Comparative study of the discrete velocity and lattice Boltzmann methods for rarefied gas flows through irregular channels

Rooted from the gas kinetics, the lattice Boltzmann method (LBM) is a powerful tool in modeling hydrodynamics. In the past decade, it has been extended to simulate rarefied gas flows beyond the Navier-Stokes level, either by using the high-order Gauss-Hermite quadrature, or by introducing the relaxation time that is a function of the gas-wall distance. While the former method, with a limited number of discrete velocities (e.g., D2Q36), is accurate up to the early transition flow regime, the latter method (especially the multiple relaxation time (MRT) LBM), with the same discrete velocities as those used in simulating hydrodynamics (i.e., D2Q9), is accurate up to the free-molecular flow regime in the planar Poiseuille flow. This is quite astonishing in the sense that less discrete velocities are more accurate. In this paper, by solving the Bhatnagar-Gross-Krook kinetic equation accurately via the discrete velocity method, we find that the high-order Gauss-Hermite quadrature cannot describe the large variation in the velocity distribution function when the rarefaction effect is strong, but the MRT-LBM can capture the flow velocity well because it is equivalent to solving the Navier-Stokes equations with an effective shear viscosity. Since the MRT-LBM has only been validated in simple channel flows, and for complex geometries it is difficult to find the effective viscosity, it is necessary to assess its performance for the simulation of rarefied gas flows. Our numerical simulations based on the accurate discrete velocity method suggest that the accuracy of the MRT-LBM is reduced significantly in the simulation of rarefied gas flows through the rough surface and porous media. Our simulation results could serve as benchmarking cases for future development of the LBM for modeling and simulation of rarefied gas flows in complex geometries.

Figure 1

The marginal VDF $\int v_1\mathrm{\Phi }(x_2=0,{\mathbf{v}}_{2\mathrm{D}})f_{\mathrm{eq},2\mathrm{D}}d{v}_1$ in the force-driven Poiseuille flow between two parallel plates. The rarefaction parameter $\delta$ in the linearized BGK equation is (a) $\delta =0.01$, (b) $\delta =0.1$, (c) $\delta =1$, and (d) $\delta =10$. (Solid lines) The marginal VDF obtained from DVM with $N_v=320$ in Eq. (16). Note that $N_v=32$ is enough to get accurate results; we use $N_v=320$ here just to show the detailed structure of the VDF. (Squares) Results from the high-order Gauss-Hermite quadrature with 32 abscissas in each velocity direction.

Figure 2

(a) The velocity profile in the Poiseuille flow between two parallel plates, obtained from the DVM with $N_v=32$ in Eq. (16). For symmetry, only half of the spatial domain is shown. (b) The ratio between the effective viscosity and real viscosity of gas. Symbols are the numerical results according to Eq. (25). Solid lines are the analytical results given by Eq. (22). Also note that according to Eq. (23), ${\mu }_e/\mu$ is 0.9186, 0.8494, 0.7383, 0.3607, and 0.0534 when $\delta$ is 20, 10, 5, 1, and 0.1, respectively.

Figure 3

The geometry of the rough channel as used in Couette flow. Shaded regions are solid plates, while the gas flows in the white region.

Figure 4

The convergence test of the mass flow rate $Q$ obtained from the DVM, for Couette flow in a rough channel of $\epsilon =2\%$ and $D=1.5$. The influence of the number of grid points in (a) the $x_1$ direction, (b) the $x_2$ direction within the rough region, that is, $x_2\in [min\left(r\right),max\left(r\right)]$, and (c) the $x_2$ direction outside the rough region, that is, $x_2\in (max\left(r\right),1]$.

Figure 5

The convergence test of the mass flow rate $Q$ obtained from the DVM, for the Couette flow in a rough channel of $\epsilon =2\%$ and $D=1.5$. The influence of the number of discrete velocity points when Eq. (16) is used.

Figure 6

Comparisons in the average streamwise velocity profile of the Couette flow in microchannels between the DVM and MRT-LBM [36], when $D=1.5, \epsilon =2\%$, (a) $\text{Kn}=0.1$ and (b) $\text{Kn}=1$. “Smooth” here means that both plates are smooth, while “rough” means that only the bottom plate has a rough surface; see Fig. 3.

Figure 7

The ratio between the mass flow rates in the rough ($D=1.5$ and $\epsilon =2\%$) and smooth channels. Solid line, DVM; squares, MRT-LBM from [36].

Figure 8

Different porous media used in the numerical simulation [37]. The gas flows from left to right.

Figure 9

The ratio of the apparent gas permeability $k_a$ to the intrinsic permeability $k_{\infty }$ as a function of the equivalent Knudsen number ${\text{Kn}}^{*}$. Results of MRT-LBM simulations are obtained from Ref. [37].

Figure 10

The geometry for the simulation of gas flows around a square cylinder in a microchannel. Shaded regions are solids. The channel height and width are set as $H$, while the dimensions of the cylinder are set as $l\times l$.

Figure 11

The variation of the apparent gas permeability of the Poiseuille flow around a square cylinder of dimension $0.2H\times 0.2H$ in a microchannel, as a function of the Knudsen number, at three different values of the tangential momentum accommodation coefficient. Results indicated by “Cercignani” are calculated using the dimensionless flow rates by Cercignani and Daneri [43].

Figure 12

The horizontal velocity profile in the Poiseuille flow around a square cylinder of dimension $0.2H\times 0.2H$ in a microchannel; the influence of the Knudsen number and tangential momentum accommodation coefficient $\alpha$. Solid lines, $\alpha =1.0$; dotted lines, $\alpha =0.5$. The Knudsen numbers in the first and second rows are $\text{Kn}=0.1$ and 1, respectively. The velocity profiles for $\alpha =0.8$ are not shown, as they are located between those for $\alpha =1$ and $\alpha =0.5$.

Figure 13

The mass flow rate of the Couette flow around a square cylinder of dimension $0.2H\times 0.2H$ in a microchannel.

Figure 14

The horizontal velocity profile in the Couette flow around a square cylinder of dimension $0.2H\times 0.2H$ in a microchannel. Solid lines, $\alpha =1$; dotted lines, $\alpha =0.5$. The Knudsen numbers in the first and second rows are $\text{Kn}=0.1$ and 1, respectively. For clarity, the results of $\text{Kn}=0.8$ are not shown, but they lie between the results of $\text{Kn}=1$ and 0.5.

Figure 15

The apparent gas permeability of the Poiseuille flow (left) and the mass flow rates of the Couette flow (right) around a square cylinder of dimensions $0.5H\times 0.5H$ in a microchannel.