Recent progress in the modeling of Recent progress in the modeling of non-linear free surface phenomena non-linear free surface phenomena
in ocean engineeringin ocean engineering
FRAUNIE, P. (1)
GRILLI, S.T. (2) ;
(1) L.S.E.E.T, Université de Toulon(2) Department of Ocean Engineering, University
Of Rhode Island
L.S.E.E.T University Of Rhode Island
Photos: www.pvergnaud.free.fr
• Objectives :
- Coastal morphodynamics, sediment transport
- Dammages on coastal areas and structures
- Ocean-atmosphere interactions
• Tools :
- Laboratory/in situ Experiments
- Numerical Simulation : Numerical wave tank
WAVE BREAKING IN COASTAL ZONE
Free surface flows
• Flows containing several fluids/phases • Several examples :
- wave breaking- cavitation- slushing of fuel in satellite tanks …
better understanding of the physical phenomena
• Tool : numerical simulation using interface tracking methods
Mathematics and numerical modeling
Assumptions : what is to be modeled ?
• Flows : – Fully 3-D – Unsteady– Non hydrostatic– Laminar/turbulent– Single or two-phase flows
• Fluids : – Newtonian – Incompressible or not
Fluid 1 Fluid 2
Interface
n
t
Mathematical formulation : conservation equations
TPVVdivV
Vdiv
)(
)
0
Mass conservation
Momentum conservation
Surface tension (N.mSurface tension (N.m-3-3))
Body forces (m.sBody forces (m.s-2-2))
Density (kg.mDensity (kg.m-3-3))
Viscous stress tensor (N.mViscous stress tensor (N.m-2-2))
ft
(
Velocity (m.sVelocity (m.s-1-1))
PressPressureure (N.m (N.m-2-2))
… in the fluid domain
Mathematical formulation : Interface boundary conditions
nnnpptutunVnunu
).().(. .
. . .
221121
21
21
Velocity continuity at the interfaceVelocity continuity at the interface
Viscous fluids onlyViscous fluids only
Stress balance at the interfaceStress balance at the interface
Normal to the interfaceNormal to the interface
Interface velocityInterface velocity
Surface tension coefficientSurface tension coefficient
Interface Interface curvaturecurvature
Fluid 1Fluid 2
Interface
Mathematical formulation : boundary conditions
• Solid boundaries : slip condition (Euler) or no-slip Solid boundaries : slip condition (Euler) or no-slip condition (Navier-Stokes), pressure extrapolationcondition (Navier-Stokes), pressure extrapolation
• Open boundaries : Open boundaries :
- Dirichlet condition (fixed velocity and pressure) on inlet - Dirichlet condition (fixed velocity and pressure) on inlet boundaries boundaries
- Neumann condition (normal derivative of the velocity - Neumann condition (normal derivative of the velocity imposed to zero) for the velocity and fixed pressure on outlet imposed to zero) for the velocity and fixed pressure on outlet boundaries boundaries
Mathematical formulation : summary
Resolve the system composed by:
Conservation equations
Interface conditions
Boundary conditions
Equation governing the evolution of the interface
Let be C the binary function so
that :
C(x,t) = 1 if x fluid 1
C(x,t) = 0 if x fluid 20 .
