# Analysis of the Propulsion Performance and Internal Flow Field of an Underwater Launcher
## 3 Numerical Analysis
### The Modeling and Verification
#### 3.1.2 Mathematical Model
Motion equation: \[\frac{dl}{dt}=v\] (7)The base pressure and the head pressure calculated by the above jet equations are fed back to the User-Defined Functions (UDF), which are adopted to calculate the projectile velocity, displacement, and pressure. These quantities allow us to realize the coupling of the jet equations and the interior ballistics equations. The continuity, momentum, energy conservation, state, and turbulence equations for unsteady combustion-gas jet flows are as follows. (8) Mass conservation equations:  
\[ \begin{array}{l}\frac{\partial\left(\alpha_{g}\rho_{g}\right)}{\partial t}+ \
abla\cdot\left(\alpha_{g}\rho_{g}\
u\right)=0\\ \frac{\partial\left(\alpha_{l}\rho_{l}\right)}{\partial t}+\
abla\cdot\left( \alpha_{i}\rho_{l}\
u\right)=0\end{array}  \tag{8}\]  
These mass conservation equations are for each phase. Here, \(\rho_{g}\) and \(\rho_{l}\) denote the density of the gas and liquid, respectively. The volume fractions of each phase satisfy \(\alpha_{g}+\alpha_{l}=1\), where \(\alpha_{g}\) is the volume fraction of the gas phase and \(\alpha_{l}\) is the volume fraction of the liquid phase. \(\
u\) is the velocity vector. (9) Mixture momentum conservation equation:  
\[\frac{\partial}{\partial t}(\rho\
u)+\
abla\cdot\left(\rho\
u\
u\right)=- \
abla p+\
abla\cdot\left[\mu\left(\
abla\
u+\
abla\
u^{T}\right)\right] \tag{9}\]  
The momentum conservation equation is for the two-phase mixture. The density and viscosity of this two-phase mixture are given by \(\rho=\alpha_{g}\rho_{g}+\alpha_{l}\rho_{l}\) and \(\mu=\alpha_{g}\mu_{g}+\alpha_{l}\mu_{l}\). \(p\) is the fluid pressure in the flow field. (10) Mixture energy conservation equation:  
\[\frac{\partial}{\partial t}(\rho E)+\
abla\cdot\left[\
u(\rho E+p)\right]= \
abla\cdot\left(k_{\text{eff}}\
abla T\right) \tag{10}\]  
The energy conservation equation is for the two-phase mixture. The volume-of-fluid model treats energy \(E\) and temperature \(T\) as mass-averaged variables: \(E=\left(\alpha_{g}\rho_{g}E_{g}+\alpha_{l}\rho_{l}E_{l}\right)/\left(\alpha_{ g}\rho_{g}+\alpha_{l}\rho_{l}\right)\), \(T=\left(\alpha_{g}\rho_{g}T_{g}+\alpha_{l}\rho_{l}\right)/\left(\alpha_{g} \rho_{g}+\alpha_{l}\rho_{l}\right)\). \(k_{\text{eff}}\) is the effective thermal conductivity. (11) Turbulence equation:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]-\rho\overline{u_{i}^{\prime }u_{j}^{\prime}}\frac{\partial u_{j}}{\partial x_{i}}-\rho\epsilon \tag{11}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}(\rho \epsilon u_{i})=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}} {\sigma_{\epsilon}}\right)\frac{\partial\epsilon}{\partial x_{j}}\right]-C_{ \epsilon 2}\rho\frac{\epsilon^{2}}{k}+C_{\epsilon}\frac{\epsilon}{k}\rho\overline{u _{i}^{\prime}u_{j}^{\prime}}\frac{\partial u_{j}}{\partial x_{i}} \tag{12}\]  
where \(k\) and \(\epsilon\) are the kinetic energy and dissipation rate, respectively. The constants \(\sigma_{k}=1.0\) and \(\sigma_{\epsilon}=1.3\) are the Prandtl numbers corresponding to the turbulent kinetic energy and the dissipation rate, respectively. \(\mu_{t}=C_{\mu}k^{2}/\epsilon\) is the turbulent viscosity. The constants \(C_{\epsilon l}=1.44\), \(C_{\epsilon 2}=1.92\), and \(C_{\mu}=0.08\) are empirical coefficients.