On a simple model of isothermal phase transition

Transcription

1 On a simple model of isothermal phase transition Nicolas Seguin Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie Paris 6 France Micro-Macro Modelling and Simulation of Liquid-Vapour Flows Bordeaux On a simple model of isothermal phase transition p.1/21

2 Outline of the talk Modelling of isothermal phase transition Optimization of the entropy The Homogeneous Relaxation Model (HRM) The Homogeneous Equilibrium Model (HEM) The zero-relaxation limit The Riemann problem for the HEM Multiple waves and the Liu entropy condition Main results : existence, uniqueness and L 1 loc -continuity Numerical tests Some shock tubes Two-dimensional cavitation Conclusion On a simple model of isothermal phase transition p.2/21

3 Context of this study Collaboration between the Laboratoire Jacques-Louis Lions C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, N. Seguin and the CEA Saclay A. Ambroso, S. Kokh, J. Segré. Goal: Interfacial coupling of thermohydraulic models Hyperbolic systems Theory and numerics Two-phase flows On a simple model of isothermal phase transition p.3/21

4 Bifluid model and entropy optimization See [Barberon, Helluy 4], [Helluy, Seguin 6] and [Allaire, Faccanoni, Kokh 6]. (and [Coquel, Perthame 98], [Jaouen 1], [Chanteperdrix, Villedieu, Vila 2]...) Classical Euler equations for the mixture t ρ + x ρu = t ρu + x (ρu 2 + p) = t ρe + x ((ρe + p)u) = t Y + u x Y = λ(y eq Y) where Y is the vector of fractions of volume, mass and energy Thermodynamical behaviour Mixture laws for thermodynamical variables (ρ, p, T, s...) Thermodynamical equilibrium defined by entropy optimization w.r.t. Y: Y eq (τ, ε) = max Y [,1] 3 s(τ, ε, Y) On a simple model of isothermal phase transition p.4/21

5 The isothermal Homogeneous Relaxation Model The isothermal HRM we consider here is (in Lagrangian coordinates (t, x) ) t τ x u = t u + x P R (τ, α) = t α = λ(α eq (τ) α) where α is the mass fraction. The equilibrium α eq is given by the equality of chemical potentials. Proposition The HRM model is strictly hyperbolic. Its eigenvalues are τ P R (τ, α),, τ P R (τ, α). On a simple model of isothermal phase transition p.5/21

6 The isothermal Homogeneous Equilibrium Model The equilibrium pressure is given by P E (τ) = P R (τ, α eq (τ)), τ >, and the isothermal HEM is { t τ x u =, t u + x P E (τ) =. p P E (τ) p m Liquid Mixture Vapour τ 2 τ 1 τ On a simple model of isothermal phase transition p.6/21

7 The isothermal Homogeneous Equilibrium Model Proposition In Ω p = ((, τ 2 ) (τ 1, )) R, the HEM is strictly hyperbolic (λ ± (u) = ± P E (τ) ). In Ω m = [τ 2, τ 1 ] R, the HEM is nonstrictly hyperbolic (λ (u) = λ + (u) and loss of basis). The model locally becomes the system of pressureless gases. p P E (τ) p m Liquid Mixture Vapour τ 2 τ 1 τ On a simple model of isothermal phase transition p.7/21

8 From HRM to HEM Since HEM is not strictly hyperbolic, it does not enter in classical frameworks. ([Liu 87], [Chen, Liu, Levermore 94], [Hanouzet, Natalini 3], [Yong 4]...) Study of relaxation shock profiles [Yong, Zumbrun 4]: Take two states (τ L, u L ) and (τ R, u R ) and a speed σ satisfying the Rankine-Hugoniot jump relations: { σ(τ R τ L ) (u R u L ) =, σ(u R u L ) + (P E (τ R ) P E (τ L )) =. Parametrize the solutions of HRM by ξ = λ(x σt): τ(ξ), u(ξ), α(ξ). Solve the system of ODE with (τ L, u L, α eq (τ L )) for ξ = and (τ R, u R, α eq (τ R )) for ξ = +. On a simple model of isothermal phase transition p.8/21

9 From HRM to HEM Relaxation shock profiles [Kokh, Seguin 7]: First case: if α eq (τ L ) = α eq (τ R ) (= or 1), the relaxation term is inactive. = Use of the Lax entropy condition Second case: α eq (τ L ) = α eq (τ R ) We have to solve the 2 2 ODE system { σ 2 τ (ξ) + d ξ P R (τ(ξ), α(ξ)) =, α (ξ) = (α eq (τ(ξ)) α(ξ)). If the equations of state of each phase are perfect gas pressure laws: Proposition A relaxation shock profile exists for the discontinuity The discontinuity statisfies the Liu entropy condition On a simple model of isothermal phase transition p.9/21

