Conference on Modelling Fluid Flow (CMFF’03) The 12 th International Conference on Fluid Flow Technologie s

Budapest, Hungary, September 3 - 6, 2003

VERY LARGE EDDY SIMULATION FOR THE PREDICTION OF UN STEADY VORTEX MOTION

Albert RUPRECHT Head of Fluid Mechanics Group*

Thomas HELMRICH Research assistant

Ivana BUNTIC PhD Student

Institute of Fluid Mechanics and Hydraulic Machinery University of Stuttgart

*Corresponding author: Pfaffenwaldring 10, D-70550 Stuttgart, Germany

Tel.: (+49) 711 685 3256, Fax: (+49) 711 685 3255, Email: ruprecht@ihs.uni-stuttgart.de

ABSTRACT An new turbulence model for Very Large Eddy

Simulation, based on the extended k-ε model of Kim and Chen is developed. Introducing an adaptive filtering technique the model can distinguish between numerically resolved and unresolved parts of flow. The model is applied to the unstable vortex motion in a pipe trifurcation. This flow phenomenon could not be predicted with classical RANS methods and usually used turbulence models. By applying the VLES method with the new turbulence model the phenomenon was well predicted and the results agree quite reasonable with measurement data.

Key Words: Adaptive turbulence model, Pipe trifurcation, Very Large Eddy Simulation.

NOMENCLATURE f filter function hmax local grid size k turbulent kinetic energy L Kolmogorov length scale Pk production term U local velocity α model constant ∆ resolved length scale ∆t time step ε dissipation rate ν kinematic viscosity νt turbulent viscosity

1. INTRODUCTION Industrial calculations are usually based on the

Reynolds-averaged Navier-Stokes (RANS) equations. This means that the complete turbulence behavior is expressed by means of an appropriate turbulence model. The necessary turbulence model has to take into account all the turbulent scales ranging from the largest turbulent eddies down to the Kolmogorov scale. Consequently the model has to be very sophisticated and it is impossible to define a model suitable for all flow phenomena, especially for unsteady vortex motions.

A direct numerical simulation (DNS), where all scales down to the Kolmogorov scale are resolved by the computation is impossible. As the smallest scales strongly decrease with increasing Reynolds number, extremely high computational effort (i.e. large grid size) is required for high Reynolds number flows. Therefore DNS cannot be applied for flow of practical relevance in the foreseeable future.

A promising compromise is the Large Eddy Simulation (LES). In a “real” LES (from the turbulence research point of view) all anisotropic turbulence structures are resolved in the computation and only the smallest isotropic scales are modeled. Therefore the used turbulence models can be simpler compared to RANS models, since they only have to describe the influence of the isotropic scales on the resolved anisotropic flow field. Unfortunately with increasing Reynolds

numbers the anisotropic scales decreases and cannot be resolved in the computation of flow of practical relevance, although there are many “LES” applications in the literature. However from the turbulence research point of view these simulations are mostly unsteady RANS (URANS), since they only resolve the unsteady mean flow but do not resolve any turbulence structures.

In order to apply a “classical” URANS there must be a gap in the turbulence spectrum between the unsteady mean flow and the turbulent flow. Only then the classical turbulence models (e.g. k-ε) can be applied, as they are developed for modeling the whole range of turbulent scales. If it is not possible to distinguish between mean flow and turbulence these models cannot be applied properly.

In this case a Very Large Eddy Simulation (VLES) can be applied. Contrary to URANS there is a requirement for the turbulence model, that it can distinguish between numerically resolved unsteady movement and not resolved turbulent fluctuations which have to be modeled. VLES is similar to LES, only that a smaller part of the turbulence spectrum is resolved in the unsteady simulation. The model must express the influence of a greater part of the spectrum and therefore it has to be more sophisticated.

As mentioned above DNS and LES requires a much too high computational effort for complex industrial problems. RANS is not suitable for unsteady vortex phenomena. For this type of problem today VLES seems to be a promising way. In Table 1 the availability of the different approaches for the simulation of the flow around an aircraft is summarized. This data apply similarly for other complex flow problems.

Table 1: Application of the different approaches to the flow around an aircraft [1].