CUtC
Equation governing the interface evolution
Numerical resolution of the conservation equations :
CFD code : EOLE (PRINCIPIA R&D)
Navier-Stokes (or Euler) equations in a curvilinear formulation (ξ,η,χ) :
JT
JRHGF
tW
J 1
F,G,H : flux terms (convective, diffusive, pressure)
J : Jacobian of the transformation
0
;
0 ;
0
z
y
x
z
y
x
fff
R
KnKnKn
T
wvu
W
With :Surface tension Body
forces
• Space discretization : Centerred Finite Volume scheme (fields computed at the cell center)
• Time discretization : second order implicit scheme
Numerical resolution of the conservation equations :
Pseudo-compressibility method (Chorin 1967)
11111111
2
431~
1
nnnnnnnnn
J
T
J
RHGF
t
WWW
J
W
J
avec
w
vuW
~
~~
~~
Concept : introduction of a time-like variable τ, the pseudo-time and of pseudo-unsteady terms
New unknown introduced in the pseudo-unsteady terms, the pseudo-density
Additional equation : pseudo equation of state giving the pressure as a function of the pseudo-density (Viviand):
~ 122
01
~ln)(
n
nn UUp
• Adding artificial viscosity terms avoids numerical oscillations
• Integration step by step in pseudo-time thanks to a five step Runge-Kutta scheme (« dual time stepping »)
Convergence : solution independent on τ corresponding to the numerical solution at time level n+1
Robust method allowing to deal with high density ratios
Pseudo-compressibility method
Algorithm : Time iteration
pseudo-time iterations : pseudo time step calculation
Runge-Kutta steps:
flux computation, new velocity and pseudo-density, fixing the boundary conditions
End of Runge-Kutta steps
End of pseudo-time iterations
Interface tracking method
End of time iteration
Development and validation of a 3-D Larangian V.O.F method
Interface tracking method : aims
• 3-D method allowing to deal with large deformations of the interface (large curvatures, reconnections, deconnections …)
• Accuracy
• Fast compared to classical V.O.F methods
extension of the SL-VOF method to 3-D flows
Volume Of Fluid (V.O.F) conceptThe interface is tracked thanks to the volumic fraction of the denser
fluid (fluid 1) :
• C = 1 in a full cell of fluid 1• C = 0 in a full cell of fluid 2• 0 < C < 1 if a cell an interface occurs in the cell
c: example ofinterface representation
0.3 0.1 0
a : initial interface
0.41 0.8
1 1 0.9
b: values of C In each cell
Fluid 1
Fluid 2
C function is advected with the fluids and verifies the transport equation :
0 . CUtC
Classical discretization schemes (centred, upwind, Quick …) are diffusing the interface and are not accurate
Alternative : methods with interface reconstruction. Several possibilities :
• SOLAVOF method (Hirt & Nichols, 1981)
• CIAM (Li, Zaleski, 1994)
• SL-VOF (Guignard, 2001, Biausser 2003)
V.O.F methods
SL-VOF 3-D method (B. Biausser, 2003)
• Interface reconstruction
• Interface advection
• Computation of the new V.O.F field
3 steps allowing to update the interface during a time step :
Step I : interface reconstruction
Piecewise Linear Interface Calculation (Li 1994)
In each cell, the interface is represented by a plane portion (intersection of a plane with the computational cell)
n
(1,0,0)
(0,0,0)(0,1,0)
(0,0,1)
Interface
Two steps to calculate the interface plane portion :
• Definition of the plane direction
• Translation of the plane in order to verify the volume of the cell
Calculation of the plane direction Calculation of the plane direction : the normal to the plane (orientated from denser the fluid towards the less dense fluid) is :
Cn
Step I : interface reconstruction
Evaluation of n by finite differences from the V.O.F of the neighbouring cells :
a) Computation of normal vectors at the 8 corners of the cell:
b) Normal vector is the mean of the 8 normal vectors at the corners
i
jk
i
jk
Step I : interface reconstruction
Plane translation Plane translation : the normal to the plane and the volumic fraction Cijk of the cell determine a unique plane
Translation of the plane so that the volume contained under this plane is Cijk
If the equation of the plane of normal nijk (nx, ny, nz) is
nxx+nyy+nzz = , the problem is equivalent to calculate
(Cijk,nijk)
Step I : interface reconstruction
The calculation of (Cijk,nijk) provides a unique plane portion : polygon from 3 to 6 sides whose corners are known
A
B G
H
(a) (b) (c)
(d) (e) (f)
