Entropieendynamiquedesﬂuides Entropy in Fluid Dynamics

Entropie en dynamique des fluidesEntropy in Fluid Dynamics

Habilitation à Diriger des Recherches - Defense

Hélène MathisDecember 7th 2020Université de Nantes

Hélène Mathis Habilitation defense Université de Nantes 1 / 40

Many thanks to...

Marianne Bessemoulin-Chatard Christophe Berthon Clément Cancès AchyllesFrançois James Hala Ghazi Exprodil Jean-Marc Hérard Philippe Helluy Olivier Hurisse

Claire Chainais-Hillairet Giulia Lissoni LMJL Nicolas Therme Frédéric HérauMatthieu Hillairet MoHyCon Anaïs Crestetto Nicolas Seguin François Jauberteau

Edwige Godlewski Frédéric Coquel Gloria Faccanoni Needs Khaled Saleh Nicolas ThermeRodolphe Turpault Sabrina Carpy Matthieu Bachmann Siegfried Müller Jean Bussac


Research activities

Analysis and numerical approximation of models arising in complex multiphase flows

Hyperbolic systems of conservation laws and relaxation systems

Finite volume approximation

® Intent to develop a global approach: modelling to computation, as well as analysis ofPDE’s, numerical analysis and simulation

® Entropy is a central toolI Thermodynamic entropyI Lax entropy and extensionI Relative entropy


Outline of the presentation

1 Relative entropy in finite volume approximationSystem of conservation laws: error estimateRelaxation and parabolic limit: convergence rateRelaxation and hyperbolic limit: model adaptation

2 Construction of thermodynamic entropy for multiphase flowThe case of miscible mixtures or immiscible mixturesThree phase flows: a liquid, its vapor and a gasCapturing metastable states in van der Waals fluids

3 Some perspectives


Outline



3 Some perspectives


System of conservation laws [Lax, 57; Dafermos, 10]

Cauchy problem in Rd

∂tu(t, x) +

d∑α=1

∂αfα(u)(t, x) = 0 x ∈ Rd, t ∈ [0, T ] (E)

with u(0, x) = u0(x) ∀x ∈ Rd

u ∈ Ω ⊂ Rn convex set of admissible statesAssume the system endowed with a Lax entropy-flux pair (η, ψ)I η convex function of uI Strong solution u of (E) satisfies

∂tη(u) +d∑

α=1

∂αψα(u) = 0 in Rd × R+

Natural setting of weak entropy solutions

∂tη(u) +d∑

α=1

∂αψα(u) ≤ 0 in Rd × R+


Weak-strong uniqueness and relative entropy method

Weak-strong uniqueness [Dafermos, 79; DiPerna, 79]∫|x|<r

|u(T, x)− u(T, x)|2dx ≤ C(T, u)

∫|x|<r+LfT

|u0(x)− u0(x)|2dx

Relative entropyη(u|u) = η(u)− η(u)− (∇uη(u))> (u− u)

By convexity of η, behaves like |u− u|2

Satisfies a relative entropy inequality

∂tη(u|u) +d∑

α=1

∂αψα(u|u) ≤ −d∑

α=1

(∂αu)>Zα(u|u)

Zα(u|u) same quadratic behavior as η(u|u)

L2loc estimate obtained using a Grönwall lemma

Use this tool in the discrete setting


Finite volume approximation [Eymard, Gallouët, Herbin, 00]

Define a time step ∆t and tn = n∆t.Mesh T of cells KI h = supm(K),K ∈ T <∞I σKL: interface between K and LI N (K) neighboring cells of K

σKL

KL

nKL

Principle of conservation on a cell K between tn and tn+1

∫K

u(tn+1, x)dx−∫K

u(tn, x)dx+

∫ tn+1

tn

∫∂K

f(u(t, s)) · ndsdt = 0

First order explicit Finite Volume scheme with two-point flux approximation

m(K)(un+1K − unK) + ∆t

∑L∈N (K)

m(σKL)GKL(unK , unL) = 0, ∀K ∈ T , n ≥ 0


Numerical flux and discrete entropy

[Harten, Lax, van Leer, 83; Eymard, Gallouët, Herbin, 00; Bouchut, 04]

The numerical scheme reads as a convex combination

un+1K =

∑L∈N (K)

m(σKL)

m(∂K)unKL

σKL

KL

nKL

withunKL = unK −

∆t

m(K)m(∂K) [GKL(unK , u

nL)− f(unK) · nKL]

Assumptions on the numerical fluxConsistency and conservationUnder a CFL condition ∆t ≤ CCFLhI Preservation of admissible states by interface: uKL ∈ ΩI Entropy inequality by interface: it exists ψKL s.t.

