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Page 1 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

The Impact of Different Approaches to Imposing Pressure Equilibrium in Multimaterial Godunov Methods The Impact of Different Approaches to Imposing Pressure Equilibrium in 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

LA-UR-02-5574

Page 2 Compressible Multimaterial, 21-23, Sept. 2002, Paris 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

Page 4 Compressible Multimaterial, 21-23, Sept. 2002, Paris 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)

Page 5 Compressible Multimaterial, 21-23, Sept. 2002, Paris 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)

Page 6 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

0.55 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7

X Density

Eq Multi−Fluid Ghost Fluid NonEq Multi−Flui

1 1 2 3 4 5 6 7

X Density

Eq Multi−Fluid Ghost Fluid NonEq Multi−Flui

Equilibrium -vs- Nonequilibrium MultiFluid

tracked contacts

Page 7 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

MultiFluid Equations

Governing Eqns. in 1-D Uses a HiRes Godunov method (like MUSCL) Pred.-Corr. Method p e e E u

k k k k k

r , ;

( )

=

• 1

2 2

f f u f u

tk k x k k x

+(

)

= g g f f u

k k t k k x

r r

( ) +( )

= 0 f E f uE f up pf

k k k t k k k k x t

r r

( ) +

+

( ) -

= 0 r r u u p

t x

( ) +

+

( )

=

2

Page 8 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Is this a Hyperbolic System?

Yes! Eigenvalue analysis provides the expected results e.g., for the corresponding linearized Lagrangian system, The sound speeds are With a complete set of left and right eigenvectors L =

• (

)

0 0 0 0 , , , , , c c c p = g r V f u p p

T

= (

)

, , , , , r r

1 2 1 2

Page 9 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

MultiFluid Predictor Step

Nonlinearly interpolate (slope limited) Time-center example Predict any volume tracking f u e p

k k k k k k

, , , , , , r r g

( )

G V V tV V t AV S

n n t n x +

= + =

• (

)

1 2

2 2 D D r r r g g

t k k x k k x

u u +(

)

=

• (

)

1 u* u* u W u W u p p W W

l l r r r l l r * =

+

• (

)

+

Page 10 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

MultiFluid Interface Treatment

Naturally uses a volume tracking method - VOF, PLIC, Youngs’ method In 1-D the SLIC method suffices:

uDt 1 2 3 4

Page 11 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Adaptive Two-Shock Riemann Solver

Up pressure L R

• 0.2

0.2 0.4 0.6 0.4 0.6 0.8 1.2

C

rarefaction shock

Two-shock approximation

Shock Contact Rarefaction

g ∂ ∂ r = - = V p p V c p

S 2

G = V p p V

S 2 2 2

2 ∂ ∂ p p c u

* =

+ r D +

( ) + ( )

r 2

2 3

G D D u O u G = G g g

2

2 f k k k k

( )

Â

Page 12 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

MultiFluid Closure

Basic constaints Alternate energy equation (CGF) f E f uE pu up

k k k t k k k x k x k x

r r g g r r

( ) +( ) +

+ = 0 f k

k

 = 1 r r =  f k

k k

g g = Â Ê Ë Á ˆ ¯ ˜

• f k

k k 1

p f p

k k k k

= Â Ê Ë Á ˆ ¯ ˜ g g u f u

x k x k

= (

)

Â

• r p

f p

k k k

= Â

Page 13 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

A MultiFluid Equilibrium Treatment

Use basic definitions: g g = Â Ê Ë Á ˆ ¯ ˜

• f k

k k 1

p f p

k k k k

= Â Ê Ë Á ˆ ¯ ˜ g g d g f f p p p

k k k k

=

• Ê

Ë Á ˆ ¯ ˜ f f f

k k k

:= + d f e f e p f

k k k

r r d

( )

= (

) +

: r r d

k k k k

f f f = (

)

+

( )

By Colella, Glaz & Ferguson also see Miller & Puckett (earlier by LeBlanc @ LLNL?) must limit!

Page 14 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Equilibrium Treatment Beyond the Weak Shock Limit

The weak shock approximation The volume changes can be extended as the Hugoniot This is like the steps taken in the Riemann solver p p c u p c u u

h

• =

+ Æ + +

( )

r r

r

D D D

G 2 2

r d c V u u c f f

[ ] = -[ ]Æ

ª - D ˜ g g

d k k

ª + Ê Ë ˆ ¯ 1

1 2 Gk f f

k k

W c c

u c

= Æ +

( )

r r 1

1 2 GD

Previous method is ∆t limit of a Riemann solution

Æ •

Page 15 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

MultiFluid Equilibration

Relaxation can be introduced to slow approach to equilibrium

Ad hoc method Sound wave method Riemann solution method

With interface tracking the quality

• f solutions is influenced by these

choices Without interface tracking, solutions are extremely sensitive to this! p p p

k k

:= +

• (

)

a a 1 a = c t x D D

max , . df 0 1

( )

