implicit scheme for the diffusion approximation. This scheme assumed that the temperature of the radiation and the gas were equal, and also made no attempt to correct the diffusion approximation in optically thin regimes where it becomes unphysical.

This extended abstract will describe the scheme of Whitehouse et al. (2005), based on that of Whitehouse & Bate (2004). This scheme has separate temperatures for the gas and radiation, and also includes a flux-limiter (Levermore & Pomraning, 1981; Turner & Stone, 2001) to

43

constrain the propagation velocity of photons to the speed of light in optically thin regions.

2. Method Adding radiative transfer to SPH adds

an additional variable, E, the radiation energy density (though for SPH we prefer the specific radiation energy ξ), and an additional equation describing its time evolution. Also, there are additional terms in the momentum equation describing the radiation pressure, and in the (gas) energy equation, which includes the effect of energy transfer between the gas and the radiation. These form the equations of radiation hydrodynamics (see, e.g., Mihalas & Mihalas, 1984).

The new radiation energy evolution equation can be written as

where the first term on the right hand side is the diffusion of the flux F, the second the radiation pressure term, and the final term controls the exchange between the specifc gas energy u and the specifc radiation energy ξ. Here a is the radiation density constant, κ is the opacity, ρ is the density c is the speed of light, and cv is the specific heat capacity. The gas energy equation is modified by adding the exchange term above with the opposite sign.

In the diffusion approximation, the relationship between the flux F and the radiation energy density E=ξ /ρ is modelled by

( ) EF ∇−

=κρ

c

ignoring scattering.From the diffusion term above, this gives the evolution of the radiation energy density as being

⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅∇= EE

χρ3c

DtD

We define the specific radiation energy ξ to be E/ρ for SPH. This second-order spatial derivative is very sensitive to particle disorder. Work on conduction, which has a similar double spatial derivative, by Cleary and Monaghan (1999) reformulated the conduction equation to first-order, and gave results in good agreement with analytic models of tests problems, even when discontinuities were included.

Whitehouse & Bate (2004) used a similar formulation for radiative transfer in the diffusion approximation, including a flux-limiter. This was an implicit method, enabling radiative transfer (which occurs on a timescale related to the speed of light) to be done using the same timestep as the hydrodynamics (which occur on a much longer timescale related to the speed of sound). This method however was extremely slow, involving the exchange of energy between pairs of SPH particles iteratively.

Whitehouse et al. (2005) developed a new formulation using the Gauss-Seidel method to iterate towards a solution for the system of equations. The method then solved both the gas and radiation energy equations simultaneously, which involved the solution of a quartic (fourth-order polynomial) equation, as the term describing the energy exchange between the gas and the radiation depends on the temperature to the fourth power. As quartic equations can be solved analytically, we use an analytic solver rather than an iterative method like Newton-Raphson which may slow the calculation down further. This method can then be iterated to convergence.

3. Test Results Our implicit method was tested using

a one-dimensional SPH code written

44

especially for the purpose. It was first tested using a shock tube test, to investigate the behaviour at different opacities. We then performed tests based on those Turner & Stone (2001) used to test their radiation hydrodynamics module for ZEUS-2D to check the algorithm in more detail.

Figure 1 shows a shock tube test (from Whitehouse et al. 2005), where two flows travelling at the speed of sound in opposite directions collide at x = 0. The domain was 2×1015 cm long, the initial density was 10-10 g cm-3, the initial temperature was 1500 K, and there were 100 particles. Ghost particles were placed outside the boundaries and maintain the initial quantities of their respective real particles.

Figure 1: A set of shocks with differing opacity at

time t = 1.0×109 s. The plots show the gas temperature at a given position. The crosses are

the SPH results, while the solid line gives the analytic solution for an adiabatic shock, and the dashed line the solution of an isothermal shock. Opacity decreases from optically thick at the top

of the figure to thin at the bottom.

The gas and the radiation are highly coupled in this test and hence their temperatures are equal. The opacities used are 40, 0.4, 4×10-3 and 4×10-5 cm2 g-

1. The calculation took less than a minute for the optically thick case, up to twenty-three minutes for the optically thin case. The SPH results are depicted by the crosses in the figure, and the analytic solutions of the problem (e.g. Zel'dovich & Raizer, 2002) are drawn as lines in the adiabatic (solid line) and isothermal (dashed line) cases.

As the figure shows, the optically thick and optically thin extremes match their respective analytic solutions well. The temperature shows a smooth transition between the adiabatic and isothermal regimes as the opacity decreases.

Figure 2: A supercritical shock with piston

velocity 1.6×106 cm s-1 and 100 particles. The large panel shows the results of an explicit

radiative transfer calculation, with radiation (solid line) and gas (dotted line) temperatures plotted as a function of optical depth. The sub-panel

45

shows the logarithm of the difference between the explicit code and our implicit algorithm with

different timesteps. The long dashed line shows, for example, the error of the implicit algorithm

using the hydrodynamic timestep. Figure 2 shows a supercritical shock based on the one from Turner & Stone (2001), where radiation is a dominant effect. A piston is proceeding from the right, at a speed of 1.6×106 cm s-1 into a medium of density 7.78×10-10 g cm-3 and opacity 0.4 cm2 g-1. Here the shock is violent enough for radiation to preheat the matter flowing into the shock (to the left). The shock is at an optical depth of zero. The dashed line is an analytic solution of the problem from Turner & Stone (2001), the solid line is the radiation temperature, and the dotted line is the gas temperature. The sub-panel shows the logarithm of the difference between the explicit code and our implicit algorithm with different timesteps. The long dashed line shows, for example, the error of the implicit algorithm using the hydrodynamic timestep. The other linetypes show the results with a timestep of one tenth (long-dashed line) and one hundredth (short-dashed line) times the hydrodynamic timestep. As the sub-panel indicates, the results of our implicit algorithm with the hydrodynamic timestep are quite close to the explicit results.

4. Application As mentioned previously, radiative transfer is important in many areas of astrophysics. We have applied it to the collapse of a molecular cloud core (Whitehouse & Bate 2006). This collapse occurs when interstellar gas becomes gravitationally unstable, and leads to the formation of one or more stars (or brown dwarfs). The algorithm was implemented in a pre-existing SPH code used for star formation, and included physically realistic opacities for both gas and dust, and specific heat capacities.

Figure 3: We show the evolution of the

maximum gas (solid line) and radiation (dotted line) temperatures at the maximum density

during a simulation of the collapse of a molecular cloud core (see Whitehouse & Bate 2006; the initial conditions are from Boss & Myhill 1992).

For comparison, the short-dashed line gives the temperature-density relation of the barotropic

equation of state used by Bate 1998. Figure 3 shows the evolution of the maximum temperatures (the solid line is the gas temperature, and the dotted line the radiation – the temperatures are equal for most of the simulation) during a three-dimensional simulation of the collapse of a molecular cloud core in a simulation of star formation. This was performed using 500,000 particles. Previous simulations of this nature have typically relied on a barotropic equation of state to mimic the effect of radiative transfer (dashed line).

This figure shows how the evolution of the maximum temperature differs when our algorithm is used for radiative transfer. The barotropic equation of state underestimates the maximum temperature by a factor of two or three. Additionally, the use of a barotropic equation of state only successfully approximates the temperature evolution of the particle with the maximum

46

temperature. The three-dimensional structure given by our code is significantly different from that given by calculations with a barotropic equation of state, due to the radiation of energy from higher temperature regions to lower ones. The temperature away from the maximum can be up to an order of magnitude hotter than a barotropic equation of state would suggest.

5. Conclusions An algorithm for including radiative transfer in the diffusion approximation will be explained in the talk. In test calculations this implicit method shows good agreement with both analytic solutions and an explicit version of the algorithm. The algorithm involves the simultaneous solution of both of gas and radiation energy equations, and is implicit, so the hydrodynamic timestep can be used. The algorithm is fast enough to run an astrophyical simulations of the collapse of a molecular cloud core in a reasonable amount of computer time.

6. Acknowledgments This work formed the PhD thesis of

SCW at the University of Exeter under the supervision of Prof Matthew R. Bate. This PhD was funded by the UK's Particle Physics and Astronomy Research Council. Preparation for and travel to this workshop has been funded by Swiss National Science Foundation grant number PP002-106627.

7. References BASTIEN, P., CHA, S.-H., VIAU, S., 2004, Revista Mexicana de Astronomia y Astrofisica Conference Series BASTIEN, P., CHA, S.-H., VIAU, S., 2006, ApJ,

639, 559 BOSS, A. P., MYHILL, E. A., 1992, ApJS, 83, 311 BROOKSHAW, L., 1985, Proc. Astron. Soc. Austr., 6, 207 BROOKSHAW, L. 1986 Proc. Astron. Soc. Austr., 6, 461 LEVERMORE, C. D., POMRANING, G. C., 1981, ApJ, 248, 321 LORRIMER, G. S., 1983, PhD thesis, Monash University LUCY, L. B., 1977, AJ, 82, 1013 MIHALAS, D., MIHALAS, B. W., 1984, Foundations of Radiation Hydrodynamics, Oxford University Press TURNER, N. J., STONE, J. M., 2001, ApJS, 135, 95 VIAU, S., 2001, PhD thesis, University of Montreal WHITEHOUSE, S. C., BATE, M. R., 2004, MNRAS, 353, 1078 WHITEHOUSE, S. C., BATE, M .R., MONAGHAN, J. J., 2005, MNRAS, 364, 1367 WHITEHOUSE, S. C., 2005, PhD Thesis, University of Exeter WHITEHOUSE, S. C., BATE. M. R., 2006, MNRAS., 367, 21. ZEL'DOVICH, Y. B., RAISER, Y. P., 2002, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, ed. Hayes & Probstein. Dover, New York.

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SPHERIC benchmark test case number 5: sensitivity analysis to numerical and physical parameters

Damien VIOLEAU1 and Réza ISSA1

1 Research Engineers, LNHE / EDF R&D, 6 quai Watier 78400 Chatou, France

Abstract SPHERIC test case number 5 has been successfully simulated. Results are globally in a good agreement with experiments, although the shape of the free surface cannot be exactly reproduced after breaking. A sensitivity analysis was carried out to investigate the effect of turbulent closure and numerical approximations of a few physical terms.

1. Introduction Benchmark test cases have been proposed to the SPHERIC community in order to validate codes and study the ability of the available models to predict different types of flows. The test case number 5 consists of a water column collapsing in a tank with a wet bed, and seems interesting to study complex free-surface flows, since it is well documented with accurate observations of the free surface.

When performing computations, it is sometimes uneasy to decide which set of parameters is better. Besides, one should clearly distinguish physical and numerical parameters. In this study we investigate the effects of different SPH models on case number 5. We firstly examine different turbulent closure models, then various approximations of physical terms.

2. Numerical setup

2.1. Model description

We use the Spartacus-2D code, developed at EDF R&D and based on the classical SPH model proposed by Monaghan (1994) applied to weakly compressible free surface flows. It considers conservative (i.e. symmetric) forms of the momentum equation with viscous terms. Pressure forces between particles a and b are modelled through the following two possible forms:

abab

b

a

aba

pab w

ppmm ∇⎟

⎟⎠

⎞⎜⎜⎝

⎛

ρ+

ρ−=→ 22F (1)

ababa

baba

pab w

ppmm ∇

ρρ+

−=→F (2)

while viscous forces are modelled through the following two approaches:

abaab

abab

ba

baba

vab w

rmm ∇

⋅ρ+ρν+ν

=→ 28ru

F (3)

abaabab

ab

ba

bbaaba

vab w

rmm ∇⋅

ρρνρ+νρ

=→ ru

F 2 (4)

Pressure is estimated from a state equation based on a speed of sound. Solid boundaries are based on boundary and fictitious particles whose motion is prescribed by the user in case of moving walls.

Turbulent effects are also considered through several models, hereafter listed by increasing complexity:

- constant eddy viscosity model; - mixing length model; - one-equation model; - two-equation model (k-ε); - non-linear k-ε model.

48

All these models consist of considering Reynolds-averaged velocities and pres-sure, Reynolds stresses being modelled with an eddy viscosity assumption. The latter is estimated differently in each model (see Violeau and Issa 2007). An important parameter for all of them is the scalar mean rate-of-strain S, defined as

SS :2=S (5)

in which S is the symmetric part of the velocity gradient tensor. We use two models to estimate this quantity, the first of which consists of calculating the velocity gradient components by

( ) ∑ ∇⊗ρ

−=∇b

abaabba

a wm uu 1 (6)

then applying (5). The second method gives directly S as

∑ ∇⋅

ρρρ+ρ

−=b

abaabab

ab

ba

baba w

ru

mS r2

2

21 (7)

2.2. Test conditions

The case is described in Janosi et al. (2004) and on SPHERIC website. It consists of a 1.18 m long tank initially filled with a water layer of d = 1.8 cm or 3.8 cm at rest. A gate contains a water column of length 38 cm and depth 15 cm (figure 1) and opens at time t = 0 to let the water collapse into the tank. We chose an initial particle spacing of 1 mm, so that the total number of particles approaches 80,000. The maximum velocity in the flow was estimated (and checked) to be 2.4 m/s; hence the numerical speed of sound was set to 24 m/s.

15 cm

38 cm

1.18 m

d Gate

Figure 1: Case geometry.

Results were output every 0.062 s to match experimental measurement times. Experimental results concern the shape of the free surface. A small gap in time (about 43 ms) exists between the experiments and the computed results; for example the first presented graph (figure 2) was numerically obtained at t = 0.176 s (instead of t = 0.219 s, as indicated on the experimental picture).

3. Results

3.1. Sensitivity to turbulence closure

We firstly tested the case with an initial water layer of thickness 1,8 cm. Generally speaking, the model performs well, as shown by figure 2 with the one-equation turbulent model using models (2) for pres-sure, (3) for viscosity and (7) for the rate-of-strain (this choice will be considered in the following as a reference set of parameters).

Figure 2: Shape of the flow at different stages with a one-equation turbulent model

49

(reference case). The solid lines mimic the experimental free surface.

However, during the last stages of the

flow, discrepancies occur between observations and simulations: our model fails in representing correctly the splash-up after breaking.

A zoom at two different times is presented on figure 3 for the reference set of parameters, only changing the turbulent closure model. Generally speaking, in this case turbulence has a small effect; however, increasing the model complexity (from top to bottom) tends to improve the quality of predictions, especially in the vicinity of the jet just before breaking. Similar conclusions concern the shape of the void bubble appearing after the first breaking. This is confirmed with the case d = 3.8 cm presented on figures 4 and 5.

Figure 3: Effect of the turbulent model. From top to bottom: experiment; constant

eddy viscosity; mixing length; one-equation model; k-ε model; non-linear k-ε model.

Figure 4: Case with d = 3.8 cm. From top to bottom: experiment; constant eddy

viscosity; one-equation model; non-linear k-ε model.

Figure 5: Same legend as figure 4 at a later stage.

It seems that the one-equation model presents the best compromise between accuracy and simplicity. Thus, in the next section we will consider it as the reference choice.

3.2. Sensitivity to numerical parameters

50

Starting from the reference case, we successively tested model (1) for pressure, model (4) for viscosity and model (6) for the scalar rate-of-strain. The results are presented on figure 6, showing only small effects. Only model (6) for the rate-of-strain seams to improve significantly the results.

Figure 6: Effect of numerical models on the reference case. From top to bottom:

experiment; reference parameters; eq. (1) for pressure forces; eq. (4) for viscous

forces; eq. (6) for rate-of-strain.

4. Conclusions The SPHERIC test case number 5 has been successfully simulated with our SPH code Spartacus-2D. A sensitivity analysis indicates that the best turbulent model in this case is probably a one-equation model (Violeau and Issa 2007). The choice of numerical model for pressure and viscous forces, as well as rate-of-strain computation, has a rather little effect.

The present tests cannot capture correctly the splash-up. It is still not clear whether this lack of accuracy is due to

physical or numerical modelling. One could invoke, as physical reasons, the following:

- renormalization is not used; - discretization is maybe too poor; - we used a weakly compressible

model, while exact incompressible algorithms exist (see e.g. Lee et al. 2007).

On the other hand, in terms of

physical modelling, the following processes could be examined:

- air entrainment during breaking; - surface tension modelling (the scale

being here rather small); - free-surface boundary conditions

should be improved. In the future, we will investigate air

entrainment and renormalizations, which both seem important feature for this type of applications of SPH.

5. References JANOSI I.M., JAN D., SZABO K.G. and TEL T. (2004). Turbulent drag reduction in dam-break flows. Experiments in Fluids 37: 219-229. MONAGHAN J.J. (1994). Simulating free surface flows with SPH. J. Comput. Physics, 110: 399-406. LEE E.S., MOULINEC C., VIOLEAU D., LAURENCE D. and Stansby P. (2007). Comparisons of weakly compressible and truly incompressible SPH algorithms for 2D flows. Submitted to J. Comput. Phys. VIOLEAU D. and ISSA R. (2007). Numerical modelling of complex turbulent free surface flows with the SPH Lagrangian method: an overview. Int. J. Num. Meth. Fluids 53(2): 277-304.

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51

Interactive Visualization and Exploration of SPH Data John Biddiscombe1, David Graham2, Pierre Maruzewski3

1 Swiss National Supercomputing Centre (CSCS), Manno, Switzerland 2 School of Mathematics and Statistics, University of Plymouth, UK

3 Laboratory for Hydraulic Machines, Ecole Polytechnique Fédérale Lausanne, Switzerland

Abstracti Advances in graphics hardware in recent years have led not only to a huge growth in the speed at which 3D data can be rendered, but also to a marked change in the way different data types can be displayed. In particular, point based rendering techniques have benefited from the advent of vertex and fragment shaders on the GPU which allow simple point primitives to be displayed not just as dots, but rather as complex entities in their own right. We present a simple way of displaying arbitrary 2D slices through 3D SPH data by evaluating the SPH kernel on the GPU and accumulating the contributions from individual particles intersecting a slice plane into a texture. The resulting textured plane can then be displayed alongside the particle based data. Combining 2D slices and 3D views in an interactive way improves perception of the underlying physics and speeds up the development cycle of simulation code. In addition to rendering particles themselves, we can improve visualization by generating particle trails to show motion history, glyphs to show vector fields, transparency to enhance or diminish areas of high/low interest and multiple views of the same or different data for comparative visualization. We combine these techniques with interactive control or arbitrary scalar parameters and animation through time to produce a feature rich environment for exploration of SPH data.

1. Introduction The rapid development of the power of GPUs in desktop computers in the last few years has produced an explosion in the number of triangles that can be rendered per second. In addition, advances in the architecture of graphics processors have led also to programmable shaders which allow a far more flexible approach to the generation of images. Instead of representing particles as collections of triangles, it is possible to render them directly to screen by supplying only a position, radius and colour (or other combination of scalar parameters of interest). The shader which resides as a small program on the GPU can evaluate a sphere function and perform an intersection between an eye ray through the screen with the sphere (for each pixel) to produce an exact image of the particle data. Observation of this capability leads naturally to the idea of evaluating more sophisticated functions such as the SPH kernel itself on the GPU and producing an even more useful image. In fact evaluating the kernel on the GPU has been done many times [3,4,5], but in these cases, the SPH field equations are evaluated for the purpose of animating the particles themselves rather than displaying the results of the simulation. In [6], the kernel is evaluated and the particles are re-used directly from the GPU to generate the surface using point splatting. The work of Sigg et al [2] provides the starting point for our implementation of a particle renderer and for an SPH slicing algorithm on the GPU. Their key

52

development is to represent a quadric (sphere, cylinder, cone, ellipsoid or even parabolic surface) as a 4x4 matrix that can be combined with the usual transform and lighting pipeline of graphics hardware. Using their approach we have developed a particle rendering tool which produces high quality spheres at interactive frame rates. Since the particles are already passed to the GPU for display, we wish to extend the capabilities by performing a second pass of the renderer which can produce slices through the data and display the field as a continuum rather than as discrete points. The SPH kernel is spherically symmetric – having one parameter of interest (from the point of view of visualization) – which is the cut-off radius, we may therefore represent our particles as spheres and render them with two clip planes positioned equidistant from the slice plane – one in front, the other behind with the distance from the slice to clip plane chosen to be exactly one radius of the kernel used. Any particle further than this distance on either side is automatically removed and no evaluation of the kernel is required. Figure 2 shows a simple schematic of the regions of interest of the particles between clip planes.

Figure 2 : Particles rendered between two clip

planes.

Since we are only interested in the field strength on the plane itself, only portions which overlap the slice plane need be evaluated.

In the following section we describe the operation of the vertex and fragment shaders which compute these regions and colour them appropriately.

2. Vertex and Fragment Shaders 2.1. Vertex Shader

The computation of the kernel must be avoided whenever possible and we therefore wish to limit all calculations to regions where the particle projects onto the slice plane as shown in Figure 2. This calculation is performed in the vertex shader - the calculation of the bounding box projection for an arbitrary quadric is given in [2], but we can make the following assumptions • The projection onto the slice is

orthographic and therefore we need not consider any perspective correction

• The particles are spherical and the bounding box is simply the scaled transformation of the particle radius from world coordinates to the pixel coordinates of the slice.

• The slice has isotropic rectilinear pixels and we can ignore ellipsoidal terms.

This considerably simplifies the expression for the necessary pointsize R and reduces it to

ws

wk

wllrR 2

=

Where rk is the kernel radius, lw is the distance in world space along the primary axis of the slice, ls is the length in pixels of the slice, and ww is the homogenous clip coordinate of the transformed point. The transformation used for the points is given by placing the eye e, and viewpoint v, at

,, zyx cccv =r ,

ksrnve r−=

Where c represents the slice centre and

snr is the slice plane normal. All that

remains is to specify the front and rear clip planes at 0, 2rk and the ‘up’ vector is along the second slice axis. The vertex

Clip planes

Slice plane

Clipped points

rk

dp

Intersection region

Overlap region

Projection

53

shader computes the transformed coordinate and any points lying outside the clip region are removed before being passed to the fragment shader. To save time in the fragment shader we compute the (non-varying) distance

wzp wwd /= from the point to the plane

using the coordinate wz in clipping space.

2.2. Fragment Shader

The fragment shader operates on a square region of pixels given by the pointsize computed in the vertex shader. For each pixel we must compute the distance from the point centre and plug this into the kernel if it lies inside a unit circle inscribed within the box – since the box has been generated using the kernel smoothing radius, we know that the incircle exactly fits the smoothing radius. Off axis particles have a smaller intersection with the slice plane and we must only consider pixels with distance di within a smaller ring within rp

xxd ii −= , 22 1 pp dr −=

For di>rp the fragment is outside the intersection ring and is discarded. For di<=rp we must use the true distance dk to the kernel centre given by 222

pik ddd +=

To correctly display the results of the kernel evaluation, we must sum the values from overlapping kernels. This is done by rendering the scene as described into an OpenGL FrameBuffer object using a floating point texture and enabling blend mode to sum the incoming pixels. We have currently implemented cubic spline and cusp kernels, others can easily be added. The result of the fragment shader is a correctly smoothed representation of the particles along the slice. Compare the images of Figure 3 which show the same data rendered using a triangulation of the points and the output of the SPH slice render. The flaws in the triangulated version are obvious and the smoothed one is clearly superior.

Figure 3 : (a) Continuous field plot using

Delaunay triangulation of the raw points. (b) Particles overlayed on a field plot produced by

SPH smoothing.

3. GUI Implementation The output of the shaders is a correctly smoothed representation of the particles along the slice. We have embedded the tool within the ParaView [1] and sparticles GUIs using a PlaneWidget to interactively allow the user to freely move the slice plane through the data at any orientation. The output of the shader is stored in a texture which is mapped onto the plane and displayed interactively with the data. This gives the user the ability to view any slice through the data. Figure 4(b) shows an example of a large injection dataset (1 million points) rendered using transparency to give a volumetric effect. For volumetric plots, the opacity, colour and even radius of particles may be controlled on a per particle basis using any sets of scalar variables. The plot has been enhanced with a slice plane (the interactive handles are also shown) which in this case reveals features away from the primary jet. We have also implemented particle trails which display the pathlines of individual particles. The trail length is configurable and a subset of particles may be used for clarity of presentation. Figure 4(a) shows particle pathlines for the intermediate stages of a lid-driven cavity flow simulation (Re=1000). The pathlines are not quite closed, indicating that the solution has not yet reached a steady-state. The lack of pathlines in the bottom corners is an indication of the very weak recirculation in these areas.

54

Figure 4 : (a) Particles with pathlines (b)

Transparent rendering of particles. ParaView has many useful features in visualising SPH results. For example, comparisons of results using different numbers of particles can be displayed simultaneously. This is useful in determining whether the higher resolution is necessary. Figure 5(a) displays three aspects of the same computation at the instant when a wave impacts against a vertical wall. It shows that high values of pressure, turbulence viscosity and density are found at the moment of impact. Figure 5(b) shows particle mixing as a result of paddle motion and wave breaking, displaying results at three different times. Such figures and associated animations can be generated easily in ParaView. Many other features such as vector plots, glyphs, streamlines, and contouring can be found within the extensive library of filters available in ParaView.

Figure 5: (a) Simultaneous view of (top to

bottom) pressure, viscosity and density at wave impact. [Red=high, blue=low]. (b) Particle mixing during breaking of 2s waves (t=0, t=c5s, t=c10s)

4. Conclusions We have implemented a particle renderer which produces high quality output and enhanced it with capabilities for producing SPH specific plots. We have built this functionality into a custom rendering tool, sparticles, and also into a customized

version of the ParaView visualization package making it not only accessible to many users, but also providing a range of other visualization algorithms and animation tools that allow the SPH results to be combined with other conventional data and visualizations. Acknowledgements The second author acknowledges Dr Jason Hughes at the University of Plymouth for some of the computations and Dr Songdong Shao at the University of Bradford for the original I-SPH code.

5. References A.H. SQUILLACOTE, The ParaView Guide: A Parallel Visualization Application, Kitware Inc. 2006; www.paraview.org CH. SIGG, T. WEYRICH, M. BOTSCH, M. GROSS, GPU-based ray-casting of quadratic surfaces, Eurographics Symposium on Point-Based Graphics 2006 T. AMADA, M. IMURA, Y. YASUMURO, Y. MANABE and K. CHIHARA, Particle-Based Fluid Simulation on GPU, ACM Workshop on General-Purpose Computing on Graphics Processors and SIGGRAPH 2004, LA. California, 2004. ANDREAS KOLB, NICOLAS CUNTZ . Dynamic Particle Coupling for GPU-Based Fluid Simulation. Proc. 18th Symposium on Simulation Technique, 2005 M. MULLER, B. SOLENTHALER, R. KEISER, M. GROSS. Particle-Based Fluid-Fluid Interaction SIGGRAPH/Eurographics Symposium on Computer Animation 2005 MÜLLER M., CHARYPAR D., GROSS M.: Particle-based Fluid simulation for interactive applications. In SIGGRAPH/ Eurographics Symposium on Computer Animation 2003, pp. 154-159.

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Pressure measurement in 2D sloshing simulations with SPH

L. DELORME1, M.A. CELIGUETA2, E. OÑATE2, A. SOUTO-IGLESIAS1

1 Model Basin Research Group (CEHINAV), Naval Architecture Dpt (ETSIN), Technical University of Madrid (UPM). Avda Arco de la Victoria s/n, 28040 Madrid, SPAIN.

2 International Center for Numerical Methods in Engineering (CIMNE), Universidad Politécnica de Cataluña, Gran Capitán s/n, 08034 Barcelona, Spain

Abstract

Sloshing for low filling level resonant pitch motion is studied experimentally and numerically using SPH. Special attention is paid to the pressure fields on the tanks. Comparisons are made with experimental data and with Particle Finite Element Method (PFEM) calculations.

1. Introduction Extensive experimental programs

aimed at a better comprehension of the sloshing loads have been conducted for the last 30 years (Bass, 1985, Berg, 1987). The reason for this interest lies mainly in the influence of these loads in the design and operation of LNG tankers. CFD technologies are helping in the understanding of these loads, usually tracing the free surface evolution by VOF techniques (Kleefsman, 2005), but to date, it is difficult for these techniques to model fragmentation and compressibility effects, which are crucial during the impact. Meshless methods like SPH (Monaghan, 2005) can be especially appropriate when modelling the highly non linear free surface flows with impact and fragmentation that appear in violent

sloshing flows. This short paper focuses on the assessment of these local loads, following a previous one from the same group (Souto-Iglesias, 2006), in which global loads were successfully reproduced. SPH results are compared with experiments and with monophasic PFEM results (Idelsohn, 2007) for the same case.

2. Experimental results The case studied is a 2D longitudinal

section of a tank that belongs to a 138 000 m3 LNG membrane tanker in operation, at scale 1:50. Model dimensions are 90 x 58 x 5 cm and water depth is 9.3 cm (depth ratio ≈ 0.1). The tank is excited with a sinusoidal type motion (θmax = 4º) whose period matches the first sloshing period T0 = 1.9 s.

The flow is composed by a main wave, travelling from one side of the tank to the other, forming a plunging-type breaker at half way that impacts on the structure. The dissipation due to breaking is high and the experiments demonstrate that the water motion in the tank is qualitatively periodic, including the breaking process.

Figure 1: Experimental angle and pressure versus time (non-dimensional values)

56

Figure 1 shows the angle and pressure time series. In the following, the angle is made non-dimensional with θmax, the time with T0 and the pressure with the hydrostatic one. The pressure sensor is located at the unperturbed free-surface height. The pressure register is qualitatively repetitive at each cycle. However, the maximum value of the pressure is not equal in each cycle. These peaks result from the impact of the wave on the tank, presenting a random behavior. This can be explained by the very short duration of the impact and the extreme sensitivity of the impact pressure to the shape of the wave just before impact. Other physical parameters, such as the compressibility of the air and water mixture as well as the ullage pressure, have also a very important effect (Bass, 1985, Berg, 1987) and are very difficult to model

A zoom of the time series over one impact event is shown in figure 2. Frames F1 to F6 have been located on the pressure curve representing the most interesting instants regarding the pressure history. Pressure register and videos

demonstrate this process to be qualitatively repetitive.

3. Simulations A standard SPH formulation has been

used for the simulation (Monaghan, 2005). Free slip boundary conditions have been imposed with boundary particles (Monaghan, 2005). In order to calculate the pressure at the sensor position, the forces exerted by all the boundary particles within a distance h to the center of the sensor have been averaged, h being the smoothing length. The standard viscosity term is used with α=0.02. Numerical integration has been performed with a leap-frog scheme.

Simulations have been performed with 5 different resolutions: 3043, 4928, 8970, 12924 and 20205 fluid particles. Figure 3 presents the pressure time series for three resolutions. The graph shows that the trends in the experimental curve are qualitatively reproduced with SPH. However, numerical instabilities appear that need further study.

Figure 2: Experimental register above one impact event with the corresponding frames

57

Figure 3: Non-dimensional pressure over one impact event. SPH results PFEM results for the same case

are presented in figure 4. The shape of the pressure curve is qualitatively reproduced too. PFEM results present numerical instabilities of greater amplitude and frequency. Pressure maxima at the impact are greater and this can be explained by the incompressibility of the fluid, imposed when using PFEM.

Figure 4: Non-dimensional pressure over one impact event. PFEM results

The compressibility of the fluid

plays an important role in the impact phenomena (Bass, 1985). This has been investigated performing SPH simulations with different numerical sound speeds. Sound speed is typically chosen such that the Mach number is 0.01. SPH simulations have been

performed using sound speeds 10, 20, 30 and 40 m/s but the variations found in the values of the pressure peaks were not significant.

It has been demonstrated (Peregrine, 2005) that the impulse given by a wave is a more useful information than the pressure in assessing its impact. The pressure impulse (integral of the pressure through the impact) can be calculated from the pressure time series and compared with the experiments (figure 6). After the third cycle, the variations of the impulse are small and it can be noticed that both SPH and PFEM overestimate the experimental value. The biphasic nature of the impact could explain the lower experimental values but further investigation has to be done.

Figure 6: Pressure impulse.

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Figure 7: SPH simulation with 20205 fluid particles. F1 to F6 refer to figure 2

The global dynamics of the flow,

including breaking waves, is well reproduced by both methods. Figure 7, for instance, presents the frames of figure 2 obtained with SPH, showing good agreement, even after more than eight cycles.

4. Conclusions Numerical computations of long

impact pressure sequences for a 2D low filling sloshing case have been performed both with SPH and PFEM codes. Good agreement has been found in the general dynamics but unphysical oscillations in the time series of the pressure appear for both methods. Pressure impulse has been compared and reasonable but overestimated values have been found, regardless of the resolution and of the SPH numerical sound speed. So far, the influence of the gas phase on the pressure history has not been assessed with enough quality to discriminate the origin of the numerical errors. Further work has yet to be done.

5. Acknowledgments This work has been partially funded

by the program PROFIT 2007 of the Spanish Ministerio de Educación y Ciencia through the project STRUCT-LNG (file number CIT-370300-2007-12) leaded by the Technical University of Madrid(UPM).

6. References BERG, A. (1987). Scaling laws and statistical distributions of impact pressures in liquid sloshing. Det Norske Veritas DNV, Report no. 87-2008.

BASS, .L., BOWLES, E.B., TRUDELL, R.W., NAVICKAS, J., PECK, J.C., ENDO, N. and POTS. B.F.M. (1985). Modeling criteria for scaled LNG sloshing experiments, Trans. ASME, 107, 272—280.

IDELSOHN,S.; DEL PIN,F.; ONATE,E.; AUBRY,R. (2007).The ALE/Lagrangian Particle Finite Element Method: A new approach to computation of free-surface flows and fluid-object interactions. Comput. Fluids, 36, 1, 27-38,

KLEEFSMAN, K.M.T.; FEKKEN, G.; VELDMAN, A.E.P.; IWANOWSKI, B.; BUCHNER, B. (2005). A Volume-of-Fluid based simulation method for wave impact problems. J.Comp.Phys., 206, 1, 363-393,

MONAGHAN, J.J. (2005). Smoothed Particle Hydrodynamics, Rep. Prog. Phys., 37, 1703—1759.

PEREGRINE, D.H. (2003), Water-wave impact on walls. Ann. Rev. Fl. Mech, 37.

SOUTO-IGLESIAS, A., DELORME, L., ABRIL-PÉREZ, S. and PÉREZ-ROJAS, L. (2006). Liquid moment amplitude assesment in sloshing type problems with SPH, Ocean Eng., 33, 11—12.

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Enforcing boundary conditions in SPH applications involving bodies with right angles.

A. Colagrossi1, G. Colicchio1 and D. Le Touzé2

a.colagrossi@insean.it, g.colicchio@insean.it, david.letouze@ec-nantes.fr

1 INSEAN, Italian Ship Model Basin, Dpt. of Seakeeping and Manoeuvrability, Rome (Italy) 2 Laboratoire de Mécanique des Fluides, Ecole Centrale Nantes, Nantes (France)

Abstract

The viscous flow around a moving square is characterized by the flow separation induced by the geometry of the body, and by the generation of strong vorticity associated with strong pressure gradients as shown by the reference results in the benchmark test n.6 [1]. The Lagrangian nature of SPH makes the modelling of these features very difficult even at low Reynolds numbers. This work analyzes the problems associated with the Lagrangian motion of the particles, and how they disappear in an “Eulerian” SPH formulation. The stiff problem of the Lagrangian motion of the particles around the corner is partially made milder by the use of a background pressure, an anti-clamping correction and the coupling with a local analytical solution.

1. Introduction At the last SPHERIC workshop an extension of the ghost technique for bodies with right angles has been shown. It was based on the contemporary mirroring of three different families of particles at the internal corner of the body. It was applied to model the impact of the flow after the dam break against blunt obstacles. The results were satisfying in the comparison with the experimental data but their closer analysis has shown that, while flowing around the corner, the particles are affected by steep changes of pressure.

These had little influence in the case of a violent impact with a free surface, but they cannot be neglected when viscosity and vorticity are the leading terms in the force acting on the body. The study of the simple viscous flow around a fully submerged square cylinder makes easier the analysis of the flow details and highlights the problems connected with the used formulation. The application of the technique described in [2] gave poor results because: 1) unphysical cavities appear at the corners at the initial stages 2) consequently the pressure field is affected by severe errors and 3) the generated vorticity field fluctuates randomly even at low Reynolds numbers. Other solid-boundary techniques, different from the ghost one, have been applied without significant improvements. In the end, the main cause of these drawbacks were to attribute either to the Lagrangian nature of the solver or to the weakly compressible approach. The results shown in the following develop the adopted strategy of analysis used to spot the cause of the unsuitability of the solver.

2. Ghost technique for right angles

Before discussing the fundamental effects of some of the SPH basic features, a short description of the ghost technique used for modelling right angles is given here. Figure 6 shows how the copies of the three quadrants indicated as 1, 2 and 3 in

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the left plot, around the corner, are mirrored inside the body. To avoid the excess of ghost mass, a fluid particle sees the mirrored zones weighted with the values shown in the right plot of the same figure. The weights are harmonic functions of the angle θ. The velocity components of the ghost particles are prescribed according to the boundary condition chosen [2]. For sake of space they will not be discussed here.

