# Experimental and Numerical Study of Orifice Coefficient of Cargo Tank Design of LNG Vessels
## 4 Numerical Simulation
### Simulation Model  
A node-centered finite volume method (CVFEM) based on the Eulerian-Eulerian model was used to simulate the behavior of the fluid flow in the leakage simulation. The fluid flow in the experimental conditions was modeled using a commercial Navier-Stokes VOF-CFD code in the CFX [13]. The CFD code converts the governing differential equation to a set of algebraic equations by discretizing the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen No.** & **Dimensions [mm]** & **Load Case** & **Fluid Weight Per Min [kg]** \\ \hline \multirow{3}{*}{1} & & LC1 & 0.86 \\  & \(0.5\times 7\) & LC2 & 1.62 \\  & & LC3 & 2.40 \\ \hline \multirow{3}{*}{2} & & LC1 & 3.51 \\  & \(1.0\times 7\) & LC2 & 5.87 \\  & & LC3 & 7.02 \\ \hline \multirow{3}{*}{3} & & LC1 & 7.58 \\  & \(1.0\times 14\) & LC2 & 12.49 \\ \cline{1-1}  & & LC3 & 14.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Load case for leakage experiment. fluid domain using CVFEM. The fluid leak simulation is based on the solution of the unsteady fluid flow assuming that the fluid flow is in steady state and mass conservation [14; 15]. Leakage phenomena have been simulated considering the flow of fluid from the desired leakage location. The simulation model was prepared in consideration of the flow rate at the same crack sample location, reflecting the experimental conditions in Figure 4. In the leakage simulation, a steady-state leak rate is assumed as a part of the unsteady pipe flow simulation. Then, the initial steady-state head at the leak location is evaluated in the computational time step, and the leakage coefficient is evaluated. The procedure is carried out to compute the leak rate and the head at the leak location. Before discussing the analogy between the orifice and the leakage, it would be useful to understand how the resulting equation for the leak rate computation can be used to simulate a leak along the pipeline. The leakage must be considered at a computational section. Due to the apparent similarity between a leak from a pipe through a crack and an orifice flow with respect to the static hydro pressure and internal pressure, the continuity equation at the leak location takes the following form expressed in Equation (6) [16]. \[Q_{p}-Q_{Leak}\ =\ 0 \tag{6}\]  
where \(Q_{p}\) is the flow rate in the pipe line, and \(Q_{Leak}\) is the leakage flow rate. The cross-sectional shape of the fluid flow with crack specimen is shown in Figure 9. The fluid domain of the CFD model was set around the cross section of the crack specimen. The geometry and boundary condition of the constructed CFD model are shown in Figure 10. The model was simplified and constructed with different crack areas for each specimen. The chosen mesh was composed of tetrahedral elements. To improve the accuracy of the calculation within the domain across the leak, a finer mesh was adopted in the neighborhood of the rectangular orifice as shown in Figure 11.  
Figure 9: Geometry of leakage specimen section. To define the incompressible fluid flow in the CFD model, the boundary conditions are reflected in the fluid region of Figure 9, as shown in Figure 10. To reflect the pressure conditions inside the pressure tank from the experiment, the inflow part of the fluid was set to the inlet condition, and the pressure of each load condition was reflected. Furthermore, the region after the crack region was set so that the fluid passing through the crack specimen could be freely discharged by reflecting the opening condition to reflect the phenomenon of the fluid being discharged through the crack region. Additionally, it is assumed that the roughness factor of the crack surface is taken into account to prevent free slip from occurring, reflecting the rough wall conditions of the crack. In the CFD model, only areas filled with water were modeled under experimental conditions to prepare the model considering the flow of single-phase flow fluids.