Step I : interface reconstruction
Step II : interface advection
Calculation of the velocity at the polygon corners Calculation of the velocity at the polygon corners : bilinear interpolation from the velocities computed by the solver at the cell center
Corners advection Corners advection : first order (in time) Lagrangian scheme Xn+1 = Xn + U.t
U.t
Xn+1
Xn
Interface before advection
Interface after advection
After advection, the advected polygon corners are not After advection, the advected polygon corners are not necessarily coplanar so that a mean plane to these corners necessarily coplanar so that a mean plane to these corners is defined :is defined :
Normals to triangular
subdivisions
P1
P2
P3
P4
PmPolygon
corners after advection
Pm
nm Mean plane of the corners after advection (nm : mean normal , Pm : mean point)
Mean normal of the normals to triangulars
subdivisions
Mean point : iso-barycentre of the
corners
Step II : interface advection
Step III : computation of the new V.O.F field
Two configurations after advection :Two configurations after advection :
• Cells containing polygons portions (A type)
• Cells without interface (B type) type A cells
after advection
type B cells before
advection
type A cells treatment
Calculation of the mean plane to all polygons parts in the cell :
Mean plane defined by averaging the normals to the plane parts and their centres (weighted with the portions surface)
n1 n2 nm
The new VOF of the cell is the volume generated by the averaged plane and is calculated by inversion of the formulae giving as a function of Cijk and nijk
n
New VOF : Cijk
n+1
type A cells treatment
type B cells treatment
Two configurationsTwo configurations are possible : are possible :
• Cells loosing the interface during the time step : such cells become full (C = 1) or empty (C = 0) following the stream direction
• Cells without interface before advection : the value of C remains the same
Cell filled up during advection
Cell without interface
before and after advection
SL-VOF 3-D : summary
• 3-D V.O.F. method with geometrical reconstruction of the interface
• PLIC modeling (more precise than Hirt&Nichols) allowing to deal with large deformation of the interface
• Lagrangian advection (possibly use of larger time steps than with classical methods)
Evaluation of the method’s Evaluation of the method’s performancesperformances
Evaluation of the method’s performances
• Comparison with a classical 3-D V.O.F method already developed in the EOLE code : FLUVOF (Hirt & Nichols kind)
• Aability to deal with large changes of the interface
• Ability to use large time steps
Comparison with FLUVOF 3-D
• Pure advection test-case (imposed velocity) : allows to test the methods performances without NS solver
• Advection to a wall : the analytic velocity is known
• A sphere advected in such a flow is progressively changed into ellipsoïds
• Comparison with a Hirt&Nichols method using a constant piecewise reconstruction of the interface (0 order)
0
20
40
60
80
100
Z
0
20
40
60
80
100
X0
1020
30Y
X Y
Z
Domaine de calcul 3D
z
x
Solid wall
Main direction of the flow
0 100
100
Point A
In each transverse plane y = constant
Comparison with FLUVOF 3-D : computational domain
Sphere advection in a distorting velocity field
SL-VOF 3-D
simulation
Comparison with FLUVOF 3-D
0
20
40
60
80
100
Z
0
25
50
75
100
X0
Y
X Y
Z
0
20
40
60
80
100
Z
0
25
50
75
100
X0
Y
X Y
Z
FLUVOF
SL-VOF
X
Y
0 25 50 75 1000
10
20
30
40
50
60
70
80
90
X
Y
0 25 50 75 100 1250
10
20
30
40
50
60
70
80
90
Comparaisons SL-VOF (traits noirs) /SOLUTION ANALYTIQUE (traits rouge)
Comparaisons FLUVOF (traits noirs) /SOLUTION ANALYTIQUE (traits rouge)
Mesh 100X30X100
Comparison with FLUVOF 3-D
• When the curvature is maximum, accuracy is better with SL-VOF 3-D than using FLUVOF : advantage of the P.L.I.C discretization of the interface
• The SL-VOF simulation is 4 times faster than FLUVOF : advantage of the Lagrangian advection
• Volume conservation is quite good : 0.13 % of loss compared to the initial fluid volume after 70 time steps
The method’s approximates (mean plane) are not penalizing the volume conservation
Coupling with the NS solver :Rayleigh-Taylor instability
• Stratified fluids of different densities (the denser is above)
• Initial perturbation characteristic instability involving local vortices