3. Results and discussion The upper part of Figure 7 shows the particles displacement when the SPH solver [3] is applied to model the flow around the square. Unphysical cavities are created at the sides of the square, generated by the corners. Their presence jeopardizes the pressure field all around the square. Successively the flow field is so ruined that no physical statement can be derived from its analysis. As shown in the bottom part of the same plot, even the use of an MLS technique for the interpolation of the gradients only slightly improves the result without avoiding the presence of the cavities. To prove that even with the weak compressibility assumption physical results can be obtained, the SPH formulation has been rewritten in Eulerian form. That is the particles are kept fixed in space and the advective terms are discretized in the framework of the meshless formulation. To prevent troubles with moving boundaries, the problem has been rearranged as the one of a fixed square invested by an unsteady current. Figure 8 shows the pressure field generated around the square at Re=50 and at the time t=5L/U∞. The results are in good agreement with the reference data shown in the lower part of the same plot. In Figure 9 the drag force, in its pressure and viscous components (respectively top and bottom plots), obtained for the Eulerian

weakly compressible meshless solver is reported by the black dash-dotted line. It is to be compared with the Navier-Stokes solution, solid red line. The weak compressibility causes a short delay of the pressure peak and small oscillations around the mean value of the Navier-Stokes solution. Nonetheless the agreement is satisfactory. The flow field obtained with the Eulerian formulation has been used to track Lagrangian particles in the domain. Figure 10 shows the particles distribution at a given time both for right (top) and rounded corners (bottom). In both cases, the distribution is very anisotropic, so that the use of a standard meshless interpolation technique there would give poor results. In the case of rounded corners the anisotropy is only slightly less intense. In fact, when running a Lagrangian SPH with such geometry of the body, unphysical cavities are created as shown in Figure 11. There the comparison between the Lagrangian and Eulerian formulations is proposed and only the latter solution is comparable with the reference data. These results show that the Lagrangian moving particles miss the right displacements in the regions of intense and localized vortex sheets, because of the numerical errors. To reduce the effects of these errors in the Lagrangian formulation, it is necessary to introduce a local solution. So the analytical trajectory of the potential flow around the corner has been prescribed in the neighbourhood of the edge (see left plot of Figure 6). This is however not enough to obtain a satisfactory full Lagrangian solution since, as shown above, the anisotropic distribution of particles increases the numerical errors. To diminish their effect an anti-clamping force has been introduced in the SPH equations. The application of these two techniques gives the results shown in Figure 12 in which the pressure field shown is noisy but

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the global pressure field is well captured as well as the forces acting on the square, plotted by a green dotted line in Figure 9.

Acknowledgments. The present research activity is supported by the Ministero dei Trasporti in the framework of the “Programma Sicurezza (2006-2008)” and of the “Programma di Ricerca INSEAN 2007-2009”.

3.1. Figures

Figure 6: Left: sketch of the technique used for the trajectories at the corners of the square.

Right: weight function of mirrored particles from sectors 1, 2 and 3 as a function of the angle θ.

Figure 7: Pressure field around the moving

square at the time t=1.05L/U∞. Top: pressure field obtained using a classical SPH. Bottom: the one obtained using an MLS interpolation for the

gradient (Re = 50).

Figure 8: Pressure field around the moving

square with two Eulerian solvers: SPH (top) and Navier-Stokes (bottom) (Re = 50).

Figure 9: Time evolution of the pressure (top) and viscous (bottom) components of the drag

force (Re = 50).

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Figure 10: Particle population around the moving right angles square (top) and the rounded edges

square (bottom) (Re = 50).

Figure 11: Pressure field around the moving

square for the moving box. Top: Lagrangian SPH formulation with rounded edges, bottom: Eulerian formulation with the same body shape (Re = 50).

Figure 12: Top: SPH with: 1) an MLS

interpolation for the gradients, 2) an anti-clamping force, 3) a local solution for the corners.

Bottom: Navier-Stokes reference solution.

4. References

SPHERIC (SPH European Research Interest Community) WEBSITE: http://wiki.manchester.ac.uk/ LE TOUZÉ D., COLAGROSSI A. and COLICCHIO G. (2006) Ghost Technique for Right Angles applied to the Solution of Benchmarks 1 and 2, 1st SPHERIC workshop (Rome, Italy). http://cfd.me.umist.ac.uk/sph/meetings/1stSPHERIC_workshop/1stSPHERIC_workshop_Presentations.html COLAGROSSI A. and LANDRINI M. (2003), Numerical Simulation of Interfacial Flows by Smoothed Particle Hydrodynamics, Journal of Computational Physics, 191: 448 - 475.

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An FPGA-based hardware coprocessor for SPH computations

G.Marcus, G. Lienhart, A. Kugel, R. Mäenner1, P. Berczik, R. Spurzem2, M. Wetzstein, T. Naab and A. Burkert3

1 Lehrstuhl für Informatik V, University of Mannheim, marcus,lienhart,kugel,maenner@ti.uni-mannheim.de

2 Astronomisches Rechen-Institut, University of Heidelberg, berczik,spurzem@ari.uni-heidelberg.de

3 University Observatory, Physics Department, University of Munich, mwetz,naab,burkert@usm.uni-muenchen.de

Abstract We present a solution consisting of an SPH-specific FPGA-based board, and an advanced software library that provides a simple yet powerful interface to be used by the software. The current solution has been integrated with several astrophysical simulation environments, and provides a peak performance of 3.4 Gflops for Step1 and 4.3 Gflops for Step2, thus providing a speedup in the range of 6x-10x compared to a state of the art workstation for particle number ranging from 1k to 128k, when computing for all particles active.

1. Introduction Particle-based astrophysical

simulation determines movement of individual particles according to a model for the interaction forces. Long-range gravity and short-range hydrodynamics forces are considered most important for many systems. With particle numbers >106, the latter SPH part is also computationally very demanding. We support such calculations, which can be applied to a wide variety of practical problems besides astrophysics, by programmable hardware. Similar developments include the work described by (Hamada and Nakasato, 2005).

In the following sections, we present the mathematical formulation used by the

coprocessor, the hardware architecture, the software tools and interfaces developed, and the results obtained.

2. SPH Formulation We use the standard SPH formulation

as the base, and modify it as follows in order to reduce the actual number of computations done in hardware. As kernel functions, we use:

And the scalar part of the gradient of W as:

In Step 1 we compute the density, curl

and divergence of the velocity as:

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And in Step 2 the acceleration,

including artificial viscosity, as:

3. Hardware Architecture The coprocessor design consists of

one or more computational pipelines, external memory to store the particle data and an output FIFO at the end of the pipeline to store the results, plus the required communication and control logic. An overview is shown in Figure 1, which includes at the bottom the communication with the host and at the top the external memory modules used.

For each timestep, the coprocessor loads the particle data (position, velocity, mass, etc) into external memory. After selecting for step1 or step2 computations, neighbour lists are sent in sequence from the host in the format ip,N,jp1..jpN, being ip the index of the i-particle, N the number of neighbours, and jpX the corresponding j-particles indexes. Neighbour lists are received from the host and processed immediately at the rate of one neighbour interaction per cycle. Accumulated results are pushed into an output FIFO for later retrieval. Being the FIFO of a very limited size, results must be retrieved by the software regularly.

In order to reduce the size of the pipelines, increase their speed and being able to accommodate more particles in the available memory, the precision of the floating point operators is reduced to 16-

bits mantissa and 8-bits exponent, except for the accumulators.

Given this computational scheme, the overall performance is driven by the communication time and clock frequency of the coprocessor.

MPIF

Memory Control(MCU)

Data Flow Control (DFC)

Command Control(CCU)

Pipe 1

M Memory Interfaces

RAM 1 RAM M

Host InterfacePipe N

PCI

Figure 1: Hardware Architecture

To optimize communication, loading

the particle data is done in two parts. For step1, position, velocities, mass and smoothing length (plus sound speed) are loaded, while the additional values for step2 (including the updated density, Balsara factor and pressure) are uploaded only before starting step2 computations.

Additionally, as mentioned earlier, the pipelines are capable of switching between step1 and step2 computations, reusing several common parts of the operation flow. This allows further savings in chip area, making it possible to use a single design for all required operations.

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4. Software Tools Each floating point operator required

considerable development time, and many operations can be optimized depending on their operator size or signed/unsigned results. Other specialized operations, like square a value, can also be optimized when compared to a generic multiplier. To gather all this advantages into a coherent interface, a floating point library was developed. This library provides parameterized operators to select the desired precision (exponent and mantissa), specialized operators to optimize for speed and area, simulation capabilities to simplify hardware verification, and a range of high performance accumulators.

reset iData

A save

latchQ

A save

latchQ

A save

latchQ

I1 I2

syncQ1 Q2

rX

AX AYAZ BX BY BZ

floPVecDiffZGQXQY QZ

rY rZp

I1 I2

calcPDivRho2I1: rhoInI2: pInQ1: pDivRho2Out

Q1

riX riY riZ

rijX rijY rijZ

vrij

AX AY AZ BXBY BZ

floPVecCrossProdQX QY QZ

rijxvijY rijxvijZ

AX AY AZ

floPVecSquareQ

rij2

pDivRho2

is_pDivRho2i pDivRho2j

A save

latchQ

pDivRho2i

Figure 2: Detail of the SPH pipeline, generated by the Pipeline Generator

The pipelines for the SPH

computations contain tens of operators, and putting them together is a time consuming and error-prone task. As a way of adding flexibility and reliability, a pipeline generator was developed. This software tool receives a program-like description of the operations to perform and produces a hardware pipeline, using the floating point library as the building blocks. This makes the pipeline correct-by-design, and it is

very flexible to adjust the precision of operands or to introduce changes on the operations performed. A more complete description can be found at (Lienhart, Kugel and Männer 2006).

5. Software Library and Interfaces In order to use efficiently the cap-

abilities of the coprocessor, we developed a software library for C/C++/FORTRAN languages. This library provides the user with a clean interface to the SPH functionality while hiding coprocessor-specific details like design loading, buffer creation, data partition and FIFO flushing. In addition, an emulation core is provided, that allows the library to perform the same operations with the host CPU only.

Particular attention was taken on the interface of the library with existing applications. Being the coprocessor performance directly proportional to the communication between the host and the board, a generic buffer management scheme (Marcus et al., 2006) was imple-mented, allowing the library to access data structures of the application directly for direct conversion between the formats of the application and the coprocessor.

6. Results The current coprocessor is based in

an mpRACE-1 board equipped with a Xilinx Virtex2 XC2V3000 FPGA, 8MB of SRAM and communicating with the host computer via a PCI-X board, with a maximum bandwidth of 264MB/s. The design fits one SPH pipeline working at 62.5MHz and capable of processing up to 256k particles, for a peak theoretical performance of 3.4Gflops for Step1 and 4.3Gflops for Step2.

The design has been integrated with existing simulation codes in C and FORTRAN, including single node and

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parallel codes. Accuracy tests with and without using the hardware boards shows no significant degradation on the accuracy of the results, but significant gains in area and speed (Lienhart, 2004). Performance improvements with simple codes are in the range of 6x-10x speedup, while using tree codes is around 2x speedup, running with up to 128k particles (due to limitations of the hardware used for gravity force computations).

At present time, 4 boards are installed at the Titan cluster in the ARI-Heidelberg and performing production runs, with 32 next-generation boards being planned for the end of the year. This new boards will carry new FPGAs, allowing for faster operations, more pipelines per board and higher communication bandwidth.

7. Conclusions A complete solution for supporting

SPH simulations with specialized hardware is presented. The combination of state of the art hardware and software shows performance gains between 6x and 10x, while the software interfaces allow for easy interfacing with existing software. The use of innovative tools enables fast development cycles and flexible customization of the implemented algorithms.

8. Acknowledgments This research work has been

performed in the framework of the GRACE project and was funded by the Volkswagen Foundation and the Ministry of Science, Research and the Arts of the State of Baden-Württemberg, Germany.

9. References MARCUS, G., LIENHART, G., KUGEL, A. and MÄNNER, R. (2006). On Buffer Management Strategies for High Perfor-mance Computing with Reconfigurable Hardware. FPL2006 Proceedings: 343- 348. LIENHART, G., KUGEL, A. and MÄNNER, R. (2006). Rapid Development of High Performance Floating-Point Pipelines for Scientific Simulation. RAW Proceedings. LIENHART, G (2004). Beschleunigung Hydrodynamischer Astrophysikalischer Simulationen mit FPGA-Basierten Rekonfigurirbaren Koprozessoren. PhD Thesis, Universität Heidelberg. NAKASATO N. and HAMADA, T. (2005). Astrophysical hydrodynamics simulations on a reconfigurable system. 13th FCCM05: 279-280. HAMADA, T. and NAKASATO, N. (2005). Massively Parallel Processors Generator for Reconfigurable System. 13th FCCM05.

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Challenges related to particle regularization in SPH

Steinar Børve 1, Roland Speith 2

Marianne Omang 3 and Jan Trulsen 4 1 University of Oslo/Norwegian Research Establishment, steinabo@astro.uio.no

2 University of Tübingen, speith@tat.physik.uni-tuebingen.de 3 Norwegian Defence Estates Agency, momang@astro.uio.no

4 University of Oslo, jtrulsen@astro.uio.no

Abstract

Regularized Smoothed Particle Hydro-dynamics (RSPH) was proposed as an extension to standard SPH (Børve et al. 2001). The aim has been to achieve increased accuracy and efficiency through a strategy of particle regularization. In this work we discuss challenges related to the regularization technique. In particular, we look at conservation of angular momentum and kinetic energy, numerical diffusion and viscosity, and handling of solid boundaries and free surfaces.

1. Introduction SPH has over the last 30 years been

applied successfully to a wide range of problems. Most notably, the method ele-gantly handles complex free surfaces and interfaces. The method has therefore become widely used in astrophysical applications where SPH can be viewed as a natural extension of the N-body method for gravitationally dominated fluids. However, due to a lack of reproducibility, SPH solutions obtained from increasingly more irregular particle distributions will exhibit an increasing amount of numerical errors. In an attempt to produce a method less vulnerable to irregular particle distributions, in particular in the presence of shocks, Regularized Smoothed Particle Hydrodynamics (RSPH) was proposed (Børve et al. 2001). Since then, the method has been shown to give good results on hydrodynamic and magnetohydrodynamic shock problems when compared with both standard SPH and Eulerian methods (Børve et

al. 2005; 2006). Similar ideas has been applied to viscous and heat conduction flows (Chaniotis et al. 2002). In Fig. 1 we show preliminary results from simulating a planet-accretion disk problem used in a comparison study on codes for this type of problem (de Val-Borro et al. 2006).

Figure 1: Logarithmic density plotted in polar coordinates after 100 orbital periods of the planet-disk interaction problem. In addition, some initial work was done in 2002 to apply RSPH to incompressible water waves. This work was not brought to a conclusion, but the findings from that work will be discussed in Sect. 4. Results from a breaking waves test are illustrated in Fig. 2. In Sect. 2 we briefly explain the regularization technique and discuss con-servation of physical quantities. In Sect. 3 we take a closer look at numerical diffusion and viscosity introduced when regularizing. In Sect. 4 we look at boundaries and free surfaces in RSPH. A short summary is provided in Sect. 5.

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Figure 2: 2D breaking waves simulation.

2. Particle regularization The particle distribution is redefined at

temporal of typically 40-200 time steps. After generating a new set of particles, we use a Voronoi scheme to accurately es-timate the overlap between a given pair of old and new particles. This overlap de-termines how much of the physical proper-ties are transferred from the old to the new particle. This is done so that mass, linear momentum, and magnetic and total energy is conserved. Angular momentum, kinetic and potential energy is not exactly conserved in the current scheme. On the other hand, the regularization procedure enables us to use highly flexible criteria for achieving an adaptive, AMR-type evolution of the smoothing length (h) profile. Similar ideas have recently been presented elsewhere (Bergdorf et al. 2005).

In (Børve et al., 2004) it was noted that regularization introduces higher har-monics in the simulation of linear waves. This is because a simple coupling between particle displacement and velocity no longer exist. When simulating a plane MHD shock tube test in (Børve et al., 2005) it was noted that errors of the order 1-2% was found in a region with strong compression. This error was shown to be connected with the lack of exact kinetic energy conservation. By locally increasing the number

of particles per h whenever compression had occurred, the error was reduced by at least 50%. As for errors in angular momentum and potential energy, this has so far been viewed as a smaller problem since both mass and linear momentum are conserved in a local manner. However, it is uncertain whether the presence of free surfaces might complicate this picture. In the viscous ring test (described in Sect. 3), the loss rate for angular momentum was found to be roughly 2-3 % per viscous time unit.

3. Numerical diffusion and viscosity Regularization reduces the need for

artificial viscosity in the traditional sense. However, regularization also introduces resolution dependent numerical diffusion and viscosity much like grid codes do. The question is when this is critical and not. As a first test, we set up a 1D, force free problem where the initial density profile is square shaped, as shown by the solid curve in Fig. 3. All particles have a velocity equal to 0.07Dt/Dx. Where Dt is the time step and Dx is the particle separation. The dashed and dotted lines show the density profile after 24 and 48 regularizations. The observed diffusion is a result of the old and new particles not being aligned when performing the regularization. As a result, the mass is gradually transported out-wards in both directions.

To study the numerical viscosity due to regularization, we simulate a viscously spreading, cold accretion ring, a problem to which an analytic solution exist (Pringle 1981). This model has been widely used to evaluate new viscosity schemes (Speith 2007). In RSPH, we have included kine-matic viscosity (Meglicki et al. 1993). In Fig. 4 we compare the results from 4 different runs with the analytic solution at viscous time 0.27. Runs (a-c) correspond to regularization intervals of 320, 640, and 1280 time steps, while run (d) has no regularization at all. All the runs were performed with h=0.02 and an initial particle spacing of h/1.4. The effective viscosity was estimated to be 36% (a), 11% (b), and (1%) higher than the analytic viscosity. Deviations at

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small radii would in part be resulting from problems with capturing the inner free surface. The large noise level in run (d) is partly caused by the large particle separation at the end of the simulation.

Figure 3: Diffusion due to regulari-zation in

force free, square-profile test.

Figure 4: Viscous ring test .

4. Boundaries and free surfaces Faced with solid boundaries or free sur-faces that cannot be accurately described by particles on Cartesian grid structuring, one is faced with the possibility that regularization will result in increased errors near the boundaries. Normally, we have applied the mirror particle technique to solid boundaries, shifting the center of mass position of particles initially

lying partially outside the domain. In the wave breaking simulation (see Fig. 2), an interpolation technique where particles are placed uniformly across the solid boundary was used to handle the beach floor. The free surface was handled by simply removing new particles that, had they been included, would only inherit a smaller fraction of the typical particle mass. For this reason, the free surface is not as smooth as in a normal SPH simulation. In the viscous ring test we have problems with that the radial interval where particles are found to increase much faster than expected when regularization is performed. This affects the efficiency rather than the accuracy since the density is so low on the edges of the ring. To see if the treatment of both curved, solid boundaries and free surfaces in RSPH can be improved, we are currently experimenting with techniques to modify the particle distribution slightly near boundaries and surfaces. The aim is to achieve as uniform as possible distri-butions near boundaries and surfaces. To illustrate this work in progress, two different ways of setting up particles for the viscous ring test is shown in Fig. 5. In plot (a), all particles have been placed on Cartesian grids. In plot (b), the positions of particles found near the free surfaces have been automatically shifted to better fit the shape of the free surface. Because the second distribution is a modification to the first, one can see that some problems with non-uniform particle spacing exist. This is most pronounced tangentially to the highly curved boundaries or surfaces since particles are only shifted normally to the surface. For arbitrarily complex free surfaces, the surface is defined by single particles identified as being at the free surface. The identification is currently achieved in the incremental Voronoi scheme.

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Figure 4: Current (a) and new (b) approach to representing curved boundaries and free surfaces.

5. Summary In this work we have discussed some of the challenges one is facing by introducing particle regularization as a means to avoid highly disordered particle distributions in SPH simulations.

6. References

BERGDORF, M., COTTET, G.-H., and KOUMOUTSAKOS, P. (2005). Multilevel adaptive particle methods for convection-diffusion equations. Multiscale Model. Simul., 4:328-357.

BØRVE, S., MONAGHAN, J.J., OMANG, M., and TRULSEN, J. (2003). New developments in regularized smoothed particle hydrodynamics. 5th Int. Cong. Industrial Appl. Math., Sydney, #7860. BØRVE, S., OMANG, M., and TRULSEN, J. (2001). Regularized smoothed particle hydrodynamics: A new approach to simulating magnetohydrodynamic shocks. Astrophys. J., 561: 82- 93. BØRVE, S., OMANG, M., and TRULSEN, J. (2004). 2D MHD smoothed particle hydro-dynamics stability analysis. Astrophys. J. Suppl. Ser., 153: 447-462.

BØRVE, S., OMANG, M., and TRULSEN, J. (2005). Regularized smoothed particle hydrodynamics with improved multi-resolution handling. J. Comput. Phys. 208, 345-367. BØRVE, S., OMANG, M., and TRULSEN, J. (2006). Multidimensional MHD shock tests of regularized smoothed particle hydro-dynamics. Astrophys. J., 652: 1306-1317. CHANIOTIS, A.K., POULIKAKOS, D., and KOUMOUTSAKOS, P. (2002). Remeshed particle hydrodynamics for the simulation of viscous and heat conducting flows. J. Comput. Phys., 182:67-90. DE VAL-BORRO, M. et al. (2006). A com-parative study of disc-planet interaction. Mon. Not. R. Astron. Soc., 370:529-558. MEGLICKI, Z., WICKRAMASINGHE, D., and BICKNELL, G.V. (1993) 3D structure of truncated accretion discs in close binaries. Mon. Not. R. Astron. Soc., 264:691-704. PRINGLE, J.E. (1981). Accretion discs in astrophysics. Annu. Rev. Astron. Astrophys. , 19: 137-162. SPEITH, R. (2007). Improvements of the numerical method smoothed particle hydrodynamics. Habilitation thesis (in prep.), Tübingen.

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SPH with Improved Ghost Particle Boundary Treatment

Mehmet Yildiz 1 and Afzal Suleman 2

1, 2 Ph.D, Mechanical Engineering Department / Advanced Composite Research Laboratory, University of Victoria, Victoria, BC, Canada

Abstract In this article, we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics (SPH) method. Previously reported boundary treatments may suffer from particle penetration, and excessive flow distortion, hence producing results that numerically blow up near solid boundaries. As well, current SPH boundary approaches do not properly treat curved boundaries. The new boundary treatment method presented in this article, called the multiple boundary tangent (MBT) approach, remedies these drawbacks. In this article, we present two important benchmark problems to validate the developed algorithm and show that the MBT treatment produces results that agree with known numerical and experimental solutions. The two benchmark problems chosen are the lid-driven cavity problem, and flow over a cylinder. The SPH solutions using the multiple boundary tangent approach and the results from literature are in very good agreement.

1. Introduction In most engineering problems, the domain of interest is, in general, bounded. The SPH formulations being valid for all interior particles are not necessarily accurate for particles close to the domain boundary since the distribution function (kernel) is truncated by the boundary. Therefore, the application of boundary conditions is problematic in the SPH technique, since SPH approximation no longer produces the second order accuracy. Consequently, the proper and correct boundary treatments have

been an ongoing concern for an accurate and successful implementation of the SPH approach in the solution of engineering problems with bounded domains. Improper boundary treatment has two important consequences. The first originates from the penetration of fluid particles into boundary walls, which then leave the bounded domain. The second is that kernel truncation at the boundary will produce errors in the solution. Hence, over the last decade, several different approaches have been suggested for the boundary treatment such as specular reflections, or bounce-back of fluid particles with the boundary walls, Lennard-Jones Potential (LJP) type force as a repulsive force, and ghost particles. Even though the current status of ghost particle implementation is limited to relatively simple geometries, we have observed that the ghost particle approach is the most stable and effective among all the approaches mentioned previously. Therefore, we suggest an improved solid boundary treatment, as an extension to the ghost particle approach, which eliminates many shortcomings of the aforementioned boundary treatments. We call this approach the multiple boundary tangent (MBT) method, and the various steps in the MBT technique, as depicted in Figure 1, are as follows: (1) At each time step, for all boundary particles, tangent lines are computed. (2) Given that each boundary particle has fluid particles in its influence domain as neighbors, these fluid particles are mirrored with respect to the tangent line of the corresponding boundary particle. Neighbors are computed using the standard box-sorting algorithm. Using the cell array structure (the Fortran 90 derived data type), every boundary particle is associated with its corresponding mirrored

72

particles. Spatial coordinates and particle identification numbers of mirrored particles are stored in the cell array. To be more precise, mirrored particles are associated with the particle identification number of the fluid particle from which they are originated (referred to as the “mother” fluid particle). For example, for a fluid particle indexed with i = 21, the ghost particle mirrored about a boundary particle tangent line (for example, boundary particle 11) will also be associated with i = 21. Note that fluid and boundary particles have numerical identifications that are permanent, whereas mirrored particles have varying (dummy) indices, throughout the simulation. (3) Similarly, using the cell array data structure, fluid particles with boundary truncations (near boundary fluid particles) are associated with their neighbor boundary particles. For example, near boundary fluid particle 30 has three boundary particles as neighbors, as illustrated in Figure 1-c. Storing these particles allows computation of the overlapping contributions of mirrored particles from each boundary particle, as well as associating mirrored particles with near boundary fluid particles. (4) In a loop over all particles, if a fluid particle has a boundary particle or multiple boundary particles as neighbor(s), then the fluid particle will become a neighbor of all mirrored particles associated with the corresponding boundary particles, on the condition that the mirrored particles are in the influence domain of the fluid particle in question, and for a mirrored particle, its mother particle has to be within the influence domain of the fluid particle in question. (5) During the creation of ghost particles, there is an over-creation of ghost particles due to the fact that the influence domain of neighboring boundary particles overlaps.

Figure 1: MBT method.

The overlapping contributions of mirrored particles can be eliminated by determining the number of times a given fluid particle is mirrored into the influence domain of the associated fluid particle with respect to a boundary particle’s tangent line. Near boundary fluid particles hold the information of spatial coordinates and fluid particle identity numbers, boundary particle identity numbers (i.e. the particle number for a boundary particle to which mirrored particles are associated initially), and over-creation number for mirrored particles in the cell array format. For example, the ghost particles with the index 1 and 5 are mirrored from the same fluid particle 45 about the tangent line of the boundary particle 11 and 12, respectively. (6) During the SPH summation over ghost particles for a fluid particle with a boundary truncation, the mass of the ghost particles are divided by the number of corresponding over-creations. The MBT technique suggested in the present article has the following advantages over its counterparts: (a) since ghost particles are created for a liquid particle with a missing contribution due to the boundary truncation by using its boundary particles, the approach has the potential to treat complex geometries, and (b) it allows the creation of ghost particles totally

73

conforming the shape of the boundary, thereby taking into account the effect of boundary curvature. In this article we present two important benchmark problems to validate the developed algorithm and show that the MBT treatment produces results that agree with known numerical and experimental solutions. The two benchmark problems chosen are the lid-driven cavity problem, and flow over a cylinder. The SPH solutions are obtained using incompressible (projective) SPH approach (for details, refer to [1]). Solutions produced with the MBT approach and results from the literature are in very good agreement.

2. Results and discussion 2D Lid-Driven Cavity Flow: The no-slip boundary conditions are employed within the cavity for bottom (y =0), left (x = 0), and right (x = L) walls. The top of the cavity (y = H) has the boundary condition vx = vo = 10−3 m/s and vy = 0. The initial condition for all interior fluid particles is taken as vx = vy = 0. The parameters H = L = 0.1 m, ρ = 1000 kg/m3, and µ = 10−3 kg/ms were selected for this simulation, producing a Reynolds number Re = 100. The computational domain consisted of an array of 121 × 121 particles, and produced a velocity vector plot (which is plotted using a reduced 31 × 31 array, for clarity) in Figure 2. For the lid-driven cavity benchmark problem, the numerical results have been reported by Ghia et al. [1] and are referenced for comparison. For the sake of space, we present only the normalized vertical velocity component which is plotted at the horizontal domain centerline y = H/2 in Figure 3, which also demonstrates the heuristic convergence of the solution as the number of SPH particles is increased. SPH results agree very well with the numerical findings of Ghia et al.

Figure 2: Velocity vector plot for Re=100.

Figure 3: vy/vo versus x/L at y = H/2 for Re =

100.

Flow over a cylindrical obstacle: The flow over a cylindrical obstacle was first studied by Takeda et al. as compressible flow [3] and then by Morris et al. as weakly compressible flow [4]. The reported results from Takeda et al. are for Reynolds numbers between 6 and 55. The study by Morris focused on Reynolds number (calculated based on cylinder radius) of 0.03 and 1. In this work, two-dimensional flow over a cylindrical obstacle is solved using projective (incompressible) SPH on a rectangular domain with the length of L = 0.9 m, a height of H = 0.6 m, and a cylinder diameter D = 0.04 m. The center of the cylinder is located at Cartesian coordinates (L/3,H/2). The simulation parameters are taken as ρ = 1000 kg/m3, µ = 10−3 kg/ms. The total number of all particles (including boundary and fluid particles) is 19, 997. The particles start moving from rest with zero initial velocities, and the adaptive time

74

stepping which satisfies the CFL condition is implemented. The periodic boundary condition is applied for inlet and outlet particles in the direction of the flow. The no-slip boundary condition is implemented for the cylindrical obstacle. For upper and lower walls that bound the simulation domain, the symmetry boundary condition for the velocity is applied. Wake sizes for the Reynolds number ranging from 10 to 50 are compared with the experimental results [1], which is not presented here for the sake of efficient usage of the space. The length of the wakes for each of the Reynolds numbers are measured from the trailing edge of the cylindrical obstacle to the location in the wake where the x-component of the velocity vector is zero, or nearly zero (see velocity vector plots in Figure 4 for the horizontal line used for this measurement).

Figure 4: Wake Velocity Vector Plot for Re =

10, 20, 30, 40, and 50

3. Conclusions In this article we have presented solutions for two important benchmark problems to validate that the proposed MBT method produces results that agree with known numerical and experimental solutions presented in literature. It was found that the SPH results and the results from the literature were in very good

agreement. Presently, the technique proposed herein has been tested only for solid boundaries. Further testing of the multiple boundary tangent method on sharp changes in solid boundary geometries, as in the case of flow over a backward and forward-facing step, has been initiated. As well, our future work includes flow simulations over more complex solid boundary geometries, such as flow over an airfoil. We have not tested the algorithm for treating time-evolving, free-surfaces at this point.

4. Acknowledgments Funding provided by the Natural Sciences

and Engineering Research Council NSERC CRD project no. CRDPJ 261287-02 is gratefully acknowledged.

5. References M.YILDIZ, R.ROOK, A.SULEMAN, Incompressible SPH with improved boundary treatment for simulating benchmark flow Problems, J. Comput. Phys (revised version is under review) U. GHIA, K.N. GHIA, C.T. SHIN, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid Method, J. Comput. Phys.48 (1982) 387. H. TAKEDA, S.M. MIYAMA, M. SEKIYA, Numerical simulation of viscous flow by Smoothed Particle Hydrodynamics, Prog. Theor. Phys. 92 (1994) 939-960. J.P. MORRIS, P.J. FOX, Y. ZHU, Modeling low Reynolds number incompressible flows using SPH, J. Comput. Phys. 136 (1997) 214-226.

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75

Coupling between roll motion and 2D sloshing

L. DELORME1, C. LÓPEZ-PAVÓN1, E. BOTIA1 and R. ZAMORA-RODRÍGUEZ1

1 Model Basin Research Group (CEHINAV), Naval Architecture Dpt (ETSIN), Technical University of Madrid (UPM). Avda Arco de la Victoria s/n, 28040 Madrid, SPAIN.

Abstract The coupled effects between roll motion and sloshing inside a rectangular tank are studied. The flow in the tank is modelled by means of SPH and non linear potential theory. Results obtained with both methods are compared.

1. Introduction The coupling between the motion of a

structure that incorporates a tank, and the motion of an existing mass of liquid inside this tank, has many engineering applications. It can be found in the aircraft and truck industry, in the marine engineering and in the buildings design. It has been paid a particular attention over the last decades (see Ibrahim, 2005 for a good recent review). The characteristics of the coupled system depend on the ratio of the natural frequency of the primary system and the natural frequency of the fluid inside the tank. When this ratio is not close to one, the liquid motion can amplify the structure motion. On the other hand, when both frequencies match, the liquid acts like a damper for the external structure.

Both cases can be found for instance in the naval field. In the case of tanker ships, the liquid inside their tanks can affect dramatically the ship stability, whereas anti-roll tanks are devices installed onboard some vessels that use the same principle but in order to decrease in a significant way their maximum roll angles (Armenio, 1996, Souto-Iglesias, 2006). This has to be obtained with low filling levels in the tank, since high filling levels imply much bigger natural frequencies than the ones corresponding to the vessel.

Two configurations corresponding to those two cases have been studied. In order to model the flow in the tank, results obtained with nonlinear potential theory (Faltinsen, 2000) are compared with those obtained with SPH (Monaghan, 2005). Other methods like RANS solvers with MAC technique for tracing the free surface evolution have been previously described (Armenio, 1996), but the inability of these methods to cope with wave breaking events makes the present work a promising start point to face these problems.

Figure 1: Sketch of the problem. More details in

the text.

2. Case Study The case study is a generic box-barge with an onboard tank (figure 1). The barge is 2 m long, 0.333 m wide and has a depth of 0.14 m. For simplicity, the center of gravity of the ship, its center of rotation, and the center of the unperturbed free surface are supposed to be both located at the same point O. The heel angle of the barge (φ) is the only degree of freedom of the motion and is assumed to satisfy equation (1).

flwdamprestaddS MMMMMJ ++++=••

ϕ (1)

In this equation, JS is the ship moment of inertia with respect to the rolling axis, Madd the

76

moment due to added mass, Mrest the restoring moment, Mdamp the moment due to damping forces, Mw the moment of the wave and Mfl the moment exerted by the fluid on the tank. Coefficients for added mass coefficient, and linear damping have been calculated using a linear BEM code. The roll resonance frequency of the barge is ω0 ≈ 6.5 rad/s.

Two different configurations for this same barge have been studied, whose characteristics are presented in table 1.

Table 1: Configurations. Case A B

Tank length [m] 0.166 0.333

Water depth / TankLength

0.34 0.17

First sloshing frequency [rad/s]

12.0 6.69

Liquid mass / Barge mass

0.2 0.02

Wave steepness 7.3 10-3 4.9 10-3

For both configurations, a range of wave moment frequencies has been considered. For each of these cases, the motion of the barge has been compared with the “ballast condition” i.e. the motion of the barge, as obtained from equation (1) when no liquid force is considered.

3. Tank Flow Model. In order to calculate the term Mfl in equation

(1), two techniques have been used. On the one hand, non-linear potential solution, following Faltinsen et al. (2000). On the other hand, a “classical” SPH formulation (Monaghan, 2005) has been used.

3.1. Potential multimodal solution

The fluid is considered incompressible and the flow irrotational. The amplitude of roll motion is supposed to be small. The flow is described by means of a velocity potential satisfying the Laplace equation in the fluid domain and free-slip conditions on the solid boundaries. The free surface is decomposed into Fourier series with

time dependent coefficients βi (Faltinsen, 2000). A system of nonlinear ordinary differential equations is derived for the evolution of the βi (Faltinsen, 2000). Since the hydrodynamic moment is a function of the βi, this system has to be coupled with equation (1) to calculate the motion of the tank. This is done faster than real time with a standard CPU.

The limitations of the multimodal solution come from the restrictions in the amplitude of the roll angle, the impossibility to deal with wave-breaking phenomena and with shallow waters. For this last reason, only the configuration A has been studied with this approach.

3.2. SPH

A standard SPH formulation has been used (Monaghan, 2005). Integration in time has been done with a reversible predictor corrector scheme. Artificial viscosity and XSPH correction have been used with factors 0.02 and 0.5, respectively. Simulations have been performed using 3200 fluid particles.

Boundary conditions have been imposed by using the ghost particles technique. An extra-force term acting on the fluid particles closest to the boundaries has been added. The aim of this term is to avoid that any fluid particle gets out of the computational tank.

The moment over the rolling axis is calculated with the equation (2):

⎟⎠⎞

⎜⎝⎛ −×⋅= ∑ dt

dvgrmM ii iifl

rr (2)

where the sum is over all the fluid particles, ri is the position of particle i, and mi is the mass of particle i.

77

Figure 2: Case A. Roll angle versus time. ω =

4.61 rad/s.

4. Results

4.1. Case A

Figure 2 presents the heel angle time series obtained with an excitation frequency of 4.61 rad/s with both multimodal solution and SPH. Results show very good agreement between both.

Steady state amplitude versus excitation frequency has been plotted in figure 3. It can be observed that the resonance frequency of the coupled system is different from the one of the ship without liquid tank, this with both multimodal and SPH. However, the steady state amplitude is different with both methods near the excitation frequency ω= 6.52 rad/s.

Figure 3: Case A. Steady roll amplitude Vs

excitation frequency

The moment over the rolling axis obtained with both methods has been plotted in figure 4. They are both in phase but the multimodal solution leads to a greater moment. This may be due to the inviscid character of the potential solution.

Figure 4: Case A. Moment exerted by the fluid on the tank. ω =4.61 rad/s. SPH (solid line) and

multimodal (dashed line)

4.2. Case B

Figure 5: Case B. Particles’ position.

ω = 6.91 rad/s

In the configuration B, the roll resonance frequency of the barge matches the first sloshing frequency of the fluid. As a consequence, violent motion appears in the tank, even for small motion amplitude, as illustrated in figure 5.

Figure 6: Case B. Steady roll amplitude Vs

excitation frequency Steady roll amplitude versus excitation

frequency is plotted on figure 6. Here, the liquid acts like a damper for the ship roll motion. At resonance, there is an important roll reduction that can be observed in figure 7.

78

Figure 7: Case B. Roll angle versus time. ω =

6.91 rad/s

Figure 8 shows the different moments acting over the rolling axis. The wave moment has been multiplied by a ramp function in order to reach more rapidly a steady state. For this low angle, the restoring moment is almost linear with the heel angle. Thus, a 90 degrees phase lag can be observed between the motion of the tank and the moment exerted by the fluid on it, illustrating the efficiency of the roll reduction in this case.

Figure 8:

Moments from equation (1) versus time. ω = 6.91 rad/s. Restoring moment is the one of greater amplitude; its value has been divided

by 5 for visibility.

5. Conclusions Coupling between 1-DOF roll motion and sloshing in tanks has been studied in this article. Two different configurations have been analysed, simulating the fluid with SPH method

and, when possible, by means of a non-linear potential solution.

