## 3 Experimental Test and Validation
### Validation  
The CFD simulation model which we built up was validated by comparing with the experimental data, and applying the Fast Fourier Transform (FFT) analysis method on the collected data in order to identify main frequencies. Figure 5 is the time domain and spectrum of frequencies obtained through the FFT method of the experiment pressure signal, which was taken from the inlet pressure \(P_{in}\), which was shown in Figure 1. The frequencies contain broadband spectrum with a series of peaks, \(f_{e}(1)=2894\,\rm Hz\approx 2.89\) kHz, the sub-harmonic frequency is \(f_{e}(2)=5992\,\rm Hz\approx 5.99\,\rm kHz\), the fundamental frequency is \(f_{e}(3)=1.196\times 10^{4}\,\rm Hz\approx 11.96\) kHz, and the super-harmonic frequency is \(f_{e}(4)=1.788\times 10^{4}\,\rm Hz\approx 17.88\,\rm kHz\). The most important external factor affecting the structures of the waterjet is the hydrodynamic characteristics at the outlet of the nozzle. For this reason, during the simulation, the pressure oscillation characteristic at different locations of the nozzle was monitored. The location of the monitoring points in the simulation were shown in Figure 6, case 1 was any point selected from the domain 1, case 2 was any point selected from the domain 2, and case 3 was any point selected from the domain 3. Among them, case 1 and case 2 were used for monitoring cavitation frequencies, and case 3 was used for monitoring the acoustic natural frequency. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\rm\bf Ds/mm\) & \(\rm\bf D/mm\) & \(\rm\bf L/mm\) & \(\rm\bf d1/mm\) & \(\rm\bf I/mm\) & \(\rm\bf\Theta\) (\({}^{\circ}\)) \\ \hline
23 & 10 & 24 & 2 & 0.7 & 21 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Self-resonating nozzle structure parameters. Additionally, through the FFT method, the simulation frequencies containing broadband spectrums with a series of peaks is shown in Figure 7, \(f_{s}(1)=2934\) Hz \(\approx\) 2.93 kHz, \(f_{s}(2)=4961\) Hz \(\approx\) 4.96 kHz, \(f_{s}(3)=9942\) Hz \(\approx\) 9.94 kHz, \(f_{s}(4)=1.49\times 10^{4}\) Hz \(\approx\) 14.9 kHz, \(f_{s}(5)=5.735\times 10^{4}\) Hz \(\approx\) 57.35 kHz. Theoretically, the value of the acoustic natural frequency of the nozzle is about 14 kHz, and the simulation result \(f_{s}(4)\ =\ 1.49\ \times 10^{4}\) Hz \(\approx\) 14.9 kHz from Figure 7c was the most closest to the acoustic natural frequency. In order to verify the simulation modal, we were attending on the acoustic natural frequency of the nozzle, so that the comparative studies focus on the results of Figures 5 and 7c. Figure 5: Experimental pressure signals and spectrum distributions. Figure 6: The location of the monitoring points in the simulation. Figure 7: Pressures and spectrum distributions at different positions of the nozzle outlet. (a) Case 1. (b) Case 2. (c) Case 3. As we know, the self-resonating water jet has the characteristics of the pulse and cavitation, which is a typical nonstationary, time-varying signal [36]. The cavitation also has the modulating function on the mechanism [37]. The modulation signals model can be expressed as Equation (18) [38], the signals of different positions at the outlet of the nozzle can be considered as a modulation model as Equation (18), the frequencies bands with the experimental and simulation are compared in Table 3, and the results are in a good agreement. \[x(t)=\sum_{j=1}^{n}A_{i}(t)\cos\varphi_{i}(t), \tag{18}\]  
Error evaluation was provided according to different frequency bands of the experimental and simulation results. Define the absolute error \(E_{(a)}(i)\) as Equation (19), where \(f_{s}(i)\) are the simulation frequencies and \(f_{e}(i)\) are experimental frequencies, then we can take the absolute error at different frequencies bands from the results that are shown in Table 3. The absolute error \(E_{(a)}(i)\) ranges from 1.3% (very good) to 17.2% (satisfactory).