## 5 Results and Discussion  
### Turbulence Decay and Integral Lengthscale Development without Disk  
The first step of the investigation consists of the calibration of the parameters of the synthetic turbulence. This is finding TI\({}_{\rm B}\) and \(L_{\rm I,B}\) so that when a turbulence box is introduced in the computational domain, the desired target values are attained after a distance of \(5D\), at \(x_{D}\). The parameters of the synthetic turbulence for all boxes are shown in Table 3. These were computed longitudinally and averaged over the whole volume. It is immediately noticed that high TI values were necessary to reproduce the evolution of the turbulence intensities reported by the experiments. Consequently, the approach followed could be seen as rather crude, on account of the Taylor approximation. However, the results reproduce, for the most part, the longitudinal evolution of turbulence predicted also by the empirical relations found in the literature. This can be seen in Figure 3, which shows the free (no disk) homogeneous turbulence decay in each TI case obtained with LES. There, every value represents the average of the TIs computed from time-series stored in nine probes distributed in a crosswise plane, in turn located at every longitudinal position indicated by the marks in the curves. LES results are compared to the values measured in the wind tunnel and to the least-squares fit with Equation (4). As mentioned in Section 3, experimental values from [16] are used in the Ti3 case. To complement HWA data (which in the Ti12 case has only 3 points), LDA measurements are used, obtained in the experiment with the low thrust disks but outside the wake envelope (\(y=\pm 1\)D). It can be seen that for both Ti3 and Ti12 the decay predicted by the LES follows fairly well the experimental values. The subframes in Figure 3 represent the same data but plotted in a log-log scale, so the power law decay of the TI (of slope \(-n\) in Equation (4) can be better appreciated. This permits seeing that, after some distance, the decay rate is approximately equal for both TIs. Furthermore, it is also seen that the LDA (measurements outer wake) in the Ti3 case deviates from a constant decay rate and therefore from the LES. Conversely, the LDA data in the Ti12 compares very well with the LES prediction. Equation (4) is an empirical relation first proposed by [41] to describe the decay in homogeneous turbulence seen in a wind tunnel. The applicability of this relation has been proven in a wide range of Re flows in later work [39; 42; 50; 51]. In most of the results reported in the literature, a fit is produced setting the virtual origin \(x_{0}\) to zero in the equation, which neglects the agreement close to the grid or the place where turbulence originates as the stations where measurements or calculations are reported are generally far from such region. However, as the complete evolution of TI is monitored, a better fit is obtained by setting \(x_{0}\) to a position different from where the turbulence is introduced (in particular, to an upstream location). The fit of Equation (4) for the curves shown in Figure 3 yields the results shown in Table 4. If the fit is made using \(x_{0}=0\), the parameters are closer to those reported in the literature (see Section 2), although the curve would display a much higher TI at the inlet than the one given with \(x_{0}\
eq 0\), this is, \(\sim\)60% for Ti3 and \(\sim\)100% for Ti12. The mesh spacings used for the fits are \(M=0.0225\) m for Ti3 and \(M=0.20\) m for Ti12. The problem of setting \(x_{0}\) has been discussed by [52], which show that when \(x_{0}\) is properly determined, its value and the exponent of the power law decay \(n\) is Re-independent, while \(A\) is indeed a function of initial conditions (including Re), so the turbulence decay takes a universal self-similar behaviour. Table 3 also shows the magnitude of the \(L_{1,\text{B}}\) in the synthetic turbulence.