2.1. Experimental set-up

The experiments were carried out in the water channel facility at the Norwegian University of Science and Technology. The details of the facility are described by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021) in Appendix A of their article. The test section provides a free-surface open-channel test bed filled to a height of 0.22 m ($H$) for the present investigation and measures 1.8 m in width and 11 m in length. To trip the boundary layer, a strip of multiscale grip tape with a length of 50 mm in the streamwise direction was placed at the bottom wall immediately downstream of the active grid, spanning the whole width of the test section. To effectively dampen the surface waves created by the motion of the grid bars, two slanted surface plates were installed on both the upstream and downstream sides of the active grid. The plate installed inside the contraction measured 1.5 m in the streamwise direction and was cut to the shape of the contraction out of a 6-mm-thick marine-grade aluminium sheet. The surface plate placed inside the test section was a rectangular 10-mm-thick acrylic plate, measuring 1.8 m in width and extending 3 m downstream (see figure 1 for a schematic).

Figure 1. Schematic of the water channel, active grid, and measurement set-up.

Since the introduction of active grids, pioneered by the seminal work of Makita (Reference Makita1991), they have become powerful tools for researchers to effectively tailor the turbulent characteristics of flows. Applying minor modifications, the same active grid utilised by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021) was used to set the incoming flow conditions in the present investigation (see Jooss et al. Reference Jooss, Li, Bracchi and Hearst2021, Appendix B, for detailed information regarding the design and operation of the grid). In short, the grid consists of 18 vertical and 10 horizontal bars each of which was connected to a stepper motor, allowing for the independent control of each bar. The agitator wings were built in a square shape out of 1-mm-thick stainless steel sheets with holes cut out of them to reduce the blockage ratio. In order for the grid to fit the cross-section of the flow, the configuration of the wings was modified compared with that used by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021). This included using only two horizontal bars and installing an additional row of half-wings on the vertical bars (figure 1). The grid mesh length ($M$), i.e. the distance between the grid bars, was 100 mm. Fully random sequences of the rotational period, acceleration and velocity as described by Hearst & Lavoie (Reference Hearst and Lavoie2015) were fed to the motors. The mean rotational velocity was changed between test cases to create different incoming flow conditions as described in § 2.3.

2.2. Experimental measurements

Planar PIV measurements were performed in streamwise–wall-normal planes in the spanwise centre of the channel. To evaluate the streamwise evolution of the flow, four streamwise planes were examined located at $X = 55M$, $65M$, $72M$ and $85M$, where $X$ represents the streamwise axis of the global coordinate system, whose origin is at the active grid location. Polystyrene particles with a mean diameter of 40 $\mathrm {\mu }$m (Dynoseeds TS40, with a density of 1050 kg m$^{-3}$) were mixed with recirculated water and used as tracer particles. A Litron Nano L PIV laser (dual-pulse Nd:YAG with a maximum energy of 200 mJ per pulse) was utilised to illuminate the fields of view (FOVs). Before entering the channel through the glass floor, the laser beam was reflected upwards by a Thorlabs Laser Line mirror and transformed into a sheet by a series of two spherical and one cylindrical lenses (LaVision Sheet Optics). The laser sheet was focused so that its thickness was less than 1 mm throughout the FOV. A LaVision Imager MX 25 MP camera ($5120 \times 5120$ pixels with a pixel size of 4.5 $\mathrm {\mu }$m), fitted with a Sigma 180 mm $f$/2.8 EX DG lens, captured double-frame images of the flow fields. The camera was set to capture FOVs of approximately $200\ {\rm mm} \times 200\ {\rm mm}$ in the streamwise and wall-normal directions, resulting in a magnification factor of 0.115. An $f$-stop of 5.6 was used throughout the measurements, yielding particle image sizes of more than 2 pixels (according to the Gaussian fits to the autocorrelation of particle images and consistent with the approximate equation found in Smith & Neal (Reference Smith and Neal2016) and Adrian & Westerweel (Reference Adrian and Westerweel2011)). A programmable timing unit (LaVision PTU X) synchronised the laser pulses with the exposure period of the camera. Double-frame particle images were recorded at 2 Hz using LaVision DaVis 10.1 software, providing independent samples for the targeted analyses. The same software was used later to process the image pairs, employing an iterative cross-correlation process with a square $128 \times 128$ window for the first pass and a $48 \times 48$ window as the final pass with an overlap of 75 %. The resulting nominal spatial resolution and vector spacing were 2 and 0.5 mm, respectively. The preprocessing of the image pairs, i.e. subtracting the background, smoothing the particle images, intensity normalisation and subtracting the sliding background, was performed with special care to optimise the signal-to-noise ratio and reduce PIV uncertainties. The resulting velocity uncertainties, estimated by DaVis based on the cross-correlation fields (Sciacchitano et al. Reference Sciacchitano, Neal, Smith, Warner, Vlachos, Wieneke and Scarano2015; Wieneke Reference Wieneke2015), were approximately $2\,\%$ of $U_\infty$ next to the bottom wall of the channel and dropped below $1\,\%$ in the regions away from the edges of the FOV. A sufficient particle image size together with the image preprocessing procedure circumvented pixel-locking issues (Hearst & Ganapathisubramani Reference Hearst and Ganapathisubramani2015), i.e. no spurious peaks were observed in histograms of subpixel displacements. To calculate vorticity fields, the velocity fields were filtered over a $3 \times 3$ window, and then, the gradients were estimated using a three-point centred scheme. The uncertainty of the instantaneous vorticity was estimated to be approximately 2 s$^{-1}$ using the PIV uncertainty propagation method (Sciacchitano & Wieneke Reference Sciacchitano and Wieneke2016). Utilising a 25 MP camera in this experiment provides an adequate number of vorticity data points per each PIV field without compromising on the data quality. PIV fields were fitted with a local coordinate system, where $x$- and $y$-axis point in the streamwise and wall-normal directions, respectively. The $x$-axis origin was located at the upstream edge of the domain, whereas the $y$-axis origin was at the bottom wall.

