2. Experimental apparatus and methods

2.1. Wind tunnel

Experiments were performed in the TFT boundary-layer wind tunnel at a nominal velocity of $U_{ref} = 25\ \textrm {m}\,\textrm {s}^{-1}$. The blowdown wind tunnel is described in detail in Mohammed-Taifour et al. (Reference Mohammed-Taifour, Schwaab, Pioton and Weiss2015) and Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016). As illustrated in figure 1, its test section is 3 m in length and 0.6 m in width. In the first half of the test section, a zero-pressure-gradient (ZPG) boundary layer develops on the upper surface and separates because of the adverse pressure gradient (APG) imposed by the diverging test-section floor. The boundary layer subsequently reattaches due to the favourable pressure gradient (FPG) that occurs when the floor converges again. The use of a bleed slot ensures that the boundary layer on the lower surface stays attached to the contoured part of the test-section floor. This slot connects directly to the atmosphere, while the interior of the test section is maintained at a slightly elevated pressure by a mesh positioned at the exit. Also shown in figure 1 is the pressure distribution measured on the centreline ($z=0\ \textrm {m}$) and at two symmetric spanwise positions ($z={\pm }0.15\ \textrm {m}$), as well as a contour plot of the average longitudinal velocity field on the centreline in the region of the TSB. For reference, at $x=1.3\ \textrm {m}$ the incoming boundary-layer thickness is $\delta =33\ \textrm {mm}$ and the momentum thickness is $\theta =3.26\ \textrm {mm}$, which implies a Reynolds number $Re_{\theta } \simeq 5000$.

Figure 1. (a) Average wall-pressure coefficient $c_p$ measured on the test-section centreline ($z=0\ \textrm {m}$) and at two symmetric spanwise positions ($z={\pm }0.15\ \textrm {m}$). (b) Profile sketch of the test section with contour plot of the longitudinal velocity (the numbers 1 to 5 refer to the particle image velocimetry measurement stations). Solid white and dashed blue lines denote the dividing streamline and the $\delta _{99}$ boundary-layer thickness, respectively. Inset: settling chamber and contraction of the PJA system (§ 2.2).

The position of reattachment on the test surface is strongly influenced by the presence of an FPG. This so-called suction-and-blowing configuration was specifically chosen to generate a TSB that mimics the region of separated flow in turbulent SBLIs, where the shear layer in the aft part of incident-shock or compression-ramp interactions is usually deflected towards the wall (Babinsky & Harvey Reference Babinsky and Harvey2011; Threadgill & Bruce Reference Threadgill and Bruce2020). Thus, the flow development in the current configuration is different than in flows that use a suction-only boundary condition to separate the boundary layer, and where reattachment occurs because of turbulent diffusion only (Wu et al. Reference Wu, Meneveau and Mittal2020). LeFloc'h et al. (Reference LeFloc'h, Weiss, Mohammed-Taifour and Dufresne2020) discuss the pressure distribution generated in the TFT boundary-layer wind tunnel in relation to other references from the literature.

