## 2 CFD Simulations
### Numerical Simulations  
At hypersonic speeds, the air density adopts ideal gas for the reasons below: (1) According to Ref [33], we adopt the pressure far field boundary condition to model a free stream condition at infinity for the aerodynamic application, and this boundary type is available only for compressible flows. For compressible flows, the ideal gas is the appropriate density relationship. In the ANSYS fluent, the pressure far field boundary condition is applicable only when the density is calculated using the ideal gas. Using it for other flows is not permitted. (2) The ideal gas has the limitation that it is approximately valid for the low-pressure gas region of the PT and PV diagrams. The ideal gas behavior can be expected when \(P\)/\(P_{c}\ll 1\), where \(P_{c}\) is the critical pressure of air flow and the value is 3.77 MPa. If the flow conditions correspond to the case, the ideal gas could be adopted in the simulation. Otherwise, it could not be used. In this paper, the initial pressure value of free stream is 576 pa and the maximum pressure value is about 73728 pa. The maximum value of \(P\)/\(P_{c}\) is about 0.019, which corresponds to the above case. The viscosity adopted the three-coefficient Sutherland Law, and the specific heat at constant air pressure is 1006.43 j kg-k. One should note that in this paper, for the SST k-omega turbulence model, the turbulence specification method of the input boundary adopts the intensity and viscosity ratio, the values of which are 1% and 1%. The effective diffusivities \(\Gamma_{k}=\mu+\frac{\mu_{t}}{\alpha}\), \(\Gamma_{\omega}=\mu+\frac{\mu_{t}}{\epsilon\omega}\). The turbulent viscosity \(\mu_{t}=\frac{\epsilon\omega}{\omega}\frac{1}{\max\left[\frac{1}{\omega^{ \alpha}+\mu_{0}}\right]}\), where \(\alpha^{*}=\alpha_{\infty}^{*}\left(\frac{\alpha_{0}^{*}+R_{\ell}/R_{k}}{1+R _{\ell}/R_{k}}\right)\), \(R_{\ell}=\frac{\rho k}{\mu\omega}\), \(\alpha_{0}^{*}=\beta_{i}/3\), \(F_{2}=\tanh(\phi_{2}^{2})\)and \(\phi_{2}=\max\left[2\frac{\sqrt{k}}{(0.08\omega)}\frac{500\mu}{\rho\omega}\right]\). The applied model constants are as fol \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**v/L (m)**} & \multirow{3}{*}{
\begin{tabular}{c} **Reference** \\ **Data** \\ \end{tabular} } & \multicolumn{6}{c}{**Far-Field Distances**} \\ \cline{2-9}  & & & & & **Type of Upwind Order** & & \\ \cline{2-9}  & & & & **Numerical Results Of P/P\({}_{\text{t}}\), T/T\({}_{\text{t}}\) and Error Ratios** & & & \\ \cline{2-9}  & & & **12 m** & & **16 m** & & **20 m** \\ \cline{2-9}  & & & **First-Order** & **Second-Order** & **First-Order** & **Second-Order** & **First-Order** & **Second-Order** \\ \hline \multirow{3}{*}{\(-0.002\)} & \multirow{3}{*}{122.10} & 112.04 & 125.88 & 125.16 & 115.19 & 110.22 & 126.47 \\ \cline{2-9}  & & 8.25\% & 3.09\% & 2.51\% & 5.66\% & 9.74\% & 3.58\% \\ \cline{2-9}  & & 20.79 & 20.55 & 20.53 & 20.47 & 20.02 & 20.11 \\ \cline{2-9}  & & 0.63\% & 0.54\% & 0.64\% & 0.94\% & 3.10\% & 2.70\% \\ \hline \multirow{3}{*}{\(-0.001\)} & \multirow{3}{*}{123.39} & 112.79 & 134.72 & 134.83 & 113.54 & 118.13 & 136.72 \\ \cline{2-9}  & & 8.59\% & 9.18\% & 9.27\% & 7.99\% & 4.27\% & 10.80\% \\ \cline{2-9}  & & 20.39 & 20.64 & 20.44 & 20.10 & 20.44 & 20.29 \\ \cline{2-9}  & & 3.12\% & 1.91\% & 2.85\% & 4.48\% & 2.89\% & 3.59\% \\ \hline \multirow{3}{*}{0} & \multirow{3}{*}{123.83} & 112.42 & 136.43 & 136.33 & 107.39 & 122.88 & 137.17 \\ \cline{2-9}  & & 9.22\% & 10.17\% & 10.09\% & 13.28\% & 0.77\% & 9.96\% \\ \cline{1-1} \cline{2-9}  & & 19.33 & 20.71 & 20.07 & 15.72 & 14.71 & 19.16 \\ \cline{1-1} \cline{2-9}  & & 9.34\% & 2.91\% & 5.89\% & 26.30\% & 31.03\% & 10.16\% \\ \hline \hline \end{tabular}
\end{table}