In addition to the regular PIV processing, a single-pixel approach (Westerweel, Geelhoed & Lindken Reference Westerweel, Geelhoed and Lindken2004; Oldenziel, Sridharan & Westerweel Reference Oldenziel, Sridharan and Westerweel2023) was utilised in the regions below the regular PIV domain to resolve the mean velocity profile in the near-wall region. Using images of a 2-D calibration target, a third-order polynomial function was fitted to map the image plane to the object plane; this function was then used to dewarp the particle images. This procedure corrects the misalignment between the pixel rows in the image and the bottom wall of the channel and rectifies the perspective and lens diffraction effects. The dewarped image pairs were intensity normalised using a min–max filter (Adrian & Westerweel Reference Adrian and Westerweel2011). Using the fact that the streamwise evolution of the flow is negligible across the FOV, one row in the first image frame (4924 pixels) was cross-correlated with the rows of pixels (4924 pixels) in the second frame in a domain of 8 pixels in the wall-normal direction. The resulting correlation fields were then summed over all the recorded frames. A 3-point Gaussian fit was utilised to estimate subpixel mean displacements. This procedure yielded the mean velocity data in the near-wall region with a spatial resolution of 1 pixel ($\approx$40 $\mathrm {\mu }$m) along the wall-normal axis. The single-pixel and regular PIV results were merged utilising a weighting function that was a combination of a linear and an error function. The resulting mean velocity profiles are shown in figure 2(a).

Figure 2. (a) Inner normalised mean velocity profiles for all the test cases; the inset shows the streamwise evolution of the wake region for case B. (b) Inner normalised turbulent fluctuations profiles for the active cases measured at $X/M = 55$ and REF. (c) Inner normalised turbulent fluctuation for case B measured at the different streamwise locations. REF, $-\blacklozenge -$, grey; case A, $-\blacksquare -$, blue; case B, $-\bullet -$, green; case C, $-\blacktriangle -$, red; with lighter colours indicating increasing streamwise distance from the grid. The solid and the dashed black line are DNS data of a canonical ZPG-TBL (Sillero et al. Reference Sillero, Jiménez and Moser2013) and open channel flow (Yao et al. Reference Yao, Chen and Hussain2022), respectively. In (b) and (c), standard deviations of the streamwise (filled symbols) and wall-normal (open symbols) velocities are shown with filled and open symbols, respectively, and the symbols with black border indicate the Reynolds shear stress profiles.

2.3. Test cases and their characteristics

Three different active grid rotational velocity distributions, i.e. top hat distributions centred around mean rotational velocity ($\varOmega$) with a spread of $\omega = \varOmega /2$, were used to set the incoming flow conditions. These active grid cases, called ‘active cases’ hereafter, were tested at four different streamwise locations, where 3000 PIV fields were collected for each case. A reference case, REF, with the active grid removed from the facility was tested separately at $X/M = 95$ with 2000 recorded PIV fields; this case is used as a benchmark for measurements in the facility and was only acquired at one very far downstream position such that its $Re_\tau$ was comparable to the other cases. The test cases are listed in table 3 together with their parameters. By adjusting the pump's operating frequency, $U_\infty$ was kept constant for the active cases at $U_\infty = 0.61 \pm 0.1$ m s$^{-1}$. Here $U_\tau$ was estimated using a composite velocity profile fit to the mean velocity profiles as described by Rodríguez-López, Bruce & Buxton (Reference Rodríguez-López, Bruce and Buxton2015). Esteban et al. (Reference Esteban, Dogan, Rodríguez-López and Ganapathisubramani2017) showed that the method yields reliable estimates of $U_\tau$ for TBLs subjected to FST. The values of $U_\tau$ are listed in table 3.

Table 3. The freestream and boundary layer parameters of the cases at the different streamwise locations.

