3.1. Time-averaged boundary layer characteristics

This section documents the effect of the spanwise buffer layer blowing on the mean characteristics of the TBLs over a range of momentum thickness Reynolds numbers from $4760 \leq \textit{Re}_{\theta } \leq 10\,386$ . The measurements were taken with a hot wire that was located on the test section spanwise centreline, 1 cm downstream of the drag measuring platen. The actuator array was located on the platen but not operated for the baseline boundary layer measurements.

The conditions in the wind tunnel test section were monitored with a Pitotstatic probe and a temperature sensor located in the free stream. Their voltage outputs were simultaneously recorded with the voltage time series from the drag platen load cell, and hot-wire sensors. The hot-wire anemometer voltage time series was sampled at 25 kHz for 30 s. Multiple records were taken to build up converged statistical averages. The measurements were taken with the plasma actuator sequentially off and on without changing the free stream velocity. The actuator off condition represents the baseline condition. This was repeated for each momentum thickness Reynolds number case.

Figure 5. Mean velocity profiles of baseline TBLs at different momentum thickness Reynolds numbers with velocity and wall-normal values scaled by inner variables. Solid curve denotes the log-law of the wall.

Figure 6. Comparison between shear velocity of the baseline TBLs determined from the drag measuring platen and with the Clauser method for different momentum thickness Reynolds numbers.

Figure 5 shows mean velocity profiles of the baseline TBLs at different momentum thickness Reynolds numbers with the velocity and wall-normal values scaled by inner variables, $\nu$ and $u_{\tau }=(\tau _w/\rho )^{1/2}$ . The baseline shear stress, $\tau _w$ , used in the scaling in the figure was derived from the drag measuring platen. For this, the total force measured by the load cell was divided by the surface area of the platen to obtain $\tau _w$ .

Alternatively, the Clauser method was applied to the mean profiles to obtain the shear stress used in determining the values of $u_{\tau }$ . A comparison of the shear velocities obtained by the two methods is shown in figure 6. Over the full range of TBL cases, the average difference in the $u_{\tau }$ values obtained by the two measurement approaches was $\pm 0.85$ %. As will be shown, the effect the spanwise blowing has on the mean velocity profile will prevent the use of the Clauser method for determining the wall shear stress. Therefore, in those cases, the wall shear stress is determined with the drag measuring platen.

As previously discussed, Yao et al. (Reference Yao, Chen, Thomas and Hussain2017) had emphasized the importance of having the control flow specifically targeting the buffer layer of the TBL. With that motivation, Duong (Reference Duong2019) had performed experiments with the pulsed-DC plasma actuator over a range of DC voltages to determine the velocity and spatial distribution of the induced mean flow in still air. These velocity measurements were obtained in a quiescent environment with a glass total pressure Pitot probe that was mounted on a two-component ( $x,y$ ) micrometer traversing mechanism in a separate actuator test facility. The measurement tip of the glass probe was drawn to an inside diameter of approximately 0.15 mm. This could measure to a minimum distance from the surface of $y_{min}=0.2$ mm. The total pressure was measured using a differential pressure transducer with the reference port open to the room atmosphere. A representative example of the mean velocity spatial distribution from the pulsed-DC actuator is shown in figure 7(a). For this, the DC voltage was 4 kV and the pulsing frequency was 500 Hz. This shows constant level contours of the mean $x$ -component velocity as a function of the 2-D spatial dimensions. The $x$ dimension is the distance from the plasma-forming edge of the actuator exposed electrode. It is aligned parallel to the surface. The $y$ -dimension is aligned perpendicular to the surface. The normalized wall-normal profiles shown in figure 7(b) document that the pulsed-DC actuator serves to produce a compact, self-similar wall jet when operated in a quiescent environment.

Figure 7. Constant level velocity contours (a) and normalized wall-normal profiles (b) of mean velocity field generated by a pulsed-DC plasma actuator in still air: $V_{DC}=4$ kV; $f=500$ Hz.

It is evident in figure 7 that the majority of the induced velocity field resides close to the surface and quickly decays a short distance from the exposed electrode. The peak velocity, $\overline {U}_{max}$ , is centred at $y \approx 0.25$ mm and $x \approx 1.75$ mm. These dimensions and peak velocity, now indicated as the spanwise velocity $\overline {W}_{max}$ appropriate to our application, are displayed in table 2 when normalized by inner variables for the baseline TBLs at the different free stream velocities and corresponding momentum thickness Reynolds numbers. This indicates that for these cases, the majority of the plasma actuator induced velocity is confined within the buffer layer, and that the peak velocity is of the order of the friction velocity, $u_{\tau }$ .

Table 2. Pulsed-DC plasma actuator induced spatial velocity field characteristics normalized by inner variables.

Figure 8. Mean velocity profiles with spanwise buffer layer blowing in the TBLs at different momentum thickness Reynolds numbers with velocity and wall-normal values scaled by inner variables. Actuator voltage was 8 kV. The line is from baseline boundary layers in figure 5.

Having confirmed that the induced mean flow from the pulsed-DC actuator array is confined to the buffer layer of the turbulent boundary, the effect this had on the inner-variable scaled mean velocity profiles is shown in figure 8. The actuator voltage for each of these profiles was fixed at 8 kV. The inset label in the figure lists the corresponding momentum thickness Reynolds number, which can be equated to the respective free stream velocities in table 2. Also listed for each profile are the effective spanwise spacing between electrodes in inner variables, $\lambda _z^+$ . We note that these spacings are generally an order of magnitude larger than the $z^+$ distances ( $\lambda _z^+$ in the boundary layer) of the peak actuator induced velocities listed in table 2.

As shown in figure 8, the spanwise buffer layer blowing resulted in a reduction of the wall shear stress that is reflected in a lowering of the wall shear velocity, $u_{\tau }$ . This produced an upward shift of the mean velocity profiles when scaled by inner variables. The solid line corresponds to the log region for the baseline boundary layers in figure 5. Additionally, the slope of the log layer is also affected by the spanwise buffer layer blowing. This effectively changes the von Kármán coefficient, which prevents the use of the Clauser method to determine the wall shear stress from the mean velocity profiles of the actuated boundary layers. As a result, with the spanwise blowing cases the wall shear stress was determined from the drag measuring platen.

Further evidence of the effect of the spanwise buffer layer blowing on the mean velocity profiles is shown in figure 9. This compares the outer-variable scaled mean velocity profiles for the baseline (0 kV) and spanwise buffer layer blowing (8 kV) boundary layers at the lowest $\textit{Re}_{\theta }=4760$ , where the boundary layer thickness was sufficient to allow hot-wire measurements close to the wall. Figure 9(a) shows profiles to the full extent of the boundary layer. The portion of the profile closest to the wall is shown in figure 9(b). It is evident when normalized by outer variables, that the effect of the spanwise buffer layer blowing is primarily confined to the near-wall region. There, it has resulted in a reduction in the near-wall mean strain rate that is consistent with a lowering of the wall shear velocity.

Figure 9. Mean velocity profiles for the baseline (0 kV) and spanwise sublayer blowing (8 kV) boundary layers across the layer (a) and with a magnified view near the wall (b) for the case with $\textit{Re}_{\theta } =4760$ .

