# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 2 Numerical Method
### Gas-Kinetic BGK Scheme for the Time-Dependent Solution \(f\)  
Within the finite volume framework, the BGK scheme aims at evaluation of the time-dependent gas distribution function \(f\) at each cell interface, from which the numerical fluxes can be computed. For a general structured mesh, we take the \(i-\) direction face \(x_{i+1/2,j}=(x_{i+1/2,j},y_{i+1/2,j})\) between cells \((i,j)\) and \((i+1,j)\) as an example, and for convenience, the \(x-\) and \(y-\) directions are assumed to be the normal and tangential directions to the local cell interface, respectively. The generalized solution \(f\) of the BGK model with the presence of a source term (Equation (3)) at the cell interface \(x_{i+1/2,j}\) and at time \(t\) can be written as  
\[ \begin{array}{ll}f(x_{i+1/2,j},t,\xi,\xi^{2})&=\tfrac{1}{2}\int_{0}^{t}e^{ -(t-t^{\prime})/\tau}\big{(}g(x(t^{\prime}),t^{\prime},\xi,\xi^{2})+\tau \omega s(x(t^{\prime}),t^{\prime},\xi,\xi^{2})\big{)}dt^{\prime}\enspace,\\ &+e^{-t/\tau}f_{0}(\mathbf{x}(0),\xi,\xi^{2})\end{array} \enspace, \tag{12}\]  
where the particle trajectory is described by \(x(t^{\prime})=x_{i+1/2,j}-\xi^{\prime}(t-t^{\prime})\), and the vectors of absolute and relative particle velocities \(\xi\) and \(\xi^{\prime}\) are defined as  
\[\xi=(\xi_{x},\xi_{y})^{T},\quad\quad\xi^{\prime}=(\xi^{\prime}_{x^{\prime}}, \xi^{\prime}_{y})^{T}\enspace=(\xi_{x}-u_{e},\xi_{y}-v_{e})^{T}. \tag{13}\]The solution \(f\) describes the gas evolution process, starting from an initial gas distribution function \(f_{0}\) and approaching the equilibrium state \(g\). The source term \(\tau\omega s\) seems to have an effect of driving \(f\) away from the equilibrium state, which is similar to the preparation of the macroscopic initial data in the reconstruction stage. Below we will show how to evaluate the three unknowns \(f_{0}\), \(g\) and \(s\). For simplicity, the location of the cell interface \(\mathbf{x}_{i+1/2,j}=(0,0)\) is used. In the study of natural convection, the flow is usually supposed to be smooth, i.e., no discontinuities exist. Therefore, based on the Chapman-Enskog expansion and to the second order accuracy [12], the initial state \(f_{0}\) can be constructed as  
\[ \begin{array}{ll}f_{0}(\mathbf{x}(0),\xi,\zeta^{2})&=g_{0}+\frac{\partial g_{0}}{ \partial x}x+\frac{\partial g_{0}}{\partial y}y-\tau\Big{(}\frac{\partial g_{0 }}{\partial t}+\xi^{\prime}_{\ x}\frac{\partial g_{0}}{\partial x}+\xi^{ \prime}_{\ y}\frac{\partial g_{0}}{\partial y}-\omega s_{0}\Big{)}\\ &=g_{0}\Big{[}1-a\xi^{\prime}_{\ x}t-b\xi^{\prime}_{\ y}t-\tau(A+a\xi^{ \prime}_{\ x}+b\xi^{\prime}_{\ y})\Big{]}+\tau\omega s_{0}\end{array} \,, \tag{14}\]  
where \(a\) and \(b\) are related to derivatives of \(g_{0}\) in space and \(A\) is related to the derivative of \(g_{0}\) in time. Both the equilibrium state \(g_{0}\) and the source term \(s_{0}\) are defined at the cell interface and at time \(t=0\), which can be calculated from Equations (4) and (9) and the reconstructed macroscopic variables. For example, a second-order interpolation for conservative variables \(\mathbf{W}=\big{[}\rho,\rho u,\rho v,\rho E\big{]}^{T}\) is used here:  
\[W_{i+1/2,j}=\frac{1}{2}(\mathbf{W}_{ij}+\mathbf{W}_{i+1,j}), \tag{15}\]  
from which all the necessary macroscopic quantities at the cell interface are obtained. Based on the Taylor expansion of the Maxwellian distribution function, the slopes \(a\), \(b\) and \(A\) can be expanded as a linear combination of the collision invariants \(\varphi_{\alpha}\) (\(\alpha=1\sim 4\)):  
\[ \begin{array}{ll}a=a_{\alpha}\varphi_{\alpha}=a_{1}+a_{2}\xi_{x}+a_{3}\xi_{ y}+\frac{1}{2}a_{4}(\xi_{x}^{2}+\xi_{y}^{2}+\zeta^{2})\\ b=b_{\alpha}\varphi_{\alpha}=b_{1}+b_{2}\xi_{x}+b_{3}\xi_{y}+\frac{1}{2}b_{4}( \xi_{x}^{2}+\xi_{y}^{2}+\zeta^{2})\\ A=A_{\alpha}\varphi_{\alpha}=A_{1}+A_{2}\xi_{x}+A_{3}\xi_{y}+\frac{1}{2}A_{4}( \xi_{x}^{2}+\xi_{y}^{2}+\zeta^{2})\end{array} \,, \tag{16}\]  
where all the unknowns \(a_{\alpha}\), \(b_{\alpha}\) and \(A_{\alpha}\) are local constants.