10 The Riemann problem for HEM The Riemann problem for HEM: [Godlewski, Seguin 6] p P E (τ) t τ x u =, t u + x P E (τ) { =, (τ (τ, u)(, x) = L, u L ) if x <, (τ R, u R ) if x >. p m Liquid τ 2 Mixture τ 1 Vapour τ We consider self-similar solutions u(x, t) = u(x/t). t u 1 u 2 u L u R x On a simple model of isothermal phase transition p.1/21

11 The Riemann problem for HEM A discontinuity is admissible if it satisfies the Liu entropy condition: σ(u l, u r ) σ(u l, u) for all u S (u l ) with τ (min(τ l, τ r ), max(τ l, τ r )). The speed of an admissible discontinuity is smaller than the speed of any discontinuity included in it. p P E (τ) p P E (τ) p P E (τ) τ L τ2 τ τ τ2 τ1 τ2 τ L τ1 τ (τ L ) τ (τ L ) τ1 τ L τ < τ L < τ 2 τ 2 < τ L < τ 1 τ 1 < τ L On a simple model of isothermal phase transition p.11/21

12 Main results Theorem 1 For all u L, u R Ω, the self-similar solution of the Riemann problem exists and is unique, allowing the occurrence of vacuum {τ = + } when necessary. Theorem 2 Let W (x/t; u L, u R ) be the self-similar solution of the Riemann problem. For any L > and initial data (u L, u R ), (v L, v R ) Ω 2, there exists a constant C > such that L L W (ξ; u L, u R ) W (ξ; v L, v R ) dξ C( u L v L + u R v R ) where stands for the Euclidean norm of R 2. On a simple model of isothermal phase transition p.12/21

13 Nonpositive multiple waves Nonpositive multiple waves are given as a succession of nonpositive rarefaction waves and admissible discontinuities (geometric interpretation) = U (u L ). u u u u L u L τ L τ 2 u τ1 τ τ τ τ2 τ 1 τ L τ < τ L < τ2 u τ 1 < τ L u u L τ 2 τ τ1 L τ τ τ 2 τ L τ 1 On a simple model of isothermal phase transition p.13/21

14 An example of Riemann solution Construction (by symmetry) of U (u R ) and find the intersection I (u L, u R ) = U (u L ) U + (u R ). A solution with multiple waves: u u R u (u R ) u (u L) u 2 (u L) t I (u L, u R ) u (u R ) u L τ 2 τ 1 u 2 (u L) u (u L) τ I (u L, u R ) u L u R x On a simple model of isothermal phase transition p.14/21

15 Uniqueness of the Riemann solution The uniqueness is given if I (u L, u R ) = U (u L ) U + (u R ) is a singleton. u t u R (τ 2, u ) u τ 2 τ 1 τ u L u L u R x On a simple model of isothermal phase transition p.15/21

16 Some properties of the exact solution Remark If two states are separated by a phase transition discontinuity, at least one of them is at saturation. Remark If the initial data is composed only by pure phases, no mixture zone can occur in the solution. Remark The Cauchy problem can be ill-posed if the initial data is in the mixture zone. [Lagoutière] = Instable behaviour of the mixture zone On a simple model of isothermal phase transition p.16/21

17 Numerical schemes Any classical Finite Volume scheme can be used. Two approaches: Compute HRM + instantaneous equilibrium α = α eq after each time step Directly compute HEM Here, we used two different numerical schemes: Suliciu s relaxation method ([Coquel et al. 99], [Bouchut 4]) The Rusanov scheme 1D and 2D tests in the Eulerian setting. On a simple model of isothermal phase transition p.17/21

18 Numerical tests Two rarefaction waves in the liquid with occurence of vapor 1.5 u 4x4 - Rusanov 4x4 - Relaxation HEM - Rusanov HEM - Relaxation rho 4x4 - Rusanov 4x4 - Relaxation HEM - Rusanov HEM - Relaxation On a simple model of isothermal phase transition p.18/21

19 Numerical tests Two rarefaction waves with one state in the liquid and the other in the mixture 1.5 u 4x4 - Rusanov 4x4 - Relaxation HEM - Rusanov HEM - Relaxation rho 4x4 - Rusanov 4x4 - Relaxation HEM - Rusanov HEM - Relaxation On a simple model of isothermal phase transition p.19/21

20 Numerical tests Box in a hypervelocity liquid flow (mesh: 3x6) On a simple model of isothermal phase transition p.2/21

21 Conclusion Study of a model of isothermal phase change Relaxation Riemann problem Numerical approximations Remaining points Zero-relaxation limit for rarefaction waves and smooth solutions The Cauchy problem More numerical tests More complex modelling Realistic equations of state More complex phase change On a simple model of isothermal phase transition p.21/21