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MURI REVIEW APR 2011 Charbel Farhat , Alex Main and Kevin Wang

MURI REVIEW APR 2011 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. ( rf ). @. ( r v f ). @. + = 0.

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MURI REVIEW APR 2011 Charbel Farhat , Alex Main and Kevin Wang

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  1. MURI REVIEW APR 2011 CharbelFarhat, 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. (rf) @ (rvf) @ + = 0 @t @x COMPUTATIONAL FRAMEWORK • Eulerian based • Comprehensive and robust embedded CFD method - fluid, fluid-fluid, fluid-structure, fluid-structure-fluid interfaces - robustness with respect to shock and contact strengths - blending of computational and analytical approaches - flow through cracks • Generalized level-set equation (conservation form) • Rigorous, conservative, and intersection-free method for enforcing • the Neumann transmission conditionsbased on the concept of a • surrogate material interface

  3. OUTLINE • Review of FVM-ERS method • Implementation of exact Riemann solver (ERS) for Tait • -JWL interfaces • Implementation of programmed burn method in AERO-F • Theory • Numerical Examples • Implementation of one dimensional simulations

  4. 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 • MUSCL-based solver with Roe Flux j + 1/2 j j + 1 • Interface capturing via the level-set equation (conservation form)

  5. 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)

  6. 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 EXACT 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

  7. EQUATIONS OF STATE • Stiffened Gas p = re(g - 1)+gp • Also support Tait EOS for compressible liquids p = Arb + B • Jones-Wilkins-Lee equation of state for explosive • products • Perfect Gas (PG) is a subset of SG (with p = 0)

  8. 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 - Highlynonlinearfunctionp(r,e) - Presence of exponentials

  9. uL + FL(rL, pL;rIL) = uIL = uIR = uR + FR(rR, pR; rIR) GL(rL, pL; rIL) = pIL pIR= GR(rR, pR; rIR) = TAIT-JWL ERS • FVM-ERS needs an exact Riemann solver for problems • involving interfaces between fluids modeled as Tait and • fluids modeled as JWL • 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 EOS • shock algebraic equation • rarefaction differential equation

  10. 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 TAIT-JWL ERS • 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

  11. JWL EOS • Riemann invariants can be tabulated for the JWL • equation of state • Tait-JWL is available for explicit and implicit time • stepping • Tabulation also works for implicit time stepping for • Tait-JWL (as well as SG-JWL) - Computederivatives of tabulatedquantitiesfromsparse grid tabulation directly

  12. SHOCK TUBE PROBLEM • 1D Shock tube with TNT gaseous products to the right at • high pressure, and water at low pressure to the left • Water modeled using Tait equation of state • ( r0 = 1000 kg/m3, p0 = 100 kPa, B = 331 MPa ) • TNT gaseous products modeled using JWL equation • of state • A = 548.4 GPa • B = 9.375 Gpa • R1 = 4.94 • R2 = 1.21 • w = 0.28 • r0 = 1630 kg/m3

  13. r = 1630 (kg/m3) r = 1000.0 (kg/m3) u= 0.0 (m/s) u= 0.0 (m/s) p= 7.81 x 109 (Pa) p= 105(Pa) SHOCK TUBE PROBLEM • pR = 7.81 GPa, rR= 1630 kg/m3, uR= 0 pL = 100 kPa, rL= 1000 kg/m3, uL= 0 Water TNT products • Simulation to t = 1.25 x 10-4s in 3D AERO-F code

  14. SHOCK TUBE PROBLEM • Compare to Fedkiw (1999)

  15. SHOCK TUBE PROBLEM • Compare to Fedkiw (1999)

  16. PROGRAMMED BURN • Simulate detonation of high explosives • Motivated because detonation products generally have • varying fluid state

  17. CHAPMAN-JOUGUET THEORY • The simplest model of detonation waves is • Chapman-Jouguet (CJ) theory. Unburned HE r0, p0, u0=0 Burned Products rCJ, pCJ, uCJ s - Detonationwave has constant velocity (s) - Assumes thatgaseousproductsjustbehind the detonationwave move atsonicvelocity - Implies s = uCJ + cCJ

  18. CHAPMAN-JOUGUET THEORY • Post-detonation state ( rCJ, pCJ, uCJ) then comes from • the Rankine-Hugoniot relations: rCJ (uCJ– s) = - r0 s pCJ+ rCJ(uCJ– s)2 = p0+ r0s2 eCJ+ + (uCJ– s)2= e0 + + s2 pCJ p0 1 1 rCJ r0 2 2 • Where e0 is the detonation energy associated with the • unburned high explosive

  19. PROGRAMMED BURN • Numerical method • Similar to “Change in EOS” method in GEMINI • Extended to handle more general cases • General idea – prescribe constant speed of detonation • front • When the detonation front passes a cell, change the • equation of state for that cell from unburned to burned

  20. PROGRAMMED BURN • Fluxes are computed with FVM-ERS • What about interface tracking? • Possible for 3 phases to meet at a single point

  21. INTERFACE TRACKING • We only track interface between explosive (burned AND • unburned) and water • Already know where interface between burned and • unburned explosive is!

