Fluid-Structure Coupling in a Water-Wedge Impact Problem Nicolas AQUELET, Mhamed SOULI, Nicolas COUTY
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Fluid-Structure Coupling in a Water-Wedge Impact Problem Nicolas AQUELET, Mhamed SOULI, Nicolas COUTY ASME/JSME PVP Division Conference - San Diego - July 25 - 29, 2004 Plan What ’s the purpose of this approach? How to make the modeling? Fluid-Structure Coupling Application to Slamming problem Conclusion Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 2 What ’s the purpose of this approach? Why to modelize the impact between a wedge and a free surface? Answer: SLAMMING! But what’s SLAMMING?… Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 3 What ’s the purpose of this approach? 2D-model Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 4 What ’s the purpose of this approach? Bibliography: Some theoretical results Assumptions: 2D Problem : ( x , y , t ) Rigid wedge Constant drop velocity Incompressible and no rotational fluid No cushioning ?? p x Free surface a Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 5 What ’s the purpose of this approach? Bibliography: Some theoretical results Wagner (1932), Zhao et Faltinsen (1993): (Mpa) Asymptotical Approach valid for a < 40 Pressure = f(time) for a=30 Pressure = f(time) for a=10 at a fixed point of the at a fixed point of the wedge wedge (sec) Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 6 What ’s the purpose of this approach? Bibliography: Some theoretical results Dobrovol ’skaya (1969), Garabeddian (1953): If infinite wedge , the flow is self-similar (x,y,t) 3 unknowns x ( Vt , y ) Vt V: Constant drop velocity x 2 unknowns at t=t3 p Free surface at t=t2 Free surface at t=t1 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 7 What ’s the purpose of this approach? Bibliography: Some theoretical results Dobrovol ’skaya (1969), Garabeddian (1953): If infinite wedge , the flow is self-similar 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 Free surface at t = 0sec Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 8 Plan What ’s the purpose of this approach? How to make the modeling? Fluid-Structure Coupling Application to Slamming problem Conclusion Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 9 How to make the modeling? Material movement Modeling problems: Large fluid deformations Fluid-Structure Interactions State n Two solutions: Lagrangian Formulation Lagrangian Modeling of Fluid Fluid /Structure Contact Eulerian Formulation Eulerian Modeling of Fluid Fluid/Structure Coupling Performed by LS-DYNA: Explicite Finite Element code State n+1 State n+1 ASME/JSME PVP Conference - July 25-29, 2004 10 How to make the modeling? Lagrangian Formulation STRUCTURE: Dynamic equations of the structure FLUID: Lagrangian Formulation of Navier-Stokes Equations vi = ij , j t vi = ij , j t e = ij . ij t e = ij . ij t Mass conservation is automatically verified Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 11 How to make the modeling? Lagrangian Formulation Modeling of water flow with a Lagrangian Formulation: Strong distortions of fluid meshes Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 12 How to make the modeling? Eulerian formulation STRUCTURE: FLUID: Dynamic equations of the structure Eulerian Formulation of Navier-Stokes Equations vi = ij , j t = .div( v ) v j . t x j e = ij . ij t vi vi = ij , j .v j t x j e e = ij . ij .v j t x j Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 13 How to make the modeling? Split operator : 1st phase : Lagrangian cycle Material movement vi = ij , j t State n e = ij . ij t Intermediate state Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 14 How to make the modeling? Split operator : 2nd phase : cycle of advection Equations of transport V . = 0 t (0, x) = Lagrangian solved by Godunov ’s methods Performed by LS-DYNA: Explicite Finite Element code Intermediate state State n+1 ASME/JSME PVP Conference - July 25-29, 2004 15 How to make the modeling? Approche Eulérienne multi-matérielle Introduction of a new unknown: the volume fraction : Volume Fraction = Volwater Volelement V . = 0 t (0, x) = Lagrangian Performed by LS-DYNA: Explicite Finite Element code air water Intermediate state 1 0.7 Etat n+1 ASME/JSME PVP Conference - July 25-29, 2004 16 How to make the modeling? Free surface tracking by Young method (VOF: Volume Of Fluid) air 0.71 0 ? 0 1 0.3 0 1 1 0.5 Volume Fractions for 9 cells are used to compute the slope of the material interface in the centre cell water Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 17 How to make the modeling? Lagrangian Formulation Eulerian Formulation contact >>> Transmission of Interaction forces: Coupling>>> Transmission of Interaction forces: structure nodes to fluid nodes structure nodes to fluid particles Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 18 How to make the modeling? Lagrangian Formulation Geometric interface Eulerian Formulation Material interface contact >>> Transmission of Interaction forces: Coupling>>> Transmission of Interaction forces: structure nodes to fluid nodes structure nodes to fluid particles Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 19 Plan What ’s the purpose of this approach? How to make the modeling? Fluid-Structure Coupling Application to Slamming problem Conclusion Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 20 Fluid-Structure Coupling At t = t n-1/2 n , no yet coupling and At t = t , the velocity field is computed: F is added to the forces applied to the fluid particle Structure penetration Fluid particle at the structure zoom node position zoom Computation of the relative distance Vf Vs n d =d n-1 +(Vs-Vf).dt (here d n-1 Performed by LS-DYNA: Explicite Finite Element code =0) n F = -k.d k ASME/JSME PVP Conference - July 25-29, 2004 21 Fluid-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 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 22 Fluid-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 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 23 Fluid-Structure Coupling At t = t n as M n Fs n : n af K Mf s n n Ff d n n n with a = s, f Ma aa = Fa Kd n1/2 n1 n n1/2 Vf d = d Vs .dt = 0 n n-1/2 n-1/2 n n F M M V V F d s f s f Kdn = f s 2 Ms Mf