# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 3 Numerical Results and Discussions
### Nusselt Numbers  
In heat transfer problems, the Nusselt number is an important dimensionless parameter to measure the intensity of convection, which is defined as the ratio of the total heat transfer rate to that due to conduction alone. According to Equation (29), the local Nusselt number \(Nu_{local}\) at the hot (outer) wall is given by  
\[Nu_{local}=\frac{-k\Big{(}\frac{\partial T}{\partial r}\Big{)}_{r=r_{o}}}{-k \Big{(}\frac{\partial T_{cond}}{\partial r}\Big{)}_{r=r_{o}}}=\frac{\ln(r_{o} /r_{i})r_{o}}{T_{h}-T_{c}}\Big{(}\frac{\partial T}{\partial r}\Big{)}_{r=r_{o }}, \tag{30}\]  
where the temperature gradient is calculated using a simple finite difference approximation. The global Nusselt number \(Nu\) can be obtained by integrating \(Nu_{local}\) over the cylinder surface, which gives  
\[Nu=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\ln(2)r_{o}}{T_{h}-T_{c}}\Big{(}\frac{ \partial T}{\partial r}\Big{)}_{r=r_{o}}d\theta. \tag{31}\]  
Figure 3 shows the computed variations of \(Nu\) with time \(t\) at different Rayleigh numbers. Also presented are the results from the N-S solver. It is observed that for \(Ra\geq 10^{4}\) a convective flow has been developed from the initial pure conduction and pure rotating flow, since all the predicted Nusselt numbers are reaching final values larger than 1 after a short transition process. The convective flow can be seen more clearly from the flow patterns shown in the next section. In fact, the critical value  
Figure 2: Radial distributions of temperature \(T\) for the pure conduction case. of \(Ra\) for this rotating annulus case is between \(10^{3.2}\) and \(10^{3.3}\), which is close to the value of 1708 for the Rayleigh-Benard convection between horizontal plates. Figure 3: Computed variations of \(Nu\) with time \(t\) for different \(Ra\). It is also observed from Figure 3 that both numerical methods predict a time-independent \(Nu\) for \(Ra\leq 10^{6}\) and a time-dependent one for \(Ra\geq 10^{6.1}\). This transition range is very close to the result of [PERSON] et al. [4]. For each case with \(Ra\leq 10^{6}\), the converged Nusselt numbers from the two methods agree well with each other, and for each case with \(Ra\geq 10^{6.1}\), the computed time-averaged values show also good agreement. Particularly, approximately periodic oscillations occur for \(Ra=10^{6.1}\) and \(Ra=10^{7}\), and the oscillations change to irregular states for \(Ra\geq 10^{8}\). Although the computations started from the same initial condition, the predicted convective flow from the BGK scheme seems to develop earlier. Figure 4 shows the computed variation of (time-averaged) \(Nu\) with \(Ra\), together with those from literatures. Note that the computed results from the BGK scheme and the N-S solver are very close to each other. For clarity, we only show the results from the BGK scheme, and their consistency can be clearly seen in Figure 3 (time-independent values for \(Ra\leq 10^{6}\) and time-averaged values for \(Ra>10^{6.1}\)). This indicates that, for the current rotating annulus case, the accuracy of the BGK scheme is comparable to that of the N-S solver; both are based on the full compressible N-S equations. However, when comparing the present results (from both methods) with that from [PERSON] et al. [6] by using an incompressible solver with the Boussinesq approximation, obvious discrepancies are found for \(Ra>10^{6.5}\). Currently, these discrepancies are not fully understood. As far as the authors are concerned, besides the numerical reasons, the density effect can be a possible physical reason. As we all know, the density is treated as a constant in the Boussinesq approximation-based solver, while in the current flow solvers, the density is calculated from the ideal gas law. For small \(Ra\), the velocity and kinetic energy are small, thus very little difference is found. However, with the increase of \(Ra\), the compressible effect may become more and more important. The semi-empirical correlation from [PERSON] et al.