# Frequency Modulation and Erosion Performance of a Self-Resonating Jet
## 2 Research Method  
### Experimental Method  
The experiment was conducted with the use of a multifunctional water jet test bench (Figure 2) developed by our research team independently. The flow was supplied from a high-pressure plunger pump with a maximum flow rate of 120 L/min. A pressure transducer (Model: CY100) was installed immediately next to the nozzle inlet, and the inlet pressure of each test could be accurately controlled. The other pressure transducer was positioned on the nozzle to measure the pressure fluctuations in the oscillation chamber. All pressure transducers used in experiments had been calibrated by the manufacturers, and the main uncertainty in this experiment was the accuracy of the pressure transducer, which was less than \(\pm\)0.5% FS (full scale). Each process of pressure acquisition was 60 s and the sampling frequency was set to 100 Hz. Bessel lowpass was used to eliminate the electronic signals caused by the motor with a frequency of 50 Hz. The Hilbert-Huang transform (HHT) combined with empirical mode decomposition (EMD) [28] were adopted in filtering and analyzing the pressure signals, as the oscillations were characterized by being non-linear and non-stationary. The specimens were clamped into a holder, and then mounted on the moving table. The moving table had both X and Y motions with an accuracy of 0.1 mm. In each erosion test, the moving table was moved away from the jet impact region before the pressure was maintained at the setting value. The moving table was then moved to adjust the specimen to be collinear with the axis of the nozzle. During each test, the specimen was exposed to the jet for 90 s. The standoff distance for the nozzle could be varied by sliding the input pipe vertically and clamping it into place at the desired setting. However, a fixed standoff distance of 60 mm was used for all tests. The specimen adopted in the present study had the dimensions of \(100\times 100\times 70\)mm. The compressive strength was measured by a material testing machine (Figure 3). The mean value of the compressive strength was 2.561 MPa within the tolerance scope from \(-4.76\%\) to \(7.53\%\). Figure 3: Material testing machine. Figure 2: Schematic diagram of the experimental setup. The volume removed was measured by filling the generated cavities with fine-grained salt with a density of \(\rho_{\mathrm{s}}=1.452\) g/cm\({}^{3}\). These salts were then removed and weighed with the use of an electronic balance (Figure 4) with resolution to 0.1 mg. The volume removed \(V_{e}\) was then calculated through Equation 1.  
\[V_{e}=\frac{m_{s}}{\rho_{\mathrm{s}}}\,\mathrm{cm}^{3} \tag{1}\]  
where \(m_{s}\) is the salt mass used to fill the erosion cavity. The erosion area, which gad an irregular shape with a slightly serrated border (Figure 5), could be achieved by the method of line tracing in Photo Shop. To be more specific, the erosion area \(A_{e}\) was encircled by creating a spline in Photo Shop and the pixels of erosion \(N_{pe}\) were obtained. The real areas were calculated as follows:  
\[A_{e}=\frac{N_{pe}}{N_{p}}\,\mathrm{cm}^{2} \tag{2}\]  
where \(N_{p}\) is the number of pixels per square centimeter in reality. To reduce the impact of casual factors, all results were averaged for further analysis. The Helmholtz nozzle used in the experiment is shown in Figure 6, and was designed according to the optimal structure with the JMP method by [PERSON] [29]. The upstream nozzle had an inlet diameter of \(d_{in}=13\) mm, a convergent angle of \(\alpha=14^{\circ}\), and an outlet diameter of \(d_{1}=5.9\) mm. The chamber diameter was constant with \(D_{C}=72\) mm and the chamber length, \(L_{C}=36.6\) mm. A downstream nozzle was used with \(d_{2}=10.7\) mm. There were four suction holes of \(d_{s}=2\) mm with a uniform distribution around the circumference of the chamber according to the simulation results by [PERSON] [30]. A fitting was used for connecting the jet bench and the nozzle.