# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 2 Numerical Method  
### BGK Model for Natural Convection in a Rotating Annulus  
For the sake of simplicity, let us consider a constant Cartesian rotating frame of reference where the rotation axis coincides with the \(z-\) coordinate axis. For the 2D convective flow in radial-tangential plane, the following compressible full N-S equations are derived from the conservation laws:  
\[ \begin{array}{l}\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{v}_{r})= 0\\ \frac{\partial\rho v_{r}}{\partial t}+\
abla\cdot(\rho\mathbf{v}_{r}\mathbf{v}_{r}+p \mathbf{I})=\
abla\cdot\overline{\mathbf{\overline{r}}}-\rho\mathbf{\omega}\times(\mathbf{ \omega}\times\mathbf{r})+2\rho\mathbf{\omega}\times\mathbf{v}_{r}\\ \frac{\partial\rho\mathbf{r}}{\partial t}+\
abla\cdot((\rho E_{r}+p)\mathbf{v}_{r})= \
abla\cdot(k\
abla T+\overline{\mathbf{\overline{r}}}\cdot\mathbf{v}_{r})\end{array}  \tag{1}\]  
where \(\rho\), \(p\), \(E_{r}\), \(T\), \(k\) are the density, pressure, relative total energy, temperature and thermal conductivity coefficient, \(\mathbf{v}_{r}\) is the vector of relative velocities, \(\mathbf{\omega}\) is the vector of angular velocities, \(\mathbf{r}\) is the position vector, and \(\overline{\mathbf{\overline{r}}}\) is the viscous stress tensor. The last two terms on the right side of momentum equations represent the centrifugal and Coriolis forces, respectively. Note that the Boussinesq approximation is not used here and these two source terms are related to buoyancy effects. Although many numerical methods are based on solving Equation (1), it is not convenient for us to construct the BGK model because the relative quantities are used as conservative variables and an extra \(E_{r}\) needs to be introduced. For this reason, a simple relation between absolute and relative velocities \(\mathbf{v}=\mathbf{v}_{r}+\mathbf{\omega}\times\mathbf{r}\) is substituted into Equation (1) and then we get the N-S equations for the absolute velocity formulation:  
\[ \begin{array}{l}\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{v}_{r} )=0\\ \frac{\partial\rho v}{\partial t}+\
abla\cdot(\rho\mathbf{v}_{r}+p\mathbf{I})=\
abla \cdot\overline{\mathbf{\overline{r}}}-\rho\mathbf{\omega}\times\mathbf{v}\\ \frac{\partial\rho v}{\partial t}+\
abla\cdot(\rho\mathbf{E}\mathbf{v}_{r}+p\mathbf{v})= \
abla\cdot(k\
abla T+\overline{\mathbf{\overline{r}}}\cdot\mathbf{v})\end{array} , \tag{2}\]where \(E\) is the total energy. In this formulation, the centrifugal and Coriolis forces are simplified to a single source term \((-\rho\mathbf{\omega}\times\mathbf{v})\). Although the above two equations are mathematically identical, the present BGK scheme is developed for Equation (2). As demonstrated in [16; 17], it is necessary to include the source effects in the BGK model. In our previous work [20] for simulating 2D flows around arbitrarily moving bodies [20], a moving reference frame-based BGK scheme was used and the source effects were accounted for by including a particle acceleration term. This extra term was mainly reflected in three aspects: particle trajectory, construction of the initial state \(f_{0}\), and construction of the equilibrium state \(g\). Therefore, the final flux function could be quite complex. Also, because the calculations for the particle acceleration-related terms are indirect, it is difficult to obtain the explicit expression of \(f\). In this work, we adopt an alternative way by introducing a simple source term \(\omega s\) (\(\omega\) is the magnitude of the vector \(\mathbf{\omega}\)), which is computationally more efficient. From a physical point of view, the use of \(\omega s\) indicates that an external force is directly applied to \(f\) at the cell interface. In other words, at each time step, starting from an initial state \(f_{0}\), the distribution function \(f\) first approaches to a local equilibrium state \(g\) under the combined effects of particle transport and collision. Then the extra term \(\omega s\) drives \(f\) to be close to a \"new\" local equilibrium state.