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High-order gas evolution model for computational fluid dynamics

High-order gas evolution model for computational fluid dynamics. K un Xu Hong Kong University of Science and Technology. C ollaborators: Q.B. Li, J. Luo , J. Li, L. Xuan , …. Experiment. Theory. Scientific Computing. Fluid flow is commonly studied in one of three ways:

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High-order gas evolution model for computational fluid dynamics

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  1. High-order gas evolution model for computational fluid dynamics Kun Xu Hong Kong University of Science and Technology Collaborators: Q.B. Li, J. Luo, J. Li, L. Xuan,…

  2. Experiment Theory Scientific Computing Fluid flow is commonly studied in one of three ways: • Experimental fluid dynamics. • Theoretical fluid dynamics. • Computational fluid dynamics (CFD).

  3. Contents • The modeling in gas-kinetic scheme (GKS) • The Foundation of Modern CFD • High-order schemes • Remarks on high-order CFD methods • Conclusion

  4. Computation: a description of flow motion in a discretized space and time Collision Mean Free Path The way of gas molecules passing through the cell interface depends on the cell resolution and particle mean free path

  5. Gas properties Continuum Air at atmospheric condition: 2.5x1019 molecules/cm3, Mean free path : 5x10-8m, Collision frequency  : 109 /s Gradient transport mechanism Navier-Stokes-Fourier equations (NSF) Martin H.C. Knudsen (1871-1949) Danish physicist Rarefaction Typical length scale: L Knudsen number: Kn=/L High altitude, Vacuum () , MEMS (L ) Kn 

  6. Physical modeling of gas flow in a limited resolution space f : gas distribution function, W : conservative macroscopic variables Fundamental governing equation in discretized space: Take conservative moments to the above equation: For the update of conservative flow variables, we only need to know the fluxes across a cell interface! PDE-based modeling:use PDE’s local solution to model the physical process of gas molecules passing through the cell interface

  7. The physical modeling of particles distribution function at a cell interface

  8. Modeling for continuum flow: : constructed according to Chapman-Enskog expansion

  9. Smooth transition from particle free transport to hydrodynamic evolution Hydrodynamics scale Discontinuous (kinetic scale, free transport)

  10. Numerical fluxes: • Update of flow variables: • Prandtl number fix by modifying the heat flux in the above equation

  11. Gas-kinetic Scheme ( ) Central-difference Upwind Scheme Kinetic scale Hydrodynamic scale

  12. M. Ilgaz, I.H. Tuncer, 2009

  13. High Mach number flow passing through a double ellipse M6 airfoil

  14. M=10, Re=10^6, Tin=79K, Tw=294.44K, mesh 15x81x19

  15. Hollow cylinder flare: nitrogen Mesh 61x105x17

  16. temperature pressure

  17. The Foundation of Modern CFD

  18. Modern CFD (Godunov-type methods) Governing equations: Euler, NS, … Introduce flow physics into numerical schemes (FDS, FVS, AUSM, ~RPs) Spatial Limiters(Boris, Book, van Leer,…70-80s) (space limiter)

  19. A black cloud hanging over CFD clear sky (1990- now) Carbuncle Phenomena Roe AUSM+

  20. M=10 GKS GRP

  21. Godunov’s description of numerical shock wave Is this physical modeling valid ?

  22. Physical process from a discontinuity Gas kinetic scheme Godunov method Particle free transport ? collision NS NS Riemann solver Euler Euler (infinite number of collisions)

  23. High-order schemes(order =>3)

  24. Reconstruction + Evolution The foundation of most high-order schemes: 1st-order dynamic model: Riemann solver inviscid viscous

  25. High-order Kinetic Scheme (HBGK-NS) BGK-NS (2001) HBGK (2009)

  26. High-order gas-kinetic scheme (HGKS)

  27. Comparison of gas evolution model: Godunov vs. Gas-Kinetic Scheme (b): Riemann solver evolution (a): gas-kinetic evolution Space & time, inviscid & viscous, direction & direction, kinetic & Hydrodynamic, fully coupled ! High-order Gas-kinetic scheme: one step integration along the cell interface. Gauss-points: Riemann solvers for others

  28. Laminar Boundary Layer

  29. Viscous shock tube

  30. 500x250 mesh points 5th-WENO 6th-order viscous Reference solution 4000x2000 mesh points Sjogreen& Yee’s 6th-order WAV66 scheme 500x250 mesh points 5th-WENO-reconstruction +Gas-Kinetic Evolution

  31. 1000x500 Sjogreen& Yee’s 6th-order WAV66 scheme 1000x500 Gas Kinetic Scheme

  32. 1400x700 Gas-kinetic Scheme Osmp7 (4000x2000)

  33. Remarks onhigh-order CFD methods

  34. ? Mathematical manipulation physical reality (weak solution) There is no any physical evolution law about the time evolution of derivatives in a discontinuous region !

  35. Even in the smooth region, in the update of “slope orhigh-order derivatives” through weak solution, the Riemann solver (1st-order dynamics) does NOT provide appropriate dynamics. Example: Riemann solver only provides u, not at a cell interface

  36. Huynh, AIAA paper 2007-4079 Unified many high-order schemes DG, SD, SV, LCP, …, under flux reconstruction framework Riemann Flux Interior Flux Z.J. Wang

  37. STRONG Solution from Three Piecewise Initial Data Update flow variables at nodal points ( , ) at next time level, And calculate flux Solution at t= Reconstructed new initial condition from nodal values Initial condition at t=0 Generalized solutions with piecewise discontinuous initial data W(x)=

  38. Control Volume PDE’s local evolution solution (strong solution) is used to Model the gas flow passing through the cell interface in a discretized space. PDE-based Modeling

  39. Different scale physical modeling quantum Boltzmann Eqs. Newton Navier-Stokes Euler Flow description depends on the scale of the discretized space 44

  40. Conclusion • GKS is basically a gas evolution modeling in a discretized space. This modeling covers the physics from the kinetic scale to the hydrodynamic scale. • In GKS, the effects of inviscid & viscous, space & time, different by directions, and kinetic & hydrodynamic scales, are fully coupled. • Due to the limited cell size, the kinetic scale physical effect is needed to represent numerical shock structure, especially in the high Mach number case. Inside the numerical shock layer, there is no enough particle collisions to generate the so-called “Riemann solution” with distinctive waves. The Riemann solution as a foundation of modern CFD is questionable.

  41. In the discontinuous case, there is no such a physical law related to the time evolution of high-order derivatives. The foundation of the DG method is not solid. It may become “a game of limiters” to modify the updated high-order derivatives in high speed flow computation.

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