The drag reduction indicated by the force balance was also verified by comparison of the TBL momentum thicknesses measured directly upstream and downstream of the actuator platen. This was performed for the case at $\textit{Re}_{\theta }=4760$ that was shown in figure 9. Mean velocity profiles from that case were input to the appropriate form of the momentum integral equation to verify that the reduction of wall shear stress was consistent with that measured directly from the drag measurement platen, which recorded for this case a 71 % drag reduction. This level of drag reduction corresponds to a reduction in friction velocity of $u_{\tau _{plasma}}/u_{\tau _{baseline}}=0.54$ . Measurement of the momentum and displacement thicknesses just upstream of the actuator platen gave $\theta =5.4$ and $\delta ^*=7.2$ mm for a shape factor of $H=1.33$ . The corresponding values just downstream of the platen were $\theta =5.5$ and $\delta ^*=7.6$ mm for a shape factor of $H=1.38$ . Application these momentum thickness values to the appropriate form of the Kármán momentum integral equation produced a value of $u_{\tau _{plasma}}= 0.37$ m s⁻¹. As indicated in table 1, in the lowest Reynolds number case, $u_{\tau _{baseline}}=0.67$ m s⁻¹. It then follows that based on the integral momentum balance, $u_{\tau _{plasma}}/u_{\tau _{baseline}}=0.37\, {\rm (m\, s^{-1})}/0.67\, {\rm (m\, s^{-1})}=0.54$ , which is in perfect agreement with the value measured from the force balance. Again we note that Jasinski & Corke (Reference Jasinski and Corke2020) using the same force balance with a smooth platen surface matched the Coles–Fernholz relation (Baars et al. Reference Baars, Squire, Talluru, Abbassi, Hutchins and Marusic2016) to within 1 % over a wide range of $\textit{Re}_{\theta }$ .

Figure 10 documents the change in the wall shear velocity as a function of the pulsed-DC voltage for each of the boundary layers at the different momentum thickness Reynolds numbers. At the higher Reynolds numbers, the trend was a general lowering of shear velocity with increasing actuator voltage. However, at several of the lower Reynolds numbers, the highest actuator voltage causes the shear velocity to increase above a minimum. For our cases, this trend begins to occur for $\textit{Re}_{\theta } \leq 8735$ . Reviewing the values of $\overline {W}_{max}/u_{\tau }$ in table 2, which are representative of the 8 kV actuator voltage, would suggest that an optimum condition that minimizes $\tau _w$ might be where the scaled spanwise blowing is $\overline {W}_{max}/u_{\tau } \approx 0.5$ .

Figure 10. Ratio of wall shear velocities with buffer layer blowing on and off for different actuator voltages and momentum thickness Reynolds numbers.

Figure 11. Ratio of wall shear velocities with buffer layer blowing on and off from figure 10 along with data from Wei & Zhou (Reference Wei and Zhou2024) shown as a function of the actuator electrode spacing scaled by inner variables. Line is a logarithmic fit.

The magnitude of the spanwise blowing is, however, not the only design parameter that affects the change in the wall shear stress. The other is the spanwise spacing between the pulsed-DC actuator electrodes, that are ‘blowing sites’ which determines the number of low-speed streaks under simultaneous control. This is illustrated in figure 11 which replots the change in the wall shear velocity from figure 10 now as a function of the spanwise spacing in inner variables, $\lambda _z^+$ , between the actuator electrodes (indicated by the ND data). This is observed to collapse all of the data onto a single curve suggesting that $\lambda _z^+$ is a primary scaling parameter for the spanwise blowing effect on $u_{\tau }$ . The line is a logarithmic fit to the results. The top abscissa corresponds to the actuator array electrode spacing divided by 100, which is the average spanwise wavelength of the near-wall streak structure observed by Kline et al. (Reference Kline, Reynolds, Schraub and Runstadler1967). Thus, the upper abscissa can be read as being the number of near-wall streaks between electrodes. Based on this, figure 11 indicates that the reduction in $u_{\tau }$ increases as the number of low-speed streaks contained in the interelectrode spacing (i.e. blowing sites) decreases. The curve fit indicates that the degree to which $u_{\tau }$ is reduced by the spanwise blowing scales logarithmically with the spacing between the electrodes, i.e. blowing sites. This logarithmic variation implies that the reduction in $u_{\tau }$ due to the actuation is inversely proportional to the spanwise interelectrode spacing.

As a precursor to the next section that focuses on the effects of the spanwise buffer layer blowing on the unsteady characteristics of the TBLs, figure 12 shows spectra of streamwise velocity fluctuations measured at different heights in the boundary layers without and with the buffer layer blowing for the case with $\textit{Re}_{\theta }=4760$ . Comparing the spectra, they reveal that an effect was to uniformly lower the streamwise velocity fluctuations throughout the buffer and log layers. This reduction will later be correlated with a reduction in the turbulence production that results from the buffer layer blowing.

In comparing the spectra with and without the spanwise blowing, we also note that there is no evidence of a peak at the plasma actuator DC pulsing frequency (500 Hz) or its harmonics. This reveals two aspects. The first is that the flow does not sense any unsteadiness in the velocity induced by the DC pulsing. The second is that any electronic noise produced by the plasma actuator is too low to be detectable in generally sensitive hot-wire signals. The spurious downward spike in the spectra at approximately 1300 Hz only occurred at the spectra at the lowest sensor location at this lowest Reynolds number, and did not appear at any of the higher $\textit{Re}_{\theta }$ cases.

Figure 12. Power spectral density of streamwise velocity fluctuations at different heights in the boundary layer at $\textit{Re}_{\theta }=4760$ without and with buffer layer blowing.

3.1.1. Actuator scaling

In recent experiments, Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) and Wei & Zhou (Reference Wei and Zhou2024) utilized a wall-mounted array of dielectric-barrier-discharge plasma actuators that followed our approach of generating a near-wall spanwise flow. The actuator arrangement followed our designs that were shown in figures 2 and 3, with the exception that rather than the unidirectional blowing illustrated in figure 2, they utilized a spanwise-opposed wall jet configuration that we have also utilized (Duong Reference Duong2019; Thomas et al. Reference Thomas, Corke, Duong, Midya and Yates2019, Reference Thomas, Corke and Duong2023) and have shown to also be effective in lowering the viscous drag. Another difference is that they operated the plasma actuators with an AC waveform rather than in our case, a pulsed-DC waveform, with the latter being more energy efficient.

As in our experiments, Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) and Wei & Zhou (Reference Wei and Zhou2024) performed direct drag measurements using a floating-element force balance having a 20 cm $^2$ platen area. The platen was located just downstream of the plasma actuator array. In the Wei & Zhou (Reference Wei and Zhou2024) experiments, $520 \leq \textit{Re}_{\tau } \leq 747$ , which are considerably lower than our range of $\textit{Re}_{\tau }$ values with $2489 \leq \textit{Re}_{\tau } \leq 4810$ .

