# Simulation Study on Thermal Wake Characteristics of Underwater Vehicle under Rotary Motion
## 2 Mathematical Models and Numerical Simulation Methods  
### Mathematical Model  
In this paper, the scaled-down model of underwater vehicle is taken as the research object, and the rotary motion of underwater vehicle is simulated. To comply with the conservation law of physics, the following equations should be satisfied:  
Equation of continuity:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{\rm i})}{\partial x_{ \rm i}}=0 \tag{1}\]  
Consider the effect of thermal wake lift force, momentum equation:  
\[\rho\frac{\partial(u_{\rm i})}{\partial t}+\rho u_{j}\frac{\partial(u_{\rm i })}{\partial x_{\rm i}}=-\frac{\partial P}{\partial x_{\rm i}}+\frac{\partial} {\sigma x_{\rm i}}\Bigg{[}\mu\left(\frac{\partial u_{\rm i}}{\partial u_{j}} +\frac{\partial u_{j}}{\partial u_{\rm i}}\right)\Bigg{]}-\frac{2}{3}\frac{ \partial}{\partial x_{\rm i}}\Bigg{[}\mu\frac{\partial u_{j}}{\partial x_{j}} \Bigg{]}+(\rho_{0}-\rho)g_{\rm i} \tag{2}\]Energy equation:  
\[\frac{\partial(\rho c_{\rm p}T)}{\partial t}+\frac{\partial(\rho u_{i}c_{\rm p}T)} {\partial x_{i}}=u_{i}\frac{\partial P}{\partial x_{i}}+\frac{\partial}{ \sigma x_{i}}\bigg{(}k\frac{\partial T}{\partial x_{i}}\bigg{)} \tag{3}\]  
Among them, the unit of density \(\rho\) is kg/m\({}^{3}\), the unit of temperature T is K, the unit of viscosity \(\mu\) is Pa-s, the unit of thermal conductivity \(\lambda\) is W/m\(\cdot\)K, the unit of velocity component along the X and Y directions \(\mu_{i},\mu_{j}\) is m/s, the unit of gravity acceleration g\({}_{i}\) is m/s\({}^{2}\), and the unit of specific heat capacity at constant pressure c\({}_{\rm p}\) is J/(kg\(\cdot\)K).  
Due to the influence of disturbance of hull and enclosure, cooling water discharge and boundary conditions, the upwelling and diffusion process of submarine wake is a very complicated turbulent motion. The vortex motion can be better simulated by using the realizable k-\(\epsilon\) turbulence model. The equation of turbulent energy is as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho k\mu_{i})}{\partial x _{i}}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu+\frac{\mu_{i}}{ \sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\bigg{]}+G_{k}+G_{b}-\rho \varepsilon-Y_{m}+S_{k} \tag{4}\]  
The equation of specific dissipation rate is  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho\varepsilon u _{i})}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu+ \frac{\mu_{i}}{\sigma_{\xi}}\bigg{)}\frac{\partial\varepsilon}{\partial x_{j }}\bigg{]}+G_{1\varepsilon}\frac{\varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b}) -C_{2\varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{5}\]  
\[C_{1\varepsilon}=1.44,=1.92,\sigma_{k}=1.0,\sigma_{\varepsilon}=1.3,C_{3 \varepsilon}=0,G_{b}=0,Y_{m}=0,S_{k}=0,S_{\varepsilon}=0 \tag{6}\]  
Among them, G\({}_{\rm k}\) is the turbulent kinetic energy caused by the average velocity gradient and G\({}_{\rm b}\) is the turbulent kinetic energy caused by buoyancy. \(\sigma_{k}\) is the turbulence Pr number of turbulent kinetic energy, and \(\sigma_{\varepsilon}\) is the turbulence Pr number of the dissipation rate. The expression is as follows:  
\[G_{k}=-\rho\overline{u^{\prime}u^{\prime}}_{j}\frac{\partial u_{j}}{\partial x _{i}},G_{b}=\beta g_{i}\frac{u_{i}}{\Pr_{t}}\frac{\partial T}{\partial x_{i} },\ C_{1}=\max\bigg{[}0.43,\frac{1}{\eta+5}\bigg{]},\eta=S\frac{k}{\varepsilon },S=\sqrt{2S_{\rm ij}S_{\rm ij}} \tag{7}\]