2. Experimental methodology

The experiments took place in the large closed-loop wind tunnel at the Norwegian University of Science and Technology, which has a test section with dimensions 11.1 m  $\times$  2.71 m $\times$  1.8 m. The height of the roof is gradually adjusted to 1.91 m in order to compensate for boundary layer growth and to keep the pressure gradient in the streamwise direction approximately zero. All experiments were conducted at an inflow velocity $U_{\infty } \simeq \mathrm{10}$ m s⁻¹ and a background turbulence intensity ${\lt}\,0.3$ %. Figure 2 displays schematics of the wind tunnel set-up. These schematics are to scale, however, the roof height adjustment has been neglected for simplicity.

Figure 2.

Wind tunnel schematics. (a) The wind tunnel seen from above. The spanwise measurement locations are marked with $\times$ in the $y$ -direction. The disc spacing $\varDelta$ is measured centre to centre. (b) The wind tunnel seen from the side. The spanwise measurements are marked by $\times$ . The schematics are to scale except the discs, and the automated traverse is represented as the dark grey figures farthest downstream. The sloping roof is not explicitly shown.

Figure 3.

(a) Photograph of the hot-wire; (b) photograph of the non-uniform disc (N) in the wind tunnel with the traverse; (c) photograph of the mesh disc (M); (d) schematic of the M disc; (e) schematic of the N disc. The black rod through the discs represents the mounting rod. Note that a combined study, such as suggested by (b) and (c), is not part of this investigation, the discs are always of the same type.

The two PDs used in this study both have a diameter $D = \mathrm{200}$ mm, resulting in a Reynolds number of $Re = {U_{\infty }D}/{\nu } \approx \mathrm{1.25 \times 10^5}$ , where $\nu$ is the kinematic viscosity. For comparison, the experiments of Obligado et al. (Reference Obligado, Klein and Vassilicos2022) were performed at $Re \approx \mathrm{0.43 \times 10^5}$ . The discs are the same as those used in Vinnes et al. (Reference Vinnes, Gambuzza, Ganapathisubramani and Hearst2022a , Reference Vinnes, Neunaber, Lykke and Hearst2023), where their individual wakes are documented (see also figures 4(e) and 4( f), figures 6(e) and 6( f), and schematics and a photograph of the set-up can be seen in figure 3. Both discs have a solidity of 57% and a drag coefficient of $C_D \approx \mathrm{0.8}$ (Vinnes et al. Reference Vinnes, Gambuzza, Ganapathisubramani and Hearst2022a ). The non-uniform disc (N) has been cut from a 5 mm acrylic sheet, and has decreasing blockage with increasing distance from the centre; the technical drawing can be found in Vinnes et al. (Reference Vinnes, Gambuzza, Ganapathisubramani and Hearst2022b ). The mesh disc (M) was made out of wire mesh with a thickness of 0.8 mm and a mesh size of 2.3 mm, and it has uniform blockage. A significant phenomenological difference in the wakes of the two PDs is that the N-type disc exhibits vortex shedding, while the M-type disc does not. The lack of vortex shedding has been reported for porous plates with uniform blockage below 0.72, cf. Castro (Reference Castro1971) and Cicolin et al. (Reference Cicolin, Chellini, Usherwood, Ganapathisubramani and Castro2024). Recently, the vortex shedding of non-uniform discs with similar design has been investigated by Bourhis & Buxton (Reference Bourhis and Buxton2024). Despite a similar solidity and drag coefficient, the wake of the M-type disc recovers significantly more slowly than that of the N-type disc (Vinnes et al. Reference Vinnes, Gambuzza, Ganapathisubramani and Hearst2022a , Reference Vinnes, Neunaber, Lykke and Hearst2023), both due to the small openings that do not shed energetic structures, yielding smaller integral length scales and less turbulent kinetic energy which hinders the recovery process (Kurelek et al. Reference Kurelek, Piqué and Hultmark2023), and because the lack of vortex shedding causes less mixing and a very late onset of self-similarity (Steiros et al. Reference Steiros, Obligado, Bragança, Cuvier and Vassilicos2025).

In the present study, the PDs were mounted on rods spanning the entire height of the wind tunnel, unlike the studies of Vinnes et al. (Reference Vinnes, Gambuzza, Ganapathisubramani and Hearst2022a , Reference Vinnes, Neunaber, Lykke and Hearst2023), where the discs were mounted on top of masts to mimic actuator disks or wind turbines. The present configuration prevents the wake of the discs from drifting down into the wake of the mast (Pierella & Sætran Reference Pierella and Sætran2017). We expect the impact of the mounting rod on the measurements to be negligible for studying wake merging, as both discs act like flow homogenisers, as they are mounted downstream of the rods. Therefore, there are no large structures from the rod, and the turbulence generated by the discs diminishes the impact of the velocity deficit from the rod’s wake rapidly downstream. In addition, for the N disc, at the vertical centreplane, the disc has a 100% blockage core and the disc was oriented in a way such that bars are aligned with the mounting rod to minimise additional blockage through the mounting. Therefore, we expect the effect of the mounting rods to be secondary, particularly for the focus on wake merging.

The discs were aligned with respect to the wind tunnel and each other by utilising an alignment laser with a wide opening angle positioned upstream by the mounting of the PDs and two reference points farther downstream. The model solid blockage of two discs in the wind tunnel is 1.3%, and the blockage of the whole set-up, including the discs, rods and support structures is 6%.

