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IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER

IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER Charbel Farhat, Alex Main and Kevin Wang Department of Aeronautics and Astronautics Department of Mechanical Engineering Institute for Computational and Mathematical Engineering Stanford University

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IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER

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  1. IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER Charbel Farhat, Alex Main and Kevin Wang Department of Aeronautics and Astronautics Department of Mechanical Engineering Institute for Computational and Mathematical Engineering Stanford University Stanford, CA 94305

  2. OUTLINE • Design and implementation of implicit time-stepping for • fluid-fluid interaction • Numerical results and timing for the fluid-fluid solver • Shock tube problem • Turner Implosion problem • Design and implementation of implicit time-stepping for • fluid-structure interaction • Numerical and timing results for 2D version of IMP45

  3. 1 1 Fj,j+1 = Fj+1/2 (nj,j+1) = (Fj + Fj+1 )- | F’ |j+1/2 (Wj+1 – Wj) = Roe (Wj, Wj+1, gs, ps) (stiffened gas) 2 2 @(rf) (ruf) @ + = 0 @t @x COMPUTATIONAL FRAMEWORK • MUSCL-based solver with Roe Flux j + 1/2 j j + 1 • Interface capturing via the level-set equation (conservation form)

  4. FVM-ERS • FVM with exact local Riemann solver for multi-phase flows Wjn W*n W*n Wj+1n j - 1 j - 1/2 j j + 1/2 j + 1 - Fj,j+1 = Roe (Wjn, W*n, EOSj) Fj+1,j = Roe (Wj+1n, W*n, EOSj+1) - W*n and W*n determined from the exact solution of local two-phase Riemann problems C. Farhat, A. Rallu and S. Shankaran, "A Higher-Order Generalized Ghost Fluid Method for the Poor for the Three-Dimensional Two-Phase Flow Computation of Underwater Implosions", Journal of Computational Physics, Vol. 227, pp. 7674-7700 (2008)

  5. Wnpj+1 Wnpj • Exact solution of the analytical problem (Tait’s EOS) 1 1 (RR(pI; pR,rR) - RL(pI; pL,rL)) uI = (uL + uR) + 2 2 RL(pI; pL,rL) + RR(pI; pR,rR) + uR – uL = 0 pI, rIL, rIR, uI - Newton’s method LOCAL RIEMANN SOLVER • Wave structure and Riemann problem rIL,pI,uI ,rIR contact discontinuity rarefaction shock t gas water x j j + 1/2 j + 1 rLuL pL rRuR pR

  6. FVM-ERS (EXPLICIT) • GFMP with exact local Riemann solver j - 1/2 j + 1/2 j - 1 j j + 1 • If fjnfj+1n> 0 then Fj,j+1 = Fj+1,j = Roe (Wjn, Wj+1n, EOSj= EOSj+1) If fjnfj+1n< 0 then Fj,j+1 = Roe (Wjn, WjRn(rIL, pI, uI), EOSj) Fj+1,j = Roe (Wj+1n, W(j+1)Rn(rIR, pI, uI), EOSj+1) ~ Dt - Wjn+1 = Wjn - (Fj,j+1 - Fj,j-1) (forward Euler) Dx ~ - Unpack Wn+1 using fn and solve the level-set equation to get fn+1 - Pack Wpn+1 using fn+1 to get the updated solution Wn+1

  7. FVM-ERS (IMPLICIT) • Implicit Extension of FVM-ERS method j - 1/2 j + 1/2 j - 1 j j + 1 • If fjnfj+1n> 0 then Fj,j+1 = Fj+1,j = Roe (Wjn+1, Wj+1n+1,EOSj= EOSj+1) If fjnfj+1n< 0 then Fj,j+1 = Roe (Wjn+1, WjRn+1,EOSj) Fj+1,j = Roe (Wj+1n+1, W(j+1)Rn+1, EOSj+1) ~ Dt - Wjn+1 = Wjn - (Fj,j+1 - Fj,j-1) (backward Euler) Dx ~ - Unpack Wn+1 using fn and solve the level-set equation to get fn+1 - Pack Wpn+1 using fn+1 to get the updated solution Wn+1

