## 2 Experimental Methods
### Flow-Field Velocimetry Using Schlieren Images  
In this study, an optimized optical flow algorithm is used to obtain the velocity field based on schlieren images. Optical flow is considered as the distribution of the apparent velocities of the movement of brightness patterns in an image [32; 33]. The basic assumption in optical flow techniques is that the gray levels of objects in subsequent frames do not change over time. One constraint can be established based on the change in image brightness at a point on the image plane due to motion, which can be written as:  
\[I_{x}u+I_{y}v+I_{t}=0 \tag{1}\]  
As shown in Figure 2, \(I_{x}=\frac{\partial I}{\partial x}\) represents the brightness change in the horizontal direction; \(I_{y}=\frac{\partial I}{\partial y}\) represents the brightness change in the vertical direction; \(I_{t}=\frac{\partial I}{\partial t}\) represents the brightness change of the same position at different times; \(I\) denotes the luminance intensity; \(t\) is the time between frames and \(u=\frac{dx}{dt}\) and \(v=\frac{dy}{dt}\) are velocities. The data conservation term H\({}_{1}\) can be written as:  
\[H_{1}=f_{1}(I_{x}u+I_{y}v+I_{t}) \tag{2}\]  
where \(f_{1}\) is the penalty function. The nonconvex generalized Charbonnier penalty function \(f_{1}(x)=\left(\sigma^{2}+x^{2}\right)^{\epsilon}\) is chosen to improve the robustness, where \(\sigma=0.001\) and \(\epsilon=0.45\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & No. & 1 & 2 & 3 & 4 & 5 \\ \cline{2-7} Acoustic & Frequency (Hz) & / & 10 & 30 & 50 & 100 \\ \cline{2-7} excitations & Velocity amplitude (m/s) & / & 0.39 & 0.50 & 0.41 & 0.18 \\ \cline{2-7}  & Relative amplitude & / & 0.54 & 0.69 & 0.56 & 0.25 \\ \hline \multirow{3}{*}{Flow conditions} & \multirow{3}{*}{Air} & Flow rate = 41 slm, average velocity = 0.73 m/s, & \multicolumn{3}{c}{\(\text{Re}=1110\)} \\ \cline{2-7}  & & & Flow rate = 3 slm, average velocity = 0.99 m/s, & \multicolumn{3}{c}{\(\text{Re}=480\)} \\ \hline Camera & Camera 1 & Frame rate = 2000 fps, exposure time = 1/6600 s \\ \cline{2-7} setting & Camera 2 & Frame rate = 2000 fps, exposure time = 1/2000 s \\ \hline \hline \end{tabular}
\end{table}
Table 1: The test conditions and parameters of experiment setup.  
Figure 2: The principle of optical flow methods.  
In addition, spatial consistency should be considered, which refers to the assumption that points in small neighborhoods have similar velocities. One way to express this constraint is to minimize the square of the magnitude of the optical flow velocity gradient, which can be written as applying the penalty function to the first-order velocity space smoothness [32]:  
\[H_{2}=f_{1}\left(\frac{\partial u}{\partial x}\right)+f_{1}\left(\frac{\partial u }{\partial y}\right)+f_{1}\left(\frac{\partial v}{\partial x}\right)+f_{1} \left(\frac{\partial v}{\partial y}\right) \tag{3}\]  
Then, the global cost function \(E\left(H_{1},H_{2}\right)\) can be written as:  
\[E(H_{1},H_{2})=\int H_{1}dxdy+\beta\int H_{2}dxdy \tag{4}\]  
In most cases, these two assumptions cannot be fulfilled simultaneously. Thus, there is competition between the two assumptions, which leads to an optimal compromise to minimize the global cost function. The relative importance of each hypothesis is controlled by a weight parameter \(\beta\). The density displacement field is estimated by the variational method in which all model assumptions are formulated by the cost-optimization problem. The use of variational methods allows flexibility in combination with modern optimization techniques. In the current study, the optimized optical flow method proposed by [PERSON] et al. [32,33] is adopted. The error of the optical flow algorithm is about 0.319 pixel/frame, estimated by the Middlebury benchmark test database, which corresponds to 0.064 m/s in the current study. For more details of the algorithm, please refer to Ref. [32].