## 3 Results
### Distribution of Gas Holdup  
Gas holdup is one of the basic parameters characterizing the flow characteristics of bubble plumes, which affects the aeration efficiency by affecting the contact area of two phases, gas and liquid. The gas holdup is affected by conditions such as the inlet flow rate, structural parameters, liquid phase properties and bubble size. [PERSON] et al. (2019) used experiments to study the variation law of gas-liquid hydraulic properties in non-Newtonian fluids. The results show that the gas holdup in non-Newtonian fluids is lower than that in Newtonian fluids. [PERSON] et al. (2019) thought that the change in gas holdup is affected by the distribution of bubble size, and it is helpful to increase the gas holdup by  
Figure 4: (**a**,**c**) are the turbulent kinetic energy at the central axis and exit, respectively; (**b**,**d**) show the distribution of turbulent kinetic energy data at the central axis and exit, respectively.  
Figure 3: Turbulent kinetic energy (TKE) clouds under the condition of four different nozzle diameters.  
the reasonable selection and arrangement of aeration equipment to stabilize the bubble size distribution. [PERSON] et al. [37] increased the gas holdup in the container by 120% by changing the nozzle. When the cone angle is 115 degand the nozzle diameter is 2 mm, the inlet pressure has a great influence on the gas holdup in the flow field (Figure 5a). The area with gas holdup less than 100% increases first and then decreases with the increase in inlet pressure, indicating that there is a great difference in the ability of jet entrainment. The larger the area with gas content less than 100%, the more favorable the coverage of the jet. The gas holdup variation along the central axis exhibits good similarity (Figure 5b). The gas holdup inside the jet is essentially consistent under different pressures during the initial stage of the jet. The pressure differences begin to manifest in the characteristics of gas holdup near the outlet. Specifically, the gas holdup in the jet is significantly higher than the other three sets of operating conditions when the pressure is 0.18 MPa. To further analyze the mass transfer and mixing effects of the gas-liquid two-phase flow, Figure 5c shows the variation in gas holdup at the outlet of jet field. The gas holdup generally exhibits a bimodal structure. The gas holdup is significantly lower than inside the jet at the jet-air boundary, indicating a strong entrainment effect between the gas and liquid phases within the jet. The gas holdup changes most dramatically at the gas-liquid interface when the pressure is 0.45 MPa. However, the extent of air entrainment does not show significant variations with increasing pressure, suggesting a limited lateral influence range of the jet. If the gas holdup is too high, the lateral spreading of the jet is more severe, leading to a higher degree of droplet dispersion and reducing the spraying effectiveness. The entrainment effect between the jet and air is weak when the gas holdup is too low, which impacts the spraying effectiveness. Therefore, the best spraying effectiveness is achieved at pressures of 0.27 MPa and 0.36 MPa when the cone angle is 115 degand the nozzle diameter is 2 mm.