As in our experiments shown in figure 10, Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) and Wei & Zhou (Reference Wei and Zhou2024) investigated the effect of the plasma actuator voltage ( $E_{p-p}$ in their case) on the friction drag reduction. They found there to be an optimum voltage that was a function of $\textit{Re}_{\tau }$ , similar to our case put in terms of $\textit{Re}_{\theta }$ in figure 10. At their optimum power levels, the viscous drag reduction they obtained ranged from 20 % to 75 % which is comparable to the present results (indicated by the ND data) shown in figure 11, which is expressed in terms of the viscous shear velocity, $u_{\tau }$ .

Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) documented an increase in the von Kármán coefficient that we previously documented in figure 8. Other documented characteristics noted by Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) were a reduction in the [ $u,v$ ] Reynolds stress and turbulence energy production that accompanied the drag reduction that has been previously documented by Duong (Reference Duong2019) and Duong, Corke & Thomas (Reference Duong, Corke and Thomas2021). They similarly obtained ensemble-averaged streamwise velocity fluctuation time series based on a ‘variable-interval time-averaged’ (VITA) detection in which they observed a reduction in the ‘ejection’ portion of the VITA signature that accompanied the spanwise blowing, also first documented by Duong (Reference Duong2019) and Duong et al. (Reference Duong, Corke and Thomas2021).

Following these similar results, the large difference in the range of $\textit{Re}_{\tau }$ between those (Cheng et al. Reference Cheng, Wong, Hussain, Schroder and Zhou2021; Wei & Zhou Reference Wei and Zhou2024) and the present experiments offers an opportunity to further investigate the actuator scaling relations. With that objective, figure 11 also presents data from Wei & Zhou (Reference Wei and Zhou2024) figure 2(a) of the change in the viscous drag as a function of $\textit{Re}_{\tau }$ (expressed here as the ratio of the shear velocity with the actuator on and off) for their highest plasma actuator voltage, $E_{p-p}=5.6$ and 6 kV. For their fixed physical spacing of the electrodes, the different $\textit{Re}_{\tau }$ values resulted in different effective electrode spacing when normalized by inner variables, $\lambda _z^+$ . This is represented in figure 11 along with a logarithmic fit to their data. Again, the logarithmic variation with $\lambda _z^+$ shows that the reduction in friction drag is inversely proportional to the number of low-speed streaks under simultaneous control.

The data in figure 11 are cross-plotted in figure 13. In order to account for the difference between the opposed wall jets used in Wei & Zhou (Reference Wei and Zhou2024) and the unidirectional wall jets used in the present study, the effective gap width used for opposed wall jets is taken as the distance from the surface electrode to the centre of the interelectrode gap (which forms a stagnation point for the opposed wall jets). In figure 13 the two sets of data representing the level of viscous drag reduction for two disparate ranges of shear-stress Reynolds numbers, collapse reasonably when the inner variable electrode spacing, $\lambda _z^+$ , is normalized by their respective baseline $\textit{Re}_{\tau }$ values. As indicated, this normalization is equivalent to normalizing the effective interelectrode spacing, $\lambda _z$ , by the TBL thickness, $\delta$ . This provides strong evidence that the underlying mechanism of drag reduction is the same in both experiments and is dominated by the spanwise, plasma-induced blowing.

Figure 13. Cross-plot of data from Wei & Zhou (Reference Wei and Zhou2024) and the present results (ND) shown in figure 11 with the inner variable electrode spacing of each scaled by their respective boundary layer baseline $\textit{Re}_{\tau }$ values.

Thomas et al. (Reference Thomas, Corke, Duong, Midya and Yates2019) had shown a linear relation between the reduction in the viscous drag and the buffer layer maximum blowing velocity, $\overline {W}_{max}$ . In experiments similar to those that produced the results presented in figure 7, Thomas et al. (Reference Thomas, Corke, Duong, Midya and Yates2019) (their figure 17) determined the plasma-induced velocity as a function of the applied pulsed-DC voltage. The induced velocities ranged from 0.40 to 1.6 m s⁻¹. In contrast, Wei & Zhou (Reference Wei and Zhou2024) reported having plasma actuator generated velocities with a considerably larger range of 0.01 –6.87 m s⁻¹. With that, they determined that for a given baseline boundary layer $\textit{Re}_{\tau }$ , the viscous drag reduction actually varied logarithmically with the actuator blowing velocity, $\overline {W}_{max}$ .

This difference in the actuator blowing velocity between those of Wei & Zhou (Reference Wei and Zhou2024), and those cited by Thomas et al. (Reference Thomas, Corke, Duong, Midya and Yates2019), which are the same for the results presented in this section and listed in table 2, provided another means to investigate actuator scaling. This is embodied in figure 14 in which the data from Wei & Zhou (Reference Wei and Zhou2024) are replotted for their highest actuator voltage, $E_{p-p}=6$ kV, with the highest blowing velocity, $W_{max}=6.87$ m s⁻¹, at their two highest $\textit{Re}_{\tau }$ values. Included in the plot are the data from the present study (figure 10) at the two pulsed-DC actuator voltages of 7 and 8 kV for the two lowest $\textit{Re}_{\tau }$ cases. Despite the approximate order of magnitude lower actuator blowing velocity, and five-times larger $\textit{Re}_{\tau }$ used in the current study, a comparable reduction in $u_{\tau }$ is produced. The logarithmic trend of the viscous drag reduction documented by Wei & Zhou (Reference Wei and Zhou2024) is readily apparent in the plot. The present data in figure 10 are highlighted by the dashed ovals. The are observed to fall on the Wei & Zhou (Reference Wei and Zhou2024) trends when the actuator blowing velocity in inner variables, $W_{max}^+$ , on the abscissa is scaled by the corresponding $\textit{Re}_{\tau }$ . This scaling is equivalent to a spanwise blowing velocity Reynolds number, $W_{max} \delta /\nu$ .

Figure 14. Cross-plot of data from Wei & Zhou (Reference Wei and Zhou2024) that documented the effect on actuator blowing velocity on their viscous drag reduction, and corresponding data from figure 10 at two pulsed-DC voltages.

Comparisons like these are extremely important not only in the application of this technology, but also in further understanding the flow physics underlying TBLs. Additionally, the two different types of plasma actuators (unidirectional or opposed blowing) used in these studies represents alternate methodologies to produce the spanwise buffer layer blowing. Knowing the scaling with Reynolds number and blowing velocity, opens the opportunity for other approaches such as compressed air tangential jets investigated by Zhang et al. (Reference Zhang, Wei, Zhang, Wang and Zhou2024).

3.2. Unsteady boundary layer characteristics

This section documents changes in the structural characteristics of the TBL that result from the unidirectional spanwise buffer layer blowing. These experiments were performed in the Mach 0.15 Wind Tunnel with the TBL conditions given in table 1. The experimental set-up for these measurements is shown in figure 15.

Figure 15. Photograph showing plasma actuator array mounted onto the surface of the drag measurement platen, and pair of X hot-wire probes (a) and coordinate for conditioned spatial velocity reconstructions (b).

