## 4 Inflow
### Turbulent Inflow  
For LES simulations, the Mann turbulence generator is used with input parameters based on results found in the literature for offshore conditions at varying stability. Although the Mann model offers less accurate inflows for non-neutral atmospheric boundary layers, it provides a controlled and consistent method for generating turbulence that is practical for scenarios where computational resources are limited. It is also widely accepted and applied for modeling atmospheric boundary layers, particularly within wind energy research. However, due to its limitations, validating the generated inflow using either a precursor LES simulation or field measurements would be valuable to this study. The present study has not performed such a validation but has instead used values for \(\alpha_{k}\epsilon^{2/3}\), \(L\), and \(\Gamma\) found by [PERSON] et al. [30], where the Mann model parameters were fit to wind fields simulated using LES. Three different wind speeds were simulated for the unstable, neutral, and stable conditions. [PERSON] and [PERSON] [39] found the parameters to be largely independent of wind speed but not elevation. Values at hub height were therefore chosen and are presented in Table 3.  
For the \(y\)- and \(z\)-direction, the length of the turbulence box exceeds the size of the mesh, and the cell lengths are equal to the maximum chord of the turbine blade. Total length and cell size in the \(x\)-direction correspond to the total time \(T_{mann}\) and time step \(dt_{mann}\) defined in the Mann model. They were chosen so that a total time of 3000 s could be run, 500 s to get past the statistically unsteady ramp-time, followed by 2500 s of simulation  
\begin{table}
\begin{tabular}{l l l} \hline \hline  & **Vestas V80** & **Samsung S7.0-171** \\ \hline
**Unstable** & & \\ \(\alpha_{k}\epsilon^{2/3}\) & 0.0419 & 0.0429 \\ \(L_{Mann}\) & 80 & 86.6 \\ \(\Gamma\) & 4.4 & 3.5 \\ \hline
**Neutral** & & \\ \(\alpha_{k}\epsilon^{2/3}\) & 0.0257 & 0.0171 \\ \(L_{Mann}\) & 27.2 & 33.1 \\ \(\Gamma\) & 3.5 & 3.45 \\ \hline
**Stable** & & \\ \(\alpha_{k}\epsilon^{2/3}\) & 0.0129 & 0.0115 \\ \(L_{Mann}\) & 16.6 & 13.8 \\ \(\Gamma\) & 1.75 & 1.075 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mann inflow parameters.  
time to reach a converged solution. \(dt\) is 0.25 s. The effect of cell size and time step of the turbulence box on the solution was not explored in this study, but the values used are similar to other studies such as [PERSON] et al. [30] and [PERSON] et al. [40].  
The generated turbulence was scaled to give the desired \(TI\), and the shear profile was superimposed on it. Again, results from the LES simulations in [PERSON] et al. [30] were used. Table 4 gives their results for \(\alpha\), \(TI\), and \(U_{Inub}\).  
Also given in Table 4 is the Richardson number, \(Ri\), a parameter used to define stability. The given values were used by [PERSON] et al. [30] to define inflow stability. Along with the Monin-Obukhov length, the Richardson number has been found to perform well in classifying atmospheric stability conditions; see [PERSON] and [PERSON] [41]. It relates buoyancy-generated turbulence with turbulence produced by shear, as seen in Equation (6), where \(\Theta\) is the potential temperature, \(T\) is the mean temperature, \(\Delta u/\Delta z\) is the velocity gradient, and \(g\) the gravity constant.  
\[Ri=\frac{g}{T}\frac{(\Delta\Theta/\Delta z)}{(\Delta u/\Delta z)^{2}} \tag{6}\]