LA-UR-02-5574 LA-UR-02-5574 The Impact of Different Approaches The Impact of Different Approaches to Imposing Pressure Equilibrium in to Imposing Pressure Equilibrium in Multimaterial Godunov Methods Multimaterial Godunov Methods Bill Rider Bill Rider rider@lanl.gov Los Alamos National Laboratory 23 September, 2002 Numerical Methods for Compressible Multimaterial Flows Paris, France Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 1
LA-UR-02-5574 Overview � To start to assess the capability of different approaches to achieving closure in multimaterial flows � One wants to allow for more general behavior than simple models: � independent energies (differing temperatures) � Non-equilibrium on cell level � Differing pressures and velocities Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 2
LA-UR-02-5574 Introduction � Interface behavior is fundamental - tracking is often necessary (desired) � Commonly used approaches: � Volume of fluid - Multi-Fluid Method � Front Tracking � Level sets - Ghost Fluid Method � Outstanding issues are related to conservation, entropy & convergence (especially as associated with unstable phenomena, K-H, R-T & R-M) Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 4
LA-UR-02-5574 MultiFluid Method � Used to compute several fluids in a - shocked flow - single velocity and (pressure), multiple temperatures � Defined by a system of conservation laws (when summed over fluids) � Determined closure assumptions: � Pressure-velocity equilibrium via relaxation (needed for well-posed evolution) � Interface behavior (physical & numerical) � Temperature non-equilibrium (maintained) Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 5
LA-UR-02-5574 Equilibrium -vs- Nonequilibrium MultiFluid 7 Eq Multi−Fluid tracked contacts 6 Ghost Fluid 5 NonEq Multi−Flui Density 4 7 3 Eq Multi−Fluid 2 6 Ghost Fluid 1 5 0 NonEq Multi−Flui 0 1 X Density 4 3 2 1 0 0.55 0.6 0.7 0.8 0.9 X Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 6
LA-UR-02-5574 MultiFluid Equations � Governing Eqns. in 1-D + ( ) k g tk k k = f f u f u � Uses a HiRes x g x Godunov method ( ) + ( ) k k k k r r = 0 (like MUSCL) f f u t x ( ) � Pred.-Corr. ( ) + 2 r r + = u u p 0 Method t x ( ) + ( ) - k k k k k k k r r + = 0 f E f uE f up pf t t x ( ) k k k k k 2 r , = - 1 p e e E u ; 2 Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 7
LA-UR-02-5574 Is this a Hyperbolic System? � Yes! � Eigenvalue analysis provides the expected results e.g., for the corresponding linearized Lagrangian = ( ) T system, r r V f , , , , u p , p 1 2 1 2 ( ) L = - 0 0 0 0 , , , , c c , = g r � The sound speeds are c p � With a complete set of left and right eigenvectors Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 8
LA-UR-02-5574 MultiFluid Predictor Step � Nonlinearly interpolate (slope limited) ( ) k k k k k k r r g f u e p , , , , , , G � Time-center + ( ) n n n 1 2 = + D = - D - V V tV V t AV S 2 2 t x example + ( ) ( ) k k k k r r = r g g - u 1 u t x x � Predict any volume tracking ( ) + - - u * u * W u W u p p * = l l r r r l u + W W l r Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 9
LA-UR-02-5574 MultiFluid Interface Treatment � Naturally uses a volume tracking method - VOF, PLIC, Youngs’ method � In 1-D the SLIC method suffices: 3 2 u D t 4 1 Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 10
LA-UR-02-5574 Adaptive Two-Shock Riemann Solver Rarefaction ∂ r 2 V p c g = - = Contact ∂ p V p S Shock * = + r D p p c u r Two-shock approximation ( ) + ( ) + G D D 2 3 u O u 2 ( ) 1.2 2 rarefaction f k k k 2 g g L Â G = G -0.2 0.2 0.4 0.6 U p 0.8 k pressure C ∂ 2 2 G = V p 0.6 R shock ∂ 2 0.4 p 2 V S Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 11
LA-UR-02-5574 MultiFluid Closure - 1 Ê ˆ f k k g =  g Á ˜ � Basic constaints Ë ¯ k f k =  f k k = 1 r r Â Ê ˆ k k k = g g  k k Á ˜ p f p = ( ) Ë ¯ k k  u f u x k k =  x or p f p k k � Alternate energy equation (CGF) ( ) + ( ) + k k g r k k k k k k r r + = 0 f E f uE pu up x x g r t x Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 12