Some studies have shown that, for TBLs subjected to FST, using $\delta _{99}$ to determine the boundary layer thickness yields dubious results (Dogan et al. Reference Dogan, Hanson and Ganapathisubramani2016; Hearst et al. Reference Hearst, de Silva, Dogan and Ganapathisubramani2021). Thus, Dogan et al. (Reference Dogan, Hanson and Ganapathisubramani2016) utilised the iterative integral approach, originally presented by Perry & Li (Reference Perry and Li1990), to calculate the boundary layer thickness ($\delta$). The same method is employed here, where $\delta = \delta ^* U_\infty /(C_1 U_\tau )$, $\delta ^* = \int _0^\infty (1-U/U_\infty )\,\mathrm {d}\kern 0.05em y$ denotes the boundary layer displacement thickness, and $C_1 = \int _0^1 ((U_1-U)/U_\tau )\,\mathrm {d}\kern 0.05em y/\delta$. The resulting $\delta$ values (see table 3) were typically 20 % greater than $\delta _{99}$. It is worth mentioning that resolving the mean velocity profiles all the way down to the viscous sublayer using the single-pixel approach improved the estimates of $U_\tau$ and $\delta ^*$. Considering each active case, $\delta$ increases in the streamwise direction, whereas a decreasing trend is observed for $U_\tau$. The growth rate of $\delta$ is higher than the decay rate of $U_\tau$, resulting in increasing $Re_\tau$ values with the development of the boundary layer downstream. These trends are in agreement with that observed for a canonical ZPG-TBL (Marusic et al. Reference Marusic, Chauhan, Kulandaivelu and Hutchins2015) and a ZPG-TBL interacting with FST (Jooss et al. Reference Jooss, Li, Bracchi and Hearst2021). Comparing different cases at the same streamwise location indicates a marginal increase in $U_\tau$ by increasing $u'_\infty /U_\infty$ in line with previous studies (Hancock & Bradshaw Reference Hancock and Bradshaw1989; Sharp, Neuscamman & Warhaft Reference Sharp, Neuscamman and Warhaft2009; Dogan et al. Reference Dogan, Hanson and Ganapathisubramani2016; Esteban et al. Reference Esteban, Dogan, Rodríguez-López and Ganapathisubramani2017; Jooss et al. Reference Jooss, Li, Bracchi and Hearst2021); however, in contrast to these studies, $\delta$ is not considerably affected by FST intensity for the flows tested here. This yields similar $Re_\tau$ for different cases tested at the same downstream location, which is reminiscent of observations made by Asadi et al. (Reference Asadi, Kamruzzaman and Hearst2022a) regarding the robustness of $Re_\tau$ to the incoming turbulence in a closed channel flow. Compared with the aforementioned literature, this is likely related to the limited height of FST flow above the TBL; the former is exposed to the dynamic boundary condition at the free surface which, in turn, limits the expansion of the boundary layer typical for FST-TBL flows. The evidence for this effect is presented later in § 3. Furthermore, the spatial two-point autocorrelation of the streamwise velocity fluctuations (see Dogan et al. Reference Dogan, Hearst, Hanson and Ganapathisubramani2019 and Asadi, Kamruzzaman & Hearst Reference Asadi, Kamruzzaman and Hearst2022b for examples) in the freestream region was examined (not shown here), revealing approximately equivalent dimensions of the FST scales for the different active cases. The above evidence indicates that the flows examined in this study are, in fact, developing open channel flows.

Figure 2(a) shows mean streamwise velocity profiles of all the test cases normalised by inner variables. Two DNS datasets are included for reference, i.e. a canonical ZPG-TBL (Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013) at $Re_\tau \approx 1990$ and a fully developed open channel flow (Yao, Chen & Hussain Reference Yao, Chen and Hussain2022) at $Re_\tau \approx 2000$. The current results agree with previous studies (Blair Reference Blair1983; Hancock & Bradshaw Reference Hancock and Bradshaw1983; Thole & Bogard Reference Thole and Bogard1996; Dogan et al. Reference Dogan, Hanson and Ganapathisubramani2016; Hearst et al. Reference Hearst, Dogan and Ganapathisubramani2018; Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022a) in that the only significant effect of incoming turbulence on the mean velocity profile is in the wake region, where the wake is suppressed by FST. The inset in figure 2(a) illustrates the streamwise evolution of the outer region for case B (which is representative of all active cases in this study), similar to observations made by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021), the wake suppression effect is relieved downstream with the decay of the FST intensity.

Figure 2(b) presents the statistics of the fluctuating velocity components of cases A, B and C at $X/M = 55$. The REF case and DNS data are included for comparison. Similar to the results of Hearst et al. (Reference Hearst, Dogan and Ganapathisubramani2018), Dogan et al. (Reference Dogan, Hearst, Hanson and Ganapathisubramani2019), You & Zaki (Reference You and Zaki2019), Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021) and Asadi et al. (Reference Asadi, Kamruzzaman and Hearst2022a), streamwise fluctuations within the boundary layer are considerably amplified in the outer regions by incoming turbulence, demonstrating visible deviations from the REF and DNS profiles. FST also penetrates into the TBL and amplifies the streamwise fluctuations in the inner regions, which is demonstrated more explicitly for the extreme case C. The freestream wall-normal fluctuations are suppressed due to the dynamic boundary condition at the free surface, which results in relatively high $u_\infty '/v_\infty '$ ratios in the freestream ($1.4 < u_\infty '/v_\infty ' < 2.6$ for all the test cases). This is in line with our previous observation (Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022a) in a turbulent channel flow, where inlet turbulence did not affect the wall-normal fluctuations profiles (see Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022a, figure 3b); however, the results of Dogan et al. (Reference Dogan, Hearst, Hanson and Ganapathisubramani2019) showed that a significant increase in wall-normal fluctuations of FST above a ZPG-TBL penetrates deeper in the boundary layer and the profiles exhibit deviations in the inner regions (see Dogan et al. Reference Dogan, Hearst, Hanson and Ganapathisubramani2019, figure 4). Nevertheless, figure 2(b) demonstrates negligible Reynolds shear stress values in the freestream, indicating uncorrelated streamwise and wall-normal fluctuations of FST. In addition, the Reynolds shear stress profiles match inside the boundary layer. This can be explained through the argument made by You & Zaki (Reference You and Zaki2019) that the wall-normal fluctuations of FST permeate into the boundary layer, work against the mean shear and, hence, produce Reynolds shear stresses. Accordingly, since FST has a negligible effect on $v'$ in the inner region, i.e. $y^+ \lessapprox 1000$, for the flows tested here, no significant effects on the Reynolds shear stress could be seen.

Lastly, figure 2(c) illustrates the streamwise evolution of the second-order statistics for case B as the representative of other cases. Moving downstream, the decay of streamwise and wall-normal fluctuations in the freestream is visible. Interestingly, the wall-normal fluctuations grow across the inner region of the boundary layer. Although marginal, the same trend is also present in the Reynolds shear stress profiles. This indicates that the boundary layer develops and matures downstream, while the FST decays. This is in line with the observation made by Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021) regrading the streamwise fluctuations in the inner regions and complements it by providing the wall-normal and Reynolds shear stress measurements.