Good agreement between both methods has been found for small roll angles, but differences appear close to resonance. Dissipation plays an important role and further work has to be done, by comparing with experimental values, not yet available.

6. Acknowledgments This work has been partially funded by the

program PROFIT 2007 of the Spanish Ministerio de Educación y Ciencia through the project STRUCT-LNG (file number CIT-370300-2007-12) leaded by the Technical University of Madrid(UPM).

7. References ARMENIO, V., FRANCESCUTTO, A. and LA ROCCA, M. (1996). On the roll motion of a ship with partially filled unbaffled and baffled tanks - Part 2: Numerical and experimental analysis, Int. Jour. of Offsh. and Pol. Eng., 6(4): 278-290. FALTINSEN, O.D., ROGNEBAKKE, O.F., LUKOVSKY, I.A. and TIMOKHA, A.N. (2000). Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J. Fluid Mech., 407: 201-234. IBRAHIM, R. (2005). Liquid Sloshing Dynamics Theory and Applications. Cambridge University Press. MONAGHAN, J.J. (2005). Smoothed Particle Hydrodynamics, Rep. Prog. Phys., 37, 1703—1759.

SOUTO-IGLESIAS, A., DELORME, L., ABRIL-PÉREZ, S. and PÉREZ-ROJAS, L. (2006). Liquid moment amplitude assesment in sloshing type problems with SPH, Ocean Eng., 33, 11—12.

SPHERIC

79

A New Stable and Consistent Version of the SPH Method in Lagrangian Coordinates

A. FERRARI, M. DUMBSER, E.F. TORO, A. ARMANINI

Laboratory of Applied Mathematics, Department of Civil and Environmental Engineering University of Trento, via Mesiano 77, 38050 Trento, Italy. ferraria@ing.unitn.it

Abstract The purpose of this study is to solve the problem of the consistency conditions of the SPH method. In fact, consistency of SPH is in general not always guaranteed because it depends on the distribution of the observation points inside the kernel support. We focus on the Euler equations of compressible gasdynamics and propose an alternative stable and consistent approach. It consists of first-order Godunov-type SPH schemes following the work of Vila (1999), but we write the equations in Lagrangian coordinates. In this approach, the artificial viscosity term is removed and monotone inter-particle fluxes are introduced, thus assuring the stability of the method. Moreover, starting from an initially equidistant distribution of points in Lagrangian coordinates, the consistency conditions are always satisfied automatically, providing also an improvement of the accuracy. Three different Riemann solvers have been implemented for the new scheme, namely the Godunov flux based on the exact Riemann solver, the Rusanov flux and, following the work of Munz (1994), a modified Roe flux. Some well-known numerical 1D test cases (Toro 1999) are solved and compared with the first-order Godunov finite volume method in Eulerian coordinates and the standard SPH scheme with Monaghan’s viscosity term and adaptive kernel estimation (Sigalotti et al. 2006).

1. Introduction The SPH scheme is one of the earliest particle methods in numerical mechanics (Li and Liu 2002) and it is based on the Lagrangian approach. The basic idea of the SPH scheme is to replace the generic continuum by a know distribution of points, which can be placed randomly. The classical SPH method suffers from several well-known problems, namely the stability and the consistency issues. Generally, the stability problems are dealt with the use of an artificial viscosity term used by Monaghan (1994) in the motion and thermal energy equations. Balsara (1995) carried out a von Neumann analysis on the SPH method with Monaghan's artificial viscosity term and found out that only a small range of ratios of smoothing length to particle distance for a specified choice of kernel function lead to stable continuum behaviour. In spite of that, unto this day, the viscosity term proposed by Monaghan is mainly used. Recently, an alternative approach has been proposed by Vila (1999) and Moussa & Vila (2000). It consists of computing a numerical flux between each pair of interacting particles using exact or approximate Riemann solvers, without the artificial viscosity term but preserving the stability of the SPH method. In the present work, the focus is put on the solution of the 1D Euler equations in Lagrangian coordinates for an ideal gas. The studied numerical schemes are Godunov-type SPH methods with three different Riemann solvers. The per-formance of the proposed numerical

80

methods is verified by some well-known 1D test cases (Toro 1999).

2. Euler equations in Lagrangian coordinates

As seen in the literature (Liu et al. 2003) and (Rodriguez-Paz & Bonet 2003), we know that the accuracy of the SPH method depends on the distribution of the observation points inside the kernel support. In particular, it is well-known that the consistency conditions are always satisfied for a uniform distribution of elements. Following these remarks, the new idea consists of using Godunov type schemes in Lagrangian coordinates. In this way, the position of the particles are fixed during time evolution in Lagrangian coordinates (computational space) while they are moving in Eulerian coordinates (physical space). This alternative has a great advantage over the standard SPH approach: for an initial uniform distribution of elements in Lagrangian coordinates, the scheme is consistent.

2.1. Governing equations

The Euler equations for an ideal gas in Lagrangian coordinates (ξ,t) in one dimension space read as:

,d1dtdx,0FtU

ξρ

+==∂∂

+∂∂ u

ξ (1)

where U is the vector of the conserved variables and F is the flux:

.p

pF,E

VU

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

u

uu

(2)

Here V denotes the specific volume, p is the pressure, u corresponds to the velocity and E is the specific total energy:

.pV1

121E 2

−γ+= u (3)

The system (1) is in conservative form.

2.2. Numerical schemes

The numerical schemes consist of the SPH method evaluating the numerical flux at the aid of Riemann solvers:

( ) ,W

U,UF2VtUUj

ijnj

ni

nij

N

1jj

ni

1ni ξ∂

∂∆−= ∑

=

+ (4)

where N is the total number of interpolation points, W is the SPH kernel (cubic B-spline) and is a numerical flux vector, evaluated using an exact or approximate solution of the Riemann problem between the states and . In this paper, we make use of three different Riemann solvers in Lagrangian coordinates:

• the Godunov scheme; • the Rusanov scheme:

( ) ( ),UUcFFF2 ni

nj

maxnj

ni

nij −−+= (5)

with cmax the maximum celerity in Lagrangian coordinates:

( ),c,cmaxc nj

ni

max = (6)

nj

njn

jnjn

i

nin

ini

pc,

pc

ργρ=

ργρ=

(7)

• a modified Roe scheme:

( ) ( )( ),UUR~RFFF2 ni

nj

1nj

ni

nij −Λ−+= − (8)

with Λ~ and R the eigenvalues and the corresponding right eigenvectors of the Roe matrix. In the original version (Munz 1994), the eigenvalues are:

.c~,0~,

c~ni

ni

32ni

ni

1 ρ=Λ=Λ

ρ−=Λ

(9) We add an

artificial viscosity term in 2

~Λ to avoid spurious oscillations for the contact wave:

( )nj

ni2 ,max~ uu⋅α=Λ (10) where α

is a parameter 10 ≤α≤ . Note that α=0 reproduces the original Roe flux and α=1 is equivalent to the amount of the numerical viscosity introduced by the Roe scheme in Eulerian coordinates.

2.3. Numerical results

nijF

niU n

jU

81

We have selected different test problems for the one-dimensional, time dependent Euler equations for ideal gases with γ=1.4 (Toro 1999). Due to a lack of space, only one test case is presented in this section. The initial condition consists of two constant states on the left and on the right.

Table I: The two constant states on the left (L) and on the right (R).

ρ u p

L 5.99924 19.5975 460.894 R 5.99242 -6.19633 46.095

The two constant states are separated by a discontinuity at the position x=0.4. The spatial domain is discretised by N=100 points placed inside the computational interval [0,1]. The boundary conditions are transmissive and the numerical solution is referred to at time t = 0.035 units. In the following Figs. the numerical results of five different numerical schemes are compared with the exact solution. The implemented schemes are the proposed Godunov type SPH method in Lagrangian coordinates using the three alternative Riemann solvers of section 2.2, the standard SPH method with adaptive kernel estimation following the work of Sigalotti et al. (2006) and the first-order Godunov finite volume method in Eulerian coordinates. The solution of the test contains a slowly moving shock wave on the left, a right-moving contact discontinuity and a right shock (Toro 1999). Figs. 1 and 2 show the solution profiles for density and pressure.

x

Den

sity

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

10

15

20

25

30

35

40

Exact solutionEulerian first-order FV, Godunov, CFL=0.9Standard SPH [Sigalotti et al. 2006], CFL=0.9SPH, Godunov, CFL=1.9SPH, Rusanov, CFL=1.9SPH, modified Roe, α=0.75, CFL=1.9

Figure 1: Profile for density, obtained by five different numerical schemes.

x

Pre

ssur

e

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Exact solutionEulerian first-order FV, Godunov, CFL=0.9Standard SPH [Sigalotti et al. 2006], CFL=0.9SPH, Godunov, CFL=1.9SPH, Rusanov, CFL=1.9SPH, modified Roe, α=0.75, CFL=1.9

Figure 2: Profile for pressure, obtained by five

different numerical schemes.

Comparing the numerical solutions, the most accurate profiles for density and pressure are obtained by the Godunov type SPH scheme in Lagrangian coordinates with the modified Roe flux. In particular, it solves the contact wave very precisely. Only a small glitch is present in physical space (x,t) due to the separate integration of the ODE for position (1), but it is

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not present in Lagragian mass coordinates. Spurious oscillations near the discontinuity appear for the SPH method with the exact Riemann solver, which is a well-known phenomenon in Lagrangian coordinates (Munz 1994). The finite volume method in Eulerian coordinates is more diffusive for the contact wave, as expected. Sigalotti’s SPH scheme solves the contact discontinuity not very well because it is not conservative and not monotone. It produces unphysical oscillations in the numerical profile for pressure and wrong post-shock values in the star region of the flow for density. The resolution of the left shock wave is similar for all schemes, except of the SPH scheme of Sigalotti et al. (2006) which obtains a wrong shock position.

3. Conclusions In this paper, the focus has been on the one-dimensional Euler equations and their discretization. We have proposed a new approach, always consistent for an initially uniform distribution of points. It consists of using Godunov type SPH scheme in Lagrangian coordinates, which provides an improvement in the accuracy and stability over the standard SPH method. Note that this approach is able to handle large deformations of the computational domain without an algorithm to rezone the mesh. Possible future developments of the proposed SPH scheme is to increase the order of the method and to extend the approach to multidimensional applications, such as free surface flows, impact problems and the interactions between fluid (water) and solids.

4. References BALSARA, D.S. (1995) Von Neumann stability analysis of smoothed particle hydrodynamics --

Suggestions for optimal algorithms. J. Comp. Phys., 121: 357-372. LI, S. and LIU, W.K. (2002) Meshfree and particle methods and their applications. Applied Mechanics Reviews, 55: 1-34. LIU, M.B., LIU, G.R. and LAM, K.Y. (2003) Constructing smoothing functions in smoothed particle hydrodynamics with applications. J. Comput. Phys. Appl. Math., 155: 263-284. MONAGHAN, J.J. (1994) Simulating free surface flows with SPH, J. Comp. Phys., 110: 399-406. MOUSSA, B.B. and VILA, J.P. (2000) Convergence of SPH Method for scalar nonlinear conservation laws. J. Numer. Anal. 37: 3, 863-887. MUNZ, C.D. (1994) On Godunov-Type Schemes for Lagrangian Gas Dynamics. SIAM J. Num. Analysis, 31: 1, 17-42. RODRIGUEZ-PAZ, M.X. and BONET, J. (2003) A Corrected Smooth Particle Hydrodynamics Method for the Simulation of Debris Flows. Numer. Meth. Partial Diff. Eqns. 20, 140-163. SIGALOTTI, L.Di G., LOPEZ, H., DONOSO, A., SIRA, E. and KLAPP, J. (2006) A shock-capturing SPH scheme based on adaptive kernel estimation. J. of Comp. Physics, 212: 124-149. TORO, E.F. (1999) Riemann solvers and numerical methods for fluid dynamics: a practical introduction. 2nd. ed. Berlin [etc.] Springer. VILA, J.P. (1999) On Particle Weighted Methods and Smooth Particle Hydrodynamics. Math. Models and Methods in Appl. Sciences, 9: 2 161-209.

SPHERIC

Degradation and Instability of Incompressible SPH Computations of Simple Viscous Flows

Jason P. Hughes and David I. Graham

School of Mathematics and Statistics

University of Plymouth, UK

Abstract

This paper reports the results of computations using incompressible smoothed particle hydrodynamics (I-SPH) to predict three elementary shear flows, (i) Couette flow between parallel plates (impulsively-started and steady-state), (ii) Pressure-driven flow between parallel plates (impulsively-started and steady-state), and (iii) lid-driven cavity flow. In all cases, a steady-state solution was looked for. However in some cases a steady-state solution could not be obtained, with the computations becoming unstable before a fully-developed state was reached. In other cases it was found that progressing the solution beyond the ‘steady-state’ led to a degradation in the solution and, eventually, to instability in the computations. In order to establish the cause of the stability, many I-SPH simulations were carried out with different parameters/variables. It is found that the instability is dependent on a number of factors, such as (i) formulation of the pressure Poisson equation (PPE), (ii) inter-particle spacing, and (iii) particle smoothing length.

1. Introduction

Incompressible smoothed particle hydrodynamics (I-SPH) is a two-step predictor-corrector method that tries to ensure that SPH velocity fields are incompressible. The predictor step solves a purely viscous problem, neglecting the pressure gradient. The correction step then updates the predictor

velocity by subtracting away a zero-curl part which is proportional to the pressure gradient, see Shao and Lo (2003).

The pressure is found by solving the discrete form of the Pressure Poisson equation:

∗∆+∗

∇∆

=∇ρ

⋅∇

[1]

subject to zero-gradient conditions on solid boundaries, where ∗ρ and ∗

are respectively the density and velocity after the prediction step. This is an implicit equation which is discretized in matrix form and the matrix equation solved using a biconjugate gradient method to find ∆+ , the particle

pressure at time ∆+ . Three different SPH approximations for

the divergence term in equation [1] have been considered. These are:

(i) ∇⋅ρ

=⋅∇

[2]

(ii) ∇⋅−ρ

=⋅∇

[3]

(iii) ∇⋅ρ

+ρ

ρ=⋅∇

!! !!!! !!!! !! [4]

In equations [2]-[4], the approximations are defined at particle a, whose neighbours are particle b, m is the particle mass, ρ is the density and u is the velocity. A cubic spline kernel function "$#%

is used (Shao and Lo

(2003)). We refer to the SPH approximations for divergence in equations [2], [3] and [4] as

83

div0, div1 and div2 respectively. Alternatively, by using the equation for

mass conservation, the PPE can be expressed as

&'''' ()*+,

∆ρρ−ρ

=∇ρ

⋅∇ ∗∆+

∗

[5]

where -ρ is the particle density at time t and

∗ρ is the intermediate particle density after the prediction step. We refer to this formulation of the PPE as dent. Some workers replace the particle density at time t, -ρ , in equation [5] with the initial constant

density of the particles. This formulation is referred to as den0.

Throughout, the timestep, dt, used satisfied the viscous diffusion criterion

( ν≤ ./10234/65 7) and Courant condition

( 819:;<9 ≤∆ ), where dx is the initial particle

spacing, ν is the kinematic viscosity and ∆x is the instantaneous increment in particle position.

2. Results and discussion

2.1 Couette flow and pressure-driven flow For the fully-developed Couette and

pressure-driven flows, particle velocities should reach steady-state values and remain there indefinitely. However, this did not happen and the solutions became unstable, even when the initial conditions for the simulations corresponded to steady-state conditions. Initially the particles were in a regular array and the starting velocity of each particle was set to be that of fully-developed flow (i.e. => = ). Note that the top lid, at ?@ = , moves with constant velocity and the

bottom lid, at AB = , remains fixed. As the simulations are continued in time ,

the particles below the moving lid become disordered as seen in figure 1, where the particle positions at 19s are shown. Figure 2 shows a large variation in the u-velocity of

particles below the moving lid. The solutions continue to degrade with time, eventually becoming unstable. We observed similar behaviour when using the div2, den0 and dent PPE formulations, though the latter two remained stable for longer.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 1: Particle positions at 19s.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0y (m)

U-v

elo

city

(m

/s)

Figure 2: Variation of velocity with y at 19s.

Similar behaviour occurs when the initial velocity of the fluid particles is zero and the flow develops due to the motion of the lid. A steady-state solution is reached, and the time-dependent behaviour of the flow is in good agreement with the analytical series solution (Morris et al (1997)). However progressing the simulations beyond the steady-state leads to a similar breakdown in the solution as shown in figures 1 and 2.

Simulations for pressure-driven flow between parallel plates have shown similar behaviour to that observed for Couette flow. The transient behaviour is in good agreement with the analytical series solution and a steady-state solution can be reached. However, continuing the simulations in time

84

leads to errors in the velocities of particles near the walls, and eventually instabilities.

2.2 2D lid-driven cavity flow Initial calculations were carried out using

the div2 formulation of the PPE (equation [4]), but they eventually became unstable. In order to improve the stability we considered (i) time-step reduction, (ii) methods for implementing boundary conditions of PPE (Hughes and Graham (2007)), (iii) increasing the number of particles and (iv) varying the particle smoothing length. Although some of these variables do have an effect on the stability in I-SPH, it was found that if the div2 PPE formulation is used then the computations would always become unstable eventually.

With the correct particle spacing and smoothing length the other PPE formulations (div0, div1, dent, den0) can be used to produce stable solutions. However, when solving the PPE it is necessary to set a reference pressure at one particle. If the same particle is used on each time step, then the computations were unstable. This is related to the solvability constraint on a system with gradient boundary conditions (Cummins and Rudman (1999)). Various methods were used to satisfy this constraint, but the most successful was to randomise the reference particle on each timestep.

The I-SPH method has been used to solve the lid-driven cavity flow at two values of Reynolds number, CDFE

= and GIHJHKHLFM= .

Initially, the velocities of wall and dummy particles on the top lid are set equal to (1,0) and the velocities of all other particles are set to be zero. The lid then drives the flow, which in theory should develop to a steady state. Figure 3 shows the development of the velocity profiles for GIHJHKHLFM

= , using 50×50 particles with a smoothing length N6OPQRS

= and the div1 PPE formulation. Fully-developed solutions are found at around 60s (results were computed beyond 60s and showed little change). The results presented

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0U-velocity (m/s)

y (m

)

t = 10

t = 20

t = 40

t = 70

t = 100

Figure 3 : Development of velocity profiles at TUVW = ( GIHJHKHLFM

= ).

in figure 3 were obtained using a smoothing length of N6OPQRS

= . Initial calculations were carried out with a smoothing length of

N6ORQXS= , which is widely used, but the

velocity profiles obtained were less accurate and less smooth.

When using the div2 PPE formulation, the computations become unstable in a short time. Figure 4 shows the large velocity vectors that occur in the corners below the lid, which lead to gaps occurring between particles, at a time of 0.248s.

Figure 4: Velocity vectors at t=0.248s for div2

PPE formulation (Re=1000).

Stable solutions were obtained using the div0, den0 and dent PPE formulations, though the results were less accurate than with the div1 formulation. Figure 5 shows

85

a comparison of the velocity profiles for the four different formulations ( GIHJHKHLFM

= , 50x50 particles, N6OPQRS

= ).

-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.050.000.050.100.150.200.25

0.0 0.2 0.4 0.6 0.8 1.0x (m)

V-v

elo

city

(m

/s)

div0

div1

den0

dent

Figure 5: Velocity profiles at YZ[\ = for

various PPE formulations ( GIHJHKHLFM= ).

For a low Reynolds number ( CDFE= ), the

div2 PPE formulation reaches a steady-state solution, which occurs at approximately 0.1s. However, if the simulations progress beyond this time, then eventually the method becomes unstable. This does not occur with the other PPE formulations. Figure 6 shows the development of velocity profiles with time for CDFE

= , using 50×50 particles with N6OPQRS

= and the div2 PPE formulation. It is seen that a steady-state solution is reached at 0.1s and the solution remains unchanged until 0.169s. Then in a short time the solution degrades rapidly and the computations become unstable.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0U-velocity (m/s)

y (m

) t = 0.1

t = 0.169

t = 0.1697

t = 0.1698

t = 0.1699

Figure 6: Development of velocity profiles at TUVW = with time for div2 PPE formulation

( CDFE= ).

Using the other PPE formulations stable solutions for ]^F_

= were obtained, with the velocity profiles remaining virtually unchanged beyond `abcd

= . However, for coarser resolutions of particles (e.g. 25×25) the calculations were unstable for efGg6hij

= , and stable for kl6mno

= . As expected, the accuracy improved with finer resolutions of particles (resolutions of 10×10, 25×25, 35×35, 50×50 and 70×70 particles were used).

3. Conclusions

I-SPH has been used to solve moderate Re viscous flows. It has been shown that stability and accuracy depend on a number of factors, including PPE formulation, its boundary conditions and the method used for setting a reference pressure, as well as the h/dx ratio and particle resolution.

4. Acknowledgments

The authors thank Dr. Songdong Shao for providing the initial I-SPH code.

5. References

CUMMINS S.J. & RUDMAN M. (1999). An SPH projection method. J. Comp. Phys., 152 (2): 584-607.

HUGHES J.P. GRAHAM D.I. (2007). Accuracy of low Reynolds number computations using incompressible SPH. IAHR2007 congress, Venice.

MORRIS J.P., FOX P.J. and ZHU Y. (1997). Modeling low Reynolds number incompressible flows using SPH. J. Comp. Phys., 136: 214-226.

SHAO S. and LO E.Y.M. (2003). Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. In Water Resources, 26: 787-800.

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SPHERIC

87

SPH for Cold and Low Viscous Shear Flow

Yusuke IMAEDA,1 Toru TSURIBE,2 and Shu-ichiro INUTSUKA 3

1 Kobe University, Department of Earth and Planetary Science, Imaeda@kobe-u.ac.jp 2 Osaka University, Department of Earth and Space Science, tsuribe@vega.ess.sci.osaka-u.ac.jp 3 Kyoto University, Department of Physics, inutsuka@tap.scphys.kyoto-u.ac.jp

Abstract SPH is a powerful tool to investigate the fluid

dynamics, especially in astrophysics. However, the application of standard SPH to the cold and strong shear, which is occasional in the astrophysical problem, gives the large errors in density calculation. Although shear flow is divergent free, sufficient amount of artificial viscosity is required in order to prevent this density error. This indicates that one should be careful to reduce the artificial viscosity, even if the flow is not divergence dominant. A new solution for this density problem, which is based on the particle rezoning, is also introduced.

1. Introduction Smoothed Particle Hydrodynamics (SPH)

is one of the powerful calculation methods that are widely used in many scientific fields, especially in the astrophysics. It is invented to calculate the multidimensional fluid in astrophysics (Lucy (1977), Gingold & Monaghan (1977)), and many researchers developed SPH to improve the calculation ability. However, less attention has been paid to a shear flow. Sear flow is one of the fundamental motions in fluid dynamics, and has to be checked in detail to understand the calculation method. For example, we are interested in the formation of Jupiter, which is born in a cold (cs=0.05) shearing gas disk. Imaeda and Inutsuka (2002) (II02) showed that SPH has a flaw in the long-term evolution of such a cold shear flow, where the large numerical error arises in the density after the dynamical time. Monaghan (2006) also tested the effect of the shear flows on SPH, but they did not find such density errors in their investigated model. Therefore, we revisit this

issue in more detail. In this paper, we especially focus on the initial particle distribution, the strength of AV, and the difference of temperature, which are not investigated precisely in our previous work.

2. Shear test with standard SPH with artificial viscosity

Two dimensional test calculations are shown in this paper. We use two types of initial particle distributions. One is the aligned particle distribution and the other is the iterative particle distribution. We use 4900 particles. The particles are placed on the Cartesian grid for the aligned particle distribution. For the iterative particle distribution, we prepare it as follows: The particles are randomly placed at first. Then we iteratively move the particles

by 212

i ix a t∆ = ∆r uv

, where iauv

is the pressure

from the neighbor particles. The velocity is reset to 0 in every step, which act as a strong viscous effect to damp the small scale fluctuations. Fig. 1 shows these initial distributions and the corresponding density. The density errors are about 1% in the iterative case. Gaussian kernel is used and the kernel width h is defined by

12( )pmh η

ρ= where 1η = . The domain of

calculation is -0.5<x<0.5 and -0.5<y<0.5, and periodic boundary condition is adopted. Both initial distributions represent the constant density of unity quite well.

88

Figure 1: The initial particle distribution and

the density

Then we shift the particle position according to sin(2 )y xπ∆ = . If the continuous fluid is deformed by the above equation, it is obvious that the density does not change. Fig. 2 shows the SPH density after the above shift. While the aligned case gives the good results (left panel), the iterative case has large errors in density (right panel) after the shift. This result indicates that SPH density in shearing shift is quite sensitive to the initial particle distribution. If the particles are aligned with the shear direction, the result is fine. There is no chance that one particle approaches the other particles too much. However, if the particles are not aligned with the shear direction, SPH gives the large errors in density. In general, it is not necessary that the particle aligns with the shear direction, since our interested flow is frequently complicated. Therefore, to start with such a non-aligned initial particle distribution is important to test the numerical method.

Figure 2: Density after the shearing shift

The above density errors can be seen in the evolutional calculation, too. We assume the initial velocity field of 0xv =

and sin(2 )yv xπ= . The equation of state is

isothermal and the sound speed is 0.05sc = .

The artificial viscosity is included

as sig .| |

ab abab

ab ab

Kv v rrρ

Π = −

uur uur

uur ,

where 0.5K = , sig sa sbv c c= + and the

equation of motion is calculated numerically. The results at t=1.0 are shown in Fig.3. The comparison between Fig.2 and Fig.3 clearly shows that the density errors are numerical ones, not physical ones, though the shear flow is unstable in general; if there is no restoring force to suppress the Kelvin Helmhorz instability.

Figure 3: The dependence on the initial particle

distribution

89

. Figure 4: The effect of AV parameter K

Fig.4 shows the effect of AV controlling

parameter K on the results. We start the calculation from the same iterative initial particle distribution. The results at t=1.0 are shown. The sound speed is 0.1 for all these calculations. While K=1.0 and 0.5, the errors are less than several tens of %, but it becomes comparable to the initial density when we reduce the AV parameter K

We also test the effect of the sound speed in Fig. 5. The AV parameter K is 0.5 and the results at t=1.0 are shown. While the sound speed is large (temperature is high), the density error is about 10% or 20%, but it becomes as large as the initial density itself when the sound speed is low.

The difference arises around cs=0.01 in these tests and it corresponds to the typical velocity difference between two particles (i.e. 0.09). If the sound speed is small, the pressure repulsing force between two particles becomes less effective to prevent the particle approach, and large density errors arise. Therefore, even if the flow is almost divergent free, a sufficient artificial viscosity should be included in the SPH calculation, or we need sufficient number of particles in order to keep the velocity difference between two particles smaller than the sound speed. However, many researchers in astrophysics tend to reduce the AV term since it

causes the artificial (unphysical) angular momentum transfer in the result. The correction introduced by Balsara (1995) reduces the AV term where rot v is dominant. But our results indicate that one should be careful to reduce the AV term.

Figure 5: The effect of sound speed.

3. "Particle rezoning" method As seen in a previous section, in cold and

low viscous shear flow, standard SPH with artificial viscosity is not sufficient to keep smooth density unless special distribution of particle position is kept. One possible solution to overcome this problem is suggested in previous work (II02) in non-conservation form with constant smoothing length. Here we introduce different method to overcome the difficulty. The present new method is much simpler, faster, and in conservation form. The main reason of the density error in cold and low viscous shear flow is that one particle approaches another particle too close due to the lack of sufficient pressure or viscous force. However, this close encounter of SPH particles can be avoided by ''particle rezoning" which we newly introduce. Since SPH particle spreads in a finite region with overlapping with another SPH particle,

90

some part within a kernel in one SPH particle is closer to the center of another SPH particle (Fig.6.). In ''particle rezoning" method, a part of mass within a kernel is exchanged between two SPH particles and position of mass center of each SPH particle is modified consistently. The mass center of a SPH particle is modified without any artificial term in equation of motion.

Figure 6: Mass exchange and the correction of

the particle position

Figure 7: Particle position and the density plot

with particle rezoning method

Therefore, it is also available with zero temperature without artificial viscosity. Mass of SPH particle can be kept constant if desired. Momentum and energy can be treated consistently in conservation form. Fig.7 shows the results of the new ''particle rezoning" method. Although with very small sound velocity (cs=0.0) without any viscosity, the density distribution is kept to be smooth during long-term evolution in shear flow.

4. Conclusions

Cold and strong shear calculation is tested by standard SPH with artificial viscosity. When the artificial viscosity is strong and the temperature is low, the intrinsic SPH density errors arise in the calculation. This indicates that although shear flow is divergent free, sufficient amount of artificial viscosity is required in order to prevent the density errors from arising. The large sound speed reduces the error, too. One should be careful to reduce the artificial viscosity, when the flow is cold and shear is strong, which is occasional in the astrophysical problem. A new solution to overcome this problem that is based on the particle rezoning method also reduces the density errors even if the temperature is low.

5. References GINGOLD, R. A. and MONAGHAN, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non- spherical stars. Mon. Not. R. Astr. Soc., 181: 375- 389. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. IMAEDA, Y. and INUTSUKA, S. (2002), Shear Flows in Smoothed Particle Hydrodynamics, The Astrophysical Journal, 569, 501-518 MONAGHAN, J. J. (2006), Smoothed particle hydrodynamic simulations of shear flow. Monthly Notices of the Royal Astronomical Society, 365, 199-213. BALSAR, D. S. (1995). von Neumann stability analysis of smooth particle hydrodynamics--suggestions for optimal algorithms, Journal of Computational Physics, 121, no. 2, 357-372

SPHERIC

91

SPH simulation of moderate Reynolds number flows

Libor LOBOVSKÝ1 and Jan VIMMR1

1 Department of Mechanics / University of West Bohemia in Pilsen, Univerzitní 22, 30614 Plzeň, Czech Republic, lobo@kme.zcu.cz, jvimmr@kme.zcu.cz

Abstract Within this study, the standard smoothed particle hydrodynamics (SPH) method is applied to the elementary problems of incompressible fluid dynamics. The implemented SPH code is used to solve the Poiseuille flow of viscous fluid in two and three dimensions. The SPH boundary condition implementation as well as the performance of the SPH code during the modelling of confined channel flows at moderate Reynolds numbers are discused.

1. Introduction The article compiles a knowledge gained

from the published literature on SPH. Three well-known elementary types of SPH solid boundary conditions are implemented and applied to a two dimensional as well as three dimensional laminar flow problems. The results are compared to the analytic solution.

2. SPH model The standard SPH method is applied within

this study, i.e. a continuum is discretised by a single set of interpolating points (particles) where the values of pressure, velocity and density are updated. A symetric form of the SPH equations is implemented as follows.

The conservation of mass is assured by the continuity equation

ijijiji Wm

dtd

∇⋅−= ∑ )(j

vvρ (1)

where ρ is the density, m is the mass, v is the velocity vector and W is the smoothing function with a continuous derivative ∇iW. The

subscripts i, j denotes the variables at the particles i, j, respectively, and ∇i denotes a derivative according to ri, where r is the position vector. The cubic B-spline smoothing function is applied, Monaghan (1992). The smoothing function support is defined by 2h, where h is the smoothing length.

The motion of the discretised continuum is governed by the Navier-Stokes equation

iiji

dtd fv

++= VPij

(2)

where Pij is the pressure term, Vij represents the viscous forces and fi is the body force. The pressure term is derived so that

22ij ∑ ∇⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

jiji

j

j

i

ij W

ppmρρ

P

(3)

where p is the pressure. The viscous forces are modelled by a term derived by Morris et al. (1997)

∑ −+−

∇⋅−+=

jji

ji

ijijiji

ji

jij

Wm)(

))((22 vv

rr

rr

η

µµρρ

V

(4)

which combines a standard SPH and a finite difference approximation of the first derivative. The symbol µ stands for the dynamic viscosity and η=0.01h2 is a corrective constant avoiding a creation of singularity when particles are approaching each other.

The presented SPH model implies the quasi-incompressible representation of the incompressible fluid using the equation of state published by Batchelor (1967)

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+= 10

0γ

ρρ

i

iii Kpp

(5)

where K is the bulk modulus

92

γρ

200 cK=

(6)

where 0c is the initialy defined sound speed and γ is a constant parameter. A constant 0p, 0ρ respectively, indicates the initial pressure, the initial density respectively. When the sound speed value is set so that the resulting Mach number is less than 0.1, the fluid is assumed to be quasi-incompressible. The parameter γ=7 is usually considered.

Furthermore, two commonly used stabilising terms are also considered. The von Neumann-Richtmeyer SPH artificial viscosity term is added to the pressure term in (3), Monaghan & Gingold (1983),

2

22

2 )()()(21

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+−

−⋅−

+

+=Π

ηρρβ

ji

jiji

ji

jiij

hh

rr

rrvv

(7)

for (vi-vj)⋅(ri-rj)<0, where β is the artificial viscosity parameter, and XSPH correction is implemented, Monaghan (1994),

ijij

ij

jji

i Wmdtd

)(21 ρρ

ε+

−+= ∑

vvvr

(8)

A parameter ε is in an interval <0;1>.

The time integration of SPH equations is performed using a predictor-corrector scheme while determining the timestep according to the conditions published in Morris et al. (1997).

3. Boundary conditions In order to model the flow through straight

channels, the periodic boundary conditions are applied in the flow direction. In order to model no-slip and anti-penetration boundary conditions, there are three commonly used types of solid boundary conditions implemented.

3.1. Solid b.c. type I

Solid boundary conditions type I utilise the ghost particles which are created at every timestep by symetric reflection of the fluid particles accross the boundary surface, Libersky et al. (1993). The width of the reflected boundary region is defined by the smoothing function support 2h. The ghost particles have

assigned the same density and pressure as the reflected fluid particles. In order to satisfy the no-slip and the anti-penetration boundary conditions, the ghost particle velocity vector has the same magnitude but the opposite direction than the velocity vector of corresponding fluid particle.

3.2. Solid b.c. type II

Solid boundary conditions type II are represented by boundary particles which fill in the boundary wall region to the width of the smoothing function support 2h. The position of the boundary particles is constant. In order to model the no-slip and the anti-penetration boundary condition, the boundary particles have assigned a virtual velocity which is updated at every timestep. Its value is extrapolated from the fluid particle velocity across the boundary tangent plane. The velocity at the tangent plane is assumed to be zero. Thus the resulting virtual velocity vector has an opposite direction than the fluid particle velocity, Morris et al. (1997).

3.3. Solid b.c. type III

Solid boundary conditions type III combine the boundary particles which line the solid boundary surface and the ghost particles which reflect the fluid particles and their velocities across the surface, Liu and Liu (2003). The boundary particles position is constant and their velocity is zero. The properties of the ghost particles are updated at every timestep according to the properties of fluid particles which are being reflected, as described above.

93

Figure 1: SPH and analytic solution of the

Poiseuille flow for Reynolds number equal to 1, 3, 5, 10 and 20.

4. Results and discussion The implemented SPH code is tested on

the Poiseuille flow between two parallel infinite plates and is confronted with the analytic solution. The channel width is 4⋅10-3 m and the length of the simulated channel section is 2⋅10-

3 m. The fluid density is 1000 kgm-3, its dynamic viscosity is 8.9⋅10-4 Pa⋅s and its initial velocity is 0 ms-1. The 2D simulation setup consists of 800 regularly distributed fluid particles with the initial interparticle distance of 10-4 m. There are additional 2 sets of 80 boundary particles representing the channel walls for the b.c. II and 2 sets of 20 boundary particles for the b.c. III. Various Reynolds number flows driven by body forces are modelled, Figure 1. The viscosity term (4) is tested during a flow driven by the body force 5⋅10-4 ms-2 for various values of dynamic viscosity, Figure 2.

The differences in results of calculations, which are performed for all three solid boundary implementations, are negligible. The computed fluxes differ in less than 1% from the analytical data and the volume variations are less than 0.1% for all three boundary conditions.

Figure 2: SPH and analytic solution of the

Poiseuille flow for dynamic viscosity equal to 10-

2, 5⋅10-3, 2⋅10-3, 10-3 and 5⋅10-4 [Pa⋅s].

Naturally, the SPH model is capable of

simulating the same problem in 3D as well, Figure 3. The 3D model consists of 18800 regularly distributed fluid particles with initial spacing of 10-4 m. The periodic b.c. are applied in both x and z direction, while the flow is driven in x direction.

The solid b.c. type II is further applied to model the Poiseuille flow in a pipe with radius 2⋅10-3 m, Figure 4. The simulation consist of 6090 regularly distributed particles with spacing 2⋅10-4 m. Results of both presented 3D flow simulations for the body force 5⋅10-4 ms-2 agrees well with the analytic solution.

Figure 3: SPH solution of the 3D Poiseuille flow between two paralel infinite plates

5. Conclusions While comparing the computational

performance of the boundary conditions, the type I and III are more efficient for the problems with a simple geometry (such as the Poiseuille flow). In such case, it is sufficient to create the ghost particles and the virtual velocities during the predictor step. The boundary type II requires calculation of virtual velocity for every pair of interacting fluid and boundary particles separately. Thus its performance is significantly slower than the performance of the other two conditions. On the contrary, utilising the boundary conditions type II can be advantegeous when complex geometry is being

94

modelled. In such case, reflecting the ghost particles across the boundary surface may produce an undesirable increase, decrease, respectively, in the boundary particle mass in the regions of sudden shape changes (for convex shape, concave shape, respectively). Such a change of boundary mass may corrupt the entire calculation.

For simulations of the Poiseuille flow with higher Reynolds number value, the solution of the presented SPH model becomes unstable for all three boundary implementations. The computation can be stabilised by decreasing the sound speed value which encreases the artificial compressibility of the model.

