Fluid-Structure Coupling in a Water-Wedge Impact Problem

ASME/JSME PVP Division Conference - San Diego - July 25 - 29, 2004

Fluid-Structure Coupling

in a Water-Wedge

Impact Problem

Nicolas AQUELET, Mhamed SOULI, Nicolas COUTY

2ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code

PlanPlan

What ’s the purpose of this approach?

How to make the modeling?


Application to Slamming problem

Conclusion


What ’s the purpose of this What ’s the purpose of this approach?approach?

Why to modelize the impact

between a wedge and a free surface?

Answer: SLAMMING!

But what’s SLAMMING?…



2D-model



Bibliography: Some theoretical results

2D Problem : ( x , y , t )

Rigid wedge

Constant drop velocity

Incompressible and no rotational fluid

No cushioning

Assumptions:

p

x

??

Free surface



Wagner (1932), Zhao et Faltinsen (1993):

Asymptotical Approach valid for

(Mp

a)

(sec)

Pressure = f(time) for

at a fixed point of the wedge


Pressure = f(time) for

at a fixed point of the wedge



Dobrovol ’skaya (1969), Garabeddian (1953):

If infinite wedge , the flow is self-similar

p

x

Free surface at t=t1

Free surface at t=t2

at t=t3

( x , y , t ) x y

Vt Vt ,( )

3 unknowns 2 unknowns

V: Constant drop velocity




Free surface at t = 0sec

For a finite wedge, this property is valid away from the edges

Away from the leading edge :

Incompressibility?

Away from trailing edge

Free surface when the jet leaves the wedge

Bibliography: Some theoretical resultsDobrovol ’skaya (1969), Garabeddian (1953):

If infinite wedge , the flow is self-similar


PlanPlan





Conclusion


State n+1

State n+1

How to make the modeling?How to make the modeling?

Modeling problems: Large fluid deformations Fluid-Structure Interactions

Two solutions:Lagrangian Formulation Lagrangian Modeling of Fluid Fluid /Structure ContactContactEulerian Formulation Eulerian Modeling of Fluid Fluid/Structure Coupling Coupling

Material movement

State n



STRUCTURE:

Dynamic equations of the structure

j,iji

t

v

ijij.t

e

FLUID:

j,iji

t

v

ijij.t

e

Lagrangian Formulation

Mass conservation is automatically verified

Lagrangian Formulation of Navier-Stokes Equations



Modeling of water flow with a Lagrangian Formulation:

Strong distortions of fluid meshes

Lagrangian Formulation



FLUID:

jj x.v)v(div.

t

j

ijj,ij

i

x

vv.

t

v

jjijij x

ev..

t

e

Eulerian formulation

STRUCTURE:Dynamic equations of the structure

j,iji

t

v

ijij.t

e

Eulerian Formulation of Navier-Stokes Equations



j,iji

t

v

ijij.t

e

Split operator :

1st phase : Lagrangian cycle

State n

Intermediate state

Material movement



State n+1

2nd phase : cycle of advection

Intermediate state

Lagrangianx

Vt

),0(

0.

Equations of transport

solved by Godunov ’s methods

Split operator :



Etat n+1

Introduction of a new unknown: the volume fraction

Intermediate state

Lagrangianx

Vt

),0(

0.

Approche Eulérienne multi-matérielle

water airVolume Fraction = Volwater

Volelement

1 0.7



0.3

0.5

0.71

1

11

0 0

0

water

air

Free surface tracking

by Young method (VOF: Volume Of Fluid)

Volume Fractions for 9 cells are used to compute the

slope of the material interface in the centre cell

?



Lagrangian Formulation Eulerian Formulation

Coupling>>> Coupling>>> Transmission of Interaction forces:

structure nodes to fluid structure nodes to fluid particlesparticles

contact >>>contact >>> Transmission of Interaction forces:

structure nodes to fluid nodes


Geometric interface Material interface


Lagrangian Formulation Eulerian Formulation

Coupling>>> Coupling>>> Transmission of Interaction forces:

structure nodes to fluid structure nodes to fluid particlesparticles

contact >>>contact >>> Transmission of Interaction forces:

structure nodes to fluid nodes


PlanPlan





Conclusion


Computation of the relative distance

d = d +(Vs-Vf).dt

(here d =0)

n n-1

Fluid-Structure CouplingFluid-Structure Coupling

At t = t , no yet coupling and

the velocity field is computed:

Structure

VsVf

zoom

Fluid particle

at the

structure

node position

n-1/2

n-1

nAt t = t ,

F is added to the forces

applied to the fluid particle

zoom

kF = -k.d

penetration

n



K???Which K to respect the physical solution

of the interaction problem?