Hot-wire anemometry was used for all velocity measurements. A Dantec Dynamics 55P01 single-wire probe was used, with a sensing element of 1.25 mm. A Dantec Dynamics StreamLine Pro constant temperature anemometer operated the hot-wire. An automated traverse was employed for moving the hot-wire to different measurement positions. The accuracy of the traversing system is approximately $\pm 0.5$ mm in both the $x$ - and $y$ -directions. The sampling time at each measurement point was $t_s =$ 180 s, which corresponds to 9000 flow turnovers based on the mean velocity and disc size. A sampling frequency of $f_s =$ 75047 Hz was used with a hardware low-pass filter of 30000Hz. In order to reduce the remaining high-frequency noise, an iterative digital low-pass filtering method (Mi et al. Reference Mi, Deo and Nathan2005, Reference Mi, Xu and Du2011) was used with a digital filtering frequency of $f_c\,{=}\,1.1 f_k$ , where $f_k$ is the Kolmogorov frequency. The Kolmogorov frequency was estimated by a fifth-order differential scheme in the same iterative process (Hearst et al. Reference Hearst, Buxton, Ganapathisubramani and Lavoie2012). The hot-wire measurements were temperature corrected as per Hultmark & Smits (Reference Hultmark and Smits2010). The hot-wire was calibrated approximately every 4–5 hours, and a weighted average of the calibrations was used for the measurements taken in between the calibrations. In order to reduce temperature drift during measurements, the wind tunnel was run at high velocity for approximately an hour before conducting experiments. The presented measurement procedure results in a maximum random error, calculated by the method of Benedict & Gould (Reference Benedict and Gould1996), of the mean velocity of 2.5% or less, depending on the turbulence intensity, but can increase to around 4.6% for single cases with very high turbulence intensities ( $u'/U\gt 0.25$ ), e.g. for the N1.5 case at the centreline; the maximum random error of the turbulence intensity is 1% or less. Mean velocities are converged within $\pm 0.5$ % after 45 s; standard deviations are converged within $\pm 1.0$ % after 120s or earlier if the turbulence is developed at the respective position, if not (e.g. at the wake borders or close to the discs where the standard deviations are also very low), the convergence is within $\pm 2.0$ %. Note that in general convergence and uncertainty are better in regions that are not subject to high intensity intermittent flow, i.e. the shear layers are more difficult to capture than the wake cores.

The wakes of two side-by-side M-type discs and two side-by-side N-type discs were investigated for three different spacings $\varDelta = 1.5D$ , 2.0 $D$ and 3.0 $D$ from centre to centre. These spacings allow us to investigate the merging of the wakes in different stages of their individual evolution, while also minimising interactions with the wind tunnel walls. The naming convention for the remainder of this work will refer to the disc configuration and the lateral spacing between the two discs. For example, M1.5 refers to the case with two M discs with a 1.5 $D$ spacing, and N3.0 refers to the case with two N discs with a 3.0 $D$ spacing. In addition, the wakes of both individual discs were measured in order to compare this with the dual-disc wakes. The wakes behind the PDs were measured at 8 $D$ and 30 $D$ downstream in the spanwise direction, as well as along the centreline in the streamwise direction from 0.5 $D$ to 40 $D$ . In other words, a total of 24 measurement series were conducted for this study. For the streamwise scans, the hot-wire was positioned so that it coincided with the centre between the two discs. The spacings of the measurements were adjusted for each case to capture relevant turbulence phenomena while simultaneously balancing the total acquisition time with the period over which the facility and calibration were thermally stable. Table 1 provides an overview of the test cases, as well as which discs were used with which disc spacing.

Table 1.

Summary of test configurations. Here, $x_i$ denotes the wake-interaction length determined by the shape parameter, i.e. where the wakes start interacting with each other, $x_s$ denotes the downstream position where the centreline wavenumber spectrum reaches a near –5/3 slope in the inertial region, an indication that the wakes have merged, $x_m$ denotes the downstream position where the wakes are merged with fully developed turbulence. A ‘–’ indicates that the specific value was not obtained within the measurement domain. Specific definitions for $x_i$ and $x_m$ are provided in § 5, and for $x_s$ in § 4.

3. Wake characteristics

Before engaging in a detailed discussion of the wake merging process, we will first provide details on the first- and second-order statistics of the wake of the two-disc system to understand the context in which wake merging occurs. Note that, since a single-wire is used for the measurements, all presented quantities are based on the streamwise velocity. From these statistics, it is also possible to get an initial notion of where the wakes start to interact, i.e. ‘feel’ each other, and merge, characterised by turbulence that has reached a certain level of maturity.

