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)