## 3 Geometry and Operation of the Wind Tunnel
### Operation of the Wind Tunnel  
Due to factors such as machining accuracy and installation error, the actual Mach number and operation pressure may deviate from the theoretical calculation and numerical simulation. It was necessary to calibrate the actual operation parameters. Mach number and local sound velocity were calculated using the isentropic formula and sound velocity formula:  
\[\frac{\mathrm{P}_{0}}{\mathrm{P}_{\mathrm{s}}}=(1+\frac{\mathrm{\gamma}-1}{2 }\mathrm{Ma}^{2})^{\frac{\mathrm{\gamma}}{\mathrm{\gamma}-1}} \tag{4}\]  
\[\frac{\mathrm{T}_{0}}{\mathrm{T}_{\mathrm{s}}}=1+\frac{\mathrm{\gamma}-1}{2} \mathrm{Ma}^{2} \tag{5}\]  
\[\mathrm{a}=\sqrt{\mathrm{\gamma RT}_{\mathrm{s}}} \tag{6}\]  
where \(\mathrm{P}_{0}\) is the total pressure, \(\mathrm{P}_{\mathrm{s}}\) is the static pressure at the nozzle outlet, \(\mathrm{\gamma}\) is the specific heat ratio (for air \(\mathrm{\gamma}\) = 1.4), \(\mathrm{Ma}\) is the Mach number at the nozzle exit, T\({}_{0}\) is the total temperature, T\({}_{\mathrm{s}}\) is the static temperature at the nozzle exit, a is the local sound velocity, and R is the gas constant (R = 287 J/(kg\(\cdot\)K)). The total pressure P was measured by the pressure tap in the stagnation chamber, and \(\mathrm{P}_{\mathrm{s}}\) was measured by the wall static pressure tap at the nozzle exit. By calculating the local sound velocity and the Mach number of the primary and secondary layers, the velocity of airflow was calculated and the convective Mach number was further calculated. The main purpose of the test was to ensure that the pressure of the primary and secondary airflow was the same in the test section. The total and static pressure of the three cases in operation was determined, which were different from the designed ones. Figure 6 shows the pressure of the primary and the secondary airflow measured by the pressure scanner. Figure 6a-c corresponds to case 1, 2 and 3, respectively. The total operation time of the wind tunnel was about 15-24 s. The pressure of the primary airflow gradually decreased to about 700-800 kPa and then remained constant, and the pressure of the secondary airflow was finally stabilized at about 200, 300, and 600 kPa, respectively. The Mach numbers of the three cases were calculated according to the isentropic formula, as shown in Figure 7. It can be seen that although the total pressure changed gradually, the Mach numbers remained stable throughout the operations and were close to the design value. In the stable stage of the total pressure, the actual Mach numbers of the primary airflow of the three cases were 2.99, 2.95, and 2.95, and the Mach numbers of the secondary airflow were 1.99, 2.46, and 2.85, respectively. The parameters are shown in Table 2. The experimental Mach number of the supersonic mixing layer wind tunnel were close to the designed Mach numbers, and the error was about 1.5-1.7%. The actual convective Mach numbers of the three cases were 0.23, 0.10, and 0.02, respectively. The Mach numbers of the simulations were within 0.10 of the experimental results, and the convective Mach number was within 0.01. Therefore, the facility was able to generate the desired Mach number and the large eddy simulation had good accuracy. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **Ma** & **Mc** & **U/m/s** & **P/kPa** & **P/kPa** & **T0/K** & **T\_**/K** \\ \hline \multirow{2}{*}{Case 1} & 2.99 & \multirow{2}{*}{0.23} & 619 & 804 & 23.6 & 293 & 107 \\  & 1.99 & & 512 & 194 & 23.4 & 293 & 165 \\ \hline \multirow{2}{*}{Case 2} & 2.95 & \multirow{2}{*}{0.10} & 611 & 712 & 20.9 & 293 & 107 \\  & 2.46 & & 567 & 330 & 20.5 & 293 & 132 \\ \hline \multirow{2}{*}{Case 3} & 2.95 & \multirow{2}{*}{0.02} & 611 & 699 & 20.8 & 293 & 107 \\  & 2.85 & & 604 & 612 & 20.8 & 293 & 111 \\ \hline \end{tabular}
\end{table} 
Table 2: Operation conditions of the wind tunnel. Figure 6: Total and static pressure in operation (normalized by local atmospheric pressure p\({}_{\rm I}\)) (**a**) case 1, (**b**) case 2, (**c**) case 3.