## 2 Nozzle Design and Numerical Verification  
A nozzle was used to accelerate the airflow to achieve the desired Mach number. The supersonic nozzle was designed as a convergent-divergent contour, which ensures good airflow quality in both sections to avoid flow separation. A bicubic curve and the Foelsch method [20] were applied to design the convergent and divergent sections, respectively. Boundary layer correction was carried out on the upper and lower curves to eliminate the influence of the boundary layer and shock waves in the nozzle. The design Mach number of the primary flow was 3, and the design Mach numbers of the secondary flow were 2.0, 2.5, and 2.9, respectively. The convective Mach numbers of the three nozzles were 0.25, 0.1, and 0.01, respectively. The throats of the nozzles were 3.31 3.64, 5.36, and 8.32 mm high for Mach numbers of 3.0, 2.9, 2.5, 2.0, respectively.  
Before carrying out the construction of the wind tunnel, it was essential to evaluate the performance of the design contours of the nozzles. Therefore, large eddy simulation was performed based on Ansys Fluent and solved by a density-based implicit method. A subgrid model of the large eddy simulation is the WALE model. The convective term was the Roe vector flux splitting scheme, the pressure term was the three-order MUSCL scheme, the momentum equation and energy equation were discretized by the three-order MUSCL scheme, and the time step was 1 \(\times\) 10\({}^{-7}\) s.  
To improve computational efficiency, a two-dimensional model was used. The air was set as an ideal gas and the viscosity was modeled using the three coefficients Sutherland law. The inlet condition was a pressure inlet. The wall was set as non-slip adiabatic. A total of 2.4 million grids was used to solve the flow domain, and the grids were clustered in the region of the mixing layer and boundary layer. The conditions of the simulation are shown in Table 1, where P\({}_{0}\), P\({}_{8}\) is the total and static pressure, and subscripts 1 and 2 represent the primary and the secondary flow, respectively. The total temperature was set as 293 K according to the local environment.  
Figure 2a-d shows the contours of nozzles with Mach numbers of 3.0, 2.9, 2.5, and 2.0, respectively, and Figure 2e,f show the Mach number and velocity profiles at the outlet of the four nozzles. The distribution of the Mach numbers at the exit of the nozzles was uniform and the Mach number of the simulations were within 0.05 compared to the design numbers. In Figure 2f, the core flow velocities of Mach numbers of 3.0, 2.9, 2.5, 2.0 are about 625, 618, 584, and 526 m/s, respectively, which are somewhat larger than those calculated by the isentropic formula. The overall variation of the velocity was less than 5 m/s in the core region of the flow. A decrease in velocity could be seen near the wall due to the existence of the boundary layer, and the boundary layer was about 2.5 mm on a single side. According to the simulation, the designed curves of the nozzles were satisfactory to generate the desired flow.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline Ma\({}_{1}\) & 3.0 & 3.0 & 3.0 \\ Ma\({}_{2}\) & 2.0 & 2.5 & 2.9 \\ Mc & 0.25 & 0.10 & 0.01 \\ P\({}_{01}\)/(kPa) & 800 & 800 & 800 \\ P\({}_{02}\)/(kPa) & 170 & 372 & 688 \\ P\({}_{81}\)/(kPa) & 21.7 & 21.7 & 21.7 \\ P\({}_{82}\)/(kPa) & 21.7 & 21.7 & 21.7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of the wind tunnel.