# CFD-Based Metamodeling of the Propagation Distribution of Styrene Spilled from a Ship
## 2 Mathematical Representations
### Numerical Details  
Among various HNSs, the present study used a styrene material for the simulation based on knowledge of detailed scenarios with various current-velocity conditions. Figure 1 shows the schematic of the model that represents the current-velocity profile. It is known that the current-velocity profile consists of a tide-induced current, a density-driven current, and a wind-driven current [17; 18]. Because of the complexity of atmospheric conditions, we modeled the current-velocity profile in terms of the surface current velocity (\(v_{s}\)), deep current velocity (\(v_{d}\)), and depth of surface layer (\(d_{s}\)). The present study considered four parameters, as follows: i) the surface velocity (\(v_{s}\)), ii) the ratio of the deep current velocity to surface current velocity (\(v_{d}\)/\(v_{s}\)), iii) the depth of the surface layer (\(d_{s}\)), and iv) the crack positions as the side (collision) and bottom (grounding). The surface velocity was set in the range from 0.1 to 1 m/s, and the deep current velocity was taken to be 30% to 100% of the surface current velocity (\(v_{s}\)). Furthermore, the depth of the surface layer was in the range of 10 to 100 m. The simulation conditions were finally set for 80 cases. The present simulation ignored the interfacial waves between the seawater and air for simplicity because it would have required tremendous computational resources and complicated data about atmospheric conditions. Hence, the slip condition at sea level was taken as the boundary condition. Furthermore, the pressure out condition was used for all surfaces except for the bottom surface, upon which the wall boundary condition was applied. The density, viscosity, and diffusivity properties of styrene were set to 906 (kg/m\({}^{3}\)), 0.76 \(\times\) 10\({}^{-3}\) (kg/(m-s)) and 2.0 \(\times\) 10\({}^{-9}\) (m\({}^{2}\)/s), respectively. It was also important to establish the accident scenario. This simulation was conducted under the scenarios considering two cracks from the side (collision) and bottom (grounding) [19; 20]. Based on information provided by KRISO (Korea Research Institute of Ships & Ocean Engineering), a mass flow rate from a crack was set as 500 kg/s. With a time step of 0.05 s, the transient simulation was performed for a total simulation time of 1000 s. The corresponding averaged Courant number was estimated to be 0.1231, and the maximum Courant number of 0.9232 was found near the crack position. The estimated \(y^{+}\) values were in the range of 100 to 220, showing an acceptable range for predicting turbulent stresses appropriately.  
Figure 1: The schematic of the representative current profile.  
As shown in Figure 2, the computational domain had a size of 1000 m \(\times\) 1000 m \(\times\) 100 m. The detailed configuration of the ship was provided by KRISO. The accident ship had a length of 160 m, a width of 30 m, and a draft of 14.5 m. The crack with its area of 0.25 m\({}^{2}\) was positioned at 80 m from the front of the ship. Furthermore, another scenario considered a side crack that could form owing to collision-induced damage. The side-crack was located at 10 m below sea level, and the bottom crack due to grounding was located at the center of the bottom surface of the ship. We used the tetrahedral grid system and built much finer grids around the ship, as shown in Figure 2b. For the grid convergence test, we compared the velocity profile behind the ship (\(x=290\), \(z=0\), 450 m \(<y<\) 550 m) in the case of \(v_{s}=1\), \(v_{d}/v_{s}=0.5\), and \(d_{s}=30\) m. As shown Figure 3a, the maximum deviation between grid numbers of 1,770,958 and 2,482,851 was estimated to be 0.7% at \(y=500\) m. Therefore, the present study adopted 1,770,958 elements and 322,925 nodes after the grid convergence tests. We set the convergence criterion for the continuity as \(10^{-4}\), for the energy as \(10^{-6}\), and for others as \(10^{-3}\).