## 2 Models
### Thrust Calculation  
The thrust of a hypersonic vehicle is generated by the ramjet. Air flow is captured by the inlet, then mixed with fuel, and burns to generate thrust. The air mass flow rate captured by the engine inlet, denoted by \(\dot{m}_{inlet}\), is evaluated:  
\[\dot{m}_{inlet}=\varphi_{inlet}\cdot\rho\cdot Ma\cdot c\cdot A \tag{10}\]  
where \(A\) denotes the inlet area and \(\varphi_{inlet}\) is the inlet flow capture coefficient, which reflects the ability to capture air flow under different Mach numbers and angles of attack.  
The calculation of \(\varphi_{inlet}\) is shown in Equation (11), and the values of coefficients are given in Table 3.  
\[\varphi_{inlet}=K_{1}\cdot\alpha^{2}+K_{2}\cdot Ma^{2}+K_{3}\cdot\alpha+K_{4} \cdot Ma+K_{5}\cdot Ma\cdot\alpha+K_{6} \tag{11}\]  
Since the kind of fuel in the scramjet is fixed, there is a relationship between the fuel mass flow rate and the air mass flow rate in complete combustion, which is displayed in  
Figure 2: The change of \(C_{L}\) and \(C_{D}\) with angle of attack and Mach number.  
Equation (12), where \(k\) is a proportion coefficient with a value at 0.07 and \(\dot{m}_{fst}\) denotes the fuel mass flow rate required for complete combustion.  
\[\dot{m}_{fst}=k\cdot\dot{m}_{inlet} \tag{12}\]  
The ratio of current fuel mass flow rate to the fuel mass flow rate required for complete combustion is called the equivalence ratio, which is denoted by \(\Phi\)[34]:  
\[\Phi=\frac{\dot{m}_{fuel}}{m_{fst}} \tag{13}\]  
When \(\Phi=1\), it indicates that current amount of air flow can be completely consumed; when \(\Phi<1\), it indicates that the current combustion in the engine is in an oxygen enriched state. However, when \(\Phi<0.45\), it leads to insufficient fuel, and the engine cannot work normally. Therefore, the range of \(\Phi\) is between 0.45 and 1.  
Combining Equations (12) and (13), the fuel mass flow rate is:  
\[\dot{m}_{fuel}=\Phi\cdot k\cdot\dot{m}_{inlet} \tag{14}\]  
Then, the thrust is:  
\[T=\dot{m}_{fuel}\cdot I_{sp}=\Phi\cdot k\cdot\dot{m}_{inlet}\cdot I_{sp} \tag{15}\]  
where \(I_{sp}\) denotes the specific impulse, a parameter related to equivalence ratio and Mach number:  
\[I_{sp}=f(\Phi)\cdot g(Ma) \tag{16}\]  
\(f(\Phi)\) and \(g(Ma)\) are shown in Equation (17), and the values of coefficients are given in Table 4.  
\[\begin{array}{c}f(\Phi)=(1+\frac{1-\Phi}{2})\\ g(Ma)=\sum\limits_{i=1}^{4}C_{i}\cdot Ma^{i}\end{array} \tag{17}\]  
\(f(\Phi)\) reflects the influence of equivalence ratio on specific impulse. If the equivalence ratio is lower, fuel is more likely to burn completely due to more adequate oxygen; thus, larger thrust can be generated by a unit mass of fuel, and the specific impulse is larger as a result. Compared with the HL-20 aircraft model, whose impulse is independent from equivalence ratio [4; 6], this model is closer to the actual situation.  
\(g(Ma)\) reflects the influence of Mach number on the specific impulse. When Mach number increases, specific impulse will decrease.  
Figure 3 shows the comparison between the parametric thrust model and the original data at an angle of attack of \(5^{\circ}\). It can be seen that the parametric model can accurately reflect the trend of thrust with Mach number and equivalence ratio: with the increase of Mach number, the thrust increases gradually; when the equivalence ratio increases, more fuel is burned and more thrust is generated.  
\begin{table}
\begin{tabular}{c c c c} \hline \(C_{4}\) & \(C_{3}\) & \(C_{2}\) & \(C_{1}\) \\ \hline
62.50 & \(-933.84\) & \(3391.2\) & \(2024.4\) \\ \hline \end{tabular}
\end{table}
Table 4: Fitting coefficients of \(g(Ma)\).