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NUMERICAL MODELING OF COMPRESSIBLE TWO-PHASE FLOWS Hervé Guillard

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## NUMERICAL MODELING OF COMPRESSIBLE TWO-PHASE FLOWS Hervé Guillard

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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

Some examples of two-phase flows :

Steam generator in a nuclear power plant

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,

- 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

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

Let us consider 2 unmiscible fluids described by the Euler eq

Let X_k be the characteristic function of the fluid region k

where σ is the speed of the interface

Introduce averaging operators

Let f be any regular enough function

Multiply the eq by X_k and apply Gauss and Leibnitz rules

Define averaged quantities :

etc

Models for :

How to construct these models ?

Use the entropy equation :

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

- 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

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 ld

Entropy

Some examples : Assume - pressure equilibrium

- velocity equilibrium

- temperature equilibrium

Multi-component Euler equations :

eos : solve : p1 = p2, T1=T2

Model # eqs complexity hyperbolic conservative contact

respect

total non

equilibrium 7 +++ yes no yes

pressure

equilibrium 6 +++ no no ?

pressure and

velocity

equilibrium 5 ++ yes no yes

pressure and

velocity and

temperature 4 + yes yes no

equilibrium

Why the 4 equation conservative model cannot compute a contact

1

u

p

Ti

0

u

p

Ti+1

1

u

p

Ti

1

u

p

Ti

0

u

p

Ti+1

1

u

p

Ti

Y

u

p

T

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

Goal : Introduce some effects related to non-equlibrium

One example of “parabolic” two-phase flow model

Is a relative velocity (drift – flux models)

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

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 coming

from the non-equilibrium modell

ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW

ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW

Rankine-Hugoniot Relations :

Infinite drag term (gas and liquid velocities are equal)

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

- 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

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