## 3 Results and Discussion
### Turbulence Statistics
#### 3.2.2 Turbulent Kinetic Energy (k)  
As can be seen from Figure 13, the turbulent kinetic energy of the hydrothermal plume presents a mushroom-like spatial distribution. Its value reaches its maximum in the plume stem area near the vent orifice and decreases sharply along the radial direction. The maximum turbulent kinetic energy calculated by rke_case2 and sst_case2 are 0.069 J kg\({}^{-1}\) and 0.083 J kg\({}^{-1}\). According to the computational relationship between turbulent kinetic energy and velocity, there is a positive correlation between the velocity field and turbulent kinetic energy. When the hydrothermal plume diffuses laterally in the neutrally buoyant plume height, there are strong anisotropic flows such as vortices. The velocity contour line in Figure 13 indicates that the rke model is sensitive to velocity variation in the far field region, while the sst model is not suitable for free shear flow in the late plume period because of its large magnitude variation in velocity prediction. Compared with the previous turbulent kinetic energy nebulae distribution, the prediction error of the sst model on the transverse diffusion of the plume neutral buoyancy layer is larger [12; 14; 15].  
The changes in turbulent kinetic energy of nozzle center line with plume height under different working conditions are shown in Figure 14a. Since there is no measurement data of turbulent kinetic energy, the conclusion of [PERSON] et al. was taken as reference [12]. The  
Figure 12: Turbulent viscosity of the center line of the quasi-steady hydrothermal plume ([PERSON] et al. 2014 [14]). (**a**) The normalized turbulent viscosity and the rise height (the black scatter represents [PERSON]’s calculation); (**b**) The relationship between maximum turbulent viscosity and buoyancy flux and buoyancy frequency (the black trend line represents the results of [PERSON] et al.). Dotted lines of the same color are obtained by fitting corresponding scatter points.  
Figure 13: Turbulent kinetic energy distribution of the quasi-steady hydrothermal plume.  
curves in the area near the vent orifice show that the difference between turbulent kinetic energy calculated by the rke model and the sst model is small, and the change trends of the two models are similar.  
Similarly, it is determined that there is a functional relationship among turbulent kinetic energy, buoyancy frequency and buoyancy flux by dimensional analysis. Figure 14b shows the linear fitting of maximum turbulent kinetic energy of the vent orifice center line with the buoyancy flux and buoyancy frequency. The turbulent kinetic energy scaling law of the rke model and the sst model is as follows:  
\[\left\{\begin{array}{l}k_{\max,rke}=22.646\ B_{ext}^{1/2}\cdot N^{1/2}\\ k_{\max,sst}=24.291\ B_{exit}^{1/2}\cdot N^{1/2}\end{array}\right.. \tag{15}\]  
From the numerical analysis of maximum turbulent kinetic energy, under the same working conditions, the maximum turbulent kinetic energy calculated by the sst model is slightly greater than that calculated by the rke model, but the difference is not significant. Except for the difference in turbulent kinetic energy at transverse diffusion, the numerical values of turbulent kinetic energy predicted by the rke model and the sst model are similar to the curve change trend, indicating that these two models have similar effects on the prediction of turbulent kinetic energy of hydrothermal plume.