η(unKL) ≤ η(unK)−∆t

m(K)m(∂K) [ψKL(unK , u

nL)− ψ(unK) · nKL]

Discrete entropy inequality

m(K)(η(un+1K )− η(unK)) + ∆t

∑L∈N (K)

ψKL(unK , unL) ≤ 0


Main results: weak-BV estimate

Weak-BV estimate [Cancès, M., Seguin, 16]

Under the assumptions on the numerical flux, assume that there exists ζ ∈ (0, 1) suchthat

∆t ≤ (1− ζ)β0

β1CCFLh

where β1 ≥ β0 > 0 s.t. spec(∇2uη(u)

)⊂ [β0;β1]. Then there exists a constant C s.t.

NT∑n=0

∆t∑

K∈T ∩B(0,r)

∑L∈N (K)

|σKL| |GKL(unK , unL)− f(unK) · nKL| ≤

C√h

Analogous to the scalar case [Eymard, Gallouët, Herbin, 00]: (1− ζ) guaranteesnumerical diffusion

[Fjordholm, Käppeli, Mishra, Tadmor, 15] Similar result with a ad hoc numericaldiffusion, continuous in time

Remove the BV assumption in [Jovanović, Rohde, 05]


Main results: L2 error estimate

Error estimate between

A strong solution u of (E)

Its approximate solution uh(t, x) = unK , (t, x) ∈ [tn, tn+1), x ∈ KFrom the weak-BV estimate, one deduces

Discrete entropy inequality by interface

Weak-BV estimate on the numerical entropy flux

Relative entropy inequality and control of error terms expressed in terms of Radonmeasures

Theorem [Cancès, M. Seguin, 16]

Let u be a strong solution to (E) and uh its approximation. Under the assumptions onthe numerical flux and a restrictive CFL condition, there exists a constant C > 0 s.t.∫ T

0

∫B(0,r+Lf (T−t))

|u− uh|2dx dt ≤ C√h

Recover a convergence rate in h1/4, similar to the scalar case [Chainais-Hillairet, 99]

Holds for standard entropy satisfying numerical schemesHélène Mathis Habilitation defense Université de Nantes 11 / 40

Outline



3 Some perspectives


Outline



3 Some perspectives


Hyperbolic systems with relaxation [Chen, Levermore, Liu, 94]

Hyperbolic systems with u ∈ Rk and v ∈ Rn−k

∂tu+

d∑α=1

∂αf1,α(u, v) = 0

∂tv +d∑

α=1

∂αf2,α(u, v) =1

εr(u, v)

(R)

Equilibrium manifold parametrized in terms of the k conserved quantities u

r(u, v) = 0⇔ v = veq(u)

As ε→ 0, the dynamics is asymptotically described by the equilibrium model

∂tu+

d∑α=1

∂αf1,α(u, veq(u)) = 0 (E)


Entropy structure

The relaxation system (R) is endowed with a Lax entropy pair (H,Ψ)

∂tH(u, v) +d∑

α=1

∂αΨα(u, v) =1

ε∇H(u, v) ·

(0k

r(u, v)

)≤ 0

I Dissipation ensures stabilityI Relaxation may preserve smoothness of the initial condition under the

Kawaskima-Shizuta condition, algebraic condition on f1, f2 and r(u, v) [Hanouzet,Natalini, 03; Yong, 04]

The restriction of the entropy flux pair (H,Ψ) on the equilibrium manifold gives aentropy pair (η, ψ) for the equilibrium model (E)

η(u) := H(u, veq(u)), ψ(u) := Ψ(u, veq(u))

I The equilibrium system is hyperbolic

Hierarchy of hyperbolic models


Chapman-Enskog expansion

Up to terms of order ε2, the smooth solutions of the relaxation system (R) satisfy (atleast formally)

An intermediate model

∂tu+

d∑α=1

∂αf1,α(u, veq(u)) = −εd∑

α=1

∂α (∇vf1,α(u, veq(u))v1)

v = veq(u) + εv1

where

v1 =(∇vr(u, veq(u))

