A comparative study of discrete velocity methods for low-speed rarefied gas flows

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

Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

“…Also displayed in this figure is the result obtained by DUGKS. 23 It can be seen from Figure 3B that the simulation results of three different methods for solution reconstructions at the cell interface are basically the same and they compare well with that of DUGKS in the free molecular flow regime, transition flow regime and slip flow regime. However, in the continuum flow regime, the results of improved methods (Scheme I and Scheme II) are better than that of the original model as shown in Figure 3A, and the improvement becomes more evident as the Knudsen number is decreased.…”

“…Figure shows the apparent gas permeability for Poiseuille flow in different flow regimes. Also displayed in this figure is the result obtained by DUGKS . It can be seen from Figure B that the simulation results of three different methods for solution reconstructions at the cell interface are basically the same and they compare well with that of DUGKS in the free molecular flow regime, transition flow regime and slip flow regime.…”

“…The convergence criterion is set by the condition that the error of at two adjacent iteration steps does not exceed 10 −10 . Consistent with previous study, 23 in our simulation, the hard-sphere (HS) model is utilized to calculate the dynamic viscosity, where the exponent w in Equation (27) is 0.5. The Knudsen number is varied from Kn = 10 −4 to 10, which covers the free molecular flow regime (Kn ⩾ 10), transition flow regime (10 −1 ⩽ Kn < 10), slip flow regime (10 −2 ⩽ Kn < 10 −1 ), and continuum flow regime (Kn < 10 −2 ).…”

“…The first test case is the force-driven Poiseuille flow with different Knudsen numbers. 23,45,46 This problem can be considered as a viscous fluid flow between two infinite parallel plates with temperature of T 0 = 273 K separated by a distance of L in the y-direction. An external force is exerted in the x-direction, so that Equation (1) can be rewritten as…”

Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?

“…Note that unified gas-kinetic scheme (UGKS) [9,10,11,12,13] can remove the restrictions on cell size and time step by simultaneously handling free streaming and collision of gas molecules during transport processes. However, as information exchanging relays on the evolution of velocity distribution function, UGKS still needs a large number of iterations to obtained steady-state solutions in near-continuum flows [14,15].…”