2.2. Pulsed-jet actuators

The flow is controlled with a series of PJAs that can be placed at two streamwise positions on the test surface and that are distributed across the complete test-section span. Those actuators produce rectangular jets of velocity $U_j(t)$ ejecting from the test-section wall at a specific frequency. They were chosen because of their relatively large control authority compared to other types of actuators, the possibility of independently controlling their forcing amplitude, frequency and duty cycle as well as their proven success in the active control of two-dimensional flow separation (Petz & Nitsche Reference Petz and Nitsche2007; Cattafesta & Sheplak Reference Cattafesta and Sheplak2011). The pulsed jets are generated using a supply line of compressed air with a pressure $P_j$ that imposes a given mass-flow rate. The total mass flow is controlled by eight solenoid valves which simultaneously feed a settling chamber that is mounted on top of the test surface (see inset in figure 1). The settling chamber is filled with light foam to homogenize the air flow and ensure a constant forcing amplitude across the test-section span. The flow from the settling chamber is then accelerated through a third-order polynomial shaped contraction with an area ratio of $50$ before reaching a series of equally spaced slots perforated on the test surface. Each slot has a width of $h=1\ \textrm {mm}$ in the main streamwise direction of the flow and a length of $6h$ in the spanwise direction. The slots are spaced $2\ \textrm {mm}$ apart and are built with a slant angle in the test surface so that the jets exit at $45^\circ$ from the horizontal direction (an image of the system is shown in Mohammed-Taifour et al. Reference Mohammed-Taifour, Le Floc'h and Weiss2020). The control signal of the solenoid valves is a square wave of frequency $F_j$ that can be varied between $0$ and $300\ \textrm {Hz}$. The duty cycle was fixed at $25\,\%$ for the duration of the experiments. The amplitude of the pulsations is controlled by an analogue pressure reducer through which the pressure-supply level $P_j$ can be manually controlled between 1 and 7 bar. The jet velocity $U_j$ corresponding to each $P_j$ was measured experimentally with a single hot-wire probe positioned $1\ \textrm {mm}$ below the test surface at the centre of slot closest to the test-section centreline. An example of time traces of $U_j(t)$, measured when the wind tunnel was turned on at $U_{ref}=25\ \textrm {m}\,\textrm {s}^{-1}$, is shown in figure 2 for a frequency of 100 Hz and four amplitudes ranging from 1 to 6 bar. Repeatability studies demonstrated that the jet velocity can be reproduced within $0.5\ \textrm {m}\,\textrm {s}^{-1}$ by manually controlling the supply pressure $P_j$. Further measurements across the span of the wind tunnel demonstrated the homogeneity of the jets outflow within a few per cent. No evidence of negative velocity (i.e. air ingestion) was observed, either when the actuators pulsed air in the main wind-tunnel flow or in a quiescent environment.

Figure 2. Exemplary time traces of the jet velocity for $F_j = 100\ \textrm {Hz}$.

The pulsed jets are characterized by their maximum velocity $U_{j, max}$ when the solenoid valves are open, from which we define the velocity ratio $\lambda _j=U_{j,max}/U_{ref}$. The momentum coefficient is defined as $C_\mu =(U_{j, eff} / U_{ref})^2 (h / \delta )$, where $U_{j, eff}$ is the effective jet velocity $U_{j, eff}=\sqrt {\bar {U}_j^2+(U_{j, std})^2}$, $h=1\ \textrm {mm}$ is the width of the control slots and $\delta =33\ \textrm {mm}$ is the boundary-layer thickness at $x=1.30\ \textrm {m}$. In the definition of the jet effective velocity, $\bar {U}_j$ and $U_{j, std}$ are the mean value and the standard deviation of the jet velocity signal $U_j(t)$ during a full period, respectively. Defined as such, $C_\mu$ represents the average momentum input of the forcing system normalized by a reference momentum in a fully incompressible flow (Greenblatt & Wygnanski Reference Greenblatt and Wygnanski2000). We further define a mass-flow coefficient $C_m=\dot {m}_j/ \dot {m}_{ref} = (\bar {U}_j / U_{ref})(h/h_{ts})$, where $h_{ts}=0.15\ \textrm {m}$ is the height of the test section at $x=0$ and where $\dot {m}_j$ and $\dot {m}_{ref}$ are the mass-flow rates of the jet and the main stream in the wind tunnel, respectively. Table 1 recaps the values of $\lambda _j$, $C_\mu$, and $C_m$ as a function of the supply pressure $P_j$ for frequency $F_j=100\ \textrm {Hz}$ (figure 2). Note that, at a fixed value of the supply pressure $P_j$, $\lambda _j$, $C_\mu$ and $C_m$ vary slightly as a function of $F_j$.

Table 1. PJA parameters for $F_j=100\ \textrm {Hz}$.