)−1[ d∑α=1

∂αf2,α(u, veq(u))

−∇veq(u)>d∑

α=1

∂αvf1,α(u, veq(u))

]

Smooth solutions of the intermediate model solves (R) up to ε2

As ε→ 0, recover the equilibrium model (E)εv1 is a measure of the deviation with respect the equilibrium


The idea of model adaptation

Make use of the hierarchy of models for computational facilitiesI Model (R) contains more informations, more complicated to approximateI Model (E) smaller, does not see the fine structures

Select the better model in time and spaceI Partition of the computational domain DI Numerical indicator of disequilibrium E

An idea of dynamic model adaptationA Domain decomposition with a tolerance θ

I Equilibrium domain DE(t) = 1E≤θI Relaxation domain DR(t) = D \ DE(t)

B Solve (E) in DE(t) and (R) in DR(t)C At interfaces DE(t) ∩ DR(t), adopt a coupling strategy


The discrete setting: a hierarchy of numerical schemes

A splitting in time approximation of the model (R)I From time tn to tn+1,−, convective part with wnk = (unK , v

nK)

un+1,−K = unK −

∆t

m(K)

∑L∈N (K)

m(eKL)F1(wnK , wnL,nKL)

vn+1,−K = vnK −

∆t

m(K)

∑L∈N (K)

m(eKL)F2(wnK , wnL,nKL)

I From time tn+1,− to tn+1, implicit time integration

un+1K = un+1,−

K

vn+1K = vn+1,−

K +∆t

εr(un+1

K , vn+1K )

(R∆)

Projection on the equilibrium for (E)

un+1K = unK −

∆t

m(K)

∑L∈N (K)

m(eKL)F1(wnK , wnL, nKL)

vn+1K = veq(u

n+1K )

(E∆)


Indicator and the adaptation algorithm

Numerical indicator [M., Cancès, Godlewski, Seguin, 15; Cancès et al., 16]

Performing a Chapman-Enskog expansion in the numerical scheme

En+1K := max(ε‖vn+1

1,K ‖, ‖vnK − veq(unK)‖)

Algorithm of model adaptation [M., Cancès, Godlewski, Seguin, 15; Cancès et al., 16 ]

Let wnK the given approximate solution in the cell K at time tn and a threshold θ.

A For any cell K ∈ T , compute numerical indicator of disequilibrium En+1K

B For any cell K ∈ T , if [|En+1K | > θ]

Then K ∈ DR(tn)Else K ∈ DE(tn)

C At this stage, DR(tn) ∪ DE(tn) = DFor any cell K ∈ T :

- If K ∪N (K) ∈ DR(tn), then wn+1K is computed by (R∆)

- If K ∪N (K) ∈ DE(tn), then wn+1K is computed by (E∆)

- Else, compute wn+1K by a coupling condition [Caetano, 06; Ambroso et al., 07]


Indicator and the adaptation algorithm

Numerical indicator [M., Cancès, Godlewski, Seguin, 15; Cancès et al., 16]

Performing a Chapman-Enskog expansion in the numerical scheme

En+1K := max(ε‖vn+1

1,K ‖, ‖vnK − veq(unK)‖)

[Degond, Dimarco, Mieussens, 07; Jin, Liu, Wang, 12] Kinetic models

3 The continuous in space method is entropy satisfying [M., Cancès, Godlewski, Seguin,15] ∑

K∈D

H(wn+1K ) ≤

∑K∈D

H(wnK)


Application to a hierarchy of two-phase flow models

Vapor bubble surrounding by a liquid submitted to a left going shock

Relaxation model (R) Equilibrium model (E)

Adaptation with θ = 10−2 Adaptation with θ = 1

Mass fraction of the vaporHélène Mathis Habilitation defense Université de Nantes 20 / 40

Application to a hierarchy of two-phase flow models

Domain partition: (R) in blue and (E) in red

First interaction


Second interaction



Outline



3 Some perspectives


Extensive setting - assumptions [Callen, 85]

Fluid of mass M ≥ 0, energy E ≥ 0, occupying a volume V ≥ 0, entirely described by itsentropy function

S : W = (M,V,E)→ S(W )

Concave on C := W ∈ (R+)3, S(W ) > −∞Positively Homogeneous of degree 1 (extensive)

∀λ ∈ R+∗ , ∀W ∈ C, S(λW ) = λS(W )

