Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation

Abstract

SUMMARYBased on the Bhatnagar-Gross-Krook (BGK) Boltzmann model equation, the uniÿed simpliÿed velocity distribution function equation adapted to various ow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non-linear source terms. Based on the unsteady time-splitting technique and the non-oscillatory, containing no free parameters, and dissipative (NND) ÿnite-di erence method, the gas kinetic ÿnite-di erence second-order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic ow parameters at each point in the physical space. As a result, a uniÿed simpliÿed gas kinetic algorithm for the gas dynamical problems from various ow regimes is developed. To test the reliability of the present numerical method, the one-dimensional shocktube problems and the ows past two-dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related ows indicate that both high resolution of the ow ÿelds and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained.

Gas-kinetic numerical method for solving mesoscopic velocity distribution function equation

“…Then they are solved in a coupling feasible way. On the basis of the developed finite difference methods [14,15,18], the finite difference second-order scheme is adopted to directly solve the molecular velocity distribution function by using the improved Euler method and the NND-4(a) scheme [19] which is a two-stage scheme with secondorder accuracy in time and space…”

“…Then the gas-kinetic numerical schemes can be constructed by developing a time-splitting method for unsteady equation and finite difference technique. The gas-kinetic numerical methods were presented and successively applied to one-dimensional, two-dimensional and simple gas flows with low Mach numbers from various flow regimes [13][14][15][16]. The present study aims at extending and developing this gas-kinetic method for three-dimensional complex flows and aerodynamic phenomena with high Mach numbers and simulating Poiseuille-like micro-scale gas flows in micro-channels occuring in Micro-Electro-Mechanical Systems (MEMS).…”

Hybrid Multiscale Methods for Hyperbolic and Kinetic Problems

“…Clearly the details of the schemes are rather problem dependent [1,10,18,20,29,38,42,51,52,62,63,65]. We quote the recent works by Weinan E and Bjorn Engquist for a general approach to heterogeneous multiscale methods in scientific computing [22][23][24].…”

A multidimensional approach to rarefied and transitional flows with the CIP method

“…YABE AND Y. OGATA accurate and stable nonlinear kinetic-type models [5][6][7] are indispensable for a wide variety of rarefied gas and transition regime flows.…”