The non-dimensional forcing frequency $F^+=F_j L_b/U_{ref}$, where $L_b=0.42\ \textrm {m}$ is the average length of the unforced backflow region on the test-section centreline, is provided in table 2, together with the ratios of several forcing frequencies used in the experiments to the characteristic frequencies of the breathing and shedding modes of the unforced flow (see § 3).

Table 2. Non-dimensional forcing frequency $F^+=F_j L_b/U_{ref}$ and ratio of several forcing frequencies to the characteristic frequencies of the breathing and shedding modes of the unforced flow: $f_{breath} \simeq 0.6\ \textrm {Hz}$ and $f_{shed} \simeq 15\ \textrm {Hz}$, respectively.

2.4. Three-dimensional effects in the average flow

Because the test section of the wind tunnel is rectangular, three-dimensional effects in the average flow are unavoidable. Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016) used oil-film visualizations to draw a consistent topological map of the skin-friction lines in the unforced TSB. They showed the strongly three-dimensional nature of the near-wall flow but argued that wall-normal measurements near the centreline can be considered as quasi-two-dimensional since the flow is symmetric and features a narrow band around the centreline where the wall streamlines are straight. Mohammed-Taifour, Dufresne & Weiss (Reference Mohammed-Taifour, Dufresne and Weiss2019) and LeFloc'h et al. (Reference LeFloc'h, Weiss, Mohammed-Taifour and Dufresne2020) further investigated the three-dimensional nature of the average (unforced) flow through Reynolds-averaged Navier–Stokes simulations and interpreted the directions of the wall streamlines by classical secondary-flow arguments in the sidewall boundary layers.

Oil-film visualizations on the test surface are shown in figure 3 for the unforced flow $(a)$ and two forced cases: a relatively mild forcing where the actuators were placed at $x=1.45\ \textrm {m}$ $(b)$ and a stronger forcing with actuators placed at $x=1.55\ \textrm {m}$ $(c)$. In both cases the forcing frequency was 20 Hz and the supply pressure was set to 4 bar. It can be seen that the overall structure of the wall streamlines is not changed by the actuation, and that the flow remains symmetric, thus precluding any significant average spanwise velocity through the centreline. On the other hand, for relatively strong forcing, and despite the spanwise-homogeneous forcing amplitude, the separation line is strongly curved, which implies a vanishing band of quasi two-dimensional flow. This can be explained as follows: as will be discussed in § 4, forcing does not have a strong effect on the average pressure distribution in the test section, nor, consequently, on the potential streamlines. Therefore, the secondary flows on the sidewalls and the associated corner effects remain relatively unaffected. On the other hand, as can be observed in figure 3, forcing strongly changes the position of flow separation on the test-section centreline. This difference in behaviour between the centreline and the sidewalls leads to stronger three-dimensionality of the average wall streamlines when forcing is more effective.

Figure 3. Sample of oil-film visualizations. $(a)$ Unforced flow, $(b)$ forcing at $x=1.45\ \textrm {m}$ and $(c)$ forcing at $x=1.55\ \textrm {m}$ for a supply pressure of 4 bar and a forcing frequency of 20 Hz. In $(c)$ the upstream part of the test surface is blocked by the actuator set-up.

Despite this caveat, the measurements reported in this article were, for practical reasons, performed on the centreline of the test section. The underlying assumption is that the flow physics reported herein is locally two-dimensional, which can be substantiated by the fact that the separation line is only slightly curved between $z=-0.1$ and $z=0.1\ \textrm {m}$, even when forcing is strong. Nevertheless, the consequence of this hypothesis will be further discussed in § 6.

3. The unforced flow

In this section we provide salient features of the TSB in its uncontrolled state. The presentation is limited to the material that is relevant for a discussion of the controlled cases in the next sections. Further details can be obtained in Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016).