The influence of the artificial viscosity term (7) and the XSPH correction (8) on the profile development were also examined. The results of the performed simulations did not show any significant influence neither on the resulting flux development nor on the stability of the calculation.

6. Acknowledgment This work is supported by the research

project MSM 4977751303 of the Ministry of Education, Youth and Sports of the Czech Republic.

Figure 4: SPH solution of the 3D Poiseuille flow through the cylindrical pipe

7. References BATCHELOR, G.K. (1967). Introduction to fluid dynamics. Cambridge University Press, Cambridge. LIBERSKY, L.D., PETSCHEK, A.G., CARNEY, T.C., HIPP, J.R., ALLAHDADI, F.A. (1993). High strain Lagrangian hydrodynamics. J. Comp. Phys., 109: 67-75. LIU, G.R., LIU, M.B. (2003). Smoothed Particle Hydrodynamics. World Scientific Publishing, Singapore. MONAGHAN, J.J., GINGOLD, R.A. (1983). Shock simulation by the particle method SPH. J. Comp. Phys., 52: 374-389. MONAGHAN, J.J. (1992). Smoothed Particle Hydrodynamics. Ann. Rev. Astron. Astrophys., 30: 543-574. MONAGHAN, J.J. (1994). Simulating free surface flows with SPH. J. Comp. Phys., 110: 399-406. MORRIS, J.P., FOX, P.J., ZHU, Y. (1997). Modeling low Reynolds number incompressible flows using SPH. J. Comp. Phys., 136: 214-226.

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Extension of the Finite Volume Particle Method to Higher Order Accuracy and Viscous Flow

R. M. Nestor1 and N. J. Quinlan2

Dept. of Mechanical and Biomedical Engineering, National University of Ireland, Galway, Ireland 1 malachy.nestor@nuigalway.ie 2 nathan.quinlan@nuigalway.ie

Abstract The Finite Volume Particle Method (FVPM) is

a mesh-free method for Computational Fluid Dynamics (CFD). The method is formulated from the integral conservation form of the governing equations, and particle interactions are described in terms of fluxes between the particles. To date, these fluxes have been computed using first order upwind methods. In the present work, the accuracy of the FVPM is improved by using a higher order upwind method. In particular, a consistency-corrected Smoothed Particle Hydrodynamics (SPH) approximation is used in the implementation of the higher order method. The improved accuracy of the method is demonstrated using shock tube simulations in one dimension. In addition, a viscous implementation of the method is presented, and the improved accuracy of the high order extension of the FVPM for viscous flow is demonstrated using a Poiseuille flow test case.

1. Introduction The Finite Volume Particle Method (FVPM)

was introduced by Hietel et al. [1], and subsequently developed by Keck [2], Teleaga [3], and others. As in SPH, the fluid in FVPM is represented by particles, which in turn are associated with smooth, overlapping, compactly supported kernel functions. The particles are viewed as discrete volumes to which the integral conservation form of the governing equations apply. A particle-particle interaction can then be defined in terms of a flux,

which depends on the overlap of kernel supports. The finite volume-based formulation allows the use of upwind numerical flux functions, which eliminate the need for artificial viscosity in shock capturing simulations. Particle-boundary interaction can also be described using fluxes, allowing boundary conditions to be enforced without the use of fictitious particles. In addition, Teleaga [3] has shown that FVPM inherits the conservation properties of the finite volume method, regardless of any variation in smoothing length.

Previous implementations [1-3] of the FVPM for inviscid compressible flow have been limited to the use of first order upwind methods for computing particle interactions. Higher order upwind methods have been developed for mesh-based CFD which lead to improved accuracy while still producing oscillation-free solutions in the presence of discontinuities. In this work, a higher order upwind method is implemented in the FVPM, and the improved accuracy is demonstrated using a shock tube test case. In addition, the FVPM is extended to the solution of viscous flow, and the results are validated for a Poiseuille flow test case.

2. Methods In this section, the basic formulation of the

FVPM is outlined, and some of the features of the method are highlighted. The formulation of higher order and viscous extensions of the method are also presented.

96

2.1. The Standard FVPM

The FVPM is derived from the integral form of the governing equations:

,0=∫ ⋅∇+∫∂

∂dV

VF

VdVU

t

rrr (1)

where U and F are the vector of conserved variables and the flux vector respectively: where ρ is the

density, T is the temperature, E is the total energy, p is the pressure, τ is the viscous stress tensor, k is the thermal conductivity, ur is the fluid velocity vector, and I is the identity matrix. The derivation of the method is given by Hietel et al. [1] and will not be repeated here. The semi-discrete FVPM can be written as:

( ) ( ) ,),( bGbjUiUGj ijdt

iUiVdββ −∑−=

rrrr

(2)

where Vi is the particle volume and G represents a numerical approximation (for example an upwind or central discretisation) to the relative flux xUF &

rr⋅− .

x& is the average particle velocity, as distinct from the fluid velocity. Equation 2 is similar to the corresponding equation for the Finite Volume Method, except that the area between a pair of elements is replaced by a geometric coefficient βij. This coefficient is a weighting for the interaction between a pair of particles, and is defined by:

.

1)(

)()()()(∫

∩ ∑=

∇−∇=

⎟⎠⎞⎜

⎝⎛jViV

dxN

kxkW

xiWxjWxjWxiWij

rrrβ

(3)

Equation 3 is evaluated using numerical integration in the shaded particle overlap region of Figure 1. Since βij = –βji, the interaction between a pair of particles is symmetric regardless of the variation in smoothing length. This is significant because it ensures that the method is conservative.

The βbGb term in Equation 2 represents particle-

boundary interaction, and is simply a boundary geometric coefficient times a flux contribution. Boundaries can therefore be implemented in a non-ambiguous way, and without the need for fictitious particles. Another interesting feature of the method is that the particle velocity x& need not equal the fluid velocity, e.g. the particles may be stationary. In addition, the convective terms of the governing equations can be discretised using any upwind finite volume scheme, eliminating the need for empirically determined artificial viscosity coefficients in shock-capturing applications.

2.2. Higher Order Accurate Extension of the FVPM

The standard upwind discretisation of the flux function F is only first order accurate. This is equivalent to computing the fluxes between particles using a zero order reconstruction of particle values to the particle-particle interface, as shown in Figure 2.

Upwind methods can be extended to higher orders of accuracy by using linear, quadratic, or higher order reconstruction of the particle values to the particle-particle interface. For a linear reconstruction, the gradient of solution values within each particle is required. Barth and Jespersen [4] have suggested that the numerical approximation to this gradient must be exact when the solution has linear variation. This ensures that analytic solutions with linear variation are reproduced by the numerical scheme.

Figure 1: Particle interaction in FVPM

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

∇−⋅−+−+⊗==

TkupEuIpuu

uF

EuU

rrr

rr

r

rrr

τρτρ

ρ

ρρρ

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This requirement can be satisfied in the mesh-

free framework of the FVPM using the first order consistent SPH gradient approximation of Bonet and Lok [5]:

( ) ,~

∑ ∇−=∇j jWijjVi

rrφφφ (4)

where:

( ) .1~

jWj ixjxjWjVjW ∇

−∑ −⊗∇=∇ ⎟

⎠⎞

⎜⎝⎛ rrrrr

Equation 4 will reproduce the gradient of a linearly varying function exactly, even when the particles become disordered.

The fully discrete form of the higher order FVPM may be implemented using a second order Runge-Kutta method, omitting the boundary terms for brevity:

( ) ( ) ( )( ) ( ) ( ) ,*

)(,*)(

1

,2

*

RULUGj ijtn

iUiVniUiV

njUn

iUGj ij

tniUiViUiV

∑∆−=+

∑∆

−=

β

β

r

r

w

here U(L) and U(R) denote the extrapolated solution values to the left and right of the particle-particle interface respectively.

2.3. Viscous Extension of the FVPM

Extension of the method to viscous flows involves approximation of the viscous stress and thermal conductivity terms of the flux function F

r.

These terms are usually centrally discretised, as opposed to the convective terms which are

discretised using upwind methods. The current approach is to again use the consistency corrected SPH gradient approximation of Equation 4 for computing the required velocity and temperature gradients.

3. Results and Discussion The higher order upwind FVPM for inviscid

flow is validated using a one-dimensional shock tube test case. An initial pressure ratio of 4 between the left and right states is selected. The velocity in the tube is initially set to zero and 400 Lagrangian particles are initially uniformly distributed along the length of the tube. Instantaneous pressure distributions in the tube are shown in Figure 3 at a time of 4.6 ms. The higher order upwind method shows an enhanced ability to resolve the pressure discontinuity compared with the first order version.

The viscous implementation of the FVPM is

validated for Poiseuille flow. The Poiseuille channel flow is a well known incompressible viscous test case which has an analytic solution for both transient and steady states. The channel is modelled as a rectangular domain with periodic boundary conditions at the inlet and outlet, and no-slip walls on the other two sides. The fluid is modelled as an ideal gas, and the flow is driven by a uniform and constant body force. The particles are Lagrangian, initially at rest, and are initialised in a cartesian pattern with 28 particles across the channel. The Reynolds number based on the theoretical steady state mean velocity is Re≈230. The density variation in the channel using these

Figure 2: Zero (a) and first order (b) reconstruction of an arbitrary variable Φ to the particle-particle interface

Figure 3: Instantaneous pressure distribution in shock tube with (a) first order upwind method and (b) higher order

upwind method

(a) (b)

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parameters is less than 1%, so the flow is effectively incompressible and the FVPM results can be compared with the analytic solution.

The explicit, compressible solver would require

a large number of timesteps to approach the steady state solution at the low Mach number encountered in this test case. For this reason, only transient results for Poiseuille flow are shown in Figure 4. The first order upwind FVPM exhibits excessive diffusion and the velocity profile across the channel is strongly smeared. The higher order version performs better, with the computed results closely matching the analytic solution.

4. Conclusions The objective of this work was to extend the

mesh-free FVPM to both a higher order of accuracy and to viscous flow. The higher order extension of the method was achieved using solution reconstruction methods. Linear solution reconstruction was implemented using a consistency-corrected SPH gradient approximation. The improved accuracy of the higher order upwind FVPM for inviscid flow was demonstrated by the enhanced shock-capturing ability of the method for a shock tube test case.

In addition, a viscous implementation of the FVPM was presented, and the results have been validated for Poiseuille flow. The results showed the diffusive nature of the first order upwind FVPM, but the higher order upwind FVPM showed good agreement with the analytic solution.

5. Acknowledgments This research is supported under the Embark

Initiative by IRCSET Research Grant RS/2005/95.

References HIETEL, D, STEINER, K., and STRUCKMEIER, J. (2000): A Finite Volume Method for Compressible Flows, Mathematical Models and Methods in Applied Science, 10: 1363-1382

KECK, R. (2005): A projection technique for incompressible flow in the meshless finite volume particle method, Advances in Computational Mathematics, 23: 143-169.

TELEAGA, D. (2005): A Finite Volume Particle Method for Conservation Laws, PhD Thesis, Universitat Kaiserslautern.

BARTH, T.J., and JESPERSEN, D.C. (1989): The Design and Application of Upwind Schemes on Unstructured Meshes, AIAA 27th Aerospace Sciences Meeting, Reno, NV, USA.

BONET, J., AND LOK, T.-S. L. (1999): Variational and momentum preservation aspects of Smooth Particle Hydrodynamics formulations, Computer Methods in applied mechanics and engineering, 180: 97-115.

Figure 4: Transient velocity profiles for Poiseuille flow test case. (a): first order FVPM, (b): higher order FVPM.

(a) (b)

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A new parallelized 3D SPH model: resolution of water entry problems and scalability study

Guillaume Oger 1, David Le Touzé 1, Bertrand Alessandrini 1 and Pierre Maruzewski 2

1 Ecole Centrale Nantes (ECN), Nantes, France 2 Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

Abstract SPH is a rather young method, which still needs various development concerning namely its interpolation procedures, the exact comprehension of the necessary conditions to ensure stability, or the imposition of free surface conditions as non-exhaustive examples. However, it is also necessary to develop a model as generic as possible, allowing to include easily any improvement. Such a model should be three-dimensional, thus able to answer most of the engineering topics. Due to strong CPU and memory requirements, this 3D SPH model is preferably to be parallelized, so as to allow computations on a cluster of processors. This paper deals with the introduction of such a parallelized scheme. The main principles adopted for communications between processors are presented, followed by an evaluation of the efficiency of the proposed model, on the test case of a solid sphere impacting the free surface. This study has been realized on the 8092 processor Blue Gene computer of EPFL.

1. Introduction By its ability to treat efficiently problems

involving more and more complicated physics, SPH represents a very interesting alternative method to “standard” mesh-based methods such as Finite Volume, Finite Difference or Finite Element Methods. Presently, most of the models present in the SPH related literature are two-dimensional, and thus do not really suffer from high computational cost difficulties. But

physics and engineering fields of research are indeed three-dimensional, which dramatically increases computational costs, and finally limits the use of the SPH scheme. A parallelized model integrating some very specific optimization tools is proposed here. The implemented model has been tested in terms of acceleration and efficiency of the calculation with regards to the number of processes used, with various total particle numbers. Very encouraging results have been found on the ECN Cray XD1 cluster using up to 32 processors. In particular, a mean efficiency of about 90% has been obtained. However, the efficiency of the parallelized model has to be validated at larger scale. This scalability study is realized on a water entry problem, namely on a test case involving a sphere impacting the free surface at high velocity.

Figure 1: Billard ball impacting the free surface at 4,8 m/s (free fall).

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Figure 2: Illustration of a near-equivalent SPH model simulation (4,8 m/s imposed velocity).

2. Core Implementation The core of the SPH scheme presented

hereafter is based on the use of exact Riemann solvers (Godunov type scheme), extracted from the works of Vila et al. Indeed, this method offers various advantages such as dispensing with the artificial viscosity, decreasing numerical dissipations and increasing stability in a very original way, inspired from Compressible Finite Difference and Finite Volume schemes developped from the sixties until now. Boundary conditions are imposed using ghost particles, which is the most accurate frontier treatment in our experience. On the other hand, this scheme integrates a variable smoothing length scheme, necessary to apply this model to various fields of application. Actually, this technique is particularly adapted to the test case of water entry, providing accuracy in the impact area (where a constant smoothing length is used). The discretization is then slowly relaxed from the limit of this zone up to the tank border.

3. Parallelization of the SPH solver The extension from a two-dimensional to a three-dimensional SPH model is pretty immediate. Nevertheless, for computational cost saving and memory requirement reasons, the practical implementation of such a three dimensional code implies its parallelization, in order to make possible the calculation on a cluster. This parallelization is achieved here using the standard MPI (Message Passing

Interface) for inter-process communications (used conjointly with FORTRAN 90). We chose the domain decomposition as parallelization strategy, this method consisting in splitting the whole fluid domain into sub-domains, each sub-domain corresponding to one dedicated processor. An example of such a procedure is illustrated in figure 3 for the case of a two-dimensional dam breaking, for which the calculation involves eight processors. In order to make this code as simple as possible, the creation of rectangular sub-domains of optimized shape is adopted. Interactions between these various sub-domains are then achieved using MPI, by systematic particle data communications.

Figure 3: Successive domain splitting for a calculation involving eight processors.

In this paper, for the sake of simplicity and clarity, this parallelization is discussed from a two-dimensional point of view, but its extension to the three-dimensional case is immediate and is implemented in our 3D model.

3.1. Parallelization description

First, note that each domain is determined at the beginning of the calculation , its sizes being adapted during the simulation so that each process always has approximately the same number of particles to treat, in order to make the

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global calculation as efficient as possible. Let us consider a given process of interest. This process owns a large number of particles for which it is able to compute the interactions (discrete convolutions) everywhere except near its limits. Indeed, its neighbor processes contain the missing particles, that we call now “foreign particles”. Its neighbor processes must therefore communicate to it the foreign particle data (position, velocity, pressure...) in order to allow the process of interest to complete its calculation, namely to account for all of the neighbors of its own particles. Thus, the process of interest has first to communicate to its neighbor processes the limits of the areas (dotted zones in figure 4) containing the foreign particles that interact with its own particles.

Figure 4: Process of interest with its neighbor processes.

These limits (dashed lines) are determined from the particles being the closest to the process of interest limits (see figure 5), and using the interaction radius R=2h.

Figure 5: Limits of the foreign particle areas. The MPI standard allows non-blocking communications. This particularity is of interest since it is possible to achieve simultaneously any operation (dependent or not on the parallelization itself) and the communications. The present scheme widely uses this advantage, which provides large optimization opportunities.

3.2. Scalability results

This parallelized model gave successful results in terms of efficiency for a limited number of processes (about 90% on 32 processes). The next step then consists in applying this model on larger process numbers (up to 256 processes). Such large process number tests are quite challenging, because of difficulties due to sub-domain size heterogeneities, larger foreign particle numbers (responsible for increased communication delays and memory requirements), and subtleties appearing from the variable-H scheme. These new scalability tests have been carried out on the 8092-processor Blue Gene computer of the EPFL for various particle and process numbers; acceleration and efficiency results are summarized in figures 6 and 7 respectively.

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Figure 6: Acceleration of the calculation.

Figure 7: Efficiency of the calculation. In these tests, the time reference value is based on 2 processes for 51,717 and 151,905 particles, on 8 processes for 1,672,179 particles, and 128 processes for 10,161,291 particles. As expected, for a given particle number, the efficiency decreases as the number of processes increases, due to the importance of communication delays compared to the effective calculation time, so that the use of a number of processes bigger than 16 is useless for 150,000 particles or less. For about 1,600,000, a correct efficiency is preserved with a maximum of 64 processes (about 75%). These results also reveal an efficency higher than 80% for 256 processes with 10,000,000 particles (this efficiency should be taken into account carefully because the time reference value chosen is for 128 processes).

4. Conclusions In this paper, a study based on a three-dimensional parallelized SPH model applied to free surface impact simulations has been presented. A special attention has been paid to the description of the parallelization itself, before a brief analysis of the first scalability results obtained on the 8092-processor Blue Gene computer of the EPFL through the use of a maximum of 256 processors up to now with 50,000 to 10,000,000 particles. These results are very encouraging, presenting interesting efficiency values. This work also reveals the need for improvements concerning namely the memory consumption, which remains quite subtle because of the variable-H scheme constraints. Further development are still needed, before to pursue these scalability tests, to finally reach possible calculations on 8092 processors. The resulting model should be a powerful tool based on SPH.

5. References VILA, J. P. (1999). On particle weighted methods and SPH. Mathematical Models and Methods in Applied Sciences, 9: 161-210. LAVERTY, S. M. (2003). Experimental hydrodynamics of spherical projectiles impacting on a free surface using high speed imaging techniques. Master in sciencein ocean engineering at the MIT.

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Symmetry assumptions in SPH

M. Omang1, S. Børve2, 3, S. O. Christensen1, J. Trulsen2

1 Norwegian Defence Estates Agency, momang@astro.uio.no, svein.christensen@forsvarsbygg.no 2 Institute of Theoretical Astrophysics, University of Oslo, jtrulsen@astro.uio.no

3Norwegian Defence Research Establishment, steinabo@astro.uio.no

Abstract In this paper the problem of symmetry assumptions in SPH is revisited. We present our new approach, which is based on basic SPH interpolation theory applied directly to a chosen symmetry; planar, spherical or cylindrical. A formulation is developed with a new kernel function and a modified set of SPH equations of motion, specific for each type of symmetry. In the present work we concentrate on the cylindrically symmetric description, but the method is similar for both planar and spherical symmetries. The assumption of axi-symmetry is quite useful for shock and shock reflection problems, and the capability of the formulation is demonstrated through the presentation of numerical and experimental results of a blast wave test. We find satisfactory agreement between the numerical and experimental results.

1. Introduction The SPH method was originally developed for a problem description in terms of Cartesian coordinates. Different approaches have been suggested as to how specific symmetry assumptions can be explicitly taken into account. These developments often met problems related to singularities at the symmetry axis. In this paper we discuss an alternative approach based on applying the proper symmetry assumption to the basic interpolation theory foundation in SPH. Previous work (Omang et al. 2005, 2006a, b, 2007) has

shown that the formulation is capable of handling the symmetry axis to a high degree of accuracy at a low numerical noise level. In the next section we give a closer description of our cylindrically symmetric SPH formulation. The formulation is implemented in Regularized Smoothed Particle Hydrodynamics (RSPH), an extension to SPH (Børve et al 2001, 2004, 2005), but the description should be applicable to any SPH method. We use RSPH to study a blast wave problem presented in section 3. Summary and discussions are given in section 4.

2. An axi-symmetric SPH description Our approach to developing kernel functions in cases where symmetry assumptions apply is based on fundamental SPH interpolation theory. Any field function f , can be expressed as an integral interpolant of itself for a suitable kernel function W ,

f = dr ' f (r' )W (r'−r,h)∫ . (1)

The generic kernel function, which is used in the current description, is chosen to give simple analytical solutions to the integral interpolant, and has the following form:

WA =1

Mδ hδ

(4 − q 2 )5 0 ≤ q ≤ 20 q > 2,

⎧ ⎨ ⎩

(2)

where⎭⎬⎫

⎩⎨⎧=

90098388608,

32048 ππ

δM for the

dimension δ = 2,3 .

If cylinder symmetry is assumed, the integral interpolant of Eq (1) is modified to

104

f (r,z) = dr' r' dz' f (r' ,z' ) dϕ∫∫ WA (q)

≡ dr' r' dz' f (r' ,z' )WA3C (r' ,z' ,r,z;h)∫ (3)

where q 2 = A + B sin(ϕ / 2) , B = 4ri rj / h 2andC = 4 − A . The resulting kernel function is of the form,

WA3C =4

M 3h 3 (C − B sin2 v)5 dv0

vmax

∫ . (4)

The indices A3C are used to indicate the chosen generic kernel, in this case, A , the spatial dimension, 3, and the symmetry assumption, C, for cylindrical. Integral limits are given such that vmax = π / 2 if B ≤ C and vmax = arcsin( C / B ) if B > C . The resulting kernel function is given by

WA3C =1

M 3h 3 w1 0 ≤ B ≤ Cw2 0 ≤ C < B

(5)

where

w1 =

πC 5

128ki

BC

⎛

⎝ ⎜

⎞

⎠ ⎟

i=0

5

∑i

(6)

and

w2 =B5

960arcsin( C

B ) pii=0

5

∑ CB

⎛

⎝ ⎜

⎞

⎠ ⎟

5−i

i

ii B

ClBC

BCB

−

=∑ ⎟

⎠

⎞⎜⎝

⎛−+

44

0

5

1960

. (7)

The parameters ki , pi , and li are given in Table 1. For B > 10C computational speed can be gained by approximating the kernel function w2 by a rapidly converging asymptotic series expansion as discussed in Omang et al 2005. On the basis of Lagrangian formalism, the following equations of motions are obtained;

dx i

dt= v i , (8)

ρ = m jj

∑ WA3C , (9)

dv i

dt= − m j (

Pi

ρ i2 +

Pj

ρ j2 + Πij )

j∑ ∇ iWA3C , (10)

dei

dt= m j (

Pi

ρ i2 +

12

Πij )j

∑

(vrj∂

∂rj+ vri

∂∂ri

+ vzij∂

∂zi)WA3C , (11)

and

Pi = (γ − 1)ei ρ i . (12)

Here mi , ρi , ei , x i = (ri ,zi ), and v i ≡ (vri ,vrx ) are the mass, density, internal energy, position and velocity for particle i , respectively. The summation is over all particles j within two smoothing lengths distance, and vzij = vzi − vzj . We use artificial viscosity Πij of the form given by Monaghan and Lattanzio (1985).

3. Blast wave simulations with RSPH

In this paper we present results from a set of experiments with the high explosive (HE) C-4. This is a plastic HE, which can easily be formed to any shape, with the implication that inhomogeneities in the charge density may occur. In the current test, a 2 litre spherical plastic container is filled with C-4 and mounted on a small wooden platform at different heights above ground. Pressure gauges on ground, positioned at increasing radial distances from the charge centre, allow direct comparison between numerical and experimental data. In the literature different approaches are given as to how high explosives can be represented for numerical purposes, ranging from solving chemical reaction equations to implementing empirical formulas. In the current work, we use a constant volume approach, which implies that we assume the detonation front to propagate through the HE rapidly enough for the volume to be left unaltered and without energy loss. The initial HE data used in these simulations are given in Table 2. Atmospheric pressure for a height above sea level of approximately 500 m was used in these simulations. In the initial phase of the detonation, we assume spherical symmetry, and use a 1D spherically symmetric RSPH description to allow high resolution at a relatively low computational

105

cost. The results at t=0.28 ms are saved, since the position of the shock is now approaching the ground, and the result is used as input for the axi-symmetric code. We look at two different tests for which the charges are positioned at 1 and 2 meters above ground, respectively. Pressure sensors are positioned at 4, 5, 6 and 10 meters radial distances from the charge ground zero. Standard SPH summations formulas of the form Eq (9) are used to compute field variables for these positions. In Figure 2 we have plotted the pressure for t=3.5 ms after detonation. At this time the spherical shock has been reflected from the ground and a complex shock reflection structure has formed, a Mach reflection, consisting of the incident shock, a reflected shock and a triple point. The highest pressure levels are observed in the area just behind the Mach reflection. The pressure sensors allow for continuous measurements of the pressure at specific positions on the ground. In Figure 3 we have plotted the results for increasing radial distances. The empirical results are plotted in grey, whereas the numerical results are plotted in black. The Figure illustrates the satisfactory agreement between the results. Both the time of arrival and the pressure peak levels are well resolved. In the experimental data we observe secondary peaks, which are not as well represented by the numerical results.

Figure 2 Pressure plot for t=3.50 ms

Figure 3 Pressure time history at r = 4, 5, 6 and 10 meters distance.

4. Summary and Conclusions In this paper we have revisited our new SPH formulation for symmetry assumptions in SPH. We use the formulation to study blast waves generated from high explosives. The results are compared to experimental results from pressure gauges. The numerical results manage to reproduce the experimental results to a high degree of accuracy. Secondary pressure peaks, however, are less accurately reproduced. The discrepancy may be due to the simplified explosive model used.

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Table 1: Table of constant for the kernel function WA3C

Table 2: HE charge data

E (MJ/kg)

Q (kg)

ρ (kg/m3)

R0

(mm) 5.62 2.95 1475 78

5. References

BØRVE, S. and OMANG, M. and TRULSEN, J. (2001), Regularized Smoothed Particle Hydrodynamics: A new approach to simulating Magnetohydrodynamic shocks. Astrophys. J., 561:82-93

BØRVE, S. and OMANG, M. and TRULSEN, J. (2004), Two-dimensional MHD Smoothed Particle Hydrodynamics Stability Analysis, J. Astrophys. Suppl. Ser. 153:447-462

BØRVE, S. and OMANG, M. and TRULSEN, J. (2005), Regularized Smoothed Particle Hydrodynamics with improved multi-resolution handlingJ. Comput. Phys 208,1:345-367 CHRISTENSEN, S. O. (2006) Free field pressures from thermobaric charges, trials with large magnesium particles. Technical report FoU 31/2006, Norwegian Defence Estates Agency. MONAGHAN J. J., and LATTANZIO, J. C. (1985). A refined particle method for astrophysical problems, Astron. Astrophys 149,:135-143

OMANG, M. and BØRVE, S. and TRULSEN, J. (2005), Alternative kernel functions for Smoothed Particle Hydrodynamics in cylindrical

symmetry. Shock Waves 14, 4: 293-298. OMANG, M. and BØRVE, S. and TRULSEN, J. (2006a), SPH in spherical and cylindrical coordinates. J. Comput. Phys 213, 1: 391-412,

OMANG, M. and BØRVE, S. and TRULSEN, J. (2006b), Numerical simulations of shock wave reflection phenomena in non-stationary flows using regularized smoothed particle hydrodynamics (RSPH) Shock Waves Journal 16, 2:167-177 OMANG, M. and BØRVE, S. and TRULSEN, J. (2007), Shock collision in 3D using an axi-symmetric Regularized Smoothed Particle Hydrodynamics (RSPH) Shock Waves Journal in press.

Index 0 1 2 3 4 5

k 256 -640 960 -800 350 -63

p 3840 -9600 14400 -12000 5250 -945

l 4384 -8768 9004 -4620 945

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DNS SPH simulation of 2D wall-bounded turbulence

Martin ROBINSON 1&2 , Joseph MONAGHAN2 and John MANSOUR 1&2

1 PhD Student 2 School of Mathematical Sciences, Monash University, Melbourne, Australia

firstname.lastname@sci.monash.edu.au

Abstract Ensemble statistics from SPH simulations of decaying 2D wall-bounded turbulence have been compared with published results from a pseudo-spectral code. While the results are qualitatively similar, the dissipation of kinetic energy is significantly higher for the SPH results and the velocity is noisier at the kernel length scale. Results using higher resolutions and with forced 2D turbulence are currently being generated.

1. Introduction The combination of SPH and turbulence is

poorly understood. There have been a number of SPH turbulence models proposed, based on versions of existing turbulence models (e.g. Monaghan 2002, Violeau 2002). While these turbulence models show promising results, there is little knowledge of how successful SPH is at modelling the full range of turbulent scales via a Direct Numerical Simulation (DNS). We have implemented 2D wall-bounded decaying turbulence simulations using the numerical pseudo-spectral simulations by Clercx (1999 & 2005) as a benchmark. Two-dimensional turbulence was chosen due to the relative ease of computation and the availability of experimental and simulation data for this domain. Bounded turbulence was chosen over the classical periodic box due to the additional computational effort required to make the turbulence truly isotropic, since correlations between points near the periodic boundary

cause the turbulence to become anisotropic at the larger length scales.

2. SPH Method This work uses the standard SPH method

outlined by Monaghan (2005). The cubic spline is used for the kernel. The viscosity term is calculated using the Morris (1997) form and the continuity equation is integrated to find the density for each particle.

The particle positions and velocity are integrated using a Verlet second order integrator. In order to preserve the reversibility of the simulation (in the absence of viscosity), dρ/dt is calculated using the particle positions and velocity at the end, rather than the middle, of the timestep.

The no-slip walls are modeled using three layers of immovable SPH particles. These boundary particles are identical to the other fluid particles in every way except for the fact that their positions are constant.

3. Initial Conditions The initial conditions are chosen to match

those by Clercx et al. (1999). The SPH particles were initialized on a grid, with the velocity field calculated from a 2D 65x65 Chebyshev series, with the coefficients randomly generated from a zero-mean Gaussian pdf with variance σnm, given by

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

=44

2

811

811 m

m

n

nnmσ

64,0 ≤≤ mn (1)

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In order to provide a quiet start at the no-

slip walls, the velocity field is reduced down to zero at the boundaries using the smoothing function f(x)f(y), where

[ ]))1(100exp(1)( 22xxf −−−= (2)

The resultant velocity field v is made

divergence-free by taking advantage of the fact that any vector field can be separated into its divergence-free component vd and the gradient of a scalar φ.

φ∇+= dvv (3)

Taking the divergence of both sides gives

φ2∇=⋅∇ v (4)

Since the initial particles are on a grid, Equation (4) is solved using a second order finite difference relaxation technique.

The divergence-free velocity field is then normalized so that the total kinetic energy KE(0)=1 per unit mass per unit area.

Twelve randomly initialized simulations with Re=1500 were calculated in order to provide ensemble statistics. The Reynolds Number is defined as Re=UL/ν, where U is the rms velocity of the initial flow field, L is the half-width of the box, and ν is the kinematic viscosity. Each simulation uses 250x250 particles, which is comparable to the 216x216 Chebyshev modes used by Clercx.

4. Results and discussion Figure 1 shows a time sequence from one

of the ensemble simulations. The SPH particles are coloured by vorticity, which is calculated

Figure 1. SPH particles colored by vorticity for one of the ensemble simulations. Starting at top left: t = 1, 2, 4, 16, 32.

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from a least-squares fit of a linear velocity field around each particle. As opposed to 3D turbulence, where energy flows down to smaller length scales, energy flows up to larger length scales as the vortices spread and interact with each other to form progressively larger vortices. These images, in particular the approximate size of the vortices, are qualitatively similar to Clercx’s example time series, with the most significant difference being the additional noise at the SPH kernel length scale.

Figure 2 shows a plot of normalised total kinetic energy (normalised by the initial kinetic energy, which is identical for each simulation) versus time for each simulation. From these plots, it can be seen that the SPH simulations are more dissipative than the results by Clercx. While it takes about 70 time units for Clercx’s kinetic energy to drop to 0.01 of its initial value, our simulations drop by this amount in just 30 time units (the time is made non-dimensional using L/U). This might be caused by these simulations being under-resolved, with the SPH kernel smoothing out too much of the small-scale motion of the particles. Additional simulations at higher resolutions would need to be calculated.

Figure 3 shows a plot of the total enstrophy Ω(t) divided by the total kinetic energy KE(t), averaged over the ensemble simulations (i.e. <KE(t)/Ω(t)>). This can be interpreted as the

mean squared wave number <k2> of the flow (Clercx 1999). Clercx found that between t=1 and t=10, the decay rate of <KE(t)/Ω(t)> is very close to t-0.63 (dashed reference line). The SPH simulation gives a similar but slightly lower decay rate of t-0.5 (full reference line).

Figure 4 shows plots of normalised total angular momentum (normalised by the angular momentum of an identical fluid-filled box with kinetic energy KE(t), in solid body rotation) versus time for each simulation. Ten out of the twelve simulations experienced a spontaneous spin-up of the fluid in the box (the fixed boundary particles provide the additional angular momentum). This is the exact ratio reported by Clercx.

5. Conclusions and Future Work The two most significant areas of error for

the SPH simulations are the higher rate of kinetic energy decay and the noisy motion of the particles at the kernel length scale. These effects are probably related, due to the kernel dissipating the unphysical build-up of energy at the kernel length scale, and might be solved using either XSPH or higher resolutions.

The results obtained so far are of decaying turbulence, but it would be informative to run SPH simulations of forced turbulence as well. Clercx (2005) has performed experiments and numerical simulations of forced turbulence using the same 2D box, where energy is

Figure 3. <Ω(t)/KE(t)> versus time Figure 2. Normalised kinetic energy versus time

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injected at a certain length scale through sinusoidal perturbations of the box’s angular velocity. We have set up a similar SPH simulation and are currently generating results.

Figure 4. Total normalised angular momentum versus time. The highlighted plots are those runs that experienced little or no spin-up

One of the aims of these simulations is to provide fully resolved SPH simulations that can act as a benchmark for SPH simulations incorporating a turbulence model. Specifically, we plan to compare these results to those obtained using the SPH-α model, developed by Monaghan (2002).

Finally, the Lagrangian nature of SPH makes it naturally suited to simulating and studying the Lagrangian properties of turbulence, such as mixing. Richardson's (1926) theory of turbulent particle pair dispersion will be tested using the forced turbulence results, and Finite-Time Lyapunov Exponents will be used to identify the Lagrangian Coherent Structures (Haller, 2000) that are responsible for the mixing processes.

6. References CLERCX, H.J.H., MAASSEN, S.R. and VAN HEIJST, G.J.F. (1999) Decaying two-dimensional turbulence in square containers

with no-slip or stress-free boundaries. Physics of Fluids, 11(3):611-626 CLERCX, H.J.H., VAN HEIJST, G.J.F., MOLENAAR, D. and WELLS, M.G. (2005) No-slip walls as vorticity sources in two-dimensional bounded turbulence. Dynamics of Atmospheres and Oceans, 40(1-2):3–21

HALLER, G. and YUAN, G. (2000). Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D. 147, 352-370 MONAGHAN, J.J. (2002). SPH compressible turbulence. Mon. Not. R. Astr. Soc., 335:843–852 MONAGHAN, J.J. (2005). Smoothed particle hydrodynamics. Reports of Progress in Physics, 68:1703–1759 MORRIS, J.P., FOX, P.J. and ZHU, Y. (1997). Modeling low Reynolds Number incompressible flows using SPH. Journal of Computational Physics, 136, 214-226 RICHARDSON, L. (1926). Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. A., 110, 709-737 VIOLEAU, D., PICCON, S. and CHABARD, J.P. (2002) Two attempts of turbulence modeling in smoothed particle hydrodynamics. In Proc. 8th Symp. on Flow Modeling and Turbulence Measurements, Advances in Fluid Modeling and Turbulence Measurements, 339–346

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Development of a Parallel SPH code for free-surface wave hydrodynamics

Benedict D. ROGERS 1 , Robert A. DALRYMPLE2, Peter K. STANSBY1 and Dominique L.

LAURENCE1 2

1 School of Mech. Aero. & Civl Engng, University of Manchester, U.K. 2 Dept of Civil Engng, The Johns Hopkins University, Baltimore, U.S.A.

1. Introduction Smoothed Particle Hydrodynamics (SPH) is a very attractive numerical technique for the simulation of free-surface flow phenomena, such as occurs in shallow water bodies or the violent motions present in many coastal engineering problems. Due to its meshless formulation, many problems such as wave breaking that were very difficult, become tractable and possible. However, a closer examination of even some of the simplest problems soon reveals that three-dimensional simulations are essential and mandatory in order to incorporate and capture important three-dimensional effects, for example vortex stretching in turbulent flows. As soon as a three-dimensional approach is required the problem becomes much more expensive computationally. SPH, when simulating incompressibility water flows, uses very small time steps. Therefore, in order to capture a particular physical phenomenon that may not occur for several wave periods necessitates both a huge number of particles, O(106) and greater, in addition to a very large number of timesteps. This naturally leads to excessively long runtimes when run on simple desktop computers, e.g. 2months to simulate only 10 waves in a 3-D basin with only 250,000 particles. Thus, when using SPH to simulate

water wave mechanics at a beach (e.g. as a wave breaks), the use of a parallel framework is essential to both speeding up the computation time, and to facilitate the simulation of problems with massive particle numbers. In this work we will present the initial results for converting the Johns Hopkins University (JHU) 2-D and 3-D SPH codes for use with the Message Passing Interface (MPI). The serial JHU-SPH code has already been demonstrated to simulate breaking waves on both 2-D and 3-D simple planar beaches (Dalrymple and Rogers 2006). However, the size and resolution was limited by computational resources so that the total number of water particles over the water depth in the surf zone was often on the order of only 10 particles, which given the extended influence of solid wall boundary conditions, is nowhere near sufficient to undertake convergence tests or to capture reliably the coherent turbulent structures that can be found in the surf and breaking zones.

