## 3 Results and Discussion
### Feature Height and Entrainment Coefficient  
#### 3.3.1 Feature Height (\(\mathrm{Z_{max}}\), \(\mathrm{Z_{neutral}}\))  
Based on the vertical velocity and inner and outer density difference in the hydrothermal plume, the maximum plume rising height of the plume and the neutrally buoyant plume height are obtained (Table 6). The plume neutral buoyancy zone inferred from the detection data indicates that the neutrally buoyant plume heights calculated by rke_case2 and sst_case2 are consistent with the observation results of this scientific investigation. The rke model and the sst model have a height ratio \(\mathrm{Z_{neutral}}/\mathrm{Z_{max}}\) of 0.72-0.74 in all working conditions, with an average value of 0.7304. These are close to the empirical value of 0.761 of the classical plume theoretical model [9]. The results show that RANS numerical simulation method can effectively reproduce the macroscopic dynamic process of hydrothermal plume in the Longqi hydrothermal field. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Simulation**} & \multicolumn{1}{c}{\(\mathbf{B_{exit}}\)} & \multicolumn{1}{c}{\(\mathbf{B_{asymp}}\)} & \multicolumn{1}{c}{\(\mathbf{Z_{neutral}}\)} & \multicolumn{1}{c}{\(\mathbf{Z_{max}}\)} & \multicolumn{1}{c}{\(\mathbf{Z_{max,MTT}}\)} & \multicolumn{1}{c}{\(\mathbf{\gamma_{\mathrm{\
u,max}}}\)} & \multicolumn{1}{c}{\(\mathbf{k_{max}}\)} & \multicolumn{1}{c}{\(\mathbf{\epsilon_{max}}\)} \\ \multicolumn{1}{c}{**Run**} & \multicolumn{1}{c}{**(m\({}^{4}\) s\({}^{-3}\))**} & \multicolumn{1}{c}{**(m\({}^{4}\) s\({}^{-3}\))**} & \multicolumn{1}{c}{**(m)**} & \multicolumn{1}{c}{**(m)**} & \multicolumn{1}{c}{**(m)**} & \multicolumn{1}{c}{**(m\({}^{2}\) s\({}^{-1}\))**} & \multicolumn{1}{c}{**(m\({}^{2}\) s\({}^{-2}\))**} & \multicolumn{1}{c}{**or \(\mathbf{\omega}\) (s\({}^{-1}\))**} \\ \hline \multirow{5}{*}{rke} & case1 & 0.008183 & 0.000871 & 215.931 & 298.798 & 228.012 & 0.088830 & 0.045862 & 0.098373 \\  & case2 \({}^{*}\) & 0.016365 & 0.001742 & 237.444 & 324.296 & 271.154 & 0.096543 & 0.064709 & 0.182601 \\  & case3 & 0.032729 & 0.003484 & 285.216 & 386.447 & 322.458 & 0.098900 & 0.101572 & 0.292038 \\  & case4 & 0.016365 & 0.001742 & 154.578 & 213.540 & 169.320 & 0.093237 & 0.073890 & 0.213846 \\  & case5 & 0.016365 & 0.001742 & 121.113 & 164.538 & 140.079 & 0.083291 & 0.064936 & 0.183805 \\  & case6 & 0.008248 & 0.001200 & 216.910 & 299.595 & 247.052 & 0.091085 & 0.045349 & 0.088810 \\  & case7 & 0.010119 & 0.001442 & 227.484 & 315.133 & 258.649 & 0.095439 & 0.053072 & 0.120054 \\ \hline \multirow{5}{*}{sst} & case1 & 0.008182 & 0.000871 & 204.377 & 280.073 & 228.012 & 0.069245 & 0.048405 & 29.112717 \\  & case2 \({}^{*}\) & 0.016365 & 0.001742 & 237.045 & 320.710 & 271.154 & 0.090891 & 0.075377 & 38.509010 \\  & case3 & 0.032729 & 0.003484 & 288.355 & 391.499 & 322.458 & 0.100053 & 0.104194 & 47.898373 \\  & case4 & 0.016365 & 0.001742 & 149.798 & 209.158 & 169.320 & 0.067635 & 0.075532 & 38.452026 \\  & case5 & 0.016365 & 0.001742 & 124.301 & 170.514 & 140.079 & 0.060755 & 0.075374 & 38.525658 \\  & case6 & 0.008248 & 0.001200 & 231.468 & 313.539 & 247.052 & 0.075987 & 0.046407 & 29.675610 \\  & case7 & 0.010119 & 0.001442 & 240.631 & 326.288 & 258.649 & 0.082737 & 0.055771 & 32.198055 \\ \hline \hline \end{tabular} \({}^{*}\) Reference environment group.  
\end{table} 
Table 6: Hydrothermal plume simulation results of rke model and sst model. Figure 15: Turbulent dissipation rate distribution of the quasi-steady hydrothermal plume. As there is a certain gap between the numerical simulation and the actual environment buoyancy flux, the calculation method of the initial buoyancy flux is modified [37]. Buoyancy flux \(B_{0}\) can be calculated in two ways: the initial buoyancy flux \(B_{exit}\) and the corrected progressive buoyancy flux \(B_{asymp}\).