## 3 Experimental Test and Validation  
### Experimental Setup  
The experimental setup was shown in Figure 3, which mainly consists of a water tank, a plunger pump, which supplies a maximum operating pressure of 60 MPa and a maximum flow rate of 60 L/min, a flowmeter, a pressure gauge, a relief valve, a high-pressure tank, a nozzle, and a hit plate were installed inside the high-pressure tank. The Piezotronics ICP model 102B03 pressure sensor, with a resonant frequency of 500 kHz, has a non-linearity of less than 1%FS. In our experiment, we used two sensors, one was arranged on the pipeline outside the high-pressure tank to record the inlet pressure \(P_{in}\), while the other was placed at the drainage of the high-pressure tank to record the confining pressure \(P_{a}\). An RHS-20 hydrophone with the flat response 120 kHz and \(\pm\)3 dB, with oscillations up to 200 kHz, as well as a LMS SCADAS multi-functional data acquisition system.  
In the experiment, the temperature of water is about 10 \({}^{\circ}\)C, the mean value of inlet pressure was \(\overline{P_{in}}=15.51\) MPa, the mean value of measured confining pressure was \(\overline{P_{a}}=1.503\) MPa, the mean value of measured flow rate of the pump was \(\overline{Q}=19.98\) L/min, the distance between the hit plate and the nozzle was 6 mm, and the sampling frequency was \(f_{a}=204,800\) Hz.  
Self-resonating nozzle geometry schematic description is shown in Figure 4, the structure parameters are shown in Table 2. The nozzle consists of three parts, an upstream domain contraction (_D_s/_D_), a downstream domain contraction (_D_/_d_1), and a resonant chamber with a length of L with a diameter of \(D\).  
Figure 4: Self-resonating nozzle geometry schematic description (cut view).  
Figure 3: Schematic diagram of the test system.  
For the sudden changes of flow velocity at the downstream domain contraction, pressure waves will be formed when high-speed flow passes through the nozzle. As a result, fluid resonance formed at the upstream and downstream contractions. According to the fluid resonance, the peak resonance will be achieved when the self-excited frequency of the water jet \(f_{\rm s}\) was matched with the acoustic natural frequency \(f_{n}\) of the nozzle [13]. The fundamental frequency of the water jet structuring \(f_{\rm s}\) corresponds with the Strouhal number \(S_{d}\) expressed as (13), and the average flow is defined as (14). The acoustic natural frequency of the chamber \(f_{n}\) is determined by the length of the chamber \(L\), which is expressed as (15). When (\(D_{S}\)/D) > 1, the \"mode parameter\" \(K_{N}\) is given by (16).  
\[f_{\rm s}=S_{d}\frac{V_{out}}{d}, \tag{13}\]  
\[\rm Q= \frac{\pi d_{1}{}^{2}}{4}V_{out}, \tag{14}\]  
\[f_{n}=K_{N}\frac{c}{L}, \tag{15}\]  
\[K_{N}=\frac{2N-1}{4}\quad\rm N=1,2,3.\ldots\ldots, \tag{16}\]  
\[f_{n}=f_{s}, \tag{17}\]  
where \(d1\) is the nozzle orifice diameter, \(V_{out}\) is the jet velocity, \(\rm m/s\), \(\rm N\) is mode number of the organ-pipe chamber of the nozzle, and the sound wave velocity \(c\) = 1450 \(\rm m/s\) is given by the data from [34,35]. Joint Formulas (13)-(17), combined parameters in Table 2 and experiment parameters, and the acoustic natural frequency of the nozzle is about 14 kHz.