4.1. Intermittency and the interface location

As shown in figure 6(a,b), increasing the FST intensity resulted in increased variations of the velocity and vorticity thresholds identified to mark the interface. The distributions of the velocity and vorticity thresholds for case B at different streamwise locations are depicted in figure 6(c). P.d.f.s of the velocity thresholds, represented by the solid lines in the figure, exhibit a sharper peak and a narrower distribution by moving in the streamwise direction, i.e. the lighter colours. A more condensed range of vorticity thresholds is also noticed in this figure, where, generally, there is a decline in the likelihood of finding high vorticity threshold values as one moves in the streamwise direction. As the intensity of FST decreases farther downstream from the grid, the trends observed in figure 6(c) suggest a resurgence of the threshold distribution shown in figure 6(a,b) for REF. The same trend is also confirmed for all three cases (not shown here). This observation is significant in that it demonstrates that the effect of the turbulence is not to affect a permanent change on the TBL that is transported downstream but rather is local.

The effects of FST on the intermittency and the location of the interfaces were discussed in § 3. The $\overline {y_i}$ values increases in the streamwise direction for each active case (tables 4 and 5), indicating the movement of the interface away from the wall. Although the trend in the fluctuations is less clear, $\sigma (y_i)/\overline {y_i}$ tangibly decreases moving in the streamwise direction. This is different than what was observed by You & Zaki (Reference You and Zaki2019) for the material boundary of the TBL, where the $\sigma (y_i)/\overline {y_i}$ value increased abruptly after a sudden injection of homogeneous-isotropic turbulence on top of the TBL and then remained constant farther downstream. The progressively more chaotic behaviour of the interface observed by You & Zaki (Reference You and Zaki2019) might be attributable to the artificial addition of FST in their DNS and/or the way the boundary layer fluid was tracked in their work by ignoring the entrainment process. Overall, the interface statistics presented in the current study, similar to the velocity and vorticity threshold distributions, show a return to the undisturbed state.

Figure 8 shows the streamwise evolution of the intermittency and $y_i$ distribution obtained using the velocity and vorticity interfaces for cases A, B and C. The REF case is also included for comparison. The wall-normal coordinate is normalised by $\overline {y_i}$ for each test case as it provides a more physical scaling basis compared with $\delta$ which solely depends on the mean velocity profile. Moving in the streamwise direction, the intermittency increases in the inner regions, corresponding to the movement of the interface away from the wall. This is also visualised in figure 8(g–l), where the peaks of the p.d.f.s are located farther from the wall for downstream measurement locations, i.e. lighter colours. As shown by case REF, for a Gaussian distribution of $y_i$, the peak is located at $y/y_i = 1$, and $\gamma = 0.5$ at $y/y_i = 1$. The streamwise trends in the profiles are in the same direction, where the peak location moves away from the wall and the intermittency grows at $y/y_i = 1$. Thus, similar to the velocity thresholds, a tendency to recover the canonical characteristics is observed when moving downstream of the active grid.

Figure 8. Streamwise evolution of (a–f) intermittency profiles and (g–l) p.d.f.s of the interface location for cases A (blue), B (green) and C (red) in the left, the middle and the right panels, respectively, with lighter colours indicating increased streamwise distance from the grid. (a–c) and (g–h) are derived using the velocity interfaces, whereas (d–f) and (j–l) are for the vorticity interfaces.

4.2. Conditional averages

Conditional averaging is a common tool in boundary layer studies to examine flow statistics in the vicinity of an interface, e.g. TNTIs (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014; Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015), UMZs (de Silva et al. Reference de Silva, Philip, Hutchins and Marusic2017; Chen et al. Reference Chen, Chung and Wan2020), the core boundary of turbulent channel flow (Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014; Yang et al. Reference Yang, Hwang and Sung2016; Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022a) and the boundary of a ZPG-TBL subjected to FST (Wu et al. Reference Wu, Wallace and Hickey2019a; Hearst et al. Reference Hearst, de Silva, Dogan and Ganapathisubramani2021). The same technique is employed here to probe the flow statistics in the proximity of the identified interfaces. A local wall-normal coordinate is defined whose origin is always on the interface, i.e. $y-y_i$. This allows us to move along the interface and average the statistics towards the boundary layer and FST, i.e. $y-y_i < 0$ and $y-y_i>0$, respectively. The conditionally averaged profiles are then calculated by averaging the vertical profiles over the instantaneous PIV fields for each test case. A prerequisite for this analysis is that the interface should be a single-valued function of $x$. This is not always the case since the interface folds back on itself occasionally. To circumvent the issue, whenever the interface takes on several wall-normal locations at a specific $x$, the closest point to the wall is chosen as the interface location for the velocity interface, ensuring that no FST is included as a boundary layer region in the analysis. For the vorticity interface, the farthest point from the wall is selected, representing the projection of the freestream on to the interface. Hereafter, the conditional averaging is denoted by angle brackets, i.e. $\langle \,\rangle$.

Figure 9 shows the conditionally averaged profiles derived using the velocity and vorticity interfaces for the active cases measured at the most upstream measurement location, i.e. $X/M = 55$. The results of the REF case are also included for comparison. Figure 10(a) demonstrates the conditionally averaged streamwise velocity profiles. The profiles are offset by the mean conditional velocity at the interface location ($\langle u_i \rangle$). The conditionally averaged profiles across both the velocity and the vorticity interface show a relatively uniform streamwise velocity inside the FST and a positive shear within the TBL in all the cases. The transition between these regions is different for the velocity and vorticity interfaces. When considering the velocity interfaces, a sharp jump in the streamwise velocity profile occurs across the interface, where the magnitude of the jump increases with increasing FST intensity. This is consistent with the observations of Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) and Asadi et al. (Reference Asadi, Kamruzzaman and Hearst2022a). Using the approach utilised by Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014) and Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) to quantify the magnitude of the jump ($D[U_\infty ]$) for the conditionally averaged profiles across the velocity interface shown in figure 9(a) yields $D[U_\infty ] \approx 0.03$, 0.05, 0.06 and 0.07 for REF, A, B and C, respectively. In contrast, a smooth transition links the uniform velocity profile on the freestream side of the vorticity interface to the sheared velocity profile on the TBL side, and there is thus no explicit velocity jump across the vorticity interface.