S of class C2 s.t.∀W ∈ C, ∂S

∂E=

1

T> 0

I Intensive potentials: temperature T , pressure p, chemical potential µ

TdS = dE + pdV − µdM Gibbs relation


Mixtures of K phases

Consider a state W = (M,V,E) and K phases of masses Mk ≥ 0, volumes Vk ≥ 0 andenergies Ek ≥ 0, described by an entropy Sk(Wk), k = 1, . . . ,K




Extensive constraints∑Kk=1 Mk = M,

∑Kk=1 Ek = E

Immiscible mixtures∑Kk=1 Vk = V

Miscible mixturesV = Vk, ∀k = 1, . . . ,K





∑Kk=1 Ek = E



Extensive mixture entropy at thermodynamical equilibrium

S(W ) =K

k=1

Sk(Wk) S(W , V ) =K

(V )k=1

Sk(Wk, V )





∑Kk=1 Ek = E




S(W ) =K

k=1

Sk(Wk) S(W , V ) =K

(V )k=1

Sk(Wk, V )

Characterization of the thermodynamical equilibriumT = Tk, µ = µk, k = 1, . . . ,K

p = pk p =∑Kk=1 pk Dalton law





∑Kk=1 Ek = E




S(W ) =K

k=1

Sk(Wk) S(W , V ) =K

(V )k=1

Sk(Wk, V )

Characterization of the thermodynamical equilibriumT = Tk, µ = µk, k = 1, . . . ,K

p = pk p =∑Kk=1 pk Dalton law

Intensive mixture entropy at thermodynamical equilibrium

τ = V/M , e = E/Ms(τ, e) = concave hull(sk) Not necessarily strictly concave

ρ = M/V , ε = E/V

σ(ρ, ε) =K

k=1

(σk)

Strictly concaveHélène Mathis Habilitation defense Université de Nantes 23 / 40

Outline



3 Some perspectives


Industrial applications

Three phase applicationsNuclear safety demonstration for pressurized water reactorsI Loss of coolant accident [Bartak, 90]I SUPERCANON experiment [Riegel, 78]

Compressible flowsI Brutal depressurizationI Phase transition wave

Contact with the ambiant air

Fluid composed of a liquid l, its vapor v and a gas g[Bachmann, Helluy, M., Müller, 10; M., 19; Hérard, M., 19]


Extensive vs. intensive settings

Fluid composed of a liquid l, its vapor v and a gas g



Fluid composed of a liquid l, its vapor v and a gas gExtensive setting

For a given state (M,V,E)Phase k described by Sk(Wk)

Intensive setting

For a given state (τ, e)Phase k described by sk(τk, ek) with fractionsϕk = Mk/M , αk = Vk/V , zk = Ek/E





Intensive setting


Set of constraints

Ωext

∑kMk = M∑k Ek = E

Vl + Vg = V

Vg = Vv

Ωint

∑k ϕk = 1, ϕk ∈ [0, 1]∑k zk = 1, zk ∈ [0, 1]

αl + αg = 1, αk ∈ [0, 1]

αg = αv





Intensive setting


Set of constraints

Ωext

∑kMk = M∑k Ek = E

Vl + Vg = V

Vg = Vv

Ωint

∑k ϕk = 1, ϕk ∈ [0, 1]∑k zk = 1, zk ∈ [0, 1]

αl + αg = 1, αk ∈ [0, 1]

αg = αv

Mixture entropy at equilibrium

S(M,V ,E,Ml,Mg)

= max(Wk)k∈Ωext

∑k

Sk(Wk)

s(τ, e, ϕl, ϕg)

= max((αk)k,(zk)k)∈Ωint

∑k

ϕk sk(τk, ek)

Previous tools of convex analysis useless here


Intensive mixture entropy

Intensive mixture entropy at thermodynamical equilibrium

s(τ, e, ϕl, ϕg) = max((αk)k,(zk)k)∈Ωint

∑k

ϕk sk(τk, ek)

Theorem [M., 19]

For given mass fractions ϕk, k = l, g

s is strictly concave with respect to (τ, e)

Gibbs relationTds = de+ pdτ

Thermodynamical equilibrium characterized by

Thermal equilibriumT := Tl = Tg = Tv

Partial Dalton’s lawp := pl = pv + pg


Homogeneous Relaxation Model (R)