In this paper the development of a VLES

turbulence model is presented. The model is based on the extended k-ε model of Chen and Kim [2]. By adding an appropriate filtering techniques - which depends on the local grid spacing and the computational time step - the new turbulence model distinguishes between resolved and modeled part of

the turbulent spectrum. It has an adaptive characteristic, in such a way that it can be applied for the whole range of approaches from RANS up to DNS.

The presented application of the new adaptive turbulence model is the flow in a pipe trifurcation of a water power plant. In the spherical geometry an unsteady not periodic vortex motion has been observed. This phenomenon is predicted by the adaptive model.

2. SIMULATION METHOD

2.1 Very large Eddy Simulation (VLES) “Real” Large Eddy Simulation (LES) from the

turbulence research point of view require an enormous computational effort since all anisotropic turbulence structures have to be resolved in the computation and only the smallest isotropic scales are modeled. Consequently this method also can not be applied for industrial problems today.

Today’s calculations of flows of practical relevance (characterized by complex geometry and high Reynolds number) are usually based on the Reynolds-averaged Navier-Stokes (RANS) equations. This means that the influence of the complete turbulence behavior is expressed by means of an appropriate turbulence model. To find a turbulence model, which is able to capture a wide range of complex flow effects quite accurate is impossible. Especially for unsteady flow behavior this method often leads to rather poor results. The RANS and LES approach can schematically be seen in Figure 1, where a typical turbulent spectrum and its division in resolved and modeled parts is shown.

Figure 1: Modelling approach for RANS and LES.

The recently new established approach of Very Large Eddy Simulation can lead to quite promising results, especially for unsteady vortex motion. Contrary to URANS there is a requirement to the

applied turbulence model, that it can distinguish between resolved unsteady motion and not resolved turbulent motion which must be included in the model. It is similar to LES, only that a minor part of the turbulence spectrum is resolved (schematically shown in Figure 2). VLES is also found in the literature under different other names:

- Semi-Deterministic Simulation (SDS), - Coherent Structure Capturing (CSC), - Detached Eddy Simulation (DES), - Hybrid RANS/LES, - Limited Numerical Scales (LNS).

Figure 2: Turbulence treatment in VLES.

2.2 Numerical Method The calculations are carried out using the

program FENFLOSS which has been developed at the institute for more than a decade [3,4].

The partial differential equations are solved by a Galerkin Finite Element Method. The spatial discretization of the domain is performed by 8-node hexahedral elements. For the velocity components and the turbulence quantities a tri-linear approximation is applied. The pressure is assumed to be constant within each element. For advection dominated flow a Petrov-Galerkin formulation of 2nd order with skewed upwind orientated weighting functions is applied. The time discretization is done by a three-level fully implicit finite difference approximation of 2nd order.

For the solution of the momentum and continuity equation a segregated solution algorithm is used. Each momentum equation is handled independently. The momentum equations are linearized by successive substitution. The linear systems are solved by the BICGSTAB2 algorithm of van der Vorst [5] with an incomplete LU decomposition (ILU) for preconditioning. The pressure is treated by a modified Uzawa type pressure correction scheme [6]. The pressure correction is carried out in a local iteration loop without reassembling the system matrices until the continuity error is reduced by a given order (usually 6-10 iterations needed).

After the solution of the momentum and continuity equations the turbulence quantities are calculated and a new turbulence viscosity is obtained. The k- and ε-equations are also linearized by successive substitution and the linear systems are solved by the BICGSTAB2 algorithm with ILU preconditioning. The whole procedure is carried out in a global iteration until convergence is obtained. For unsteady simulations the global iteration has to be carried out in each time step.

The parallelization of the code is introduced by domain decomposition using double overlapping grids. The linear equation solver BICGSTAB2 is carried out in parallel and the data exchange between the domains is organized on the level of the matrix-vector multiplication in the BICGSTAB2 solver. The preconditioning is carried out locally on each domain. The data exchange is carried out using MPI (Message Passing Interface) on machines with distributed memory. On shared-memory-computers the code runs also parallel by applying OpenMP. For details on the numerical procedures and parallelization the reader is referred to [7,8].