Figure 9. (a) Streamwise velocity, (b) spanwise vorticity and (c) streamwise turbulent fluctuations profiles conditionally averaged across the velocity interface (——) and the vorticity interface ($-\bullet -$) for cases A (blue), B (green) and C (red) measured at $X/M = 55$ as well as case REF (grey).

Figure 10. Streamwise evolution of (a–c) streamwise velocity, (d–f) spanwise vorticity and (g–i) streamwise turbulent fluctuations profiles conditionally averaged across the velocity interface (——) and the vorticity interface ($-\bullet -$) for cases A (blue), B (green) and C (red) in the left, the middle and the right panels, respectively, with lighter colours indicating increased streamwise distance from the grid.

Figure 9(b) presents the conditionally averaged spanwise vorticity profiles. It is clear that the mean spanwise vorticity level inside the TBL is negative, whereas it is zero in the freestream. This is true for both the velocity and vorticity interfaces and provides strong evidence for the validity of the method, which was supposed to identify an interface separating the vortical structures of the TBL from those of the FST. The previously observed difference in the behaviour of the conditionally averaged velocity profiles is also represented here, where sharp peaks in the vorticity profiles are indicative of high shear events at the velocity interface, similar to the previous observations for TNTIs (Chauhan et al. Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014; Eisma et al. Reference Eisma, Westerweel, Ooms and Elsinga2015), the uppermost UMZ edge of a TBL subjected to FST (Hearst et al. Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) and the core boundary of turbulent channel flows (Kwon et al. Reference Kwon, Philip, de Silva, Hutchins and Monty2014; Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022a), while the sharp jumps of the mean vorticity are observed across the vorticity interface similar to the previous observation for TNTI identified using a spanwise vorticity threshold (Borrell & Jiménez Reference Borrell and Jiménez2016; Watanabe et al. Reference Watanabe, Zhang and Nagata2018).

The streamwise fluctuations are evaluated with respect to the conditionally averaged velocity profiles, and the resulting streamwise-normal Reynolds stress profiles are shown in figure 9(c). A relatively uniform fluctuation level is exhibited within the freestream, whereas the fluctuations increase on the TBL side of the interface when moving towards the wall, i.e. $(y-y_i)<0$. The bulge in the profiles obtained using the velocity interface is an artefact of using constant velocity contour lines in instantaneous velocity fields and were similarly visible in the results of Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014) (see figure 6 in their work), who used a constant KED threshold to identify the TNTI.

The results presented above exhibit the characteristics expected from an interface that separates the FST region with a uniform mean velocity and negligible mean spanwise vorticity from the TBL region with a sheared velocity profile and mean negative spanwise vorticity, affirming the validity of the method. It should be noted that the physical characteristics of the conditionally averaged properties have been taken into account in this study to confirm the validity of the method. This provides a more vigorous standard than the conventional logic of merely observing step changes in the conditional statistics (see Appendix B for an example).

To assess the streamwise evolution of the conditionally averaged properties, the profiles measured at different streamwise locations are depicted in figure 10 for each grid case, i.e. case A, B and C in the left, the middle and the right column, respectively. To avoid inclusion of the streamwise changes in the normalisation of the vertical coordinate, it is normalised by $H$, which is a constant. When considering the profiles obtained using the velocity interface, a similar streamwise trend is observed for all the tested cases, where the magnitude of the streamwise velocity jump across the the velocity interface decreases in the streamwise direction (figure 10a–c); consequently, the peaks in the conditionally averaged vorticity profiles appear to generally diminish downstream of the grid. On the boundary layer side of the interface, the mean spanwise vorticity magnitude drops by moving farther downstream, whereas on the freestream side, it retains an approximately zero value regardless of the streamwise location (figure 10d–f). The former can be explained by the movement of the interface away from the wall, and the latter is an inherent feature of the freestream as discussed earlier. Moreover, figure 10(g–i) exhibit similar trends as seen in figure 9(c), where the level of streamwise fluctuations scales with the FST intensity and thus diminishes downstream of the grid. The trends observed in the profiles acquired using the vorticity interface are generally similar to those found using the velocity interface, where the streamwise velocity profiles in figure 10(a–c) display reduced shear within the TBL when moving downstream of the grid. This corresponds to a decrease in mean vorticity magnitude on the TBL side of the profiles in figure 10(d–f). The uniformity of the velocity and the absence of vorticity within the FST remain consistent in all examined scenarios at various streamwise positions. The only deviation is in the scaling of the streamwise velocity fluctuations conditionally averaged across the vorticity interface, where no meaningful pattern is seen. This indicates that the vorticity interface does not accurately track the momentum events in the fields when FST is present, as shown explicitly in figure 5( f) for a specific field. Overall, when considering the streamwise development of conditionally averaged profiles, a similar recovery effect is observed, with the statistics tending to return to the conditional averages across the interface as seen in case REF.

