# Imaging Characteristics of Disturbance Flow Field Surrounding a Hypersonic Target
## 2 Methods
### Flow Refractive Index Computation and Ray Tracing Algorithm  
The refractive index of the flow can be derived from the flow density via the Gladstone-Dale relationship [19]  
\[n=1+\rho K_{\text{GD}} \tag{1}\]  
where \(n\) is the refractive index, \(\rho\) is the density in kg/m\({}^{3}\), and \(K_{\text{GD}}\) is a coefficient which can be expressed as [8]  
\[K_{\text{GD}}=2.23\times 10^{-4}\bigg{(}1+\frac{7.52\times 10^{3}}{\lambda^{2}} \bigg{)} \tag{2}\]  
where \(\lambda\) is the light wavelength in nm, and \(K_{\text{GD}}\) varies weakly with the wavelength. The refractive index field calculated by Equation (2) is shown in Figure 2b, where \(\lambda\) is selected as 532 nm. The propagating path of light in any medium with a nonuniform distribution of the refractive index can be expressed by ray equation [8]  
\[\frac{\text{d}}{\text{d}s}\bigg{(}n\frac{\text{d}\text{r}}{\text{d}s}\bigg{)}= \
abla n \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Figure Serial Number** & **Flight Altitude (km)** & **Mach Number** \\ \hline (a) & 20 & 10 \\ (b) & 30 & 8 \\ (c)/(e) & 30 & 10 \\ (d) & 30 & 12 \\ (f) & 50 & 10 \\ \hline \end{tabular}
\end{table} 
Table 1: Flight conditions. Figure 1: The geometry model of target. Figure 2: The density and refractive index distribution of the disturbance flow field: Flight altitude \(=\) 20 km; Mach number \(=\) 10 Ma; (**a**) density distribution; (**b**) refractive distribution. where \(s\) is the arc length of the ray propagation path, \(\mathbf{r}\) is the position vector of the ray propagation, and \(\
abla n\) is the refractive index gradient. Under the assumption of the ray vector \(\mathbf{T}=n(\mathrm{d}\mathbf{r}/\mathrm{d}s)\), Equation (3) can be expressed as first-order differential equations. \[\left\{ \begin{array}{l}\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s}=\
abla n\\ \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}=\frac{\mathbf{T}}{n}\end{array} \right. \tag{4}\]  
Equation (4) can only be solved analytically when the refractive index satisfies a particular distribution. The approximate solution for the disturbance flow field with random changes the in refractive index is typically obtained by numerical solutions. By introducing a new parameter \(\mathrm{d}t=\mathrm{d}s/n\), Equation (4) can be expressed as  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}t}=n(\
abla n)=f(t, \mathbf{T})\\ \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}=\mathbf{T}=n[\cos\alpha,\cos\beta,\cos\gamma] =g(t,\mathbf{r})\end{array} \right. \tag{5}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are the angles between the light and the \(x\)-axis, \(y\)-axis, and \(z\)-axis, respectively. Assuming \(\mathbf{D}(\mathbf{r})=n\
abla n\), the fourth order Runge-Kutta method [20] could be carried out to solve Equation (5). The approximate solution is expressed as  
\[\left\{ \begin{array}{l}\mathbf{T}_{i+1}=\mathbf{T}_{i}+\frac{h}{6}(\mathbf{K}_{1}+2\mathbf{ K}_{2}+2\mathbf{K}_{3}+\mathbf{K}_{4})\\ \mathbf{r}_{i+1}=\mathbf{r}_{i}+\frac{h}{6}(\mathbf{L}_{1}+2\mathbf{L}_{2}+2\mathbf{L}_{3}+\mathbf{L}_ {4})\end{array} \right. \tag{6}\]  
\(\mathbf{K}_{j}\) and \(\mathbf{L}_{j}\,j=(1,2,3,4)\) can be obtained by  
\[\left\{ \begin{array}{l}\mathbf{K}_{1}=f(t_{i},\mathbf{T}_{i})=\mathbf{D}(\mathbf{r}_{i})\\ \mathbf{L}_{1}=g(t_{i},\mathbf{r}_{i})=\mathbf{T}_{i}\end{array} \right. \tag{7}\]  
\[\left\{ \begin{array}{l}\mathbf{K}_{2}=f\Big{(}t_{i}+\frac{h}{2},\mathbf{T}_{i}+\frac {h}{6}\mathbf{K}_{1}\Big{)}=\mathbf{D}\Big{(}\mathbf{r}_{i}+\frac{h}{2}\mathbf{T}_{i}\Big{)} \\ \mathbf{L}_{2}=g\Big{(}t_{i}+\frac{h}{2},\mathbf{r}_{i}+\frac{h}{2}\mathbf{L}_{1}\Big{)}= \mathbf{T}_{i}+\frac{h}{2}\mathbf{D}(\mathbf{r}_{i})\end{array} \right. \tag{8}\]  
\[\left\{