3.1. Mean velocity

Figure 4 displays the normalised mean streamwise velocity measured across the spanwise direction at 8 $D$ and 30 $D$ downstream of the discs for all cases. If the normalised mean velocity at the centreline, $y/D = 0$ , is lower than that of the free flow, $\bar {U}/U_{\infty }\lt 1$ , this is a momentum-based indication that the wakes are interacting. Figures 4(a) and 4(c) show these profiles for the side-by-side N and M configurations, respectively, with different spacing between the discs, and figure 4(e) displays the mean velocity profiles for the single-disc cases for reference, all for $8D$ . Figure 4(a) shows that the double wake of the N discs is wider than that of the M discs. Comparing the wakes in figures 4(a) and 4(c) with their respective single-disc counterparts in figure 4(e) shows that the wake merging will not only be influenced by the disc spacing, but it is also heavily influenced by the disc design as well, as the wake of the N disc is both wider and less pronounced than that of the M disc, which is in agreement with Vinnes et al. (Reference Vinnes, Neunaber, Lykke and Hearst2023). As mentioned above, this point is significant, as these two discs have the same drag coefficient, $C_D\approx 0.8$ , but they still have different wakes and therefore wake merging. Using the drag coefficient as the only significant characteristic when replacing a wind turbine with a PD is therefore a limited approach, which was one of the main findings of Helvig et al. (Reference Helvig, Vinnes, Segalini, Worth and Hearst2021) and later Kurelek et al. (Reference Kurelek, Piqué and Hultmark2023). From the mean velocity profiles, we find that the side-by-side M disc profiles interact minimally at 8 $D$ and can be represented well by two single M disc wakes superimposed with different disc spacings; basic superposition of the wakes is illustrated in Appendix A. The same cannot be said for the N disc velocity profiles, where only the 3 $D$ disc spacing case, N3.0, appears as two single N disc wakes superimposed, cf. Appendix A. For the N2.0 case, it is still possible to discern two separate wakes, but the lower centre velocity suggests they have already started interacting. With N1.5, the two N disc wakes have already merged to a single velocity profile at $8D$ downstream with one distinct mean velocity minimum at $y/D = 0$ . This combined velocity deficit is higher than the maximum velocity deficit of the single N disc. This demonstrates that the dual-disc configuration results in greater flow retardation than the single-disc case.

Figure 4.

Evolution of the normalised mean streamwise velocity measured across the spanwise direction at 8 $D$ and 30 $D$ for all cases. The panels show the wakes of the N disc configurations at (a) 8 $D$ and (b) 30 $D$ ; the M disc configurations at (c) 8 $D$ and (d) 30 $D$ ; and the single-disc cases at (e) 8 $D$ and ( f) 30 $D$ . Note that the $y$ -axes in the 8 $D$ and the 30 $D$ cases are scaled differently.

Figures 4(b), 4(d) and 4( f) show the same velocity profiles as discussed above, but at 30 $D$ downstream from the discs. As expected, all wake velocity profiles are wider and have smaller gradients farther downstream due to entrainment. By comparing the single M disc wake in figure 4( f) with the M3.0 case in figure 4(d), we can conclude based on the mean velocity profiles that these wakes have yet to interact even at 30 $D$ since they are identical to the single-disc wake both in shape and velocity magnitude. The M1.5 and M2.0 cases in figure 4(d) both show that the wakes have started to interact and merge, since the mean velocity at $y/D=$ 0 is decreasing. Using the same reasoning, the wakes in the N3.0 case in figure 4(b) interact but have yet to fully merge as well. As indicated, a simple possibility to find where turbulence interaction plays an important role in the merging process is to compare the superposition of the mean velocity profiles of the single discs with the measured profiles (see Appendix A). For the mean velocity profiles, superposition is a reasonable approximation at both 8 $D$ and 30 $D$ for cases where the wakes do not meet close to the PDs, i.e. with the exception of the N1.5 and N2.0 cases.

Figure 5.

Evolution of the normalised mean streamwise velocity measured in the streamwise direction for all dual-disc cases. (a) The evolution for the N cases; (b) the evolution of the M cases.

Figure 5 displays the mean velocity, normalised by the velocity at the first measurement position, along the centreline, with the N disc cases in figure 5(a) and the M disc cases in figure 5(b). The N1.5 case in figure 5(a) has a strong gradient just downstream of the discs, and the mean velocity slowly recovers, but even at 40 $D$ downstream, a pronounced velocity deficit remains. The N2.0 case has a weaker gradient than the N1.5 case, while the N3.0 case does not exhibit this behaviour but has a weak velocity decline downstream of the discs. As seen in figure 4, the N configurations have wide wakes, which means they will meet earlier in their spatial evolution. For the N1.5 case, the discs are placed close enough together that their individual wakes start interacting almost immediately downstream of the discs. At this point in the flow, there is a strong shear layer between the wake and the free stream, and the turbulence production within the individual wakes is still high. It is worth reiterating that each wake has a characteristic evolution on its own, including turbulence build-up and decay, cf. figure 1. Depending on the disc spacing, the two wakes will meet either in a region with high turbulence production, such as in the N1.5 case, as will be discussed in §§ 3.2, or in a region where the single wakes are already decaying. We see a strong gradient in figure 5 for the N1.5 case, since the wakes meet while the turbulence production is still high. This argument can be extended to the N2.0 case, although it has a weaker gradient than the N1.5 case, the individual disc wakes still meet early in their evolution where the wakes have not decayed significantly. For the N3.0 case, however, the disc spacing is large enough that, despite a wide wake width, the wakes meet sufficiently far downstream such that the wakes have initiated their decay before they start interacting such that we observe a ‘soft’ merging, indicated by a gradually decreasing velocity. In figure 5(b), the M1.5 case shows a similar trend to the corresponding case in figure 5(a), only farther downstream and with a less steep gradient. The same can be said for the M2.0 case, but for the M3.0 case there are no significant changes in the magnitude of the mean velocity, further confirming that in this case the wakes of the two M discs have, for this parameter, not started to significantly interact within the measurement domain. In other words, the M configurations follow the same trend as the N configurations, but differ in that the M disc wakes meet farther downstream.