3. ADAPTIVE TURBULENCE MODEL Classical turbulence models, which are usually

applied in engineering flow predictions, contain the whole turbulent spectrum. They usually show a too viscous behavior and very often damp out unsteady motion to early. As discussed above the turbulence model for the VLES have to distinguish between the resolved and unresolved part of the turbulent spectrum (Figure 2). Therefore an adaptive model is developed, which adjusts its behavior according to the approach (schematically shown in Figure 3). This means that this model can be applied for all approaches.

Figure 3: Adjustment for adaptive model.

The advantage of the adaptive model is, that with increasing computer power the resolved part of the turbulent spectrum increases and the modeled part decreases and consequently the accuracy of the calculations improves.

There are several filtering techniques in the literature [e. g. 9-11]. Here a filtering techniques similar to Willems [12] is applied. In the following

the new adaptive turbulence is presented. The nomenclature of resolved and modeled parts can be seen in Figure 4.

Figure 4: Distinguishing of turbulence spectrum for VLES.

The basis of this adaptive model is the k-ε model of Chen and Kim [2]. This model has been chosen, because it is quite simple and its results are much better – especially for unsteady flows –compared to the standard k-ε model. The transport equations for k and ε are given as

ε−+

∂∂

σν

+ν∂∂=

∂∂+

∂∂

kjk

t

jjj P

xk

xxk

Utk

(1)

4434421term additional

kk

3

2

2

k1jk

t

jjj

Pk

Pc

kc

Pk

cxxx

Ut

⋅

+ε−

ε+

∂ε∂

σν

+ν∂∂=

∂ε∂+

∂ε∂

εε

ε

(2)

These equations prescribe the whole turbulence spectrum and therefore a filtering procedure has to be implemented.

According to Kolmogorov theory it can be assumed that the dissipation rate is equal for all scales. This leads to

ε=ε ). (3)

However the turbulent kinetic energy needs a filtering

∆−⋅=L

f1kk)

. (4)

A suitable filter can be obtained to

∆>

∆−

≥∆=

L for L

1

L for 0f

3/2 (5)

where

∆=

∆⋅

⋅α=∆3D for V

2D for ∆Vh with

h

tumax

3maxmax

(6)

with a model constant (α= 2 to 5). ∆V is the volume of the local element, u is the local velocity and ∆t is the time step. The Kolmogorov scale L is given as

ε=

2/3kL . (7)

The filtering procedure leads to the final equations

ε−+

∂∂

σν

+ν∂∂=

∂∂+

∂∂

kjk

t

jjj P

xk

xxk

Utk ))

(8)

and

kk

3

2

2

k1jk

t

jjj

Pk

Pc

kc

Pk

cxxx

Ut

))

))

⋅

+ε−

ε+

∂ε∂

σν

+ν∂∂=

∂ε∂+

∂ε∂

εε

ε

(9)

with the production term

j

i

i

j

j

itk x

U

x

U

x

UP

∂∂

∂∂

+∂∂

=ν))

. (10)

For details on the model and its development

the reader is referred to Ruprecht [4]. The simulation of vortex shedding, which can

be considered as a very convenient test case for CFD computations, shows very often great difficulties when applying URANS. For example applying URANS with the standard k-ε model to the flow behind a bluff trailing edge (Figure 5) leads to a steady state solution. The vortex shedding is suppressed by the too diffusive turbulence model. Applying a more sophisticated turbulence model, e. g. the extended k-ε model of Kim and Chen [2], the vortex shedding is obtained. The comparison is shown in Figure 6, where the streamlines at a certain time step are plotted.

Figure 5: Flow behind bluff trailing edge.

Figure 6: Streamlines of the vortex shedding behind bluff trailing edge.

Figure 7: Vortex shedding behind a bluff trailing edge, pressure distribution, comparison of Kim & Chen model and adaptive VLES model.

Furthermore by more detailed observation of the obtained results, a relatively strong damping of vortex shedding in the downstream flow is noticed applying the Kim and Chen model.

Using the VLES method the results severely improve. In Figure 7 the pressure distribution behind the bluff trailing edge clearly shows a reduced damping of the vortex motion, which coincided better with experimental observations.

4. APPLICATION

4.1 Problem description The application presented is the flow in a pipe

trifurcation of a water power plant. The water passage consists of the upper reservoir, channel, surge tank, penstock, trifurcation and three turbines.