The actuator array in this set-up was fabricated in the same manner as previously discussed in § 2. However, in this set-up, the actuator array consisted of seven plasma actuators spaced 28 mm apart compared with nine actuators spaced 22 mm apart in the Mach 0.6 wind tunnel experiments presented in § 3.1. Based on the $u_{\tau }$ value for the baseline boundary layer, the 28 mm electrode spacing corresponds to a $\lambda _z^+ \approx 300$ . Based on figure 11, we anticipated that this would result in a sizable reduction in the wall shear stress that would produce tangible changes in the characteristics of the turbulent boundary that could aid in identifying the underlying viscous drag reduction mechanisms.

The set-up included two dual hot-wire sensor probes. One [ $u,v$ ] dual-sensor hot-wire was fixed in space at a location that was 1 cm, or $x^+=115$ downstream of the actuator array. Previous measurements (Duong Reference Duong2019; Duong et al. Reference Duong, Corke and Thomas2021) indicated that conditions there were representative of those over the plasma array. The spanwise location of the fixed sensor was aligned with the edge of an exposed electrode facing the covered electrode of one of the plasma actuators in the array. This spanwise location of the fixed [ $u,v$ ] sensor is defined as $z=0$ . The wall-normal position of the fixed [ $u,v$ ] sensor corresponded to $y^+=15$ .

The second sensor involved interchangeable [ $u,v$ ] and [ $u,w$ ] dual-sensor hot-wires that were mounted to a traversing mechanism and moved in 3-D space throughout the boundary layer. The motion resolution of the traversing mechanism in each direction was $\unicode{x1D6E5} _z=0.0254$ mm, which when normalized by TBL inner wall variables corresponded to $\unicode{x1D6E5} _z u_{\tau }/\nu =0.27$ . All of the hot-wires were operated in a constant temperature mode using the previously described floating ground design.

Figure 16(a) documents the spanwise uniformity of the baseline boundary layer mean velocity profile measured 1 cm downstream of the passive plasma actuator array. The profiles were measured at spanwise locations in $z^+=50$ intervals over a range of $z^+=600$ . This physically corresponded to a span of 5.22 cm in increments of 4.25 mm that encompassed a pair of plasma actuator electrodes in the array. A comparable spanwise uniformity occurred for the $u$ , $v$ and $w$ turbulence intensity profiles (Meyers Reference Meyers2023). Figure 16(b) compares the baseline $u$ , $v$ and $w$ component turbulence intensity profiles with those of Klebanoff (Reference Klebanoff1954). While these profiles are for the baseline boundary layer, as documented by Meyers (Reference Meyers2023), similar spanwise uniformity existed for the boundary layer with the spanwise buffer layer blowing.

Figure 16. (a) Mean velocity profiles at different spanwise locations in the measurement region for the baseline TBL and (b) comparison of the $u$ , $v$ and $w$ component turbulence intensity profiles measured at the $z^+=50$ to those of Klebanoff (Reference Klebanoff1954).

In these experiments, it was necessary to first determine the operation settings for the plasma actuator array. The object was to find the condition that would provide the largest reduction in the wall shear stress, with the assumption that this would provide the best opportunity to examine the effect this had on the underlying structural characteristics of the boundary layer. For this, the DC pulsing frequency and duty cycle remained fixed at the original 500 Hz and 1 %, respectively. Only the DC voltage was varied.

The effect of varying the plasma actuator array DC voltage on the per cent of drag reduction is shown in figure 17. The plot includes results from the present experiment as well as from the previous experiment of Duong et al. (Reference Duong, Corke and Thomas2021). The plasma actuator array had been rebuilt in the interim between the previous and current experiments and including the previous results provided an indication of the high degree of repeatability of the measurements. The agreement between the two experiments is shown to be excellent. A similar result for an opposed wall jet plasma actuator voltage that maximized the viscous drag reduction was documented by Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) (their figure 10).

As previously specified in § 2, the load cell for these experiments had a full-scale rating of 0.5 N (51 g) and a total uncertainty of 0.05 %, which equated to a force measurement uncertainty of 0.025 g. In this set of experiments, the time-averaged shear force exerted on the platen was obtained from a necessary number of records, each containing 30 s of contiguous load cell voltages, that were needed to reach a stable mean value. Based on the results in figure 17, the pulsed-DC actuator voltage used in this phase of the experiments was fixed at 5 kV.

Figure 17. Viscous drag reduction as a function of the Pulsed-DC actuator array DC voltage in experiments focusing on the unsteady TBL characteristics produced by spanwise buffer layer blowing.

The effect of the buffer layer spanwise blowing on the three fluctuating velocity components and Reynolds stresses is presented in figure 18. These correspond to the $z^+=50$ spanwise location. These document a significant reduction in the fluctuation levels and Reynolds stresses across a majority of the boundary layer. Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) found comparable reductions in the $uv$ -Reynold stress that accompanied their viscous drag reduction. These results are particularly significant since these turbulent stress components are important in the turbulence production near the wall (Corino & Brodkey Reference Corino and Brodkey1969).

Figure 18. Wall-normal profiles of the buffer layer blowing to baseline ratios of the three fluctuating velocity components and Reynolds stresses measured at the $z^+=50$ location.

Figure 18 established a reduction in the turbulent stress values through most of the boundary layer. In establishing the scaling with inner-wall variables, Duong (Reference Duong2019) and Duong et al. (Reference Duong, Corke and Thomas2021) had demonstrated a relation between the reduction in the $u$ component of turbulence intensity and the reduction in $u_{\tau }$ that occurs with the spanwise buffer layer blowing. This involved defining a ratio $R^2_{u_i}$ where

(3.1) \begin{equation} R_{u_i}^2 =\frac {\left(\overline {u_i}^2/u_{\tau }^2\right)_{5 \,{\textrm{kV}}}}{\left(\overline {u_i}^2/u_{\tau }^2\right)_{0 \,{\textrm{kV}}}} \end{equation}

in which $u_i$ refers to a fluctuating velocity component, $u$ , $v$ or $w$ , and the subscripts $0 \,{\rm kV}$ and $5 \,{\rm kV}$ that, respectively, refer to the baseline and spanwise buffer layer blowing boundary layers. A similar approach was followed here in order to examine if the scaling applies to the $v$ and $w$ velocity components. For this, values of $R^2_{v}$ and $R^2_{w}$ were calculated at every height in the boundary layer at all of the spanwise locations. The result is shown in figures 19(a) and 19(b).

Figure 19. Wall-normal profiles of $R^2_{v}$ (a) and $R^2_{w}$ (b) given by (3.1), and wall-normal profiles of $v_{5 \,{\textrm{kV}}}/v_{0 \,{\textrm{kV}}}$ (c) and $w_{5 \,{\textrm{kV}}}/w_{0\,{\textrm{kV}}}$ (d) at each measured spanwise position in the TBLs.

The $R^2_{v}$ and $R^2_{w}$ profiles are observed to reasonably collapse across the full $z^+$ span of the measurements. Spatially average values based on the region from the wall up to the edge of the log layer are indicated in each plot by the vertical dashed lines. The values are given in table 3. For reference, $R^2_{u}$ is also included in the table.

Table 3. Turbulence intensity component scaling with $u_{\tau }$ .

