A Unified Gas-Kinetic Scheme for Continuum and Rarefied Flows II: Multi-Dimensional Cases (Communications in Computational Physics, 2012)
With discretized particle velocity space, a multi-scale unified gas-kineticscheme for entire Knudsen number flows has been constructed based on the kineticmodel in one-dimensional case [J. Comput. Phys., vol. 229 (2010), pp. 7747-7764]. Forthe kinetic equation, to extend a one-dimensional scheme to multidimensional flow isnot so straightforward. The major factor is that addition of one dimension in physicalspace causes the distribution function to become two-dimensional, rather than axiallysymmetric, in velocity space. In this paper, a unified gas-kinetic scheme based on theShakhov model in two-dimensional space will be presented. Instead of particle-basedmodeling 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 andthe 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 ofdirectly 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 fromthe free molecule to the Navier-Stokes solutions. In comparison with the DSMC andNavier-Stokes flow solvers, the current method is much more efficient than DSMC inlow speed transition and continuum flow regimes, and it has better capability thanNS solver in capturing of non-equilibrium flow physics in the transition and rarefiedflow regimes. As a result, the current method can be useful in the flow simulationwhere both continuum and rarefied flow physics needs to be resolved in a single computation. This paper will extensively evaluate the performance of the unified schemefrom free molecule to continuum NS solutions, and from low speed micro-flow to highspeed non-equilibrium aerodynamics. The test cases clearly demonstrate that the unified scheme is a reliable method for the rarefied flow computations, and the schemeprovides an important tool in the study of non-equilibrium flow.