The trifurcation distributes the water from the penstock into the three branches to the turbines. It has a spherical shape because of structural reasons. The water passage is partly shown in Figure 8.

In the power plant severe power oscillations were encountered at the outer turbines 1 and 3, in the range of +/-10% of nominal power. A vortex instability could be discovered as a reason for that. A vortex is forming in the sphere, starting at the top and extending into a side branch. After a certain period it changes its behavior and extends to the opposite side branch, again for a certain time period. Then it jumps back again. The two vortex positions are schematically shown in Figure 9. This unstable vortex motion, which is not periodic, causes the power fluctuations, since the vortex produces very high losses in the branch, in which it is actually located, because of strong swirling inlet flow to the branch. These losses reduce the head of the turbine and consequently the power output. In the following the simulation of this vortex movement is presented and the results are compared with measurement in a model test.

Figure 8: Water passage with trifurcation.

Figure 9: Unsteady vortex structures.

4.2 Unsteady vortex simulation For the simulation the geometry is discretized

by approximately 500.000 elements. The computational grid, shown in Figure 10, is decomposed into 32 partitions and the computations are run on a CRAY T3E.

Applying URANS with the standard k-ε model leads to a steady state solution. The obtained vortex structure is shown in Figure 11. The vortex ranges from one side branch to the other. It is completely stable and the swirl component is severely underpredicted. This also leads to an underprediction of the losses in the side branches. The URANS is not able to predict the unsteady flow phenomenon.

Figure 10: Computational grid.

Applying VLES with the adaptive turbulence model the unstable vortex movement can be predicted. In Figure 12 the flow at a certain time step is presented. In this time step the vortex extends into the first side branch. It starts at the top of the sphere. The vortex is prescribed by an iso-pressure surface and by instantaneous streamlines.

Some time later, Figure 13, the vortex has “jumped” to the opposite side branch. Since the geometry is not completely symmetric the vortex stays longer in branch 3 than in branch 1. This is shown in the simulation as well as in model tests.

Figure 11: Vortex structure obtained by URANS with k-ε turbulence model.

Figure 12: Flow in the trifurcation, vortex position in branch 1.

Figure 13: Flow in the trifurcation, vortex position in branch 2.

Due to the strong swirl at the inlet to the side branch, in which the vortex is located, the inlet losses into this branch are much higher compared to the others. Consequently the discharge through this branch is reduced. In Figure 14 the discharge characteristic is shown. It is obvious, that the discharge varies alternatively between the two side branches. Low discharge corresponds with the location of the vortex in this branch. If the vortex is located in the other side branch the discharge is high. The discharge in the middle branch only shows much smaller oscillations.

Figure 14: Discharge through the different branches.

Dis

char

ge in

%

Left Middle

Right

Time in s

Figure 15: Loss coefficients of the three branches

In the simulation a free outflow boundary condition is applied at the end of the branches. In reality, however, the turbine is located there. Consequently the discharge variation is rather small since the flow rate through the different branches is prescribed by the turbines. For comparison with experimental data therefore the loss coefficients are calculated and compared with the model test data. In Figure 15 the loss coefficients for the three branches are presented. It can be observed, that the branch on the right hand side shows the highest values and has a strong variation. The loss coefficient of the middle branch is nearly constant.

4.3 Model tests Model tests were carried out at ASTRÖ in Graz,

Austria. For details on the measurements the reader is referred to [13]. In Figure 16 the test rig is shown, looking down from the penstock to the trifurcation. The trifurcation was build in acrylic glass so that the vortex motion could be observed.

In Figure 17 the view from a top of the trifurcation is shown. One can see the vortex starting at the top and expanding into the side branch. The vortex is made visible by reducing the pressure level, that small cavitation bubbles appear in the vortex center but without disturbing the flow structure severely. In the model tests the vortex also moves from one side to the other, as described above.

From the pressure and discharge measurement the loss coefficients of the different branches were calculated. In Figure 18 the variations are shown.

Figure 16: Test rig at ASTRÖ, Graz.

Figure 17: Vortex in the trifurcation, measurements of ASTRÖ.

