## 2 CFD Simulations
### Numerical Methods
#### 2.2.2 Flux Type  
At the hypersonic speed, we adopt the density-based solver employing time-derivative preconditioning [29]. The system governing equations in vector form is as follows:  
\[\Gamma\frac{\partial}{\partial t}\int_{V}QdV+\oint[F-G]\times dA=\int_{V}HdV \tag{14}\]where \(V\) and \(A\) represent an arbitrary control volume and surface area, the source terms like energy sources and body forces are contained in vectors \(H\), the vectors \(\Gamma\), \(F\), and \(G\) are defined as follows and the parameter \(\Theta\) is [29]:  
\[\Gamma=\left\{ \begin{array}{ccccc}\Theta&0&0&0&\rho_{T}\\ \Theta u&\rho&0&0&\rho_{T}u\\ \Theta v&0&\rho&0&\rho_{T}v\\ \Theta w&0&0&\rho&\rho_{T}w\\ \Theta H-\delta&\rho u&\rho v&\rho w&\rho_{T}H+\rho C_{p}\end{array} \right\}, \;F=\left\{ \begin{array}{c}\rho v\\ \rho vu+p^{\dagger}_{T}\\ \rho vv+p^{\dagger}_{T}\\ \rho vw+p\hat{k}\\ \rho vE+pv\end{array} \right\},\;G=\left\{ \begin{array}{c}0\\ \tau_{xi}\\ \tau_{yi}\\ \tau_{zi}\\ \tau_{ij}v_{j}+q\end{array} \right\}\Theta=\left(\frac{1}{U_{r}^{2}}-\frac{ \rho_{T}}{\rho C_{p}}\right) \tag{15}\]  
Generally speaking, the flux vector \(F\) could be evaluated in two common ways:  
* ROE flux-difference splitting scheme [30]. At each face, the discrete flux is expressed as: \[\mathrm{F}=\frac{1}{2}\big{(}F_{R}+F_{L}\big{)}-\frac{1}{2}\Gamma|\hat{A}|\delta Q\] (16)
* AUSM flux-vector splitting scheme [31], which provides a numerical flux form to avoid using explicit artificial dissipation. \[\mathrm{F}=m_{f}\varphi+p_{i} \tag{17}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Reference**} & \multicolumn{5}{c}{**Far-Field Distances**} \\  & \multicolumn{2}{c}{**Data**} & \multicolumn{5}{c}{**Type of Upwind Order**} \\ \cline{2-9} x/L (m) & \multicolumn{2}{c}{**P/P\({}_{\mathrm{t}}\)**} & \multicolumn{5}{c}{**Numerical Results Of P/P\({}_{\mathrm{t}}\) T/T\({}_{\mathrm{t}}\) and Error Ratios**} \\ \cline{2-9}  & \multirow{2}{*}{**T/T\({}_{\mathrm{t}}\)**} & \multicolumn{2}{c}{**12 m**} & \multicolumn{2}{c}{**16 m**} & \multicolumn{2}{c}{**20 m**} \\ \cline{3-9}  & & \multicolumn{2}{c}{**First-Order**} & \multicolumn{2}{c}{**Second-Order**} & \multicolumn{2}{c}{**First-Order**} & \multicolumn{2}{c}{**Second-Order**} & \multicolumn{2}{c}{**First-Order**} & \multicolumn{2}{c}{**Second-Order**} \\ \hline  & \multirow{2}{*}{103.63} & 81.68 & 93.15 & 63.99 & 106.92 & 107.09 & 102.75 \\ \cline{3-9} \(-\)0.007 & & 21.18\% & 10.11\% & 38.25\% & 3.18\% & 3.34\% & 0.85\% \\ \cline{3-9}  & \multirow{2}{*}{18.15} & 15.34 & 17.55 & 12.08 & 19.57 & 19.59 & 20.47 \\ \cline{3-9}  & & 15.51\% & 3.29\% & 33.47\% & 7.81\% & 7.95\% & 12.78\% \\ \hline  & \multirow{2}{*}{114.55} & 106.49 & 112.16 & 101.22 & 112.50 & 106.96 & 106.52 \\ \cline{3-9} \(-\)0.006 & & 7.04\% & 2.09\% & 11.64\% & 1.79\% & 6.63\% & 7.01\% \\ \cline{3-9}  & \multirow{2}{*}{20.19} & 18.71 & 19.13 & 17.15 & 20.03 & 19.92 & 20.33 \\ \cline{3-9}  & & 7.31\% & 5.24\% & 15.03\% & 0.79\% & 1.32\% & 0.68\% \\ \hline  & \multirow{2}{*}{117.87} & 114.80 & 117.71 & 118.76 & 115.21 & 105.16 & 113.59 \\ \cline{3-9} \(-\)0.005 & & 2.60\% & 0.13\% & 0.76\% & 2.26\% & 10.78\% & 3.63\% \\ \cline{3-9}  & \multirow{2}{*}{20.28} & 20.36 & 20.06 & 20.02 & 20.26\% & 19.86 & 20.20 \\ \cline{3-9}  & & 0.36\% & 1.10\% & 1.28\% & 0.09\% & 2.11\% & 0.41\% \\ \hline  & \multirow{2}{*}{119.40} & 110.43 & 119.13 & 117.55 & 115.48 & 105.66 & 117.72 \\ \cline{3-9}  & & 7.51\% & 0.22\% & 1.55\% & 3.28\% & 11.50\% & 1.41\% \\ \cline{3-9}  & \multirow{2}{*}{20.43} & 20.78 & 20.17 & 20.13 & 20.33 & 19.62 & 20.06 \\ \cline{3-9}  & & 1.72\% & 1.24\% & 1.47\% & 0.50\% & 3.95\% & 1.78\% \\ \hline  & \multirow{2}{*}{121.74} & 110.71 & 122.20 & 120.54 & 115.45 & 106.74 & 122.04 \\ \cline{3-9}  & & 9.07\% & 0.37\% & 0.99\% & 5.17\% & 12.33\% & 0.25\% \\ \cline{3-9}  & \multirow{2}{*}{20.52} & 20.86 & 20.30 & 20.30 & 20.31 & 19.58 & 20.08 \\ \cline{3-9}  & & 1.67\% & 1.09\% & 1.07\% & 1.03\% & 4.58\% & 2.17\% \\ \hline \hline \end{tabular}
\end{table}