## 2 Methodological Framework
### Boundary Conditions  
The simulations are performed by the commercial CFD code WindSim. The RANS equations are solved via a general collocated velocity method (GCV) [34] in a body-fitted coordinate grid formulation (BFC). Therefore, the variables are stored on a collocated mesh. The GCV method, however, uses a segregated pressure-based solver strategy, thus a momentum interpolation algorithm is performed to evaluate mass fluxes on the control volume faces. This algorithm is a variant of the Rhie-Chow  
Figure 2: Schematic representation of the [PERSON] wake model. (\(D\) is the rotor diameter, \(U_{\infty}\) the undisturbed wind velocity, \(x\) the downstream distance of the wake from the rotor, \(V\) the velocity within the wake and \(k\) the wake expansion coefficient).  
momentum interpolation method [35; 36], which avoids the undesired side effect of producing solutions that depend on the relaxation factors used [37]. The \(k-\varepsilon\) two equation turbulence model is used to close the equations. Full mathematical formulation of the model can be found in [38], whereas a detailed description of the computational domain and mesh can be found in Section 2.3.1. The hybrid method of [39] is used to discretize the convective terms. The diffusion terms, on the other hand, are discretized by using the central differencing scheme. At the inlet of the domain, a log law velocity profile is set with the turbulence parameters obeying the following equations  
\[U(z)=\frac{U_{i}^{*}}{\kappa}\ln\left(\frac{z}{z_{0}}\right),\qquad k=\frac{U _{i}^{*2}}{0.3}\qquad\text{and}\qquad\varepsilon=\frac{U_{i}^{*3}}{\kappa z}, \tag{13}\]  
where \(U(z)\) is the streamwise wind velocity at height \(z\), \(U_{i}^{*}\) is the inlet friction velocity, \(\kappa\) is [PERSON]'s constant set to \(0.41\), \(z_{0}\) is the effective roughness height, \(k\) is the turbulent kinetic energy and \(\varepsilon\) is the dissipation rate. The ambient turbulence intensity at hub height (\(TI_{h}\)) assuming an isotropic normal stress approximation is given by  
\[TI_{h}=\frac{\sqrt{\frac{2}{3}k}}{U_{h}}\Rightarrow TI_{h}=\frac{\kappa\sqrt{ \frac{2}{3}}}{\ln\left(\frac{z_{h}}{z_{0}}\right)\sqrt[4]{C_{\mu}}}, \tag{14}\]  
where \(U_{h}\) is the streamwise wind velocity at hub height \(z_{h}\) and \(C_{\mu}\) is an empirical constant equal to \(0.09\). This value for \(C_{\mu}\) was recommended by [40] after researching free turbulent flows. By knowing the values for the turbulence intensity at hub height (see Section 3), using Equation (14) the appropriate value for the effective roughness height (\(z_{0}\)) is found. The wall function of the ground surface is described by the following equations  
\[U_{r}=\frac{U_{w}^{*}}{\kappa}\ln\frac{z_{r}}{z_{0}},\qquad k=\frac{U_{w}^{*2} }{0.3}\qquad\text{and}\qquad\varepsilon=\frac{C_{\mu}^{0.75}k^{1.5}}{\kappa z_ {r}}, \tag{15}\]  
where \(U_{r}\) is the absolute value of the velocity parallel to the wall at the first grid node and \(z_{r}\) is the normal distance of the first grid node from the wall. Here the wall friction velocity \(U_{w}^{*}\) is calculated from \(U_{w}^{*}=\sqrt{\tau_{w}/\rho}\), the wall shear stress is \(\tau_{w}=s\rho U_{r}^{2}\) and \(s=\left(\frac{\kappa}{\ln\frac{\kappa}{z_{0}}}\right)^{2}\). The lateral walls of the domain are impermeable and frictionless. The outlet and top plane are treated as a fixed pressure boundary. Moreover, a diffusive link is set at the top plane to help preserve the inlet boundary profile calculated via Equation (13) ([41]). The pressure value at the outlet is set to zero.