2. Parallelizing the JHU-SPH Code

In comparison to mesh-based

numerical codes, parallelizing a meshfree particle code presents some unique challenges. Most notably, the number of particles held on each

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processor will not remain the same, and one individual particle may exist on many processors during its lifetime. Furthermore, the simulation of free-surface hydrodynamics includes the possibility of moving bodies (both with and without feedback). These bodies must of course be allowed to move seamlessly across the interface between processes.

To overcome these issues, the JHU-SPH code uses some features of linked lists and pointers to accommodate the varying number of particles on each processor. The

connectivity of the particles is achieved by splitting the computational domain up into boxes of dimension 2h, where h is the smoothing length. Since the kernel has compact support, only those particles in the neighbouring boxes need to interact with each other. This is essentially the Particle-in-Cell (PIC) technique.

To parallelise the code, the workload needs to be distributed amongst the available processors. Figure 1 displays a typical situation where the domain has been split amongst three different processors.

processor 1 processor 2 processor 0 Figure 1 Subdivision of the domain across processors

_________________________________________________________________ However, when the domain is split

up amongst several processors and box ii is located on the boundary of the processor, the code needs to know the contents of neighbouring box-position

ii+1, i.e. East which lies on a different processor. The easiest method to accomplish the necessary transfer of information is to use a column of ghost cells as shown in Figure 2.

processor 1

processor 2 processor 0

Ghost cells Ghost cells

Transfer of particle data

Transfer of particle data

Figure 2 Transfer of particle data between neighbouring processes

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The particle information in these ghost cells must therefore be imported from the neighbouring processor every time step.

3. Initial Results

Initially, the code was tested using a simple problem which requires no load balancing compared to the flows at beaches. Figure 3 shows 2-D recirculating flow past a square cylinder for Re = 100. It can be seen that the flow behind the cylinder is now exhibiting classical vortex shedding behaviour. This simulation has only

16000 particles. The straight vertical lines indicate the location of the partitions between each processor, in this case 4 processors. Cleary this geometry is very simple and enclosed, allowing us to see how the parallel code performs on a problem that has no need for load-balancing. Figure 4 displays the speed-up for this problem for a small number of processors. It can be seen that the as the number of processors increases the potential speed up decreases (8 processors gives a speed-up of 6) demonstrating that the communication between each process is becoming more influential.

Figure 3 Initial validation case of flow past a square cylinder

SPEED-UP

0123456789

0 2 4 6 8 10processors

S

S_ideal16000 Particles

Figure 4 Speed-up for 16000 particles

SPEED-UP

02468

1012141618

0 5 10 15 20processors

S

S_ideal250,000 Particles

Figure 5 Speed-up for 250,000 particles

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This serves to illustrate an important feature of parallelization of a code which is particularly evident when using the PIC structure. Not only must communication between processors be minimised, but the proportional area of each processor that is involved in communication must also be minimised. Thus only exchanging information between adjacent 2h boxes across processor interfaces serves to our advantage for very large problems where there is a limited number of processors available.

The same test case was run with

250,000 particles. In Figure 5, we can see the speed-up, where it is clear that with far more particles being held on each processor, the speed-up is much more efficient up to 16 processors which was the limit of access for the authors. This immediately confirms that our approach to parallelizing and opening up the possibility of attempting massive problems is valid, and that a more sophisticated division of the domain is not necessary.

4. Problem with Beach Figure 6 shows a far more typical

problem which involves a wedge sliding a planar beach. It is very clear that if the domain is split up into equal geometric parts that the one processor will be full utilized leaving the others hanging while they wait which is clearly a waste of computational resources. Using the case of using 4 processors for a small simulation of approximately 120,000 particles, with no load balancing, we achieve a speed up of only 1.5 (40% efficiency). However, when we apply a very primitive form of static load balancing so that there are approximately the same numbers of particles on each processor, we increase the efficiency by over 30% to

obtain a speed-up of 2.7 (67% efficiency). Clearly, these initial results indicate that there are benefits to be gained by balancing the load across the processors. This is for a small 2-D case so the benefits are expected to be better for larger 3-D cases to be demonstrated later. Improving the efficiency for this problem towards unity is still under investigation.

Figure 6 2-D Sliding Wedge down a planar beach: no load balancing

5. Future Work The same techniques have yet to

applied to the 3-D parallel code and formulated into an algorithm that adapts automatically to the simulation. This will be the topic of the final presentation which will also show other potential test cases.

6. Acknowledgments The authors wish to acknowledge

the support of EPSRC (Grant GR/S28310) for funding this work. The lead author is also very grateful for support of Manchester Computing in particular the help and advice from Dr Jon Gibson and Dr Kevin Roy.

7. References DALRYMPLE, R.A. and ROGERS, B.D. (2006) Numerical Modeling of Water Waves with the SPH Method, Coastal Engng, Conference, 53(2-3), 141-147.

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SPH analysis of a planing surface

Luca Savio, Stefano Brizzolara, Michele Viviani

Department of Marine Technology (DINAV), University of Genoa, Via Montallegro 1, 16145 Genova, Italy,

Phone: +39 010 3532438; fax: +39 010 3532127; e-mail: savio@dinav.unige.it Phone: +39 010 3532386; fax: +39 010 3532127; e-mail: brizzolara@dinav.unige.it

Phone: +39 010 3532547; fax: +39 010 3532127; e-mail: viviani@dinav.unige.it

Abstract In the present study SPH method was

applied in order to study the peculiar flow around a 2D flat planing surface. Results were compared with analytical solutions and RANSE calculations. For the application of the method to this particular case an SPH code with an inlet and an outlet of particles was developed.

Introduction The CFD modeling of the flow of water

around a planing hull is still an open topic, also nowadays, because of its highly non-linear behavior. In this paper, the incoming flow, with a constant velocity V0, is divided in two region by a stagnation point; the first region, named spray, moves upward before falling down, the second one moves downward following the surface and forming the wake field.

This paper presents a first study on the validation of a SPH method in the simplified 2D problem of a flat plate (figure 1), with authors final aim to generalize the method to the real 3D planing hull case.

Many solutions using conformal mapping can be found in literature (Sedov 1937 and Pierson and Leshnover 1948), but they generally assume that the pressure distribution is imposed and that the spray extends to the infinity. On the other hand, calculations with SPH are rather straight forward, once a description

of the inlet and outlet of particles is achieved.

Figure 1 Flat plate problem

Calculations setup

Calculations were carried out for a flat 2D surface, with an angle of attack of 16°, with an initial draught T0 equal to 3 cm (figure 2) and an incoming uniform flow with two speeds of 5.65 m/s and 2.0 m/s. The first condition results in a complete and stable planing regime, while the second one never reaches a steady planing condition.

Figure 2 Generic calculation setup

SPH in this paper The SPH method implemented at

DINAV for studying problem under exam is a standard SPH method, modified to let particles entering and leaving. In this code, the so-called VSLGP (Variable smoothing

116

length ghost particles) boundary treatment approach is used (Viviani et al. 2007).

In order to save computational time and reduce number of particles involved in calculation without loosing spatial resolution, a coupled SPH–Eulerian solver was developed, which simulates inlet and outlet. Fluid domain is divided in three regions by the code. The first region (inlet), shows the subsequent features: • It is a control area, so if empty a new

vertical row of particles is created. • Particles velocity inside this area is

imposed, while their density varies. • When a new row of particles is created

all its variables are imposed. • Region span is two smoothing lengths.

The second region, the central part, is where the SPH method is fully applied.

The last region (span = 4h) is the outlet region where particles are moved according to an Eulerian solver for N-S inviscid equations, before being destroyed. The pressure gradient is imposed, while the diffusive term is calculated by using the mean value of velocity of the connected cells.

Particles were initially placed on an hexagonal grid, with smoothing length set to be 6 mm (eight times smaller than spray theoretical thickness), resulting in a about 55000 particles (increasing to about 66000 during calculations). Sound speed adopted is 130 m/s and 70 m/s for inlet speeds of 5.65 m/s and 2 m/s respectively. Artificial viscosity was employed in order to ensure numerical stability.

RANSE Computations A 2D CFD model was created using a

CFD solver based on the VOF method of Hirt and Nichols (1981) which uses a fixed Eulerian rectangular grid and a particular technique to represent solid moving obstacles called the Fractional-Area-Volume-Obstacle-Representation (Hirt and Sicilian, 1985), which effectively is able to overcome the solid boundary

representation problems typical of Cartesian grid based VOF methods.

The numerical model has a bottom at -2.2m under the TE of the plate; the inlet condition is given at 2.0 in front of the TE end the outlet condition (zero gradient condition) is set at about 1 m aft of TE. The total number of cell used is 88000, with local refinements (4mm).

Calculations results For what regards 5.65 m/s flow speed,

calculations showed good agreement between SPH, RANSE solution and conformal mapping. SPH seems to better capture the particular flow of particles that, after being in spray, fall down and join the incoming flow, while RANSE produce a more stable vertical force.

The kinematics field is well described by SPH, as visible from figure 3, where a plot of the particles distribution is presented. Maximum pressure point and spray thickness at time t= 0.84 s are indicated. For comparison, flow pattern simulated at same time step by RANSE solver is reported in figure 4.

Figure 3 SPH solution for t=0.84 s

Figure 4 RANSE solution for t = 0.84 s

A good qualitative agreement between two methods was achieved. In particular stagnation point position (0.88 m from

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trailing edge in RANSE, 0.94 m in SPH) is very much in agreement.

Another interesting comparison can be achieved by considering the pressure distribution over the flat surface at the same time step. The selected time step at t=0.32s is an intermediate one, half way of the transient flow evolution.

In figure 5, pressure distribution obtained through SPH is compared with the one predicted by RANSE; shapes are similar and mean values comparable, while the SPH simulation shows distinct high frequency pressure oscillation around the average value; in figure 6, ideal conformal mapping solution is reported for comparison.

Figure 5 SPH and RANSE pressure

distributions

Figure 6 – Ideal Pressure distributions

In SPH pressure on boundary particles

is measured as the mean value of real particles pressure inside a control box (height and span equal to 8 and 4 smoothing lengths) (Vivani et al., 2006).

It is worth underling that no data smoothing was applied to pressure distribution predicted by the SPH method.

The resulting vertical force suffers of some form of oscillation in both methods, as figure 7 shows. RANSE method seems to find the steady solution at a later time step. (influence of different initial set up). Asymptotic value, though, is comparable in terms of mean values, measuring 6700 N/m for RANSE and 7500 N/m for SPH.

Figure 7 SPH (left) and RANSE (right)

vertical force

The 2 m/s case is peculiar, because the steady planing state is never reached. At this low speed a jet is formed during the transient, but it is not stable and after some time the jet totally collapses and joins the incoming flow. As it can be seen in figure 8, spray region extension is limited when compared with the 5.65 m/s case.

Figure 8 Final flow pattern–SPH–V0 2 m/s

This condition is difficult to simulate

numerically due to its intrinsic highly non-linear behavior.

Solution with RANSE showed strong instabilities, which prevented to reach a valid solution; collapsing jet falling down on the free surface creates a very unstable inflow.

In figure 9 a plot of the vertical force time history is reported. The oscillations are present only in the first part of the simulation, while they are reduced when

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-50

0

50

100

150

200

250

300

350

time [s]

Force [N/m]

Vertical Force 2 m/s

the jet collapses and flow presents only a small re-circulation region.

Figure 9: Vertical force– V0 = 2 m/s (SPH)

Conclusions and future work

The study showed that SPH is able to

fully represent the peculiar flow inherent to planing surfaces. Moreover, SPH has shown a higher flexibility than RANSE, being able to carry out successfully considerably different cases, while RANSE resulted in unstable solutions.

The technique adopted for inlet and outlet of particles is simple and could be further developed in order to have more complex inlet flow conditions.

Pressures and vertical forces oscillations are still to be investigated in a higher detail. In SPH, they seem to be related to pressure waves that are generated close to stagnation point and subsequently are propagated inside fluid domain. One solution to this problem could be to let planing surface free to move vertically. A similar problem is being studied at present for what regards slamming simulation, using the above-mentioned technique, with promising results; it has been noted that letting body free to move produces an intrinsic smoothing, thus strongly dampening oscillations.

This preliminary study will be further developed with comparisons with experiments that will be carried out at

DINAV on flat planing surfaces at various angles of attacks.

References

HIRT, C. W. and NICHOLS, B. D. (1981), “Volume of Fluid (VOF) method for the dynamics of free boundaries”, J. Comp. Physics, 39, 201–225. HIRT, C. W. and SICILIAN, J. M.: 1985, ‘A Porosity Technique for the Definition of Obstacles in Rectangular Cell Meshes’, Proc. Fourth Int. Conf. Ship Hydrodyn., National Academy of Science, Washington, DC. LIU, G.R., LIU M.B., 2003, “Smoothed Particle Hydrodynamics – A Meshfree Particle Method”, World Scientific Publishing Co. Pte. Ltd. MONAGHAN, J.J. , 1994, “Simulating free surface flows with SPH, Journal of computational Physics, 110, 399-406 PIERSON, J. D., LESHNOVER, S. (1948), “An Analysis of the Fluid Flow in the Spray Root and Wake Regions of Flat Planing Surfaces”, Stevens Institute of Technology, Report SIT-DL-48-335. SEDOV, L.I. (1937), “Two Dimensional Problem of Planing on the Surface of a Heavy Fluid” Trudy Conferentsii Po Volnovomu Soprotiveleniyu [Works of the Conference on Wave Drag], Izd. Tsagi, 1937. VIVIANI M., SAVIO L., BRIZZOLARA S. 2007, Evaluation of slamming loads on ship bow section adopting SPH and RANSE method’, to be presented at IAHR 07 VIVIANI, M., SAVIO, L., BRIZZOLARA, S., 2006 “Evaluation of slamming loads on V-shape ship sections with different numerical methods”, Proc. Numerical Towing Tank Symposium.

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Water wave propagation using SPH models.

P.-M. Guilcher1, G. Ducrozet1, B. Alessandrini1 and P. Ferrant1.

1 Laboratoire de Mécanique des Fluides, Ecole Centrale Nantes, Nantes (France)

Abstract The propagation of water waves is characterized by a slow dynamics, with few energy dissipation. This phenomenon thus largely differs from the classical application of free-surface SPH to fluid-structure impact flows or violent flows involving breaking waves and interface reconnections, to which the SPH method is well adapted. The present work analyzes the capability of the SPH method to deal with propagation of such wave trains by comparing various SPH formulations. Results on damping and phase shifting show that the use of Riemann solvers and renormalization techniques brings significant improvements to the standard SPH scheme.

1. Introduction The SPH method is a rather recent numerical method and as a matter of fact, lots of formulations can be found in the literature, pointing out the different ways adopted to improve the robust initial scheme proposed by Monaghan in [1]. The difficulty is then to select among them a scheme fulfilling the properties of consistency, convergence and stability. This is particularly crucial to simulate by SPH wave propagation over a long duration. This problem is very challenging for any numerical method, since almost no dissipation occurs and the wave train must be correctly advected for a long time. The wave propagation problem is therefore suitable to determine the schemes verifying the three properties mentioned above. We will present here the main features of four different numerical schemes based on the SPH

method. More details of the standard SPH method can be found in [1] for example. In the SPH method, the problem is solved at a set P of points. Each of these “particles” i of position ix and elementary volume of fluid iw carries its own characteristics, such as its velocity iv , density iρ or pressure ip . The particle distribution is then updated in time with

( )

ii

ii j j i ij

j P

dx vdtdw w w v v Adt ∈

⎧ =⎪⎪⎨⎪ = −⎪⎩

∑

1.1. Lax-Wendroff scheme

The Euler equations for mass and momentum conservation are then discretized at the particle locations to obtain the standard SPH scheme (written in a slightly different manner)

0

( )

i

i ii i j i j i j ij ij

j P

dmdt

dm v m g w w p p Adt

ρ ρ π∈

⎧ =⎪⎪⎨⎪ = − + +⎪⎩

∑

where i i im wρ= represents the mass associated to the point i. Pressure is related to density through the Tait's equation of state

20 0

0

1 , 7apγ

ρ ρ γγ ρ

⎛ ⎞⎛ ⎞⎜ ⎟= − =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

where 0ρ is the density at the free surface and

0a is a speed of sound chosen to ensure low variations of the density. An artificial viscosity term ijπ is required to stabilize the scheme

(with α=0,03 here).

·

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1.2. Godunov scheme

Alternatively, stability can be achieved by introducing a Riemann solver. We refer the reader to [2] for more explanations on the use of Riemann solvers in SPH. Thus the scheme reads

( )

0

0

2 ( ( ))

2 ( ( )) 1

ii j E E ij ij

j P

i ii

i j E E E ij E ijj P

dm w w v v x Adt

dm v m gdt

w w v v v x p A

ρ

ρ

∈

∈

⎧ = − −⎪⎪⎪

=⎨⎪⎪− ⊗ − +⎪⎩

∑

∑

where 0v is a relative velocity, equal to the fluid velocity v in a Lagrangian framework. The terms Eρ and Ev are the solution of the exact Lagrangian Riemann problem solved at the middle of each couple of particles i and j (see [3] for the Riemann problem adapted to the Tait's equation of state).

1.3. Renormalization

The evaluation of gradients for the standard scheme is given by

ij ijA W= ∇

where W is the kernel function, the cubic spline kernel from Monaghan [1] is used here. With this assumption, two conditions [2] are required to achieve convergence of the numerical results

0xhx

∆ →⎧⎪⎨

→ ∞⎪∆⎩

To relax the second condition and obtain convergence with a finite number of neighbouring particles, the standard evaluation of gradients can be replaced by:

( )12ij i j ijA B B W= + ∇

where iB is the renormalization matrix at point i (see [2]).

1.4. Boundary conditions

Wall conditions are enforced using the technique of ghost particle, which mirrors the fluid particle and its characteristics.

2. Results and discussion Four numerical schemes have been tested in the case of the generation and propagation of small-amplitude waves. A regular wave train has been generated by a piston-type wavemaker whose motion is prescribed by a simple law (see Figure 1). Figures 2 and 3 show the free surface profile at final time t=20s of the simulation for the SPH method and the reference solution (a spectral potential model with fully nonlinear free surface conditions). On one hand a phase shifting is present for both standard Lax-Wendroff (Figure 2-top) and standard Godunov (Figure 3-top) schemes. This phase shifting disappears with the renormalized estimation of gradients (Figure 2-bottom and Figure 3-bottom). On the other hand, the use of a Riemann solver decreases dramatically the damping of the free surface. The Godunov scheme compares much better to the spectral reference solution than the standard scheme with artificial viscosity (α=0,03) for the amplitude of the wave train. Indeed, the upwinding provided by the Riemann solver is self-adaptive and succeeds in stabilizing the scheme without causing too much numerical dissipation, contrary to the use of an artificial viscosity. Various calculations have been performed by varying the artificial viscosity coefficient α but lower values of α have led to instabilities near the wavemaker and higher values to an increased damping. So, the renormalized Godunov SPH scheme gives the best results both in terms of phase shifting and damping.

·

·

·

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In order to study the convergence of the latter scheme, the fundamental mode of the Fourier decomposition obtained with a time moving window of broadness equal to the wavemaker period has been calculated for a probe located at the middle of the tank (see Figure 4). One can observe that this SPH solution actually converges towards the reference potential solution, when refining only the particle distribution (the numbers of neighbors is kept constant). The relative error for the fundamental mode is plotted in Figure 5 as a function of the number of particles, showing the convergence of the scheme, of order ~1,5. The damping of the fundamental mode in the wave tank is also plotted (Figure 6), showing a linear evolution of the damping along the wave tank. As a conclusion, the standard SPH scheme is not able to model wave propagation. Some improvements are required, namely a Riemann formulation to increase the stability properties, and the renormalization to obtain the convergence of the scheme.

3. Figures

Figure 4: Left: sketch of the wave tank with the piston-type wavemaker. Right: wavemaker law of motion with

L=10m, H=0,135m, A=0.0035368m, τ=0,25s and ω=6,5222 rad/s.

Figure 5: SPH and spectral free surface profiles at t=20s. Top: standard Lax-Wendroff scheme (standard

SPH scheme). Bottom: renormalized Lax-Wendroff SPH scheme.

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Figure 6: SPH and spectral free surface profiles at t=20s. Top: standard Godunov SPH scheme. Bottom: renormalized Godunov SPH scheme.

Figure 7: Fundamental mode of the Fourier

decomposition (time moving window of broadness the wave period). Comparison of the SPH results

(renormalized Godunov) to the spectral reference.

Figure 8: Convergence of the fundamental mode of

the Fourier decomposition as a function of the number of particles per wavelength.

Figure 9: Damping of the fundamental mode of the

Fourier decomposition as a function of the probe location.

4. References MONAGHAN J.J. (1992), Smoothed Particle Hydrodynamics, Annu. Rev. Astron. Astrophys., 30: 543-574. VILA J.P. (1999), On Particle Weighted Methods and Smooth Particle Hydrodynamics, Mathematical Models and Methods in Applied Sciences, 9(2): 161-209. IVINGS M.J., CAUSON D.M. and TORO E.F. (1998), On Riemann Solvers for Compressible Liquids, International Journal of Numerical Methods in Fluids, 28(3): 395-418.

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Exact Kernel Integration in SPH

Mihai BASA 1, Nathan J. QUINLAN 2, Martin LASTIWKA 3

Department of Mechanical and Biomedical Engineering, National University of Ireland, Galway. 1 mihai.basa@nuigalway.ie

2 nathan.quinlan@nuigalway.ie 3 martin.lastiwka@nuigalway.ie

Abstract

We introduce a simple and computationally efficient method to improve the accuracy of Smoothed Particle Hydrodynamics (SPH) gradient and interpolation approximations, under certain conditions.

1. Introduction The standard SPH gradient and

interpolation approximations on a function F(x) can be viewed as the result of two steps of approximation: a smoothing and a discretising operation.

We show that the second step in the SPH approximations is unnecessarily coarse. The value of the function F(x) is by definition known only at the particles. However, the kernel function is known over all space and there is no need to approximate its integral. The integration can be performed numerically with arbitrary accuracy, and in some cases (Gaussian [1] or 1D spline kernels, for example) it can even be calculated analytically.

2. Method In evaluating a function F at a particle

location a, standard SPH can be considered to perform the following two approximations, using information from neighbour particles b:

∑∫ ≈≈b

abbb WVFdxWFF (1)

Exact Kernel Integration (EKI) modifies the approximation made in the second step to:

∑ ∫≈b V

ab

b

dxxWFF )( (2)

where we consider the volume Vb as centred around each associated neighbour b, and having a spherical shape (circular or linear, in lower dimensions). The volume of each particle is considered to be determined by the ratio of its mass and density.

In what follows we will refer to the terms VbWab in (1) and ∫Wa(x)dx in (2) as the “weighting terms” of the interpolation operation for standard SPH and EKI, respectively.

Figure 1: EKI weighting terms as the exact integrals of the kernel over particle volumes (shown for the interpolation operation in 1D).

According to this formulation, standard

SPH can be said to approximate the integral weighting term with a midpoint approximation: the value of the integrand at the volume’s centre multiplied with the volume Vb. EKI avoids this level of approximation.

volume of particle outside

compact support

volumes of particles inside compact support

Kernel EKI weighting terms proportional

to areas under kernel

124

3. Implementation A value for the integral term in Eq. (2) can

be obtained through analytical or numerical integration. However, simple analytic integrals can generally only be found for one-dimensional or simple (single-part) kernels, such as Gaussians.

Numerically computing integrals is a relatively costly operation. To offset this cost, the values of integrals over typical ranges of parameters are pre-computed and then looked-up from recorded tables at runtime.

Figure 2: Integration regions (dark grey) form as intersections between particle volumes and

the compact support (shown for 2D). In a non-dimensionalised form, the value of

the integral for a given kernel shape can be expressed in terms of only two parameters: distance to neighbour and radius of neighbour volume. Results for a useful range of distances and volumes can therefore be recorded in two-dimensional look-up tables.

Figure 3: Look-up tables of EKI weighting terms

for interpolation (a), and gradients (b), for a cubic spline kernel.

A suitable interpolation method (e.g.

bilinear) can then be used for value extraction. The interpolation can be made arbitrarily accurate by increasing the resolution of the look-up tables, or using higher-order interpolation methods.

4. Theoretical analysis In order to analyse the difference between

standard SPH and EKI, we perform a truncation error analysis in 1D.

The Taylor series expansion of the kernel function W of particle a, around neighbour particle b is:

...2

)()(''

))((')()(2

+−

+

+−+=

bb

bbb

xxxW

xxxWxWxW (3)

or, in a non-dimensional form:

...2

)()(''ˆ1

))(('ˆ1)(ˆ)(ˆ

2

+−

+

+−+=

bb

bbb

sssWh

sssWh

sWsW (4)

where s = x / h represents a non-dimensio-nalised distance from the centre of particle a, and Ŵ(s) = h W(x), with h representing smoothing length.

a b

b b

kernel compact support

particle volume Va

neighbour volumes Vb

a)

b)

∫Wab

(x)d

x

∫∇W

ab(x

)dx

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We integrate both sides of equation (4) over the neighbour volume Vb, and multiply by h to obtain:

...2

)()(''ˆ

))(('ˆ)(ˆ)(ˆ

2

+−

+

+−+=

∫

∫∫

b

bb

V

bb

Vbbbb

V

dsss

sW

dssssWVsWhdssW (5)

We can recognise in the LHS and the first

term on the RHS the two interpolation weighting terms of EKI and SPH, respectively. The remaining terms in the above equation represent, therefore, the difference between the two.

It can be shown that, for a symmetric kernel function W and centred particle volumes, all terms involving odd powers of (s - sb) are zero. The difference between standard SPH and EKI non-dimensional interpolation weighting terms is obtained as:

...23

1)(''ˆ2

)(ˆ)(ˆ

22 +⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−=

=−∫

hVssW

hV

VsWhdssW

bbb

b

Vbb

b (6)

The next term in this series would include a

fourth-order derivative of the kernel function. Such higher-order terms appear only for quartic (and higher) polynomial kernel functions. For a cubic spline kernel (as used in this work), the first term of the series on the RHS of (6) is the only one that appears. In this particular case the difference between standard SPH and EKI weighting terms is proportional to the curvature of the kernel at the neighbour location.

5. Empirical error analysis EKI and standard SPH gradient

approximation were compared in a synthetic test in 3D. A number of particles were distributed in a uniform 3-dimensional

Cartesian lattice. Their positions were randomised, with a normal distribution and a standard deviation of 2% or 20% of inter-particle distance. The particles were then assigned values corresponding to a 3-dimensional sinusoidal wave, and both standard SPH and EKI were used to determine the three-dimensional gradient of the function. The error, ε, in the gradient was obtained by comparison against the known analytic gradient. This operation was performed for a wide range of ratios of smoothing length to test function wavelength, h/λ, as well as inter-particle spacing to smoothing length, ∆x/h.

The difference in error between the two methods is plotted on logarithmic scales in Figure 3. An increase on the horizontal axis, log10(∆x/h), corresponds to a decrease in the number of neighbours to a particle. An increase on the vertical axis, log10(h/λ), corresponds to an increase in the frequency of the evaluated function (relative to smoothing length). Common values of log10(∆x/h) for 3D simulations lie between 0 and -0.2 (i.e. approximately 34 to 133 neighbours to a particle).

2% randomisation

log10ε SPH – log10ε EKI

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Figure 4: Non-dimensionalised error difference between standard SPH and EKI, for a range of

test parameters, in logarithmic units. As can be seen in the plot, there is a small

range of shorter wavelengths for which typical standard SPH performs better than EKI. This however only occurs with very low particle randomisations, which are not generally maintained in real flows. In practically all other cases EKI performs as well or better than standard SPH.

6. Test case The behaviour of the new method was

compared against standard SPH in a realistic flow problem. A 2D transient lid-driven cavity was simulated using both methods in identical conditions. Differences between the two methods stand out most when dealing with lower neighbour numbers, as presented here.

This test case involves viscous flow inside a square cavity in which one of the walls is sliding tangentially at a constant speed. A pseudo-incompressible formulation was used, together with the Cleary [2] viscosity model, at a Reynolds number of 100.

A typical number of neighbours per particle for this two-dimensional problem is 30, but for the purpose of this comparison the number was reduced to 20. The speed of sound was set to

20 times the lid velocity, and the initial particle arrangement was a 28x28 regular array.

Time

To

talk

inet

icen

ergy

0 1 2 3 4 5 60

0.01

0.02

0.03

Standard SPHExact Kernel IntegrationReference FVM solution

Figure 5: Total kinetic energy for the lid-driven test problem (nondimensional).

The total kinetic energy of the particles is

monitored during the simulation, and plotted comparatively against time. The expected behaviour, represented by a high-resolution finite-volume solution, is an asymptotic increase towards a steady-state value. Due to the relatively low resolution and small number of neighbours, both methods produce an uneven kinetic energy curve. The EKI method’s evolution is however more stable and significantly closer to the reference.

In the same test case, a plot of particle paths, in Figure 6, also reveals smoother trajectories obtained with the EKI method.

7. Conclusions A modification to improve the accuracy of

standard SPH with little or no performance penalty has been presented. Theoretical and empirical accuracy characteristics of the new method were investigated. The new method was shown to improve the accuracy of SPH in most flow conditions. Results quantifying the improvement in accuracy were obtained in synthetic tests, as well as in real flow cases.

20% randomisation

log10ε SPH – log10ε EKI

127

a) b)

Figure 6: Detail of particle trajectories in the time interval 2-4 for standard SPH (a) and EKI (b). Moving lid is at top, sliding from left to right.

8. Acknowledgments The authors would like to thank Padraic

Dooley for his help in obtaining a reference FVM solution.

This work was supported by Basic Research Grant SC/2002/189 from the Irish Research Council for Science, Engineering and Technology under the Embark Initiative, funded by the National Development Plan.

9. References TAKEDA H, MIYAMA S, SEKIYA M (1994), Numerical Simulation of Viscous Flow by Smoothed Particle Hydrodynamics, Progress of Theoretical Physics, 92:939-960. CLEARY P W (1998), Modelling confined multi-material heat and mass flows using SPH, Applied Mathematical Modeling, 22:981-993

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Some Industrial SPH Applications Undertaken at the BAE

Systems Advanced Technology Centre Robert Banim 1

1 BAE SYSTEMS Advanced Technology Centre, Filton, Bristol, BS34 7QW, UK. robert.banim@baesystems.com

Abstract

The BAE Systems Smoothed Particle Hydrodynamics code (BAESPH) and pre and post processor (SPHERE) have been developed over a number of years to the stage that these tools are now being used to undertake sloshing and materials processing simulations for the aerospace and automotive industries. Fuel tank, oil tank and gearbox sloshing simulations have been undertaken for McLaren Racing where the liquid location as the car completes a race lap is the main output. For Airbus, the loading data on the walls of a wing fuel tank have been computed when the wing is subject to limit gust loading. Materials processing applications for BAE Systems have included simulation of a highly viscous flow through a static mixer. A brief description of these simulations and the developments made to BAESPH and SPHERE is given.

1. Introduction The SPH meshless method was first proposed in 1977 simultaneously by Lucy [1] and Gingold and Monaghan [2]. Recent reviews of the SPH method are given in Li and Liu [3] and Monaghan [4]. The application of SPH to incompressible, free surface flows was first undertaken by Monaghan [5], with validation against a bore and wave problems. Since, the SPH technique has been used to model free surfaces flows by Cleary et al [6] and Roubtsova and Kahawita [7], amongst many others

and specifically sloshing by Colagrossi [8] and Colagrossi et al [9,10]. BAESPH uses the SPH method taken from Monaghan [5], details are given by Cleary et al, [11]. In summary, the strong form of the governing equations (1,2) is discretised by a collocation technique using an interpolant or smoothing kernel W(r,h), where r is position and h the smoothing length.

( . )D

Dt

ρρ= − ∇ v (1)

D

PDt

ρ ρ= −∇ +− ∇ ⋅v

gτ (2)

The continuous and discrete definitions of the SPH averaging operator on some function A, are given in (3) and (4).

( ) ( ) ( ),IA A W h d′ ′ ′= −∫r r r r r (3)

( ) ( ),bb

b b

AA m W h

ρ≈ ∇ = ∇ −∑ br r r

(4) The interpolating points may be thought of as particles each carrying a mass m, and a velocity v. The SPH kernel used here is spline based and vanishes for separations > 2h. This means that summations involve only near neighbours. By implementing an efficient grid based search algorithm that scales linearly with the number of particles, fast solutions are obtained, vindicating the approach under the criteria put forward by Idelsohn and Oñate, [12].

129

The SPH discrete equations of motion are given by (5-6), where the notation W(ra – rb, h) = Wab has been used.

( )ab a b a abb

dm W

dt

ρ= − ⋅ ∇∑ v v (5)

2 2

a a bb ab a ab a

b a b

d P Pm W

dt ρ ρ= − + + Π ∇ +

⎛ ⎞⎜ ⎟⎝ ⎠

∑vF

(6) ∏ab is the viscous term given by Cleary, [13]. The form of the continuity equation used in (5) ensures that there is no smoothing of density at the free surface whilst the momentum equation (6) is written in symmetrised from to conserve linear and angular momentum. The equation of state used in this simulation is given in (7).

1o

o

P Pγ

ρ

ρ= −

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

(7)

where γ = 7 for the fuel. Po is defined to limit the maximum fluid compression to less than 1% as described by Monaghan, [5]. BAESPH uses Lennard-Jones forces to implement fixed wall boundaries, has inflows and outflows and is a shared memory parallel code.

2. Sloshing for Mclaren Racing A number of pieces of sloshing work for Mclaren Racing have been undertaken. Main fuel tank sloshing work is described here. There is a requirement to keep the fuel centre of gravity as low as possible to improve car handling. Fuel sloshing in the main fuel tank was simulated for a complete lap of the Jerez circuit. Results from these simulations allowed Mclaren designers to optimise the internal tank design.

CAD geometry of the fuel tank was provided in CATIA format and a surface mesh of tri-elements rapidly generated in Altair Hypermesh at a target 20mm edge length. This mesh was read into SPHERE. For “parts” from finite element surface meshes, SPHERE converts the nodes into SPH particles. The conversion process accounts for the treatment of wall forces in the BAESPH code. Each wall particle can have multiple normal directions assigned to it, each associated with a different repulsion direction for a Lennard Jones force. The weighting algorithm applied to each direction has been tuned such that complex convex and concave corners are handled without generating instabilities from large boundary forces. SPHERE automatically creates these multiple normals from a surface mesh by assigning them from the normal directions of all attached elements. In this way, SPH simulations in components with complex CAD geometry are rapidly setup.

Fig. 1 SPH boundary particles on fuel tank User defined parts can also be created within SPHERE from primitives and normals added or removed. Two such parts were created to model the upper and lower tank baffles. Using the tools with SPHERE, various holes were “cut” out of the baffles to represent different configurations. Baffle surfaces have

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normals pointing out in each direction and holes have additional normals pointing towards the hole centres.

Fig. 2 Multiple normals on slice through tank walls. 6650 SPH particles representing 40 kg of fuel were automatically generated within SPHERE. A filling algorithm is employed that creates a set of uniformly spaced particles inside a closed part. Accelerations corresponding to a complete lap of the Jerez circuit were then applied to the walls during a transient simulation. SPHERE was used to generate animations of the fuel particles, Figure 3.

Fig. 3 Fuel sloshing during lap, from SPHERE

The animations were used to determine the effectiveness of 5 different tank geometries. Quantitative data was output from SPHERE giving the variation in the centre of gravity and moments induced by the sloshing on the tank walls about the longitudinal axis through the car’s centre of mass.

3. Sloshing pressures for Airbus The purpose of this work was to calculate the wall pressures due to fuel sloshing in a typical outer wing tank. A 2-dimensional model of a port wing tank was created from a cross section in a vertical plane through the fuel tank along the span wise direction. The tank is split into 10 compartments by vertical wing ribs as shown in Figure 4.

Fig. 4 2-d tank geometry Airbus provided accelerations based on a generic interpretation of the accelerations encountered in a limit gust. Flow paths between the compartments at the upper and lower stringer castellations are modelled by introducing gaps of representative size where the ribs meet the upper and lower tank surfaces. An SPH representation of the tank was created in SPHERE and particles generated for two test cases of 30% and 80% fill levels, requiring 49333 and 141306 particles respectively. Figures 5a,b & c show the evolution of the fuel location at time intervals; t = 0.6 s, 0.9913 s and 1.41 s. Only the 6 inboard compartments are shown as the fuel does not reach any further outboard.

Fig 5a 30% full tank. Flow solution at t = 600ms

z-direction

outboard

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Fig 5b 30% full tank. Flow solution at t = 991ms

Fig 5c 30% full tank. Flow solution at t = 1410ms SPHERE was modified to compute the pressure on the upper and lower tank walls by spatially averaging over the segments making up each compartment with a sampling period of 25 ms. In Figure 6, the pressures on the upper (U) and lower (L) walls are shown for each compartment. The compartments are numbered from C01 at the inboard end to C10 at the outboard end.

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

0 250 500 750 1000 1250 1500 1750 2000Time [ms]

Pres

sure

LC01UC01LC02UC02LC03UC03LC04UC04LC05UC05

Fig. 6 Fuel tank pressure plot. The analytical solution for the pressures on the full tank compartments was found to correspond well with the SPH solution, giving confidence in the obtained results.

4. Flow through a static mixer The flow of a highly viscous mixture through a complex geometry static mixer has been simulated. SPHERE was modified to automatically create the outward pointing normals on particles derived from the edge nodes of the shell elements defining the mixer geometry.

