L -P - R L S P EDP-N 2011 Large time-step and asymptotic-preserving numerical schemes for hyperbolic systems with sources C. Chalons ∗ , M. Girardin ∗∗ ∗ Université Paris Diderot-Paris 7, France ∗∗ CEA-Saclay, France 25-26 Octobre 2011 1 / 41
L -P - R L S P I This work is motivated by the study of two-phase flows involved in nuclear reactors in nominal, incidental or accidental conditions Several models and approaches : "Micro"-scale models : fine description of liquid / vapor interface topologies "Macro"-scale models : two-phase flow described as a mixture at thermodynamical equilibrium "Middle"-scale models : the so-called bi-fluid approach, takes into account desequilibrium between both phases We are interested in the numerical approximation of one particular bi-fluid averaged model, the so-called 7 -equation or Baer-Nunziato like model 2 / 41
L -P - R L S P T 7- In one space dimension, the model reads ∂α k ∂α k ∂ x = Θ ( p k − p l ) , ∂ t + u I ∂ t ( α k ̺ k ) + ∂ ∂ ∂ x ( α k ̺ k u k ) = 0 , ∂ t ( α k ̺ k u k ) + ∂ ∂ ∂α k ∂ x ( α k ( ̺ k u 2 k + p k )) − p I ∂ x = α k ̺ k g − Λ ( u k − u l ) , ∂ t ( α k ̺ k e k ) + ∂ ∂ ∂α k ∂ x ( α k ( ̺ k e k + p k ) u k ) − p I u I ∂ x = α k ̺ k gu k − p I Θ ( p k − p l ) − u I Λ ( u k − u l ) with α 1 + α 2 = 1 u I , p I : interfacial velocity and pressure (to be precised) We note that the system is nonconservative , with short form ∂ U ∂ t + ∂ ∂ x F ( U ) + B ( U ) ∂ U ∂ x = S ( U ) 3 / 41
L -P - R L S P T 7- In 1 D and dimensionless form, the model reads ∂α k ∂α k ∂ t + u I ∂ x = Θ ( p k − p l ) , ∂ t ( α k ̺ k ) + ∂ ∂ ∂ x ( α k ̺ k u k ) = 0 , ∂ t ( α k ̺ k u k ) + ∂ ∂ ∂α k ∂ x ( α k ( ̺ k u 2 k + p k )) − p I ∂ x = α k ̺ k g − Λ ( u k − u l ) , ∂ t ( α k ̺ k e k ) + ∂ ∂ ∂α k ∂ x ( α k ( ̺ k e k + p k ) u k ) − p I u I ∂ x = α k ̺ k u k g − p I Θ ( p k − p l ) − u I Λ ( u k − u l ) We assume that the drag force and pressure relaxation coe ffi cients are given by Θ = θ ( U ) Λ = λ ( U ) ǫ 2 | u 1 − u 2 | ǫ 2 for a small parameter ǫ . Then we have p 2 − p 1 = O ( ǫ 2 ) , u 2 − u 1 = O ( ǫ ) 4 / 41
L -P - R L S P A Following the Chapman-Enskog method, assume that � p r = p 1 − p 2 = 0 + ǫ p 1 r + O ( ǫ 2 ) u r = u 1 − u 2 = 0 + ǫ u 1 r + O ( ǫ 2 ) and set ρ = α 1 ρ 1 + α 2 ρ 2 ρ u = α 1 ρ 1 u 1 + α 2 ρ 2 u 2 ρ e = α 1 ρ 1 e 1 + α 2 ρ 2 e 2 ρ Y = α 2 ρ 2 Theorem. A first-order approximation w.r.t. ǫ of the 7-equation model is given by the following di ff erential drift-flux model ∂ t ρ + ∂ x ρ u = 0 ∂ t ρ Y + ∂ x ( ρ Yu + ρ Y (1 − Y ) u r ) = 0 ∂ t ρ u + ∂ x ( ρ u 2 + p + ρ Y (1 − Y ) u 2 r ) = ρ g ∂ t ρ e + ∂ x ( ρ eu + pu + ρ Y (1 − Y ) u 2 r u ) = ρ gu with u r given by the (Darcy-like) di ff erential closure relation | u r | u r = ρ Y ( ρ − ρ Y ) ( 1 − 1 ) ∂ x p Λ ρ 1 ρ 2 ρ See Ambroso-Chalons-Coquel-Galié-Godlewski-Raviart-Seguin, CMS 2008 5 / 41
L -P - R L S P M Eigenvalues of the Jacobian matrix F ′ ( U ) + B ( U ) are always real and given by u I u k u k ± c k k = 1 , 2 where c k is the sound speed of phase k t u 1 u I u 2 u k − c k u k + c k U L U R x Riemann solutions are not known and di ffi cult to calculate / approximate Pressure laws may be strongly non linear, even tabulated Resonance occurs if u I = u k ± c k The model is not conservative... 6 / 41
L -P - R L S P O Note that : Time-step CFL restrictions are naturally based on acoustic waves (that are not predominant here, too bad !) k , u ( | u k ± c k | , | u k | , | u I | ) ∆ t ∆ x ≤ 1 max 2 Flows are subsonic and / or with low Mach number in nuclear reactors Our objective is to propose a numerical scheme : able to deal with any equation of state and any choice ( u I , p I ) stable under a more adapted time-step CFL restriction based on transport waves (that are predominant here so that accuracy is required) k , u ( | u k | , | u I | ) ∆ t ∆ x ≤ 1 max 2 and asymptotic-preserving 7 / 41
L -P - R L S P T - As a first-step, we consider the following model ∂ t ̺ + ∂ x ̺ u = 0 , ∂ t ̺ u + ∂ x ( ̺ u 2 + p ) = ̺ g − ̺α ǫ u , ∂ t ( ̺ E ) + ∂ x ( ̺ Eu + pu ) = ̺ ug − ̺α ǫ u 2 for a small parameter ǫ . Then we have u = O ( ǫ ) Recall that : Eigenvalues are given by u − c u u + c where c is the sound speed Pressure laws may be strongly non linear, even tabulated, which makes di ffi cult the resolution Time-step CFL restrictions are naturally based on acoustic waves in Godunov-type schemes u ( | u ± c | , | u | ) ∆ t ∆ x ≤ 1 max 2 8 / 41
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