Presentation on theme: "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."— Presentation transcript:
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
Fluid flow is commonly studied in one of three ways: – Experimental fluid dynamics. – Theoretical fluid dynamics. – Computational fluid dynamics (CFD). TheoryExperiment 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
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 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 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 The physical modeling of particles distribution function at a cell interface
8 : constructed according to Chapman-Enskog expansion Modeling for continuum flow:
Smooth transition from particle free transport to hydrodynamic evolution Discontinuous (kinetic scale, free transport) Hydrodynamics scale
10 Numerical fluxes : Update of flow variables: Prandtl number fix by modifying the heat flux in the above equation
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 !
Mathematical manipulation physical reality ? There is no any physical evolution law about the time evolution of derivatives in a discontinuous region ! weak solution)
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
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
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 Different scale physical modeling Boltzmann Eqs. Navier-Stokes Euler quantum Newton Flow description depends on the scale of the discretized space
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 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.