4.3. Zonal averages

In boundary layer studies, zonal averaging is utilised to investigate the differences between the flow features when the flow is situated inside different regions at different wall-normal locations. For example, the wall-normal profiles of flow statistics were examined by Kwon et al. (Reference Kwon, Philip, de Silva, Hutchins and Monty2014) inside and outside the core region of a turbulent channel flow, Chen et al. (Reference Chen, Chung and Wan2020) inside different UMZs of a turbulent pipe flow and You & Zaki (Reference You and Zaki2019) inside and outside the material boundary of a TBL subjected to FST. We employ the same technique here to examine the streamwise velocity statistics zonally averaged inside the TBL and FST. The details of the technique were described by You & Zaki (Reference You and Zaki2019) (see § 3.1 of their work entitled ‘conditional sampling’). In short, the same binary values used for calculating the intermittency profiles are used here as a weighting factor to calculate the zonal averages inside the TBL and FST.

The streamwise velocity profiles zonally averaged inside the boundary layer ($U_{BL}$) and freestream ($U_{FS}$) for REF together with active cases at $X/M = 55$ are depicted in figure 11(a,d) derived using the velocity and the vorticity interfaces, respectively. It should be noted that $U_{BL}$ and $U_{FS}$ can be combined using $\gamma$ to reproduce the mean velocity profile, i.e. $U = \gamma U_{BL} + (1-\gamma ) U_{FS}$. The zonally averaged profiles of the active cases inside FST are only shown for the regions with $\gamma < 0.9$ since FST is seldom present beyond this range. The same is applied to the zonally averaged TBL profiles for the regions with $\gamma > 0.1$. In both figure 11(a) and 11(d), the boundary layer profiles are matched in the log region for the different cases and closely follow the logarithmic law of the wall, whereas the wake suppression effect by increasing the FST level is visible in the outer region. In fact, the wake suppression effect induced by FST results from a combination of a more frequent presence of freestream flow in the regions closer to the wall (as shown in figure 7) and a fuller $U_{BL}$ profile. When demarcated by the velocity interface, the freestream exhibits a more uniform streamwise velocity level (figure 11a) than when demarcated by the vorticity interface (figure 11a). Figure 11(b,e) illustrates zonally averaged fluctuations of streamwise velocity relative to the zonal mean obtained using the velocity and the vorticity interface, respectively. The boundary layer profiles in the inner regions follow the general scaling trend observed in figure 2(b), indicating amplification of turbulent fluctuations within the boundary layer by FST. In figure 11(b), freestream fluctuation profiles maintain a constant level in the wall-normal direction, whereas in figure 11(b), elevated fluctuation levels are observed within the freestream in the inner regions. The spanwise vorticity profiles zonally averaged inside the TBL and FST are illustrated in figure 11(c, f) derived using the velocity and the vorticity interface, respectively. The vorticity interface ensures a consistent zero mean vorticity level within the FST independent of the wall-normal location, whereas the zonally averaged profiles in the FST, obtained from the velocity interface, show non-zero vorticity levels in the inner areas. The results presented in figure 11 elucidate the underlying difference between the velocity and the vorticity interface in distinguishing the freestream flow from the TBL based on the momentum and the vorticity content, respectively, demonstrating the validity of the method.

Figure 11. Profiles of (a) and (d) the inner normalised mean velocity, (b) and (e) streamwise turbulent fluctuations and (c) and ( f) spanwise vorticity zonally averaged inside the boundary layer (the solid lines as well as the lines with filled symbols) and FST (the dashed lines as well as the lines with open symbols) for cases A (blue), B (green) and C (red) measured at $X/M = 55$ as well as case REF (grey). (a–c) are derived using the velocity interface, whereas (d–f) are derived using the vorticity interface.

The streamwise development of the zonally averaged statistics obtained using the velocity and the vorticity interface are illustrated in figures 12 and 13, respectively, for cases A, B and C. The inner normalised velocity profiles (figures 12a–c and 13a–c) illustrate a gradual recovery of the wake region with the decay of FST intensity in the streamwise direction, as noted before in the mean velocity profiles shown in figure 2(a) and similar to the results of Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021). The log region remains intact at various streamwise locations, corroborating the robustness of the logarithmic law of the wall to freestream perturbations, as previously illustrated in figure 2(a) and observed in other studies (Hancock & Bradshaw Reference Hancock and Bradshaw1989; Dogan et al. Reference Dogan, Hanson and Ganapathisubramani2016; Hearst et al. Reference Hearst, Dogan and Ganapathisubramani2018; You & Zaki Reference You and Zaki2019; Jooss et al. Reference Jooss, Li, Bracchi and Hearst2021). The FST fluctuation profiles in figures 12(d–f) and 13(d–f) are arranged in a decreasing order moving in the streamwise direction, indicating the decay of the grid generated turbulence. For cases A and B, the boundary layer fluctuation profiles collapse in the inner regions for different streamwise locations, whereas the FST intensity level separates them in the outer regions. In case C, however, the FST impact on the boundary layer profiles in the outer regions extends closer to the wall, indicating a deeper penetration of the FST effects on the streamwise fluctuations of the boundary layer in this case with a significant level of turbulence in the freestream. This distinction is similar to the findings of Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021) (see figure 5e, f in their paper). The spanwise vorticity profiles zonally averaged inside the FST, depicted in figures 12(g–i) and 13(g–i), generally collapse, indicating that the streamwise development has no impact on the FST profiles. The TBL profiles exhibit minor deviations for $y/\delta <0.5$, where the vorticity magnitude decreases slightly when moving in the streamwise direction. The TBL profiles generally match for $y/\delta >0.5$, indicating an outer scaling of the vorticity in this region.

Figure 12. Streamwise evolution of (a–c) the inner normalised mean velocity, (d–f) streamwise turbulent fluctuations and (g–i) spanwise vorticity profiles zonally averaged inside the boundary layer (the solid lines) and FST (the dashed lines) obtained using the velocity interfaces for cases A (blue), B (green) and C (red) in the left, the middle and the right panels, respectively, with lighter colours indicating increased streamwise distance from the grid.