With these modifications, the SPH geometry for the problem was rapidly created in SPHERE. The inflow boundary condition was then setup to model the flow into the mixer at a fixed flow rate, Figure 7 and animations generated of the transient pressure field, Figure 8.

Fig. 7 Flow into the mixer.

Fig. 8 Particles coloured by pressure at 34.9s

5. Conclusions The BAE Systems SPH code, BAESPH, and SPHERE pre/post processor have been developed to the stage where they are used solve a variety of industrial flow problems. Problem setup is straight forward with no requirement to generate a volume mesh. Run times are also fast in comparison to Eulerian methods with interface tracking algorithms.

6. Acknowledgments Tim Goss, Mark Ingham, Paddy Lowe, Nick Butler, John Sutton & Dick Glover, Mclaren Racing, Woking, UK.

Melissa Bergeon and Rob Lamb, Airbus, Filton, Bristol, UK.

Martin Johnson, Land Systems, Glascoed, UK.

7. References

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LUCY L.B. A Numerical Approach to the Testing of the Fission Hypothesis. Aston. Jn. No. 82, pp 1013-1024, 1977. GINGOLD R.A. & MONAGHAN J.J. Smoothed Particle Hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc., No. 181, pp 375-389, 1977. LI S. & LIU W.K. Meshfree and particle methods and their applications. Appl. Mech. Rev., Vol. 55, No. 1, pp 1 – 34, 2002. MONAGHAN J.J. Smoothed Particle Hydrodynamics. Rep. Prog. Phys. No. 68, pp 1703 – 1759, 2005. MONAGHAN J.J. Simulating Free Surface Flows with SPH. Journal of Computational Physics No. 110, pp 399-406, 1994. CLEARY P., PRAKASH M., HA J., STOKES N. & SCOTT C. Smooth Particle Hydrodynamics: Status and future Potential. Proc. 4th Int. Conf. on CFD in the Oil and Gas, Metallurgical & Process Industries, SINTEF/NTNU, Trondheim, Norway, pp 1 – 21, 2005. ROUBTSOVA V. & KAHAWITA R. The SPH technique applied to free surface flows. Computers & Fluids, in press May 2006 COLAGROSSI A. Dottorato di Ricerca in Meccanica Teorica ed Applicata CICLO. A meshless Lagrangain method for free-surface and interface flows with fragmentation. PhD Thesis, Universita di Roma, La Sapienza, 2004. COLAGROSSI A. & LANDRINI M. Numerical Simulation of interfacial flows by SPH. J. Comp. Phys. No 191, pp 448-75, 2003.

COLAGROSSI A., LUGNI C., DOUSET V., BERTRAM V. & FLATINSEN O. Numerical and experimental study of sloshing in partically filled rectangular tanks. 6th Numerical Tank Towing Symp. Rome, Italy, 2003. CLEARY P., STOKES N., MONAGHAN J. & PRAKASH M. Smoothed Particle Hydrodynamics Theory Manual Issue 2. BAE SYSTEMS Australia SPH Report. No SPH/1377/010, 2002. IDELSOHN S.R. & OÑATE E. To mesh or not to mesh. That is the question…Comp. Meth. In applied mechanics and engineering. In Press May 2006. (available on-line at www.elseviercom/locate/cma) CLEARY P., SPH Tech Note #8 New Implementation of viscosity. CSIRO Division of Mathematics and Statistics Tech Report. No. DMS – C96/32, 1996.

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Modeling Star Formation with SPH

Ralf S. KLESSEN and Paul C. CLARK

Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany

Abstract

Smoothed particle hydrodynamics in connection with stellar dynamics is the method of choice when modeling formation and evolution of star clusters in the Milky Way.

Star Formation Stars form by gravoturbulent fragmentation

of interstellar gas clouds. The supersonic turbulence ubiquitously observed in Galactic molecular gas generates strong density fluctuations with gravity taking over in the densest and most massive regions. Collapse sets in to build up stars and star clusters. Turbulence plays a dual role. On global scales it provides support, while at the same time it can promote local collapse. Stellar birth is thus intimately linked to the dynamical behavior of parental gas cloud, which governs when and where protostellar cores form, and how they contract and grow in mass via accretion from the surrounding cloud material to build up stars. Slow, inefficient, isolated star formation is a hallmark of turbulent support, whereas fast, efficient, clustered star formation occurs in its absence. For more details see Mac Low & Klessen (2004), or Ballesteros-Paredes et al. (2007).

Modeling Star Formation To adequately describe star formation in

interstellar gas clouds, it is necessary to follow the fragmentation of self-gravitating turbulent

gas and resolve localized gravitational collapse over several orders of magnitude in density. Due to the stochastic nature of supersonic turbulence, one cannot predict where and when stars will form within the cloud. To compute the time evolution of the system SPH is the method of choice. The method is able to resolve high density contrasts as particles are free to move, and so the particle concentration increases naturally in

Figure 10: Formation of a star cluster in turbulent self-gravitating interstellar gas. The yellow dots indicate the location of sink particles, which we identify as direct progenitors of individual stars or binary stellar systems.

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high-density regions. We use the publically available parallel code GADGET (Springel et al. 2001). It is modified by including a turbulent driving scheme (Mac Low et al. 1998). In addition, once the central region of a collapsing gas clump exceeds a limiting density contrast we introduce a “sink particle”, which is able to accrete gas from its surrounding while keeping track of the mass and linear and angular momentum of the infalling material (Bate et al. 1997). Replacing high-density cores by accreting sink particles allows us to follow the dynamic evolution of the system over many local free-fall timescales. We identify sink particles as the direct progenitors of individual stars (Wuchterl & Klessen 2001). For a more detailed account of the sink-particle method and its implementation we refer the reader to Jappsen et al. (2005).

Mass Spectra

It can be shown that molecular cloud regions without turbulent support form dense clusters of stars, regardless of the initial density structure, within a few global free-fall timescale. When gravitational contraction has sufficient time to act, the clump mass spectrum is well approximated by a power law dN/dM Mx. The mass distribution of protostellar cores, however, is better described by a combined log-normal – power-law distribution with properties similar to the observed IMF of multiple stellar systems for low and intermediate-mass stars. Studies of decaying turbuence such as illustrated in Figure 1 lead to clustered star formation much like in the case of pure gravitational contraction. Supersonic turbulence, even if it is strong enough to compensate gravity on large scales, will provoke local collapse in shock compressed regions. As efficiency and timescale of star formation depend sensitively on the strength and the spatial scale of energy input into the system, large-scale turbulence leads to clustered star formation on short timescales, whereas for small-scale turbulence stars form in isolation and with low efficiency.

References BALLESTEROS-PAREDES, J., KLESSEN, R. S., MAC LOW, M.-M., VAZQUEZ-SEMADENI, E., 2006, in Protostars and Planets V, eds. B. Reipurth, D. Jewitt, & K. Keil (University of Arizona Press, Tucson) BATE, M. R., BONNELL, I. A., PRICE, N. M. 1995, MNRAS, 277, 362 JAPPSEN, A.-K., KLESSEN, R. S., LARSON, R. B., Y., L., MAC LOW, M.-M. 2004, A&A, 435, 611 MAC LOW, M.-M. & KLESSEN, R. S., 2004, Rev. Mod. Phys., 76, 125 MAC LOW, M.-M., KLESSEN, R. S., BURKERT, A., SMITH M. D., 1998, Phys. Rev. Lett., 80, 2754 SPRINGEL, V., YOSHIDA, N., WHITE, S. D. M. 2001, New Astronomy, 6, 79

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Smoothed Particle Hydrodynamics model for multiphase flow in porous media

Alexandre M. Tartakovsky1, Paul Meakin 2 and Andy Ward 3

1 PhD, Computational Mathematics Technical Group, Pacific Northwest National Laboratory, Richland, Washington, 99352-9999, USA, alexandre.tartakovsky@pnl.gov

2 PhD, Center for Advanced Modeling and Simulation, Idaho National Laboratory, Idaho Falls, Idaho, 83415-2211, USA , paul.meakin@pnl.gov

3 PhD, Hydrology Technical Group, Pacific Northwest National Laboratory, Richland, Washington, 99352-9999, USA, Andy.Ward@pnl.gov

Abstract A numerical model based on smoothed particle hydrodynamics (SPH) was used to simulate pore-scale liquid and gas flow in synthetic two-dimensional porous media consisting of non-overlapping grains. The model was used to study effects of pore scale heterogeneity and anisotropy on relationship between the average saturation and the Bond number. The effect of the wetting fluid properties on drainage was also investigated. It is shown that pore-scale heterogeneity and anisotropy can cause saturation/Bond number and entry (bubbling) pressures to be dependent on the flow direction suggesting that these properties should be described by tensor rather than scalar quantities.

1. Introduction Unsaturated flow in porous media is an example of multiphase flow in domains bounded by geometrically complex boundaries. Besides the non-linearity due to the dynamically changing boundaries separating aqueous and gaseous phases, simulations of coupled liquid-air flow are complicated by large density and viscosity ratios. As a result, traditional grid based methods have not been widely applied to these processes. Lattice Boltzmann (LB)

method has been extensively used to model multiphase flow in porous media but the LB solution becomes unstable when the density and viscosity ratios becomes larger than 50 (Pan et al., 2004).

An alternative approach is to use mesh-free Lagrangian particle methods such as smoothed particle hydrodynamics (SPH). An advantage of SPH is that it does not require explicit interface tracking or contact-angle models. SPH, originally developed in the context of astrophysical applications by Lucy (1977) and Gingold and Monaghan (1977), has been applied for the simulations of various subsurface flow problems including pore-scale modeling of saturated flow (Zhu et al., 1999), non-reactive (Zhu and Fox, 2002, Tartakovsky and Meakin, 2005a) and reactive (Tartakovsky et al., 2007; Tartakovsky et al., in press) transport and free surface flows (Tartakovsky and Meakin, 2005a, b). Tartakovsky and Meakin (2006) developed a SPH model for multiphase flows where a combination of pair-wise molecular-like interaction forces and hydrodynamic forces is used to simulate fluid-fluid and fluid-fluid-solid interactions.

In this paper granular fractured media consisting of grains with varying aspect ratios and a micro-fracture were generated. The multiphase SPH model was used to predict flow normal and

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perpendicular to the fracture alignment to study effects of pore scale heterogeneity and anisotropy on relationship between fluid saturation and the Bond number.

2. SPH multiphase flow equations Using SPH interpolation scheme

( ) ( , ) /s j j jjA A W h n= −∑r r r the continuity

and Navier-Stokes equations are discretized and reduced to a set of ordinary differential equations governing the motion of each particle. Using formal similarities between SPH and MD simulations Tartakovsky and Meakin (2006) proposed a multi-scale SPH model in which the momentum conservation equation for each particle is written in the form:

interactionii i idmdt

= +v F F (1)

In equation (1) the total force acting on each particle is a combination of the ‘hydrodynamic’ force Fi calculated from the SPH discretization of the Navier-Stokes equation:

( )2 2 ,j ii i i j

j j i

P P W hn n

⎛ ⎞= − + ∇ −⎜ ⎟⎜ ⎟

⎝ ⎠∑F r r

( ) ( )2

4,i j ij

ij i ij ij i j i j ij

W r h mn n r

µ µµ µ

+ ∇ ++∑

vr g , (2)

and an interaction force interactioni ijj

=∑F F ,

resulting from the molecular-like pair-wise fluid-solid and fluid-fluid particle-particle forces, ( )cos 1.5 / /ij ij ij ij ijs r h rπ= −F r for rij<h

and 0 otherwise, that control the phase separation and wetting behavior. sij is the strength of the pair-wise particle-particle force Fij and the particle number density is given by:

( ),i ijj

n W r h=∑ . (3)

In SPH simulations solid phase is represented by particles that are fixed in space but enter into the summation in

equations (2) and (3) to avoid a large nonphysical decrease in fluid density near the fluid-solid boundary. The forces entering equation (2) are anti-symmetric. As a result, linear momentum is exactly conserved. The particle-particle interaction forces Fij are repulsive at short separation distances, attractive at mid-range distances and, for computational efficiency, have a limited interaction range, h. Beyond these requirements, the exact form of the particle-particle interactions is not critical for reproducing the correct immiscible behavior with a surface tension that satisfies the Young-Laplace equation. The interaction strength sij between particles i and j depends on which fluid components or solid phases the particles represent. If the interaction strength, s, between dissimilar particles is smaller than the interaction strength between like particles, the particle-particle interaction forces prevent the fluids represented by the dissimilar particles from mixing and create a surface tension. Various wetting conditions are modeled by making the interaction strength between solid or boundary particles and wetting fluid particles larger than the interaction strength between solid particles and nonwetting fluid particles. The strength of wetting is determined by the strength of the fluid-boundary particle-particle interactions relative to the strength of the fluid-fluid particle-particle interactions. The short range repulsive part of Fij, for the fluid-boundary interparticle interactions also simulates no-flow boundary conditions. The equation of state Pi=kni is used to close the system of flow equations (1)-(3). Equation (1) is integrated using explicit “velocity Verlet” algorithm (Allen and Tildesley, 2001). The multi-scale SPH model has been extensively validated against the Young-Laplace equation under both static and dynamic conditions (Tartakovsky and Meakin, 2005c, 2006) and found to accurately produce the effects of surface tension.

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3. Flow in Fractured Porous Media

The effect of pore scale heterogeneity and anisotropy on unsaturated flow was studied in synthetic granular porous media with vertically (Figure 1a) and horizontally (Figure 1b) aligned micro-fractures. The fractured porous medium was drained by incremental increase in the gravitational force. Our simulations mimic the approach commonly used in the laboratory to measure the capillary pressure, Pc, / saturation, S, relationship. The continuum scale capillary pressure for the liquid-gas system under static conditions in a porous medium at elevation H above a level at which the liquid pressure is equal to the gas pressure, is defined as cP gHδρ= (δρ is the difference in fluid densities) and the saturation is the ratio of the liquid volume to the pore volume. Pc/S relationship is needed for upscaling multiphase flow equations from the pore scale to the Darcy

Figure 1: Drainage of a wetting liquid from

a porous medium containing a microfracture. Equilibrium distribution of fluids for three different Bond numbers

(dimensionless). (continuum) scale. Figures 1a and 1b show the equilibrium distribution of the strongly wetting liquid for three Bond numbers ( )2 /cBo l P Hσ= (dimensionless capillary number) during drainage for perpendicular and parallel fracture

orientations. The orientation of the micro-fracture has a large effect on the equilibrium fluids distribution for all values of the Bond number used in the simulations. We also found that the fracture orientation has a substantial impact on S/Bo relationship (Figure 2). The S/Bo curves for the vertical fracture exhibits a ‘dual-domain’ behavior. One part corresponds to the drainage from the fracture with a smaller entry pressure, and the second part corresponds to drainage from the porous matrix with a larger entry pressure. The horizontal micro-fracture creates a local capillary barrier effect because the liquid is unable to enter the wider micro-fracture from the narrower pores.

Figure 2: Effect of flow direction on Bo (or

Pc)/ saturation relationship during the drainage of the strongly wetting liquid in

the fractured medium.

4. Conclusions A smooth particle hydrodynamics model was used to simulate gas-liquid flows in porous fractured media and to investigate the effects of pore-scale heterogeneities on the saturation / Bond number, S/Bo, relationship. The behaviors obtained from the simulations showed directional differences that call into question the validity of effective properties derived by linear averaging of nonlinear relationships. The simulations presented in this paper demonstrate that smoothed particle

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hydrodynamics is a powerful and flexible method for modeling multiphase flow in fractured and porous media

5. Acknowledgments This work was supported by the Multiscale Mathematics Research and Education program, Advanced Scientific Computing Research and Environmental Management Science Program of the U.S. Department of Energy Office of Science. The Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle under Contract DE-AC06-76RL01830.

6. References GINGOLD, R.A. and MONAGHAN, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non- spherical stars. Mon. Not. R. Astr. Soc., 181: 375- 389. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. ZHU, Y., FOX P.J. and MORRIS, J.P. (1999). A pore-scale numerical model for flow through porous media, International Journal for Numerical and Analytical Methods in Geomechanics, 23: 881. ZHU, Y. and FOX, P.J. (2001). Smoothed particle hydrodynamics model for diffusion through porous media. Transport in Porous Media, 43: 441. ZHU, Y. and FOX, P.J. (2002). Simulation of pore-scale dispersion in periodic porous media using smoothed particle hydrodynamics. Journal of Computational Physics, 182: 622.

TARTAKOVSKY, A. M. and MEAKIN, P. (2005a). A smoothed particle

hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh–Taylor instability. Journal of Computational Physics, 207: 610.

TARTAKOVSKY, A.M., MEAKIN, P., SCHEIBE T.D. and EICHLER WEST, R. (2007). Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. Journal of Computational Physics, doi: 10.1016/j.jcp.2006.08.013 .

TARTAKOVSKY, A. M. and MEAKIN, P. (2005b). Simulation of free-surface flow and injection of fluids into fracture apertures using smoothed particle hydrodynamics. Vadose Zone Journal, 4: 848.

TARTAKOVSKY A. M. and MEAKIN P. (2005c). Modeling of surface tension and contact angles with smoothed particle hydrodynamics. Physics Review E, 72, 026301.

TARTAKOVSKY, A.M. and MEAKIN, P. (2006). Pore-scale modeling of immiscible and miscible flows using smoothed particle hydrodynamics. Advances in Water Resources, 29: 1464.

ALLEN, M.P. and TILDESLEY, D.J. (2001) Computer simulation of liquids. Oxford University Press, Oxford.

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Reactive transport and biomass growth in porous media

Alexandre M. Tartakovsky1, Paul Meakin 2 and Timothy D. Scheibe 3

1 PhD, Computational Mathematics Technical Group, Pacific Northwest National Laboratory, Richland, Washington, 99352-9999, USA, alexandre.tartakovsky@pnl.gov

2 PhD, Center for Advanced Modeling and Simulation, Idaho National Laboratory, Idaho Falls, Idaho, 83415-2211, USA , paul.meakin@pnl.gov

3 PhD, Hydrology Technical Group, Pacific Northwest National Laboratory, Richland, Washington, 99352-9999, USA, Tim.Scheibe@pnl.gov

Abstract A Lagrangian multi-scale particle model based on smoothed particle hydrodynamics (SPH) was used to simulate pore-scale flow, reactive transport and biomass growth in fractured porous media. The biomass growth was controlled by double Monod kinetics. The deformation and attachment and detachment of biomass was modeled through a combination of pair-wise short range repulsive and medium range attractive forces and hydrodynamics forces resulting from the fluid flow.

1. Introduction Artificially enhanced biomass growth emerges as an important tool in many subsurface engineering applications. Bio-organisms are used in remediation to immobilize non-organic contaminants and to convert organic contaminants to innocuous end products. In the petroleum industry biomass can be used to plug high permeability regions and preferential flow paths to facilitate extraction of oil from low permeability regions in oil reservoirs. Numerical models have emerged as an important tool in understanding the mechanisms of biomass growth and spreading in porous media. Cellular automata (CA) models are widely used to model biomass growth. CA models are based on a simple discretization of the

computational domain with uniform cells, and a set of rules to fill unoccupied cells, empty occupied cells or move biomass between cells. Knutson et al. (2005) partially included effects of fluid-biomass interactions by requiring that biomass can spread only into cells with low fluid shear stresses. In this paper, a smoothed particle hydrodynamics (SPH) method (Lucy, 1977; Gingold and Monaghan, 1977) is used to simulate pore-scale flow, reactive transport and biomass growth and spreading. The Lagrangian particle nature of SPH allows complex biomass-fluid and biomass-solid interactions to be modeled through simple pair-wise interaction forces. We propose a multi-scale model that combines pair-wise interaction forces with hydrodynamic forces to describe growth and spreading of biomass in fractured porous media. Both a CA model and a multiscale model were used to simulate biomass growth. Results of the two different models are compared and discussed in the Conclusions section.

2. SPH transport equations This work assumes that the changes in the biomass are governed by the dual Monod kinetics (Knutson et al. 2005),

s dA B

dM A BYk M k Mdt K A K B

= −+ +

, (1)

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where M is the biofilm concentration; A and B are the concentrations of electron donors and electron acceptors respectively; Y is the yield coefficient; kS is the maximum growth rate; kd is the degradation rate; and KA and KB are the half-saturation constants for A and B. The advection and diffusion of electron donors and acceptors are governed by the combination of the continuity, momentum conservation equations (Navier-Stokes equations) and equations of state coupled with diffusion reaction equations. The system of equations was solved using a Lagrangian particle method based on SPH. SPH uses meshless discretization of the computational domain and an interpolation scheme, ( ) / ( )i i ii

A A nW= −∑r r r , allowing approximation of a continuous field A(r) using the values of A at a set of discretization points. W is the SPH weighting function and ri is the positions of discretization point i. ni = ρi / mi is the particle number density and ρi and mi are the density and the mass of a phase associated with point i. Because each point possesses a mass and volume it is natural to think about discretization points as physical particles. The SPH approximation of continuous fields allows the mass and momentum conservation equations to be written in the form of a system of ordinary differential equations (ODEs) (Tartakovsky et al. 2007),

( )i j ij

n W= −∑ r r (2)

and N Si

i idmdt

−=v F (3)

( )2 2jN S i

i i i jj j i

P P Wn n

−⎛ ⎞

= − + ∇ −⎜ ⎟⎜ ⎟⎝ ⎠

∑F r r

( )( )( )

( )2

4 i ji ji j

j i j i j i jn n

µ µµ µ

−+ −

+ −∑

v vr r

r r

( ),i i jW h∇ − +r r g . (4)

Both mobile fluids and solids (soil grains) are represented by particles.

j∑ indicates summation over all

particles. Particles representing soil grains are frozen in space, their velocity is set to zero and they enter into the calculation of densities of fluid particles (eq. (4)) and forces acting on the fluid particles (eq. (6)). Fi

N-S is the hydrodynamic force acting on fluid particles calculated from the Navier-Stokes equation, v is the fluid velocity vector, P is the pressure, g is the gravitational acceleration vector, and µ is the dynamic viscosity of the solution. In general, the mass mi associated with fluid particle i and the viscosity µi depends on the fluid composition, which may change with time. In this work it was assumed that biomass density is equal to the fluid density and that the viscosity of fluid containing biomass is ten times greater than the viscosity of fluid containing no biomass. Following Tartakovsky et al. (2007) the system of diffusion/reaction equations (1) – (3) can be cast in the form of a system of ODEs:

( )( )( )

( )21 A i i j j i ji

i jj fluidi i j i j

D m n m n A AdAdt m n n∈

+ −= −

−∑ r r

r r

( ), i ii i j s

A i A i

A BW h Yk MK A K B

∇ − −+ +

r r , (5)

( )( )( )

( )21 B i i j j i ji

i jj fluidi i j i j

D m n m n B BdBdt m n n∈

+ −= −

−∑ r r

r r

( ), i ii i j s

A i A i

A BW h Yk MK A K B

∇ − −+ +

r r (6)

and

i i is i d i

A i A i

dM A BYk M k Mdt K A K B

= −+ +

, (7)

where j fluid∈∑ indicates summation over

all the fluid particles. Excluding solid particles from the summations enforces no-diffusion no-reaction conditions at solid-

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fluid interfaces. Initially biomass was randomly distributed on the surfaces of the porous medium by randomly selecting fluid particles near the solid particles and assigning biomass concentration M0 to these particles. Zero biomass concentration was assigned to the rest of the fluid particles. Evolution of the biomass was calculated according to equation (9). Once the concentration of the biomass at any particle i exceeded M0, the excess of the biomass was moved to the nearest fluid particle with a biomass concentration less than M0. Biomass growth is known to be also controlled by the forces exerted on the biomass by the flowing fluid. In this paper we investigated two models to account for the effect of fluid shear stress on the spreading of the biomass. Model 1 is similar to the CA model used by Knutson et al. (2005). Particles with non-zero biomass concentration are assumed to be immobile, and excess biomass is allowed to move only to the nearest particles with shear stresses, ( )/ /x yv y v xτ µ= ∂ ∂ + ∂ ∂ ,

less than critical stress τcr. Model 2 is based on the ideas that biomass is kept together by attractive forces mediated by biopolymer, and that attractive forces between biomass and soil grains keep biomass attached to the solid surface. In Model 2 the excess biomass can be transferred to any neighboring fluid particle with M < M0 and the behavior of the biomass (deformation, attachment, detachment and splitting) is modeled through a combination of hydrodynamic forces and a short range repulsive and medium range attractive pair-wise forces that act between all pairs of particles including particles representing soil grains and fluid particles with and without biomass

N Sii i ij

j

dmdt

−= +∑v F F . (8)

In this work we used pair-wise interaction forces ( )cos 1.5 / /ij ij ij ij ijs r h rπ= −F r for rij<h

and 0 otherwise. sij is the strength of the force that depends on the properties of materials and is different for interaction between different types of particles. The interaction strength, sij, between pairs of particles containing biomass was set to be larger than sij for the interaction between pairs of particles with and without biomass. If sij for interactions between fluid particles containing biomass and solid particles is set to be greater than sij for interactions between two particles containing biomass the model will produce biomass growth in the form of continuous biofilms. Otherwise, patchy biofilms will be preferentially formed. In this work the interaction strength, sij, for interactions between fluid particles containing biomass and solid particles was set to be greater than sij for interactions between two particles containing biomass. The interaction strength, sij, between fluid particles with zero biomass and solid particles was set to be the same as sij for two fluid particles containing zero biomass. Since solid particles are immobile, interactions between pairs of solid particles were not considered and sij for forces between pairs of solid particles were not specified. The system of flow and diffusion equations was integrated using the explicit “velocity Verlet” algorithm (Allen and Tildesley 2001).

3. Numerical results A fractured porous medium was generated by randomly inserting non-overlapping discs with random radii on either side of the gap between two self-affine fractal curves representing a micro-fracture. The injection of electron donors and acceptors in different halves of computational domain was simulated by assigning concentrations A=1, B = 0, M = 0 to the particles entering at y = 32 in the left part of domain and A=0, B = 1, M = 0 in the right part of domain.

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Figure 1: Distribution of the biomass

produced by Model 1 and Model 2. Gray particles denote soil grains, black particles

represent fluid particle with non-zero biomass concentration, and the color scale indicates the concentration product, AB, in

the fluid particles with zero biomass concentration.

Particles exiting the flow domain at y = 0 were returned into the flow domain at y = 32 with zero concentrations of A, B and M. Figures 1a and 1b show the steady-state distribution of biomass resulting from continuous injection of solutes containing electron donors and acceptors obtained with Model 1 and Model 2 respectively.

4. Conclusions A Lagrangian particle model based on SPH was developed and used to simulate flow, reactive transport and biomass growth. Two different models for biomass growth were considered. For the set of parameters used in the simulations both Model 1 and Model 2 predicted that the biomass forms bridges connecting soil grains, oriented in the direction of flow so as to minimize the resistance to fluid flow. According to Model 1 most of the biomass was distributed near the fracture entrance where biomass growth occurred while in Model 2 biomass was distributed uniformly along the walls of the fracture. In Model 2, as in Model 1, growth of the biomass occurred mainly

near the fracture entrance where the concentration of electron donors and acceptors was the largest. However, unlike Model 1, in Model 2 biomass was allowed to be deformed by the shear stresses of the fluid, and detach and attach to the surfaces of the soil grains. As a result, the biomass spread more widely through fractured porous domain.

5. Acknowledgments This work was supported by the Environmental Research Science Program of the U.S. Department of Energy Office of Science. The Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle under Contract DE-AC06-76RL01830.

6. References ALLEN, M.P. and TILDESLEY, D.J. (2001) Computer simulation of liquids. Oxford University Press, Oxford. GINGOLD, R.A. and MONAGHAN, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non- spherical stars. Mon. Not. R. Astr. Soc., 181: 375- 389. KNUTSON, C.E., WERTH, C.J., VALOCCHI A.J. (2005). Pore-scale simulation of biomass growth along the transverse mixing zone of a model two-dimensional porous medium. Water Resour. Res. 41: W07007, doi:10.1029/2004WR003459. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. TARTAKOVSKY, A.M., MEAKIN, P., SCHEIBE T.D. and EICHLER WEST, R. (2007). Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. Journal of Computational Physics, doi: 10.1016/j.jcp.2006.08.013 .

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SPH Modeling of Forced Water Waves

Muthukumar Narayanaswamy 1, Jannette Frandsen 2

and Robert Dalrymple 1

1 Johns Hopkins University, Dept of Civil Engineering, muthu@jhu.edu 2 University of Hawaii Manoa, Dept of Ocean and Resources Engineering, frandsen@hawaii.edu

Abstract The Johns Hopkins University 2-D SPH model is used to study forced water waves. Results from this model are being compared with experimental data obtained by Dr. Jannette Frandsen at the Louisiana State University. In the experiments a 1m2 square tank was subject to harmonic forcing in surge and heave motions. The free-surface data were recorded at the tank boundaries and at the centre of the tank. A variety of forcing scenarios by changing the forcing amplitudes and frequencies were considered. Multiple laboratory tests were conducted over a range of shallow to intermediate water depths.

1. Introduction Free-surface oscillations of liquids in

partially filled containers pose engineering challenges in a wide variety of practical problems. Tuned liquid dampers are used to dissipate earthquake and wind induced vibrations in tall buildings. They need to be designed to withstand large hydrodynamic forces at the walls due to resonant seismic or wind forcing. Wave induced ship motions such as heave/pitch/roll result in sloshing of liquids in LNG containers and ballast tanks. In these partially filled containers, slamming loads due to nonlinear forced waves are of critical interest. 3-D flow phenomena become important in shallow containers with comparable length and breadth. The free-surface motions in such tanks are characterized by bore formations, large run-up along the sidewalls and wave breaking in the middle of the domain. These characteristics are

observed even for small amplitude forcing as long as the forcing frequency is close to the resonant frequency of the tank. Typically, earthquakes induce motion in both the horizontal and vertical directions. The presence of heave motions would alter the free-surface response by introducing different physical mechanisms, for e.g., Faraday Waves (Miles and Henderson 1990). Another example is earthquake-induced water waves in reservoirs and basins. Large waves were observed at the Los Angeles dam during the 1994 Northridge earthquake (Ruscher and Synolakis 1998). The run-up and overtopping induced by these waves can pose a significant danger to life, infrastructure, and environment. Hence, understanding these fluid-structure interactions is a necessary precursor to developing accurate, predictive tools for engineering and design use.

Smoothed Particle Hydrodynamics is a

Lagrangian method, originally developed by Lucy (1977) and Gingold and Monaghan (1977), to simulate non-axisymmetric problems in astrophysics. Since then, this method has been improved to simulate a wide variety of problems in solid mechanics and fluid dynamics. Monaghan (2005) provides a comprehensive review of SPH and its applications.

One of the main advantages of SPH is that

it no special techniques are required to track the free-surface in interfacial flows, making it an attractive method to study water waves. Naturally, SPH has been used to study various free surface problems such as propagation of

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solitary waves (Monaghan and Kos (1999)), wave breaking (Rogers and Dalrymple (2004)), and dam break (Monaghan (2004)). Landrini et al. (2003) studied 2-D sloshing using both single phase and two-phase SPH models and obtained good comparisons with experimental data.

The Johns Hopkins University SPH model

has been used to study to a wide variety of coastal engineering problems. Dalrymple and Knio (2001) and Dalrymple et al. (2001) applied the model to simulate landslide generated waves and green water overtopping over an emerged deck. Gómez-Gesteira and Dalrymple (2003) studied the impact of a dam break generated wave on a tall cylindrical structure. In order to replace the standard viscous formulation in SPH, a LES approach to SPH was developed by Rogers and Dalrymple (2004). This model was subsequently used to study wave breaking induced turbulent structures, tsunami generation and propagation.

The goal of this study is to determine the

performance of this model in predicting free-surface oscillations due to external forcing in shallow tanks. A brief description of the SPH model, the experiments and some preliminary results are presented in the following sections.

2. Numerical Model The governing equations for the sub-

particle scale SPH model are obtained by Farve averaging the Navier-Stokes equations. The averaged mass and momentum equations are given by

u~⋅∇−= ρρdtd (1.4)

( ) τρ

ρρρ

r∇+∇∇++∇−=1~11~

0 ugu vPdtd (1.5)

where r

τ is the sub particle scale stress tensor. In the above equations, tilde represents Favre averaged quantities and the overbar represents Reynolds averaging. It is worth noting that for incompressible flow, Reynolds averaging and Favre averaging give the same set of equations for mean and turbulent flow. Detailed information on the turbulent stresses and SPH discretization can be found in Rogers and Dalrymple (2004).

3. Experimental Details

The free surface motion in a square tank is generated by a six degrees-of-freedom (DOF) shake table (1000 kg Pay load), as shown in Figure 1. Six electro-mechanical actuators supply the motion of the system. The experimental test series are carried out in a square tank (without a lid) with fresh water. The tank size has inner base dimensions of 1 m2. We have assumed that the walls are stiff (thickness of 0.25 m) and therefore assumed that no local wall displacements would interact with the free surface motions. The tank is place on a flat bed on top of the shake table.

Numerous experiments are conducted by

varying the water depth, forcing frequency, amplitude and direction. Herein only free surface behavior at a single shallow water depth (h=5cm) due to sway base excitation is explored. The acceleration of the tank under sway excitation is prescribed by

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Figure 11. Shake table with definition of excitation modes and plan view of the tank with wave gage locations(1-3)

Ay = −ahωh

2 Sin(ωht) (1.6) where ah is the displacement amplitude of

the tank and ωh is the frequency. The free surface is measured using

capacitance wave gauges. The free surface elevations are recorded at (1) the center of the wall (x, y) = (0, -b/2); (2) quarter point (0, -b/4); and (3) at tank center (0, 0) in accordance with the coordinate system shown in Fig. 1. The locations of the three wave gauges are indicated by black dots in Fig. 1. The duration of the experimental test series are approximately 50s outside resonance and typically twice as long during resonance. Details of the experiments can be found in Frandsen and Tubbs (2005).

4. Preliminary Results In the test case presented here, the tank is

forced with ah = 6 mm and ωh = 1.88 rad/s. The parameter kh = ahωh

2

g is used as a measure of the nonlinearity of the system. For this test the kh = 0.002 which corresponds to weak nonlinearity. In the

Figure 12. Free surface comparison at wave gage 1.

experiments, initial sloshing motions followed by traveling waves were observed.

The free surface comparison between

experimental data and SPH results for the initial transient phase are shown in Figure 2. It can be seen in the figure that while the comparisons are the crests are reasonable, the model results seem to underpredict the troughs.

Further tests are being undertaken to test

the effects of viscous and turbulent dissipation rates on the comparisons. Results from these tests and comparisons for different forcing parameters will be presented at the workshop.

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5. References DALRYMPLE , R. A. and O. KNIO (2001) SPH modeling of water waves. Proc. Coastal Dynamics 2001, ASCE, 779-787, Lund, Sweden, 2001. DALRYMPLE, R. A., O. KNIO, D.T. COX, M. GESTEIRA, and S. ZOU (2001). Using a Lagrangian particle method for deck overtopping, Proc. Ocean Wave Measurement and Analysis, ASCE, 1082-1091, 2001.

FRANDSEN, J. B., K. R. TUBBS, and W. PENG (2005). Free surface Lattice Boltzmann simulation of shallow water in horizontally moving tanks. The Fifth International Symposium on Ocean Wave Measurement and Analysis, WAVES 2005.

GINGOLD, R. A. and MONAGHAN, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non- spherical stars. Mon. Not. R. Astr. Soc., 181: 375- 389. GÓMEZ-GESTEIRA, M. and R. A. DALRYMPLE, Using a 3D SPH Method for wave impact on a tall structure, J. Waterways, Port, Coastal, and Ocean Engineering, ASCE, 130(2), 63-69, 2004. LANDRINI, M. and COLAGROSSI, A. and FALTINSEN, O. M. (2003). Sloshing in 2-D flows by SPH Method. The 8th International Conference on Numerical Ship Hydrodynamics. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. MILES, J. and D. HENDERSON. (1990). Parametrically forced surface waves. Annual Review of Fluid Mechanics, 22, 143–165. MONAGHAN J.J. and KOS, A. (1999) Solitary waves on a Cretan beach, J. Waterway, Port, Coastal, and Ocean Engg, ASCE, 125(3).

MONAGHAN J. J (2005). Smoothed Particle Hydrodynamics. Rep. Prog. Phys. 68, 1703–1759

ROGERS B. and DALRYMPLE R. A. (2004). SPH modeling of breaking waves. In Proc. 29th International Conference on Coastal Engineering, Lisbon, September 2004. RUSCHER, C. and C. E. SYNOLAKIS (1998). The sloshing of the Los Angeles Dam during the 1994 Northridge earthquake, Proceedings of the NEHRP Conference and Workshop on Research on the Northridge, California Earthquake of January 17, 1994, Volume III, 821–828.

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An SPH model of wave breaking: quantitative comparisons to laboratory observations

Ely, A.C. and Swan, C.

Department of Civil and Environmental Engineering, Imperial College of Science, Technology and Medicine, London, SW7 2AZ, UK

Abstract This paper aims to provide a study of solitary waves breaking on an impermeable plane slope using the SPH method. The entire breaking process has been considered including: the steepening of the wave form; generation of the overturning wave; impact of the jet in front of the advancing wave crest; run-up and wash down on the slope; and the development of a hydraulic jump due to the sheet flow representing the super-critical wash down. The SPH model has been developed using current modelling techniques which have been selected and tested for their effectiveness in modelling such free-surface flows. The numerical results are then compared quantitatively to the laboratory observations, thus highlighting both the successful aspects of the adopted SPH model and also those areas that require further improvement.

