# The Experiments and Stability Analysis of Hypersonic Boundary Layer Transition on a Flat Plate
## 3 Experimental Results and Stability Analyses
### Instability Characteristics along Streamwise Direction  
The inhouse LST code written by FORTRAN used in this paper is shown in reference [34]. Figure 8 illustrates the variations in disturbance growth rates within the \(\omega\)-\(\beta\) plane at \(\text{Rex}\) = 6.5 \(\times\) 10\({}^{5}\) under the freestream conditions of case 3. Two distinct modes coexist in the boundary layer--the lower frequency corresponds to the first mode, while the higher frequency characterizes the second mode. Notably, the bandwidth of the first mode surpasses that of the second mode. The diagram emphasizes the prevalence of the second mode in the hypersonic boundary layer, notably featuring a predominantly two-dimensional disturbance. Conversely, the first mode, exhibiting the highest amplification, predominantly occurs near \(\beta\) = 0.8, signifying its three-dimensional perturbation nature. The evolution of instability waves along the streamwise direction was obtained and analyzed. Figure 9 shows the PSDs of the boundary layer instability waves at several streamwise positions under different unit Reynolds numbers by PCB. As illustrated in Figure 9, for a unit Reynolds number (\(Re_{\text{unit}}\)) of 6.38 \(\times\) 10\({}^{6}\), the absence of a discernible peak characterizing second-mode waves in the power spectral density (PSD) before the streamwise location of \(x\) = 180 mm indicates that second-mode disturbances have not yet developed, signifying a laminar boundary layer state at this location. However, a subtle peak with broadband characteristics around 100 kHz emerges  
Figure 8: Growth rate in \(\omega\)-\(\beta\) plane: (**a**) \(Re_{\text{unit}}\) = 6.38 \(\times\) 10\({}^{6}\); (**b**) \(Re_{\text{unit}}\) = 8.20 \(\times\) 10\({}^{6}\). Figure 9: Comparison of PCB-measured spectra along the streamwise direction: (**a**) \(Re_{\text{unit}}\) = 6.38 \(\times\) 10\({}^{6}\); (**b**) \(Re_{\text{unit}}\) = 8.20 \(\times\) 10\({}^{6}\). in the PSD at \(x=240\) mm, suggesting the initiation and growth of second-mode waves. Subsequently, the PSD of the second mode remains consistent between \(x=330\) mm and \(x=360\) mm, indicating saturation of the second-mode disturbances. In the case of a higher unit Reynolds number, i.e., \(Re_{\text{unit}}=8.20\times 10^{6}\), the peak in the PSD at the streamwise location of \(x=240\) mm surpasses that of \(Re_{\text{unit}}=6.38\times 10^{6}\), indicating the earlier appearance and growth of second-mode disturbances before \(x=240\) mm. Beyond \(x=330\) mm, the energy associated with second-mode waves exhibits a gradual increase and disperses into neighboring frequency domains, signaling the imminent breakdown of second-mode waves and the transition of the boundary layer. According to previous studies, when the Ma \(>4\), the second-mode wave is more unstable, always leading to transition. Figure 10 shows the evolution of the growth rate of second-mode disturbances in different frequencies along the flow direction by LST. The second modes at different frequencies start to grow from different stations, and the high-frequency modes usually begin to arise and grow at first. Figure 10b indicates that, at the same unit Reynolds number condition, the higher frequency of the second mode wave can achieve a higher growth rate than that at low frequency. However, due to the attenuation of the high-frequency disturbance, it develops to a very short distance in the direction of flow, as shown in Figure 11. As the high-frequency modes begin to decay, the low-frequency modes in the boundary layer begin to appear and grow along the flow direction. Although the growth rate of lower-frequency disturbance is smaller than that of higher frequency, it usually becomes key to promote transition because of its larger propagation distance in the boundary layer. Figure 11: _Cont._  
Figure 10: The growth rate of second-mode waves along flow directions predicted by LST: (a) \(Re_{\text{unit}}=6.38\times 10^{6}\); (b) \(Re_{\text{unit}}=8.20\times 10^{6}\).