## 3 Experimental Verification  
To verify the feasibility of numerical simulations in extracting vortex acoustic signal characteristics, in this paper, a vertical-axis vortex acoustic signal measurement experiment is conducted under laboratory setting [25]. Compared to the experiments in [23], the field of vertical-axis vortex is more stable with simpler equipment setup. In the experiment, the measurements of acoustic signals passing through the vortex field using a tri-element ultrasonic sensor array are compared with the results of numerical simulations. As shown in Figure 2a, the vertical-axis vortex is generated by opening a hole at the bottom of an acrylic open-loop water tank, with a water storage area on the left and a vortex zone on the right. A tri-element sensor array is arranged on both sides of the vortex zone, with a transmit-receive spacing of 25 cm. The sensors operates at a frequency of 500 kHz, a sharpness angle (beam width) of \(5^{\circ}\pm 1^{\circ}\) (\(-3\) dB), and a usable range of 0.03\(\sim\)2 m.  
Figure 1: Time-reversal extraction of vortex acoustic signals. (**a**) Initial acoustic wave emission; (**b**) re-emission after time reversal. The data logger has a maximum sampling rate of 100 MHz, and the arbitrary waveform generator can output user-defined arbitrary waveforms. The dual-channel sampling rate is 2.4 GSa/s. As shown in Figure 2, taking the center point O of the vortex zone as the origin, a coordinate system is established. The middle sensor b in the TRM1 array serves as the reference coordinate for the array. By horizontally moving both TRM arrays along the y-axis by 4 cm in both positive and negative directions, flow field data within a range of \(-6\) cm to 6 cm can be measured. Fourier Transform [26] is applied to the signals with and without vortex presence after collecting vortex acoustic signal data based on time-frequency analysis theory. The cross-power spectral density method in the frequency domain is used to calculate the time delay difference in vortex acoustic signals, \(\Delta\phi\), at the same array element position for both conditions [27]:  
\[\Delta\phi=\phi_{i}-\phi_{0} \tag{2}\]  
[PERSON] [28] derived the velocity distribution of the vertical-axis vortex based on experimental data:  
\[v(r)=\frac{\Gamma}{2\pi r}\bigg{(}1-\exp\bigg{(}-1.25\frac{r^{2}}{r_{0}{}^{2} }\bigg{)}\bigg{)} \tag{3}\]  
The formula above provides the basis for numerical simulations on the sound signals passing through the vertical-axis vortex, where the vortex characteristic radius \(r_{0}=4\) cm, the Mach number \(Ma=\)3.33 \(\times\) 10\({}^{-4}\), and the circulation \(\Gamma=2.8\pi r_{0}Ma=\)0.1759 m\({}^{2}\)/s, and the numerical simulation results of the vortex acoustic signal phase are obtained. As can be seen from Figure 3, a phase curve can be drawn by connecting the phase data points of the vortex acoustic signals received by the simulated array elements. The phase difference between the phase curve with vortex and the phase curve without vortex is the phase difference curve. Figure 3: Vortex acoustic signal phase characteristics. Figure 2: Vertical-axis vortex acoustic signal measurement. (**a**) Measurement top view and instrument connection. (**b**) Measurement front view. The maximum change in the vortex acoustic signal phase difference is defined as the phase jump \(\Delta\Phi\), calculated as follows:  
\[\Delta\Phi=\max(\Delta\phi_{i}(x))-\min(\Delta\phi_{i}(x)) \tag{4}\]  
Using least squares fitting, the phase difference data obtained from the experiment after reversals of five times are fitted and compared with the phase difference data obtained from numerical simulations, as shown in Figure 4.  
As seen from Figure 4a-c, the fitted phase difference data within the vortex characteristic radius range (\(\pm\)4 cm) align well with the ones obtained by numerical simulation. Figure 4d shows that at time reversal \(N\) = 5, both experimental and numerical simulation results exhibit a characteristic of linear amplification with increased time reversals.