# A Gas-Kinetic BGK Scheme for Natural Convection in a Rotating Annulus
## 2 Numerical Method
### BGK Model for Natural Convection in a Rotating Annulus
Through the updated \(f\), the source effects can be naturally included in the flux evaluation. The determination of \(s\) is described below by using the method of undetermined coefficients. To start with, the modified Boltzmann equation is first written as  
\[\frac{\partial f}{\partial t}+\xi_{x}{}^{\prime}\frac{\partial f}{\partial x }+\xi_{y}{}^{\prime}\frac{\partial f}{\partial y}=-\frac{1}{\tau}(f-g)+\omega s, \tag{3}\]  
where \(f\) is the gas distribution function, \(g\) is the equilibrium distribution function, and \(\tau\) is the particle collision time. The parameters \(\xi_{x}{}^{\prime}\) and \(\xi_{y}{}^{\prime}\) are defined as \(\xi_{x}{}^{\prime}=\xi_{x}-u_{e}\) and \(\xi_{y}{}^{\prime}=\xi_{y}-v_{e}\), where \(\xi_{x}\) and \(\xi_{y}\) are (absolute) particle velocities, \(u_{e}\) and \(v_{e}\) are components of the vector of entrainment velocities \(\mathbf{v}_{e}=\mathbf{v}-\mathbf{v}_{r}=\mathbf{\omega}\times\mathbf{r}\). All \(f\), \(g\) and \(s\) are functions of space \((x,y)\), time \(t\), particle velocities \((\xi_{x},\xi_{y})\) and internal variables \(\zeta_{i}(i=1,2,\cdots,K)\). The number of dimensions \(K\) is equal to 3 for diatomic gas in the 2D case. The equilibrium state \(g\) is usually defined as a Maxwellian:  
\[g=\rho\bigg{(}\frac{\lambda}{\pi}\bigg{)}^{\frac{K+2}{2}}e^{-\lambda((\xi_{x }-u)^{2}+(\xi_{y}-v)^{2}+\zeta^{2})}, \tag{4}\]  
where \(\lambda\) is related to \(p\) and \(\rho\) through the relation \(\lambda=\rho/2p\). The notation \(\zeta^{2}=\zeta_{i}\zeta_{i}\) is used. From the Chapman-Enskog expansion, the source term \(s\) should be designed to satisfy the following equation in order to ensure that the BGK model (Equation (3)) correctly recovers the governing macroscopic equations (Equation (2)):  
\[\int\omega s\mathbf{\varphi}d\Xi=[0,-\rho(\mathbf{\omega}\times\mathbf{v}),0]^{T}, \tag{5}\]  
where the notation \(d\Xi=d\xi_{x}d\xi_{y}d\zeta_{1}d\zeta_{2}\cdots d\zeta_{k}\) has been used, and \(\mathbf{\varphi}\) is the vector of the collision invariants  
\[\mathbf{\varphi}=\bigg{[}1,\xi_{x},\xi_{y},\frac{1}{2}(\xi_{x}^{2}+\xi_{y}^{2}+ \zeta^{2})\bigg{]}^{T}. \tag{6}\]  
Let us rewrite Equation (5) in components:  
\[\int\omega s\left[ \begin{array}{c}1\\ \xi_{x}\\ \xi_{y}\\ \frac{1}{2}(\xi_{x}^{2}+\xi_{y}^{2}+\zeta^{2})\end{array} \right]d\Xi=\left[ \begin{array}{c}0\\ \rho\omega v\\ -\rho\omega u\\ 0\end{array} \right]. \tag{7}\]On the right side of the above equation, \(\omega\) is a constant. The momentum densities \(\rho u\) and \(\rho v\) can be related to \(f\) or \(g\) from the following relation:  
\[W=\left[ \begin{array}{c}\rho\\ \rho u\\ \rho v\\ \rho E\end{array} \right]=\int f\left[ \begin{array}{c}1\\ \xi_{x}\\ \xi_{y}\\ \frac{1}{2}(\xi_{x}^{2}+\xi_{y}^{2}+\xi^{2})\end{array} \right]d\Xi=\int g \left[ \begin{array}{c}1\\ \xi_{x}\\ \xi_{y}\\ \frac{1}{2}(\xi_{x}^{2}+\xi_{y}^{2}+\xi^{2})\end{array} \right]d\Xi. \tag{8}\]  
The comparison between Equations (7) and (8) indicates that \(s\) probably has a form similar to \(f\) or \(g\). Obviously, constructing the extra source term \(s\) related to the gas distribution function \(f\) is closer to the physical reality. However, from computational (mathematical) perspective, difficulties would be encountered in numerically solving the BGK model, since it will be difficult to explicitly construct \(s\). Instead, the assumption that \(s\) could be expressed as Maxwellian-type functions like \(g\) is used in the present work, although this is not the only choice. This treatment was also motivated from [PERSON]'s work for the Euler equations with heat transfer [21], where the extra source term was constructed as the difference between two Maxwellian-type functions. Physically, it should be noted that this assumption could be appropriate for the concerned smooth flow problem; however, in cases where non-equilibrium effects dominate, it may have a significant effect.