A contour plot of the average longitudinal velocity $\bar {U}(x,y)$ in the unforced TSB with representative velocity vectors is shown in figure 4. As expected, the flow detaches from the wall and reattaches further downstream, thereby creating a recirculation region with negative average longitudinal velocity. The degree of detachment may be quantified by streamwise positions referring to a specific threshold of the near-wall forward-flow fraction $\gamma$ (Simpson Reference Simpson1989). Incipient detachment (ID) is the position where the near-wall flow is in the positive direction exactly 99 % of the time ($\gamma =99\,\%$) and marks the start of the separation process. At transitory detachment (TD) and transitory reattachment (TR), the flow goes in the positive and negative directions with equal probability ($\gamma =50\,\%$). These two positions delimit the average recirculation zone of the TSB and correspond to the usual definitions of turbulent separation and reattachment with $c_f=\tau _w/\frac {1}{2} \rho U^2_{ref}=0$, where $\tau _w$ is the average wall shear stress and $\rho$ is the fluid density (Coleman, Rumsey & Spalart Reference Coleman, Rumsey and Spalart2018). Finally, complete reattachment (CR) marks the very end of the separation process with $\gamma =99\,\%$ again. In the present case, $x_{ID}\simeq 1.59\ \textrm {m}$, $x_{TD} \simeq 1.75\ \textrm {m}$, $x_{TD} \simeq 2.17\ \textrm {m}$ and $x_{CR} \simeq 2.30\ \textrm {m}$, where $x=0$ marks the entrance of the test section (figure 1). Based on these values, we define two characteristic lengths of the unforced TSB: $L_{50}=x_{TR}-x_{TD}=0.42\ \textrm {m}$ and $L_{99}=x_{CR}-x_{ID}=0.71\ \textrm {m}$. For clarity, in the remainder of the article we will use $L_b$ to denote the length $L_{50}$ of the unforced flow, as in Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016).

Figure 4. Averaged velocity field along the test-section centreline. The white solid line delimits $\bar {U}=0$.

Streamwise distributions of the average wall-pressure coefficient $c_p=(p-p_{ref})/\frac {1}{2} \rho U^2_{ref}$, the fluctuating wall-pressure coefficient $c_{p^{\prime }}=p_{rms} / \frac {1}{2} \rho U^2_{ref}$ and the near-wall forward-flow fraction $\gamma$, are shown in figure 5 (here, $p_{rms}$ is the root mean square of the pressure fluctuations). The slope of the $c_p$ distribution is very steep between $x=1.45\ \textrm {m}$ and $x = 1.65\ \textrm {m}$ and decreases afterwards when the shear layer detaches from the wall. The maximum of $c_p$ is reached at $x = 2.25\ \textrm {m}$, between TR and CR, and the wall pressure subsequently decreases because of the flow acceleration induced by the convergence of the tunnel floor. The fluctuating pressure coefficient $c_{p^{\prime }}$ shows a bi-modal character with a first local maximum close to ID and a second, global maximum close to CR. As demonstrated by LeFloc'h et al. (Reference LeFloc'h, Weiss, Mohammed-Taifour and Dufresne2020), the first maximum is caused by the superposition of two separate phenomena occurring at approximately the same streamwise position: first, the pressure signature of the low-frequency breathing motion of the TSB and second, the effect of the APG on the turbulent structures responsible for the pressure fluctuations in the attached boundary layer, which shifts the energy of the pressure fluctuations to lower frequencies. The second maximum of the $c_{p^{\prime }}$ distribution is associated with the convection of large structures within the shear layer and their subsequent shedding downstream of the TSB (Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016; LeFloc'h et al. Reference LeFloc'h, Weiss, Mohammed-Taifour and Dufresne2020).

Figure 5. Averaged and fluctuating pressure coefficients and forward-flow fraction along the test-section centreline.

