# A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
## 4 Numerical Tests
### 3D Natural Convection in a Cubic Cavity  
To further test the performance of USTLBM in three-dimensional problems, here we present the example of 3D natural convection in a cubic cavity. The physical set-up of the problem is a cubic cavity with its left and right walls, respectively, fixed on constant temperatures of \(T_{h}\) and \(T_{c}\). On the other four walls, Neumann boundary condition with normal temperature gradient equal to zero is implemented. The non-dimensional parameters that determine the converged flow pattern are still the Rayleigh number and Prandtl number as defined in Equations (66) and (67). The average Nusselt number on the wall is defined as  
\[Nu_{w}=\frac{1}{L^{2}}{\int_{0}^{L}}\int_{0}^{L}\frac{\partial T}{\partial x} \mathrm{d}y\mathrm{d}z \tag{77}\]  
Considering the numerical efficiency, non-uniform Cartesian mesh is adopted in this example. Two cases with high Rayleigh numbers of \(Ra=10^{5}\) and \(10^{6}\) are simulated.  
Tables 7 and 8 give the average Nusselt number on the wall, the maximum horizontal velocity along the vertical centerline and the maximum vertical velocity along the horizontal centerline at different mesh sizes and Rayleigh numbers. It is observed that, be refining the mesh resolution, both USTLBM and STLBM can converge to the reference data [30; 31; 32; 33; 34]. Such performance indicates the robustness of USTLBM and STLBM on non-uniform mesh in three-dimensional scenarios. Careful comparisons reveal that, compared to our two-dimensional test of the natural convection in a square cavity (Section 4.1), the advantage of USTLBM over STLBM is not so obvious in this three-dimensional example. The possible reason is that, since the non-uniform mesh is adopted here, errors are inevitably introduced in the interpolation process, due to which the advantage of USTLBM in lowering numerical errors is counteracted and the results of USTLBM and STLBM are quite close. If uniform mesh is applied, the present USTLBM performs better than STLBM under the same mesh size, especially on coarser meshes (see Table 9). Such evidence indicates that USTLBM can lower the numerical error of STLBM while maintaining the same order of accuracy in space.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{\(Ra\)} & \multicolumn{3}{c}{**Inner Cylinder, \(\overline{k_{eqi}}\)**} & \multicolumn{3}{c}{**Outer Cylinder, \(\overline{k_{eqo}}\)**} \\ \cline{2-7}  & **[PERSON] and** & **[PERSON] [29]** & **Present** & **[PERSON] and** & **[PERSON] [29]** & **Present** \\  & **[PERSON] [28]** & & **Present** & **[PERSON] [28]** & **[PERSON] [29]** & **Present** \\ \hline \(10^{3}\) & \(1.081\) & \(1.082\) & **1.091** & \(1.084\) & \(1.082\) & **1.079** \\ \(6\times 10^{3}\) & \(1.736\) & \(1.715\) & **1.705** & \(1.735\) & \(1.715\) & **1.719** \\ \(10^{4}\) & \(2.010\) & \(1.979\) & **1.960** & \(2.005\) & \(1.979\) & **1.986** \\ \(5\times 10^{4}\) & \(3.024\) & \(2.958\) & **2.877** & \(2.973\) & \(2.958\) & **2.964** \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of average equivalent heat conductivity at different Rayleigh numbers.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameters** & **Grid** & **41 \(\times\) 21 \(\times\) 41** & **61 \(\times\) 31 \(\times\) 61** & **81 \(\times\) 41 \(\times\) 81** & **101 \(\times\) 51 \(\times\) 101** & **Reference [30; 31; 32; 33; 34]** \\ \hline \multirow{2}{*}{\(Nu_{w}\)} & STLBM & 4.349 & 4.367 & 4.361 & 4.355 & \multirow{2}{*}{4.337–4.464} \\  & USTLBM & 4.381 & 4.372 & 4.361 & 4.354 & \multirow{2}{*}{4.337–4.464} \\ \multirow{2}{*}{\(u_{\max}\)} & STLBM & 36.379 & 37.002 & 37.279 & 37.400 & \\  & USTLBM & 36.213 & 36.960 & 37.264 & 37.396 & \\ \multirow{2}{*}{\(w_{\max}\)} & STLBM & 63.891 & 65.129 & 65.351 & 65.431 & \multirow{2}{*}{63.952–65.817} \\  & USTLBM & 64.487 & 65.262 & 65.380 & 65.446 & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparisons of flow parameters for 3D natural convection in \(Ra=10^{5}\).