## 5 Results and Discussion
### Turbulence Decay and Integral Lengthscale Development without Disk
According to these results, it is seen that the requirement of two cells per \(L_{1}\) (assuming the applicability of the minimum condition to represent a wavelength [53]) is sufficiently fulfilled. In effect, for the case Ti3, \(L_{1}/\Delta\simeq 3\) is obtained. The resolution of eddies is somewhat improved in the case of Ti12, where \(L_{1}/\Delta\simeq 4\). These resolutions that seem a priori rather coarse are the result of a series of compromises that have been previously explained. In [30] a similar resolution was used for the synthetic inflow in LES computations of the wake of a rectangular channel, obtaining a good comparison with experimental results related to the flow structures. The results of the turbulence decay in the absence of the disk show that these values are enough to supply an integral lengthscale to procure the desired magnitudes \(L_{1}=0.01\) m (Ti3) and \(L_{1}=0.03\) m (Ti12) at \(x_{D}\). Figure 4 shows the development of the longitudinal and transversal integral lengthscales, \(L_{1}\) and \(L_{2}\), along the flow. These are computed using velocity time-series (recorded at the same positions for TI above) and employing the method described in Section 4.4. The measurements from [16] are also used for comparison in the Ti3 case. There, the comparison of the measurements with the LES shows a small overestimation of \(L_{1}\), although it should be considered that the difference is increased in the figure as the experimental value is below the target \(L_{1}=0.01\) m. A fit of Equation (5) is also made for \(L_{2}\) obtained with LES. In the Ti3 case, the least-squares fit method applied yields \begin{table}
\begin{tabular}{l c c c c c} \hline  & \multicolumn{2}{c}{\(x_{0}=0\)} & \multicolumn{2}{c}{\(x_{0}\
eq 0\)} \\ \hline  & **1/\(A\)** & **\(n\)** & **1/\(A\)** & **\(n\)** & \(x_{0}\) **[m]** \\ \hline Case Ti3 & 24.11 & 1.281 & 9.85 & 1.519 & \(-0.021\) \\ Case Ti12 & 28.49 & 1.15 & 11.43 & 1.661 & \(-0.0845\) \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters of the fit of TI decay of LES to Equation (4). \(x_{0}\) denotes the virtual origin of the curve with respect to the inlet. \begin{table}
\begin{tabular}{l c c c} \hline  & **TI\({}_{\text{B}}\)** [\%] & \(L_{1,\text{B}}\times 10^{-3}\)** [m] & \(k/\frac{1}{2}U_{\infty}^{2}\)** [-]** \\ \hline Case Ti3 & 35.0 & 5.82 & 0.37 \\ Case Ti12 & 60.2 & 15.3 & 1.08 \\ \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the different boxes of synthetic turbulence used as inflow for the LES. \(B_{1}=0.089\), \(n_{1}=0.392\) and an origin set at \(x_{0}=-0.188\) m (upstream) from the inlet. Therefore, according to reference values provided along Equation (5), the LES slightly overestimates \(L_{2}\) for Ti3. In the Ti12 case, the comparison with measurements is made with the only three points available so it is less decisive, although they suggest a higher growth rate. The fit of Equation (5) to the LES predicted \(L_{2}\) yields \(B_{1}=0.064\) and \(n_{1}=0.254\) with \(x_{0}=-0.068\) m, which, compared to the values in the literature, also indicates a slight overestimation by the computations. For the purposes of this work and considering the mesh restrictions, the development of the integral lengthscales in the flow predicted by the LES is deemed satisfactory. In both TI and \(L_{i}\) results, we can observe small oscillations for the first positions next to the inlet, which are likely the result of the adjustment of the synthetic velocity field to the LES. Specifically, the incompressibility conditions imposed by the solver as the original formulation of the [PERSON] algorithm does not produce divergence free fields, as pointed out by [29].