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

Abstract

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 from free molecule to continuum NS solutions, and from low speed micro-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 with moving mesh and velocity space adaptation

“…The philosophical principal underlying the UGKS is to use the partial-differential-equation (PDE) to get a local solution around a cell interface [28], the so-called PDE-based modeling. One of the main reason to use this principal is that the modeled physical scale of the kinetic equation and the numerical scale of a mesh size and time step may be different significantly.…”

“…Solving the above equation, the f nþ1 is determined explicitly in the next time level. More detailed formulation can be found in [26,28].…”

Numerical investigation of the heat and mass transfer analogy in rarefied gas flows

“…Analytic expressions for this breakdown parameter in the continuum and free-molecular flow limits are given. Then, several test cases are investigated using the Unified Gas Kinetic Scheme (UGKS) of Xu et al [22][23][24] modified here with the addition of a simple surface adsorption model based on the Langmuir kinetics. The UGKS is chosen since it solves efficiently a discrete approximation to the Boltzmann equation appropriate for flows spanning the range from continuum to rarefied.…”

Numerical investigation of gaseous heat and mass transfer: the effect of rarefaction

“…To investigate the method under more rare ed these conditions are also simulated with comparison to a solution by the UGKS [138] which will be described in detail in Sec. 5.…”

“…Unfortunately not all ow cases of interest t into this category and so an alternative method is required. The core method selected was the UGKS method, as developed by K. Xu [75,137,138], with additional boundary conditions as outlined in Sec.…”