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Diapositive 1

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Title: Diapositive 1


1
NUMERICAL MODELING OF COMPRESSIBLE TWO-PHASE
FLOWS Hervé Guillard
INRIA Sophia-antipolis, Pumas Team, B.P. 93,
06902 Sophia-Antipolis Cedex, France, Herve.Guilla
rd_at_sophia.inria.fr Thanks to Fabien Duval,
Mathieu Labois, Angelo Murrone, Roxanna Panescu,
Vincent Perrier
2
Some examples of two-phase flows
Crossing the wall of sound
Granular medium HMX
3
Some examples of two-phase flows Steam
generator in a nuclear power plant
4
Multi-scale phenomena
Need for macro-scale description and averaged
models
5
Example of Interface problems Shock-bubble
interaction
Non structured tet mesh 18M nodes

128
processors
3h30 h
6
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7
TWO-PHASE MODELS
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,
8
OVERVIEW OF THIS TALK
  • 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

9
HOMOGENIZED MODELS 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 s is the speed of the interface
Introduce averaging operators
10
Let f be any regular enough function
Multiply the eq by X_k and apply Gauss and
Leibnitz rules
Define averaged quantities
etc
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12
THE TWO FLUID MODEL
Models for
13
How to construct these models ? Use the entropy
equation
14
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
15
Final form of the entropy equation
16
Simplest form ensuring positive entropy
production
17
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
- .............
18
One example Bubble column AMOVI MOCK UP (CEA
Saclay)
Pressure relaxation time
Velocity relaxation time
Temperature relaxation time
Bubble transit time
19
  • 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

20
CHAPMAN-ENSKOG EXPANSION
21
CHAPMAN-ENSKOG EXPANSION
22



23
Some examples Assume pressure equilibrium
classical two-fluid model (Neptune)
eos solve p1 p2 for the volume fraction
Non-hyperbolic !
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25
Some examples Assume - pressure equilibrium
-
velocity equilibrium
one-pressure, one velocity model
(Stewart-Wendroff 1984, Murrone-Guillard,
2005)
26
one-pressure, one velocity model
(Stewart-Wendroff 1984, Murrone-Guillard,
2005) Hyperbolic system
u-c, uc gnl, u,u ld Entropy
27
Some examples Assume - pressure equilibrium
-
velocity equilibrium
- temperature equilibrium
Multi-component Euler equations
eos solve p1 p2, T1T2
28
A Small summary
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
29
Why the 4 equation conservative model cannot
compute a contact
1 u p Ti
0 u p Ti1
1 u p Ti
1 u p Ti
0 u p Ti1
Y u p T
1 u p Ti
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
30
Parabolic reduced system
Goal Introduce some effects related to
non-equlibrium
31
One example of parabolic two-phase flow model
Is a relative velocity (drift flux models)
32
Mathematical properties of the model
First-order part hyperbolic Second-order
part dissipative
33
Comparison of non-equilibrium model (7 eqs) Vs
Equilibrium model (5 eqs) with dissipative
Terms (air-water shock tube pb)
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35
Sedimentation test-case (Stiffened gas state
law) Note velocities of air and water are of
opposite sign
36
Sedimentation test-case (Perfect gas state
law) Note velocities of air and water are of
opposite sign
5 eqs dissipative model
Non-equilibrium model
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38
Non equilibrium Model (7 eqs)
Equilibrium Model (5 eqs)
39
Two-phase flows models have non-conservative
form Non-conservative models Definition of
shock solution
Traveling waves
40
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
41
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
42
Convergence of travelling waves solutions of
the 5eqs dissipative model toward shock
solutions
43
ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW
44
ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW
ISOTHERMAL MODEL with DARCY-LIKE DRIFT LAW
Rankine-Hugoniot Relations
45
NUMERICAL TESTS
Infinite drag term (gas and liquid velocities are
equal)
46
TRAVELLING WAVE SOLUTIONS

47
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
48

Pressure
velocity
Gas Mass fraction
Drag Coeff 10000 kg/m3/s
Drag Coeff 5000 kg/m3/s
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50
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
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