5.1. 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
and
so that when a turbulence box is introduced in the computational domain, the desired target values are attained after a distance of
, at
. 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 (
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
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
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
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
, 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
, this is, ∼60% for Ti3 and ∼100% for Ti12. The mesh spacings used for the fits are
m for Ti3 and
m for Ti12. The problem of setting
has been discussed by [
52], which show that when
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
in the synthetic turbulence. According to these results, it is seen that the requirement of two cells per
(assuming the applicability of the minimum condition to represent a wavelength [
53]) is sufficiently fulfilled. In effect, for the case Ti3,
is obtained. The resolution of eddies is somewhat improved in the case of Ti12, where
. 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
m (Ti3) and
m (Ti12) at
.
Figure 4 shows the development of the longitudinal and transversal integral lengthscales,
and
, 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
, although it should be considered that the difference is increased in the figure as the experimental value is below the target
m. A fit of Equation (
5) is also made for
obtained with LES. In the Ti3 case, the least-squares fit method applied yields
,
and an origin set at
m (upstream) from the inlet. Therefore, according to reference values provided along Equation (
5), the LES slightly overestimates
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
yields
and
with
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
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 Mann algorithm does not produce divergence free fields, as pointed out by [
29].
5.2. Velocity Deficit
The results for the wake simulations obtained with the AD are now shown. The first comparison is made from the results of the streamwise velocity deficit along the vertical direction at different longitudinal positions, normalized by the freestream velocity at
. For these and other quantities extracted across the wake field shown in
Section 5.2,
Section 5.3,
Section 5.4 and
Section 5.5, the values are sampled at every cell centre along a line in the
y-direction (at the mid-plane
) at different
positions. These quantities correspond to time averages made during 20 LFT (13.33 s). No further spatial averaging is made. Note that LES results at
have been added to the available
x-positions of the LDA data as this is a position where values computed from HWA are later shown.
Figure 5 shows the results for the high and low solidity disks under the inflow Ti3. The agreement to the experimental results is very good, with the larger difference observed around the shear layer (i.e., the wake envelope at
) from the disk edges, especially for the disk with higher thrust.
In the case of the Ti12 inflow,
Figure 6 shows a minor reduction in the agreement of the LES with the measurements. The predictions commence to differ when moving further into the far wake, around the centreline and shear layer regions. Remarkably, the results of RANS are almost identical to those from LES in both TI cases.
5.3. Turbulence Kinetic Energy in the Wake
It is expected that the wake created by the disks augments the turbulence level with respect to the ambient value. It is now investigated how the computations of the added turbulence compare to the experimental results within the wake.
Figure 7 shows the profiles of
k (this is,
for the LES) at different downstream positions along the wake, when the inflow of the case Ti3 is used. There, it is observed that the LES results match quite well the measured (LDA data) turbulence levels behind both disks. However, it is noticed that, except for the nearest position to the disk, the LES predicts a higher diffusion of shear turbulence in the crosswise direction, an effect that is increased with the disk thrust.
For the disks in the Ti12 case in
Figure 8, the LES compare mostly well with the experimental data, although a small overestimation of
k can be seen just behind the disk (
). It is also observed that the shear layer originating at the edges of the disk is mixing faster with the ambient turbulence compared to the Ti3 inflow. Indeed, the effect of shear prevails deeper into the wake in the LES with the highest thrust disk, whereas it is mixed faster into the ambient turbulence when the thrust is lower. Results from the RANS computations are discussed in the next section.
It is also noticed that some inhomogeneities appear along the curves of LES, very noticeable in the simulations with the Ti12 inflow. These seem to indicate a footprint of the turbulence structures of the inflow turbulence. Although this feature could provide evidence of the need of performing averages in the azimuthal direction or creating synthetic turbulence that would cover longer simulation periods, it is thought that the results shown in the figures are sufficient for the purpose of these comparisons.
5.4. Turbulence Dissipation in the Wake
In the LES computations, the dissipation shown corresponds to
. The dissipation is calculated from the HWA data using Equations (
2) and (
3), which was only available in the Ti12 case. Therefore, results of Ti3 are shown only for completeness in
Figure 9, where it can be seen that, as before, differences between LES and RANS are small, as the curves differ only at
where RANS predicts a higher dissipation within the shear layer.
For the results with the Ti12 inflow in
Figure 10, it is seen that, for the disk
, the LES predictions compare well with measured values. For the disk
, the measurements reveal a large increase of dissipation within the shear layer, compared to the data of computations with the lower thrust AD. Furthermore, at least within the three longitudinal positions available, measured dissipation in the shear layer is more or less maintained. The computations with
also display a somewhat stronger mixing of turbulence from
, where dissipation becomes more uniform and less predominant in the shear layer.