The spectral content of the wall-pressure signatures at $x=1.60\ \textrm {m}$ (ID) and $x=2.30\ \textrm {m}$ (CR) is shown in figure 6. A medium-frequency activity is apparent as a broad peak in the p.s.d. at approximately 15–20 Hz in the reattachment region. This unsteadiness is linked to the convective roll-up and shedding of vortical structures in the separated shear layer that are responsible for the second peak of $c_{p^{\prime }}$ in figure 5. The structures are initiated between ID and TD and most likely grow downstream through a pairing-like process similar to that occurring in turbulent free shear layers. As they are convected downstream at a velocity of approximately $U_c = 0.3 U_{ref}$, the structures are then accelerated towards the wall in the FPG zone and produce a maximum wall-pressure signature at the edge of the separation bubble near CR (Weiss et al. Reference Weiss, Mohammed-Taifour and Schwaab2015; Mohammed-Taifour & Weiss Reference Mohammed-Taifour and Weiss2016). When scaled with the average length of the recirculation zone, the shedding frequency of the structures is of the order of $St_s=fL_b/U_{ref}=0.25\text {--}0.35$. Given the convective velocity of $U_c \simeq 0.3 U_{ref}$, this implies that there is on average only approximately one large structure within the streamwise length $L_b$ of the TSB.

Figure 6. The p.s.d. ($a$) and pre-multiplied p.s.d. ($b$) of wall-pressure fluctuations at the local and global maxima of $c_{p'}$.

The low-frequency unsteadiness of the TSB is most apparent at $x=1.60\ \textrm {m}$, where it emerges as a hump with a central frequency of 0.6 Hz in the pre-multiplied spectrum. This low-frequency pressure signature is also present in the reattachment zone, albeit at a lower amplitude: Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) showed that when computed in a narrow frequency range characteristic of the low-frequency unsteadiness, the filtered fluctuating wall-pressure coefficient shows two distinct peaks located at the streamwise positions of maximum APG and FPG (their figure 11). With the use of two thermal-tuft probes, and following a methodology similar to that of Eaton & Johnston (Reference Eaton and Johnston1982), Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) then showed that these pressure fluctuations are the signature of a low-frequency random cycle of contraction and expansion, dubbed breathing, of the TSB. In a subsequent experiment, Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016) demonstrated that this breathing motion is well illustrated by the first proper orthogonal decomposition mode of the velocity field (see also LeFloc'h et al. Reference LeFloc'h, Weiss, Mohammed-Taifour and Dufresne2020). Using the same scaling as above, the breathing frequency is of the order of $St_b=fL_b/U_{ref} \simeq 0.01$.

Both the thermal-tuft measurements of Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) and the PIV experiments of Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016) revealed that during the random cycles of low-frequency contraction and expansion, the instantaneous detachment moves over a distance approximately twice as large as the instantaneous reattachment. However, no attempt was made to investigate a possible time lag between the low-frequency motion in the detachment and reattachment zones. Mohammed-Taifour (Reference Mohammed-Taifour2017) subsequently performed a lead–lag analysis of the wall-pressure signal between $x_1=1.60\ \text {m} \simeq x_{ID}$ and $x_2=x_{CR}=2.30\ \textrm {m}$. The results are presented in figure 7. The magnitude-squared coherence function $C_{p'_1p'_2}$ is significantly larger than zero in the low-frequency part of the spectrum and the phase angle shows a linear dependency with the frequency, with a slope of $\theta _{p'_1p'_2}/2{\rm \pi} \simeq 0.044f$. This indicates that the two signals are highly correlated and that there is an average delay of 44 ms between the pressure signal at CR compared to the signal at ID (Bendat & Piersol Reference Bendat and Piersol2010). In other words, the instantaneous (low-frequency) reattachment moves after the instantaneous detachment. The time delay of 44 ms corresponds to the time that it takes for a fluid particle to follow an average streamline between the streamwise positions of ID and CR in the potential flow outside of the boundary layer. From the average velocity field shown in figure 4, the length of such a streamline starting at ($x=1.60\ \textrm {m}$, $y=0.04\ \textrm {m}$) and integrated until $x=2.30\ \textrm {m}$ is 0.80 m. The average potential-flow velocity between ID and CR can be estimated from an average pressure coefficient $\widetilde {c_p}$ between these two positions. From figure 5 we obtain $\widetilde {c_p} \simeq 0.42$, which implies a potential velocity of $U_{pot} \simeq U_{ref} \sqrt {1-\widetilde {c_p}}=19\ \textrm {m}\,\textrm {s}^{-1}$. Dividing the length of the streamline (0.80 m) by this velocity gives an estimated convection time of 42 ms, which is indeed very close to the time delay obtained from the lead–lag analysis. This convection time is much shorter than the characteristic period of the low-frequency breathing motion, which, in this flow, is of the order of 1 s.

