Abstract
In the study of rarefied gas dynamics, the discrete velocity method (DVM) has been widely employed to solve the gas kinetic equations. Although various versions of DVM have been developed, their performance, in terms of modeling accuracy and computational efficiency, is yet to be comprehensively studied in all the flow regimes. Here, the traditional third-order time-implicit Godunov DVM (GDVM) and the recently developed discrete unified gas-kinetic scheme (DUGKS) are analysed in finding steady-state solutions of the low-speed force-driven Poiseuille and lid-driven cavity flows. With the molecular collision and free streaming being treated simultaneously, the DUGKS preserves the second-order accuracy in the spatial and temporal discretizations in all flow regimes. Towards the hydrodynamic flow regime, not only is the DUGKS faster than the GDVM when using the same spatial mesh, but also requires less spatial resolution than that of the GDVM to achieve the same numerical accuracy. From the slip to free molecular flow regimes, however, the DUGKS is slower than the GDVM, due to the complicated flux evaluation and the restrictive time step which is smaller than the maximum effective time step of the GDVM. Therefore, the DUGKS is preferable for problems involving different flow regimes, particularly when the hydrodynamic flow regime is dominant. For highly rarefied gas flows, if the steady-state solution is mainly concerned, the implicit GDVM, which can boost the convergence significantly, is a better choice.
Highlights
Multi-scale flows, where different temporal and spatial scales are presented, are often found in nature and engineering, which represent a modeling and simulation challenge
Our results show that both the Godunov DVM (GDVM) and discrete unified gas-kinetic scheme (DUGKS) can accurately reproduce the results in all the flow regimes, provided that the mesh resolution is sufficient
For the GDVM, the convection term of the kinetic model is approximated by the upwind scheme with the underlying assumption of molecular free streaming between two grid points, while in the DUGKS the collision and transport processes are coupled physically by using the discrete characteristic solution of the kinetic equation
Summary
Multi-scale flows, where different temporal and spatial scales are presented, are often found in nature and engineering, which represent a modeling and simulation challenge. In the slip regime (10−3 < Kn < 0.1), the NS solvers and the DSMC method become either inaccurate or inefficient: the NS equations are inappropriate to describe rarefied (non-equilibrium) gas flows because they are derived based upon the near equilibrium hypothesis, while the particle nature of the DSMC method restricts its application in near hydrodynamic regime [2], as the temporal and spatial resolutions must be smaller than the molecular collision time and mean free path, respectively. For multi-scale gas flows, it is intuitive to use continuum-particle hybrid methods that solve the flow fields in different regimes by appropriate solvers [3,4,5,6]. Hybrid methods may encounter great difficulties for flows with a continuous and complex variation of flow physics [7]
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