In theory:

the bigger the stiffness K,

the smaller the penetration d



However:

If the stiffness K is too bigger,

the run becomes unstable

And:

If the stiffness K is too smaller,

the penetration becomes unacceptable



0.

,

dtVVdd

fsKdFaM1/2n

f

1/2n

sn1n

nnn with

Zhong’s work for contact-impact (1993): Defence node Algorithm

2

n1/2n-f

1/2n-s

f

nf

s

ns

fs

fsn

dt

d

dt

V-V

M

F

M

F

MM

MMKd

d

KMs Mf

FfFs

as af

n

n n

n n

At t = t :n



Numerical example:Impact of water column

water

Rigid Wall

V0Excepted Pressure

in the Eulerian cells near the wall:

MPacV 7.70

10 .5 smV

3.1026 mkg

1.1500 smc



Pressure at the impact for different timestep:

dt=1e-6secdt=6e-7sec

dt=6e-8sec



Comparison with a model of reference :

Rigid Wall

Model with coupling Model of reference

Eulerian nodes are blocked

V0 V0






An other numerical Example:Piston

Structure

Fluid

V0

V0: constant



Model of reference Model with coupling

V0 is imposed on the Fluid boundary

V0 is imposed on the Structure





PlanPlan





Conclusion


Application to Slamming problemApplication to Slamming problem

30°

V0=6m/s

Reference theoretical pressure plotted away from the edges

element 50



Comparison theory/coupling

The results disagree and the numerical curve is perturbed



Comparison theory/coupling by decreasing the time step

The results still disagree and the perturbations are stronger than previously



Comparison of pressures applied on two neighbouring structure elements

The curves are almost

« symmetrical »

Influence between the structure element pressures

self-similarity is not respected


Application to Slamming problemApplication to Slamming problem Interessant approach: Impulse = momentum

transmitted to the structure (by unit area)

Impulse:

tt0

t

t

dtpI0

.



Comparison of impulses applied on two neighbouring structure elements

Influence between the structure element impulses



2

n1/2n-f

1/2n-s

f

nf

s

ns

fs

fsn

dt

d

dt

V-V

M

F

M

F

MM

MMKd pf

A penalty factor is introduced in the Zhong ’s formula: 0<pf<1

The coupling force computed by the Zhong ’s approach seems to be too strong

The previous pressure and impulse curves are plotted again by decreasing the penalty factor




Influence between the structure element pressures

pf=0.1




Influence between the structure element impulses

pf=0.1




Small Influence between the structure element pressures

pf=0.01




Small Influence between the structure element impulses

pf=0.01




Very small Influence between the structure element pressures

pf=0.001

Self-similarity is respected




Very small Influence between the structure element impulses

pf=0.001



Comparison theory/coupling with

The theoretical and numerical pressures agree

pf=0.001

The jet reaches the trailing edge



Reference theoretical pressure plotted away from the edges

element 20

element 50

A mesh refinement enables to converge more quickly

pf=0.001



Comparison of impulses for small penalty factor:

The momentum received by the structure changes little



Deformable wedge: Comparison

Von mises stress history

Displacement history

pf= 0.1 / pf=0.01



Von Mises stress history for pf= 0.1 / pf=0.01



Displacement of node 132 for pf= 0.1 / pf=0.01



Displacement of node 170 for pf= 0.1 / pf=0.01


PlanPlan





Conclusion


ConclusionConclusion

Zhong ’s coupling converge to solutions for simple problems

However, a penalty factor is required for slamming problem

The less the penalty factor is, the less the oscillations in the coupling forces are. A good agreement with the theory is obtained.

The impulse for different penalty factor is almost identical. Thus, the deformations of the structure for two different stiffness are close.

Documents

Fluid-Structure Coupling in a Water-Wedge Impact Problem