## 5 Numerical Simulations
### Les
#### 5.2.2 Verification  
A simulation was run on a finer mesh with the number of cells increased by a factor of \(r=1.2\) in the \(x\), \(y\), and \(z\) directions. The two solutions were then used to calculate the grid convergence index for the velocity deficit downstream of the turbine. Along with values for turbine output, the GCI study shows that a finer mesh is not needed; see Figure 12 and Table 8.  
A total run time of 3000 s proved to give a converged solution with respect to wake, thrust, and power. A case with a halved time step of \(dt=0.05\) was run to test the model's sensitivity to this parameter. This also had the effect of lowering the residuals. Results show that \(dt=0.1\) and the corresponding residuals are sufficiently low, which is shown in Appendix A.  
Both the shear profile and turbulence intensity of the simulation were checked for positions upstream of the turbine. It is shown in Figure 13 that shear profiles for all positions and stabilities gave good agreement with the desired shear-profile shape. Upon  
Figure 11: Mesh used in LES simulations for the Vestas turbine, (**a**) \(yz\)-plane, (**b**) zoomed in on \(xz\)-plane, and (**c**) wide view of \(xz\)-plane.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Thrust [kN]** & **Power [MN]** \\ \hline V80\({}_{medium}\) & 151.2 & 0.728 \\ V80\({}_{fine}\) & 151.2 & 0.729 \\ Diff [\%] & 0 & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Turbine thrust and power for LES refinement study.  
inspection, turbulence intensity was found to decrease substantially from the inlet to the turbine. To correct for this, intensity at the inlet was increased. Table 9 gives an overview of the achieved and target values for turbulence intensities.  
In addition to the parameters presented above, a number of other simulations were run in order to examine the numerical models sensitivity and reli  
Figure 12: Plot of downstream velocity deficit with error bars from the GCI study for LES simulations of the V80 turbine.  
Figure 13: Inflow shear profiles at locations downstream of inlet for V80 LES simulations in (**a**) unstable, (**b**) neutral, and (**c**) stable atmospheres.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Inlet** & **Turbine** & **Target** \\ \hline Unstable & 12.4 & 9.73 & 9.23 \\ Neutral & 7.75 & 4.64 & 4.79 \\ Stable & 5.1 & 2.81 & 2.91 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Average turbulence intensity in the inflow of the Vestas V80 turbine for LES simulations; all values are given in [%].  
This included comparing the Smagorinsky and a dynamic one equation eddy-viscosity turbulence models, testing for different values for the Smagorinsky coefficient, examining different damping functions within the Smagorinsky model and running a simulation with a larger domain. Of these, it was found that the turbulence model, the Smagorinsky coefficient, and the extended domain influenced the solution to a very small degree. The choice of damping function provided in openFOAM did impact the solution more, and among the Prandtl, van Driest, and cube-root volume damping functions, the simpler cube-root volume damping functions were shown to give results most representative of the validation case.