  8. IMPLICIT FLUID-FLUID • Backward Euler advancement requires the solution of a • nonlinear equation • Use Newton’s method, which requires Jacobians of the • flux functions dFj,j+1 dFj,j+1 dWjn dFj,j+1 dW*n + = dpj dWjn dpj dW*n dpj dFj,j+1 dFj,j+1 dW*n = dpj+1 dW*n dpj+1 • Need Jacobians of two-phase Riemann problems

  9. STIFFENED GAS • Local two phase Riemann solver for stiffened gas (SG)- • stiffened gas requires the solution of the equation uL + FL(rL, pL;pI) = uIL = uIR = uR + FR(rR, pR; pI) • Taking the total differential yields derivatives of pI , uI dFL dFL dFL duL + drL dpL dpI + + drL dpL dpI dFR dFR dFR = duR + drR dpR dpI + + drR dpR dpI • Derivatives of rIR , rIL then come from the Riemann • invariants

  10. OTHER EOS • Also support Tait EOS for compressible liquids p = Arb + B • Jacobians for Tait-Tait, SG-Tait follow the same derivation • Perfect Gas (PG) is a subset of SG (with p = 0)

  11. p = A(1 - )e-R1+ B(1 - )e-R2 + wre wr wr R1r0 R2r0 r0 r0 r r JWL EOS • Jones-Wilkins-Lee (JWL) equation of state for modeling explosive products of combustion (and in particular Trinitrotoluene — a.k.a. TNT) where A, B, R1, R2, w and r0 are material constants - Highly nonlinear function p(r,e) - Presence of exponentials

  12. uL + FL(rL, pL;rIL) = uIL = uIR = uR + FR(rR, pR; rIR) GL(rL, pL; rIL) = pIL pIR= GR(rR, pR; rIR) = JWL EOS • Solution of exact Riemann problem involves a • system of two nonlinear equations (1) (2) • FL and GL depend on the nature of the interaction in the • phase modeled by the JWL EO • shock algebraic equation • rarefaction differential equation

  13. rIR,uIR ,pIR t rarefaction c(r,p) rR,uR ,pR = r x du + _ dr = s rw+1 p - Ae-R1+ Be-R2 r0 r0 r r SG-JWL RIEMANN SOLVER • Rarefaction wave in a JWL medium (k) • The isentropic evolution in the • rarefaction fan between two • constant states is given by (1) (2) complex Riemann problem • Algebraic entropy (s) formula for the JWL EOS • No obvious algebraic Riemann invariants for the JWL EOS • No analytical Jacobians of the invariants either

  14. JWL EOS • Riemann invariants are tabulated for the explicit • time stepping scheme • For implicit time-stepping, where Jacobians are • required, they are not tabulated; rather they are • computed on-line by solving an ODE • Relatively cheap compared to other aspects of • the simulation • Support both SG-JWL and JWL-JWL

  15. TIME INTEGRATORS • We support two different time integrators • Backward Euler • Three Point Backward Difference (3BDF) • Backward Euler estimates the time derivative at time • n+1 at node i by ~ dWi Win+1 - Win = (1) dt Dt • The integration of the fluid equations at time step n+1 • assumes that node i is of the same phase; thus • there is no problem

  16. 3BDF • 3BDF approximates the derivative at time step n+1 as ~ dWi a0Win+1 -a1Win + a2Win-1 = (2) dt Dt • But node i at time n-1 may be of a different phase • Because density can be discontinuous across a fluid • interface, Win-1 and Win are not necessarily related in • this case

  17. 3BDF • When node i has changed phase between time step • n and n+1, replace Win-1 with W*n-1 • Where W*n-1 is the exact solution of the two phase • Riemann problem on the upstream side of the interface • at node i at time step n-1 n-1 n W*n-1 i+1 i-2 i i-1

  18. 2fin+1 - 2fin = Dt d fi dt LEVEL SET 3BDF • A similar issue arises when we use the 3BDF integrator • on the level set • 3BDF requires fn+1, fn , and fn-1 • After reinitialization fn-1no longer exists • Solution is to use a special integrator 1 dfin - 2 dt • The final term can be estimated from the spatial • fluxes at time step n

  19. LIMITATIONS • The fluid interface may cross no more than one cell • per time step - Required to handle phase change • AERO-F automatically ensures this is not violated by reducing the time step as necessary

