# Influence of Incident Shock on Fuel Mixing in Scramjet
## 2 Numerical Method
### Reynolds-Averaged Navier-Stokes (RANS)  
The RANS simulation has been widely used in engineering and academic research due to its excellent characteristics such as strong robustness and low computational cost [27]. In the simulation of jet mixing in a scramjet, it has been proven to be a reliable calculation method due to its accuracy [28; 29]. After performing Favre averaging on the instantaneous system of Equations (1), (2), (5), and (7), the RANS equation can be obtained as follows [30]:  
\[\frac{\partial\overline{p}}{\partial t}+\frac{\partial\overline{\rho}\overline {u}_{j}}{\partial x_{j}}=0 \tag{10}\]  
\[\frac{\partial\overline{\rho}\overline{u}_{i}}{\partial t}+\frac{\partial \overline{\rho}\overline{u}_{i}\overline{u}_{j}}{\partial x_{j}}=-\frac{ \partial\overline{p}}{\partial x_{i}}+\frac{\partial\overline{\tau}_{ij}}{ \partial x_{j}}+\frac{\partial}{\partial x_{j}}\big{(}\overline{\rho} \overline{u}_{i}\overline{u}_{j}-\overline{\rho}\overline{u}_{i}\overline{u}_ {j}\big{)} \tag{11}\]\[ \begin{array}{c}\frac{\partial\overline{\rho}\widetilde{E}}{\partial t}+\frac{ \partial\left(\overline{\rho}\widetilde{E}\widetilde{u}_{j}+\overline{\rho} \widetilde{u}_{j}\right)}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}( \overline{\eta}_{j}+\widetilde{u}_{i}\overline{\tau}_{ij})\\ -\frac{\partial}{\partial x}\left(\overline{\rho}\widetilde{E}\widetilde{u}_{j }-\overline{\rho}\widetilde{E}\widetilde{u}_{j}+\overline{\rho}\widetilde{u}_ {j}-\overline{\rho}\widetilde{u}_{j}\right)\\ +\frac{\partial^{2}}{\partial x_{j}}(\overline{u}_{i}\overline{\tau}_{ij}- \widetilde{u}_{i}\overline{\tau}_{ij})\end{array}  \tag{12}\]  
\[ \begin{array}{c}\frac{\partial\overline{\rho}\widetilde{Y}_{n}}{\partial t}+ \frac{\partial\overline{\rho}\widetilde{u}_{j}}{\partial x_{j}}=\widetilde{ \omega}_{n}-\frac{\partial}{\partial x_{j}}\left(\overline{\rho}V_{i,n}Y_{n}+ \overline{\rho}u_{j}^{\prime}Y_{n}^{\prime}\right)\end{array}  \tag{13}\]  
The viscous stress tensor after Reynolds averaging is expressed as follows:  
\[\overline{\tau}_{ij}=2\widetilde{\mu}\left(\widetilde{S}_{ij}-\frac{1}{3}\frac {\partial\widetilde{u}_{k}}{\partial x_{k}}\delta_{ij}\right) \tag{14}\]  
The strain force tensor after Reynolds averaging is expressed as follows:  
\[\widetilde{S}_{ij}=\frac{1}{2}\left(\frac{\partial\widetilde{u}_{i}}{\partial x _{j}}+\frac{\partial\widetilde{u}_{j}}{\partial x_{i}}\right) \tag{15}\]  
\[ \begin{array}{c}\frac{\partial\overline{\rho}\widetilde{E}}{\partial t}+ \frac{\partial\left(\overline{\rho}\widetilde{E}\widetilde{u}_{j}+\overline{ \rho}\widetilde{u}_{j}\right)}{\partial x_{j}}=\frac{\partial}{\partial x_{j}} (\overline{q}_{j}+\widetilde{u}_{i}\overline{\tau}_{ij})\\ -\frac{\partial}{\partial x_{j}}\left(\overline{\rho}\widetilde{E}\widetilde{u }_{j}-\overline{\rho}\widetilde{E}\widetilde{u}_{j}+\overline{\rho}\overline{u }_{j}-\overline{\rho}\widetilde{u}_{j}\right)\\ +\frac{\partial}{\partial x_{j}}\left(\overline{u}_{i}\overline{\tau}_{ij}- \widetilde{u}_{i}\overline{\tau}_{ij}\right)\end{array}  \tag{16}\]  
In Equation (15), \(\widetilde{E}\) is the mass-averaged specific total energy derived as follows:  
\[\widetilde{E}=\widetilde{e}+\frac{1}{2}\widetilde{u}_{i}\widetilde{u}_{i}+k \tag{17}\]  
where \(\widetilde{e}\) is the specific energy of the mixture expressed as follows:  
\[\widetilde{e}=\sum_{n=1}^{N}\widetilde{Y}_{n}\widetilde{h}_{n}-\frac{\overline {\rho}}{\overline{\rho}} \tag{18}\]  
The heat flux vector is expressed as follows:  
\[\overline{q}_{i}=-\overline{\lambda}\frac{\partial\widetilde{T}}{\partial x_{ i}}+\overline{\rho}\sum_{n=1}^{N}\widetilde{Y}_{n}\widetilde{h}_{n}\widetilde{V}_{i,n} \tag{19}\]  
where \(\overline{\lambda}\) is the thermal conductivity of multi-component mixed gas.