∂tρ+ ∂x(ρu) = 0

∂t(ρu) + ∂x(ρ|u|2 + p) = 0

∂t(ρ(e+|u|2

2)) + ∂x((ρ(e+

|u|2

2) + p)u) = 0

∂tϕk + u∂xϕk = 0, k = l, g

∂tzk + u∂xzk = Qzk , k = l, g

∂tαl + u∂xαl = Qαl

Lax entropy pair (−ρσ,−ρσu) with

σ(τ, e, (ϕk)k, (αk)k, (zk)k) =∑k

ϕksk(τk, ek)

Relaxation pressure: p = p(1/ρ, e, (ϕ)l,g, αl, (z)l,g)

Characterization of the equilibriumI Fractions Y at equilibrium

Yeq(τ, e, ϕl, ϕg) = argmax((αk)k,(zk)k)∈Ωint

σ(τ, e, (ϕk)k, (αk)k, (zk)k)

I Pressure p at equilibrium, partial Dalton’s law

peq(τ, e, ϕl, ϕg) := p(τ, e, ϕl, ϕg , Yeq(τ, e, ϕl, ϕg))


Homogeneous Equilibrium model (E)

∂tρ+ ∂x(ρu) = 0

∂t(ρu) + ∂x(ρ|u|2 + peq) = 0

∂t(ρ(e+|u|2

2)) + ∂x((ρ(e+

|u|2

2) + peq)u) = 0

∂tϕk + u∂xϕk = 0, k = l, g

The closure laws are derived from the equilibrium mixture entropyTeq(1/ρ, e, ϕl, ϕg) := Tl = Tg = Tv

peq(1/ρ, e, ϕl, ϕg) := pl = pg + pv

Since the equilibrium mixture entropy s(τ, e, ϕl, ϕg) is not concave, does not enterthe framework of [Chen, Levermore, Liu, 92]

Theorem [M., 19]

The Homogeneous Equilibrium Model (E) is hyperbolic

Proved by a Godunov-Mock-like argument based on the property of s(τ, e, ϕl, ϕg)


The whole hierarchy and its applications

Three-phase three-velocity model [Hérard, M., 19]

Velocity−−−−−−→Relaxation

Homogeneous Relaxation Model [M., 19]

eq. thermo.−−−−−−−→Relaxation

Homogeneous Equilibrium Model [M., 19]

Including phase transitionIndustrial applicationsI [Quibel, 20; Hurisse, Quibel, preprint]: Homogenous Relaxation Model, comparison

to SUPERCANON experiment, loss of coolant accident in pressurized water reactorI [Hérard, Hurisse, Quibel, 20]: Four-phase extensions for vapor explosion phenomena

(RIA)


Outline



3 Some perspectives


Metastable states and the van der Waals equation

Superheated water, two-phase framework

Cubic law: reduced van der Waals Equation of State, isotherm case

Non monotone pressure p(τ) below the critical temperature T = 1

Spinodal zone between τ− and τ+: unstable elliptic region for the Euler system,speed of sound c =

√−p′(τ)


The van der Waals model and the Maxwell construction

Non convex extensive Helmoltz free energy F (M,V ) and intensive energy f(τ)

dF = −pdV + µdM f(τ) = −τp(τ) + µ(τ)

Maxwell constructionComputation of the convex hull of fEqual area rule: define two saturation densities τ∗1 and τ∗2 s.t. p(τ∗1 ) = p(τ∗2 ) = p∗

3 Recover monotone isothermal curves, correct phase transition [Müller, Voß, 06]

7 Loss of metastable states Aim: recover both equilibria


Mixture entropy and equilibria

[Ghazi, 18] [James, M., 16; Ghazi, James, M., 19; Ghazi, James, M., 20]

Two phases k = 1, 2 of specific volume τk and mass fraction ϕk ∈ [0, 1], described by thesame nonconvex van der Waals Helmoltz free energy f

For a given mixture state of specific volume τMixture energy

F(ϕ1, ϕ2, τ1, τ2) = ϕ1f(τ1) + ϕ2f(τ2)

Set of intensive constraintsI Mass conservation: 1 = ϕ1 + ϕ2I Immiscible mixture: τ = ϕ1τ1 + ϕ2τ2

Proposition: second principle of Thermodynamics

The set of states which minimize F are

Saturation states: τ1 = τ∗1 , τ2 = τ∗2

τ1 = τ2 = τ


Mixture entropy and equilibria


Two phases k = 1, 2 of specific volume τk and mass fraction ϕk ∈ [0, 1], described by thesame nonconvex van der Waals Helmoltz free energy f

