Modelling of interfaces separating compressible

fluids and mixtures of materials

Multiphase shock relations and extra physics

Erwin FRANQUET and Richard SAUREL

Polytech Marseille, UMR CNRS 6595 - IUSTI

A Multiphase model with 7 equations

For solving interfaces problems and shocks into mixtures

)()( )(

1221111111111 uuppp

xp

x

pEu

t

EI

kI

)()²(

12111111111 uux

px

pu

t

uI

)pp(xt 2111

0x

ut

11111

2u 1PPI and Baer & Nunziato (1986)

21

121

21

2211

ZZ

pp)

xsgn(

ZZ

uZuZ

Saurel & al. (2003)

Chinnayya & al (2004)21

12211

21

1221I

ZZ

)uu(ZZ)

xsgn(

ZZ

pZpZp

Reduction to a 5 equations model

• When dealing with interfaces and mixtures with stiff mechanical relaxation the 7 equations model can be reduced to a 5 equations modelInfinite drag coefficientInfinite pressure relaxation parameter

Not conservative

2211

xu

cc

)cc(x

ut

2

222

1

211

112211

0x

ut

1111

0x

)P²u(tu

0)( xPEu

tE

0x

ut

2222

222111 EEE

• Kapila & al (2001)

To deal with realistic applicationsshock relations are mandatory

• 7 unknowns : α1, Y1, ρ, u, P, e, σ

• 4 conservation laws :cteY 1

cteum 00

20 vvmPP

02 0

00

vv

PPee

• Mixture EOS :• One of the variable behind the shock is given (often σ or P)

,,ePP

An extra relation is needed : jump of volume fraction or any other thermodynamic variable

How to determine it ?

Informations from the resolution of the 7 equations model

0

2e+010

4e+010

6e+010

8e+010

1e+011

1.2e+011

1.4e+011

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

.

x (m)

Phases pressures (Pa)

0

1000

2000

3000

4000

5000

6000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

.

x (m)

Phases velocities (m/s)

Impact of an epoxy-spinel mixture by a piston at several velocities

Fully dispersed waves

Why are the waves dispersed in the mixture ?

W1* W1

0

W2*

W20W2

0

W10 σ1 W1

0W1L

W20W2

L

(u+c)2

Dispersion mechanism

W1* W1

0

W2*

W20

σ1 W10W1

L

W20W2

L W2R W2

0

W1Rσ1 ’

Dispersion mechanism

Consequences• The two-phase shock is smooth• shock = succession of equilibrium states (P1=P2 and u1=u2)• We can use the 7 equations model in the limit :That is easier to integrate between pre and post shock states.

• In that case, the energy equations reduce to :

• And can be integrated as :

and

0dt

dvP

dt

de kk

k

0ˆ 0*0* kkkkk vvPee

As P1=P2 at each point, we have :

PPP ˆˆˆ21

The mixture energy jump is known without ambiguity :

02

000

vvPP

ee

To fulfill the mixture energy jump each phase must obey :

02 0

*0*

0*

kkkk vv

PPee

Saurel & al (2005)

Some properties

• Identifies with the Hugoniot adiabat of each phase

• Symmetric and conservative formulation• Entropy inequality is fulfilled• Single phase limit is recovered• Validated for weak and strong shocks for

more than 100 experimental data• The mixture Hugoniot curve is tangent to

mixture isentrope

Shock relations validation

3000

4000

5000

6000

7000

8000

9000

10000

0 500 1000 1500 2000 2500 3000 3500 4000Material Velocity (m/s)

Shock Velocity (m/s)

Epoxy-Spinel mixture

2500

3000

3500

4000

4500

5000

5500

6000

6500

0 500 1000 1500 2000 2500Material Velocity (m/s)

Shock Velocity (m/s)

Uranium-Molybdene mixture

30003500

4000

450050005500

6000

6500

70007500

8000

0 500 1000 1500 2000 2500 3000Material Velocity (m/s)

Shock Velocity (m/s)

Paraffine-Enstatite mixture

3000

4000

5000

6000

7000

8000

9000

10000

0 500 1000 1500 2000 2500 3000 3500 4000 4500Material Velocity (m/s)

Shock Velocity (m/s)

Epoxy-Periclase mixture

xu

cc

)cc(x

ut

2

222

1

211

112211

0x

ut

1111

0x

)P²u(tu

0)( xPEu

tE

0x

ut

2222

+ Mixture EOS+ Rankine-Hugoniot relationsConsequences :

A Riemann solver can be built

This one can be used to solve numerically the 5 equations modelThe second difficulty comes from the average of the

volume fraction inside computational cells : It is not a conservative variable

It necessitates the building of a new numerical method

The reduced model (with 5 equations) is

now closed

A new projection methodSaurel & al (2005)

)uS(tV *2/1i2/1i1

311n

i2 VVxV

)Su(tV 2:1i*

2/1i3

1ni

jj

x

V

3,..,1j

Volume fractions definition

t

xxi-1/2 xi+1/2

t n+1

u*i-1/2 u*i+1/

2

S+i-1/2 S-

i+1/2

Euler equations context

u1, P1, e1 u2, P2, e2 u3, P3, e3

V1 V2 V3

Relaxation system

Conservation and entropy inequality are preserved if :

j

jjI pp

j

jjI uYu)( Ijj

j pp

0

jj

)( jIjjjj uuYu

)()( jIjIIjjIjjj uuYupppE

This ODE system is solved in each computational cell so as to reach the mechanical equilibrium asymptotic state ( )

It can be written as an algebraic system solved with the Newton method.