Note: $'$ denotes RMS

Rewriting (3.1) as

(3.2) \begin{equation} R_{u_i}^2 = \left (\frac {u_{\tau ,0 \,{\textrm{kV}}} }{u_{\tau ,5 \,{\textrm{kV}}}}\right )^2 \overline {\left (\frac {u'_{i,5 \,{\textrm{kV}}}} {u'_{i,0 \,{\textrm{kV}}}}\right )^2} \end{equation}

separates out the ratio of the wall shear velocities for the baseline ( $0\, \,{\rm kV}$ ) and buffer layer spanwise blowing ( $5\, \,{\rm kV}$ ) cases. In the present experiment with the 5 kV actuator voltage

(3.3) \begin{equation} \left (\frac {u_{\tau ,0 \,{\textrm{kV}}}}{u_{\tau ,5 \,{\textrm{kV}}}}\right )^2=2.56 \end{equation}

so that

(3.4) \begin{equation} R_{v}^2 = 2.56 \overline {\left (\frac {v'_{5 \,{\textrm{kV}}}}{v'_{0 \,{\textrm{kV}}}}\right )^2}\qquad \mbox{and}\qquad R_{w}^2 = 2.56 \overline {\left (\frac {w'_{5\,{\textrm{kV}}}}{w'_{0 \,{\textrm{kV}}}}\right )^2}. \end{equation}

The values of $\overline {u'_{i,5 \,{\textrm{kV}}}/u'_{i,0 \,{\textrm{kV}}}}$ that are needed to satisfy (3.4) are given without parentheses in table 3.

As a check, the wall-normal profiles of $v'_{5\,{\textrm{kV}}}/v'_{0 \,{\textrm{kV}}}$ and $w'_{5 \,{\textrm{kV}}}/w'_{0 \,{\textrm{kV}}}$ at each spanwise position are shown in figures 19(c) and 19(d). As with the $R_{v}^2$ and $R_{w}^2$ profiles shown in the figure, a reasonable collapse is observed. The spatial average for each plot based on the region from the wall to the edge of the log layer is again shown by the vertical dashed line. These are indicated by the bracketed values in table 3. The excellent agreement between the predicted and actual values supports the relationship (3.1) of the levels all three components of the turbulence intensity to the level of the wall shear stress.

This scaling provides an indication of how the spanwise buffer layer blowing affects the individual turbulence components. The values of $u'_{5 \,{\textrm{kV}}}/u'_{0 \,{\textrm{kV}}}$ , $v'_{5 \,{\textrm{kV}}}/v'_{0 \,{\textrm{kV}}}$ and $w'_{5 \,{\textrm{kV}}}/w'_{0 \,{\textrm{kV}}}$ are also given in table 3. These indicate that the largest effect of the spanwise buffer layer blowing is on the $v'$ fluctuations with a 21 % reduction. This is closely followed by the $u'$ fluctuations with a 18 % reduction, and finally the $w'$ fluctuations with a 13 % reduction. Note that these values are the spatial average over the full boundary layer thickness. As shown in figure 18, the reduction in the peak velocity fluctuation levels with the buffer layer blowing are somewhat larger, and is greatest for the streamwise component.

As with Duong et al. (Reference Duong, Corke, Thomas and Disser2018), the spanwise buffer layer blowing was found to result in a reduction in the turbulence production defined as

(3.5) \begin{equation} P = p\frac {\delta }{U^{3}_{\infty }} = -\overline {uv}\frac {{\rm d}U}{{\rm d}y}\frac {\delta }{U^{3}_{\infty }}. \end{equation}

This is documented in figure 20, which shows wall-normal profiles of the production based on (3.5) without and with buffer layer blowing. The decrease in the production with the buffer layer blowing was particularly pronounced near $y^+=12$ , where an approximate 66 % reduction in the production occurs. Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) similarly documented a substantial reduction in the turbulence production, $-\overline{uv} ({{\rm d}U}\kern-2.8pt/{{\rm d}y})$ , near the wall for their drag reduced cases, which they found to be `consistent with the stabilized streaks’.

Figure 20. Comparison between wall-normal profiles of the turbulence production of the baseline and buffer layer blowing cases.

3.2.1. Conditional flow field reconstruction

In order to further investigate the effect of the flow control on the coherent structures in the near-wall region of the boundary layer, conditional-averaged reconstructions of the flow field were performed. The coordinate system for this was shown in figure 15(b). With this, the [ $u,v$ ] hot-wire probe at the fixed [ $x,z$ ] location was again positioned at $y^+=15$ . Recall that the fixed probe was located at $x^+=115$ downstream of the plasma actuator array, where previous results (Duong Reference Duong2019; Duong et al. Reference Duong, Corke and Thomas2021) had indicated conditions were representative of those over the plasma array. The fixed probe provided the [ $u,v$ ] time series used to define `events’ on which the conditional averages were based. The other [ $u,v$ ] and [ $u,w$ ] hot-wire probes were located at the same $x$ location as the fixed probe, and moved in the [ $y,z$ ] directions with the same range as used to compile results in figure 16. The time series from both the fixed and moving hot-wire probes were sampled simultaneously.

The method for defining an ‘event’ utilized the VITA technique that was pioneered by Blackwelder & Kaplan (Reference Blackwelder and Kaplan1976). This utilized the $u$ velocity time series and applied a discriminator, $D$ , such that

(3.6) \begin{equation} D(t) = \left\{\!\! {\begin{array}{c} 1 \;\;\mbox{if}\;\; \widehat {\textit{var}} \gt ku^2_{\textit{rms}}\\[4pt] 0 \;\; \mbox{Otherwise} \end{array} }\right . \end{equation}

where

(3.7) \begin{equation} \widehat {\textit{var}}(x_i,t,T)=\widehat {u^2}(x_i,t,T) - \left [\widehat {u}(x_i,t,T)\right ]^2 \end{equation}

is the localized variance of streamwise velocity fluctuations over the time interval $T$ at a spatial location, $x_i$ . With this discriminator, Blackwelder & Kaplan (Reference Blackwelder and Kaplan1976) recommended that $k=1.2$ , and that $T\kern-1ptu_{\tau }^2/\nu = 10$ . Furthermore, they applied the detection at $y^+=15$ in the boundary layer, where the turbulence intensity was a maximum. The conditionally averaged velocity field that results from VITA detection is thought to embody the initial ‘ejection’ of low momentum fluid from the wall that is followed by a faster moving ‘sweep’ of fluid that approaches from outside the wall region (Blackwelder & Kaplan Reference Blackwelder and Kaplan1976). These motions are associated with streamwise near-wall vortices. Figure 21(a) shows examples of the streamwise velocity ‘signature’ associated with VITA detections having different values of the threshold parameter, $k$ .

Once a reference time for each VITA event was determined by the detection function, $D(t)$ , the characteristics of the flow field in time and space were constructed based on a conditional average defined as

(3.8) \begin{equation} \langle Q(y_i,z_i,T)\rangle _{y^+=15} = \frac {1}{N}\sum _j^N Q(y_i,z_i,t_j+T) \end{equation}

where $Q$ is the estimated quantity which is a function of the independent variables [ $y_i,z_i$ ] that denote the position in space of the moving probe where the sampling occurred, and the subscript $y^+=15$ denotes the wall-normal position of the fixed probe where the detection occurred. The quantities $t_j$ are those points in the time series when a detection occurred. Since $D(t)$ was equal to unity only during very short time intervals, the times, $t$ , were taken to be midway between the beginning and end of the period during which $D(t)\neq 0$ . A positive or negative time delay $T$ with a range of $-4 \leq T^*= ({TU_{\infty }}/{\delta }) \leq 4$ , was used to determine the temporal behaviour of $Q$ before and after a detection occurred. Here $N$ is the total number of samples used to form the ensemble average.

