## 3 Results and Discussion
### Quantitative Analysis  
For quantitative analysis, the present study introduces three key parameters: The streamwise propagation velocity (\(\overline{v}_{x}\)), transverse propagation velocity (\(\overline{v}_{y}\)), and averaged HNS mass fraction (\(w_{s-l}\)). First, \(\overline{v}_{x}\) and \(\overline{v}_{y}\) are the time-averaged propagation velocities at the HNS interface in the \(x\) direction and the y direction, respectively. Figure 3: HNS propagation HNS behavior obtained by (**a**) different current velocity with \(\rho_{lms}=800\) kg/m\({}^{3}\) and (**b**) different HNS density with \(v_{c}=0.514\) m/s, according to crack location at 1000 s.  
\[\overline{v}_{x}=\frac{1}{\text{T}}\int_{0}^{\text{T}}v_{x}(t)dt,\ \overline{v}_{y}=\frac{1}{\text{T}}\int_{0}^{\text{T}}v_{y}(t)dt \tag{15}\]  
where T is the total time at which HNS interface reaches the outlet region in the computational domain. We also suggest a new parameter, the area-averaged mass fraction of the HNS at sea level (\(z=0\)), given by:  
\[w_{s-l}=\left(\frac{1}{A_{lms}}\int\int w(x,y,z)dA\right)_{at\,z=0} \tag{16}\]  
where \(w\) and \(A_{lms}\) represent the local mass fraction of the HNS and the area covered by the HNS at sea level. The parameters defined above would be useful for understanding the HNS propagation behavior in a quantitative manner. As far as we know, this is the first effort for a quantitative analysis of HNS propagation. Figure 4a shows the behavior of the HNS interface with respect to the current velocity for 200 s. The convective effect becomes stronger as the current velocity increases. For instance, at \(v_{c}=1.542\) m/s, the local interface velocity may be approximately 1.6 m/s, which is slightly faster than the current velocity because of the flow change induced by the shape of the transport ship, as well as the buoyancy effect. Figure 4b,\(c\) provides the estimated propagation distance (\(D_{p}\)) of the HNS interface in the streamwise and transverse directions according to the current velocity, respectively. It is seen that the streamwise propagation substantially increases, while the propagation in the transverse direction rather decreases with the increase in the current velocity because the total flow momentum is redistributed in both directions. Figure 4: _Cont._  
Figure 5a represents the effects of the current velocity on \(\overline{v}_{x}\) for different HNS densities. The results show that \(\overline{v}_{x}\) is proportional to the current velocity, and no difference can be seen according to the HNS density. This is also seen in Figure 3b. The reason for this is because the advection effect becomes more dominant in the streamwise direction compared to the buoyancy effect. The maximum propagation distance is shown in Figure 5b. From the results, there is no difference in \(\overline{v}_{x}\) between the cases of the side crack and bottom crack. This indicates that the crack position is no longer involved, and its effect is negligible. Figure 6a shows the effects of the current velocity on \(\overline{v}_{y}\) for different HNS densities. As the current velocity increases, \(\overline{v}_{y}\) decreases, because the advection effect becomes dominant with high current velocity. Similar to Figure 4c, the propagation in the transverse direction decreases owing to the increase in HNS density. Meanwhile, Figure 6b shows the maximum propagation distance according to the crack position. For the side crack, the averaged transverse propagation distance was estimated to be approximately 30 m larger than that for the bottom crack. This deviation could be made by the  
Figure 4: (**a**) Behavior of the interface at 50–200 s for the different current velocity with \(\rho_{\text{hrs}}\) = 800 kg/m\({}^{3}\), and max propagation distance over time in the direction of (**b**) streamwise and (**c**) transverse with \(\rho_{\text{hrs}}\) = 800 kg/m\({}^{3}\).