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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

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

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

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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

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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

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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

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

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

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

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

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

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

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High Mach number flow passing through a double ellipse M6 airfoil

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

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Hollow cylinder flare: nitrogen Mesh 61x105x17

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temperature pressure

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21 The Foundation of Modern CFD

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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

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

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24 M=10 GKSGRP

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

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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

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High-order schemes (order =>3)

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

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

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High-order gas-kinetic scheme (HGKS)

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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 !

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32 Laminar Boundary Layer

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Viscous shock tube

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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

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

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1400x700 Gas-kinetic Scheme Osmp7 (4000x2000

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Remarks on high-order CFD methods

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

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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

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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

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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

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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

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

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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.

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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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Gas-kinetic schemes for flow computations Kun Xu Mathematics Department Hong Kong University of Science and Technology.

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