## 3 Results and Discussion
### Metamodeling Using Kriging Model  
After the HNS accident occurs, the response must be made as fast as possible. At this time, it would be meaningless to perform a CFD simulation practically. Hence, another focus of this study is  
Figure 6: (**a**) Transverse propagation velocity of the HNS for the input variables and (**b**) difference of max propagation distance in the transverse direction between the cases of the side crack and bottom crack. Figure 7: Predicted averaged mass fraction at sea-level of the HNS according to (**a**) HNS density and (**b**) crack location. on the development of a metamodel to predict the HNS propagation. This model would be useful in making the maneuver, especially at the early stage of the HNS-spill accident. Of course, the current study has the limitation of not including all possible scenarios. In this regard, the current study is at the first step of establishing the metamodel platform based on the well-known Kriging model and the CFD results. Using Equations (11)-(14), we considered 18 numerical cases, as listed in Table 1, to obtain the parameters, such as \(\theta_{k}\) and \(\hat{\beta}\), for the input variables. Here, subscript \(k\) indicates the input variable, such as current velocity, current density, and crack location. Finally, the present study determined the coefficients involved in the prediction model, as summarized in Table 2. Here, we alternatively performed hidden point tests for evaluation because there was no experimental and numerical data for comparison. In doing so, the CFD simulations for eight cases, listed in Table 3, different from the cases listed in Table 1, were carried out. Their results were compared with the estimation of the metamodel proposed in this work. Figure 8 presents the comparison between the CFD results and the hidden point test results. The black dots represent the CFD results of 18 cases (Table 1), and the red dots indicate the additional CFD results for eight cases. Moreover, the surface was formed by the solutions of the metamodel. For evaluation, the hidden point error \(\delta\) was checked by:  
\[\delta=\left(\frac{V^{\prime}-V}{V}\times 100\right)\% \tag{17}\]  
where \(V^{\prime}\) is the result of the prediction model, and \(V\) is the result of additional cases. As a result of the comparison using Equation (17), the average error of the prediction model for each parameter was estimated to be 8.48% in the streamwise propagation velocity, 5.38% in the transverse propagation velocity, and 6.71% in the averaged HNS mass fraction, respectively. All errors are relatively small, showing that the current model is well fitted. In addition, to improve the accuracy of the prediction model, the eight additional cases were simulated and the metamodel was updated based on the results for better correction. Finally, we obtained Figure 9, which shows the results of the metamodel for the key parameters. Putting the possible inputs, such as HNS density, current velocity, and crack location, into the metamodel proposed in the present study would make the estimated outputs, such as the averaged HNS propagation velocity in the streamwise and transverse directions and the HNS mass fraction at sea level. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Key Parameters**} & \multicolumn{4}{c}{**Determined Coefficients**} \\ \cline{2-5}  & \(\overset{\alpha}{\beta}\) & \(\theta_{current}\) & \(\theta_{density}\) & \(\theta_{location}\) \\ \hline \(\overline{v}_{x}\) & 1.0904 & 0.7 & 0.1 & 0.1 \\ \(\overline{v}_{y}\) & 0.1474 & 0.4 & 0.2 & 0.1 \\ \(w_{s-l}\) & 0.8580 & 0.5 & 0.2 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients of each key parameter for the Kriging model.