## 4 Experiments Results and Discussion
### Results and Discussion
#### 4.3.3 Results of the Detonation Shock Tube Test  
Since the shock tube allows for more quantitative studies than the other tests and the free-stream flows can be accurately obtained through the shock tube theory, the calibration of the sensors was also conducted using the detonation shock tube. However, the test time in the shock tube is much  
Figure 19: Heat flux curves. (**a**) C02 (NTC-10) and C05 (NTC-2.5) for three test runs; (**b**) C05, C07 and C08 in Run 1.  
Figure 20: Heat flux curves with/without corrections. (**a**) Comparison of C02 (NTC-10) and C05 (NTC-2.5); (**b**) Comparison of C03 (NTC-10) and C06 (NTC-2.5).  
shorter than that of the other tests. Due to the test space limitation of the shock tube, only two NTC-2.5 sensors (C05 and C06) were installed on two R20 spheres; the sensor arrangement is shown in Figure 12. Five tests were conducted. The measured heat flux curves for C05 are shown in Figure 21, where the heat flux of NTC-2.5 is corrected in the same manner as described in Section 4.3.2. However, since the test time was less than \(1\) ms, the corrections were also very limited, and the discrepancy was 6% at most, as shown in Figure 8. The heat flux curves show excellent repeatability, similar to the other tests.  
The value between 9.8 and 10.4 ms in Figure 21 is used as the experimental heat flux. The heat flux results are summarized in Table 5, including the theoretical results obtained using the Fay-Riddle formula and the test flow in the shock tube. According to the Fay-Riddle formula, the heat flux at the stagnation point is [39]:  
\[q_{w}=0.763\text{Pr}^{-0.6}(\rho_{e}u_{e})^{0.4}(\rho_{w}u_{w})^{0.1}\sqrt{ \left(\frac{du}{dx}\right)_{0}}(h_{e}-h_{w}) \tag{16}\]  
Here \(\rho\), \(\mu\), and \(h\) are the gas density, viscosity, and enthalpy, respectively, and \(Pr\) is the Prandtl number. The subscripts \(w\) and \(e\) denote the conditions at the wall and the outer edge of the boundary layer, respectively. \((du/dx)_{0}\) is the velocity gradient at the outer edge of the boundary layer. The subscripts \(1\) and \(2\) in Table 5 denote the test gas in front and behind the incident shock wave, respectively. \(Ms\) is the incident shock Mach number. The experiment in the detonation shock tube was conducted under the following conditions: a mixture of \(2\text{H}_{2}+\text{O}_{2}+1.5\text{N}_{2}\) and a pressure of 0.08 MPa in the driving section. The shock tube was filled with air at a pressure of 5000 Pa.  
It is observed that the measured heat fluxes with corrections are, on average, 7.5% lower than the theoretical results, and the maximum deviation is 10.1%. The main reason is that the sensor is too large for the spherical model; the curvature of the stagnation point is changed, resulting in lower heat flux.  
Figure 21: Heat flux curves of C05 (NTC-2.5) at the stagnation point in the shock tube.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Runs**} & \multirow{2}{*}{\(M\)**s**} & \multirow{2}{*}{\(\rho_{2}\) **(kg/m\({}^{3}\))**} & \multirow{2}{*}{\(p_{2}\) **(kPa)**} & \multirow{2}{*}{\(u_{2}\) **(m/s)**} & \multirow{2}{*}{
\begin{tabular}{c} **Theoretical** \\ **Heat Flux** \\ **(MW/m\({}^{2}\))** \\ \end{tabular} } & \multirow{2}{*}{\(\sqrt{\left(\frac{du}{dx}\right)_{0}}\)} & \multirow{2}{*}{\(\sqrt{\left(\frac{du}{dx}\right)_{0}}\)} & \multirow{2}{*}{\(\sqrt{\left(\frac{du}{dx}\right)_{0}}\)} & \multirow{2}{*}{\(\sqrt{\left(\frac{du}{dx}\right)_{0}}\)} & \multirow{2}{*}{\(\sqrt{\left(\frac{du}{dx}\right)_{0}}\)} \\ \cline{1-1} \cline{5-12} \cline{4-12} \cline{4-1  
Nevertheless, the overall repeatability is excellent, and sensors with smaller size are being developed to obtain the heat flux at a location with small curvature.