## 2 Numerical Methods  
### Governing Equations  
Since the flow at the muzzle is hypersonic, compressible N-S equations are selected as the governing equations. Based on the continuum hypothesis, by neglecting body force and heat sources, the dimensionless two-dimensional axisymmetric unsteady compressible N-S equations in Cartesian coordinates can be expressed as:  
\[\frac{\partial\mathbf{U}}{\partial t}+\frac{\partial\mathbf{F}}{\partial x}+\frac{ \partial\mathbf{G}}{\partial y}+\mathbf{H}=\frac{1}{\text{Re}}\Bigg{(}\frac{\partial \mathbf{F}_{v}}{\partial x}+\frac{1}{y^{\phi}}\frac{\partial y^{\phi}\mathbf{G}_{v}}{ \partial y}+\mathbf{H}_{v}\Bigg{)} \tag{1}\]where  
\[\mathbf{U}=\left[\begin{array}{c}\rho\\ \rho u\\ \rho v\\ e\end{array}\right]_{\mathbf{F}}=\left[\begin{array}{c}\rho u\\ \rho u^{2}+p\\ \rho uv\\ (e+p)u\end{array}\right]_{\mathbf{G}}=\left[\begin{array}{c}\rho v\\ \rho uv+p\\ \rho v^{2}\\ (e+p)v\end{array}\right]_{\mathbf{H}}=\frac{\delta}{y}\left[\begin{array}{c} \rho v\\ \rho uv\\ \rho v^{2}\\ (e+p)v\end{array}\right],\] \[\mathbf{F}_{v}=\left[\begin{array}{c}0\\ \tau_{xx}\\ \tau_{xy}\\ \tau_{xx}u+\tau_{xy}v-q_{x}\end{array}\right]_{\mathbf{G}_{v}}=\left[\begin{array} []{c}0\\ \tau_{yx}\\ \tau_{yy}\\ \tau_{yx}u+\tau_{yy}v-q_{y}\end{array}\right]_{\mathbf{H}_{v}}=\left[\begin{array} []{c}0\\ 0\\ \tau_{\theta\theta}\\ 0\end{array}\right]_{\mathbf{G}}\]  
Here, \(\rho\), \(p\), \(u\) and \(v\) are the density, pressure, x-velocity, and y-velocity, respectively. The coefficient \(\delta\) determines the flow type. For two-dimensional axisymmetric flow, \(\delta=1\). Re is Reynolds number which is given by, \(\text{Re}=\frac{\rho_{0}u_{0}D}{\mu_{0}}\), where \(\rho_{0}\), \(u_{0}\) and \(\mu_{0}\) are the flow properties at the jet entrance center, and \(D\) denotes muzzle diameter. \(e\) is the internal energy per unit mass, \(e=\frac{p}{\gamma-1}+\frac{1}{2}\rho\Big{(}u^{2}+v^{2}\Big{)}\), where \(\gamma\) is specific heat ratio of gas \(q_{x}\) and \(q_{y}\) are the heat flux in the x and y directions, respectively, and are given by \(q_{x}=-k\frac{\partial T}{\partial x}\), \(q_{y}=-k\frac{\partial T}{\partial y}\), where \(k\) is the thermal conductivity. Finally, \(\tau_{xx}\Big{(}\tau_{xy},\tau_{yx},\tau_{yy},\tau_{\theta\theta}\Big{)}\) are the viscous stresses, and are given by:  
\[\left\{\begin{array}{c}\tau_{xx}=-\frac{2\mu}{3}\Big{(}\frac{ \partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{v}{y}\Big{)}+2\mu \frac{\partial v}{\partial x}\\ \tau_{yy}=-\frac{2\mu}{3}\Big{(}\frac{\partial u}{\partial x}+\frac{\partial v }{\partial x}+\frac{v}{y}\Big{)}+2\mu\frac{\partial v}{\partial y}\\ \tau_{xy}=\tau_{yx}=\mu\Big{(}\frac{\partial u}{\partial y}+\frac{\partial v }{\partial x}\Big{)}\\ \tau_{\theta\theta}=2\mu\frac{v}{y}-\frac{2\mu}{3}\Big{(}\frac{\partial u}{ \partial x}+\frac{\partial v}{\partial y}\Big{)}\end{array}\right.\]  
The viscosity coefficient \(\mu\) is given by:  
\[\mu=\mu_{L}+\mu_{T} \tag{2}\]  
where \(\mu_{L}\) is the molecular viscosity coefficient which can be obtained by [PERSON]'s law. [PERSON]'s law is given by \(\mu_{L}=T^{\frac{3}{2}}\frac{1+T_{s}/T_{\infty}}{T+T_{s}/T_{\infty}}\), where \(T_{s}\) is the Sutherland constant, which is \(T_{s}=124\)K when the medium is air. The turbulent viscosity, \(\mu_{T}\), is given by the turbulence model. The value of \(\mu_{T}\) is obtained by solving the transport equations of turbulence kinetic energy (\(k\)) and turbulence dissipation rate (\(\varepsilon\)). In this paper, the muzzle jet flow problem is solved using the realizable \(k-\varepsilon\) two-equation model, which is suitable for non-homogeneous turbulence.  
In addition to the N-S equations, a gas state equation is needed. For a perfect gas, the dimensionless gas state equation is:  
\[P=\frac{1}{\gamma Ma_{\infty}^{2}}\rho T \tag{3}\]  
where \(Ma\) is Mach number of flow and \(T\) is the temperature.