Highorder gas evolution model for computational fluid dynamics

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, …

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

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

Computation: a description of flow motion in a discretized space and time Collision ree n. F a Me th Pa The way of gas molecules passing through the cell interface depends on the cell resolution and particle mean free path

Gas properties Continuum Air at atmospheric condition: 2. 5 x 1019 molecules/cm 3, Mean free path : 5 x 10 -8 m, 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 5

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 6

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

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

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

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

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

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

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High Mach number flow passing through a double ellipse M 6 airfoil

M=10, Re=10^6, Tin=79 K, Tw=294. 44 K, mesh 15 x 81 x 19

Hollow cylinder flare: nitrogen Mesh 61 x 105 x 17

temperature pressure

The Foundation of Modern CFD 21

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 -80 s) （space limiter） 22

A black cloud hanging over CFD clear sky (1990 now) Carbuncle Phenomena Roe AUSM+ 23

M=10 GKS GRP 24

Godunov’s description of numerical shock wave Is this physical modeling valid ? 25

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

High-order schemes (order =>3)

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

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

High-order gas-kinetic scheme (HGKS)

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 ! High-order Gas-kinetic scheme: one step integration along the cell interface. Gauss-points: Riemann solvers for others

Laminar Boundary Layer 32

Viscous shock tube

500 x 250 mesh points 5 th-WENO 6 th-order viscous Reference solution 4000 x 2000 mesh points Sjogreen& Yee’s 6 th-order WAV 66 scheme 500 x 250 mesh points 5 th-WENO-reconstruction +Gas-Kinetic Evolution

1000 x 500 Sjogreen& Yee’s 6 th-order WAV 66 scheme 1000 x 500 Gas Kinetic Scheme

1400 x 700 Gas-kinetic Scheme Osmp 7 (4000 x 2000）

Remarks on high-order CFD methods

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

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

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

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

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

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

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.

• In the discontinuous case, there is no such a physical law related to the time evolution of highorder 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.