One of the issues in comparing ensemble averages based on the occurrence of VITA ‘ejection-sweep’ events is that as previously documented (Duong Reference Duong2019; Duong et al. Reference Duong, Corke and Thomas2021), the number and strength of these events is significantly reduced with the spanwise buffer layer blowing. This is demonstrated in figure 21(b) which shows the frequency of events as a function of the VITA threshold parameter, $k$ , for the baseline and spanwise buffer layer blowing boundary layers. The frequency of events was simply defined as the inverse of the number of VITA detections that occurred over the same fixed period of time. This demonstrates that at any value of the detection threshold parameter, $k$ , the frequency of events was significantly lower in the boundary layer with the buffer layer blowing. At the recommended (Blackwelder & Kaplan Reference Blackwelder and Kaplan1976) $k=1.2$ with buffer layer blowing, the frequency of VITA events was approximately 57 % lower than with the baseline boundary layer. This reduction in the frequency of VITA events with buffer layer blowing is quite comparable to the 62 % reduction in the viscous drag and the 38 % reduction in $u_{\tau }$ that had occurred with the buffer layer blowing. Such a connection between the wall shear velocity and frequency of ejection-sweep events was first reported by Kline et al. (Reference Kline, Reynolds, Schraub and Runstadler1967).

Figure 21. Sample ensemble averages for different VITA threshold parameters, $k$ , in baseline boundary layer (a) and variation in the frequency of VITA events as a function of $k$ , for the baseline and spanwise buffer layer blowing TBLs (b). Here $k=1.2$ used for conditional measurements.

The relevance of the VITA events to the turbulence Reynolds stresses and thereby turbulence production (3.1), is illustrated in figure 22. This shows ensemble averaged time series of the [ $u v$ ]- and [ $u w$ ]-Reynolds stress associated with VITA events measured at $y^+=15$ for the baseline and spanwise buffer layer blowing boundary layers. The threshold parameter for these was $k=1.2$ . These time series indicate a substantial reduction in both Reynold stress components during ‘ejection-sweep’ events that results from the spanwise blowing. Thus, in addition to reducing the frequency of these events, when they do occur their contribution to the turbulence production is substantially reduced. Overall, due to the spanwise buffer layer blowing, the $uv$ -stress component associated with ‘ejection’ and ‘sweep’ events was reduced by 52 % and 67 %, respectively. Similarly, the $uw$ -stress was reduced by 32 %. In concurrence with these results, while applying the VITA technique Cheng et al. (Reference Cheng, Wong, Hussain, Schroder and Zhou2021) observed a reduction in the amplitude of the ‘ejection’ portion of the VITA ensemble velocity time trace.

Figure 22. Conditionally averaged $\langle uv \rangle$ and $\langle uw \rangle$ Reynolds stress time traces associated with VITA detections measured at $y^+=15$ for baseline and spanwise buffer layer blowing boundary layers.

3.2.2. Coherent vortical structure reconstructions

Three-dimensional reconstructions of vortical structures were obtained through ensemble averages of the velocity field in 3-D space that were correlated to the occurrence of VITA events. The time series used in the VITA detection was from the $u$ -component of the stationary hot-wire located 1 cm downstream of the actuator array at a height corresponding to $y^+=15$ . The spanwise position corresponded to $z=0$ shown in figure 15. The reconstructed velocity field was obtained from the ( $u,v$ ) and ( $u,w$ ) hot-wire sensors that were moved in the ( $y,z$ ) directions at a fixed $x$ position located 1 cm downstream of the actuator array. Velocity time series from the stationary hot-wire probe and either of the dual sensor probes were simultaneously sampled. In postprocessing, the time instants of a VITA detection in the $u$ -velocity from the stationary probe was used to mark a time instant in the ( $u,v$ ) or ( $u,w$ ) time series about which an ensemble average was constructed.

An important step in the spatial velocity reconstructions was determining the spanwise extent to which the velocity fluctuations at the location of the movable probe were correlated with the VITA events at $z=0$ . The result is presented for the baseline boundary layer in figure 23. This documents the correlation coefficient, $\overline {R}_{max}=\sigma [u(z^+_{ref},t)u(z^+,t+\tau )]/(\sigma [u(z^+_{ref})]\sigma [u(z^+)])$ , between velocity time series of the fixed hot-wire sensor and that of a second sensor spaced some spanwise distance, $z$ , apart. Here $\sigma$ is the standard deviation. The two sensors were at the same height in the boundary layer, and based on the definition $-1 \leq \overline {R} \leq +1$ .

Figure 23 indicates that velocity fluctuations associated with VITA events detected at $z=0$ ( $z^+=0$ ) are reasonably correlated with velocity fluctuations to a spanwise distance corresponding to $z^+=400$ . This was then taken as the spanwise limit over which spatial velocity reconstructions were performed.

Figure 23. Spanwise correlation coefficient, $[\overline {R}(z^+,\tau )]_{max}$ , between the fixed probe at $[x^+,y^+,z^+]= [0,15,0]$ and the moving probe at $[x^+,y^+,z^+]=[0,15,50{-}600]$ in baseline boundary layer.

Figure 24. Constant level contours of $\langle \omega _y (y^+_{15},z^+_i,x^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ for the baseline (a) and spanwise sublayer blowing (b) boundary layers.

Figure 24 presents constant level contours of the wall-normal vorticity in the $[x^+,z^+]$ plane at a height of $y^+=15$ , based on the conditionally averaged velocity field correlated with $k=1.2$ VITA events detected at $[x^+,y^+,z^+]=[0,15,0]$ , namely $\langle \omega _y (y^+_{15},z^+_i,x^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ . Figure 24(a) is for the baseline boundary layer, and the figure 24(b) is for the boundary layer with spanwise buffer layer blowing. For these plots and those to follow, the original correlation time scale was converted to a pseudospatial streamwise length scale, $x^+$ , by assuming a convection speed of $0.6U_{\infty }$ .

The coherent features of $\omega _y$ in the baseline boundary layer are consistent with the spanwise mean flow inhomogeneity caused by the counter-rotating longitudinal vortical structures associated with the wall ‘streak structure’ first observed by Kline et al. (Reference Kline, Reynolds, Schraub and Runstadler1967). The spanwise spacing between $\omega _y$ contours of opposite signs, for example denoted in figure 24(a), is consistent with the well documented $\lambda ^+_z=100$ streak spacing.

As previously discussed, our spanwise sublayer blowing approach was initially motivated in part, by the channel flow simulations of Schoppa & Hussain (Reference Schoppa and Hussain1998) who found it resulted in a reduction in the spanwise mean flow distortion that flanked the low-speed wall streak structure, as well as to produce a significant reduction in the wall shear stress. Schoppa & Hussain (Reference Schoppa and Hussain2002) later linked these observations to the control of ‘streak transient growth’ in which the wall normal vorticity, $\omega _y$ , is a critical stability parameter. The premise was that this form of flow control would weaken $\omega _y$ and thereby intervene in the autonomous cycle shown in figure 1.