Comparing the measured loss coefficients with the ones from the simulation (Figure 15) one can observe, that the maximum values are still underpredicted. For the right hand side branch the numerical simulation shows a maximum of 6 whereas in the measurements the value of 8 is obtained. The values for the left hand side branch are clearly lower (approximately 3 in the simulation and 4 in the measurements). The overall tendency of the flow, however, is well predicted. Also the quantitative prediction is quite reasonable.

Figure 18: Loss coefficients for the different branches, measurements by ASTRÖ.

The underprediction of the loss coefficient is assumed to be due to the rather coarse grid on one side. On the other side it is due to the strong anisotropic turbulence behavior, which can not be predicted accurately by a turbulence model based on the eddy viscosity assumption. Therefore it is

Time in s

Loss

coe

ffici

ent ζ

right

left

middle

intended to develop an adaptive, algebraic Reynolds-stress model in future.

4.4 Problem solution In order to solve the oscillation problem in the

hydro power plant, the trifurcation was changed. To avoid the forming of the vortex the upper and lower region of the sphere is cut off by welding in flat blades. The modified shape is shown in Figure 19. In the meantime this modification has been carried out and the power oscillations disappeared. As a side effect the power output increased severely since the losses in the trifurcation decreased without the existing of the vortices.

Figure 19: Modified trifurcation to avoid the power oscillations.

5. CONCLUSIONS An adaptive turbulence model for Very Large

Eddy Simulation is presented. The model is based on the extended k-ε model of Kim and Chen. By introducing a filtering technique the model can distinguish between numerically resolved and unresolved parts.

Applying this new model the unstable vortex motion in a pipe trifurcation is calculated. This flow phenomenon could not be predicted with classical RANS methods and usually used turbulence models. By applying the VLES method with a new turbulence model the phenomenon can be obtained and the results agree quite reasonable with measurement data. The vortex swirl, however, is still underpredicted. A further improvement is expected by applying an adaptive Reynolds-stress model to account for the anisotropy of the turbulence in swirling flows.

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[2] Chen, Y. S., Kim, S. W. (1987): Computation of turbulent flows using an extended k-ε turbulence closure model, NASA CR-179204.

[3] Ruprecht, A. (1989): Finite Elemente zur Be-rechnung dreidimensionaler turbulenter Strömungen in komplexen Geometrien, Doctorate Thesis, Uni-versity of Stuttgart.

[4] Ruprecht A. (2003): Numerische Strömungs-simulation am Beispiel hydraulischer Strömungs-maschinen. Habilitationsschrift, Universität Stutt-gart.

[5] Van der Vorst, H. A., (1994): Recent Developments in Hybrid CG Methods, Proc. High Performance Computing & Networking, München, 1994.

[6] Zienkiewicz, O.C., Vilotte, J. P., Toyoshima, S., Nakazawa, S. (1985): Iterative method for constrained and mixed finite approximation. An inexpensive improvement of FEM performance, Comp. Meth. Appl. Mech. Eng., 51.

[7] Maihöfer, M. (2002): Effiziente Verfahren zur Berechnung dreidimensionaler Strömungen mit nichtpassenden Gittern, Dissertation Universität Stuttgart.

[8] Maihöfer, M., Ruprecht, A. (2003): A Local Grid Refinement Algorithm on Modern High-Performance Computers, Parallel CFD, Moskau.

[9] Spalart, P. R., Jou, W. H. Strelets M, Allmaras, S. R. (1997): Comments on the Feasibility of LES for Wings, and on Hybrid RANS/LES Approach, 1st AFOSR International Conference on DNS/LES Rouston.

[10] Constantinescu, G. S., Squires, K. D. (2000): LES and DES Investigations of Turbulent flow over a Sphere, AIAA-2000-0540.

[11] Magnato, F., Gabi, M., “A new adaptive turbu-lence model for unsteady flow fields in rotating ma-chinery”, ISROMAC 8, 2000.

[12] Willems, W., Peters, N. (1997): Large Eddy Simulation of a Turbulent Mixing Layer Using a New Two-Level-Turbulence Model, 11th Symp. On Turb. Shear Flows, Grenoble.

[13] Hoffmann, H., Roswora, R. R., Egger, A. (2000): Rectification of Marsyangdi Trifurcation, Hydro Vision 2000, Charlotte.