Lagrange Projection strategy for the gas dynamics equations Relaxation for the Lagrangian step Source terms and notion of consistency in the integral sense Properties

Description

1/41 Lagrange-Projection strategy for the gas-dynamics equations Relaxation for the Lagrangian step Source terms and notion of consistency in the integral sense Properties EDP-Normandie 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

• see ambroso-chalons-coquel-galié-godlewski-raviart-seguin


• ?k?kg ?


• bi-fluid approach

Informations

• Publié par: pefav
• Nombre de lectures: 22
• Langue: English

Extrait

L-P    - R  L S         

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

P

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L-P    - 

R  L S         

I

This work is motivated by the study of two-phase ows involved in nuclear reactors in nominal, incidental or accidental conditions

Several models and approaches :

"Micro"-scale models: ne description of liquid/vapor interface topologies "Macro"-scale models: two-phase ow described as a mixture at thermodynamical equilibrium "Middle"-scale models: the so-called bi-uid approach, takes into account desequilibrium between both phases

We are interested in the numerical approximation of one particular bi-uid averaged model, the so-called 7-equation or Baer-Nunziato like model

P

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L-P    - R  L S         P

T7- 

In one space dimension, the model reads  ∂ ∂α t k +u I ∂α k = Θ(p k −p l ) ∂x

 

∂ )+ ∂ (α k ∂t (α k ̺ k ∂x ̺ k u k )=0

∂ ∂ t (α k ̺ k u k )+ ∂ ∂ (α k (̺u 2 +p k ))−p I ∂ α∂ x k =α k ̺ k g x k k

−Λ(u k −u l )



  ∂ ∂t (α k ̺ k e k )+ ∂ ∂x (α k (̺ k e k +p k )u k )−p I u I ∂ α∂ x k =α k ̺ k gu k −p I Θ(p k −p l )−u I Λ(u k −u l ) withα 1 +α 2 =1

u I p I : interfacial velocity and pressure (to be precised)

We note that the system isnonconservative, with short form

∂U =S(U) ∂ ∂t U + ∂ ∂ x F(U)+B(U) ∂x

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L-P    - R  L S         P

T7- 

In 1Dand dimensionless form, the model reads  ∂ ∂α t k +u I ∂α ∂ k =Θ(p k −p l ) x

 

∂ ∂ t (α k ̺ k )+ ∂ ∂ x (α k ̺ k u k )=0

∂ ∂ t (α k ̺ k u k )+ ∂ ∂ x (α k (̺ k u k 2 +p k ))−p ∂α ∂x k =α k ̺ k g I

−Λ(u k

−u l )

  ∂ ∂ t (α k ̺ k e k )+ ∂ ∂x (α k (̺ k e k +p k )u k )−p I u I ∂ α∂ x k =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 coefficients are given by

Θ= θ( ǫ 2 U) Λ λ ǫ ( 2 U) |u 1 −u 2 | = for a small parameterǫ. Then we have

p 2 −p 1 =O(ǫ 2 )

u 2 −u 1 =O(ǫ)

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