3.2. Turbulence intensity

The turbulence intensity was calculated by dividing the standard deviation of the streamwise velocity fluctuations, $u'$ , by the local mean streamwise velocity, $U$ . Figure 6 shows the spanwise $u'/U(y)$ profiles for all cases at 8 $D$ and 30 $D$ downstream. If the turbulence intensity at the centreline exceeds the value of the background flow, $u'/U(y = 0)\gt 0.3$ %, this is a turbulence-based indication that the wakes are interacting. The maximum magnitude of the spanwise $u'/U(y)$ profiles coincides with where the largest velocity deficit is for the fully merged wake profiles in the N1.5 case at 8 $D$ , and the N1.5 and N2.0 cases at 30 $D$ . For the other cases, the identities of the single wakes are still recognisable to varying degrees so that the maximum magnitude of the spanwise $u'/U(y)$ profiles is in the (inner) shear layer. The magnitude of the spanwise turbulence intensity profiles, $u'/U(y)$ , decays in all cases between the $8D$ and $30D$ scans. Note that the $y$ -axes of figure 6 for the cases at 8 $D$ and the cases at 30 $D$ are scaled differently. Comparing the dual-disc configurations with their single-disc counterparts, one can see that the magnitude of the spanwise $u'/U(y)$ profiles is roughly the same, regardless of disc spacing. The only exception is the N1.5 case, where the $u'/U(y)$ -peak at $y/D = 0$ is slightly larger than its maximum value for the single N disc at both $8D$ and $30D$ downstream. This is due to the wakes meeting relatively close to the discs in this case, where the turbulence production for the individual wakes is high as well. The degree to which simple superposition of the spanwise $u'/U(y)$ profiles of the single discs matches the measured side-by-side wakes is illustrated in Appendix A. For the spanwise $u'/U(y)$ profiles, simple superposition gives strong agreement in the case of little wake interaction (i.e. N2.0, N3.0, M1.5, M2.0, M3.0 at 8 $D$ , and M3.0 at 30 $D$ ), but overestimates the magnitude in the centre of merged wakes (N1.5 at 8 $D$ and N1.5, N2.0, N3.0, M1.5 and M2.0 at 30 $D$ ).

Figure 6.

Evolution of $u'/U(y)$ profiles measured across the spanwise direction at 8 $D$ and 30 $D$ for all cases. The panels show the the spanwise $u'/U(y)$ profiles of the N disc configurations at (a) 8 $D$ and (b) 30 $D$ ; the M disc configurations at (c) 8 $D$ and (d) 30 $D$ ; and the single-disc cases at (e) 8 $D$ and ( f) 30 $D$ . Note that the $y$ -axes in the 8 $D$ and the 30 $D$ cases are scaled differently.

Displayed in figure 7 is the evolution of the streamwise $u'/U(x)$ profiles for all dual-disc cases. Both configurations show higher values of the streamwise $u'/U(x)$ profiles for smaller disc spacings, but differ in magnitude. From 1.5 $D$ to 2 $D$ the peak magnitude of the streamwise $u'/U(x)$ profiles for both configurations is halved. Also, the maximum value of the streamwise $u'/U(x)$ profiles for the N1.5 case is almost thrice the maximum $u'/U(x)$ value for the M1.5 case.

Based on both the spanwise and centreline turbulence intensity profiles, it is possible to get a crude notion of how the wake merging occurs for the different dual-disc cases. The approach of Obligado et al. (Reference Obligado, Klein and Vassilicos2022) to quantifying the wake-interaction length through linear regression in the monotonically increasing region before the wakes merge requires a higher number of measurement points in this region than were acquired in the present study. Duplicating their approach was not an objective here, and other means of assessing wake interaction are considered in the subsequent sections. Nonetheless, the $u'/U(x)$ peaks in the centreline measurements can be considered a first indication of wake merging, as the start of the decay of the turbulence intensity indicates that the wakes have merged. For M2.0 and M3.0, the wakes start to interact and merge where the turbulence intensity of the single discs is already decaying since their wake width is smaller, which again leads to a smaller peak of the streamwise $u'/U(x)$ profiles along the centreline compared with their N disc counterparts.

Figure 7.

Evolution of the streamwise $u'/U(x)$ profiles at the centreline for all dual-disc cases. (a) The centreline profiles for the N disc cases; (b) the centreline profiles for the M disc cases.

4. Development of a spectral inertial range between the discs

A classical way to assess the development of a turbulent flow is to monitor the establishment of an inertial range in the spectrum. Close to the discs, the flow is near laminar at the centreline (cf. figure 7) and the spectra have low energy across all frequencies and do not exhibit the classical shape of a turbulence spectrum with an established inertial range. When the wakes start to interact, the turbulence also starts to develop until it is mature and develops an inertial range with the classical $-5/3$ decay. Therefore, by identifying when an inertial range with $-5/3$ decay has been established, one can conclude that the wakes have merged sufficiently. Note that this method of identifying the interaction of wakes tracks the transition to turbulence from a non-turbulent state, and thus a similar approach would be more nuanced if the background flow was already turbulent, as one would seek a change in the turbulence structure rather than the development of turbulence. As will be detailed later in § 5, fully developed turbulence would also require the flow to exhibit internal intermittency only at small scales.

As can be seen in figure 7, initially, the inflow is near laminar at the centreline (background turbulence intensity of 0.3%). Once turbulence is introduced along the centreline, it implies that the two wakes have started to interact. While turbulence along the centreline indicates the presence of wake interaction, it is not until the turbulence has had sufficient time to interact and mature (Obligado et al. Reference Obligado, Klein and Vassilicos2022) that the inertial range of the turbulence spectrum exhibits the classical $-5/3$ power-law relation.