The RANS computations with the modified
model of Sumner and Masson have been previously shown capable of reproducing the turbulence level in the wake [
17]. In the present comparison, it is seen that for the Ti3 inflow the agreement is very good for the disk
while it falls somewhat behind in the far wake of
. However, in both cases, the agreement in the computed dissipation of RANS and LES is very good except for
. Interestingly, it is the vicinity of the disk where the
k-
is often corrected by adding dissipative terms to the
equation to overcome the miscalculated turbulence stresses [
49]. The results with the Ti12 show the opposite picture with regard to the estimation of
k, as the agreement with measurements becomes better only for farther distances from the disk. For the closest position, the turbulence level is overestimated (as it is in the LES) despite the drop of the turbulence production terms near the disk (
is outside this region). Dissipation seems overestimated in the case of
when comparing to the measurements. This is less certain for the higher thrust disk, where at
the peak value of dissipation seems equal to the measured one, but much smaller in the case of
. Notably,
from RANS is always higher than any LES in the wakes of the Ti12 inflow. Previous work [
49] has shown that, in the ABL, the
k-
model overestimates the dissipation around the disk when comparing with LES. This has been observed to occur even upstream of the disk, where
has been seen to increase unlike computations of LES, where this value does not grow until
downstream from the rotor.
It should be remarked that Sumner and Masson [
17] showed that results with various turbulence closures (standard
k-
, RNG, El Kasmi and Masson model [
48] and their own model) compare, in essence, equally well to the measurements, with no apparent advantage of their proposed correction to the
k-
model (although
yielded by the different closures was not compared). The fact that all models compare well to measurements appears to contradict the otherwise inadequate results obtained in simulations of wakes in the ABL flow cited in that work. There, it is also argued that this is due to the relative decrease of the modelled turbulent viscosity
in the reproduction of wind-tunnel wakes with homogeneous inflow with respect to its proportion in the modelling of atmospheric flow. In those conditions, previous work by [
49] has successfully proved the advantages of LES to estimate the velocity deficit and turbulence levels in the wake.
5.5. LES Modelling in the Wake
The previous results for
k and
indicate that the LES running in OpenFOAM is able to predict with relative accuracy not only the velocity deficit in the wake, but also the level of turbulence and its dissipation in the case where the TI in the inflow is low (∼3%). For the high TI inflow (∼12%), the prediction becomes more imprecise, according to the comparison with the experimental data. In the absence of disks, it is observed that the fraction of the turbulence kinetic energy that is resolved by the LES with respect to the total
occurs for the most part in the resolved scales, at around 90% in both TI cases throughout the domain and only somewhat smaller close to the inlet [
54]. With the introduction of the disks, this does not appreciably change except for the shear layer nearby the AD at around
in the Ti3 case, where the resolved part is slightly reduced [
54]. In the Ti12 case, this difference is not noticeable and the value of
remains throughout.
Figure 11 shows the ratio of subgrid dissipation with respect to the total value
along the wake for the Ti3 case. Note that, as the positions of comparison are no longer restricted to those of the available experimental data, profiles are shown at different longitudinal positions from other figures. In
Figure 11, it is noticed that the LES has an appreciable increment in subgrid dissipation within the shear layer. Furthermore, this increase persists longitudinally even as far as when the wake appears to reach a state of transversally-uniform dissipation, i.e. at
with disk
. This is consistent with the hypothesis that turbulence is created at smaller lengthscales than the ambient turbulence at the disk edge. Due to the limited resolution of turbulence lengthscales in the Ti3 flow (missing in the synthetic flow as well), the increase in subgrid dissipation is produced at scales that seem absent in the incoming flow. These results also show that the wake envelope becomes the main carrier of dissipation. The subgrid dissipation part is also larger with higher thrust, yet by a small margin. It is observed that, in the absence of disks, most of the dissipation comes from the resolved fluctuations, in a proportion very similar to that shown outside the wake, at
[
54]. It is seen in
Figure 12 that when the inflow turbulence raises (which comprises better resolved lengthscales), the increment of subgrid dissipation in the region of the wake envelope is greatly reduced. As a result, the modelling ratio seen outside the wake is essentially conserved. From these results, it can be deduced that the LES modelling across the wake is largely determined by the ambient turbulence.
5.6. Integral Lengthscale in the Wake
The changes in
caused by the presence of the AD and the wake are now investigated. The computation of
is performed as described in
Section 4.4, which involves the assumption of the Taylor 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
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
computed from the LES in each code with the Ti3 inflow, from
to
, at
,
and
, 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
, next to the the shear layer, larger lengthscales can be discerned amongst the variations in the profile. Indeed, the maximum values of
at each
position are at close
in the wake of the disk
. This is consistent with the previous results with regard to the location of the shear layer along the wake (e.g.
k and
). Conversely, for the other disk the maxima of
would suggest a wake that expands to about
at
, 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
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
, towards the bottom part where
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
,
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 (
), which clearly sets a different scenario in comparison to this study. Precisely, the absence of a variation of
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. This has been studied in
Section 5.5, where it is shown that the LES modelling does not vary within the wake with respect to the external flow aside from very close to the disk (
) in both codes. Nevertheless, it has been seen that despite the limited resolution, the LES computations have been able to reproduce other principal features along the wake, such as the turbulence levels.
