# Imaging Characteristics of Disturbance Flow Field Surrounding a Hypersonic Target
## 2 Methods
### Background-Oriented Schlieren and Imaging Deviation Computation  
Background-Oriented Schlieren (BOS) uses the deflection of light to identify changes in the refractive index (or density) of a flow field. When light passes through the disturbance flow field, it undergoes deflection from its original path, causing a shift in the position of the light incident on the CCD camera. This shift results in a disparity between the background images captured by the CCD camera in the presence of a disturbance flow field (referred to as the experimental image) and in its absence (referred to as the reference image) [26]. In this study, Background-Oriented Schlieren is employed for the imaging deviation analysis.  
Figure 3 illustrates the schematic of Background-Oriented Schlieren (BOS) [26]. In the figure, \(Z_{O}\) represents the distance from the background to the center of the disturbance flow field, \(W\) denotes the length of the disturbance flow field, \(Z_{B}\) corresponds to the distance between the background and the camera lens, and \(Z_{I}\) signifies the distance from the lens to the imaging plane, which is approximately equal to the focal length \(f\). The dotted line in the figure represents the path of light in the absence of a disturbance flow field, while the solid line depicts the actual path of light propagation through the disturbance flow field. The \(y\)-direction deflection angle between these two rays is denoted by \(\varepsilon_{y}\). The parameter \(\Delta y\) represents the displacement in the \(y\) direction between the corresponding point on the reference image and the experimental image. \(\Delta y^{\prime}\) denotes the virtual shift of \(\Delta y\) relative to the background. Similarly, \(\varepsilon_{x}\) represents the deflection angle in the \(x\) direction, \(\Delta x\) is the displacement in the \(x\) direction, and \(\Delta x^{\prime}\) is the virtual displacement of \(\Delta x\).  
After ray tracing, the \(x\) direction deflection angle and \(y\) direction deflection angle can be expressed as [26]  
\[\varepsilon_{x}=\int_{Z_{O}-W/2}^{Z_{O}+W/2}\frac{1}{n}\frac{\partial n}{ \partial x}\mathrm{d}z\quad\varepsilon_{y}=\int_{Z_{O}-W/2}^{Z_{O}+W/2}\frac{ 1}{n}\frac{\partial n}{\partial y}\mathrm{d}z \tag{24}\]  
Based on the geometry in the diagram, the relationship between \(\Delta x^{\prime}\) and \(\Delta x\) and \(\Delta y^{\prime}\) and \(\Delta y\) can be expressed as [26]  
\[\frac{\Delta x^{\prime}}{Z_{B}}=\frac{\Delta x}{f}\quad\frac{\Delta y^{\prime} }{Z_{B}}=\frac{\Delta y}{f} \tag{25}\]  
For minimal deflection angle, it can be expressed approximately as [26]  
\[\varepsilon_{x}=\frac{\Delta x^{\prime}}{Z_{O}}=\frac{Z_{B}\Delta x}{Z_{O}f} \quad\varepsilon_{y}=\frac{\Delta y^{\prime}}{Z_{O}}=\frac{Z_{B}\Delta y}{Z_{ O}f} \tag{26}\]Thus, the displacement can be expressed as [26]  
\[\Delta x=\varepsilon_{x}\frac{Z_{O}f}{Z_{B}}\quad\Delta y=\varepsilon_{y}\frac{Z_ {O}f}{Z_{B}} \tag{27}\]  
The corresponding pixel shift on the CCD plane can be expressed as  
\[\Delta u=\frac{\Delta x}{a}=\varepsilon_{x}\frac{Z_{O}f}{aZ_{B}}\quad\Delta v= \frac{\Delta y}{a}=\varepsilon_{y}\frac{Z_{O}f}{aZ_{B}} \tag{28}\]  
where a is the pixel size of CCD. The relationship between the gray value of the deviated image and the original image satisfies [27]  
\[I_{o}(x+\Delta u,y+\Delta v)=I_{d}(x,y) \tag{29}\]  
Using the given image as the original image \(I_{o}\), and the displacement field \(\Delta u,\Delta v\) as the displacement label, the deviated image \(I_{d}\) can be generated using interpolation.  
Imaging characteristics of blurring and deviation were performed utilizing an ideal optical system with the parameters listed in Table 2.