## 5 Optical Diagnosis of the Flow  
We performed optical diagnosis analyses, i.e., FLEET and FLDI, on the facility and compared the results with numerical simulations. FLEET is a non-intrusive and non-particle optical technique for velocity measurements conducted by recording the trajectory of the fluorescent filaments generated by N\({}_{2}\) molecules [23].The detection area was at about x = 100 mm in the test section. According to the principle of FLEET, the measurement accuracy of speed is mainly determined by the displacement measurement accuracy and timing control accuracy. The timing control accuracy of the system was within 0.25 ns, which can be ignored. The displacement accuracy was measured in the static air. We generated the FLEET filament in the static air, captured 50 photographs, calculated the displacement error and obtained the velocity error. It was determined that the velocity error was within 5 m/s with a time delay of 5 \(\upmu\)s, and within 2.5 m/s with a time delay of 10 \(\upmu\)s. Graphs of the fluorescent filaments with 5 us and 10 us time delay are shown in Figure 12. The mean velocity profiles calculated with 5 us and 10 us time delay are plotted as red dash-dots and blue dots, respectively, in Figure 13, where the sixth velocity profile of the simulation at the same location is also plotted for comparison. The primary velocity was about 610 m/s and the secondary velocity was about 516 m/s, which were lower than the simulation with errors of about 4% and 9%. This result is reasonable and consistent with the pressure distribution in Figure 10 because the static pressure of the experiments was higher than that of the simulation, which corresponds to a slower speed than the simulation. This is attributed to a systematic error, probably caused by the setting of boundary conditions and machine error. Despite the velocity deviation, the region of the mixing layer of FLEET and simulation was the same, both from about 2.5 mm to \(-\)2.5 mm, as shown in Figure 13. A focused laser differential interferometer (FLDI) was utilized to measure the density fluctuation of the flow field. The optical layout is shown in Figure 14. The principle of the device is detecting the density fluctuation in the area between the two focuses of the laser beams in the flow, as only in this area are the two beams separated, and sensing an optical difference due to the density fluctuation [24]. In the FLDI experiment, the fluctuation was recorded by the a photodetector and transformed into an electronic signal. The detection location is labeled as a red dot in Figure 15c, where we also extract the density fluctuation during the process of the simulation. Fourier transformation was performed to obtain the power spectrum of the density fluctuation. The density power spectrum of the simulation and the experiment is shown in Figure 15. Both the FLDI and the simulation show a peak frequency of 85 KHz, which verifies the practical application of FLDI on the supersonic mixing layer. Other peaks at 1, 2.5, 4 and 10 kHz represent noise from the laser we used in the experiments.  
Figure 11: Velocity profiles of simulation in the streamwise direction of case 1.  
Figure 14: Layout of the FLDI device. (P1, P2, polarizer; H, half-wave plate; PH, pinhole; W1, W2, Wollaston prism; L1, L2, convex lens; D, fast photodetector).  
Figure 12: Fluorescent filaments of FLEET.  
Figure 13: Velocity profiles of the simulation and FLEET at x = 100 mm in the test section (simulation, solid line; FLEET, red dash-dot, 5 us delay and blue dot, 10 us delay).