  20. r = 50 (kg/m3) r = 1000.0 (kg/m3) u= 0.0 (m/s) u= 0.0 (m/s) p= 105 (Pa) p= 109 (Pa) SHOCK TUBE PROBLEM • 1D Shock tube with air to the left, water to the right. • Air modeled as a perfect gas (g = 1.4); water modeled • as a stiffened gas (g = 4.4, p = 6.0x 108) Air Water • Simulation to t=1 x 10-5 s in 3D AERO-F code

  21. SHOCK TUBE RESULTS

  22. SHOCK TUBE RESULTS

  23. VALIDATION • Turner (2007): implosion of a glass sphere (D = 0.0762 m) (0.5m, 0.5m) Air (P = 105 Pa) (0, 0) Sensor Water (P = 6.996 MPa) z (0.5m, -0.5m) x

  24. TURNER IMPLOSION • Air modeled as a perfect gas (g = 1.4); water modeled • as a stiffened gas (g = 7.15, p = 2.89 x 108Pa) • 780,000 grid points • Simulation to t=0.5 ms

  25. TURNER RESULTS • Explicit (FE), CFL=0.5 • Implicit (3BDF), CFL=100

  26. TURNER RESULTS

  27. TURNER RESULTS

  28. TURNER TIMING • Simulation performed on a Linux cluster using • 168 processors • Speedup of 4.33

  29. pI, rIRus x = x(t) contact discontinuity not involved rarefaction* t fluid 1 fluid 2 x i j Mij Wnj rRuR pR EMBEDDED FLUID-STRUCTURE • For embedded fluid structure, fluid-fluid Riemann • problem is replaced by a fluid structure Riemann problem n w(x,0) =W , if x ≥ 0 j w F (w) = 0 + x t * could also be a shock u(x(t), t) = u (Mij)∙ nG(Mij) s

  30. pI, rIRus x = x(t) contact discontinuity not involved rarefaction* t fluid 1 fluid 2 x i j Mij Wnj rRuR pR ONE-SIDED RIEMANN PROBLEM • (Fluid 2, shell) problem • Closed form algebraic solution of the problem exists us = uR + R2(pI(2); pR , rR) - Closed form Jacobians exist as well

  31. FLUX COMPUTATION • The flux across the face at Mij is then given by G Mij i j fluid 1 fluid 2 Fji= Roe (us, pI(2), Wnj , EOS(2), uji) Fij= Roe (us, pI(1), Wni , EOS(1), uij )

  32. EMBEDDED FSI (IMPLICIT) • Implicit Extension of Embedded FSI method ~ Dt - Wjn+1 = Wjn - (Fn+1j,j+1 - Fn+1j,j-1) (backward Euler) Dx - Update uncovered nodes to compute Wjn+1 • Solve for Wjn+1 using Newton’s method • Requires Jacobians of fluid-structure Riemann problem - Closed form solution exists for stiffened gas

  33. 3BDF FOR FSI • The same difficulty exist when using 3BDF for • embedded fluid-structure • When node i has been uncovered, Win-1 does not • exist • In this case, use W*n-1as Win-1 • W*n-1 is the solution of the exact two phase • Riemann problem on the upstream side of the structure • boundary at node i at time step n-1 n-1 n W*n-1 structure i+1 i-2 i i-1

  34. 2D Imp45 • 2D Implosion problem • Simplified IMP45 using a thin slice of the aluminum tube air ( p = 14.5 psi ) water ( p = 1500 psi) • Explicit simulation uses dt = 0.75 x 10-8 • Implicit simulation uses dt = 3.0 x 10-6

  35. IMP45 RESULTS • Pressure at a sensing node

  36. IMP45 RESULTS • Pressure fields at t=0.4 ms • Clockwise from left: • Explicit (RK2), Implicit (BDF), • Implicit (BE)

  37. IMP45 TIMING • Simulation performed on a Linux cluster using • 64 processors • Speedup of 13.8, 10.9

  38. SUMMARY • Implicitization of fluid-fluid interaction in AEROF • Development of new scheme for three point backward • difference integration - Validation on shock tube and implosion problems - Speedups of ~ 4-5 • Equipment of the FSI solver in AERO-F with an • implicit integrator - Validation on 2D implosion problem - Speedups of ~12

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