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?
Fluid-Structure Coupling
Application to Slamming problem
Conclusion
3ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
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?…
4ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
What ’s the purpose of this What ’s the purpose of this approach?approach?
2D-model
5ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
What ’s the purpose of this What ’s the purpose of this approach?approach?
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
6ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
What ’s the purpose of this What ’s the purpose of this approach?approach?
Wagner (1932), Zhao et Faltinsen (1993):
Asymptotical Approach valid for
(Mp
a)
(sec)
Pressure = f(time) for
at a fixed point of the wedge
Bibliography: Some theoretical results
Pressure = f(time) for
at a fixed point of the wedge
7ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
What ’s the purpose of this What ’s the purpose of this approach?approach?
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
Bibliography: Some theoretical results
8ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
What ’s the purpose of this What ’s the purpose of this approach?approach?
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
9ASME/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?
Fluid-Structure Coupling
Application to Slamming problem
Conclusion
10ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
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
11ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
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
12ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
Modeling of water flow with a Lagrangian Formulation:
Strong distortions of fluid meshes
Lagrangian Formulation
13ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
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
14ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
j,iji
t
v
ijij.t
e
Split operator :
1st phase : Lagrangian cycle
State n
Intermediate state
Material movement
15ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
State n+1
2nd phase : cycle of advection
Intermediate state
Lagrangianx
Vt
),0(
0.
Equations of transport
solved by Godunov ’s methods
Split operator :
16ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
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
17ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
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
?
18ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
How to make the modeling?How to make the modeling?
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
19ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Geometric interface Material interface
How to make the modeling?How to make the modeling?
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
20ASME/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?
Fluid-Structure Coupling
Application to Slamming problem
Conclusion
21ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
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
22ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
K???Which K to respect the physical solution
of the interaction problem?
In theory:
the bigger the stiffness K,
the smaller the penetration d
23ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
However:
If the stiffness K is too bigger,
the run becomes unstable
And:
If the stiffness K is too smaller,
the penetration becomes unacceptable
24ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
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
25ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
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
26ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
Pressure at the impact for different timestep:
dt=1e-6secdt=6e-7sec
dt=6e-8sec
27ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
Comparison with a model of reference :
Rigid Wall
Model with coupling Model of reference
Eulerian nodes are blocked
V0 V0
28ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
Comparison with a model of reference :
29ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
An other numerical Example:Piston
Structure
Fluid
V0
V0: constant
30ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
Model of reference Model with coupling
V0 is imposed on the Fluid boundary
V0 is imposed on the Structure
31ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Fluid-Structure CouplingFluid-Structure Coupling
Comparison with a model of reference :
32ASME/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?
Fluid-Structure Coupling
Application to Slamming problem
Conclusion
33ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
30°
V0=6m/s
Reference theoretical pressure plotted away from the edges
element 50
34ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison theory/coupling
The results disagree and the numerical curve is perturbed
35ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison theory/coupling by decreasing the time step
The results still disagree and the perturbations are stronger than previously
36ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
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
37ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem Interessant approach: Impulse = momentum
transmitted to the structure (by unit area)
Impulse:
tt0
t
t
dtpI0
.
38ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of impulses applied on two neighbouring structure elements
Influence between the structure element impulses
39ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
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
40ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of pressures applied on two neighbouring structure elements
Influence between the structure element pressures
pf=0.1
41ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of impulses applied on two neighbouring structure elements
Influence between the structure element impulses
pf=0.1
42ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of pressures applied on two neighbouring structure elements
Small Influence between the structure element pressures
pf=0.01
43ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of impulses applied on two neighbouring structure elements
Small Influence between the structure element impulses
pf=0.01
44ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of pressures applied on two neighbouring structure elements
Very small Influence between the structure element pressures
pf=0.001
Self-similarity is respected
45ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of impulses applied on two neighbouring structure elements
Very small Influence between the structure element impulses
pf=0.001
46ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison theory/coupling with
The theoretical and numerical pressures agree
pf=0.001
The jet reaches the trailing edge
47ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Reference theoretical pressure plotted away from the edges
element 20
element 50
A mesh refinement enables to converge more quickly
pf=0.001
48ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Comparison of impulses for small penalty factor:
The momentum received by the structure changes little
49ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Deformable wedge: Comparison
Von mises stress history
Displacement history
pf= 0.1 / pf=0.01
50ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Von Mises stress history for pf= 0.1 / pf=0.01
51ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Displacement of node 132 for pf= 0.1 / pf=0.01
52ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
Application to Slamming problemApplication to Slamming problem
Displacement of node 170 for pf= 0.1 / pf=0.01
53ASME/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?
Fluid-Structure Coupling
Application to Slamming problem
Conclusion
54ASME/JSME PVP Conference - July 25-29, 2004Performed by LS-DYNA: Explicite Finite Element code
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.