## 5 Results and Discussion
### Integral Lengthscale in the Wake  
The changes in \(L_{1}\) caused by the presence of the AD and the wake are now investigated. The computation of \(L_{1}\) is performed as described in Section 4.4, which involves the assumption of the [PERSON] hypothesis to transform the computed time-scales into lengthscales. Evidently, this supposition becomes more difficult to accept when shear is present in the flow. However, previous work has reported satisfactory results in wake studies that support the continuing applicability of the hypothesis. For instance, Ref. [16] has compared the lateral distribution of \(L_{1}\) behind the wake produced by a porous disk (in a setting similar to the experiments used in this work) computed from HWA with the one obtained from PIV. They did not find a difference in the results obtained from either technique, despite the fact that HWA uses the local mean velocity to calculate the lengthscale, compared to the direct spatial measurement offered by PIV. Making the same assumption, the evolution of the integral lengthscale behind the AD computations is studied. Figure 13 displays the values of \(L_{1}\) computed from the LES in each code with the Ti3 inflow, from \(y=0\) to \(y=1.5\), at \(3D\), \(4D\) and \(6D\), which correspond to the positions where HWA data for the Ti12 inflow is available. It is first noticed that there is not a clear influence of the shear layer in the size of the turbulence scales. However, for a region about \(0.5\leq y/D\leq 1.0\), next to the the shear layer, larger lengthscales can be discerned amongst the variations in the profile. Indeed, the maximum values of \(L_{1}\) at each \(x-\)position are at close \(y=0.5D\) in the wake of the disk \(C_{T}=0.42\). This is consistent with the previous results with regard to the location of the shear layer along the wake (e.g. \(k\) and \(\varepsilon\)). Conversely, for the other disk the maxima of \(L_{1}\) would suggest a wake that expands to about \(y=0.75D\) at \(x=6D\), which is larger than what the previous computations indicate. Results for the Ti12 inflow are shown in Figure 14. Notably, the computed values from the experimental time-series do not reveal a variation of the lengthscale values at the shear layer. In fact, there is no evident change in \(L_{1}\) within the wake. This trait is similarly observed in the LES results. The only variations in computations are observed at the upper part of the curves or, in the case of the disk \(C_{T}=0.45\), towards the bottom part where \(L_{1}\) is larger (but this effect is reduced further downstream). Previous experimental work by [16] showed that in the wake of a porous disk with a solidity of \(45\%\), \(L_{1}\) is approximately \(1.5\) times larger within the shear layer with respect to the values within the wake or outside the envelope. However, these measurements were obtained using an inflow with very low turbulence (TI \(<0.4\%\)), which clearly sets a different scenario in comparison to this study. Precisely, the absence of a variation of \(L_{1}\) in the shear layer can be explained considering the previous results, which point at a dominance of the ambient turbulence characteristics over the wake in the case of the inflow Ti12. Although the turbulence production is visibly higher when the disk thrust is larger (Figure 8), its effect does not appear to have an impact on the turbulence lengthscales. Similarly, the use of a lower turbulence inflow (Ti3) does not seem to decidedly increase the magnitude of the lengthscales in the area of turbulence production, or at least not in the computations performed for this work. In this regard, the fact that the characteristic lengthscales of the Ti12 inflow are better resolved by the mesh and the LES compared to the Ti3 cases can be a factor to consider. This is, if resolution is not adequate within the shear layer, it is to be expected that a sizeable part of the turbulence being produced would fall into the modelled part instead of being resolved, therefore affecting the magnitude of the computed scales.