Figure 24 reveals that the spanwise buffer layer blowing has, in fact, significantly reduced the levels of $\omega _y$ . Specifically, when averaged over a 3-D region within $-250\leq x^+ \leq 250$ by $0 \leq y^+ \leq 20$ by $0 \leq z^+ \leq 400$ , the $\omega _y$ was reduced by 58 % compared with the boundary layer without the spanwise buffer layer blowing. This reduction in $\omega _y$ is nearly the same as the reduction in $u_{\tau }$ , which supports the connection observed by Schoppa & Hussain (Reference Schoppa and Hussain1998). Consistent with the mechanism shown in figure 1, the reduction in $\omega _y$ due to the spanwise blowing should result in weakened sublayer streamwise vorticity, $\omega _x$ , which is examined next.

Figure 25. Constant level contours of $\langle\omega _x (y^+_{15},z^+_i,x^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ , for the baseline (a) and spanwise sublayer blowing (b) boundary layers.

The near-wall longitudinal vortical structures would appear as coherent features in the $\omega _x$ vorticity. This is presented in figure 25 as constant level contours in the $[x^+,z^+]$ plane at a height of $y^+=15$ , based on the conditionally averaged velocity field correlated with $k=1.2$ VITA events detected at $[x^+,y^+,z^+]=[0,15,0]$ , namely $\langle \omega _x (y^+_{15},z^+_i,x^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ . Similar to figure 24, figure 25(a) is for the baseline boundary layer, and figure 25(b) is for the boundary layer with spanwise buffer layer blowing. The elongated near-wall vortical structures are readily apparent for the baseline boundary layer (figure 25 a). Their spanwise spacing is again in agreement with the previously observed $\lambda ^+_z=100$ streak spacing (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967).

The spanwise buffer layer blowing has clearly reduced the overall magnitude of the streamwise vorticity in the elongated near-wall structures. In addition, it appears to have reduced the contiguous streamwise extent of these structures, which is related to VITA events of shorter duration and lower amplitude. However, when accounting for the lower $u_{\tau }$ levels that occur with the spanwise blowing, the average spanwise spacing between the elongated structures remains at $\lambda _z^+=100$ as indicated in figure 25(b).

To further illustrate the effect of the spanwise buffer layer blowing on the coherent near-wall longitudinal structures, figure 26 shows constant level contours in the $[y^+,z^+]$ (cross-flow) plane at the $x^+=0$ location, namely $\langle \omega _x (x^+_0, y^+_i, z^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ . These again are based on the conditionally averaged velocity field that was correlated with the $k=1.2$ VITA events detected at $[x^+,y^+,z^+]=[0,15,0]$ . Similarly, figure 26(a) is for the baseline boundary layer, and figure 26(b) is for the boundary layer with spanwise buffer layer blowing.

Figure 26. Constant level contours of $\langle \omega _x (x^+_0, y^+_i, z^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ , for the baseline (a) and spanwise sublayer blowing (b) boundary layers.

This view through the near-wall region of the boundary layers represents a ‘cut’ in the $[y,z]$ -plane at $x^+=0$ (VITA detection location) of the constant level contours that were shown in figure 25. This cross-flow-plane view of the $\omega _x$ vorticity presents another realization of the counter-rotating vortex pairs that make up the near-wall longitudinal vortical structures. This view dramatically illustrates the effect of the spanwise buffer layer blowing to reduce the level of streamwise vorticity. As expected, this is particularly evident in the boundary layer sublayer ( $y^+\leq 40$ ) which is delineated by the dashed line in figure 26. This very significant reduction in sublayer streamwise vorticity is consistent with the observed reduction in ejection/sweep event Reynolds stress. Furthermore, it results in reduced transfer of high momentum fluid towards the wall which reduces the near-wall mean strain rate.

Figure 27 presents constant level contours of normalized streamwise velocity fluctuations in the top-down ( $x^+,z^+$ ) plane at a wall-normal height of $y^+=15$ namely $\langle u'/U_{\infty } (y^+_{15}, z^+_i, x^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ . Figure 27(a) is for the baseline flow and figure 27(b) is for the case of plasma-induced near-wall spanwise blowing. The elongated streaks that are present maintain the streak spacing of $\lambda _z = 100$ as expected in both the baseline and drag reduced cases. However, the level of streamwise velocity fluctuations is shown to be significantly reduced for the actuated case. This lower streamwise fluctuation level in the actuated case is a direct consequence of the suppression of near-wall streamwise vorticity produced by the spanwise blowing shown previously in figures 25 and 26.

Figure 27. Constant level contours of $\langle u'/U_{\infty } (y^+_{15}, z^+_i, x^+_i)\rangle _{[x^+,y^+,z^+]=[0,15,0]}$ , for the baseline (a) and spanwise sublayer blowing (b) boundary layers.

3.3. Temporal recovery from spanwise buffer layer blowing

The ability to alter the TBL characteristics with on-demand spanwise buffer layer blowing provided an opportunity to investigate the characteristic time scales associated with turbulence production that could be documented when the boundary layer was allowed to return to its baseline condition. The approach then was to start from a state produced by the buffer layer blowing with boundary layer characteristics documented in the previous section, and then to follow the temporal development of the boundary layer characteristics following termination of the buffer layer blowing.

The experimental set-up to accomplish this is shown in figure 28. This utilized a timing circuit that after operating the pulsed-DC plasma actuators to produce the spanwise buffer layer blowing for 10 s, corresponding to a $t_1^+=18\,525$ , then terminated the actuator operation. The voltage time series from the hot-wire sensors and load cell were recorded at the start of the process and continued for an additional 20 s, corresponding to a $t_2^+=37\,050$ , which was found to be sufficient for the boundary layer conditions (mean and fluctuating) to return to an asymptotic baseline state. This process was repeated until converged statistics were achieved.

Figure 28. Schematic of the experimental set-up used in document the temporal effect of terminating the spanwise buffer layer blowing and boundary layer development to its baseline condition.

The temporal evolution of several of the TBL characteristics back to their baseline state are presented in subsequent figures. In these, the time, $t$ , following termination of spanwise blowing has been normalized using inner-wall variables. It is important to point out that the values of $t^+$ utilize the instantaneous $\tau _w(t^+)$ values that were recorded with the drag measuring platen, and which change as the boundary layer returns to its baseline state. In these figures, the subscript ‘ $b$ ’ refer to the ‘baseline’ condition.

Figure 29 shows the temporal evolution of the ratio of the boundary friction velocity, $u_{\tau }(t^+)/u_{\tau _{b}}$ , starting from the time when the spanwise buffer layer blowing was terminated. This indicated that with respect to the wall shear stress, the boundary layer reaches its baseline state after $t^+ \approx 4000$ from the termination of the spanwise blowing. This evolution time reference will be noted with respect to other boundary layer characteristics in subsequent figures.

