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Numerical Modeling of low Mach flows Asymptotics and Numerics Herv é Guillard

Numerical Modeling of low Mach flows Asymptotics and Numerics Herv é Guillard. Summary : Compressible/incompressible models Finite Volume Approximations Analysis of FV Solvers in the Low Mach # regime Preconditioned Solvers Numerical examples.

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Numerical Modeling of low Mach flows Asymptotics and Numerics Herv é Guillard

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  1. Numerical Modeling of low Mach flows Asymptotics and Numerics Hervé Guillard Summary : Compressible/incompressible models Finite Volume Approximations Analysis of FV Solvers in the Low Mach # regime Preconditioned Solvers Numerical examples Herve.Guillard@sophia.inria.fr

  2. THE TWO WORLDS of FLUID DYNAMICS Compressible Incompressible Hyperbolic models Elliptic models Wave propagation with Infinite speed of propagation Finite speed Key-words : Invariant domains, Key-words:Stokes, dissipative Entropy inequalities, positivity inf/sup (LBB), staggered grid, FV, collocated meshes, explicit EF,implicit time advancing time stepping What about the phenomena at the interface between these two worlds

  3. Incompressible Euler 2unknowns : velocity, pressure Elliptic Compressible Euler 3 unknowns : velocity, pressure density Hyperbolic Relationship between these two systems ?

  4. Simple Asymptotic Analysis : Compressible Euler contains Incompressible + Acoustic How these different phenomena organize ? Not a general answer : depends on Isentropic/non-isentropic Inviscid/viscous Unbounded/bounded domains Well prepared data/general initial data

  5. Fast component :

  6. Slow component :

  7. Behavior in the small Mach limit Generalcase sol of sol of incompressiblesystem Acoustic system Well-prepared case The oscillatory component disappears : incompressible system strong limit of compressible one

  8. Now let’s go to NUMERICAL SCHEMES

  9. FINITE VOLUME METHOD Godunov scheme :

  10. LOW MACH NUMBER and TWO PHASE FLOWS Hyperbolic FV methods for transonic, supersonic : ok ! But behavior of hyperbolic FV methods in the near incompressible regime ?

  11. UPWIND FV and LOW MACH # : ACCURACY PBS 0.1 0.01 In the near Incompressible Regime : FV upwind Schemes are not an accurate approximation They even does not converge to the sol of incompressible system ! 0.001 ROE VFROE GODUNOV

  12. Solution of Riemann pb in the low Mach limit Discrete eq. are not an approximation of incompressible system

  13. Remedy : Suppress the acoustic component in the solution ! of the Riemann Pb However Resulting scheme is centered Better : R1) Multiply the acoustic component by M* Even Better : R2) Increase the artificial viscosity in the Momentum eq by a factor 1/M* + R3) If possible, do not change the scheme in the transonic And supersonic regime Preconditioned Riemann Problems

  14. PRECONDITIONED DISSIPATION

  15. Design of the preconditioning matrix P fulfilling requirements R1), R2), R3) Simple in (s,u,p) variables : 1 0 0 0 P = 0 1 0 0 0 0 1 0 2 0 0 0 ß

  16. Preconditioned FV scheme Chebyshev Spectral approximation 81x 81 mesh 97x97 mesh

  17. Example of two-phase flow model : A Five equation model

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