## 5 Numerical Results and Discussions
### D Thermal Cavity Flow  
The last case is the thermal cavity flow induced by temperature gradients at wall. The geometric configuration and numerical setup are sketched in Figure 11. The computational domain is a \(10^{-5}\times 10^{-5}\) m\({}^{2}\) square, partitioned by structured rectangular mesh. The left and right walls are maintained at constant temperature \(T_{C}=263\) K, while at the top and bottom walls, \(T_{f}\), we introduce a linearly increasing temperature (from \(T_{C}=263\) K to \(T_{H}=283\) K) in left half of domain, and a linearly decreasing temperature (from \(T_{H}\) to \(T_{C}\)) in the right half. All the walls are treated as diffusive boundaries. The reference mean free path \(\lambda\) is calculated from the initial uniform density. The reference temperature \(T_{0}\) and the characteristic length \(L_{0}\) are set to be 273 K and \(10^{-5}\) m, respectively. Three cases corresponding to \(Kn\) = 0.01, \(Kn\) = 0.1, and \(Kn\) = 0.5 are computed. For all of these cases, the spatial space is discretized with \(101\times 101\) non-uniform points, and the discrete velocity space is discretized in the range of \(\left[-6\sqrt{2 RT_{0}},~{}6\sqrt{2 RT_{0}}\right]^{3}\) with \(64\times 64\times 24\) non-uniform points. The nondimensional temperature contours \(\overline{T}=T/T_{0}\) and streamlines as well as the nondimensional velocity \(\overline{u}_{x}\) contours are shown in Figure 12. The temperature profiles \(\overline{T}\) along the vertical center and the horizontal center lines are shown in Figure 13.  
As indicated from Figure 12, the overall agreement between the DVM and the moment methods, especially in terms of the temperature field, is quite well. For all the tested cases, four vortices are generated with two of them rotate counter-clockwise at the lower left and upper right of the cavity, and another two vortices rotate clockwise at the upper left and lower right of the cavity. Besides, we can find that the NSF equations are able to reproduce the temperature fields when \(Kn\) < 0.1, however, they can't predict the velocity fields accurately. In contrast, both the R13 and R26 moment equations can reproduce accurate temperature and velocity fields in comparison with the DVM solutions. Moreover, from the Figure 13, it is found that from the regions near solid walls to the cavity center, the gas temperature increases along horizontal lines, while decreases along vertical lines. The maximum temperature value decreases as the degree of rarefaction increases. It is because both the interactions between gas molecules and hot wall and the molecular collisions become weak when the gas is far away from the equilibrium state. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{
\begin{tabular}{c} **Computational** \\ **Memory (MB)** \\ \end{tabular} } & \multicolumn{3}{c}{**Computational Time (Min)**} \\ \cline{3-5}  & & \(Kn\) = 0.01 & \(Kn\) = 0.1 & \(Kn\) = 0.5 \\ \hline DVM & 9943.6 & 325.44 & 152.62 & 47.83 \\ NSF & 11.00 & 1.98 & 1.37 & 1.52 \\ R13 & 11.00 & 5.58 & 8.37 & 22.32 \\ R26 & 11.10 & 8.55 & 12.82 & 29.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the computational cost between the DVM and moment method. Figure 11: Numerical setup for thermally driven cavity flow. Figure 12: Comparison of \(\overline{T}=T/T_{0}\), \(\overline{u}_{x}=u_{x}/\sqrt{2 RT_{0}}\) contours and streamlines of the temperature gradients induced flow case at different Knudsen numbers: (**a,b**) \(Kt=0.01\), (**c,d**) \(Kn=0.1\), (**e,f**) \(Kt=0.5\). Due to symmetry of the problem, in each sub-figure, upper left and upper right are results obtained from the DVM and R26 moment method, respectively; lower left and lower right are results obtained from the R13 moment method and NSF, respectively.