Fig. 1. Schematics: (a) 2D lid-driven cavity discretized by using 60^2 spatial grids/cells, (b) 3D lid-driven cavity discretized by using 60^3 spatial grids/cells.
Fig. 2. Profiles of the perturbed horizontal velocity u (left column) and perturbed vertical velocity v (right column) along the vertical centreline (X = 0.5) and horizontal centreline (Y = 0.5), respectively, of the 2D lid-driven cavity. The DVM and DSBGK results are compared with the published DSMC data. Note that the flow velocity has been normalized by the lid velocity.
Fig. 3. The 3D lid-driven case with Ma=3.2×10−3, Kn=8: contours of the perturbed u, v, n and T on the planes of Z=0.5 and Y=0.5, respectively, obtained by the DVM using a 4×48×12 velocity grid (left column), and the DSBGK method using 10 simulated molecules per cell with 5000 time-averaging samples (right column).
Here, we present a comparison between the DVM and DSBGK methods. In terms of accuracy, both agree very well with the traditional DSMC method at moderate flow speed (50 m/s), where the DSMC results are available. With respect to efficiency, the DVM is much faster than the DSBGK method in 2D simulation, but the DVM becomes slower in 3D simulation if the fixed DSBGK time-averaging process (e.g., hundreds of time steps to make the velocity distribution smooth, and thousands of time steps to make the temperature distribution smooth) is unnecessary (e.g., in computing the mass flow rate and permeability, which are already averaged over the physical space and thus smooth at each moment) or negligible compared to the preceding convergence process (e.g., in large-scale simulations of complicated geometry). The number of time-averaging samples required by the DSBGK method is independent of Mach number, which is in sharp contrast to most molecular simulation methods, e.g., the DSMC method. Due to a large number of velocity grids required by the DVM in the tested 3D simulation (e.g., at least 4*24*12 by DVM versus only 10 by DSBGK for each spatial grid/cell), the DVM memory usage is about 40 times larger than that of the DSBGK simulation (about 10 GB versus 0.25 GB) when both used the same spatial grid/cell number for simplicity, although the DSBGK method can use much less spatial cells as clearly shown in Examples 10 and 13.
It is noteworthy that the DVM algorithm had been optimized in the above comparisons on efficiency and memory usage, i.e., using a fully time-implicit scheme with a large Courant-Friedrichs-Lewy (CFL) number of 10^4 for steady state flows, and the sophisticated layout of half-range Gauss-Hermite velocity grid in non-Cartesian coordinates.