LA-UR-02-5574 A MultiFluid Equilibrium Treatment � Use basic definitions: - 1 Ê ˆ Ê ˆ k k k = g g f k k  g =  g Á ˜ Á ˜ p f p Ë ¯ Ë ¯ k k Ê ˆ k - p p must limit! k k d = f f Á ˜ k Ë g ¯ p k k k : = + d f f f = ( ( ) ) + By Colella, Glaz & Ferguson k k k r r d f e f e p f : also see Miller & Puckett ( ) = ( ) k k k k r r + d (earlier by LeBlanc @ LLNL?) f f f Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 13
LA-UR-02-5574 Equilibrium Treatment Beyond the Weak Shock Limit � The weak shock approximation r G ( ) 2 = + r D Æ + r D + D p p c u p c u u h o o 2 � The volume changes can be Previous method is Æ • ∆ t limit of a extended as the Hugoniot Riemann solution D d u f [ ] = - [ ] Æ r ª - Ê ˆ c V u k d 2 G k f k k g ª g + 1 c f ˜ 1 ( ) Ë ¯ k G D f u = r Æ r 1 + 1 W c c 2 c � This is like the steps taken in the Riemann solver Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 14
LA-UR-02-5574 MultiFluid Equilibration � Relaxation can be introduced to ( ) slow approach to equilibrium d f 0 1 max , . k ( k � Ad hoc method ) : = a + - a p p p 1 � Sound wave method a = c t D D x � Riemann solution method � With interface tracking the quality of solutions is influenced by these choices � Without interface tracking, solutions are extremely sensitive to this! Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 15
LA-UR-02-5574 MultiFluid Equilibration � To address sensitivity issues � Use different assumptions about mixed cells: � Equilibration is an adiabatic process (not on compression), a time scale is r ( ) µ — ◊ proportional to t max 0 1 , u � and to material gradients l µ 1 ∂ ∂ � Overall form: f x ∂ x f r ( ) a = D — ◊ D t max 0 , u ∂ x Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 16
LA-UR-02-5574 Accuracy at Interface for Sod with Tracking Interfaces Errors at interface from exact value (100 cells) Scheme Density 1 Density 2 Pressure 1 Pressure 2 None -4.68% 0.71% 0.002% 0.002% Relax -4.69% 0.72% 0.003% 0.003% Equil. -8.64% 1.88% 0.005% 0.005% Lagr. -4.21% -3.00% -0.14% -0.12% Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 19
LA-UR-02-5574 Overall Accuracy for Sod with Tracking Interfaces L1 errors from exact value Scheme 100 cells 200 cells 400 cells Rate None 9.60e-3 4.44e-3 2.48e-3 0.84 Relax 9.58e-3 4.43e-3 2.48e-3 0.84 Equil 9.97e-3 4.87e-3 3.33e-3 0.54 L1 errors for Lagrangian solution 100 cells:1.38e-02 400 cells:3.52e-03 Rate ~0.99 Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 20
LA-UR-02-5574 Sod Errors 0.35 0.3 Lagrangian Eulerian M. F. 0.2 0.1 0 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x/t Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 21
LA-UR-02-5574 Accuracy at Interface for Sod without Tracking Interfaces Errors at interface from exact value (100 cells) Scheme Density 1 Density 2 Pressure 1 Pressure 2 None 1.93% -8.52% -13.08% 13.77% Relax 1.64% -7.97% -12.02% 12.77% Equil -7.92% 3.76% 0.03% 0.03% Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 22
LA-UR-02-5574 Overall Accuracy for Sod without Tracking Interfaces L1 errors from exact value Scheme 100 cells 200 cells 400 cells Rate None 1.37e-2 7.82e-3 4.75e-3 0.72 Relax 1.37e-2 7.77e-3 4.73e-3 0.72 Equil 1.17e-2 6.97e-3 4.40e-3 0.67 Single 1.07e-2 5.83e-3 3.37e-3 0.79 Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 23
LA-UR-02-5574 Sod Error 0.2478 Lagrangian 0.2 No Eulerian M.F. 0.1 0 0.1 0.2 0.2742 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x/t Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 24
LA-UR-02-5574 Blast Wave Accuracy with Tracking Interfaces Scheme L1 Error L2 Error Linf Error 1 Fluid 4.88e-2 1.40e-1 7.88e-1 None 2.01e-2 5.36e-2 3.18e-1 Relax 1.96e-2 5.22e-2 3.12e-1 Strong S. 2.03e-2 5.85e-2 4.61e-1 Equil 9.43e-2 6.10e-1 8.12 Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 25
LA-UR-02-5574 Blast Wave Solution Time = 3.800000000000103E 02 400 Cells 7 Lagrangian 6 100 Cells exact 5 4 3 2 1 0 0.55 0.6 0.7 0.8 0.9 X Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 26
LA-UR-02-5574 Blast Wave Solution 7 Eulerian M.F. 6 Exact 5 4 3 2 1 0 0.55 0.6 0.7 0.8 0.9 x/t Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 27
LA-UR-02-5574 Blast Wave Errors Error 0.3 Error 0.2 0.1 0 0.1 0.2 0.3 0.4 0.55 0.6 0.7 0.8 0.9 x/t Compressible Multimaterial, 21-23, Sept. 2002, Paris Page 28
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