Page 16 Compressible Multimaterial, 21-23, Sept. 2002, Paris 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 proportional to and to material gradients Overall form:

t u µ —◊

( )

max , 0 1 r l µ1 ∂ ∂ f x a ∂ ∂ = —◊

( )

D D t u x f x max , r

Page 19 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Accuracy at Interface for Sod with Tracking Interfaces

• 0.12%
• 0.14%
• 3.00%
• 4.21%

Lagr. 0.005% 0.005% 1.88%

• 8.64%

Equil. 0.003% 0.003% 0.72%

• 4.69%

Relax 0.002% 0.002% 0.71%

• 4.68%

None Pressure 2 Pressure 1 Density 2 Density 1 Scheme

Errors at interface from exact value (100 cells)

Page 20 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Overall Accuracy for Sod with Tracking Interfaces

0.54 3.33e-3 4.87e-3 9.97e-3 Equil 0.84 2.48e-3 4.43e-3 9.58e-3 Relax 0.84 2.48e-3 4.44e-3 9.60e-3 None Rate 400 cells 200 cells 100 cells Scheme

L1 errors from exact value L1 errors for Lagrangian solution 100 cells:1.38e-02 400 cells:3.52e-03 Rate ~0.99

Page 21 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Sod Errors

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.3 0.2 0.1 0.1 0.2 0.3 0.35

x/t

Lagrangian Eulerian M. F.

Page 22 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Accuracy at Interface for Sod without Tracking Interfaces

0.03% 0.03% 3.76%

• 7.92%

Equil 12.77%

• 12.02%
• 7.97%

1.64% Relax 13.77%

• 13.08%
• 8.52%

1.93% None Pressure 2 Pressure 1 Density 2 Density 1 Scheme

Errors at interface from exact value (100 cells)

Page 23 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Overall Accuracy for Sod without Tracking Interfaces

0.79 3.37e-3 5.83e-3 1.07e-2 Single 0.67 4.40e-3 6.97e-3 1.17e-2 Equil 0.72 4.73e-3 7.77e-3 1.37e-2 Relax 0.72 4.75e-3 7.82e-3 1.37e-2 None Rate 400 cells 200 cells 100 cells Scheme

L1 errors from exact value

Page 24 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Sod Error

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2742 0.2 0.1 0.1 0.2 0.2478

x/t

Lagrangian No Eulerian M.F.

Page 25 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Accuracy with Tracking Interfaces

4.61e-1 5.85e-2 2.03e-2 Strong S. 7.88e-1 1.40e-1 4.88e-2 1 Fluid 8.12 6.10e-1 9.43e-2 Equil 3.12e-1 5.22e-2 1.96e-2 Relax 3.18e-1 5.36e-2 2.01e-2 None Linf Error L2 Error L1 Error Scheme

Page 26 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Solution

0.55 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7

Time = 3.800000000000103E 02

X

400 Cells

Lagrangian 100 Cells exact

Page 27 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Solution

0.55 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7

x/t

Eulerian M.F. Exact

Page 28 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Errors

0.55 0.6 0.7 0.8 0.9 0.4 0.3 0.2 0.1 0.1 0.2 0.3

Error

x/t

Error

Page 29 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Accuracy without Tracking Interfaces

8.97e-1 1.66e-1 7.66e-2 Strong S. 7.88e-1 1.40e-1 4.88e-2 1 Fluid 6.48 7.27e-1 1.92e-1 Equil 1.01 1.68e-1 7.69e-2 Relax 1.01 1.69e-1 7.70e-2 None Linf Error L2 Error L1 Error Scheme

Page 30 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Solutions

0.55 0.6 0.7 0.8 0.9 1 2 3 4 5 6 7

x/t

Strong Relax Exact

Page 31 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Blast Wave Errors

0.55 0.6 0.7 0.8 0.9 1 0.8

x/t

Error

Page 32 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

MultiFluid Outstanding Issues

Nature of equilibrium approximation or relaxation approach Problems with results adjacent to contacts - wall heating/cooling Method for tracking slip requiring non- equilibrium velocity-pressure treatment More dissipative Riemann solvers, HLLE, Local LxF, LxF (viscosity instead)?

Page 33 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Key References

MultiFluid

Collela, Glaz & Ferguson, “Multifluid Algorithms for Eulerian Finite Difference Methods” manuscript, 1995 Miller and Puckett, J. Comp. Phys., 128(1), 134-164, 1996 Rider, Comp. & Fluids, 28(6), 741-777, 1999 Rider & Kothe, J. Comp. Phys. 141(1), 112-152, 1998.

Page 34 Compressible Multimaterial, 21-23, Sept. 2002, Paris LA-UR-02-5574

Summary

Equilibration is not a serious issue for weaks shocks… It is an essential issue to resolve for strong shocks This study shows the following:

Relaxation (strong shock) seems the best route to quality solutions Tracking versus capturing is a key quality issue Other approaches to relaxation need investigation