# Numerical Simulation and Design of a High-Temperature, High-Pressure Fluid Transport Pipe
## 2 Methodology and Implementation
### Numerical Method
#### 2.3.2 Gas-Leak Simulation  
As the cylinder, barrel, and projectile are rotating bodies, they are simplified via the application of axial symmetry. A simulation was performed using 2D axisymmetric grids. Figure 6 shows the mesh model used for the simulation. The number of elements was 63.000, and tri, and quad meshes were used. The initial conditions were determined on the basis of the empirical interior ballistics code data (i.e., the chamber pressure, projectile velocity, fluid temperature inside the chamber, and projectile travel time). The chamber pressure was at its maximum at t = 0.174 ms, after propellant ignition. At this time, as the propellant traveled 5.8 mm, the projectile did not pass the cylinder gap. Hence, the maximum pressure point in the chamber was set to the initial condition. At t = 0.174 ms, the chamber pressure was 190 MPa, fluid temperature inside the chamber was 3.200 K, and projectile velocity was 120 m/s. Owing to the short distance between the breech and the projectile base, we assumed that the projectile velocity and fluid velocity inside the chamber were largely the same. Therefore, the fluid velocity inside the chamber was set to 120 m/s in the muzzle direction. Figure 7 shows the initial and boundary conditions. Drag and mechanical friction were neglected when the projectile, weighing 7.46 g, moved in the barrel under the influence of pressure. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distance Traveled by Projectile from Barrel Entrance [mm]** & **Time [ms]** & **Pressure [MPa]** \\ \hline
6 & 0.25 & 137 \\
18 & 0.3 & 100 \\
41 & 0.48 & 63 \\
76 & 0.5 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamic pressure applied inside the barrel. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parts** & **E** & **\(\mathbf{\
u}\)** & **\(\mathbf{\alpha_{T}}\)** & **\(\mathbf{\rho}\)** & **Specific Heat, c** & **k** \\
**[GPa]** & **v** & **[/K]** & **[kg/m\({}_{3}\)]** & **[/kg\(\cdot\)K]** & **[W/m\(\cdot\)K]** \\ \hline Barrel & 210 & 0.3 & \(1.2\times 10^{-5}\) & 7850 & 434 & 60.5 \\ Barrel housing & 71 & 0.3 & \(2.3\times 10^{-5}\) & 2770 & 875 & 170 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Properties for barrel and barrel housing. Figure 6: Mesh model for gas-leak simulation. Figure 7: Schematic of the initial and boundary conditions. The density-based solver was selected for this simulation, because the density-based approach was suitable for high-speed compressible flows [24]. As the Reynolds number (Re) was greater than 2.300, turbulence effects also needed to be considered. The two most typical turbulence models used in CFD simulations were the \(k-\varepsilon\) model and the \(k-\omega\) model. These models differed in terms of their convergence times and the number of iterations [25]. The \(k-\varepsilon\) model was more suitable for fully turbulent flows. Thus, the realizable \(k-\varepsilon\) model was selected as the turbulence model and applied with an enhanced wall treatment provided by the Fluent software. The dynamic-mesh method in Fluent could be used to model flows in which the domain shape varied with time, as a result of the motion on the domain boundaries [26]. The motion of the projectile was thus computed using a user-defined function (UDF), based on Fluent's six degrees-of-freedom (DOF) UDF and dynamic mesh motion. This UDF computed the total force acting on the projectile base and its nose. Then, [PERSON]'s second law (Equation (19)) was used to compute the acceleration of the projectile [27]. The new velocity of the projectile was calculated using a finite difference. \[\overset{\rightarrow}{\mathrm{F}}=\overset{\rightarrow}{\mathrm{m}}\overset{ \rightarrow}{\mathrm{a}}, \tag{19}\]  
where \(\overset{\rightarrow}{\mathrm{F}}\) is the total force (viscous plus pressure), \(\mathrm{m}\) is the body mass, and \(\overset{\rightarrow}{\mathrm{a}}\) is the acceleration of the projectile. The dynamic-mesh method was applied for smoothing and layering. The split factor and collapse factor were both set to 0.4.