Ms Mf dt dt Zhong’s work for contact-impact (1993): Defence node Algorithm Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 24 Fluid-Structure Coupling Numerical example:Impact of water column water = 1026kg.m 3 V0 V0 = 5m.s 1 c = 1500m.s 1 Excepted Pressure in the Eulerian cells near the wall: cV0 = 7.7MPa Rigid Wall Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 25 Fluid-Structure Coupling Pressure at the impact for different timestep: dt=6e-7sec dt=1e-6sec dt=6e-8sec Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 26 Fluid-Structure Coupling Comparison with a model of reference : V0 V0 Rigid Wall Model with coupling Performed by LS-DYNA: Explicite Finite Element code Eulerian nodes are blocked Model of reference ASME/JSME PVP Conference - July 25-29, 2004 27 Fluid-Structure Coupling Comparison with a model of reference : Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 28 Fluid-Structure Coupling An other numerical Example:Piston V0 Structure V0: constant Fluid Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 29 Fluid-Structure Coupling Model of reference Model with coupling V0 V0 is imposed on the is imposed on the Fluid boundary Performed by LS-DYNA: Explicite Finite Element code Structure ASME/JSME PVP Conference - July 25-29, 2004 30 Fluid-Structure Coupling Comparison with a model of reference : Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 31 Plan What ’s the purpose of this approach? How to make the modeling? Fluid-Structure Coupling Application to Slamming problem Conclusion Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 32 Application to Slamming problem V0=6m/s element 50 30° Performed by LS-DYNA: Explicite Finite Element code Reference theoretical pressure plotted away from the edges ASME/JSME PVP Conference - July 25-29, 2004 33 Application to Slamming problem Comparison theory/coupling The results disagree and the numerical curve is perturbed Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 34 Application to Slamming problem Comparison theory/coupling by decreasing the time step The results still disagree and the perturbations are stronger than previously Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 35 Application to Slamming problem Comparison of pressures applied on two neighbouring structure elements The curves are almost « symmetrical » self-similarity is not respected Influence between the structure element pressures Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 36 Application to Slamming problem Interessant approach: Impulse = momentum transmitted to the structure (by unit area) t Impulse: I = p.dt t0 t0 Performed by LS-DYNA: Explicite Finite Element code t ASME/JSME PVP Conference - July 25-29, 2004 37 Application to Slamming problem Comparison of impulses applied on two neighbouring structure elements Influence between the structure element impulses Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 38 Application to Slamming problem The coupling force computed by the Zhong ’s approach seems to be too strong n n-1/2 n-1/2 n n F M M V V F d s f s f Kdn = pf f s 2 Ms Mf Ms Mf dt dt A penalty factor is introduced in the Zhong ’s formula: 0<pf<1 The previous pressure and impulse curves are plotted again by decreasing the penalty factor Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 39 Application to Slamming problem Comparison of pressures applied on two neighbouring structure elements pf=0.1 Influence between the structure element pressures Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 40 Application to Slamming problem Comparison of impulses applied on two neighbouring structure elements pf=0.1 Influence between the structure element impulses Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 41 Application to Slamming problem Comparison of pressures applied on two neighbouring structure elements pf=0.01 Small Influence between the structure element pressures Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 42 Application to Slamming problem Comparison of impulses applied on two neighbouring structure elements pf=0.01 Small Influence between the structure element impulses Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 43 Application to Slamming problem Comparison of pressures applied on two neighbouring structure elements Self-similarity is respected pf=0.001 Very small Influence between the structure element pressures Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 44 Application to Slamming problem Comparison of impulses applied on two neighbouring structure elements pf=0.001 Very small Influence between the structure element impulses Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 45 Application to Slamming problem Comparison theory/coupling with pf=0.001 The jet reaches the trailing edge The theoretical and numerical pressures agree Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 46 Application to Slamming problem A mesh refinement enables to converge more quickly element 50 element 20 Reference theoretical pressure plotted away from the edges pf=0.001 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 47 Application to Slamming problem Comparison of impulses for small penalty factor: The momentum received by the structure changes little Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 48 Application to Slamming problem Deformable wedge: Comparison pf= 0.1 / pf=0.01 Von mises stress history Displacement history Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 49 Application to Slamming problem Von Mises stress history for pf= 0.1 / pf=0.01 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 50 Application to Slamming problem Displacement of node 132 for pf= 0.1 / pf=0.01 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 51 Application to Slamming problem Displacement of node 170 for pf= 0.1 / pf=0.01 Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 52 Plan What ’s the purpose of this approach? How to make the modeling? Fluid-Structure Coupling Application to Slamming problem Conclusion Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 53 Conclusion 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. Performed by LS-DYNA: Explicite Finite Element code ASME/JSME PVP Conference - July 25-29, 2004 54