• Overturning of the interface occurs and the flow is computed with the full solver : good test for the method
2-D example (denser fluid in red)
Rayleigh-Taylor instability• Density ratio: 2• Perfect fluids in a cylindrical domain• Interface initially plane : sinusoidal perturbation of the velocity
Comparisons with 2-D axisymetric results
0.02 0.03 0.04
Y
00.020.04 X
0
0.025
0.05
Z
X
Z
t=0.22 s
0.02 0.03 0.04
Y
00.020.04 X
0
0.025
0.05
Z
X
Z
t=0.66 s
0 0.01 0.02 0.03 0.04
Y
00.020.04 X
0
0.025
0.05
Z
X
Z
t= 0.88 s
0 0.01 0.02 0.03 0.04
Y
00.020.04 X0
0.025
0.05
Z
X
Z
t=0.44 s
Full 3D Rayleigh-Taylor Instability (slice Y-Z)
Y
Z
0 0.01 0.02 0.03 0.040
0.025
0.05
t=0.44 s
Y
Z
0 0.01 0.02 0.03 0.040
0.025
0.05
t=0.22 s
Y
Z
0 0.01 0.02 0.03 0.040
0.025
0.05
t=0.66 s
YZ
0 0.01 0.02 0.03 0.040
0.025
0.05
t=0.88 s
Axisymetric Rayleigh-Taylor Instability
3-D on a radius 2-D axi
Conclusions about the test cases
• Compared to a classical V.O.F method : better accuracy when the curvature is increased, computational time is reduced
• Large accurately deformations are taken into account
Tool able to deal with wave breaking
Wave breaking applications
First tests of breaking
• Breaking of an unstable linear wave
• Breaking of a solitary wave on a beach of slope 1/15
Evaluation of the method’s ability to simulate wave breaking :
• Sinusoidal wave of high camber• Initial velocity field : Airy wave• Periodic boundary conditions over one wavelenght• L = 0.769 m• T = 0.86 s• D = 0.1 m• H = 0.1 m
Breaking of an unstable linear wave
Fast evolving towards a plunging breaker
Déferlement d’une onde linéaire instablePropagation
direction
Modulus of the velocity
Conclusions concerning this test-case
• Results comparable to those of Abadie (1998) on the same test-case for 2-D flows (aspect of the breaker jet, splash-up, maximal velocity about 2 times the phase celerity)
• First simulation of breaking conclusive with the method (reconnections and deconnections of the interface, curvature …)
• Artificial breaking, generated by a non-physical initial condition
Breaking of solitary waves on sloping beaches
Breaking on a beach of slope 1/15
• Solitary wave H0 = 0.5 m
• Computational domain : flat bottom and then sloping bottom
• Initialisation with Tanaka’s algorithm (1986) and computation of the initial fields with Boundary Integral Equations Method of S. Grilli : potential code using a Boundary Element Method
Boundary Integral Equations Method
• Nonlinear potential flows with a free surface
• Fast and accurate method for wave shoaling and overturning applications
• Unable to deal with breaking (no reconnection, irrotationnal and inviscid flows …)
Solitary wave: initialization
-4
-2
0
2
4
Z
510
1520
2530
35
00.10.2
XY
Z
Soliton 3D t=0 s : H/D=0.5, s=1/15
Soliton : breakingSloping bottom kinetic energy is transferred into potential energy camber breaking
202530X
X
Y
Z
253035 X
00.10.2
Y
X
Y
Z
253035 X
00.10.2
Y
X
Y
Z
202530X
X
Y
Z
20
X
Y
Z
00.10.2
YX
Y
Z
Soliton 3D : H/D=0.5 s=1/15
X (mm)
Z(m
m)
0 50 100 150 200
-20
0
20
40
60
80
100
120
140
Result of the simulation of the breaking of a solitary wave on a bottom of slope 0.0773 -
Weak coupling BEM - SL-VOF
PIV image of the breaking of a solitary wave on a bottom of slope 0.0773 – Experiment
made in the waterl tank in ISITV
Soliton : 2-D results and experimental results
SL-VOF method for 2-D flows has been tested successfully on wave breaking applications
Soliton : comparisons 2-D/3-D
• Test-case runs for 2-D flows compared to measurements and BIEM by Guignard (2001)
• Comparison 2-D SL-VOF / 3-D SL-VOF : very close results
• Few differences (delay for breaking) due to the coarser mesh for the 3-D run
Conclusions concerning this test-case
• Successful simulation of a physical breaking
• Comparisons with 2-D results ok : (similar results)
• Pseudo-3-D test-case : no variation of the slope in the cross direction same phenomenon in each transverse plane
Fully 3-D breaking
Breaking of a solitary wave on a sloping ridge
• Sloping bottom with a transverse modulation with a hyperbolic secant
• Slope 1/15 at the centre of the ridge and 1/36 on each side
• 350 cells along x, 40 along y and 65 along z
• Solitary wave : H0 = 0.6 m
• Coupled to Grilli’s BIEM