Figure 8 shows the flow development from near-laminar to interacting turbulent wakes to merged wakes with an established spectral inertial range with $-5/3$ decay in the two-disc configurations with $2D$ spacing. Figures 8(a) and 8(c) are velocity spectra in frequency space, whereas the figures 8(b) and 8(d) are compensated wavenumber velocity spectra. Plotting the wavenumber spectra in the compensated form is a way to emphasise the $k^{-5/3}$ power law which should appear as a plateau when such a range is achieved. Figure 8(b) shows compensated wavenumber spectra for N2.0, and figure 8(d) shows compensated wavenumber spectra for M2.0. At the first measurement point, $x/D =$ 0.5, both configurations exhibit spectra of near-laminar flow that have low energies and lack the characteristic developed inertial range of turbulent flows. At $x/D =$ 2 for the N2.0 case and $x/D =$ 8 for the M2.0 case, the flows have gone from the near-laminar behaviour to starting to exhibit some properties of classic turbulent flow, as can be seen by increased energy and the start of the development of an inertial range. However, the gradients in the inertial range are still too steep. At the measurement point $x/D =$ 10 for the N2.0 case and $x/D =$ 18 for the M2.0 case, the flows have an established spectral inertial range that follows the $-5/3$ power law closely in the inertial range, since they are now approximately straight lines with zero slope in the compensated wavenumber spectra. It can also be noted that spectra after these downstream positions do not alter in shape, but either collapse with each other or exhibit turbulence decay through a shorter inertial range and a decrease in energy. As can be seen in figures 8(a) and 8(b), the N discs exhibit vortex shedding as previously characterised by Vinnes et al. (Reference Vinnes, Gambuzza, Ganapathisubramani and Hearst2022a ), as identified by the peak in the spectra at low frequencies.

Figure 8.

Spectra examples for different centreline measurements downstream of the discs. Spectra for N2.0 in (a) frequency space and (b) compensated wavenumber space. Spectra for M2.0 in (c) frequency space and (d) compensated wavenumber space. The axes are left dimensional to show the increase in the separation of scales and the magnitude of the kinetic energy dimensionally.

In order to determine where the threshold between interacting wake turbulence and turbulence with an established spectral inertial range is, the evolution in the streamwise direction of the compensated wavenumber spectra is explored in more detail. First, for each case, spectra for all downstream centreline measurements were calculated and a moving average with equally spaced frequency interval in log space was applied (Fuchs et al. Reference Fuchs, Kharche, Patil, Friedrich, Wächter and Peinke2022). The approximate centre of the inertial range of a measurement was determined using the Taylor micro-scale, that was calculated by estimating the gradients using Taylor’s hypothesis and a seventh-order centred-difference scheme as described by Hearst et al. (Reference Hearst, Buxton, Ganapathisubramani and Lavoie2012). From this point in the spectrum, starting with 5 points, linear regression is performed on $\log {(E(k)\cdot k^{5/3})}= a +S\cdot \log k$ . The fit range was extended in steps of 10 points towards higher wavenumbers as long as the fit error for the slope $S$ decreases. Next, in a similar manner, the fit range was also extended to lower wavenumbers. Note that while a Taylor micro-scale can be calculated and a line can be fit to all spectra, including near-laminar cases, both only have a physical meaning in the case of turbulent flows. As is most clear in figure 8(b), the range of wavenumbers where the slope is close to zero within the inertial range depends on the downstream position. This is because the flow reaches a maximum turbulent kinetic energy value during mixing of the two individual wakes along the centreline, before the turbulent kinetic energy starts to decay downstream of the maximum point. The slope at each measurement position was then used as the parameter for identifying if an inertial range was present. A threshold on the linear regression slope, $S_{th}$ , was chosen for determining whether the turbulence at a measurement point fulfilled the $-5/3$ criterion in the inertial range or not. This threshold was used for all cases, and was chosen to be $S_{th} =\pm 0.15$ which is approximately 10% of $-5/3$ . The threshold was chosen to be small, while also inclusive of the final state of each case at the most downstream position.

The results of this endeavour can be seen in figure 9. The figure displays the slope within the inertial range as a function of downstream distance of the discs for both configurations and with the three different disc spacings. The first downstream measurement point where the slope threshold criterion is fulfilled is marked with a symbol with blue filling. Generally, the side-by-side N disc configurations reach an established spectral inertial range with an average extension of half a decade and with $-5/3\pm 0.15$ decay for all disc spacings in a short distance downstream of the bodies, i.e. within 10 $D$ downstream. This is in stark contrast to the side-by-side M disc configurations, which require up to 24 $D$ downstream to achieve a mature state as determined by the appearance of an inertial range that also has an average extension of approximately half a decade. A summary of the identified locations is provided in table 1 as $x_s$ . These values were found through linear interpolation between the first measurement point that fulfilled the slope criterion and the closest measurement point upstream of that. It should be noted that the slopes determined in the near field close to the discs should be interpreted with care as there is no clear inertial range, see also figure 8.

Figure 9.

Slope within the inertial range of the compensated wavenumber spectra for (a) the N configurations and (b) the M configurations. Thresholds ( $\pm 0.15$ ) are marked with black lines. Empty symbols are used to identify positions before wake merging, and filled symbols identify where the slope is within the threshold, i.e. where the wakes have merged. The first downstream position where the slope is within the threshold is marked with a blue filled symbol. The root mean square error at each measurement point is contained within the marker size.