For a given mixture state of specific volume τMixture energy

F(ϕ1, ϕ2, τ1, τ2) = ϕ1f(τ1) + ϕ2f(τ2)

Set of intensive constraintsI Mass conservation: 1 = ϕ1 + ϕ2I Immiscible mixture: τ = ϕ1τ1 + ϕ2τ2

Proposition: second principle of Thermodynamics

The set of states which minimize F are

Saturation states: τ1 = τ∗1 , τ2 = τ∗2

τ1 = τ2 = τIncludes non stable states of the spinodal zone


Dynamical system of phase transition


Build a dynamical system in order toI Get rid of the spinodal zoneI Capture asymptotically the equilibria of the minimization problemI Dissipate the Helmoltz free energy F along trajectories

A good candidate:For a given mixture state of specific volume τI Describe the mixture by the volume and mass fraction of the phase 1

r = (α1, ϕ1) =: (α,ϕ)I Consider r = −∇rF(r)

The dynamical systemα = α(1− α)(p(τ1(r))− p(τ2(r))) = Qα(r)

ϕ = ϕ(1− ϕ)(µ(τ2(r))− µ(τ1(r))) = Qϕ(r)


Asymptotically stable equilibria

Equilibria and attraction [Ghazi, 18][James, M. 16; James, Ghazi, M., 19, 20]

Saturation states r∗ = (α∗, ϕ∗) with α∗ 6= ϕ∗ ∈]0, 1[ s.t. τ∗k = τk(r∗), k = 1, 2

The phases 1 and 2 identify: r = (β, β), β ∈]0, 1[ s.t. τ1(r) = τ2(r) = τ




Saturation states r∗ = (α∗, ϕ∗) with α∗ 6= ϕ∗ ∈]0, 1[ s.t. τ∗k = τk(r∗), k = 1, 2 Attractive equilibria especially in the spinodal zone
















The phases 1 and 2 identify: r = (β, β), β ∈]0, 1[ s.t. τ1(r) = τ2(r) = τ Attractive equilibria especially in the metastable region





The phases 1 and 2 identify: r = (β, β), β ∈]0, 1[ s.t. τ1(r) = τ2(r) = τ Attractive equilibria especially in the metastable region


Dynamical system of phase transition - mixture pressure

For a τ ∈ [0.5, 4], set r(0) such that τk(r(0)) is a small perturbation of τRepresent

(τ, limt→∞

αp(τ1(r(t))) + (1− α)p(τ2(r(t))))


Dynamical system of phase transition - mixture pressure

For a τ ∈ [0.5, 4], set r(0) such that τk(r(0)) is a large perturbation of τRepresent

(τ, limt→∞

αp(τ1(r(t))) + (1− α)p(τ2(r(t))))


Coupling with the fluid dynamics

Introduce a HRM model with p(τ, r) = αp(τ1(r)) + (1− α)p(τ2(r))∂t(ρ) + ∂x(ρu) = 0

∂t(ρu) + ∂x(ρu2 + p) = 0

∂tα+ u∂xα = Qα(r)

∂tϕ+ u∂xϕ = Qϕ(r)

(R)

Undefined equilibrium model (multi-valued equilibrium pressure law)

Hyperbolicity of the convective model is not guaranteed in the spinodal zone

Theorem [James, M., 16]

Invariant domains of hyperbolicity are subsets of the attraction basins

3 Simulation of nucleation by means of a finite volume scheme with splitting approach

3 [Ghazi, James, M., 20] Extension to the non-isothermal framework


Outline



3 Some perspectives


Some perspectives and ongoing works ® ® ®

Asymptotic preserving methods, model adaptation

Better understanding of the hierarchy of three-phase flow models(Jean Bussac’s PhD, started in Oct. 2020)

Rigorous modelling of compressible multi-phase flows

Investigate real Equation of State


Some perspectives and ongoing works ® ® ®

Asymptotic preserving methods, model adaptation

Better understanding of the hierarchy of three-phase flow models(Jean Bussac’s PhD, started in Oct. 2020)

Rigorous modelling of compressible multi-phase flows

Investigate real Equation of State

® ® ® Thank you for your attention !


Date post:	06-Feb-2022
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Entropieendynamiquedesﬂuides Entropy in Fluid Dynamics

Documents