1. Introduction The process of wave breaking and the

associated fluid velocities provide an important input to many engineering designs in both deep water offshore locations and shallow water coastal regions. A range of numerical models are used to gain insights into these events. Historically, some of the most successful of these have been based on either a Fourier series representation or the Boundary Element Method (BEM). However, the former is limited by its inability to model multiple surface elevations at one spatial location and its requirement for periodic boundary conditions; while both are unable to model rotational

effects. Although the BEM can model overturning waves, it cannot capture the re-entry of the water surface once the wave has broken and is also unable to model spilling breakers. The Volume of Fluid method overcomes some of these limitations, but has the disadvantage that the precise position of the free-surface is difficult to locate and, being an Eulerian method, suffers from the inherent problems arising from numerical diffusion when the free-surface undergoes large deformations.

In an attempt to overcome these difficulties and provide improved insights into the nature of the breaking event a solution based upon Smoothed Particle Hydrodynamics (SPH) has been adopted. SPH is a fully Lagrangian method in which the fluid domain is discretised into fluid particles. These carry the material properties of the fluid and move independently of one another under the governing fluid flow equations. The underlying properties of SPH suggest that it will be a useful tool for modelling free-surface flows with large surface deformations, such as those found in breaking waves. However, rigorous comparisons to laboratory data are rare, particularly in the case of breaking waves, and as a result the accuracy of the adopted solution is unclear.

The current paper includes an analysis of two solitary waves with height to depth ratios of H/d = 0.3 and 0.45 breaking on a 5° impermeable plane slope. Numerical simulations were carried out using an SPH model and compared, quantitatively where possible, to laboratory experiments.

2. SPH Model

148

A 2-D SPH model has been developed using methods described by Monaghan (1994) as a basis. The governing equations for the SPH method are the Euler equations in Lagragian form. These are given in SPH form below:

ab

N

baabb

a Wvmdt

d ∑=

∇=1

ρ (1)

aba

N

bab

b

b

a

aa WPPdt

dv∇⎟⎟

⎠

⎞⎜⎜⎝

⎛Π++= ∑

=122 ρρ

(2)

where ρ is the fluid density of each particle, m is the particle mass, v is the particle velocity and P is the particle pressure. The kernel function, W , is based on a cubic spline function. The term, Π , refers to the artificial viscosity which is included to improve the numerical stability of simulations and is given below.

⎪⎩

⎪⎨

⎧

>⋅

<⋅−

=Π00

0

abab

ababab

abab

ab

rv

rvcρ

µα (3)

Where c is the speed of sound, α is a parameter which was set at 0.01 and,

222 abab

ababbaab hr

rvkkh

εµ

+⋅+

= (4)

where h is the smoothing length and,

hcEEvdiv

vdivk

aija

ijaa

aa 410−++

= (5)

This form, originally proposed for SPH by Colagrossi & Landrini (2003), was chosen as it proved to be less dissipative than the original form (Monaghan 1994), in which the multiplication by ( ) 2ba kk + is omitted from

equation (4). In the case of solitary wave propagation, excessive dissipation in the system leads to a reduction in the wave height. According to solitary wave theory, the amplitude and profile of the wave should remain unchanged as the wave propagates in constant water depth, so minimising the dissipation is crucial.

Pressure is found through the usual equation of state, given by Batchelor (1974), keeping density fluctuations to within 1% and using a speed of sound appropriate to the maximum estimated velocity in a solitary wave. XSPH (Monaghan 1994) is used in order to keep the particles more orderly and density re-initialisation (Colagrossi & Landrini 2003) is applied every twenty time steps to restore consistency between particle mass, density and occupied area and to produce a more regular pressure distribution. A predictor-corrector time marching scheme is used to update the density, velocity and position of the particles.

3. Numerical Simulations A numerical wave flume was designed for

the SPH simulations. This consisted of a flat left section, which was 10m long and 2m high, and to the right of this, a 5° slope extended upwards until it intersected with the top of the flume. The simulation used approximately 160,000 particles, which were initially placed on a Cartesian grid with particle spacing, p∆ = 0.625cm. Any particles placed less than one particle spacing above the beach were removed to avoid instabilities at the beginning of the simulation. The depth of water, d, was 0.5m, so particle resolution to depth ratio was d/80 and the smoothing length was set at 1.33 p∆ . The time step was 7.8125×10−5s. This was chosen to meet the SPH CFL condition (Monaghan & Kos 1999) and to coincide with the recording of experimental data.

Boundaries were represented by lines of particles that exert repulsive forces on the fluid particles. Free-slip boundary conditions were enforced using the boundary method developed by Monaghan et al. (2003). The left wall of the flume acted as a numerical wavemaker and was moved in such a way that a first order solitary wave was generated. Hague (2006) provides further details of this motion.

4. Experiment Solitary wave experiments were conducted

using a glass-walled coastal flume in the

149

Hydrodynamics Research Laboratory of the Department of Civil and Environmental Engineering at Imperial College London. The flume is 25m long, 0.5m wide and has a maximum depth of 1m . For the solitary wave experiments the water depth was set at 0.5m. A beach with a slope of 5° was built into the flume with the toe located 11.15m from the front of the wavemaker. The wavemaker was programmed to produce solitary waves according to Goring (1978).

A visualisation technique was used to measure the profile of the wave as it travelled up the slope, throughout overturning and breaking. This involved the use of two computer controlled area scan cameras and sufficient lighting to obtain good quality images of the water surface and beach. Further details of this technique are given by Christou et al. (2007). Individual frames were selected and an edge detection technique developed by Canny (1986) was used to identify the water surface profile. The pixel coordinate system obtained from the digital photographs was then transformed to a 2-D spatial coordinate system. Previous applications of this technique found that it was accurate to ±2mm (Spentza 2006).

The maximum run-up on the slope and the profile of the hydraulic jump were measured using parallel, surface-piercing, wire-resistance gauges, the accuracy of this system being ±1mm.

5. Results and discussion In order to make the numerical and

experimental wave surface profiles comparable to one another, it was necessary to consider the effects of viscous dissipation in both sets of results. Although the amount of viscous dissipation was reduced in the SPH model by using equation (4) in the artificial viscosity, preliminary simulations without a slope in the flume showed evidence that some dissipation was still occurring. In the experimental flume, dissipation arises on both the glass walls and the bed. In both cases, the wave height decays in amplitude as the first order solitary wave theory fails to fully satisfy the nonlinear

boundary conditions. In the laboratory an iterative process was used, without the slope in the flume, in order to find the amplitude required to obtain the correct height to depth ratio at the toe of the slope. Once this had been found a similar procedure was used in the numerical wave flume.

The following surface profile comparisons have been made by aligning the experimental and the numerical waves in space and time, at the waves’ breaking point - where the front face of the wave is vertical. This was chosen as the most definitive point that occurs early in the breaking procedure.

The H/d = 0.45 surface profiles are shown at four different stages of breaking in figure 1. The results show good agreement in the shape of the surface profiles between the SPH and experimental results, thus highlighting the potential of the SPH model. However, there are some discrepancies in the results which must be addressed.

The process of aligning the experimental and SPH images meant that the experimental images were all moved 0.4m to the right, along the slope. The SPH model uses free-slip boundary conditions that neglect friction on the slope. This has the effect of allowing water to move up the slope more freely, thus increasing the still water level, decreasing the height to depth ratio and pushing the breaking point of the wave further up the slope. Figure 1(a) shows that as the wave form steepens the experimental wave becomes larger than the SPH wave. For this reason, the experimental wave projects further forwards, as can be seen in figure 1(b). This results in the jet impacting further up the slope (figure 1(c)) and taking longer to complete the overturning process compared to the SPH wave.

150

Figure 1: Solitary wave with H/d = 0.45 breaking on a 5° slope.

Previous results using this SPH model

have shown some improvement in the overturning wave profile with increased resolution. Surface tension is not included in the SPH model, which may be important at this stage of breaking. Figure 1(d) shows air entrained under the overturned wave. Modelling the entrained air with SPH particles may also improve the results at this stage of breaking.

The maximum run-up was measured for the solitary wave with H/d = 0.3, giving the results as 418mm above still water level in the experiment and 753mm in the SPH simulations. Although this appears to be a significant discrepancy, this result is in large part due to the neglect of friction on the slope in the SPH model, as was noted in observations of the breaking wave above. It would also be beneficial to use the visualisation technique to better assess the experimental maximum run-up.

Images of the hydraulic jump are shown in figure (2) for the solitary wave with H/d = 0.45. Both the experimental and SPH results show the water reaching a similar height. The SPH hydraulic jump occurs further up the slope due to the reasons discussed earlier in this paper.

More results will be given in the presentation.

Figure 2: Hydraulic jump on 5° slope H/d = 0.45

6. Conclusions and Further work The SPH model described in this paper

has been used to simulate solitary waves

151

breaking on a slope. Good agreement with experiment was found in the surface profiles at all stages of breaking, however, some discrepancies existed between the timing and location of the breaking procedure. These were thought to occur because of possible differences in the height to depth ratio in combination with various SPH modelling techniques which could be improved upon in the future.

Future improvements to the SPH model are likely to include extension to 3-D; modelling of entrained air; properly including structures with appropriate boundary conditions; and introducing a turbulence model. These aspects of the model development are presently ongoing.

7. Acknowledgments The authors would like to acknowledge the

UK Engineering and Physical Sciences Research Council (EPSRC) funding and would also like to thank Jannicke Roos and Marios Christou for their experimental measurements.

8. References BATCHELOR, G. K. (1974). An Introduction to Fluid Mechanics. Cambridge University Press.

CANNY, J. (1986). A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-8 (6), pp. 679–698.

CHRISTOU, M., SWAN, C. & GUDMESTAD, O. T. (2007). The description of breaking waves and the underlying water particle kinematics. Proceedings of the 26th International Conference on Offshore Mechanics and Arctic Engineering.

COLAGROSSI, A. & LANDRINI, M. (2003). Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J. Comput. Phys. 191 (2), 448–475.

GORING, D. (1978). Tsunamis - the propagation of long waves onto a shelf. Tech.

Rep. KH-R-38, Laboratory of Hydraulics and Water Resources, California Institute of Technology.

HAGUE, C. H. (2006). Fully nonlinear computations of directional waves, including wave breaking. PhD thesis, Imperial College London.

MONAGHAN, J. J. (1994). Simulating free surface flows with SPH. Journal of Computational Physics 110 (2), 399–406.

MONAGHAN, J. J. & KOS, A. (1999). Solitary waves on a Cretan beach. Journal of Waterway Port Coastal and Ocean Engineering-ASCE 125 (3), 145–154.

MONAGHAN, J. J., KOS, A. & ISSA, N. (2003). Fluid motion generated by impact. Journal of Waterway Port Coastal and Ocean Engineering, ASCE 129 (6), 250–259.

SPENTZA, E. (2006). Laboratory study of 3D breaking waves. Imperial College Internal Report.

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152

Dynamic Boundary Particles in SPH

A. J. C. Crespo1, M. Gómez-Gesteira1 and R.A. Dalrymple2

1 Grupo de Física de la Atmósfera y del Océano, Universidad de Vigo, Ourense, Spain;

alexbexe@uvigo.es; mggesteira@uvigo.es 2 Department of Civil Engineering, Johns Hopkins University, Baltimore, USA; rad@jhu.edu

Abstract

The foundation and properties of the so called dynamic boundary particles (DBPs) are described in this paper. These boundary particles share the same equations of continuity and state as the moving particles placed inside the domain, although their positions and velocities remain unaltered in time or are externally prescribed. Theoretical and numerical calculations were carried out to study the collision between a moving particle and a boundary particle. In addition, a dam break confined in a box was used to check the validity of the approach. The good agreement between experiments and numerical results shows the reliability of DBPs.

1 Introduction Since the first applications of the SPH

method to hydrodynamical problems considerable effort has been devoted to the boundary conditions. Actually, the boundaries are constituted by particles that exert repulsive forces on fluid particles. Thus, central forces are a natural choice, although, a better approach can be obtained by means of an interpolation procedure [Monaghan and Kos (1999)].

The aim of this manuscript is the study of the role of the so called Dynamic Boundary Particles (DBPs from now on). These particles share the same properties as the fluid particles. They follow the same equations of state and continuity, but they are not allowed to move or they move according to some external input.

2 Boundary conditions

The boundary conditions do not appear in a

natural way in the SPH formalism. When a particle approaches a solid frontier, only the particles located inside the system intervene without any interaction from the outside. This contribution can generate unrealistic effects, due to the different nature of the variables to solve, since some ones, as the velocity, fall to zero when they approach the boundaries, while others, as the density, not. The different solutions to avoid boundary problems consist on the creation of several virtual particles that characterize the system limits. Basically, three different types of particles can be distinguished:

Ghost particles. Randles and Libersky

(1996) considered boundary particles whose properties, included their position, vary each time step. When a real particle is close to a contour (at a distance shorter than the kernel smoothing length) then a virtual (ghost) particle is generated outside of the system, constituting the specular image of the incident one. Both particles have the same density and pressure, but opposite velocity. Thus, the number of boundary particles varies in each time step, which complicates its implementation in the code.

Repulsive particles. This type of boundary particles is due to Monaghan (1994). In this case the particles that constitute the frontier exert central forces on the fluid particles, in analogy with the forces among molecules. Thus,

153

0.8 1 1.2 1.4 1.6 1.8 21000

1100

1200

1300

Position/h

Dens

ity (k

g/m

3 )

0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4x 104

Position/h

Pres

sure

(N/m

2 )

0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

Position/h

NPT

(a)

(b)

(c)

for a boundary particle and a fluid particle separated a distance r the force for unit of mass has the form given by the Lennard-Jones potential. In a similar way, other authors [Peskin (1977)] express this force assuming the existence of forces in the boundaries, which can be described by a delta function.

Dynamic particles. These particles verify the same equations of continuity and of state as the fluid particles, but their position remains unchanged or is externally imposed. An interesting advantage of these particles is their computational simplicity, since they can be calculated inside the same loops as fluid particles with a considerable saving of computational time. These particles were first presented in [Dalrymple and Knio, (2000)] and used in further studies on the interaction between waves and coastal structures [Gómez-Gesteira and Dalrymple (2004); Gómez-Gesteira et al. (2005), Crespo et al. (2007a)]. However, as far as we know, the properties of these particles have been not considered in detail.

3 Dynamic boundaries

3.1 Repulsion Mechanism

The boundaries exert a force to the fluid

particles when approaching. In order to analyze the fluid particles movement due to boundary particles, a schematic system composed by two particles, a boundary particle and a fluid one, was considered. This repulsion mechanism is explained in Crespo et al. (2007b)

3.2 Test problem 1: Particle movement inside a box

A simple test corresponding to the

movement of a single particle inside a box was considered to depict the main features of the interaction between moving and boundary particles. In spite of the schematic nature of the test, it proves that the particle can be kept inside the box due to the repulsive force without losses in the mechanical energy of the system.

Different tests were carried out with numerical model to study the evolution of a single particle inside a box (0.5 x 0.5 m).

In the Z axes the distance will be measured from the boundary particles. The first experiment was the fall of a particle from (X0, Z0) = (0.25, 0.3) m without initial velocity and zero viscosity (α=0). The particle was initially far from the boundaries, in such a way that gravity was the only initial force on the particle. This particle does not feel the interaction of the boundary particles until it approaches the bottom of the box. Figure 1 shows the repulsion mechanism.

Figure 1: Variation of density (a), pressure

(b) and normalized pressure term (c) for a moving particle approaching a solid boundary.

The incoming particle, a, increases the

density locally (Fig. 1a), which results in an increase in pressure (Fig. 1b) and in an increase of the pressure term ( 2ρP ) (Fig. 1c).

154

0

1

2

3

4

5

0 0.5 1t (s)

X (m

)

0

0.5

1

1.5

2

0 0.5 1 1.5t (s)

H (m

)

The normalized pressure term, ( ) ( )Rzz PPNPT 22 / ρρ= , is represented in Fig.

1c, where z refers to the distance from the incoming particle to the wall and R to the return point of the incoming particle.

Note how the fluid particle suffers the effect of the boundary when the distance particle boundary is shorter than 2h.

3.3 Test problem 2: Collapse of a

water column

Once the main properties of DBPs have been described in the previous oversimplified test case, DBPs will be used in a more realistic test. It consists in the collapse due to the gravity of a 2m high 2D water column in a tank. A complete description of the experiment is given by Koshizuka and Oka (1996). The same setup was used by Violeau and Issa (2006) to check the accuracy of their SPH code. The tank is 4m long, the initial volume of water is 1m long and its height 2m. A smoothing length, h = 0.012 m and a viscosity term, α=0.5, were considered.

This laboratory test case will allow

checking different properties of DBPs, namely, the fluid movement parallel to the left wall and bottom and the fluid collision against the right wall.

On the one hand, the movement of the fluid inside the box depends on the interaction between the fluid and the boundary apart from the geometrical constraints of the initial water parcel. Thus, a proper boundary treatment will generate a realistic water height decrease near the left wall and an accurate water velocity near the dam toe. On the other hand, the boundaries must prevent fluid escape through the right wall, which suffers the most energetic water collision in the experiment.

Water height (H) decrease near the left wall and dam toe advance (X) prove the properly behavior of boundary conditions. Fig. 2 shows how H and X fit data provided by Koshizuka and Oka (1996) experiment in an accurate way.

Experimental points were digitized from Violeau and Issa (2006).

Figure 2: Collapse of a water column in a

tank simulated with SPH model (solid line) comparing with experimental data (circles).

4 Summary The validity of the method has been

checked in an oversimplified geometry where a single particle impinges a boundary. The moving particle is observed to bounce due to the local increase of pressure terms in momentum equation. Thus, the boundaries retain the main features of the physical process.

The validity of the approach has also been checked in a dam break experiment. There, DBPs prevent fluid to leave the container and guarantee a proper water movement close to the walls.

Finally, DBPs can also be applied to mimic obstacles inside the computational domain and solid boundaries whose movement is externally imposed. In particular, DBPs have been used to generate wave mitigating dikes [Crespo et al. (2007a)], sliding doors [Crespo et al. (2007c)] and wavemakers [Crespo et al. (2007d)].

Acknowledgments

This work was partially supported by Xunta

de Galicia under proyect PGIDIT06PXIB383285PR.

155

References

CRESPO, A. J. C., GÓMEZ-GESTEIRA, M. and DALRYMPLE, R.A. (2007a). 3D SPH simulation of large waves mitigation with a dike. Journal of Hydraulic Research. In press. CRESPO, A. J. C., GÓMEZ-GESTEIRA, M. and DALRYMPLE, R.A. (2007b). Boundary Conditions Generated by Dynamic Particles in SPH Methods. Submitted to Computers, Material and Continua. CRESPO, A. J. C., GÓMEZ-GESTEIRA, M. and DALRYMPLE, R.A. (2007c). Modelling dam break behavior over a wet bed by an SPH technique. Submitted to J. Wtrwy. Port,Coastal and Ocean Engrg. CRESPO, A. J. C., GÓMEZ-GESTEIRA, M. and DALRYMPLE, R.A. (2007d). Hybridation of generation propagation models and SPH model to study extreme wave events in Galician Coast. Submitted to Journal of Marine Systems. DALRYMPLE, R.A. and KNIO, O. (2000). SPH Modelling of Water Waves. Proc. Coastal Dynamics, Lund, 779-787 GÓMEZ-GESTEIRA, M. and DALRYMPLE, R.A. (2004). Using a 3D SPH Method for Wave Impact on a Tall Structure, J. Wtrwy. Port,Coastal and Ocean Engrg.,130,63-69. GÓMEZ-GESTEIRA, M., CERQUEIRO, D., CRESPO, C. and DALRYMPLE, R.A. (2005): Green water overtopping analyzed with a SPH model, Ocean Engineering, 32: 223-238. KOSHIZUKA, S. and OKA, Y. (1996). Moving-particle semi-implicit method for fragmentation of compressible fuid. Nuclear Science Engineering. 123, 421-434. MONAGHAN, J. J. (1994). Simulating free surface flows with SPH. Journal Computational Physics, 110: 399- 406. waves. Physica D., 98: 523-533.

MONAGHAN, J. J. and KOS, A. (1999). Solitary Waves on a Cretan Beach. J. Wtrwy. Port, Coastal and Ocean Engrg., 125: 145-154. PESKIN, C. S. (1977). Numerical analysis of blood flow in the heart. Journal Computational Physics 25, 220- 252. RANDLES, P.W. and LIBERSKY, L.D. (1996). Smoothed Particle Hydrodynamics – some recent improvements and applications. Comput. Methods Appl. Mech. Eng., 138, 375- 408. VIOLEAU, D. and ISSA, R. (2006). Numerical Modelling of Complex Turbelent Free-Surface Flows with the SPH Method: an Overview. Int. J. Numer.l Meth. Fluids 53, 277-304.

SPHERIC

156

Accuracy and stability of numerical schemes in SPH

Tatiana Capone 1 Andrea Panizzo 2 Claudia Cecioni 3 and Robert A. Darlymple 4

1, 2 University of Rome “Sapienza”, DITS department, Rome, Italy. tatiana.capone@uniroma1.it

3 University of Roma Tre, DSIC department, Rome, Italy. 4 John Hopkins University, Baltimore, USA.

Abstract A correct approximation of Navier Stokes Equations (NSE) by means of the meshless SPH particle approach requires an accurate estimation of the gradient of a field variable and, in certain presented SPH models, of second derivatives. Looking at numerical accuracy, it is also important to employ a high order integration scheme to move the particles position in time. At these purposes, several different integration schemes, different type of kernel functions and also the normalization of kernel and its gradient, are here considered and compared. Simple two and three dimensional benchmark tests with periodic fluid flow are considered to assess the different employed numerical techniques. Second derivatives of the velocity field are necessary when evaluating viscous terms of the momentum equation. At this aim, this work considers three different approaches. Benchmark fluid flow considered here consist in periodic fluid flows, such as a periodic shear flow.

1. Introduction The aim of this paper is to test the

accuracy and stability of SPH in reproducing some test cases. In particular in the evaluation of: - pressure gradients (Euler equation); - second derivatives of velocity (viscous terms

in NSE). Several different kernel functions are

considered, namely the Gordon Jonhson

(Johnson et al. 1996), the Cubic Spline (Monaghan and Lattanzio, 1985), the Quintic Spline (Morris, 1996a), the Wendland (Wendland 1995), and the Gaussian (Morris, 1996b) functions. About numerical schemes, the Euler, Beeman, Verlet, Velocity Verlet are here applied. We also propose a new version of the Beeman algorithm with a predictor corrector modification that provides accuracy of the 4th order both in velocity and position, and a two step Velocity Verlet algorithm (Monaghan, 2006b). Finally the presence or not of corrected kernel gradients (CSPH, Bonet and Lok, 1999), are tested.

To test the accuracy of SPH in evaluating the gradient of a given flow variable, namely pressure, two dimensional tests are performed assuming an oscillatory motion on both x and y axes. It is prescribed that a single particle moves according to the following equations:

)sin(0 txx ω+= ; )2sin(5.00 tyy ω+= (1)

Which can be obtained from a variation of the pressure field using Euler equations, such as::

))2sin(2)sin((),( 2 tytxyxP ωωρω += (2)

Three dimensional tests are obtained imposing a motion on x, y and z axes, following the so called “Noeud de Trèfle” 3D curve, that is represented by:

)2cos(2)cos(0 ttxx ωω ++= (3) )2sin(2)sin(0 ttyy ωω −+= (4)

)3sin(20 tzz ω+= (5)

157

To get such an oscillatory movement, the pressure field must vary accordingly, such as:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+−++

⋅=)3sin(18

))2sin(8)(sin())2cos(8)(cos(

),(t

ttyttx

yxPω

ωωωω

ρωω (6)

Considering a compressible fluid the continuity equation and the Navier Stokes Equations, are:

01

=⋅∇+ vDt

Dρ

ρ (7)

( )vvgv⋅∇∇+∇++−∇= µµρρ

312P

DtD (8)

The viscous term present in equation (8), has been implemented following three different approaches for the evaluation of second derivatives, namely: 1. The first considered formulation is that proposed by Takeda et al. (1994), solving the second derivative of the velocity using a Gaussian kernel: (9)

( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅+

∂∂

⎟⎠⎞

⎜⎝⎛ +

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅−=⋅∇∇+∇

ijijijijijijij

ijij

ijijij r

Wrrrr

Wr

n

ji

jmim

ii

113

131 2

312

xvxxv

v

vvρρ

µρµ

2. The second one is to solve the second derivative of the velocity combining a standard SPH gradient estimation with a finite difference approximation of a first derivative, following the work of Brookshaw (1985) and Morris et al. (1997):

( )( )∑

+

∇⋅+=∇⋅∇

⎭⎬⎫

⎩⎨⎧

⎟⎠

⎞⎜⎝

⎛j ijhijijij

ijWiijjijm

iv

r

rv 201.02

1

ρρ

µµµ

ρ

(10)

3. The last approach avoids direct calculation of second derivatives, by using only first derivatives approximation, as introduced by Watkins et al. (1996):

( ) ( )

( )( )i

ii

v

vvv

×∇×∇−

⋅∇∇=⋅∇∇+∇ ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

µ

µµ3

4

3

12 (11)

In order to evaluate the accuracy in calculating the second derivatives, a simple two dimensional test was considered. It consists in a periodic flow in a 2x1 box with initial velocity defined as v=(sin(2πy);0.0). Periodic boundary

conditions are implemented both in x and y directions. Initial particle spacing ∆ is taken as 0.05 and density ρ0 as 1.0. To avoid initial zero pressure, initial density value is set to 1.02 (Monaghan, 2006a).

2. Results and discussion In Figure 1 and 2 are represented, in terms

of particle trajectory, respectively the worst and the best results for the 2D tests about the accuracy of different integration schemes and kernel functions. The worst result is related to the direct Euler integration scheme with the use of the Gordon Johnson kernel. The best one is related to the Beeman integration scheme with the use of Wendland kernel.

Figure 1: Particle trajectory for the worst

obtained result in 2D. Dots for analytical and lines for numerical solutions.

Figure 2: Particle trajectory for the best

obtained result in 2D. Dots for analytical and lines for numerical solutions.

About 3D tests, the worst and the best

result are represented in Figure 3 and 4, and they are related to the Verlet integration scheme with Quintic spline kernel and to the Beeman integration scheme with the aim of Wendland kernel.

158

Figure 3: Particle trajectory for the worst

obtained result in 3D. Dots for analytical and lines for numerical solutions.

Figure 4: Particle trajectory for the best

obtained result in 3D. Dots for analytical and lines for numerical solutions.

About the performed periodic shear flow,

we present results at 1 second of simulation. Figure 5 shows the particles positions, while Figures 6 and 7 present respectively the density and the velocity field as a function of y. It can be seen that there are small fluctuations in density and that the velocity field is correctly evaluated.

Results of the estimation of second derivatives of the velocity field are presented in Figures 8, 9 and 10. The best agreement with the analytical solution is obtained using the Watkins’ method, which presents a small underestimation of peak values.

Figure 5: Particles positions at t=1 sec.

Figure 6: Density as a function of y at t=1 sec.

Figure 7: Velocity field at t=1 sec.

Figure 8: Second derivative of the velocity field

at t=1 sec, Watkins method.

159

Figure 9: Second derivative of the velocity field

at t=1 sec, Morris method.

Figure 10: Second derivative of the velocity

field at t=1 sec, Takeda method.

3. Conclusions The accuracy and stability of several

numerical techniques implemented in SPH have been tested in this work. Several kernel functions and integration schemes have been compared, and obtained results show that the Wendland kernel function with the Beeman’s algorithm provide the best agreement with analytical solutions. It is to be stressed that the Wendland function does not require a tensile instability correction.

About calculation of second derivatives, three different techniques have been considered and tested. Best results have been obtained using the Watkins et al. (1996) method.

Results and suggestions presented above have been used to model further 2D flows, which will be presented at the workshop.

4. References BONET, J. and Lok, T.-S.L. (1999). Variational and momentum preservation aspect of Smooth Particles Hydrodynamic formulation. Comp. Methods Appl. Mech. Engrg, 97-115.

BROOKSHAW, L. (1985). A method of calculating radiative heat diffusion in particle simulations. Proc. Astron. Soc., Vol. 207. COLAGROSSI, A. (2004). A meshless lagrangian method for free surface and interface flows with fragmentation. PhD thesis, University of Rome “Sapienza”. JOHNSON, G. R. et al. (1996). SPH for high velocity impact computations. Comp. Meth. Appl. Mech. Eng. Vol. 139, 347-373. MONAGHAN, J. J. and Lattanzio, J. C. (1985). A Refined Method for Astrophysical Problems. Astronomy and Astrophysics 149, 135-143. MONAGHAN, J. J. (2006a). Smoothed particle hydrodynamic simulations of shear flow. Monthly Notices of the Royal Astr. Society, Vol. 365, No. 1, 199-213. MONAGHAN, J. J. (2006b). Time stepping Algorithms for SPH. Internal report, Monash University. MORRIS, J. P. (1996a). Analysis of Smoothed Particle Hydrodynamics with applications. PhD thesis, Monash Univ. MORRIS, J. P. (1996b). A study of the stability properties of Smoothed Particle Hydrodynamics. Publ. Astr. Soc. Aust. 13. MORRIS, J. P. et al. (1997). Modeling low Reynolds number incompressible flows using SPH. Journal of Computational Physics 136, 214-226. TAKEDA, H. et al. (1994). Numerical simulation of viscous flow by smoothed particle hydrodynamics. Progress of Theoretical Physics, Vol. 92, No. 5. WATKINS, S. J. et al. (1996). A new prescription for viscosity in Smoothed Particle Hydrodynamics. Astron. Astrophys. Suppl. Ser. 119, 117-187.

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WENDLAND, H. (1995). Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Advances in Computational Mathematics, Vol. 4, No. 1, 389-396.

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Application of SPH-NSWE to simulate landslide generated waves

Andrea Del Guzzo1 and Andrea Panizzo2

1 DISAT Department, University of L’Aquila , Piazzale Pontieri 1, 67040 Monteluco di Roio, L’Aquila,(IT), phone: +39 0862434554; e-mail: delguzzo.andrea@ing.univaq.it,

2 DITS Department, University of Rome La Sapienza, Via Eudossiana 18, 00185 Rome (IT) andrea.panizzo@uniroma1.it

Abstract Recent tsunamis events gained the attention of the ocean engineering scientific community on better understanding the tsunamis wave propagation from tsunamigenic sources to coastal areas. In some cases, for instance when tsunamis are generated by subaerial and underwater landslides, the generation, propagation and interaction with the shoreline are almost coincident. To properly study this topic, a model that is able to simulate contemporarily the physics of generation, propagation and runup on the coastal region is of the utmost utility. The present paper shows some results of a 1D and 2D Smoothed Particle Hydrodynamics model solving the Non Linear Shallow Water Equations. If compared to standard SPH codes, this one has the main advantage of the reduced computational time, while keeping the desired accuracy in the simulation of tsunamis flooding coastal areas. A delicate point of scientific investigation, to which the present work is intended to shed light, is the accurate simulation of the fluid structure interaction occurring when the landslide moves generating free surface oscillations. To test the implemented numerical model, experiments carried out by Watts et al. (2001) and Bellotti et al. (2006) have been taken into account, and as a reference for numerical models validation.

1. Introduction The Smoothed Particle Hydrodynamics

(SPH) approach is here used to model Nonlinear Shallow Water Equations (NSWE) with the aim of setting-up a model to properly reproduce tsunamis waves and their interaction with the coast. The present study has been conceived within the framework of a wider research program on tsunamis waves, funded by the Italian Ministry of University and Education MIUR, by the Italian Civil Protection Agency and by the National Dam Office.

In the following we present the 1D and 2D versions of the SPH-NSWE numerical model. Then, SPH-NSWE is applied to simulate benchmark cases from laboratory studies, underlining the model’s points of strength and weakness. In particular, we apply the model in simulating a phenomenon where surely 3D like models, i.e the classic SPH and the VOF, should be more properly indicated to simulate a complex three dimensional flows. Indeed we show that SPH-NSWE results reasonably agree with experimental observations, at a very low computational time. Finally, conclusions and brief remarks on ongoing researches are discussed.

2. NSWE in the SPH approach Nonlinear Shallow Water Equations can be

written in different forms. In the SPH formulation of the NSWE, each particle represents a water column with a given volume and mass, kept constant during the simulation.

162

Figure 1 shows how the water column and its physical variables are represented in the lagrangian particle approach. Particles (water columns) interaction is governed by a variable smoothing length li,n, necessary to ensure the correct number of neighbors also where the water height evolves far from the initial condition (Eq.1).

dm

nitot

oitotoini d

dll

/1

,

,,, ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛= (1)

dtoti,0 and dtoti,n are the particle’s total depths, while li,0 and li,n are the smoothing lengths at the beginning of the simulation and at the generic time n. dm is the number of dimensions. In the SPH formalism, the lagrangian form of NSWE equations for the variation in time of dtoti can be evaluated with Eq.(2), ωi being a correction factor that appears in the variable smoothing length formulation.

Figure 1: Lagrangian sketch of particles representing water columns in the NSWE-SPH approach.

( ) ( )ij,1

),( vvx −⋅∇= ∑=

iijijNj

ji

itotitot lWVdmd

dtdd

ω

ij

iij

Njji dr

dWrV∑=

−=,1

ω (2)

The lagrangian form in the SPH formalism of momentum equation in x and y direction is written as:

( ) )(.1.1

fxij

ijNj

jtot

ijij

Njj

i gSx

WhhV

dg

xW

gVDtDu

i

−∂

∂−+

∂

∂Π+−= ∑∑

==

( ) fyij

ijNj

jtot

ijij

Njj

i gSy

WhhV

dg

yW

gVDtDv

i

−∂

∂−+

∂

∂Π+−= ∑∑

==

)(.1.1

πij

being an artificial viscosity introduced by Monaghan (1992).

At the generic time step n, the updated value of the smoothing length is evaluated using Eq. (1), while the total water depth dtot is obtained using a Newton-Raphson iterative process, such as:

( ) ,)(

e

.1

n

,1,

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡∇⋅−

∆=

=

∑=

+

iijijijNj

ji

tottot

lWVtdm

ddnini

xvvω

λ

λ

. Velocity and position are finally updated using a second order accurate and explicit Newmark scheme.

3. Simulation of landslide generated waves

Here we show the application of the implemented NSWE-SPH numerical model to simulate two benchmark test cases from laboratory studies. The first is a study on underwater landslide generated water waves performed by Watts et al. (2001). The second is a study on landslide generating waves performed by Bellotti et al. (2006). The experimental study described by Watts et al. (2001), consisted of water waves generated by the movement of an underwater landslide sliding down an inclined ramp. The underwater landslide was modeled as a solid body with a semi-elliptical shape, considering a 1:1000

163

scaling factor with a potential real landslide and the Froude similarity. The equation

s(t) = soln[cos(t/to)]

describes the landslide movement, where so = ut

2/ao and to = ut/ao. ao and ut denote the landslide initial acceleration and terminal velocity respectively, while the maximum movement of the underwater landslide is equal to so and occurs in a total time to. Four wave gages were used to measure the water level oscillations, the first gage being located at x = xg, i.e. just above the center of mass of the underwater landslide when it starts moving, and the others set 0.30 m apart (300 m in real scale) with increasing x. The landslide effect has been simulated in the model modifying the continuity equation using the reduction factor of the landslide thickness, expressed as a function of the water depth as proposed by Tinti et al. (2005). Test case c2 (see Watts et al., 2001) is here considered, and presents a beach slope angle θ equal to 15° with the horizontal, the slide length b = 1.0 m, its thickness T = 0.0518 m, the starting depth of the slide d = 0.259 m, and the starting position xg = 1.166 m. This 1D test was modeled with NSWE-SPH using 238 particles with different volumes that remain constant during the simulation ensuring the same model resolution at different water depths. In fact, working with the same volume for all the particles in the domain generate a very low numerical resolution in shallow water. In our simulation we applied a quintic-spline kernel function, while a constant and small enough time step (∆t=0.0002 s) was chosen to respect the CFL condition. The gradient kernel normalization proposed by Bonet and Lok (1999) was applied to assure the conservation of angular momentum, and to get a correct evaluation of the kernel gradient. Plots in Figure (2) present the comparison between the time series of water surface elevation at x = 1.166 m, x = 1.466 m and x = 1.766 m from the shoreline at the left end of the wave flume, measured (dots), simulated with NSWE-SPH (continuous line) and 2DSPH

(Panizzo and Dalrymple, 2004). The agreement with experimental data is good considering both NSWE-SPH and 2DSPH results. About NSWE-SPH results, the fit is better at the first two wave gages, placed in shallower water with respect to the third wave gage. For the last signal, as we expected, the 2DSPH model is able of more accurately reproducing the water surface elevation than the NSWE-SPH model. Though, the carried out simulations demonstrate that it is possible to reproduce with good accuracy the problem at hand using a depth averaged model and saving as well a considerable amount of computation time. It is worth to notice that NSWE-SPH runs in few minutes on a standard laptop computer, while classic 2DSPH runs in several hours. The 2D version of the SPH-NSWE model has been applied to reproduce the physical experiment on landslide generated waves carried out at the LIAM laboratory by Bellotti et al. (2006). Some qualitative and preliminary results are reported in Fig. (3).

Figure 2: 2D test: comparison between time series of water surface elevation.

164

Figure 3: 3D test: snapshots of some preliminary results.

4. Conclusions A numerical model based on Nonlinear Shallow Water Equations (NSWE), solved by means of the Smoothed Particle Hydrodynamics (SPH) approach, has been presented. The numerical model has been developed with the aim of simulating the generation and the propagation of tsunamis waves and their run-up at the coast, and it has been implemented within the framework of a research program on tsunamis waves currently carried out at the LIAM laboratory of L’Aquila University (for more details see www.tsunamis.it). Indeed, the classic set of NSWE equations has been rewritten using the SPH formalism. Details on the implemented numerical model have been given and the model mathematics explained. The model applications to physical model tests

on landslide generated waves have been presented. Despite these phenomena are three-dimensional and may be properly studied with fully 3D numerical models such as the VOF or 3DSPH, we showed that accepting some small approximations, also a 2D Lagrangian and depth integrated numerical model is able to correctly simulate the considered phenomenon with a satisfactorily accuracy and with the main advantage related to very low computational costs.

