NUMERICAL MODELING OF COMPRESSIBLE TWO-PHASE FLOWS

Hervé Guillard

INRIA Sophia-antipolis, Pumas Team, B.P. 93, 06902 Sophia-Antipolis Cedex, France,Herve.Guillard@sophia.inria.fr

Thanks to Fabien Duval, Mathieu Labois, Angelo Murrone, Roxanna Panescu, Vincent Perrier

Crossing the “wall “ of sound

Some examples of two-phase flows

Granular medium : HMX

Some examples of two-phase flows :

Steam generator in a nuclear power plant

Multi-scale phenomena

1m3:108−1012 droplets

Need for macro-scaledescription and averaged models

Example of Interface problems : Shock-bubble interaction

Non structured tet mesh : 18M nodes 128 processors 3h30 h

model suitable for two fluid studies :

no general agreement

large # of different models : homogeneous,

mixture models, two-fluid models, drift-flux

models

number of variables,

definition of the unknowns

number of equations

large # of different approximations

conservative, non-conservative, incompressible vs incompressible techniques,

TWO-PHASE MODELS

-Construction of a general 2-phase model

- Non-equilibrium thermodynamics of two phase non-miscible mixtures

- Equilibria in two phase mixture

-Reduced “hyperbolic” models for equilibrium situations

- Technical tool : Chapman-Enskog expansion

- A hierarchy of models

- Some examples

- Reduced “parabolic” models

- First-order Chapman-Enskog expansion

- Iso-pressure, iso-velocity model

- Traveling waves and the structure of two-phase shock

OVERVIEW OF THIS TALK

HOMOGENIZED MODELS

Reference textbooks : Ishii (1984), Drew-Passman (1998)

Let us consider 2 unmiscible fluids described by the Euler eq

Introduce averaging operators

Let X_k be the characteristic function of the fluid region k

where σ is the speed of the interface

Let f be any regular enough function

Define averaged quantities :

etc

Multiply the eq by X_k and apply Gauss and Leibnitz rules

THE TWO FLUID MODEL

Models for :

How to construct these models ?

Use the entropy equation :

Assume :

Then first line :

One important remark (Coquel, Gallouet,Herard, Seguin, CRAS 2002) :

The two-fluid system + volume fraction equation is (always) hyperbolic

but

the field associated with the eigenvalue

is linearly degenerate if and only if

Final form of the entropy equation :

Simplest form ensuring positive entropy production :

Summary

- Two fluid system + volume fraction eq = hyperbolic system

the entropy production terms are positive

- This system evolves to a state characterized by

- pressure equality - velocity equality

- temperature equality

- chemical potential equality

Deduce from this system, several reduced systems

characterized by instantaneous equilibrium between

- pressure

- pressure + velocity

- pressure + velocity + temperature

- .............

One example : Bubble column : AMOVI MOCK UP (CEA Saclay)

Pressure relaxation time

Velocity relaxation time

Temperature relaxation time

Bubble transit time

Construction of reduced models :

Technical tool : The Chapmann-Enskog expansion

What is a Chapman-Enskog asymptotic expansion ?

- technique introduced by Chapmann and Enskog

to compute the transport coefficients of the Navier-Stokes

equations from the Boltzmann equations-technique used in the Chen-Levermore paper on hyperbolic

relaxation problems

CHAPMAN-ENSKOG EXPANSION

CHAPMAN-ENSKOG EXPANSION

#

#

#

Some examples : Assume pressure equilibrium :

“classical” two-fluid model (Neptune)

eos : solve p1 = p2 for the volume fraction

Non-hyperbolic !

Some examples : Assume : - pressure equilibrium

- velocity equilibrium one-pressure, one velocity model (Stewart-Wendroff 1984, Murrone-Guillard, 2005)

one-pressure, one velocity model (Stewart-Wendroff 1984, Murrone-Guillard, 2005)

Hyperbolic system

u-c, u+c gnl, u,u ldEntropy

Some examples : Assume - pressure equilibrium - velocity equilibrium

- temperature equilibrium

Multi-component Euler equations :

eos : solve : p1 = p2, T1=T2

A Small summary :

Model # eqs complexity hyperbolic conservative contact respect

total nonequilibrium 7 +++ yes no yes

pressure equilibrium 6 +++ no no ?

pressure andvelocityequilibrium 5 ++ yes no yes

pressure andvelocity andtemperature 4 + yes yes noequilibrium

Why the 4 equation conservative model cannot compute a contact

1

u

p

Ti

0

u

p

Ti+1

1

u

p

Ti

Y

u

p

T

1

u

p

Ti

1

u

p

Ti

0

u

p

Ti+1

Not possible at constant pressure

keeping constant the conservative

variables

R. Abgrall, How to prevent pressure oscillations in multi-component flow calculation: a quasi-conservative approach, JCP, 1996

“Parabolic” reduced system

Goal : Introduce some effects related to non-equlibrium

One example of “parabolic” two-phase flow model

Is a relative velocity (drift – flux models)

Mathematical properties of the model :

First-order part : hyperbolic Second-order part : dissipative

Comparison of non-equilibrium model (7 eqs) Vs Equilibrium model (5 eqs) with dissipative Terms (air-water shock tube pb)

Sedimentation test-case (Stiffened gas state law)

Note : velocities of air and water are of opposite sign

Sedimentation test-case (Perfect gas state law)

Note : velocities of air and water are of opposite sign

5 eqs dissipative model Non-equilibrium model

Non equilibriumModel (7 eqs)

Equilibrium Model (5 eqs)

Two-phase flows models have non-conservative form

Non-conservative models : Definition of shock solution

Traveling waves

Weak point of the model : Non conservative form

Shock solutions are not defined

One answer : LeFloch, Raviart-Sainsaulieu

change

into

Define the shock solutions as limits of travelling waves solution of the regularized dissipative system for

Drawback of the approach : the limit solution depends on the

viscosity tensor

How to be sure that the viscosity tensor

encode the right physical informations ?

The dissipative tensor retains physical informations comingfrom the non-equilibrium modell

Convergence of

travelling waves

solutions of the 5eqs

dissipative model

toward shock solutions

ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW

ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAWISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW

Rankine-Hugoniot Relations :

NUMERICAL TESTS

Infinite drag term (gas and liquid velocities are equal)

TRAVELLING WAVE SOLUTIONS

TRAVELLING WAVES II

If TW exists, they are characterized by a differential system of Degree 2

Isothermal case : This ODE has two equilibrium point

Stable one

unstable one

Pressure velocity Gas Mass fraction

Drag Coeff10000 kg/m3/s

Drag Coeff5000 kg/m3/s

CONCLUSIONS

- Hierarchy of two-fluid models characterized by stronger

and stronger assumptions on the equilibriums realized in

the two fluid system

- on-going work to define shock solutions for two-phase model

as limit of TW of a dissipative system characterized by a

viscosity tensor that retain physical informations on

disequilibrium