## 3 Results and Discussion
### Prediction of the Erosion Surface Geometry
#### 3.3.4 Prediction and Effect of Geometric Shape  
After determining the maximum depth of soil breaking and the maximum lateral distance, the geometry of the erosion surface was further analysed and predicted. Table 4 shows that, although the slope and jet distance of the erosion surface are different, the range of fluctuation in the depth ratio and the jet distance of ratios of the straight section of the erosion surface are small, and the shape of the erosion surface changes according to similar rules. Therefore, the morphology of the erosion surface under different jet conditions can be normalised.  
The conditions in Table 4 were selected for further tests, i.e., three jet pressures (\(p_{j}=5\), 8, 10 MPa) and six nozzle speeds (\(u_{nz}=1\), 2, 3, 4, 10, 20 cm/s), making a total of 18 working conditions. The geometric contours of the erosion surfaces obtained from each test were normalised. The processing method was as follows: The horizontal and vertical coordinates of each point on the erosion surface were divided by the maximum lateral distance \(x_{\text{max}}\) and the maximum soil-breaking depth \(h_{\text{max}}\). Thus, the erosion surface geometry under each test condition after normalisation was obtained, as shown in Figure 14. In the figure, the abscissa is the normalised transverse length, and the ordinate is the normalised depth. The degree of coincidence between the curves is high. There is a corresponding relationship between the point sets of different erosion surface contours, i.e., the shapes of the erosion surfaces have geometric similarity.  
The starting point A of the straight-line segment of the normalised erosion surface is (0,0), the slope is 1.7-2.57 with an average of 2.14, and the end point of the curved segment is (\(-1\),1). The curved segment is a decreasing function of increases in the second derivative. By comparing the inverse proportional function, logarithmic function, and exponential function, it can be seen that the error of the inverse proportional function is the smallest. The normalised erosion surface formula can be obtained by fitting according to:  
\[\left\{\begin{array}{ll}\frac{\hbar}{h_{\text{max}}}=2.14\frac{x}{x_{\text{ max}}}&,\ \left(\frac{x}{x_{\text{max}}}<\alpha_{x}\right)\\ \\ \frac{\hbar}{h_{\text{max}}}=1.153-0.153\frac{x_{\text{max}}}{x}&,\ \left(\frac{x}{x_{ \text{max}}}\geq\alpha_{x}\right)\end{array}\right. \tag{6}\]  
Figure 13: Relationship between the slope of the straight line and jet pressure and between the slope of the straight line and moving speed: (**a**) the relationship between slope and jet pressure under the premise of fixed nozzle moving speed of 10 cm/s; (**b**) the relationship between slope and nozzle moving speed under the premise of fixed jet pressure of 5, 8 and 10 MPa.  
The fitting correlation coefficient is 0.94, indicating that the normalised erosion cases are similar.  
According to the depth, the normalised shape, and the slope of the erosion surface, an erosion surface profile under any jet pressure and nozzle speed can be calculated.  
\[\left\{\begin{array}{l}h_{\max}=0.0393p_{j}^{2}+57.8/\left(u_{nz}^{0.5}+1.65 \right)-5.76\\ k_{AB}=0.108p_{j}+2.79/u_{nz}+1.17\\ x_{\max}=\alpha_{h}h_{\max}/\alpha_{x}k_{AB}\\ \frac{h_{t}}{h_{\max}}=2.14\frac{x}{x_{\max}}\\ \frac{h_{t}}{h_{\max}}=1.153-0.153\frac{x}{x_{\max}}\\ \end{array}\right.,\frac{x}{x_{\max}}<\alpha_{x} \tag{7}\]  
Taking cases T2-3 (8 MPa, 3 cm/s) and T3-1 (10 MPa, 1 cm/s) as examples, the calculated and experimental results are as shown in Figure 15. The maximum error is \(<\)11%, which is acceptable for engineering applications.