Figure 7. Magnitude-squared coherence $C$ and phase angle $\theta$ of the low-frequency component of the wall-pressure fluctuations measured at $x_1=1.60\ \text {m} \simeq x_{ID}$ and $x_2=x_{CR}=2.30\ \textrm {m}$. The blue line is a linear fit on the phase angle between $0$ and $4\ \textrm {Hz}$.

Figure 8($a$) shows distributions of the correlation coefficient between the low-pass filtered wall-pressure fluctuations measured at two reference positions ($x=1.65\ \textrm {m}$ and $x=2.30\ \textrm {m}$) and other streamwise positions on the test-section centreline. These data were obtained by low-pass filtering the pressure signal at $f =10\ \textrm {Hz}$ to concentrate on the low-frequency breathing motion. As mentioned above, the correlation coefficient is high between the two reference points, but turns slightly negative when the moving probe is located between TD and TR. This indicates that when the wall pressure is increasing at the upstream and downstream edges of the TSB, it decreases in the middle, and vice versa. This suggests a simple model of the low-frequency breathing where the modulation of the TSB size is associated with a slow, quasi-steady contraction and expansion of the average wall-pressure distribution is a manner sketched in figure 8($b$). A contraction of the TSB induces a decrease of the wall pressure at the edges of the TSB and an increase within the recirculation zone. The variation of the wall pressure that occurs near ID during this quasi-steady motion is responsible for the low-frequency pressure signature observed in figure 6.

Figure 8. ($a$) Correlation coefficient between the low-pass filtered pressure fluctuations measured at reference positions $x=1.65\ \textrm {m}$ and $x=2.30\ \textrm {m}$ and a moving point on the test-section centreline. ($b$) Conceptual model of the contraction and expansion of the average pressure distribution.

In summary, the unforced TSB investigated in this work naturally contracts and expands at a frequency of the order of one Hertz. This low-frequency ‘breathing’ motion is most likely driven by an upstream mechanism that acts on the separation position first. The shift of the separation front triggers a change of the overall pressure distribution and leads to a motion of the reattachment position in the direction opposite of separation. This proposed mechanism will be further discussed in § 6 in light of the results obtained with periodic forcing.

4. Continuous forcing

We begin with a presentation of the results obtained while the flow was continuously forced at a specific frequency and amplitude. The pulsed-jet actuators were installed at two streamwise positions in the test section: first at the beginning of the pressure rise ($x=1.45\ \textrm {m}$) and second within the APG zone ($x=1.55\ \textrm {m}$). The resulting distributions of forward-flow fraction on the tunnel centreline, measured with the MEMS calorimetric sensor, are presented in figure 9. These data were obtained for a supply pressure $P_j=4\ \textrm {bar}$, which corresponds to a nominal velocity ratio of $\lambda _j=1.4$ and a momentum coefficient of $C_{\mu }=1.4\,\%$, with variations of ${\pm }0.1$ and ${\pm }0.06\,\%$ depending on the frequency, respectively. At $x=1.55\ \textrm {m}$, steady forcing at $f=0\ \textrm {Hz}$ was also tested. For this, the supply pressure was reduced manually so as to match the nominal momentum coefficient of $C_{\mu }=1.4\,\%$.