Figure 13. Streamwise evolution of (a–c) the inner normalised mean velocity, (d–f) streamwise turbulent fluctuations and (g–i) spanwise vorticity profiles zonally averaged inside the boundary layer (the lines with filled symbols) and FST (the lines with open symbols) obtained using the vorticity interfaces for cases A (blue), B (green) and C (red) in the left, the middle and the right panels, respectively, with lighter colours indicating increased streamwise distance from the grid.

4.4. UMZs

The existence of UMZs in TBLs exposed to FST was confirmed by Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021). Given the observations of Jooss et al. (Reference Jooss, Li, Bracchi and Hearst2021) that streamwise evolution also plays a role in the system, it is natural to wonder how the UMZs evolve in space. Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) identified the UMZs using the well-established histogram method, in which the instantaneous freestream flow was eliminated in the histogram analysis to avoid concealing the peaks corresponding to the UMZs characteristic velocities (see figure 18 in their work). Following in the footsteps of de Silva et al. (Reference de Silva, Hutchins and Marusic2016) and Laskari et al. (Reference Laskari, de Kat, Hearst and Ganapathisubramani2018), Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) identified a KED-based interface as the uppermost UMZ edge of the TBL and disregarded the flow above the interface. The same histogram-based technique is employed here to identify the UMZs, where the instantaneous interfaces identified using the velocity contour lines are employed to separate the TBL flow from FST. As illustrated in the examples of figure 5, the velocity interfaces effectively isolate the momentum events associated with the TBL, which, in turn, allows for the correct identification of the UMZs present in the TBL. Apart from using the new method to identify the instantaneous boundary layer flow, the other steps are the same as stated by Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) in section 4.2 of their paper. As mentioned earlier in § 3, a subdomain of the PIV FOV in the centre with a streamwise extent of $\Delta x^+ = 2000$ is chosen for the analysis. The instantaneous streamwise velocity is isolated below the momentum-based interface whose histogram (with a bin size of $0.1U_\infty$) is examined to identify the peaks as the modal velocities of the UMZs ($u_{{UMZ}}$). The midpoint between the two neighbouring modal velocities can then be used to mark the boundary between the two adjacent UMZs. The peak detection parameters, i.e. peak prominence, peak height and minimum peak distance, are optimised to ignore spurious peaks in the histograms and kept the same for the different test cases. Although varying the streamwise domain extent, histogram bin size and peak detection parameters affects the number of the detected UMZs ($N_{{UMZ}}$), the observed trends when comparing the test cases with each other are found to be robust and similar to what was found by Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021). Figure 14 illustrates examples of the instantaneous fields with the identified UMZs for cases B and C measured at $X/M = 55$ as well as REF.

Figure 14. Examples of PIV fields with identified UMZs and their modal velocities. (a), (c), and (e) are the instantaneous PIV fields of case REF, case B measured at $X/M = 55$ and case C measured at $X/M = 55$, respectively, with the red lines indicating the velocity interface between the boundary layer and the freestream, and the blue lines indicating the UMZ edges within the boundary layer. (b), (d), and ( f) show the p.d.f. of the normalised instantaneous streamwise velocity within the boundary layer for case REF, case B measured at $X/M = 55$ and case C measured at $X/M = 55$, respectively, with the red symbols indicating the identified modal velocities of the UMZs.

The average number of identified UMZs ($\overline {N_{{UMZ}}}$) is presented in table 4 for all the test cases. Comparing REF with the active cases, similar to the results of Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021), fewer UMZs are identified when the TBL is influenced by FST. The fields in figure 14 are chosen to illustrate the general trends observed in the presence of FST, where the velocity interface approaches the wall and the number of identified UMZs decreases. According to Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021), one possible explanation would be the destruction of the UMZs in the boundary layer by FST whenever its fluctuations are comparable to those in the boundary layer at a particular wall-normal location. On the other hand, the underlying dynamics of the flow inside the boundary layer are observed to be robust to FST perturbations (Dogan et al. Reference Dogan, Hanson and Ganapathisubramani2016, Reference Dogan, Hearst and Ganapathisubramani2017; Hearst et al. Reference Hearst, Dogan and Ganapathisubramani2018), and as discussed in § 4.2, the flow characteristics inside the boundary layer were only subjected to a level change that appeared to be mainly due to the variations of the interface position. Accordingly, this might suggest that the UMZs are compressed closer to the wall and masked due to the limited spatial resolution of PIV in the near-wall region. In fact, this hypothesis is in line with the observation made by de Silva et al. (Reference de Silva, Hutchins and Marusic2016) that the instantaneous location of the TNTI did not affect the number of the UMZs present in the boundary layer; instead, when the interface was situated farther from the wall, the upper UMZs in the boundary layer were thicker. If the inner UMZs are compressed, occasionally a portion of them might be discernible in the measurement domain. In that case, it is likely that the upper boundary of the UMZ is discontinuous, i.e. the contour line marking the upper edge of the UMZ does not span the whole extent of the FOV. An example is illustrated in figure 15 for a sample PIV field of case C measured at $X/M = 55$, where the black contour lines marking the upper edge of the UMZ closer to the wall are discontinuous and reach the bottom of the FOV. The occurrence rate of these fields ($R_{d}$) containing discontinuous UMZs is listed in table 4 as a percentage of the total examined fields.

Figure 15. (a) A sample instantaneous PIV field of case C measured at $X/M = 55$, with the red line indicating the velocity interface between the boundary layer and FST, and the black lines marking the discontinuous boundary between the UMZs. (b) The histogram of the normalised instantaneous streamwise velocity within the boundary layer, with the red symbols indicating the identified modal velocities of the UMZs.

