# 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  
1. Form function of the propellant: \[\psi= \begin{cases}\chi_{0}Z(1+\lambda_{0}Z+\mu_{0}Z^{2}),&0\leq Z\leq 1\\ \chi_{\rm s}Z(1+\lambda_{\rm s}Z),&1<Z<Z_{\rm k}\\ 1,&Z=Z_{\rm k}\end{cases} \] (1) \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Projectile** & **Barrel Length** & **Projectile Weight** & **Charge Weight** & **Maximum Pressure** & **Muzzle Velocity** \\  & **Type** & **\(x\)/m** & **m/g** & **\(\omega\)/g** & **\(p_{m}\)/MPa** & **\(v_{0}\)/m\(\cdot\)s\({}^{-1}\)** \\ \hline
1 & A & 1.028 & 45 & 8.5 & 53.4 & 159.8 \\
2 & B & 1.028 & 45 & 8.5 & 41.8 & 177.5 \\
3 & C & 1.028 & 45 & 8.5 & 70.9 & 150.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Interior ballistics results. Figure 3: Chamber pressure curve under three launching conditions. where \(\psi\) is the mass fraction of burnt propellant, \(Z\) is the relative thickness of burnt propellant, and \(\chi_{0}\), \(\lambda_{0}\), \(\mu_{0}\), \(\chi_{s}\), and \(\lambda_{s}\) are form characteristic quantities of the propellant. 2. Burning law of the propellant: \[\frac{dZ}{dt}=\frac{u_{1}p^{n}}{e_{1}}\] (2) where \(e_{1}\) is half of the web thickness of the propellant, \(u_{1}\) is the burning coefficient, \(n\) is the burning exponent, and \(p\) is the mean pressure in the chamber. 3. Momentum equation of projectile: \[\int_{0}^{A}\left(p_{\rm b}-p_{\rm h}\right)\mathrm{d}A=\varphi m\frac{dv}{dt}\] (3) where \(A\) is the sectional area of the projectile, \(p_{\rm d}\) is the pressure at the projectile base, \(p_{\rm h}\) is the pressure at the projectile head, \(m\) is the mass of the projectile, \(v\) is the velocity of the projectile, and \(\varphi\) is the minor work coefficient. 4. Orifice flow equation: \[\dot{m}=\varphi_{2}S_{\rm n}p_{\rm b}K_{0}\] (4) where \[K_{0}=\left\{ \begin{array}{c}\sqrt{\frac{2k_{\rm p}}{\left(k_{\rm p}+1 \right)f_{\rm p}}\left[\left(\frac{p_{\rm h}}{p_{\rm b}}\right)^{\frac{2}{k_{ \rm p}}}-\left(\frac{p_{\rm h}}{p_{\rm b}}\right)^{\frac{kp+1}{k_{\rm p}}} \right]}},\frac{p_{\rm h}}{p_{\rm b}}>\left(\frac{2}{k_{\rm p}+1}\right)^{ \frac{kp}{k_{\rm p}-1}}\\ \sqrt{\frac{k_{\rm p}}{f_{\rm p}}\left(\frac{2}{k_{\rm p}+1}\right)^{\frac{kp+1 }{k_{\rm p}-1}}},\frac{p_{\rm h}}{p_{\rm b}}\leq\left(\frac{2}{k_{\rm p}+1} \right)^{\frac{kp}{k_{\rm p}-1}}\end{array} \right.\] \(p_{\rm b}=p/\left(1+\frac{\varphi}{3p_{\rm b}\dot{m}}\right)\) is the pressure on the projectile base, \(\varphi_{2}\) is the flow correction factor, \(S_{\rm n}\) is the cross-sectional area of the nozzle,\(f_{\rm p}\) is the impetus, and \(k_{\rm p}\) is the adiabatic exponent. \(\rho_{i}\) and \(v_{i}\) are the density and velocity of the combustion gas passing through the nozzle entrance, respectively. 5. Energy equation of interior ballistics: \[\frac{Ap(l_{\varphi}+l)}{\theta}=\frac{f\omega\psi}{\theta}-[\frac{\rho}{2}mv ^{2}+C_{p}\int_{0}^{t}\dot{m}Tdt+\int_{0}^{x}\int_{0}^{A}p_{\rm h}dA\mathrm{d}l]\] (5) where \(\omega\) is the charge weight, \(\theta=k_{0}-1\), \(l\) is the projectile displacement, \(v_{g}\) is the average velocity of the propellant gas flowing through the center nozzle inside the projectile, \(C_{p}\) is the constant-pressure specific heat capacity, and \(T\) is the gas temperature. \(l_{\varphi}\) is the diameter-shrunk length of the free volume and is given by \[l_{\varphi}=l_{0}[1-\frac{\Delta}{\rho_{\rm p}}(1-\psi)-\alpha_{\rm p}\Delta( \psi-\frac{\int_{0}^{t}\dot{m}dt}{\omega})]\] where \(l_{0}\) is the diameter-shrunk length of the chamber, \(\Delta\) is the charge density, \(\rho_{\rm p}\) is the solid propellant density, and \(\alpha_{\rm p}\) is the covolume. 6. State equation of the gas inside the chamber: \[Ap\big{(}l+l_{\varphi}\big{)}=R\bigg{(}\omega\psi T-\int_{0}^{t}\dot{m}T \mathrm{d}t\bigg{)}\] (6) where \(R\) is the gas constant and \(T\) is the temperature. 7.