## 3 Results and Discussion
### Vertical Jet Energy Dissipation Rate  
In this study, vertical jet energy dissipation refers to the energy loss of the jet in the air after the jet is ejected from the vertical pipeline. Therefore, the energy dissipation rate \(\eta\) is calculated using the energy difference between section 1# and section 2# (see Figure 3) and the energy ratio of section 1#. The formula is as follows:  
\[\eta=\frac{E_{1}-E_{2}}{E_{1}}\times 100\% \tag{7}\]  
Figure 10: Reynolds stress along the jet axis: (**a**) the Reynolds stress \(\tau\)-UV at three flows, (**b**) the Reynolds stress \(\tau\)-UW at three flows, and (**c**) the Reynolds stress \(\tau\)-VW at three flows. The Reynolds stresses are nondimensionalized by the wall shear stresses. \[E_{1}=z_{1}+\frac{p_{1}}{\rho g}+\frac{\alpha v_{1}^{2}}{2g} \tag{8}\]  
\[E_{2}=z_{2}+\frac{p_{2}}{\rho g}+\frac{\alpha v_{2}^{2}}{2g} \tag{9}\]  
where \(\eta_{1}\) is the vertical jet energy dissipation rate; \(E_{1}\) is the total energy in the vertical jet pipeline of section 1\(\#\); \(E_{2}\) is the total energy of section 2\(\#\); \(z_{1}\) and \(z_{2}\) are the position heads, that is, the height from the datum plane; \(p_{1}\) and \(p_{2}\) are the pressures at the two sections; \(v_{1}\) and \(v_{2}\) are the average flow velocities at the two sections; and \(\alpha\) is the flow velocity coefficients, and generally takes 1. In this study, the ratio r of the vertical jet energy dissipation rate under the influence of the crushing needle to the vertical jet energy dissipation rate without the influence of the crushing needle is defined as the vertical jet energy dissipation rate increase ratio. Figure 11 is the vertical jet energy dissipation rate under different working conditions. Figure 12 is the vertical jet energy dissipation rate increase ratio under different working conditions. It can be seen from the figure that  
The vertical jet energy dissipation rate is between 14.55% and 18.33% without the influence of the crushing needle. When there are crushing needles, the range of the vertical  
Figure 11: Vertical jet energy dissipation rate of each working condition: (**a**) m = 0.1; (**b**) m = 0.15; (**c**) m = 0.2, \(\eta\) is the vertical jet energy dissipation rate. Figure 12: The increase ratio of the vertical jet energy dissipation rate of each working condition: (**a**) m = 0.1; (**b**) m = 0.15; (**c**) m = 0.2. r is the increase ratio of the vertical jet energy dissipation rate. jet energy dissipation rate is between 15.18 % and 78.54%, and the increase ratio of the vertical jet energy dissipation rate is between 1.04 and 4.29, which is greatly affected by the number, size, and Reynolds number of crushing needles. When the design of the crushing needle is the same, the vertical jet energy dissipation rate is basically positively correlated with the Reynolds number; that is, the vertical jet energy dissipation rate increases with the increase in Reynolds number. This is because as the Reynolds number increases, the vertical range of the jet increases, the contact surface between the jet and the air increases, and the jet aeration concentration increases, thereby increasing the vertical jet energy dissipation rate. When the Reynolds number and the number of crushing needles are the same, the vertical jet energy dissipation rate increases with the increase in the size of the crushing needle. When the size of the crushing needle m increases from 0.1 to 0.15, the vertical jet energy dissipation rate increases from 3.4% to 19.7%, and the vertical jet energy dissipation rate increases from 0.13 to 0.86. The size of the crushing needle m increases from 0.15 to 0.2, the vertical jet energy dissipation rate increases from 0.5% to 27.35%, and the vertical jet energy dissipation rate increases from 0.73 to 1.67.