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

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

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

5. 5 Continuum Air at atmospheric condition: 2.5x10 19 molecules/cm 3, Mean free path : 5x10 -8 m, Collision frequency : 10 9 /s Gradient transport mechanism Navier-Stokes-Fourier equations (NSF) Rarefaction Typical length scale: L Knudsen number: Kn= /L High altitude, Vacuum ( ), MEMS (L ) Kn Martin H.C. Knudsen (1871-1949) Danish physicist Gas properties

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

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

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

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

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

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

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

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14. 14

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

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

18. Hollow cylinder flare: nitrogen Mesh 61x105x17

19. temperature pressure

21. 21 The Foundation of Modern CFD

22. 22 Introduce flow physics into numerical schemes (FDS, FVS, AUSM, ~RPs) Spatial Limiters (Boris, Book, van Leer, … 70-80s) Modern CFD (Godunov-type methods) Governing equations: Euler, NS, … space limiter

23. 23 A black cloud hanging over CFD clear sky (1990- now) Carbuncle Phenomena RoeAUSM+

24. 24 M=10 GKSGRP

25. 25 Godunovs description of numerical shock wave Is this physical modeling valid ?

26. 26 Gas kinetic scheme Particle free transport Physical process from a discontinuity collision NS Euler Godunov method Euler NS ? (infinite number of collisions) Riemann solver

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

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

29. 29 High-order Kinetic Scheme (HBGK-NS) BGK-NS (2001) HBGK (2009)

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

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

32. 32 Laminar Boundary Layer

34. Viscous shock tube

35. 5 th -WENO 6 th -order viscous Sjogreen& Yees 6 th -order WAV66 scheme 500x250 mesh points Reference solution 4000x2000 mesh points 5 th -WENO-reconstruction +Gas-Kinetic Evolution 500x250 mesh points

36. 1000x500 Sjogreen& Yees 6 th -order WAV66 scheme 1000x500 Gas Kinetic Scheme

37. 1400x700 Gas-kinetic Scheme Osmp7 (4000x2000

38. Remarks on high-order CFD methods

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

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

41. 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

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

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

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

45. 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.

46. 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.