Figure 9. Forward-flow fraction $\gamma$ for the unforced and forced flows with $P_j=4 \ \textrm {bar}$ ($C_\mu =1.4\,\% \pm 0.06\,\%)$. Forcing position at $x=1.45\ \textrm {m}$ ($a$) and at $x=1.55\ \textrm {m}$ ($b$). Vertical dashed lines indicate the streamwise position of actuation.

The distributions of $\gamma$ show the familiar U shape indicative of an increased amount of backflow within the TSB. For all frequencies, the distributions are narrower with forcing compared to the unforced case. This shows that forcing reduces the size of the TSB by pushing the average separation point (TD) downstream and the average reattachment point (TR) upstream. Interestingly, forcing appears to move TD approximately twice as far as TR, and that irrespective of the forcing frequency. The most effective frequency which results in the smallest TSB (minimum $L_{50}$) is $F_j=100\ \textrm {Hz}$ at both streamwise positions, although the changes in $L_{50}$ above $F_j=40\ \textrm {Hz}$ are relatively small. When scaled with the average backflow length $L_b$ of the unforced TSB, $F_j=100\ \textrm {Hz}$ is equivalent to $F^+=F_j L_b /U_{ref}=1.7$, which is of the same order of magnitude as the most effective frequencies reported by Greenblatt & Wygnanski (Reference Greenblatt and Wygnanski2000) in typical AFC applications (see also table 2). Furthermore, the results demonstrate that periodic forcing is more effective than steady blowing ($f=0\ \textrm {Hz}$) in reducing the size of the TSB, which is consistent with virtually all known results on active separation control (Greenblatt & Wygnanski Reference Greenblatt and Wygnanski2000).

Representative distributions of the average wall-pressure coefficient $c_p$ for $C_\mu = 1.4\,\%$ and $1.8\,\%$ are presented in figure 10, again for the two forcing positions. The effect of forcing on $c_p$ is much smaller than on $\gamma$ and the pressure distributions are only slightly altered. Nevertheless, the variations of $c_p$ are larger than the measurement uncertainty of the pressure transducers (§ 2.3) and than the spanwise variations in the unforced flow (figure 1). The trend is similar for both forcing positions, with a decrease of $c_p$ at the edges of the TSB and an increase between TD and TR. This behaviour is consistent with a reduction of the TSB size through forcing since a smaller TSB is associated with an increased pressure recovery in the separated region. Both the forward-flow fraction and the average pressure distributions indicate that forcing at $x=1.55\ \textrm {m}$ is more effective than at $x=1.45\ \textrm {m}$, which is most likely correlated with the larger distance between the actuators and the unforced position of TD in the latter case. Thus, in the remainder of the article we will concentrate on forcing at $x=1.55\ \textrm {m}$.

Figure 10. Average wall-pressure coefficient $c_p$ for the unforced and forced flows at varying forcing frequencies and amplitudes. Forcing position at $x=1.45\ \textrm {m}$ ($a$) and at $x=1.55\ \textrm {m}$ ($b$). Vertical dashed lines indicate the streamwise position of actuation.

The effect of the forcing amplitude is shown in figure 11 at the most effective frequency $F_j=100\ \textrm {Hz}$. Clearly, increasing $C_{\mu }$ decreases the size of the TSB with, again, more effect on separation than reattachment. At the maximum amplitude of $C_{\mu }=1.8\,\%$ ($P_j=6\ \textrm {bar}$), the average recirculation region is almost completely eliminated on the test-section centreline, with a minimum forward-flow fraction just below 50 %. It also appears that while the effect on TD and TR is relatively strong, the displacement of ID and CR is more limited. Thus, the characteristic length $L_{50}=x_{TR}-x_{TD}$ decreases more than $L_{99}=x_{CR}-x_{ID}$ when forcing is applied. These results are consistent with those obtained at other frequencies and amplitudes, which are summarized in figure 12. We conclude that the TSB length can be artificially reduced by periodic forcing in the boundary layer upstream of the average separation point. Increasing the amplitude of forcing has the strongest effect on the positions of TD and, to a lesser extent, on TR, while the positions of ID and CR are relatively unaffected. Finally, figure 12($a$) clearly demonstrates that the differences in TSB length above $F_j=40\ \textrm {Hz}$ are minimal.

