# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 3 Numerical Results and Discussions  
### Problem Description  
The computational model presented by [PERSON] et al. [6] is a closed rotating annulus with the ratio of inner radius \(r_{i}\) to outer radius \(r_{o}\) being 0.5 and the Prandtl number \(Pr\) being 0.7 for air. It is again used here for validation of the developed BGK scheme. Apart from the dimensionless parameters provided therein, in order to keep the numerical results comparable and repeatable, some constant quantities with dimensions used in our computations are given in Table 1.  
There are two important dimensionless parameters for convective flows, the Rayleigh number \(Ra\) and the rotational Reynolds number \(Re\), which are defined as  
\[Ra=\frac{\omega^{2}r_{m}(r_{o}-r_{i})^{3}\beta\Delta T}{\
u\alpha},\,Re=\frac{ \omega r_{m}d}{\
u}. \tag{27}\]  
Supposing that \(Ra\) and \(Re\) are known, from the definitions in Table 1, three unknowns exist in the above two relations, including the rotational speed \(\omega\), the temperature difference \(\Delta T=T_{h}-T_{c}\) and the axial gap between disks \(d\). With an extra assumption for the buoyancy parameter \(\beta\Delta T=1.1\,Ra^{-0.114}\), the computational state can be uniquely determined from given \(Ra\) and \(Re\). Although the assumption was originally introduced for flows obeying the Boussinesq approximation (usually \(\beta\Delta T<0.3\) is required), it is still used here for consistency. However, it is noticed that the present method is also applicable to flows with a wider range of \(\beta\Delta T\) since the ideal gas law is adopted. In the current 2D computation, \(d\) is only used to define \(Re\) and is assumed to be a constant, therefore from the above relations, different sets of \(Ra\) and \(Re\) should be specified to satisfy the relationship \(Ra^{1.114}\propto Re^{2}\). For example, the natural convection in the rotating annulus at Rayleigh number from \(10^{3}\) to \(10^{9}\) are herein simulated, corresponding to Reynolds number ranging from 67 to 148,152. \begin{table}
\begin{tabular}{c c} \hline \hline
**Quantity** & **Constant Value** \\ \hline Inner radius \(r_{i}\) & 100 mm \\ Outer radius \(r_{o}\) & 200 mm \\ Reference length \(r_{m}=0.5(r_{i}+r_{0})\) & 150 mm \\ Reference temperature \(T_{ref}=0.5(T_{h}+T_{c})\) & 288.15 K \\ Volume expansion coefficient \(\beta=1/T_{ref}\) & 3.47041 \(\times 10^{-3}\,\text{K}^{-1}\) \\ Kinematic viscosity \(\
u\) & 1.46072 \(\times 10^{-4}\,\text{m}^{2}/\text{s}\) \\ Thermal diffusivity \(\alpha=\
u/Pr\) & 2.08674 \(\times 10^{-4}\,\text{m}^{2}/\text{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Several constant quantities with dimensions used in the numerical computations. Before showing the convection results, we would like to first present two special axisymmetric cases: one is the rotating annulus without temperature difference where the induced flow is also referred to as the rotating Couette flow, the other one is the stationary annulus with only conduction occurring. Since \"exact\" solutions for these two cases are available, both of them are simulated to quantitatively verify the basic capability of the method. For the former case, the linearized theory gives \"exact\" radial distributions of circumferential velocity \(v_{c}\) and pressure gradient \(\partial p/\partial r\) as  
\[v_{c}(r)=\omega r,\partial p(r)/\partial r=\rho\omega^{2}r, \tag{28}\]  
where the radial location \(r\) falls between \(r_{i}\) and \(r_{o}\). Although the solutions are actually independent of Reynolds number, in the computation \(Re\) is taken as 67 leading to \(\omega=0.06372\) rad/s. A uniform grid with a size of \(101\times 51\) in circumferential and radial directions is used, which is fine enough through the mesh independence study. As shown in Figure 1, good agreements are found between the \"exact\" and numerical results. The difference between the computed maximum and minimum densities in the whole flow field is on the order of \(10^{-8}\) and stays nearly constant, indicating a very small compressible error.