# Performance Improvement in a Wavy Twisted Rudder by Alignment of the Wave Peak
## 2 Materials and Methods
### Computation Setup
#### 2.3.2 Numerical Method  
The solution of the governing equation was obtained using the finite volume method, by applying the numerical method of the second order as a numerical discretization method for time and space integration. The SIMPLE algorithm was applied for the velocity-pressure coupling.  
The turbulence model uses a two-layer realizable k-epsilon turbulent model that combines the realizable k-epsilon model with a two-layer approach. Unlike the standard k-epsilon model [17, 18], the realizable k-epsilon model includes a new transport equation for the turbulence dissipation rate [19]. The computation of the two-layer approach is divided into two layers. The turbulent dissipation rate and turbulent viscosity in the layer next to the wall are specified as functions of the wall distance. The values of the turbulent dissipation rate specified in the near-wall layer are blended smoothly with the values computed by solving the transport equation far from the wall [20].  
The transport equations for kinetic energy \(k\) and turbulent dissipation rate \(\varepsilon\) are as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\
abla\cdot(\rho k\overline{v})=\
abla \cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\
abla k\right]+P_{k}- \rho(\varepsilon-\varepsilon_{0})+S_{k} \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\
abla\cdot(\rho\varepsilon \overline{v})=\
abla\cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\varepsilon} }\right)\
abla\varepsilon\right]+\frac{1}{T_{\varepsilon}}C_{\varepsilon 1}P_{ \varepsilon}-C_{\varepsilon 2}f_{2}\rho\left(\frac{\varepsilon}{T_{\varepsilon}}- \frac{\varepsilon_{0}}{T_{0}}\right)+S_{\varepsilon} \tag{4}\]  
where \(\overline{v}\), \(\mu\), and \(f_{2}\) are the mean velocity, dynamic viscosity, and damping function, respectively; \(\varepsilon_{0}\) is the ambient turbulence value in the source terms that counteracts the turbulence decay; \(\sigma_{k}\), \(\sigma_{\varepsilon}\), \(C_{\varepsilon 1}\), and \(C_{\varepsilon 2}\) are model coefficients, \(P_{k}\) and \(P_{\varepsilon}\) are production terms, and \(S_{k}\) and \(S_{\varepsilon}\) are user-specified source terms. The possibility of imposing an ambient source term also leads to the definition of a specific time scale \(T_{0}\), which is defined as  
\[T_{0}=max\left(\frac{k_{0}}{\varepsilon_{0}},C_{t}\sqrt{\frac{v}{\varepsilon_ {0}}}\right) \tag{5}\]  
where \(C_{t}\) is a model coefficient. The turbulent eddy viscosity \(\mu_{t}\) is calculated as  
\[\mu_{t}=\rho C_{\mu}f_{\mu}kT \tag{6}\]  
where \(\rho\), \(C_{\mu}\), \(f_{\mu}\), and \(T\) are the density, model coefficient, damping function, and turbulent time scale, respectively. The model coefficients of the transport equation use the default values in the STAR CCM+ V.15.02 manual [21].  
The volume of fluid method was used to predict the distribution and movement of the interface of immiscible phases for free surface calculations. The high-resolution interface-capturing method was designed to track interfaces by mimicking the convective transport of immiscible fluid components [22, 23]. The flow resulting from propeller rotation was calculated using the sliding mesh method, which directly rotates the grid around the propeller. The time step for the numerical calculation was set to the time corresponding to a \(5^{\circ}\) rotation of the propeller. Table 4 presents the computational analysis conditions for the CFD analysis.