## 4 Results and Discussion  
### Flow Characteristics  
For a crack size of 10 mm, corresponding to the effective crack size of 66 mm as listed in Table 1, the velocity field at \(t=300\) s after leakage is shown in Figure 4a. As mentioned, the maximum velocity was predicted to be 237.7 m/s by using DNV PHAST, and it was used for the input boundary condition at the crack position. Rapid expansion occurs after leakage, as discussed above, and the gas phase with high velocity generated by the phase change and rapid volume expansion was considered in the CFD simulation. In Figure 4a, this CO\({}_{2}\) gas flow impinges on the bottom surface and spreads in a radial direction, similar to the impinging jet flow characteristics [25]. Owing to the flow diffusion, a rapid decrease in flow velocity was observed, e.g., the flow velocity decreased by up to 28 m/s approximately at \(y=0.3\) m. The velocity profiles at different locations from the bottom surface are presented in Figure 4b. A substantial change in the flow velocity is shown because of the rapid transport of momentum and energy through convection and diffusion in the free jet zone.  
Figure 3: Grid system and boundary conditions for CO\({}_{2}\) cargo room.  
As mentioned, the present study used the gas leakage model for appropriate inlet conditions at the prescribed crack position. The well-predicted velocity profile becomes very important because it significantly affects the flow behavior near the bottom surface. For validation, the present study conducted additional simulations under the same experimental conditions [26]. We compared the predicted velocity profile and the radius at half-maximum velocity with the experimental results [26], as shown in Figure 5. The prediction of the mean velocity profile shows very good agreement with the experimental measurements, showing that the current gas leakage model is acceptable for determining the inlet conditions.  
After a leak, the decrease of the hydraulic level in the tank leads to a gas pressure increase in the container due to the accumulated mass of leaked CO\({}_{2}\) gas. The present simulation considered the change in the pressure difference between the tank pressure and the environmental pressure, resulting in a decrease in the mass flow rate of the leaked gas. We made a controllable user-defined function embedded in the commercial code of FLUENT to control the mass flow rate varying with time.  
Figure 6 presents the predicted volume-averaged static pressure. The static pressure increased linearly with a gradient proportional to the leakage flow rates, depending on the effective crack area. The estimated pressures at \(t=300\) s were reported to be 1.04 bars, 1.14 bars, and 1.32 bars for crack sizes of 10 mm, 20 mm, and 30 mm, respectively. According to the design guidelines of the transport ship, such ventilation or decompression systems should be operated when the static pressure exceeds 1.2 bars. Thus, the estimated pressure would be useful information to have when assessing the needs of  
Figure 4: (**a**) Velocity field from crack to bottom surface, and (**b**) velocity profile in radial direction with respect to height from bottom in the case of 10 mm crack size at 300 s.  
Figure 5: Validation of (**a**) radial velocity profile and (**b**) jet radius at half-maximum velocity.  
ventilation system operation. Additionally, we considered that the local pressure near the crack would exceed the average pressure value because the CFD simulation was considering non-homogeneous conditions. However, the dynamic pressure of leaking gas on the bottom surface was less than 0.01 bars, indicating that the dynamic pressure effect was negligible, compared to the order of static pressure.