## 2 Materials and Methods  
### Turbulence Model  
The unsteady Reynolds-averaged Navier-Stokes equations are used to describe fluid motion in a two-dimensional circular hole jet with a single nozzle. The conservation equation is expressed in the form of Cartesian tensor as follows.  
Continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}=0. \tag{1}\]  
Momentum equation:  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial\big{(}\rho u_{i}u_{j} \big{)}}{\partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial \tau_{ij}}{\partial x_{j}}+\frac{\partial}{\partial x_{j}}\Big{(}-\rho\overline {u_{i}^{\prime}u_{j}^{\prime}}\Big{)}+\rho g_{i}. \tag{2}\]  
Energy equation:  
\[\frac{\partial(\rho E)}{\partial t}+\frac{\partial\big{[}u_{j}(\rho E+p) \big{]}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{(}\kappa_{eff} \frac{\partial T}{\partial x_{j}}\Bigg{)}+\frac{\partial}{\partial x_{j}} \Bigg{[}u_{i}\mu_{eff}\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{l}}{ \partial x_{l}}\Bigg{)}\Bigg{]}, \tag{3}\]  
where strain rate tensor \(\tau_{ij}\) can be expressed as  
\[\tau_{ij}=\mu\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{ j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{l}}{\partial x_{l}} \Bigg{)}. \tag{4}\]  
Through the Boussinesq hypothesis, Reynolds stress of Equation (2) can be expressed as  
\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\Bigg{(}\frac{\partial u_ {i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}-\frac{2}{ 3}\delta_{ij}\left(\rho k+\mu_{t}\frac{\partial u_{l}}{\partial x_{l}}\right). \tag{5}\]  
The turbulence model theory is used to close the above equations. At present, the k-\(\varepsilon\) model and the k-\(\omega\) model are used to calculate plume. In this paper, the Realizable k-\(\varepsilon\) (rke) model and the SST k-\(\omega\) (sst) model are selected for numerical simulation of dynamic characteristics of hydrothermal plume. The explanation of relevant physical quantities of the numerical model of hydrothermal plume in this paper is shown in Table 1.  
The turbulent kinetic energy transport equation and turbulent dissipation rate transport equation of the rke model are as follows, described in detail in Appendix A[20]:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial\big{(}\rho ku_{j}\big{)}}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_ {t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+G_{k}+G_{b} -\rho\varepsilon, \tag{6}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial\big{(}\rho \varepsilon u_{j}\big{)}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}} \Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{ \partial\varepsilon}{\partial x_{j}}\Bigg{]}+\rho C_{1}S\varepsilon-\frac{ \rho C_{2}\varepsilon^{2}}{k+\sqrt{\varepsilon v}}+C_{1\varepsilon}\frac{ \varepsilon}{K}C_{3\varepsilon}G_{b}. \tag{7}\]Similarly, the sst model can be written as [21]  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial\big{(}\rho ku_{j} \big{)}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+ \frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+P_ {k}-\beta^{*}\rho k\omega, \tag{8}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial\big{(} \rho\omega u_{j}\big{)}}{\partial x_{j}}=\frac{\partial}{\partial x_{j}} \Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\omega}}\bigg{)}\frac{\partial \omega}{\partial x_{j}}\Bigg{]}+P_{\omega}-\beta\rho\omega^{2}+2(1-F_{1}) \frac{\rho\sigma_{\omega 2}}{\omega}\frac{\partial k}{\partial x_{j}}\frac{ \partial\omega}{\partial x_{j}}. \tag{9}\]