## 3 Results and Discussion  
### HNS Propagation Characteristics  
Figure 4 shows the distributions of HNS mass fractions at \(t\) = 1000 s after leakage. After 1000 s, the flow field reached the steady state, and the shape of the HNS layer was almost constant over time. Figure 3: The comparison of the volume flow rates of the previous numerical and experimental results [20]. Figure 2: The computational domain that replaces the experimental facility [20]. The mass fraction of HNS in the wake region was measured in the range from 0.02 to 0.1. The dissolved HNS floated near the sea surface because it had a lower density than seawater. Here, the HNS layer formed behind the wake was mainly analyzed. The maximum mass fraction appeared at sea level, and it showed decreasing patterns in the depth direction. For quantitative analysis, three main parameters, namely, the HNS layer thickness (\(\delta_{layer}\)), HNS propagation velocity (\(v_{p}\)), and averaged HNS mass fraction at sea level (\(\phi_{s-l}\)) were introduced in the present study. Figure 5a represents the change in the HNS layer thickness for different current velocities and HNS densities at \(t=1000\) s. The HNS layer thickness is defined as the depth at which the integrated HNS mass reaches 90% of the total amount of spilled HNS by integrating the mass fraction of HNS (\(\phi_{lms}\)) from sea level at \(t=1000\) s, as follows. \[\left.\frac{\int_{0}^{\delta_{layer}}\int_{0}^{x_{\max}}\phi(x,y)\rho\,dxdy}{ \int_{0}^{y_{\max}}\int_{0}^{x_{\max}}\phi(x,y)\rho\,dxdy}\right|_{t=1000}=0.9, \tag{12}\]  
where \(x_{\max}\) and \(y_{\max}\) represent the length of the computational domain in the \(x\) and \(y\) directions, respectively. The denominator represents the total amount of spilled HNS. The HNS layer thickness indicates the upper limit for the \(y\)-axis of the integral in the numerator, as shown in Figure 5b. From the results, the lower the HNS density, the thinner the layer that was formed under the same current velocity conditions, because of the buoyancy effect. When the density of HNS decreases, the dissolved HNS floats rapidly to the sea surface and forms a thinner layer. The change in HNS layer thickness is presented in Figure 5a. The HNS layer thickness decreases with the increase in current velocity, but it increases after a certain value of current velocity because of the competitive role between advection and diffusion. The inflection point appears at approximately \(v_{c}\) = 0.75 m/s. When \(v_{c}\) is 0.5 m/s, the corresponding \(\delta_{layer}\) values are 4.56, 4.21, and 4.01 m for \(\rho_{HNS}\) values of 700, 800 and 900 kg/m\({}^{3}\), respectively. As \(v_{c}\) increases, the advection becomes dominant to overcome the diffusion effect and acts as the main driving force in HNS propagation. However, when \(v_{c}\) increases above a certain value (\(\sim\)0.75 m/s) at which an inflection point appears, the mixing of HNS becomes significant and makes the HNS layer less dense but thicker. Figure 6a shows the transient evolution of the HNS interface for different current velocities in the case that \(\rho_{HNS}\) = 700 kg/m\({}^{3}\). The HNS interface is defined as the location at which the HNS mass fraction is 1.0 \(\times\) 10\({}^{-4}\). Knowledge of the maximum propagation length, defined as the distance from the crack position to the interface in the mainstream direction, is important in predicting the damage region that becomes wider with time. In the case of \(v_{c}\) = 1.5 m/s, the interface moves in the down-stream direction by approximately 40 m every 25 s. The second main parameter, HNS propagation velocity (\(v_{p}\)), is defined as the time-averaged velocity of the HNS interface in the downstream direction:  
\[v_{p}=\frac{1}{T}\int\limits_{t=0}^{T}v_{x}(t)dt, \tag{13}\]  
where \(t\) and \(v_{x}\) are the time and velocity of the HNS interface in the downstream direction. \(T\) denotes the period of integration, which is adopted as the whole computation time in the present study.