5. Wake merging as identified by internal intermittency

Another physical mechanism at play during wake merging is internal intermittency. As explained in the introduction, it has been shown that wakes, both in near-laminar and in turbulent inflow, are surrounded by a ring of high internal intermittency (cf. figure 1 and Schottler et al. Reference Schottler, Bartl, Mühle, Sætran, Peinke and Hölling2018; Neunaber Reference Neunaber2019; Vinnes et al. Reference Vinnes, Neunaber, Lykke and Hearst2023), and we hypothesise that these rings are the first parts of the wake that will meet as they exist outside of typical wake width descriptions. The definition of internal intermittency employed herein refers to the probability of sudden and large changes of velocity in the flow, which can be described as the extreme events in a PDF of velocity increments. These sudden velocity fluctuations would not be detected by use of only one-point statistics, therefore two-point statistics are necessary. For calculating the degree of internal intermittency in the flow, the central velocity increment is used. It is defined as $\delta u_{\tau }(t) = u(t-\tau /2) - u(t+\tau /2)$ , and describes the velocity difference between two different points in time, their separation represented by a time scale $\tau$ . While the intermittency can be investigated through the moments of the central velocity increment (Toschi et al. Reference Toschi, Amati, Succi, Benzi and Piva1999; Pope Reference Pope2000), $\delta u_{\tau }^n$ , several investigations have estimated the internal intermittency from the Castaing parameter $\lambda ^2(\tau )$ , e.g. Peinke et al. (Reference Peinke, Klein, Kittel, Okninsky, Parisi and Roessler1993), Naert et al. (Reference Naert, Puech, Chabaud, Peinke, Castaing and Hebral1994), Schottler et al. (Reference Schottler, Bartl, Mühle, Sætran, Peinke and Hölling2018), Neunaber et al. (Reference Neunaber, Hölling, Stevens, Schepers and Peinke2020), Maus et al. (Reference Maus, Peinke and Hölling2022) and Vinnes et al. (Reference Vinnes, Neunaber, Lykke and Hearst2023). The Castaing parameter is a quantification of the deviation of the PDF of the velocity increments, $p(\delta u_{\tau }(t))$ , from a Gaussian probability distribution. If $\lambda ^2 \gt 0$ , then the flow has a higher probability of experiencing large changes in the velocity increments. It was introduced by Castaing, Gagne & Hopfinger (Reference Castaing, Gagne and Hopfinger1990) as a shape parameter of the probability density distribution of the velocity increments, and Chilla, Peinke & Castaing (Reference Chilla, Peinke and Castaing1996) showed that a simplified calculation is possible for fully developed turbulence

(5.1) \begin{equation} \lambda ^2(\tau ) = \frac {\text{ln}(F(\delta u_{\tau }(t))/3)}{4}, \end{equation}

where $F(\delta u_{\tau }(t))$ is the flatness of the velocity increments. In this work, we will use the spatial equivalent of $\tau$ , $\rho$ , by utilising Taylor’s frozen flow hypothesis to convert the time scale to a spatial scale. The two line plots in figure 10 represent the Castaing parameter $\lambda ^2(\rho )$ at two downstream positions along the centreline, $x/D = 2$ and $x/D = 5$ , for the N1.5 configuration across multiple spatial scales $\rho$ . Fully developed turbulent flows typically have both an established spectral inertial range with –5/3 decay and some internal intermittency for small scales $\rho$ , here $\rho \lesssim D$ , that decays according to a power law, and diminishes to zero for large scales, here $\rho \gtrsim D$ . This is what we see at $x/D = 5$ . For $x/D = 2$ , however, we see large values of $\lambda ^2(\rho )$ across all scales, indicating that at this downstream position the flow is not yet fully developed.

Figure 10.

The Castaing parameter across multiple spatial scales $\rho$ for N1.5 at two downstream positions. At $x/D$ = 2.0, high values of intermittency can be observed across all scales, indicating the turbulence is still developing. At $x/D$ = 5.0, there is some intermittency, but this is restricted to the smaller scales; this is typical of fully developed turbulence.

In terms of wake merging, detection of internal intermittency can be used for identifying the starting point for when the two wakes start interacting, $x_{i}$ , and the end point where the turbulence in the centre of the two wakes is fully developed, $x_{m}$ . Since the centreline measurement is done exactly in the middle of the two discs, the evolution of the initial individual wakes should be symmetric, which is confirmed in § 3 for both the mean velocity and the turbulence intensity. As shown by Vinnes et al. (Reference Vinnes, Neunaber, Lykke and Hearst2023), both disc designs show a ring of intermittency outside the momentum-defined wake, and therefore observing the intermittency rings from the two discs meeting at the midpoint between them would be the first indication of any wake interaction. Figure 11 shows the Castaing parameter contour plots for the single-disc cases at 8 $D$ and their variation in the spanwise direction. Here, $\lambda ^2(\rho )$ is the contour parameter, and increasing darkness indicates a higher probability of extreme events. The bands on the outer edges of the discs across all spatial scales indicate the presence of intermittency rings. Since Schottler et al. (Reference Schottler, Bartl, Mühle, Sætran, Peinke and Hölling2018) showed that there is a ring of intermittency, the spanwise profiles in figure 11 can be interpreted as a ring, although the measurement was done in one spatial dimension. For both single-disc cases, distinct bands can be observed, in agreement with the findings of Vinnes et al. (Reference Vinnes, Neunaber, Lykke and Hearst2023) of an intermittency ring on the outer edges of both discs’ wakes.

Figure 11.

Contour plots of the Castaing parameter $\lambda ^2(\rho )$ across spatial scales $\rho$ in the spanwise direction at 8 $D$ downstream for the single-disc cases: (a) N, and (b) M. Increasing darkness indicates a higher $\lambda ^2(\rho )$ value and thus a higher probability of extreme events. The blue solid lines indicate the centres of the discs while the blue dotted lines indicate the disc edges.

