1. IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER Charbel Farhat, Arthur Rallu, 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  Implementation of implicit time-stepping for fluid-fluid interaction  Numerical results and timing for the fluid- fluid solver  Shock tube problem  Turner Implosion  Numerical results and timing for the embedded fluid-structure solver  2D Imp mode 45  Implementation of implicit time-stepping for fluid-fluid interaction

3.  Finite volume method with MUSCL (Roe’s solver) 1 2 1 2  j,j+1 =  j+1/2 ( j,j+1 ) = (F j + F j+1 )- | F’ | j+1/2 (W j+1 – W j ) = Roe (W j, W j+1,  s, p s ) (stiffened gas) jj + 1 j + 1/2  Interface capturing via the level-set equation COMPUTATIONAL FRAMEWORK + = 0 @t@t (  ) @ @x@x @ (u)(u) (conservation form)

4.  FVM with exact local Riemann solver for multi-phase flows jj + 1j - 1j + 1/2j - 1/2 WjnWjn -  j,j+1 = Roe (W j n, W * n, EOS j ) W*nW*n  j+1,j = Roe (W j+1 n, W * n, EOS j+1 ) W * n W j+1 n FVM-ERS 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) - W * n and W * n determined from the exact solution of local two-phase Riemann problems

5.  Wave structure and Riemann problem x t rarefaction contact discontinuity shock watergas  L u L p L  IL,p I,u I,  IR jj + 1j + 1/2  R u R p R W n pj+1 W n pj R L (p I ; p L,  L ) +R R (p I ; p R,  R ) + u R – u L = 0 u I = (u L + u R ) + ( R R (p I ; p R,  R ) -R L (p I ; p L,  L ) ) 22 11  Exact solution of the analytical problem (Tait’s EOS) - Newton’s method  p   IL,  IR, u I LOCAL RIEMANN SOLVER

6. tt - W j n+1 = W j n - (  j,j+1 -  j,j-1 ) (forward Euler) xx  GFMP with exact local Riemann solver ~ - Unpack W n+1 using  n and solve the level-set equation to get  n+1 ~ - Pack W p n+1 using  n+1 to get the updated solution W n+1 jj + 1j - 1 j + 1/2j - 1/2 - If  j n  j+1 n > 0 then  j,j+1 =  j+1,j = Roe (W j n, W j+1 n, EOS j = EOS j+1 ) If  j n  j+1 n < 0 then  j,j+1 = Roe (W j n, W j R n (  IL, p I, u I ), EOS j )  j+1,j = Roe (W j+1 n, W (j+1) R n (  IR, p I, u I ), EOS j+1 ) FVM-ERS (EXPLICIT)

7. tt - W j n+1 = W j n - (  j,j+1 -  j,j-1 ) (backward Euler) xx  Implicit Extension of FVM-ERS method ~ - Unpack W n+1 using  n and solve the level-set equation to get  n+1 ~ - Pack W p n+1 using  n+1 to get the updated solution W n+1 jj + 1j - 1 j + 1/2j - 1/2 - If  j n  j+1 n > 0 then  j,j+1 =  j+1,j = Roe (W j n+1, W j+1 n+1,EOS j = EOS j+1 ) If  j n  j+1 n < 0 then  j,j+1 = Roe (W j n+1, W j R n+1,EOS j )  j+1,j = Roe (W j+1 n+1, W (j+1) R n+1, EOS j+1 ) FVM-ERS (IMPLICIT)

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 d  j,j+1 d  j,j+1 dW j n d  j,j+1 dW * n dp j dW j n dp j dW * n dp j + = d  j,j+1 d  j,j+1 dW * n dp j+1 dW * n dp j+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 u L + F L (  L, p L ; p I ) = u IL u IR = u R + F R (  R, p R ; p I ) = dF L dF L dF L  Taking the total differential yields derivatives of p I, u I d  L dp L dp I du L + d  L dp L dp I ++ dF R dF R dF R d  R dp R dp I = du R + d  R dp R dp I ++  Derivatives of  IR,  IL then come from the Riemann invariants

10.  Jacobians for Tait-Tait, SG-Tait follow the same derivation OTHER EOS  Perfect Gas (PG) is a subset of SG (with  )  Also support Tait EOS for compressible liquids p = A   + B

11. 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, R 1, R 2,  and   are material constants p = A(1 - )e -R 1 + B(1 - )e -R 2 +  e  R1R1 R2R2     - Highly nonlinear function p( ,e) - Presence of exponentials

12.  Solution of exact Riemann problem involves a system of two nonlinear equations u L + F L (  L, p L ;  IL ) = u IL u IR = u R + F R (  R, p R ;  IR ) G L (  L, p L ;  IL ) = p IL p IR = G R (  R, p R ;  IR ) = = - F L and G L depend on the nature of the interaction in the phase modeled by the JWL EO shock algebraic equation rarefaction differential equation JWL EOS (1) (2)