Figure 29. Temporal evolution of the ratio of the boundary layer wall friction velocity, $u_{\tau }(t^+)/u_{\tau _{b}}$ , following the termination of the spanwise buffer layer blowing.

As had been shown in figure 8, the spanwise buffer layer blowing that resulted in the reduction of the wall shear stress produces an upward shift of the mean velocity profiles when scaled by inner variables. With this in mind, following the termination of the buffer layer blowing, figure 30 shows the evolution of the mean velocity profiles back towards the zero-pressure-gradient canonical behaviour.

Figure 30. Temporal evolution of the mean velocity profiles scaled by inner variables following the termination of the spanwise buffer layer blowing.

The temporal development of the geometric centre of the logarithmic region of the mean velocity profile in inner variables following the termination of the spanwise buffer layer blowing is shown in figure 31. The significance of the geometric centre was highlighted by Mathis et al. (Reference Mathis, Hutchins and Marusic2009). Upon recovery to the baseline condition, the $y^+$ geometric centre asymptotes to a value of $3.9\sqrt {\textit{Re}_{\tau }}=152$ . This appears to coincide with the asymptotic time scale of $t^+=4000$ , which is noted on the plot.

Figure 31. Temporal evolution of the mean velocity log profile centre following the termination of the spanwise buffer layer blowing.

The temporal evolution of the root mean square (r.m.s.) fluctuation levels of the three velocity components following the termination of the spanwise buffer layer blowing is shown in figure 32(a). These were measured at $y^+=15$ (Reynolds stress peak wall-normal location) that corresponded to $t^+=0$ in the boundary layer. The physical distance from the wall was adjusted to maintain a constant $y^+$ as $u_{\tau }(t^+)$ increased during the recovery to the baseline condition. The time rate of change of the r.m.s. fluctuation levels are shown in figure 32(b).

The three components of the velocity fluctuation appear to reach 98 % of the baseline level at $t^+\approx 4000$ , which indicates that their development slightly lags behind that of the mean flow. With regard to the redevelopment of the respective component fluctuation levels, based on the time rate of change, the $v$ and $w$ fluctuations both lag behind the $u$ fluctuations. This is consistent with the fact that the turbulence production in the boundary layer transfers energy directly to the streamwise component, with the $v$ and $w$ components depending on pressure-strain rate terms to transfer energy to them from the streamwise component. The three components reach equal fluctuation levels at $t^+\approx 4000$ , but continue to exhibit growth beyond the $t^+\approx 4000$ . This is consistent with the fact that the characteristic turbulence time scales exceed that associated with the mean strain rate, which returned to the baseline sooner.

Figure 32. Temporal evolution of the ratios of the boundary layer velocity component fluctuation levels (a) and their rate of change (b) measured at $y^+=15$ that followed the termination of the spanwise buffer layer blowing.

As previously discussed, the objective of the spanwise buffer layer blowing was to disrupt the autonomous near-wall cycle involving liftup and breakup of coherent streamwise vorticity associated with the low-speed wall streak structure and near-wall turbulence production. A metric of merit to the approach was a reduction in the wall-normal vorticity, $\omega _y$ , which was related to the spanwise mean flow distortion that flanked the elongated near-wall structures, and was subsequently deemed a critical parameter in the autonomous cycle involving streak transient growth (Schoppa & Hussain Reference Schoppa and Hussain2002).

Considering the importance of this parameter to our objectives, figure 33 documents the temporal growth of the wall-normal vorticity, $\omega _y$ , following the termination of the spanwise buffer layer blowing. The values shown are a spatial average within a 3-D region of $0 \leq y^+ \leq 20$ by $-250 \leq x^+ \leq 250$ by $0 \leq z^+ \leq 400$ . This corresponds to the wall-normal vortical structures in the $[x^+,y^+=15,z^+]$ region that were visualized in figure 24.

The detection of VITA events was used to perform the 3-D spatial ensemble-averaged velocity field reconstructions used in the calculation of vorticity components for the baseline and spanwise buffer layer blowing boundary layers. Therefore, it was also of interest to document the temporal evolution of the number of VITA events that occurred following the termination of the spanwise buffer layer blowing. Figure 33 shows the temporal evolution of the number of VITA events along with the temporal evolution of the wall-normal vorticity component, $\omega _y$ , and the wall shear stress, $\tau _w$ . Figure 33 highlights the significant link between the level of wall-normal vorticity, the frequency of turbulence producing VITA events, and subsequently the wall shear stress. Although all of these exhibit a similar growth with $t^+$ following the termination of the spanwise buffer layer blowing, it is important to note that the temporal recovery of $\omega _y$ leads that of the number of VITA events and the increase in the wall shear stress. This observation is consistent with the autonomous near-wall mechanism shown schematically in figure 1 whereby increased levels of wall-normal vorticity are required to increase the production of streamwise vorticity which, in turn, gives rise to an increase in the number of VITA ejection-sweep events that has been shown to be tied to the level of the wall shear stress (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967).

Viewed from the standpoint of an instability mechanism of the distorted mean flow, of which $\omega _y$ is a critical parameter, one might expect a sharp delineation between stable and unstable regimes which is not apparent in figure 33. However, it must be remembered that the conditional measurements were based upon VITA event detections in both the baseline and actuated flows. Hence, it is to be expected that even in the actuated flow, the level of $\omega _y$ was within the unstable regime for streak transient growth in the sense that longitudinal streamwise buffer layer vorticity still existed in the actuated flow and produced VITA events, but at a lower frequency and level.

Figure 33. Temporal evolution of the ratios of the wall-normal vorticity component, number of VITA events and wall shear stress following the termination of the spanwise buffer layer blowing.

Another metric of practical interest is the convective distance over which the boundary layer characteristics recover from the termination of the spanwise buffer layer blowing. In channel flow DNS simulations, Schoppa & Hussain (Reference Schoppa and Hussain1998) had shown that the region of sustained drag reduction would extend to $x^+ \simeq 16\,000$ . In direct drag measurements using oil film interferometry, Thomas et al. (Reference Thomas, Corke, Duong, Midya and Yates2019) found good agreement with that prediction. However, the level of drag reduction in that early experiment used less optimized actuator conditions, producing only a 6 % reduction in $u_{\tau }$ .

As indicated, aside from the turbulence fluctuations, the recovery to the baseline boundary layer characteristics following the termination of the spanwise buffer layer blowing nominally occurred in $t^+ \approx 4000$ . The values of $t^+$ were based upon the recorded time varying values of $u_{\tau }$ , which corresponded to a physical time of 2.5 s, or a corresponding $T^*=t U_{\infty }/\delta = 82$ . Assuming a convective speed of $U_c \approx 0.7 U_{\infty }$ , the corresponding streamwise fetch associated with the return to the baseline state was $x\approx 66\delta$ . This result is encouraging in terms of practical implementation for large-scale drag reduction. The inner-variable scaled streamwise recovery distance base on $t^+=4000$ is $x^+=U_c^+t^+ \approx 86\,000$ , which is considerably larger than that reported by Schoppa & Hussain (Reference Schoppa and Hussain1998). This is not surprising since it is likely to be associated with the significantly higher reduction in the friction drag and turbulent stresses obtained in the present study which would naturally lead to a larger recovery distance.