# Numerical Modelling of a Floating Wind Turbine Semi-Submersible Platform
## 2 Materials and Methods  
### Governing Equations  
The behaviour of every incompressible Newtonian flow can be described with the mass and momentum conservation equations, Equations (1) and (2), respectively. These equations are Navier-Stokes equations. \[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}= -\frac{1}{\rho}+\frac{\partial p}{\partial x_{i}}+g_{i}+\frac{1}{\rho}\frac{ \partial v_{ij}}{\partial x_{j}} \tag{2}\]  
where \(i\) and \(j\) are the indexes for 2D flows, \(u_{i}\) is the time-averaged component of the velocity vector, \(\rho\) is the density of the fluid, \(t\) the time, \(p\) the pressure, \(g\) the gravity acceleration and \(\tau_{ij}\) the viscous stress tensor. However, transfer of these equations to computational modelling is not extended due to the high computational cost. Therefore, all the instant values obtained in the Navier-Stokes equations are decomposed in mean and fluctuating values with the Reynolds decomposition once we apply Reynolds-Averaged Navier-Stokes (RANS), Equations (3) and (4). \[\
abla\cdot\mathbf{U}=0 \tag{3}\]  
\[\frac{\delta(\rho\mathbf{U})}{\delta t}+\
abla\big{(}\rho\big{(}\mathbf{U}-\mathbf{U}_{g} \big{)}\big{)}\mathbf{U}=-\
abla p_{d}-g\mathbf{x}\
abla\rho+\
abla\Big{(}\mu_{eff} \
abla\mathbf{U}\Big{)}+(\
abla\mathbf{U})\cdot\
abla\mu_{eff}+\mathbf{f}_{s} \tag{4}\]  
where \(\mathbf{U}\) represents the flow velocity, \(\mathbf{U}_{g}\) is the velocity of the grid nodes, \(p_{d}\) is the dynamic pressure by substracting the hydrostatic component from the total pressure \(p\), and \(\mathbf{g}\) is the gravity acceleration, \(\rho\) is the mixture density of both flows, \(\mu\) is the effective dynamic viscosity, \(\
u\) is the kinematic viscosity and \(\
u_{t}\) the eddy kinematic viscosity. With this decomposition, an introduction of new terms appears, creating an open system of equations. Thus, more equations are needed to close them. In this scenario, different turbulence models have been developed throughout the years to solve this problem. To do that, these models associate the Reynolds stresses with the mean flow variables. In this study, the k-\(\omega\) SST turbulence model, Equations (5) and (6), developed by [PERSON] is used [30]. \[\frac{\partial k}{\partial t}+\mathbf{u}_{j}\frac{\partial k}{\partial x_{j}}= \frac{\partial}{\partial x_{j}}\Bigg{[}\left(\
u+\sigma\frac{k}{\omega}\right) \frac{\partial k}{\partial x_{j}}\Bigg{]}\beta^{*}k\omega+\tau_{ij}\frac{ \partial\mathbf{u}_{i}}{\partial x_{j}} \tag{5}\]  
\[\frac{\partial\omega}{\partial t}+\mathbf{u}_{j}\frac{\partial\omega}{\partial x_{ j}}=\Bigg{[}\left(\
u+\sigma\frac{k}{\omega}\right)\frac{\partial\omega}{ \partial x_{j}}\Bigg{]}-\beta k\omega^{2}+\frac{\sigma_{d}}{\omega}\frac{ \partial k}{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}+a\frac{\omega }{\kappa}\tau_{ij}\frac{\partial\mathbf{u}_{i}}{\partial x_{j}} \tag{6}\]where \(k\) is the turbulence kinetic energy, \(\mathbf{u}\) is the velocity of the flow, \(\omega\) is the specific dissipation rate, and \(\
u\) is the kinematic viscosity, while \(\sigma\), \(\beta\) and \(a\) are constants. However, in this type of problems, more than one fluid is normally simulated, and another tool is needed for the definition of each fluid and for tracking the free surface. In this study, for this aim, the Volume of Fluid (VOF) presented by [PERSON] and [PERSON] [31] is used. It is a free-surface modelling technique, i.e., a numerical technique for tracking and locating the free surface (or fluid-fluid interface). For each phase considered in the model, a variable is introduced as the volume fraction of the phase in the computational cell. Volume fractions represent the space occupied by each phase, and the laws of conservation of mass and momentum are satisfied by each phase individually. In each of the control volumes, the volume fractions of all phases sum up to unity. This model defines the percentage of volume of a certain cell that is occupied by a fluid.