13.  Rarefaction wave in a JWL medium - Algebraic entropy (s) formula for the JWL EOS - No obvious algebraic Riemann invariants for the JWL EOS -No analytical Jacobians of the invariants either x t rarefaction  R,u R,p R  IR,u IR,p IR - The isentropic evolution in the rarefaction fan between two constant states is given by complex Riemann problem  c( ,p ) +_ du dd =   p - Ae -R 1 + Be -R 2     = s (k) SG-JWL RIEMANN SOLVER (1)(2)

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 tt W i n+1 - W i n dW i dt = (1)  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 tt   W i n+1 -   W i n +   W i n-1 dW i dt = (2)  But node i at time n-1 may be of a different phase  Because density can be discontinuous across a fluid interface, W i n-1 and W i n are not necessarily related in this case ~

17. 3BDF  When node i has changed phase between time step n and n+1, replace W i n-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 i-1 i i-2 n-1n i+1 W * n-1

18. LEVEL SET 3BDF  A similar issue arises when we use the 3BDF integrator on the level set  3BDF requires  n+1,  n, and  n-1  After reinitialization  n-1 no longer exists  Solution is to use a special integrator tt  i n+1 - 2  i n d  i dt =  1 d  i n 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  AERO-F automatically ensures this is not violated by reducing the time step as necessary - Required to handle phase change

20. SHOCK TUBE PROBLEM  1D Shock tube with air to the left, water to the right.  Air modeled as a perfect gas (  ); water modeled as a stiffened gas (  x 10 8 )  Simulation to t=1e-5 s in 3D AERO-F code  = 50 (kg/m 3 ) u  = 0.0 (m/s) p  = 10 5 (Pa)  = 1000.0 (kg/m 3 ) u  = 0.0 (m/s) p  = 10 9 (Pa) AirWater

21. SHOCK TUBE RESULTS

23. TURNER IMPLOSION  Implosion of a spherical air bubble  Air modeled as a perfect gas (  ); water modeled as a stiffened gas (  2.89 x 10 8  Pa)  780,000 grid points  Simulation to t=0.5 ms Air p=0.1 MPa Water p=7 MPa

24. VALIDATION  Turner (2007): implosion of a glass sphere (D = 0.0762 m) Air (P = 10 5 Pa) Water (P = 6.996 MPa) x z (0.5m, 0.5m) (0.5m, -0.5m) (0, 0)Sensor

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

26. TURNER RESULTS

28. TURNER TIMING MethodCPU time Explicit (FE)17867 s Implicit (BE)4130 s  Simulation performed on a Linux cluster using 168 processors  Speedup of 4.33

29. EMBEDDED FLUID-STRUCTURE  For embedded fluid structure, fluid-fluid Riemann problem is replaced by a fluid structure Riemann problem * could also be a shock x t rarefaction * contact discontinuity fluid 2fluid 1 ij M ij  R u R p R WnjWnj x = x ( t ) not involved p I,  IR  u s t + = 0 w (w ) F  w(  =  W, if  ≥  j n u(x ( t ), t) = u (M ij ) ∙  (M ij ) s

30. u s = u R + R 2 (p I (2) ; p R,  R ) - Closed form Jacobians exist as well  (Fluid 2, shell) problem x t rarefaction * contact discontinuity fluid 2fluid 1 ij M ij  R u R p R WnjWnj x = x ( t ) not involved p I,  IR  u s ONE-SIDED RIEMANN PROBLEM - Closed form algebraic solution of the problem exists

31.  The flux across the face at M ij is then given by FLUX COMPUTATION  ji  = Roe (u s, p I (2), W n j, EOS (2),  ji  )  ij  = Roe (u s, p I (1), W n i, EOS (1),  ij ) ij  M ij fluid 1fluid 2

32. tt - W j n+1 = W j n - (  n+1 j,j+1 -  n+1 j,j-1 ) (backward Euler) xx  Implicit Extension of Embedded FSI method ~ EMBEDDED FSI (IMPLICIT) - Update uncovered nodes to compute W j n+1  Solve for W j n+1 using Newton’s method  Requires Jacobians of fluid-structure Riemann problem - Closed form solution exists for stiffened gas

33. 3BDF FOR FSI  In this case, use W * n-1 as W i n-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  The same difficulty exist when using 3BDF for embedded fluid-structure  When node i has been uncovered, W i n-1 does not exist i-1 i i-2 n-1n i+1 W * n-1 structure

34. 2D Imp45  2D Implosion problem air ( p = 14.5 psi ) water ( p = 1500 psi)  Simplified IMP45 using a thin slice of the aluminum tube  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 MethodCPU time Explicit (FE)12153 s Implicit (BE)882 s Implicit (3BDF)1115 s  Simulation performed on a Linux cluster using 64 processors  Speedup of 13.8, 10.9

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