5.7. Spectra behind Disks
To study the redistribution of turbulence energy along the wake, the spectra obtained at different longitudinal positions for every disk are compared with the spectrum of the free decaying turbulence. Power Spectral Density (PSD) curves are calculated from only one measuring position at centreline, so to reduce the noise in the spectral curves, the time-series of each register are divided into eight non-overlapping blocks with an equal number of samples. Then, the PSD of all blocks are averaged to produce the curve at each longitudinal position. However, noise remains along the curves that make the comparison very difficult, so a smoothing procedure is needed to be performed. To this aim, an exponential moving average is used to filter the spectra computed at each longitudinal position (a rational transfer function is employed, see [
55]). Hence, the spectra shown in the following figures have been processed with this technique, with the sole exception of that obtained from measurements without a disk, which was spatially averaged with results obtained at the other eight locations distributed crosswisely. As the spectra are calculated from data at a fixed location (sampled in time), the Taylor hypothesis is applied to transform the frequency spectra into a wavenumber spectra using
, where
f is
for measurements or
for the LES. In this way, it is possible to also compare with the PSD from the synthetic turbulence, which is calculated as the volume average of the spectra computed in the longitudinal direction.
The results for the inflow Ti3 are shown in
Figure 15. In the results without the AD, a constant decay of energy is observed as the flow moves downstream. The spectra from the synthetic box serve to mark the extension of the resolved wavenumbers (
m
−1) since the spatial resolution in the box is the same as in the LES. Note that the abrupt drop in the turbulence energy spectra is attributed to a combination of numerical diffusion and the limited spatial resolution [
5]. In case of the disk
, a gain in turbulence energy is seen immediately behind the disk, as the curves at
and
are almost identical. A small decay of energy is observed at
and, from there, an increase in turbulence energy around the highest levels (lowest
). For the disk
, the effects are accentuated, and the curves at
are the only ones displaying a decay and yet only around the inertial range. The energy of the next two longitudinal positions,
and
, increases for all wavenumbers, which represents an increment of about one order of magnitude at the lowest wavenumber, with respect to the levels displayed by the decaying turbulence without disks. Notably, the spectra of the last two positions seemingly exhibit an inertial range, characterized by the slope of
in the decay rate.
The results for the Ti12 inflow are shown in
Figure 16. In this case, the spectra computed from experimental results are also included. The spectra obtained from measurements with disks extend to larger wavenumbers than in the cases without disk, due to the use of a different frequency in the low-pass filter. For the case without disks, the energy at the lowest wavenumbers obtained from the LES proves to decay less as it has been shown before. However, they display a steady decay that adjusts well to the characteristic slope of the intertial range, also discernable in the experimental results. These observations are analogous for the results with the disk
. In contrast to the Ti3 inflow where energy is seen to increase beyond
for the disk with the same porosity, here a reduction in the contribution of shear towards the increase of energy along the wake is observed. Although the overall levels of turbulence energy in the wake are higher than in the free decaying turbulence, they maintain more or less the same relative decay from one to another. This behaviour is similar in the case of the disk
. There, only the curve at
shows an increase in energy compared to the previous disk (also matching fairly well the experimental results in the inertial range).
5.8. Vorticity Contours
Lastly, to complement all previous results,
Figure 17 shows a comparison between the the contours of vorticity obtained in the Ti3 and Ti12 cases and using the highest porosity disk,
and
, respectively. The images are taken at the
x–
y plane, at
and correspond to the vorticity field computed at the last time step of the LES runs. Make note that black bars are used to represent the disk position but do not portray the complete longitudinal region where the forces modelling the AD act, distributed using Equation (
7). This figure permits visualizing the dominant effect of the turbulence structures from the inflow of the Ti12 case, which, unlike the Ti3 inflow, prevail along the wake. With the Ti3 inflow, there is a clear shear region arising from the edges of the disk with distinct turbulence structures. Conversely, with the use of the Ti12 inflow, the shear region is substantially less noticeable in the higher ambient turbulence. Indeed, the vorticity contours from the inflow appear to dominate in the vicinity of the AD. This is consistent with the comparison of
k in
Figure 5 and
Figure 6, where its increase within the shear layer is less prominent with the increase of ambient TI. In
Figure 17, it can also be seen that, for the Ti12 inflow, structures that appear to arise from within the wake become more apparent than those outside from approximately
.