• Single phase flow in order to reduce computational time
350 cells along x, 40 along y, 65 along z
Initialisation with BIEM
-1
0Z
8
10
12
14
16
18
20
22
-2-1
01
2
X Y
Z
• First step : a part of the shoaling is computed using BIEM (accurate and faster than the VOF/Navier-Stokes solver but unable to deal with breaking)
• The solution of this first simulation is used as an initialization of the VOF/Navier-Stokes solver (free surface, velocity and pressure)
• The end of the simulation (overturning, breaking and post-breaking) is computed with the VOF/Navier-Stokes model
Initial condition for the VOF/Navier-Stokes model
Propagation direction
Overturning stage
• The breaker jet occurs : bottom variations leads to the the wave camber and overturning
• Focusing of the energy at the center of the ridge because of the steepest slope : the breaker jet firstly occurs at the center
Overturning : slices along the x-axis
3-D aspect of overturning : the wave is begining to break at the center while the breaking point is not reached on the sides
Vertical cross-section along x at y = 0 m
Vertical cross-section along x at y = 2 m
Pressure at breaking point
X (m)
Z(m
)
17 17.5 18 18.5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4Rat: 0.00 0.18 0.36 0.54 0.71 0.89 1.07 1.25 1.43 1.61 1.79 1.96 2.14 2.32 2.50
Ratio between pressure and hydrostatic pressure at breaking point (t'=9.07)vertical cross-section along x (y=0)Due to large vertical
accelerations, the pressure is not hydrostatic in front of the wave
Ratio of the computed pressure to the hydrostatic pressure in the slice y = 0 m
Velocity field
High velocities in the breaker jet + high accelerations (4.9g)
Transversal velocity (slice z= 0.3 m) : focusing
17.5 18 18.5 19 19.5 20 20.5 21 21.5 22
X
Y
Z
V0.4096490.3754660.3412830.30710.2729170.2387340.2045510.1703680.1361850.1020030.06781970.0336368-0.000546068-0.0347289-0.0689118-0.103095-0.137278-0.17146-0.205643-0.239826-0.274009-0.308192-0.342375-0.376558-0.410741
Focalisation de la vitesse au centre du domaine z=0.3 m
Breaking
Conclusions concerning this simulation
• Mesh of 900,000 cells (x = 5 cm then 2.5 cm, y = 10 cm, z 1.5 cm in the breaking zone) : CPU time 5 days and 10 h on a Digital Dec alpha bi-processor 500 MHz
• Breaking simulation with SL-VOF 3-D consistent with the BIEM simulation before breaking (focusing, values of the physical fields, interface aspect …)
• Mass conservation : loss is 0.7%, Energy conservation: loss is 10 % loss of amplitude during shoaling and delay to break with respect to the time predicted by BIEM
• Errors : numerical diffusion (coarse mesh along y and x in the shoaling zone), single phase-flow run
Breaking
-1
-0.5
0
0.5
Z
17
18
19
20
21
22
X-2
-1
0
1
2
Y
Y
Z
X
t'=11.1
3-D aspect of breaking : impact occurs firstly at the center for
x=19.85 m and progressively on the sides
Post-breaking
X
Y
Z
The wave continues to collapse, the air tube is progressively crashed : the water jet is projected with high velocity along the slope
Water jet
• Development and validation of a 3-D interface tracking method in a CFD code
• PLIC modeling and lagrangian advection accurate and fast method when compared to classical VOF methods
• Efficient method for wave breaking applications
• Loss of energy during the shoaling stage : numerical diffusion of the CFD code (mesh, artificial viscosity, single phase flow …)
Conclusions
Energy loss: numerical diffusion
• Tests sur la discrétisation et les modes diphasique/monophasique
• Shoaling d’une onde solitaire en fond plat et évaluation de la perte d’énergie totale
Discretization
E One phase flow
E 2 phase flow
Total CPU time
t (CFL=1.2) Number of time steps
10 cm 1.46 % 1.39 % 1.5 s 0.075 20
5 cm 1.35 % 0.83 % 1.5 s 0.0375 40
2.5 cm 0.66 % 0.62 % 1.5 s 0.01875 80
Numerical accuracy (pure advection)
Ln(R)
Ln
(Err
1)
1 2 3
-7.5
-7
-6.5
-6
-5.5
-5
-4.5
-4
-3.5
Mean error as a function of Rfor the sphere advection
Computational data
Linear regression slope=-1.65
Advection of a sphere
Err1:corner
k k
nR
ErrE
.1
Order : 1.65
Critical VOF
y
x
n
n ²
6
1
yx
yyxx
nn
nnnn
6
²3²3
)(2
1yx nn
y
x
n
n ²
6
1
yx
yyxx
nn
nnnn
6
²3²3
))1()1(1(6
1 33yx
yx
nnnn
C1 C2 C3 C4 C5 C6
nx+ny < nz1-C3 1-C2 1-C1
nx+ny > nz 1-C3 1-C2 1-C1
With n(nx,ny,nz)
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