Furthermore, table 4 lists the average number of the continuous UMZs ($\overline {N_{{UMZ,c}}}$) calculated by discarding the discontinuous UMZs. As presented for case REF, less than 1 % of the PIV fields contain a discontinuous UMZ, indicating that almost all the UMZs in the PIV fields were large enough for the current spatial resolution to fully cover their upper edge. Thus, discarding the discontinuous UMZs does not considerably affect the average number of the identified UMZs for this case. In contrast, $R_{d}$ increases significantly when the TBL is subjected to increasing FST intensities. Consequently, when discarding the discontinuous UMZs, $\overline {N_{{UMZ}}}$ noticeably decreases for the active cases. It should be mentioned that discontinuous UMZs are always detected in proximity to the bottom edge of the FOV, where the UMZ boundary extends all the way down. In conclusion, when FST is present, a frequent presence of UMZ segments near the bottom of the examined domain indicates the compression of the UMZs in this region. This also implies information is buried in the regions closer to the wall due to low spatial resolution, especially given the fact that the separation of the scales should be larger for a higher turbulent state. Nonetheless, further investigation of the structures in the near-wall region requires higher spatial resolution in the near-wall region or is better suited for DNS investigations. To ensure consistency when comparing cases, the discontinuous UMZs are neglected in the remainder of the present study.

Figure 16(a) illustrates the distribution of $N_{{UMZ,c}}$ identified in the PIV fields of the active cases measured at $X/M = 55$ as well as REF. When FST is present, the number of PIV fields with $N_{{UMZ,c}} \geq 2$ is still greater than those with $N_{{UMZ,c}} = 1$, indicating the existence of UMZs within the TBL for the cases tested in this study. In the presence of FST, the number of instantaneous fields with more than two UMZs in the boundary layer is significantly reduced, and the p.d.f.s exhibit a peak at $N_{{UMZ,c}} = 2$. The former is similar to the trend observed by Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) in figure 11(a) of their work. $\overline {N_{{UMZ,c}}}$ (listed in table 4) for the active cases measured at the same streamwise location demonstrates a decreasing trend as FST intensity increases. Figure 16(b) depicts the p.d.f.s of the modal velocities for the same cases as shown in 16(a). Among the active cases, the change in the distribution of the modal peaks is more noticeable than the variations of $N_{{UMZ,c}}$. Artificial peaks were observed by Hearst et al. (Reference Hearst, de Silva, Dogan and Ganapathisubramani2021) in the distribution of modal velocities in figure 11(b) (figure 11b in Hearst et al. Reference Hearst, de Silva, Dogan and Ganapathisubramani2021 has a mislabelled $x$-axis: their figure 11b shows the same information as figure 16b of the present work, i.e. the p.d.f. of the UMZ modal velocities) of their work caused by the constant KED threshold used to separate the instantaneous TBL and FST flows; since individual threshold values are selected in this study for this purpose, no spurious peaks are observed in figure 16(b). In fact, similar to the p.d.f.s of the threshold values shown in figure 6(a), the modal peaks are more evenly distributed for increased FST intensities. A greater spread in the distribution of the modal velocities with a higher FST intensity is similar to what we observed (Asadi et al. Reference Asadi, Kamruzzaman and Hearst2022a, figure 5) for modal peaks in a turbulent channel flow affected by inlet turbulence. This is a direct effect of the increased variations of the streamwise momentum inside the boundary layer induced by FST (see figure 11b). In addition, figure 16(b) shows that the probability of the occurrence of the UMZs with $u_{{UMZ}} > U_\infty$ significantly increases in the presence of FST. Based on the analysis of Heisel et al. (Reference Heisel, De Silva, Hutchins, Marusic and Guala2020), who showed that the mean velocity profile could be derived by averaging UMZ velocities, the fuller $U_{BL}$ profiles in the presence of FST, as discussed in § 4.3, can be attributed to the more frequent occurrence of UMZs with higher characteristic velocities.

Figure 16. The distribution of the (a) number of continuous UMZs and (b) normalised modal velocities identified for the active cases measured at $X/M = 55$ as well as REF: REF, $-\blacklozenge -$, grey; case A, $-\blacksquare -$, blue; case B, $-\bullet -$, green; case C, $-\blacktriangle -$, red.

Figure 17 shows the streamwise changes in the distribution of $N_{{UMZ,c}}$ and $u_{{UMZ}}$ for cases A, B and C. In general, moving in the streamwise direction, the occurrence of PIV fields with $N_{{UMZ,c}} = 1$ decreases, whereas the number of fields in which two or more UMZs are identified increases. An increasing trend is observed for $\overline {N_{{UMZ,c}}}$ with development of the flow in the streamwise direction (see table 4). This indicates that as the interface moves away from the wall with the decay of FST in the streamwise direction, more continuous UMZs are identified throughout the examined domain. The p.d.f.s of $u_{{UMZ}}$ measured at different streamwise locations (figure 17d–f) appear to change in accordance with the FST intensity, especially for $u_{{UMZ}}/U_\infty > 0.8$, where generally the peak in the p.d.f. grows sharper, whereas the probability of $u_{{UMZ}}/U_\infty > 1$ decreases with decreasing FST intensities in the streamwise direction. Overall, the UMZs within the TBL exhibit the same recovery pattern observed for the other TBL properties.

Figure 17. Streamwise evolution of the distributions of (a–c) number of continuous UMZs and (d–f) normalised modal velocities for case A ($-\blacksquare -$, blue), case B ($-\bullet -$, green) and case C ($-\blacktriangle -$, red), respectively, with lighter colours indicating increased streamwise distance from the grid.