# A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
## 4 Numerical Tests
### Natural Convection in a Concentric Annulus  
To test the flexibility of USTLBM on body-fitted mesh and on problems with curved boundaries, numerical test on natural convection in a concentric annulus is carried out in this section. Set-up of the problem is depicted in Figure 6. The gas filled between the concentric annulus is driven by the buoyancy force that is induced by the temperature gradient between the hot inner circular cylinder and cold outer circular cylinder. Non-dimensional parameters in this problem include the aspect ratio \(Ar=R_{o}/R_{i}\), the Prandtl number \(Pr=v/\chi\) and the Rayleigh number \(Ra=g\beta(T_{i}-T_{o})L^{3}/v\chi\). In our simulations, we set \(Ar=2.6\) and \(Pr=0.71\); and four cases with different Rayleigh numbers of \(10^{3}\), \(6\times 10^{3}\), \(10^{4}\) and \(5\times 10^{4}\) are considered. To accurately describe the curved boundaries encountered in this problem, body fitted mesh with the mesh size of \(251\times 81\) is adopted in all simulations. Lagrangian interpolation algorithm is introduced in computational domain to evaluate functional values on locations where particles start the streaming process.  
Isotherms at steady state of different cases are presented in Figure 7. It can be seen that our simulations keep the smoothness of the temperature field, which validates the stability of USTLBM. It is also observed that, with the increase of Rayleigh number, the isotherms are more squeezed to the lower part of the inner circular cylinder and the upper part of the outer circular cylinder. The reason is that, at higher Rayleigh number, the buoyancy force induced by the temperature gradient has larger  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(Ri\) & **[PERSON] and [PERSON] [(26)]** & **[PERSON] et al. [(27)]** & **Present** \\ \hline
10 & 4.860 & 4.848 & 4.786 \\
1 & 5.750 & 5.739 & 5.694 \\
0.1 & 12.161 & 12.138 & 12.016 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of average Nusselt number on hot wall for the mixed convection in lid-driven cavity.  
Figure 6: Illustration of the setup for natural convection in a concentric annulus.  
effect on the flow field, due to which the gas is pushed upward and compressed to the lower part of the inner cylinder and the upper part of the outer cylinder.  
To quantify the correctness of USTLBM in this problem, the following equivalent heat conductivities are defined  
\[\overline{k_{eq}}=\frac{\ln(Ar)}{2\pi(Ar-1)}\int_{0}^{2\pi}\frac{\partial T}{ \partial r}\mathrm{d}\theta,\ \ \text{for the inner surface} \tag{75}\]  
\[\overline{k_{eq}}=\frac{Ar\cdot\ln(Ar)}{2\pi(Ar-1)}\int_{0}^{2\pi}\frac{ \partial T}{\partial r}\mathrm{d}\theta,\ \ \text{for the outer surface} \tag{76}\]  
Calculated results in our simulations are tabulated in Table 6 and compared with reference data in the literature [28; 29]. It is noted that our computational results are in good agreement with the reference data. The relative errors are within 3%, which demonstrates the robustness of USTLBM in the present example on body-fitted mesh and with curved boundaries.  
Figure 7: Isotherms for natural convection in a concentric annulus.