Comparison with conventional methods

• Conventional Godunov average supposes a single pressure, velocity and temperature in the cell. In the new method, we assume only mechanical equilibrium and not temperature equilibrium.

• It guarantees conservation and volume fraction positivity• The method does not use any flux and is valid for non

conservative equations

• In the case of the ideal gas and the stiffened gas EOS with the Euler equations both methods are equivalent. Results are different for more complicated EOS (Mie-Grüneisen for example)

• The new method gives a cure to anomalous computation of some basic problems:- Sliding lines- Propagation of a density discontinuity in an uniform flow with Mie Grüneisen EOS.

• It can be used in Lagrange or Lagrange + remap codes.

Propagation of a density discontinuity in an uniform flow

with JWL EOSP = PCJ = 2 1010 Pa

u = 1000 m/s

ρ = ρCJ = 2182 kg/m3

P = PCJ = 2 1010 Pa

u = 1000 m/s

ρ = 100 kg/m3 0 0,5 1

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1

.

x (m)

Density (kg/m3)

19550

19600

19650

19700

19750

19800

19850

19900

19950

20000

0 0.2 0.4 0.6 0.8 1

.

x (m)

Pressure (MPa)

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1

.

x (m)

Density (kg/m3)

19800

19850

19900

19950

20000

20050

20100

20150

20200

0 0.2 0.4 0.6 0.8 1

.

x (m)

Pressure (MPa)

Shock tube problem in extreme conditions

Euler equations and JWL EOS

0

500

1000

1500

2000

2500

0 0.2 0.4 0.6 0.8 1

.

x (m)

Density (kg/m3)

-500

0

500

1000

1500

2000

2500

3000

0 0.2 0.4 0.6 0.8 1

.

x (m)

Velocity (m/s)

02000400060008000

100001200014000160001800020000

0 0.2 0.4 0.6 0.8 1

.

x (m)

Pressure (MPa)

P = PCJ = 2 1010 Pa

ρ = ρCJ = 2182 kg/m3

P = 2 108 Pa

ρ = 100 kg/m3

0 0,5 1

• The Godunov method fails in these conditions

Shock tube problem with almost pure fluids

Liquid-Gas interface with the 5 equations model

0

100

200

300

400

500

600

700

800

900

1000

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Mixture Density (kg/m3)

0

1e+008

2e+008

3e+008

4e+008

5e+008

6e+008

7e+008

8e+008

9e+008

1e+009

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Pressure (Pa)

0

50

100

150

200

250

300

350

400

450

500

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Normal Velocity (m/s)

0 0,8

1

P = 105 Pa

αair = 1-10-

8

P = 109 Pa

αwater = 1-10-8

ρwater = 1000 kg/m3

ρair = 50 kg/m3

Stiffened Gas EOS

Shock tube problem with mixtures of materials

Epoxy-Spinel Mixture

0

1e+009

2e+009

3e+009

4e+009

5e+009

6e+009

7e+009

8e+009

9e+009

1e+010

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Pressure (Pa)

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Normal Velocity (m/s)

1900

2000

2100

2200

2300

2400

2500

2600

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Mixture Density (kg/m3)

0 0,6

1

P = 105 Pa

αepoxy = 0,5954

P = 1010 Pa

αepoxy = 0,5954

ρepoxy = 1185 kg/m3

ρspinel = 3622 kg/m3

Stiffened Gas EOS

Shock tube problem with mixtures of materials (2)

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

0.61

0.62

0.63

0.64

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Epoxy Volume Fraction

900

1000

1100

1200

1300

1400

1500

1600

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Epoxy Density (kg/m3)

3540

3560

3580

3600

3620

3640

3660

3680

3700

0 0.2 0.4 0.6 0.8 1

abscissa (m)

Spinel Density (kg/m3)

Mixture Hugoniot testsComparison with experiments and the 7 equations

model

3000

4000

5000

6000

7000

8000

9000

10000

11000

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Material Velocity (m/s)

Shock Velocity (m/s)

New method

7 equations model

P = 105 Pa

αepoxy = 0,5954

ρepoxy = 1185 kg/m3

ρspinel = 3622 kg/m3

Up

Piston Epoxy-Spinel Mixture

2D impact of a piston on a solid stucture

RDX(Mie-Grüneisen EOS)

TNT(JWL EOS)

Copper (Stiffened Gas EOS)

U = 5000 m/s

Air (Ideal Gas EOS)