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
Hélène Mathis Habilitation defense Université de Nantes 2 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 3 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 4 / 40
Outline
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
Hélène Mathis Habilitation defense Université de Nantes 5 / 40
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+
Hélène Mathis Habilitation defense Université de Nantes 6 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 7 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 8 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 9 / 40
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]
Hélène Mathis Habilitation defense Université de Nantes 10 / 40
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
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
Hélène Mathis Habilitation defense Université de Nantes 12 / 40
Outline
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
Hélène Mathis Habilitation defense Université de Nantes 13 / 40
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)
Hélène Mathis Habilitation defense Université de Nantes 14 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 15 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 16 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 17 / 40
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∆)
Hélène Mathis Habilitation defense Université de Nantes 18 / 40
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]
Hélène Mathis Habilitation defense Université de Nantes 19 / 40
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)
Hélène Mathis Habilitation defense Université de Nantes 19 / 40
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
Adaptation with θ = 10−2 Adaptation with θ = 1
Second interaction
Adaptation with θ = 10−2 Adaptation with θ = 1
Hélène Mathis Habilitation defense Université de Nantes 20 / 40
Outline
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
Hélène Mathis Habilitation defense Université de Nantes 21 / 40
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
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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
Hélène Mathis Habilitation defense Université de Nantes 23 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 23 / 40
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
Extensive mixture entropy at thermodynamical equilibrium
S(W ) =K
k=1
Sk(Wk) S(W , V ) =K
(V )k=1
Sk(Wk, V )
Hélène Mathis Habilitation defense Université de Nantes 23 / 40
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
Extensive mixture entropy at thermodynamical equilibrium
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
Hélène Mathis Habilitation defense Université de Nantes 23 / 40
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
Extensive mixture entropy at thermodynamical equilibrium
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
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
Hélène Mathis Habilitation defense Université de Nantes 24 / 40
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]
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Extensive vs. intensive settings
Fluid composed of a liquid l, its vapor v and a gas g
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Extensive vs. intensive settings
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
Hélène Mathis Habilitation defense Université de Nantes 26 / 40
Extensive vs. intensive settings
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
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
Hélène Mathis Habilitation defense Université de Nantes 26 / 40
Extensive vs. intensive settings
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
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
Hélène Mathis Habilitation defense Université de Nantes 26 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 27 / 40
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))
Hélène Mathis Habilitation defense Université de Nantes 28 / 40
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)
Hélène Mathis Habilitation defense Université de Nantes 29 / 40
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)
Hélène Mathis Habilitation defense Université de Nantes 30 / 40
Outline
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
Hélène Mathis Habilitation defense Université de Nantes 31 / 40
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)
Hélène Mathis Habilitation defense Université de Nantes 32 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 33 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 34 / 40
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 = τIncludes non stable states of the spinodal zone
Hélène Mathis Habilitation defense Université de Nantes 34 / 40
Dynamical system of phase transition
[Ghazi, 18] [James, M., 16; Ghazi, James, M., 19; Ghazi, James, M., 20]
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)
Hélène Mathis Habilitation defense Université de Nantes 35 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 36 / 40
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 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
Hélène Mathis Habilitation defense Université de Nantes 36 / 40
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 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
Hélène Mathis Habilitation defense Université de Nantes 36 / 40
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 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
Hélène Mathis Habilitation defense Université de Nantes 36 / 40
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 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
Hélène Mathis Habilitation defense Université de Nantes 36 / 40
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 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
Hélène Mathis Habilitation defense Université de Nantes 36 / 40
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))))
Hélène Mathis Habilitation defense Université de Nantes 37 / 40
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))))
Hélène Mathis Habilitation defense Université de Nantes 37 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 38 / 40
Outline
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
Hélène Mathis Habilitation defense Université de Nantes 39 / 40
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
Hélène Mathis Habilitation defense Université de Nantes 40 / 40
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 !
Hélène Mathis Habilitation defense Université de Nantes 40 / 40