Large time-step and asymptotic-preserving numerical schemes for - PowerPoint PPT Presentation

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