  22. PHASE CHANGE • Phase change occurs when the level set at a node • changes sign - Nodeinsidedetonationwave -> burned explosive - Nodeoutsidedetonationwave-> unburned explosive • Also occurs when detonation wave passes a node • containing unburned explosive - Conservative variables are not changed - In contrast to usual FVM-ERS method

  23. EXPLOSIVE MODELING • Burned products can be modeled with either JWLor • ideal gas equations of state • Unburned explosive modeled with Tait equation of state p = Arb + B - A, B chosensothat (r0, p0) and (rCJ, pCJ) are admissible states - Internalenergy set to be e0 (detonationenergy)

  24. NUMERICAL EXAMPLES • One dimensional TNT planar burn • Three dimensional TNT spherical burn • Detonation of TNT cube • Asymmetrical (TNT) charge near flexible structure

  25. 1D PLANAR BURN • One dimensional TNT planar burn,surrounded by water • TNT gaseous products modeled using JWL equation of • state, water by stiffened gas (g = 4.4, p = 6.0 x 108) 0.25 m 0.5 m 0.25 m Explosive Water Water Explosive r0 = 1630 kg/m3 p0 = 100 kPa e0 = 3.68 MJ Water r0 = 1000 kg/m3 p0 = 100 kPa

  26. TNT MODEL • TNT modeled by JWL equation of state: • A = 548.4 GPa • B = 9.375 Gpa • R1 = 4.94 • R2 = 1.21 • w = 0.28 • r0 = 1630 kg/m3 • Chapman-Jouguet Theory says that • s = 6930 m/s • - pCJ = 20.9 GPa • - rCJ = 2100 kg/m3

  27. 1D PLANAR BURN

  28. SPHERICAL BURN • Simulation of a spherical (slice) charge in 3D 0.15 m 0.25 m Explosive r0 = 1630 kg/m3 p0 = 100 kPa e0 = 3.68 MJ Water r0 = 1000 kg/m3 p0 = 100 kPa

  29. SPHERICAL BURN

  30. SPHERICAL BURN • AERO-F burn results nearly identical to those obtained • by GEMINI • Burn pressure peak is slightly lower; AERO-F mesh is • slightly coarser • AERO-F mesh size – Dr = 0.5 cm • GEMINI mesh size – Dr = 0.25 cm • Discrepancy in the shock propagation in the water • AERO-F uses stiffened gas EOS • GEMINI uses Tillotson EOS

  31. ASYMETRICAL BURN • Detonation of a cube of TNT, surrounded by water Solid Symmetry Water Solid Symmetry Explosive • Domain is 1 m / side; charge is 0.1 m / side

  32. ASYMETRICAL BURN

  33. CHARGE NEAR STRUCTURE • Charge near thin slice of an aluminum tube water ( p = 1500 psi) air ( p = 14.5 psi ) TNT charge

  34. CHARGE NEAR STRUCTURE

  35. ONE DIMENSIONAL PROBLEMS • Spherical symmetry can be exploited for some problems • Much faster simulation times • One Dimensional solution can then be remapped onto • three dimensional domain

  36. SPHERICAL EULER EQUATIONS • Specifying a one dimensional problem in AERO-F • is almost the same as for three dimensional problems • Problem type must be set to OneDimensional • Either user specified 1D mesh, or automatically • generated uniform mesh • 1D solver saves restart data for AERO-F • 3D code “restarted” using 1D restart data

  37. ONE DIMENSIONAL PROBLEMS • Example: 1D simulation w/ AERO-F 1D results 3D domain Full 3D simulation w/ AERO-F Restart

  38. SPHERICAL EULER EQUATIONS • We use spherical shell volumes • FVM-ERS is still valid for • these volumes • Fixes appear to be • unnecessary • Simulations take a couple • of minutes

  39. NUMERICAL EXAMPLES • Spherical underwater explosion shock (TNT) • Spherically symmetric programmed burn • Remap capability

  40. TNT EXPLOSION SHOCK • Spherical underwater • explosion shock (TNT); c.f. • A.B. Wardlaw (1981) 0.32 m • Water modeled using • Tait EOS TNT r = 1630 kg/m3 p = 7.8039 MPa u = 0 • Uniform mesh, • Dr = 0.005 m 10 m • Simulation to • t=5 ms Water r = 1000 kg/m3 p = 100 kPa u = 0

  41. TNT EXPLOSION SHOCK • AERO-F (left), Wardlaw (right)

  42. PROGRAMMED BURN • Test case from GEMINI • manual • Water modeled using • stiffened gas EOS 5 m TNT r = 1630 kg/m3 u = 0 • Uniform mesh, • Dr = 0.25 m 8 m • Simulation to • t=7 ms Water r = 1000 kg/m3 p = 100 kPa u = 0

  43. PROGRAMMED BURN

  44. PROGRAMMED BURN • AERO-F burn results nearly identical to those obtained • by GEMINI • 1D pressure peak slightly higher than that for the 3D • mesh • Simulation time: ~ 30 seconds • Discrepancy in the shock propagation in the water • AERO-F uses stiffened gas EOS • GEMINI uses Tillotson EOS

  45. SOLUTION REMAP • Compute spherical solution to problem (TNT • explosion shock) • Compute spherical solution (1D) to t = 0.3 ms 16 cm 84 cm Water TNT products r = 1000 kg/m3 p = 100 kPa u = 0 r = 1630 kg/m3 p = 7.8039 MPa u = 0 One Dimensional Restart Data

  46. SOLUTION REMAP • Restart code in 3D using one dimensional restart • data 3D Mesh One Dimensional Restart Data 3D simulation w/ AERO-F 3D Results • Compute spherical solution (3D) to t = 0.5 ms

  47. SOLUTION REMAP • Three dimensional initial condition (after remap) • t = 0.3 ms

  48. SOLUTION REMAP • Three dimensional results, t = 0.481 ms • Note that this 3D mesh is very coarse (30,000 nodes)

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