A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows Ⅱ:Multi-Dimensional Cases

With discretized particle velocity space,a multi-scale unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the kinetic model in one-dimensional case[J.Comput.Phys.,vol.229(2010),pp.7747-7764].For the kinetic equation,to extend a one-dimensional scheme to multidimensional flow is not so straightforward.The major factor is that addition of one dimension in physical space causes the distribution function to become two-dimensional,rather than axially symmetric,in velocity space.In this paper,a unified gas-kinetic scheme based on the Shakhov model in two-dimensional space will be presented.Instead of particle-based modeling for the rarefied flow,such as the direct simulation Monte Carlo(DSMC)method,the philosophical principal underlying the current study is a partial-differential-equation(PDE)-based modeling.Since the valid scale of the kinetic equation and the scale of mesh size and time step may be significantly different,the gas evolution in a discretized space is modeled with the help of kinetic equation,instead of directly solving the partial differential equation.Due to the use of both hydrodynamic and kinetic scales flow physics in a gas evolution model at the cell interface,the unified scheme can basically present accurate solution in all flow regimes from the free molecule to the Navier-Stokes solutions.In comparison with the DSMC and Navier-Stokes flow solvers,the current method is much more efficient than DSMC in low speed transition and continuum flow regimes,and it has better capability than NS solver in capturing of non-equilibrium flow physics in the transition and rarefied flow regimes.As a result,the current method can be useful in the flow simulation where both continuum and rarefied flow physics needs to be resolved in a single computation.This paper will extensively evaluate the performance of the unified scheme fromfreemolecule to continuum NS solutions,and fromlow speedmicro-flow to high speed non-equilibrium aerodynamics.The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations,and the scheme provides an important tool in the study of non-equilibrium flow.

A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows III:Microflow Simulations

Due to the rapid advances inmicro-electro-mechanical systems(MEMS),the study of microflows becomes increasingly important.Currently,the molecular-based simulation techniques are the most reliable methods for rarefied flow computation,even though these methods face statistical scattering problem in the low speed limit.With discretized particle velocity space,a unified gas-kinetic scheme(UGKS)for entire Knudsen number flow has been constructed recently for flow computation.Contrary to the particle-based direct simulation Monte Carlo(DSMC)method,the unified scheme is a partial differential equation-based modeling method,where the statistical noise is totally removed.But,the common point between the DSMC and UGKS is that both methods are constructed through direct modeling in the discretized space.Due to the multiscale modeling in the unified method,i.e.,the update of both macroscopic flow variables and microscopic gas distribution function,the conventional constraint of time step being less than the particle collision time inmany direct Boltzmann solvers is released here.The numerical tests show that the unified scheme is more efficient than the particle-basedmethods in the low speed rarefied flow computation.Themain purpose of the current study is to validate the accuracy of the unified scheme in the capturing of non-equilibrium flow phenomena.In the continuum and free molecular limits,the gas distribution function used in the unified scheme for the flux evaluation at a cell interface goes to the corresponding Navier-Stokes and free molecular solutions.In the transition regime,the DSMC solution will be used for the validation of UGKS results.This study shows that the unified scheme is indeed a reliable and accurate flow solver for low speed non-equilibrium flows.It not only recovers the DSMC results whenever available,but also provides high resolution results in cases where the DSMC can hardly afford the computational cost.In thermal creep flow simulation,surprising solution,such as the gas flowing from hot to cold regions along the wall surface,is observed for the first time by the unified scheme,which is confirmed later through intensive DSMC computation.