# Aerodynamic Characteristics of Supersonic Rocket-Sled Involving Waverider Geometry
## 3 Results and Discussion
### Spectral Analysis and Acoustic Characteristics  
According to the above analysis, the shockwave oscillation under the rocket-sled is periodic. The most direct way to investigate this phenomenon is to analyze the time characteristics of pressure. Before we performed the calculation, several monitoring probes were arranged below two rocket-sleds, as shown in Figure 8e,f. The pressure change curve of each monitoring probe with time has been extracted in Figure 16. To make the results more reliable, the data for the initial running period is omitted and only the data for the stable running period is retained. The dominant frequency of the shockwave oscillations is determined by plotting the spectra of the pressure signals. The spectra are computed using the fast Fourier transform (FFT) and power spectral density (PSD) of the signals [60]. To conduct power spectrum analysis more conveniently, the pressure and time are normalized to conduct power spectrum analysis by:  
\[p^{*}=p/p_{\infty}, \tag{20}\]  
Figure 14: Pressure distribution at symmetrical section on the surface of RWRRS (The gray color describes the shape): (a) \(t_{1}=0.0180\) s, (b) \(t_{2}=0.0186\) s, (c) \(t_{3}=0.0192\) s, (d) \(t_{4}=0.0198\) s.  
Figure 15: The curves of lift and drag force over time of two rocket-sleds (The yellow color represents the fluctuation range of ASRS and the green color represents the fluctuation range of RWRRS): (a) lift force, (b) drag force. \[t^{*}=\frac{t}{D/U_{\infty}}. \tag{21}\]  
According to Figure 16a, the pressure value of P1 is always about 0.78 atm, which is consistent with the conclusion above that \"the front part of the rocket-sled is not affected by head shockwaves and reflected shockwaves\". If the probes were set below the middle of the ASRS, they could detect more pronounced periodic pressure fluctuations, such as probes P3 and P4. The mean pressure and fluctuation range of probes P3 and P4 are both larger than the other probes, which indicates that this region is directly affected by the reflected shockwave. There are some reductions in the mean pressure and fluctuation range at the rear of the rocket-sled. These results suggest that, in a rocket-sled launch test, more pressure detectors should be arranged below the middle part of the ASRS if we want to obtain more experimental data about shockwave oscillation. As shown in Figure 16b, it is satisfactory that all the pressure probes of the RWRRS show very small pressure fluctuations, and that the mean pressure of all probes are concentrated within 0.96 atm to 1.16 atm. These results show that, during operation of the RWRRS, the pressure excitation on the sled body is minimal, and that this characteristic hardly changes with time. To investigate the unsteady characteristics related to shockwave oscillation, the PSD profiles for pressure signals are analyzed in Figure 17a,b. In spectral analysis, a dimensionless criterion number, the \"Strouhal number\", is often used to characterize unsteadiness. Assuming the frequency of fluid fluctuation or oscillation is \(f\), then the Strouhal number can be defined as:  
\[St=\frac{fD}{U_{\infty}}. \tag{22}\]  
Although the mean pressure and fluctuation range of the probes on the two configurations are quite different, their characteristics of oscillation are almost identical. Several peaks can be observed in the PSD profiles, corresponding to the characteristic frequency \(St_{1}\approx 0.212\). These are associated with the interaction between the shockwave and the railway fasteners. In addition, there is another peak at \(St_{2}\approx 0.448\). The relationship \"\(St_{2}\approx 2 St_{1}\)\" indicates that the shockwave oscillation in the rocket-sled flow field has a harmonic phenomenon. This harmonic phenomenon has been observed in many experimental studies [13]. Figure 16: Pressure profile of the probes on the surface of rocket-sleds: (**a**) ASRS, (**b**) RWRRS. Figure 17: PSD profiles for pressure signals on two rocket sleds: (**a**) ASRS, (**b**) RWRRS.