5. Acknowledgments This work has been funded by MIUR-PRIN, and by the Italian Government, Department of Protezione Civile. The authors wish to thank Prof. Paolo De Girolamo, who is the scientific coordinator of the research project, dr Giorgio Bellotti and Marcello Di Risio, for the valuable suggestions.

6. References BELLOTTI, G., DI RISIO M., PANIZZO A.,

DE GIROLAMO P., “Tsunami waves generated by landslides along a straight sloping coast: new three-dimensional experiments.” Proc. of 30th Int. Conf. on Coastal Eng., ASCE, (2006).

MONAGHAN, J.J “Smoothed particle

hydrodynamics”, Annual Rev. of Astr. and Astrophysics, 30: 543-574, (1992).

WATTS P., IMAMURA F., BENGSTON A.,

GRILLI S., “Benchmark cases for tsunamis generated by underwater landslides.” Proc. Waves, ASCE, Vol. 2, 1505-1514 (2001).

PANIZZO A., DALRYMPLE, R. A., “SPH

modelling of underwater landslide generated waves.” Proc. of 29th Int. Conf. on Coastal Eng., ASCE, (2004).

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A new treatment of solid boundaries for the SPH method

J.C. MARONGIU1, F. LEBOEUF2 and E. PARKINSON3

1 PhD student, Laboratory of Fluid Mechanics and Acoustics / Ecole Centrale de Lyon Jean-christophe.marongiu@ec-lyon.fr

2 Professor, Laboratory of Fluid Mechanics and Acoustics / Ecole Centrale de Lyon Francis.leboeuf@ec-lyon.fr

3 Doctor, Head of Pelton CFD team / ANDRITZ VA TECH Hydro Company Etienne.parkinson@vatech-hydro.ch

Abstract A general treatment for solid boundary conditions for the SPH method is presented. It is adapted to complex geometries and restores mathematical consistency of the interpolation scheme in the vicinity of the boundary. The determination of boundary conditions is performed accurately using a characteristic formulation of the system of Euler equations.

1. Introduction The numerical method SPH was

initially designed to compute astrophysical phenomena and thus the standard SPH formalism assumes that the computation domain is unbounded. The use of SPH in bounded domains has hence been a key issue since its origin. Free surface condition is usually naturally satisfied thanks to the lagrangian formalism and the zero valued pressure condition, which explains the keen interest in the SPH method to simulate free surface flows. But to our knowledge a general treatment of solid boundaries has not yet been successfully addressed.

There exist three major usual treatments to model solid boundaries in SPH: repulsive wall forces, ghost particles and fictitious particles. The first model

guarantees non penetration of fluid particles through the wall but doesn’t compensate for the lack of neighboring particles in the vicinity of the frontier and thus fails to restore the mathematical consistency of the interpolation scheme. The resulting model has very poor accuracy.

The two other models aim at balancing forces across the solid boundary by appropriate symmetry of flow patterns. But they are unable to handle complex geometries with sharp edges and highly curved surfaces.

First a new numerical scheme to compute the discrete equations of motion in the vicinity of the frontier is presented. Then the field values at the solid wall are determined using the natural propagation of information from the fluid domain to the boundary.

2. Surface integral term The standard SPH interpolation

scheme to compute the gradient of a function is obtained after integration by parts and is written in the continuous form:

)1()()(

)()()(

∫∫

Ω

Ω∂

−∇−

−=∇

dyyxWyf

dSnyxWyfxf r

166

In the standard SPH formalism, the surface term is always neglected because of the choice of a kernel function with compact support, allowing its contribution to be zero for a particle inside the domain far from the boundaries. But for a particle close to the boundary and a non-zero Dirichlet boundary condition, this term can play an important role. Instead of extending the computational domain on the other side of the boundary in order to avoid that term, another approach could be to directly compute this integral surface term as we just need an appropriate discretization of the boundary surface. The numerical scheme is hence corrected by a discrete approximation using the same quadrature formula as the interior scheme, for example for the pressure gradient (using the symmetric expression proposed by Monaghan, 1992):

)2(

1

22

22

∑

∑

∈

∂∈

∇⎟⎟⎠

⎞⎜⎜⎝

⎛+−

∆⎟⎟⎠

⎞⎜⎜⎝

⎛+=∇

i

i

Djij

j

j

i

ij

Djj

j

ij

j

j

i

ij

i

WPPm

nr

WPPmP

r

r

ρρ

ρρρ

where

jr∆ is the size of the wall particle

j and iD∂ is the intersection of

iD with

Ω∂ .

3. Characteristic formulation

The expression (2) can be evaluated provided the pressure values at wall particles are known. A one-sided approach is compulsory and in order to solve correctly the interactions between fluid and wall particles, the concept of characteristics is essential. The hyperbolic nature of the system of Euler equations means that there exist some waves propagating in the fluid domain along characteristic directions in space and time and transporting characteristic information. The idea is hence to determine the field values at wall particles by using those

waves propagating from the fluid to the solid boundary. This is done in three major steps. We first compute the eigenvalues and eigenvectors of the system of equations and select the characteristic directions corresponding to waves impacting the boundary. We then perform a projection of the equations of motion on the selected eigenvectors to obtain the compatibility relations. Finally we combine the selected compatibility relations with the imposed physical boundary condition to get a fully determined set of equations.

3.1. One dimensional approach

We first consider a one dimensional approach along the direction normal to the boundary surface (Marongiu et al, 2007). For a wall particle attached to a stationary solid boundary, the compatibility relation associated to the wave traveling from the fluid to the boundary is given by:

)3(.c

ngnu

nc

dtd n

rrρρρρ−

∂∂

−∂∂

=

where the subscript n denotes the

direction normal to the boundary, gr is the vector of gravity forces and c is the local speed of sound. This compatibility relation is numerically computed through a finite difference scheme using four interpolation points regularly distributed in the fluid domain along the normal direction (figure 1). The

boundary condition 0=nu is enforced

at the wall particle. Validation This method was first applied to the case of a jet of water impinging normally a flat plate. The figure 2 gives a comparison of the pressure coefficient profiles obtained with the numerical commercial code CFX, experimental data, and SPH using

167

repulsive wall forces and this new treatment.

Figure 1: Interpolation points for the finite

difference scheme.

It is clearly seen that the new treatment gives results very close to those obtained with CFX and brings a major improvement compared to the model of repulsive wall forces. The discrepancy between CFX results and experimental data comes from the wake of the injector which is not taken into account in numerical simulations.

Pressure coefficient profiles

-0,200

0,000

0,200

0,400

0,600

0,800

1,000

1,200

0,000 0,500 1,000 1,500 2,000 2,500

Y/D [-]

Cp [-

]

experimental CFXSPH new model SPH model of boundary particles

Figure 2: comparison of pressure

coefficient profiles

The second test case consists of a jet of water impinging a real Pelton bucket. This case is much more challenging because the geometry is complex. The figure 3 gives the pressure coefficient on the bucket surface. Unphysical overestimation of the pressure value at the leading edge together with a small separation bubble downstream is observed. In other zones of the bucket, the global behaviour of the pressure coefficient is satisfactory: the maximal pressure value is obtained where the curvature of the boundary surface is maximal. But the values are overestimated again. As could be expected the one dimensional approach does not seem sufficient to predict correctly the pressure on the wall surface in a case where the main component of the velocity is not normal to the boundary surface.

Figure 3: pressure distribution in the

bucket

3.2. Complete model

The multi-dimensional approach follows the same spirit as the previous one. Compatibility relations are obtained after a projection of the equations of motion on the eigenvectors. But in order to get a numerical scheme at the boundary similar to the volumic one, we don’t project the

168

continuous equations of motion but the discrete ones. Compatibility relations are hence naturally computed in a true SPH manner.

This complete model has been tested on a two dimensional case, the well-known breaking dam. Figure 4 shows that the new model is able to ensure a continuous transition of the pressure field between the fluid domain and the boundary. But to ensure non penetration of fluid particles across the boundary surface we had to apply a small repulsive force on fluid particles situated too close to the wall. This force is based on a fictitious wall particle whose location is at the projection of the considered fluid particle on the wall and whose pressure corresponds to the initial head of the water column. This artefact appeared compulsory to avoid some local unwanted numerical errors and to run the computation successfully. We believe however that it had a very little impact on the global results.

Figure 4: numerical computation of the breaking dam using the complete model to

treat solid boundaries

4. Conclusions We propose a new treatment for the solid boundaries in SPH which uses a consistent numerical scheme at the boundary without requiring any artificial

extension of the computational domain outside the boundary surface. Field values at wall particles are determined using a method of characteristics, which ensures accuracy. Compatibility relations are computed by projecting the discrete equations of motions, which gives a complete multi-dimensional model more in adequacy with the volumic scheme. But the free slip boundary condition leads to non-zero values of the tangential components of the velocity for wall particles while these particles are at rest (for a steady wall). Unfortunately in standard SPH formalism convective terms are expected to be fully integrated in the lagrangian derivative. These terms have hence to be computed separately, which leads to a small difference between fluid and boundary numerical schemes. A possible way to overcome this latter drawback could be the use of an Arbitrary Lagrange Euler variant of SPH as proposed by (Vila, 1999).

5. References MONAGHAN, J.J. (1992). Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys., 30: 543- 74. MARONGIU, J-C. LEBOEUF, F. PARKINSON, E. (2007) Numerical simulation of the flow in a Pelton turbine using the meshless method SPH – A new simple solid boundary treatment, to be published in Journal of Power and Energy. VILA, J-P. (1999) On particle weighted methods and smooth particle hydrdynamics, Mathematical Models and Methods in Applied Sciences.

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SPH simulation of local scour processes

Stefano SIBILLA

Associate professor, Dipartimento di Ingegneria Idraulica e Ambientale, Università di Pavia via Ferrata, 1 27100 Pavia Italy

phone: +39 0382 985320; fax: +39 0382 985589; e-mail: stefano.sibilla@unipv.it

Abstract A method to predict numerically scouring processes is discussed. The method is based on an SPH description of the flow, coupled with a conventional description of the bed evolution based on the Exner equation. The method is tested against measurements of bed evolution and water velocity obtained in the case of the local scour downstream of a protection apron caused by a planar jet flowing from a sluice gate. The results show that the predicted evolution of position and depth of the maximum scour agree quite well with the experimental data.

1. Introduction The prediction of the local scour holes

which occur downstream of hydraulic structures is important in order to avoid the risk of their collapse. The maximum scour depth and position need therefore to be estimated during the design process. Engineering practice has developed several formulas, based on laboratory and field measurements, to evaluate these quantities in different complex flows (Breusers and Radukivi 1991).

One of the simplest scouring pro-cesses concerns the effect of a 2D jet on a non-cohesive granular bed. A general understanding of the process and a quantitative identification of the relevant parameters has been obtained by Rajaratnam (1981). When the ratio between the tailwater depth yT and the jet width y0 is between 1 and 10, different scouring regimes can occur, depending on

yT/y0 and on the densimetric Froude number 500 / dgUFr ∆= , where U0 is the

inlet jet velocity, dG the mean sediment size and ∆ the relative density of sediments (Johnston 1990).

A detailed experimental description of the local scour caused by a 2-D horizontal wall-jet, issuing from a sluice and developing on a rigid apron before interacting with a non-cohesive sand bed, has been obtained by Espa and Sibilla (2006): the experimental data collected during these experiments, consisting of Laser Doppler (LDA) velocity measure-ments and bed profiles obtained by digital image processing, constitute an extensive database for the validation of numerical methods for scour prediction.

Among the possible techniques which can be adopted to simulate the eroding flow, Smoothed Particle Hydrodynamics (SPH) can be interesting owing to its capability to easily reproduce the interaction between the planar jet, the submerged hydraulic jump and the free surface (Monaghan 1992).

2. Numerical method

2.1. SPH approximation

The Navier-Stokes equations are here solved by SPH, in a weakly compressible fluid approximation, for the unknown velocity v

r and density ρ of a particle i:

( )∑ ∇⋅−−=j

ijijji Wvvm

DtDρ rrr

(1)

170

+∇⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−= ∑

jij

j

j

i

ij

i Wρ

p

ρpm

DtvD rr

22

gW)ττ(ρρm

ijijj ji

j rr+∇⋅−+∑ (2)

where p is pressure, gr gravity acceleration

and the summation is extended to all the fluid particles of mass mj located at a distance hxx ij 2<−

rr.

The kernel function ,h)xxW(W jiijrr

−=

here used, characterized by a smoothing length h, is the anti-cluster kernel proposed by Gallati and Braschi (2002).

The turbulent stresses τ are evaluated by a mixing length model, calibrated in order to correctly reproduce the submerged hydraulic jump which occurs above the protection apron (Gallati and Braschi 2003).

2.2. Smoothing of flow variables

The system (1-2) is solved according to the XSPH scheme (Monaghan 1992), where a smoothed value of velocity is obtained as:

∑∑+−=j

ijj

j

jij

Tj

j

jTi

Si W

mWv

mvv

ρρββ

rrr)1( (3)

β being a smoothing parameter, from the values Tv

robtained from the explicit

solution of (2). The smoothed values are then used in the explicit solution of (1), while Tv

ris used to compute particle

trajectories. Pressure smoothing has been applied

in a peculiar way, which has been found effective in the simulation of free surface flows for long integration times.

An approximation to the difference between local and hydrostatic pressures for particles j has been first computed:

( )jiTj

Tj zhgpp −−=∆ ρ (4)

and the smoothing procedure (3) has been applied to the pressure differences. After some algebra, the final expression for pressure smoothing with hydrostatic correction is:

[ ]

∑

∑ −+

=

jij

j

j

jijiji

Tj

j

j

Si

Wm

Wzzgpm

p

ρ

ρρ

)( (5)

2.3. Bed evolution

The choice of the proper model for the numerical description of the loose bed evolution is linked to the time scale needed for a complete description of the phenomenon. The experiments (Espa and Sibilla 2006) have shown that the flow and the bed evolve slowly and an equilibrium state is not reached even hours after the beginning of the scouring process.

SPH schemes based on the description of the solid bed as a granular material, which are normally employed in the description of granular motions such as debris flows (Rodriguez-Paz and Bonet, 2004), have been successfully used also in the description of scouring processes in rapid unsteady flows such as dam break flows (Falappi, 2006). However, their use is limited by the constraints on the maximum allowable time step which arise from the high apparent viscosity of the granular material: they appear therefore to be not suitable for the description of slowly varying phenomena such as those here analysed.

A more conventional approach has therefore been adopted: the bed evolution has been described through the Exner equation which expresses the mass balance for the porous bed:

( ) 01 =+−dxdq

dtdz SBλ (5)

where λ is the porosity of the bed, zB is the bed elevation and qS is the flow rate of the granular material, evaluated by the Meyer-

171

Peter & Muller (1948) formula. Equation (5) is discretized by a finite difference scheme integrated explicitly in time.

The SPH boundary condition is imposed through a layer of fixed ghost particles of width 2h: these particles are located below the erodible bed and are moved in time according to the local zB value.

3. Results and discussion A flow at Fr = 7 and yT/y0 = 5.1 has

been analyzed, being U0 = 1.125 m/s, d50 = 1.32 mm and λ = 0.45. The flow is characterized by a submerged hydraulic jump located just above the wall jet which develops along the protection apron. The jet produces an elongated scour hole, whose depth remains relatively low.

The SPH simulation has been perfor-med with h = 7.8 mm and β = 0.01. Figure 1 shows that the main flow features and the scour hole evolution are well reprodu-ced by the simulation, while the predicted

t = 120 s

t = 600 s

t = 2700 s

VELOCITY (M/S)

Figure 1: Comparison between simulation and experiment: SPH particles colored according to the local velocity value.

dune appears to be shorter and higher than the real one (figure 2).

The simulation has been performed for 2700 s, still far from the experimental equilibrium condition found after 10 hours. The trend of the bed evolution during this phase is well estimated (figure 3).

As expected, it has been found that the correct prediction of the scour characteristics depends strongly on the correct reproduction of the submerged jump, which has been obtained here by adopting a turbulence mixing length proportional to the distance from the bed and, in any case, lower than 0.5 y0.

The comparison between the velocity profiles measured by LDA and the simulated ones, shows that the adopted model is rather effective in the reproduction of the wall jet and of the recirculation region (figure 4).

-30

-20

-10

0

10

20

30

40

50

0 100 200 300 400 500x (mm)

y (m

m)

t = 180 st = 180 s - spht = 1200 st = 1200 s - spht = 2400 st = 2400 s - sph

Figure 2: Bed profiles.

0

5

10

15

20

25

30

35

40

45

50

0 1000 2000 3000 4000t (s)

y (m

m)

Hs,maxHs,max - sphYd,maxYd,max - sph

Figure 3: Evolution of the maximum scour

depth hS,max and dune height yD,max.

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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-0.4 0 0.4 0.8 1.2u (m/s)

y (m

)

x = 0.165m - LDA

x = 0.165 m - SPH

x = 0.450 m - LDA

x = 0.450 m - SPH

Figure 4: Velocity profiles

above the protection apron.

4. Conclusions The test discussed in the present

paper has shown that the SPH method here adopted is able, despite the simplified description of the granular bed evolution, to simulate quite accurately the dynamics of this particular case of scouring caused by a planar jet flowing from a sluice gate along a protection apron.

In particular, the diffusion of the jet along the apron, the position of the submerged hydraulic jump, the evolution of the position and depth of the maximum scour appear to be predicted quite well.

Although a further validation of the method appears necessary, these preliminary results show that an SPH simulation of the eroding flow, coupled with a simple description of the bed evolution, can be a valid tool to predict scouring processes in more complex cases.

5. Acknowledgments The present research has been funded by the Italian Ministry for University and Research (MIUR) as a Research Project of National Interest (PRIN 2005).

6. References BREUSERS, H.N.C. and RAUDKIVI, A.J. (1991). Scouring. Balkema, Rotterdam. ESPA, P. and SIBILLA, S. (2006). Experi-mental study of the scour regimes downstream of an apron for intermediate tailwater depths. Proc. River Flow 2006, 2: 1715-1724, Lisboa. FALAPPI, S. (2006). Simulazioni numeri-che di flussi di fluidi viscosi e materiali granulari con la tecnica SPH. Ph.D. Thesis. Università degli Studi di Pavia. GALLATI, M., BRASCHI, G. (2002). Numerical description of rapidly varied flows via SPH method. Proc. IASTED Int. Conf. on Appl. Simulation and Modelling, 530-535, Iraklion. GALLATI, M., BRASCHI, G. (2003), Numerical description of the jump formation over a sill via SPH method. Proc. CMFF ‘03, 2: 845-852, Budapest. JOHNSTON, A.J. (1990). Scourhole deve-lopments in shallow tailwater. J. Hyd. Res., 28 (3): 341-354. MEYER-PETER, E. and MULLER, R. (1948). Formula for bed-load transport. Proc. I IAHR Congress, 39-64, Stockholm. MONAGHAN, J.J. (1992). Smoothed particle hydrodynamics. Ann. Rev. Astr. Astrophysics, 30: 543-574. RAJARATNAM, N. (1981). Erosion by plane turbulent jets. J. Hyd. Res., 19 (4): 339-358. RODRIGUEZ-PAZ, M.X. and BONET, J. (2004). A corrected SPH method for the simulation of debris flows. Num. Method Partial Differential Eq., 20: 140-163.

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An incompressible multi-phase SPH method

X. Y. Hu and N. A. Adams

Lehrstuhl für Aerodynamik, Technische Universität München 85748 Garching, Germany

Abstract An incompressible multi-phase SPH method is proposed. In this method, a fractional time-step method is introduced to enforce both the zero-density-variation condition and the velocity-divergence-free condition at each full time step. To obtain sharp density and viscosity discontinuities in an incompressible multi-phase flow a new multi-phase projection formulation, in which the discretized gradient and divergence operators do not require a differentiable density or viscosity field is proposed.

1. Introduction The smoothed particle hydrodynamics

(SPH) method is a fully Lagrangian, grid free method in which a smoothing kernel is introduced to approximate functions and their spatial derivatives originating from the interactions with neighbouring particles. Since its introduction by Lucy (1977) and Gingold & Monaghan (1977), SPH has been applied to a wide range of flow problems. The original formulation of SPH is for compressible flows and permits the evolution of fluid densities along flow trajectories. When SPH is applied to simulate incompressible flows, there are generally two approachess to impose incompressibility: one is the weakly compressible SPH formulation (Monaghan 1994) which approximates incompressi-bility by assuming a small Mach number, usually Ma< 0.1; the other is the incompressible SPH which enforces incompressibility by solving a Poisson equation with a source term proportional to the velocity divergence (Cummins &

Rudman 1999) or the density variation (Shao & Lo 2003). Compared with weakly compressible SPH the latter gives more accurate solutions and is computa-tionally more efficient for flow phenomena at moderate to high Reynolds numbers.

In this paper, a technique for a multi-phase SPH by enforcing simultaneously constraints on density variation and the velocity is developed. Also, the projection method is extended to multi-phase flow following the approach of Hu and Adams (Hu & Adams 2006).

2. Method In this paper, a technique for a multi-

phase SPH by enforcing simultaneously constraints on density variation and the velocity is developed. The essential steps are that first the intermediate particle velocities are computed at the intermediate half time step and at the full time step, respectively, and that an intermediate particle position at the full time-step is obtained from the previous time step without enforcing any constraint. In a second step the intermediate particle position at the full time step is modified iteratively to satisfy the zero-density-variation condition. At these new particle positions, the intermediate particle velocity at the full time step is modified by enforcing the velocity-divergence-free condition. As the viscous forces and surface forces are always calculated with constrained particle position and velocity at full time-steps, the velocity-divergence errors introduced by these forces are minimized.

Also, the projection method is extended to multi-phase flow following the approach of Hu and Adams (Hu & Adams

174

2006). For this new proposed gradient and divergence operators which do not involve the assumption of a differentiable density or viscosity field the density, viscosity and pressure gradient discontinuities are handled naturally. To allow for highly-efficient linear system solvers, such as the preconditioned conjugate gradient method, the Poisson operator is discretized to result in a symmetric coefficient matrix. It should be emphasized that, as similar approaches are employed to treat density and divergence constraints, the present method introduces only a minor additional complexity compared to previous incompressible SPH methods

3. Numerical Examples The following two-dimensional

numerical examples are provided to validate the proposed incompressible multi-phase SPH method. For all cases a quintic spline kernel (Morris et al. 1997) is used. A constant smoothing length, which is kept equal to the initial distance between the neighbouring particles, is used for all the test cases. As elliptic solver a diagonal or SSOR preconditioned conjugate gradient method is used.

3.1. Taylor-Green flow

We consider a case with Re = 100. The computation is performed on a domain 0<x<1 and 0<y<1 with periodic boundary conditions in both directions. In order to study the convergence properties the calculation is carried out with 900, 3600, 14400 particles, respectively. Two initial particle configurations are considered: one is starting from regular lattice positions; the other is starting from previously stored article position (relaxed configuration). Figure 1 shows calculated positions of particles and vorticity profile, respectively, at t=1 with 3600 particles.

Figure 1: Taylor-Green problem at t=1 with 3600 particles: (a) positions of particles, (b)

simulated vorticity profile (solid line) and analytical solution (dash line)

3.2. Drop deformation in shear flow

We consider a circular drop with initial radius Ro=0.02 in a Couette flow with wall velocity of v. The periodic computational domain is the region 0<x< 8Ro and 0<y<8Ro in which the drop is centred at (4Ro , 4Ro). The calculation is carried out with 9216 particles. Figure 2 shows the

175

final equilibrium stage Ca = 0.15 and the drop deformations for several capillary numbers.

Figure 2: Drop deformation in a shear flow:

(a) particles of the drop (black dots) and the shearing fluid (open circles) when Ca = 0.15, (b) relation between the deformation

parameter and capillary number.

3.3. Rayleigh-Taylor instability

We consider a Rayleigh-Taylor instability problem. The computation is performed on a domain 0<x<1 and 0<y<2.

Initially, the particles are placed on regular lattice positions. In the lower part of the domain are particles with density of 1.0. In the upper domain are particles with density of 1.8. The Reynolds number is set to Re = 420 and the Froude number is set to Fr = 1. No surface tension is included. The initial particle velocity is set to zero. The calculation is carried out with 7200 particles, which is a similar resolution as that in Cummins & Rudman (Cummins & Rudman1999). The calculated positions of particles at time t=3 and t=5 are shown in Fig. 3.

Figure 3: Rayleigh-Taylor instability:

position of particles at 2 time instance

176

Figure 3 (continue)

4. Concluding remarks We have developed an

incompressible multi-phase SPH method in which both the zero-density-variation and velocity-divergence-free constraints of the incompressiblility condition are enforced by a fractional time-step integration algorithm. A new multi-phase projection formulation in which the gradient and divergence operators are not restricted to a differentiable density and viscosity field is developed to obtain non-

smeared density and viscosity discontinities. Numerical examples are investigated and compared with analytic solutions and previous results. The results show that the method can be reliably applied to incompressible single-phase and multi-phase flows within and beyond the low Reynolds number region. In addition, since very similar approaches are employed to treat density and divergence constraints, the present method increases coding complexity only slightly.

5. References CUMMINS, S. J. and RUDMAN, M. (1999). A multi-phase SPH method for macroscopic and mesoscopic flows. J. Phys. Compt., 152: 584-607.

HU, X. X. and ADAMS, N.A. (2006). A multi-phase SPH method for macroscopic and mesoscopic flows. J. Phys. Compt., 213: 844-861.

GINGOLD, R. A. and MONAGHAN, J.J. (1977). Smoothed particle hydrodynamics: theory and application to non- spherical stars. Mon. Not. R. Astr. Soc., 181: 375- 389. LUCY, L. (1977). A numerical approach to the testing of fusion process. Journal Astronomical, 82: 1013-1024. SHAO, S. and LO E.Y.M (2003). Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Advances in Water Resources, 26: 787-800.

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Multi-phase and Multi-material Flow Modelling using Smoothed Particle Hydrodynamics

J. Ha, P.W. Cleary, M. Prakash and M. Sinnott

PhD, CSIRO Mathematical and Information Sciences, Private Bag 33, Clayton South, Victoria 3169, Australia

Abstract

This study extends the method of SPH to application areas of special effect generation, gas-solid flow, solid-liquid flow, blood flow and metal forming.

1. Introduction The flexibility and Lagrangian nature

of SPH can be exploited to model a wide range of problems involving multiple phases and materials with and without heat transfer. These extra modelling capabilities are achieved by adding the appropriate physical models into SPH.

These extra modelling capabilities enhance further the advantages SPH have over other CFD tools. The advantages include the natural handling of free surface flow, its meshless nature and the relative easy of handling history dependent variables. In the next section, SPH modellings of gas-liquid flow, liquid-solid flow, mixing, material forming processes and biomedical flows are presented.

2. Results and discussions

2.1. Free surface flow

To obtain photo-realism in the animation of fluid, CFD is increasingly used to provide the required degree of realism.

Realistic animation of fluid using SPH consists of three parts; numerical simulation of the physical flow by SPH, generation of high quality surface meshes with specific required properties from the SPH particle locations and rendering of the images. Figure 1 shows our work in animating the flooding of a street.

2.2. Gas-liquid flow

For gas-liquid flow, two approaches have been developed. The general approach is to have different SPH particles for gas and liquid. Recently, Cleary, et al. (2007) presented another approach, the continuous gas model, that is suitable for relative small gas fractions. In this model, an SPH particle represents a mixture of gas and liquid with an inter-phase superficial velocity. This means solving system of advection-diffusion equations in the fully Lagrangian particles.

178

Figure 1: The flow of water through a street. In many pyrometallurgical processes, reactive solids are added to the molten bath in the form of pellets. To assess the gas generation model explained above and buoyancy effects, the process involving reactive solids in a molten bath are simulated in 2D. The aim here is examine gas generation from the pellet at a specified rate and its transport through the liquid bath. The results are presented in Figure 2. To produce the fluid motion generated by stirring generally done in pyrometallurgical applications to improve mixing, there is a central updraft generated during these simulations. In the figure, the colour represents the volume fraction of gas. The gas motion through the liquid bath and the updraft causes the pellet motion even when the pellet and the liquid bath have the same density. The pellet motions in the left column of each frame show two weak recirculation patterns. This results mainly from the imposed central updraft. When the pellets are less dense than the liquid bath, the recirculation is less noticeable. The more buoyant pellets float rapidly and cluster near the surface. Note that the gas motion around and above each pellet creates a buoyant plume that tends to entrain fluid which in turn pushes the pellet upwards. The natural buoyancy of the pellets is enhanced by the generation of buoyant gas plumes from the reacting pellets.

Figure 2: Gas generation and its transport (left) pellets are neutrally buoyant and (right) pellets are less dense than liquid bath.

2.3. Solid-liquid flow

In many industrial systems, there is a need to suspend solids in the liquid using some form of mixing mechanism. Here, we consider the suspension of solids in a liquid mixing tank when the solid is large and have a high loading in comparison with the liquid in the tank. For such solids, they need to be modelled individually and the fluid flow around them needs to be adequately accounted for. Here, the discrete element method is used to model solid objects in flows involving liquid and solid. These solid objects could be rigid bodies or elastic bodies. The study looks at the critical impeller speed for complete pellet submergence in a liquid bath. Figure 3 shows the initial condition of the simulation.

179

Figure 3: Mixing of solid pellets in a 1 m mixing tank. The pellets are 16 mm diameter cylinders and are 22 mm long and have specific gravity of 0.5. The pellet loading is 1.5 kg. .

Figure 4: Pellet submergency at 2.5 s – left: experiment; right: SPH.

Figure 5: Average height of pellets with time.

The SPH result of impeller rotating at 180 rpm at 2.5 s is compared with experiment in Figure 4. A number of simulations of the setup using a series of impeller speed ranging from 50 to 200 rpm were carried out. For each case, the averaged height of the pellets as a function of time was obtained so that we can assess the degree of submergence of the pellets at different impeller speeds. The result of the 180 rpm case is shown in Figure 5. It shows that the degree of submergence is very slow in the first 2 seconds or so. Significant submergence occurs after about 3 seconds. This is due to the length of time for the recirculation flow to establish from the impeller motion. Simulations at various impeller speeds predicted the critical impeller speed for complete pellet submergence in agreement with experimental observation (Prakash, et al. 2007).

2.4. Blood flow

Here, we look at the pulsatile flow in a real carotid artery shown in Figure 6. The carotid arteries are located at the sides of the neck and supply blood to the face and brain. The common carotid artery (CCA) branches into the internal carotid artery (ICA), which feeds the brain, and the external carotid artery (ECA), which transports blood to the muscles of the face and head.

Figure 6: Geometry of a diseased carotid artery bifurcation. On the right, the upper daughter branch is the ECA and the lower branch is the ICA feeding blood to the brain.

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Figure 7: Boundary condition for pulsatile flow.

Following each expulsion of blood

from the heart and into the aorta, a pressure wave propagates downstream through the arterial system initiating local changes in pressure. These transients can significantly alter the flow field through the arteries. To model a 120 beats per second heart rate, we impose a 2 Hz sinusoidal velocity boundary condition at the entrance to the artery as shown in Figure 7. Figure 8 shows the velocity field of the flow in the artery. At 1.5 s, following the last pulse, the pressure has partially equalized across the lower ICA stenosis and the velocity is correspondingly reduced. As the next pulse reaches the stenosis (time = 1.55 s) pressures rise upstream, maximizing the pressure difference across the stenosis. The flow accelerates to around 2.5 m/s through the stenosis until the downstream side achieves a pressure equivalent to that for the pressure drop for the steady case (time = 1.60 s). The velocity then falls. As the pulse passes through the stenosis, the upstream pressure is relaxed and flow velocity decreases further until the downstream side once again pressurises.

Figure 8: Distribution of stream-wise velocities at 3 selected times as the pulse travels through the ICA stenosis. Dark blue represents 0.1 m/s (and greater) and red is 2.0 m/s (and greater).

2.5. Metal forming

For modelling material forming such as forging and extrusion, a plasticity model is used in conjunction with the general deviatoric stress to account for irreversible material deformation. The base material properties used in forging simulations presented in this paper are given in Table 1. Figure 9 shows a simple example of impression die forging where two dies are brought together and the work piece undergoes deformation until it takes on the shape of the cavity formed by the dies. The dies are both 1 m wide. The upper die is 0.2 m high and the lower die is 0.4 m. The work piece is 0.56 m wide by 0.4 m high. The upper moving die is located above the work piece which is sitting on the stationary lower die. The upper die is subject to an applied force and accelerates downwards. If the applied force is 3.0x105 N, then it is not sufficient to completely close the dies. The incomplete forging is shown in Figure 9 along with the final forging when using a

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more adequate forging force of 8.0x105 N. Incomplete forging due to insufficient force is one of the simplest forging defects to predict. In this case, a force of 4.0x105 N is sufficient. The use of larger forces than 4.0x105 N just affect time required to forge a component. In Figure 10, the motion of the top die with time for different applied force is shown for both inadequate and adequate forging forces. The figure shows that the forging operation took 0.06 s if the applied force is 4.0x105 N but only 0.036 s if the applied force is 8.0x105 N. Cases where the y value does not reach 0.025 m indicate that the die has not fully closed and that the force applied is inadequate. 3D forging can also be simulated easily. Figure 11 shows the early and final stages of forging of a 3D component.

Bulk Modulus (GPa) 70.0

Shear Modulus (GPa) 27.0

Initial Yield Stress (MPa) 55.2

Hardening Modulus (MPa)

1.67

Density (kg/m3) 2700.0

Table 1: Al alloy A6061 used for forging.

Figure 9: Effect of different applied force, (left) inadequate applied force of 3x105 N; (right) good forging force of 8x105 N.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Y (

m)

Time (s)

Force = 2.0E-5Force = 3.0E-5Force = 4.0E-5Force = 8.0E-5

Figure 10: Position of the top die as a function of time for different applied forging forces.

Figure 11: Early (left) and final stages of forging of a 3D component.

3. Conclusions In this paper, a wide range of

examples is presented to show the effectiveness of the various models that we have developed to extend the capability of SPH to model flows involving multiple phases and materials.

4. References P.W. CLEARY, M. PRAKASH, J. HA, N. STOKES and C. SCOTT, (2007). Smooth particle hydrodynamics: status and future potential. Progress in Computational Fluid Dynamics, 7: 70-90. M. PRAKASH, P.W. CLEARY, J. HA, M.N. NOUI-MEHIDI, H.M. BLACKBURN and G. BROOKS, (2007). Simulation of suspension of solids in a liquid in a mixing tank using SPH and comparison with physical modelling experiments. Progress in Computational Fluid Dynamics, 7: 91-100.

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AUTHORS INDEX Alessandrini, B. 99,119

Adams, N. A. 173

Armanini, A. 79

Banim, Robert 128

Basa, M. 123

Berczik, P. 5, 63

Berentzen, I. 5

Biddiscombe, J. 51

Børve, S. 67,103

Botia, E. 75

Brizzolara, S. 115

Burkert, A. 5,63

Capone, T. 156

Cecioni, C. 156

Celigueta, M.A. 55

Christensen, S. O. 103

Clark, P. C. 133

Cleary, P. W. 177

Colagrossi, A. 23,59

Colicchio, G. 23,59

Crespo, A. J. C 1,19,152

Dalrymple, R. A. 1,19,111,143,152,156

Del Guzzo, A. 161

Delorme, L. 55, 75

Duarte, R. 32

Ducrozet, G. 119

Dumbser, M. 79

Ely, A.C. 147

Falappi, S. 9

Ferrant, P. 119

Ferrari, A. 79

Frandsen, J. 143

Galindo, S. 32

Gallati, M. 9

Gariah, A. 15

Gómez-Gesteira, M 1,19,152

Graham, D. I. 51,83

Groenenboom, P. 13

Guilcher, P. M. 119

Ha, Joseph 27,177

Hu, X. Y. 173

Hughes, J. P. 83

Imaeda, Y. 87

Inutsuka, S. 87

Issa, R. 15,47

Klapp, J. 32

Klessen, R. S. 133

Kugel, A. 5,63

Lastiwka, M. 123

Laurence, D. L. 15, 37, 111

Le Touzé, D. 59,99

Leboeuf, F. 165

Lee, E-S. 15,37

Lienhart, G. 5,63

Lobovsky, L. 91

López-Pavón, C. 75

Maenner, R. 5,63

Maffio, A. 9

Mansour, J. 107

Marcus, G. 5,63

Marongiu, J.C. 165

Maruzewski, P. 51,99

Meakin, P. 135,139

Molteni, D. 23

Monaghan, J. 107

Moulinec, C. 37

Naab, T. 5,63

Nakasato, N. 5

Narayanaswamy, M. 143

Nestor, R. M. 95

Oger, G. 99

Omang, M. 67,103

Oñate, E. 55

Panizzo, A. 156,161

Parkinson E. 165

Prakash, M. 177

Quinlan, N. J. 95,123

Robinson, M. 107

Rogers, B. D. 111

Savio, L. 115

Scheibe, T. D. 139

Sibilla, S. 169

Sigalotti, L. Di G. 32

Sinnott, M. 177

Souto-Iglesias, A. 55

Speith, R. 67

Spurzem, R. 5,63

Stansby, P. 15,37,111

Suleman, A. 71

Swan, C. 147

Tartakovsky, A. M. 135,139

Toro, E. F. 79

Trulsen, J. 67,103

Tsuribe, T. 87

Vasquez, H. 5

Vimmr, J. 91

Vinogradov, S. B. 5

Violeau, D. 15,37,47

Viviani, M. 115

Ward, A. 135

Wetzstein, M. 5,63

Whitehouse, S. 42

Yildiz, M 71

Zamora-Rodríguez, R. 75