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IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER

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## IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER

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**IMPLICITIZATION OF THE MULTIFLUID**SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER CharbelFarhat, 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**Implementation of implicit time-stepping for**• fluid-fluid interaction • Numerical results and timing for the fluid-fluid solver • Shock tube problem • Turner Implosion OUTLINE • Implementation of implicit time-stepping for • fluid-fluid interaction • Numerical results and timing for the embedded • fluid-structure solver • 2D Imp mode 45**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) (stiffenedgas) 2 2 (rf) @ (ruf) @ + = 0 @t @x COMPUTATIONAL FRAMEWORK • Finite volume method with MUSCL (Roe’s solver) j + 1/2 j j + 1 • Interface capturing via the level-set equation (conservation form)**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)**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**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**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) (backwardEuler) Dx ~ - UnpackWn+1 usingfn and solve the level-set equation to getfn+1 - Pack Wpn+1 using fn+1 to get the updated solution Wn+1**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 dWjndFj,j+1dW*n + = dpjdWjndpjdW*ndpj dFj,j+1 dFj,j+1dW*n = dpj+1dW*ndpj+1 • Need Jacobians of two-phase Riemann problems**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 drLdpL dpI duL + + + drLdpL dpI dFRdFR dFR drRdpR dpI = duR + + + drRdpR dpI • Derivatives of rIR , rILthen come from the Riemann • invariants**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)**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**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**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**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**Win+1 - Win**= Dt dWi dt 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 ~ (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**a0Win+1 -a1Win + a2Win-1**= Dt dWi dt 3BDF • 3BDF approximates the derivative at time step n+1 as ~ (2) • 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 Winare not necessarily related in • this case**3BDF**• When node ihas changed phase between time step • n and n+1, replace Win-1 with W*n-1 • Where W*n-1is the exact solution of the two phase • Riemann problem on the upstream side of the interface • at node iat time step n-1 n-1 n W*n-1 i+1 i-2 i i-1**2fin+1 - 2fin**= Dt dfi 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 reinitializationfn-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**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**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 = 6.0 x 108) Air Water • Simulation to t=1e-5 s in 3D AERO-F code**TURNER IMPLOSION**• Implosion of a spherical air bubble • 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 Air p=0.1 MPa • Simulation to t=0.5 ms Water p=7 MPa**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**TURNER RESULTS**• Explicit (FE), CFL=0.5 • Implicit (3BDF), CFL=100**TURNER TIMING**• Simulation performed on a Linux cluster using • 168 processors • Speedup of 4.33**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**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) - ClosedformJacobiansexist as well**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 )**EMBEDDED FSI (IMPLICIT)**• Implicit Extension of Embedded FSI method ~ Dt - Wjn+1 = Wjn - (Fn+1j,j+1 - Fn+1j,j-1) (backwardEuler) Dx - Update uncoverednodes to computeWjn+1 • Solve for Wjn+1 using Newton’s method • Requires Jacobians of fluid-structure Riemann problem - Closedform solution exists for stiffenedgas**3BDF FOR FSI**• The same difficulty exist when using 3BDF for • embedded fluid-structure • When node ihas been uncovered, Win-1 does not • exist • In this case, use W*n-1as Win-1 • W*n-1is the solution of the exact two phase • Riemann problem on the upstream side of the structure • boundary at node iat time step n-1 n-1 n W*n-1 structure i+1 i-2 i i-1**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**IMP45 RESULTS**• Pressure at a sensing node**IMP45 RESULTS**• Pressure fields at t=0.4 ms • Clockwise from left: • Explicit (RK2), Implicit (BDF), • Implicit (BE)**IMP45 TIMING**• Simulation performed on a Linux cluster using • 64 processors • Speedup of 13.8, 10.9**SUMMARY**• Implicitization of fluid-fluid interaction in AEROF • Development of new scheme for three point backward • differenceintegration - 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