Figure 12 displays Castaing parameter contour plots for all dual-disc cases at 8 $D$ downstream and their variation in the spanwise direction. For the M2.0 and M3.0 cases in figures 12(d) and 12( f), respectively, it is possible to discern clear and individual intermittency rings for both discs. For the M1.5 case in figure 12(b), however, the outer bands of both discs are visible, and a thin and weaker band is identifiable between the discs. This is due to the fact that wakes of the M discs have already started interacting. The side-by-side N configurations show a similar trend, but with overall higher $\lambda ^2(\rho )$ across all spatial scales and over a larger spanwise length. For the N1.5 and the N2.0 cases in figures 12(a) and figure 12(c), respectively, no trace of the individual intermittency rings can be found, as the wakes in these configurations have already started to merge or, respectively, have already merged in the N1.5 case. With a disc spacing of 3 $D$ with the N discs, however, the lateral distance is large enough that at 8 $D$ , a clear band is visible between the discs in figure 12(e).

Figure 12.

Contour plots of the Castaing parameter $\lambda ^2(\rho )$ across spatial scales $\rho$ at 8 $D$ downstream for the dual-disc cases: (a) N1.5, (b) M1.5, (c) N2.0, (d) M2.0, (e) N3.0 and ( f) M3.0. Increasing darkness indicates a higher $\lambda ^2(\rho )$ value and thus a higher probability of extreme events. The blue solid lines indicate the centres of the discs while the blue dotted lines indicate the disc edges.

To create a clear link between intermittency and wake merging, we chose to look closer into $\rho /D = 1$ , i.e. a spatial scale of the same order of magnitude as the disc diameter $D$ , at the centreline. We define the start of the wake interaction, $x_{i}$ , when the Castaing parameter at this spatial scale exceeds $\lambda ^2(\rho =D)\gt 0.2$ . This value was chosen as the flow is clearly intermittent. Note that very large values of $\lambda ^2(\rho )\gg 0.3$ should be interpreted with care as this is an indication of not fully developed turbulence (cf. Appendix A in Vinnes et al. Reference Vinnes, Neunaber, Lykke and Hearst2023). Extending this argument further, $x_{m}$ is defined as when the Castaing parameter at the spatial scale $\rho /D = 1$ has decreased back to $\lambda ^2(\rho =D) = 0$ , where no internal intermittency is present at these scales. The results of this analysis can be seen in table 1. These values were found through linear interpolation between the measurement point that fulfilled the given internal intermittency criterion and the closest measurement point upstream. It should be noted that the sequence of wake merging events identified by the inertial range of the spectra and the Castaing parameter give consistent results for all cases, as can be seen in table 1. First, the intermittency rings interact, marked by $x_i$ , then the spectra exhibit an inertial range, marked by $x_s$ , and finally, the turbulence becomes fully developed, as indicated by $x_m$ .

The downstream evolution of the Castaing parameter is plotted in dependence of $\rho$ in figures 13 and 14. Figure 13 presents the Castaing parameter for the single discs, whereas figure 14 presents the Castaing parameter for the dual-disc configurations. Other spatial scales than $\rho /D = 1$ are included to emphasise the point made earlier about how high internal intermittency values across all scales is yet another indication that the flow is not fully developed. Figure 13 reveals that the disc design impacts the downstream position where the intermittency reaches its maximum, with the N disc design impacting the flow almost immediately downstream of itself, while the M disc design does not significantly influence the intermittency until approximately 7 $D$ downstream due to the narrow shear layers.

Figure 13.

Castaing parameter contour plots for downstream positions along the centreline for the single-disc cases: (a) N, and (b) M. Increasing darkness indicates a higher $\lambda ^2(\rho )$ value and thus a higher probability of extreme events.

Figure 14.

Castaing parameter contour plots for downstream positions along the centreline for the dual-disc cases: (a) N1.5, (b) M1.5, (c) N2.0, (d) M2.0, (e) N3.0 and ( f) M3.0. Increasing darkness indicates a higher $\lambda ^2(\rho )$ value and thus a higher probability of extreme events.

As can be seen in figures 14(a), 14(c) and 14(e), where the internal intermittency results for the dual-N cases are presented, the wakes start interacting almost immediately for the N1.5 case at 2 $D$ downstream, closely followed by the N2.0 case where the wakes start interacting at 3 $D$ downstream. After the wake interaction starts, the flow in these cases quickly converges downstream to fully developed turbulence with $\lambda ^2(\rho \gt D)\approx 0$ , while there is still intermittency present at the smallest spatial scales $\rho \lt D$ . This does not happen for the N3.0 case, where the wakes start interacting at 4 $D$ , and $\lambda ^2(\rho )\gt 0$ for all spatial scales $\rho$ downstream of $4D$ . With the logic previously presented in this section, the N3.0 case has therefore yet to become fully developed turbulence within the measurement domain. Figures 14(b), 14(d) and 14( f), show that the internal intermittency for the dual-M discs evolves differently. While the start of the wake interaction happens at a later downstream position compared with their N counterparts, the downstream lengths for when the Castaing parameter converges are also longer. In fact, for the M3.0 case, internal intermittency is still present within the measurement domain across all spatial scales, just like for the N3.0 case. The presence of intermittency at larger spatial scales $\rho \gt D$ indicates, as stated previously, that the turbulence has yet to reach its fully developed state. In other words, neither dual-disc wake with 3 $D$ disc spacing reaches fully developed turbulence, as assessed by internal intermittency, within the measurement domain.

6. Mapping porous disc wake merging

An interesting finding is that the internal intermittency results for when the turbulence reaches its fully developed state occur farther downstream than the corresponding turbulent spectrum indicator. In the M3.0 and N3.0 cases, the internal intermittency does not decrease back to approximately zero within the measurement domain for scales $\rho =D$ .

