# CFD-Based Metamodeling of the Propagation Distribution of Styrene Spilled from a Ship
## 2 Mathematical Representations
### CFD Model  
We needed a lot of data to develop the metamodel of the styrene propagation in the sea. Hence, the present study extensively conducted three-dimensional CFD simulations to obtain the predictions of the styrene concentration distribution by solving the Reynolds-averaged Navier-Stokes equations and the scalar transport equations at the same time. We used the commercial code of ANSYS Fluent (V.17.2) for simulations. Since the characteristic length is in the order of several hundred meters, the flow can be regarded as a turbulent flow around the ship. The present study adopted the standard \(k\)-\(\varepsilon\) model for simplicity. Numerically, we solved the mass and momentum conservation equations, which are expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\bigg{(}\ddot{u_{i}}\bigg{)}}{ \partial x_{i}}=0, \tag{8}\]  
\[\frac{\partial}{\partial t}\bigg{(}\ddot{u_{i}}\bigg{)}+\frac{\partial}{ \partial x_{i}}\bigg{(}\ddot{u_{i}}\ddot{u_{j}}\bigg{)}=-\frac{\partial p}{ \partial x_{i}}+\frac{\partial}{\partial x_{i}}\bigg{[}\mu\bigg{(}\frac{ \partial\ddot{u_{i}}}{\partial x_{j}}+\frac{\ddot{\partial u_{j}}}{\partial x _{i}}-\frac{2}{3}\delta_{ij}\frac{\ddot{\partial u_{l}}}{\partial x_{l}} \bigg{)}\bigg{]}+\frac{\partial}{\partial x_{j}}\Big{(}-\mu u_{i}^{\prime}u_{j }^{\prime}\bigg{)}, \tag{9}\]  
where \(\rho\) denotes density, \(u_{i}\) denotes the velocity vector, and \(\mu\) denotes viscosity. The present simulation used the standard \(k\)-\(\varepsilon\) model, and the wall function to predict turbulent stresses at the wall in the simulation [16]. The standard k-\(\varepsilon\) model can be expressed as:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}\bigg{(} \ddot{\rho u_{i}}k\bigg{)}=\frac{\partial}{\partial x_{j}}\bigg{[}\bigg{(}\mu +\frac{\mu_{t}}{\sigma_{k}}\bigg{)}\frac{\partial k}{\partial x_{j}}\bigg{]} +G_{k}+G_{b}-\rho\varepsilon, \tag{10}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}} \bigg{(}\ddot{\rho u_{i}}\varepsilon\bigg{)}=\frac{\partial}{\partial x_{j}} \bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{ \partial\varepsilon}{\partial x_{j}}\bigg{]}+C_{1}\frac{\varepsilon}{k}(G_{k} +G_{b})-C_{2}\rho\frac{\varepsilon^{2}}{k}, \tag{11}\]  
\[\mu_{t}=C_{\mu}\rho\frac{k^{2}}{\varepsilon}, \tag{12}\]  
where \(k\) and \(\varepsilon\) represent the turbulent kinetic energy and the dissipation rate, respectively. Furthermore, \(\mu_{t}\) denotes the turbulent viscosity and \(G_{k}\) denotes the production of turbulent kinetic energy due to the mean shear rates. Here, \(G_{b}\) indicates the turbulent kinetic energy production due to buoyancy. The model constants \(C_{1}\), \(C_{2}\), and \(C_{\mu}\) were set to \(C_{1}=1.22\), \(C_{2}=1.92\), and \(C_{\mu}=0.09\). The turbulent Prandtl numbers were set to \(Pr_{k}=1.0\) and \(Pr_{e}=1.3\). The species transport equation was adopted to describe the convection-diffusion for the \(k\)th species as follows:  
\[\frac{\partial}{\partial t}(\rho Y_{k})+\frac{\partial\bigg{(}\rho\ddot{u_{i}}Y _{k}\bigg{)}}{\partial x_{i}}=-\frac{\partial J_{jk}}{\partial x_{j}}, \tag{13}\]  
where \(J_{jk}\) and \(Y_{k}\) are the diffusion flux and the mass fraction of species \(k\), respectively.