# Evolution of Flow and Streaming in Exponential Variable Cross-Section Resonators
## Two-Dimensional GKS-BGK Model  
### Methodology of GKS-BGK  
The presented method involves a two-dimension GKS-BGK model. This section gives the main idea of this scheme. Interested readers can refer to References [23; 24] for more details about this scheme.  
The fully compressible NS equations can be recovered from the Boltzmann equation [23]. From this point, the present GKS is based on the analytical solution of the following two-dimension BGK-Boltzmann equation:  
\[f_{t}+\xi_{1}f_{x}+\xi_{2}f_{y}\ =\ (g-f)/\tau \tag{1}\]  
where \(f\) is the gas distribution function, and \(g\) is the equilibrium state function approached by \(f\). Both are functions of time (\(t\)), space (\(x,y\)), particle velocity (\(\xi_{1},\xi_{2}\)), and internal variables (\(\zeta\in R^{K}\)); \(\tau\) is the particle collision time, and the function \(g\) is supposed to be a Maxwellian distribution.  
\[g\ =\ \rho(\lambda/\pi)^{(K+2)/2}\exp\bigl{[}-\lambda\bigl{[}(\xi_{1}-u)^{2}+( \xi_{2}-v)^{2}+\zeta^{2}\bigr{]}\bigr{]} \tag{2}\]  
Here, \(\lambda\ =\ 1/2 RT\) and \(\zeta^{2}\ =\ \zeta_{1}^{2}+\zeta_{2}^{2}+\cdots+\zeta_{K}^{2}\) relate with the internal energy of the particles. For the two-dimension ideal gas, the total number of degrees of freedom is \(K=(4-2\gamma)/(\gamma-1)\), where \(\gamma\) is the ratio of specific heats; \(u\) and \(v\) are the flow velocities in directions \(x\) and \(y\), respectively. During the process of particle collisions, the conservation laws are satisfied. Then, one can impose the conservation constraint on functions \(f\) and \(g\):  
\[\int(g-f)\uppsi d\Xi\ =\ 0 \tag{3}\]  
where \(d\Xi\ =\ d\xi_{1}d\xi_{2}d\zeta\), \(d\zeta\ =\ d\zeta_{1}d\zeta_{2}\cdots d\zeta_{K}\) and  
\[\uppsi\ =\ (\psi_{1},\psi_{2},\psi_{3},\psi_{4})^{\rm T}\ =\ \bigl{(}1,\xi_{1},\xi_{2}, \bigl{(}{\xi_{1}}^{2}+{\xi_{2}}^{2}+\zeta^{2}\bigr{)}/2\bigr{)}^{\rm T} \tag{4}\]  
After taking the moment, \(\uppsi\), in Equation (1) and integrating it with respect to \(d\Xi\) in the velocity space, \(dS\) on a cell \(S_{i,j}\), and \(dt\) in a time step \([t^{n},\ t^{n+1}]\), we obtain the following:  
\[\mathbf{W}_{i,j}^{n+1}\ =\ \mathbf{W}_{i,j}^{n}+\frac{1}{S_{i,j}}\oint_{ \partial\Omega_{i,j}}\mathbf{F}\cdot\mathbf{n}\ \ dtdl \tag{5}\]  
where \(W_{i,j}^{n}\) are the average macroscopic conservative quantities over \(S_{i,j}\) with boundary \(\partial\Omega_{i,j}\) at time \(t^{n}\). \(\mathbf{F}\) is the flux across the cell interface, which is related with the function \(f\) as follows:  
\[\mathbf{W}\ =\ (\rho,\rho u,\rho v,\rho\varepsilon)^{\rm T}\ =\ \int f\uppsi\ d\Xi,\quad\mathbf{F}\ =\ \int(\xi_{1},\ \ \xi_{2})f\uppsi\ d\Xi \tag{6}\]  
where \(\rho\varepsilon\ =\ p/\left(\gamma-1\right)+\rho\Bigl{(}u^{2}+v^{2}\Bigr{)}/2\) is the total energy. Therefore, Equations (5) and (6) mean that it is critical for the GKS (Equation (5)) to obtain the function (\(f\)) at the cell interface.  
The BGK Equation (1) has the analytical solution:  
\[\begin{array}{ll}f(x,y,t,\xi_{1},\xi_{2},\zeta)=&\frac{1}{\tau}\int_{0}^{t}g \Big{(}\widetilde{x},\overline{y},\overline{t},\xi_{1},\xi_{2},\zeta\Big{)}e^{ -(t-\overline{t})/\tau}d\overline{t}\\ &+e^{-t/\tau}f_{0}(x-\xi_{1}t,y-\xi_{2}t,\xi_{1},\xi_{2},\zeta)\end{array} \tag{7}\]  
Here, \(\overline{x}~{}=~{}x-\xi_{1}\big{(}t-\overline{t}\big{)}\), \(\overline{y}~{}=~{}y-\xi_{2}\big{(}t-\overline{t}\big{)}\), and \(f_{0}\) is the gas distribution function at the beginning of each time step (\(t=0\)). The key point of the GKS-BGK is to construct the functions \(f_{0}\) and \(g\) around the cell interface by Chapman-Enskog expansion. More details about the construction process for functions \(f_{0}\) and \(g\) can be found in References [23; 24].  
It must be stressed that the inclusion of the discontinuity at a cell interface in functions \(f_{0}\) and \(g\) is a quality of the GKS, which enhances its performance in capturing flow discontinuities, such as shocks.