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

42 pages
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


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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 αtk+uI∂αk= Θ(pkpl)x
)+(αk t(αk̺kx̺kuk)=0
t(αk̺kuk)+(αk(̺u2+pk))pIαxk=αk̺kg xkk
Λ(ukul)
t(αk̺kek)+x(αk(̺kek+pk)uk)pIuIαxk=αk̺kgukpIΘ(pkpl)uIΛ(ukul) withα1+α2=1
uIpI: interfacial velocity and pressure (to be precised)
We note that the system isnonconservative, with short form
U=S(U) tU+xF(U)+B(U)x
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L-P    - R  L S         P
T7- 
In 1Dand dimensionless form, the model reads αtk+uIαk=Θ(pkpl)x
t(αk̺k)+x(αk̺kuk)=0
t(αk̺kuk)+x(αk(̺kuk2+pk))pαxk=αk̺kg I
Λ(uk
ul)
t(αk̺kek)+x(αk(̺kek+pk)uk)pIuIαxk=αk̺kukgpIΘ(pkpl)uIΛ(ukul) We assume that the drag force and pressure relaxation coecients are given by
Θ=θ(ǫ2U)Λλǫ(2U)|u1u2| = for a small parameterǫ. Then we have
p2p1=O(ǫ2)
u2u1=O(ǫ)
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