By comparing the $u'/U(x)$ plots in figure 7 for the centreline measurements with the two differing wake merging results, the streamwise $u'/U(x)$ profiles coincide with the internal intermittency prediction of the wake merging. For both disc designs, the 1.5 $D$ and 2 $D$ disc spacing cases reach a $u'/U(x)$ -peak and then slowly decay downstream. However, for the N3.0 case, the centreline $u'/U(x)$ plot in figure 7(a) reveals that there is no real peak, but rather a plateau that is reached within the measurement domain. For the M3.0 case figure 7(b) shows that $u'/U(x)$ barely increases at all within the 40 $D$ downstream that were measured, and it seems it does not reach a peak within our measurement domain. This corresponds well with the mean streamwise velocity profile in figure 5(b) for the M3.0 case, where the mean velocity does not experience a significant deficit. This further indicates that although the wakes might be interacting, they have yet to merge, at least as defined based on the intermittency criterion described herein.

These results illustrate that, while the spectra reach a slope of approximately $-5/3$ in the inertial range, there are still turbulence mechanisms at play which affect the flow. Schwarz et al. (Reference Schwarz, Ehrich, Martín and Peinke2018) showed that, for flows with identical spectra where one has intermittency present and one does not, the intermittent flow causes higher fatigue loading on wind turbines than the non-intermittent flow. Internal intermittency is thus important for understanding wake merging in general, but also specifically when discussing it in relation to wind turbines. As we have shown, internal intermittency is present for a greater downstream distance when compared with most other wake merging criteria.

Figure 15.

Conceptual wake map. The order of turbulence evolution events is independent of disc design. The three characteristics defined in the work herein are marked with ‘I’, ‘II’ and ‘III’. The map is not to scale and depends on the disc spacing, and is meant to give the reader a conceptual understanding of what happens in the flow.

Table 1 highlights that, although the downstream positions themselves may differ between the disc designs and spacings, the order of turbulence evolution events does not change. A wake map can be found in figure 15, showing the order of events within the flow. The intermittency ring of each wake lies outside the momentum-defined wake, and appears not far downstream from the discs. As Vinnes et al. (Reference Vinnes, Neunaber, Lykke and Hearst2023) noted, the occurrence of the intermittency ring is not immediately downstream of the discs. Once the individual intermittency rings meet, the Castaing parameter along the centreline between the two discs at this downstream position increases, indicating wake interaction. This is marked with ‘I’ in the wake map, and corresponds to our criterion $\lambda ^2(\rho =D) \gt 0.2$ . The shear layers of the PDs widen (Lignarolo et al. Reference Lignarolo, Ragni, Ferreira and Van Bussel2016), and once they meet, they start interacting. This causes a build-up of the turbulence intensity which eventually reaches a peak. This is similar to the evolution of turbulence in the centre of a single wake (Neunaber Reference Neunaber2020). Before this peak, the turbulence has both built up and developed sufficiently for the near –5/3 slope in the turbulence spectrum’s inertial range to occur. This is marked with ‘II’ in the wake map. After $u'/U(x)$ peaks, the turbulence decays downstream until it sufficiently resembles homogeneous isotropic turbulence for the Castaing parameter to have decreased back to approximately zero. This is marked with ‘III’ in the wake map, and corresponds to the criterion $\lambda ^2(\rho =D) \approx 0$ . The $u'/U(x)$ decay combined with both spectra that have a near $-5/3$ slope in the inertial range and intermittency only at small spatial scales indicates fully developed turbulence with features of homogeneous isotropic turbulence at this downstream position. Internal intermittency is also found at small spatial scales in the individual wake cores, and the absence of intermittency at the large scales indicates that the flow in these regions is fully developed with features of homogeneous isotropic turbulence. This has been shown for a single wind turbine by Neunaber et al. (Reference Neunaber, Hölling, Stevens, Schepers and Peinke2020) and a single PD by Neunaber et al. (Reference Neunaber, Hölling, Whale and Peinke2021). In the 1.5 $D$ cases, i.e. N1.5 and M1.5, the Castaing parameter exceeds 0.2 at approximately the same downstream position as the wavenumber spectra reach the near $-5/3$ slope.

The present investigation does not explicitly include the addition of background turbulence as it was a study objective to determine how the flow transitions to turbulence and evolves between the two wake-generating objects. Nonetheless, in many flows of practical interest, background turbulence may exist. Studies of the evolution of individual wakes of these PDs in turbulent flows, e.g. Vinnes et al. (Reference Vinnes, Neunaber, Lykke and Hearst2023), suggest that the developed conceptual map of wake merging regions is also valid for low and moderate turbulent inflows. In such cases, the expectation is that the wakes will develop more rapidly downstream and merging will occur earlier, but the sequence of events is not fundamentally different. A difficulty might arise from the identification of the inertial range in the spectra, as the turbulent inflow would already have an inertial range. For very high background turbulence intensities, entrainment may reverse Kankanwadi & Buxton (Reference Kankanwadi and Buxton2020) and the reported map may not hold. At present, we have not explicitly quantified the location of the events as these are dependent on the geometry of the wake-generating object, the spacing and perhaps the inflow conditions.

To summarise, by means of cascading turbulence events at the centreline between two discs, we identify three key indicators of the interaction and merging of two wakes, namely the build-up of intermittency, marked by $\lambda ^2(\rho =D) \gt 0.2$ , the convergence of the inertial range of velocity spectra to follow a $-5/3$ decay and the transition to fully developed turbulence with features of homogeneous, isotropic turbulence, marked by $\lambda ^2(\rho =D) \approx 0$ .