Figure 11. Forward-flow fraction $\gamma$ for the unforced and forced flows with $F_j=100\ \textrm {Hz}$ and varying jet amplitudes $C_\mu =1.24\,\%$, $1.43\,\%$, $1.75\,\%$ and $1.79\,\%$. Forcing position at $x=1.55\ \textrm {m}$ (vertical dashed line).

Figure 12. ($a$) Positions of ID/CR ($\gamma =99\,\%$) and TD/TR ($\gamma =50\,\%$) for unforced (dashed lines) and forced (solid lines) flows with $C_\mu =1.4\,\% \pm 0.06\,\%$. ($b$) Characteristic lengths $L_{50}$ and $L_{99}$ as function of $C_\mu$ and $F_j$. Forcing position at $x=1.55\ \textrm {m}$.

Contour plots of the average vorticity and the turbulence statistics measured by PIV at $C_{\mu } \simeq 1.4\,\%$ and varying forcing frequencies are presented in figure 13. These data were obtained by averaging over a total measuring time of 27 s, which led to the uncertainty brackets quoted in § 2.3. In all plots the white line delimits the region of mean backflow, which is shown to decrease in size with increasing frequency. At $F_j=100\ \textrm {Hz}$, both the length and height of the recirculation region are significantly reduced compared to the unforced case. In accordance with the MEMS calorimetric sensor data of figure 9($b$), the left intersection of the $U=0$ isoline with the $x$ axis (which corresponds to the TD point), moves more than the right intersection (TR). This shows again that separation is more displaced by periodic forcing than reattachment.

Figure 13. Contour plots of $-\overline {\omega _z}$ in s$^{-1}$, $\overline {u'u'}/U_{ref}^2$, $\overline {v'v'}/U_{ref}^2$ and $-\overline {u'v'}/U_{ref}^2$ (from (a,e,i,m,q) to (d,h,l,p,t), respectively). Unforced (a,b,c,d) and forced flow with $C_\mu =1.4\,\% \pm 0.06\,\%$ and $F_j=0$, $15$, $40$ and $100\ \textrm {Hz}$ (from (e,f,g,h) to (q,r,s,t), respectively). Reynolds stresses are multiplied by $10^3$. The white line depicts the iso-line $U=0$. Forcing position at $x=1.55\ \textrm {m}$.

Crucially, the amplitude of all turbulence statistics appears to decrease while the TSB size is reduced by increasing the forcing frequency. This is contrary to most existing results on periodic forcing in flows where the separation point is fixed by a geometric singularity or constrained by a sufficiently strong curvature, and where earlier reattachment is achieved by enhancing the turbulent kinetic energy in the shear layer. In such flows the most effective frequency is close to the unforced shedding frequency of the separated shear layer (Chun & Sung Reference Chun and Sung1996; Kiya et al. Reference Kiya, Shimizu and Mochizuki1997; Brunn & Nitsche Reference Brunn and Nitsche2003). Excitation near the shedding frequency enhances vortex merging and turbulent fluctuations in the shear layer. This increases entrainment and intensifies the momentum transfer with the outer flow, thus resulting in earlier reattachment (Bhattacherjee et al. Reference Bhattacherjee, Troutt and Scheelke1986; Brunn & Nitsche Reference Brunn and Nitsche2006; Dandois et al. Reference Dandois, Garnier and Sagaut2007). In contrast, the effect of forcing appears to be different in the present pressure-induced TSB where the separation point is free to move on the test surface. Here, periodic forcing pushes separation downstream and reattachment upstream, while the most effective frequency is a multiple of the natural shedding frequency. This creates a smaller TSB and, consequently, reduces the turbulent fluctuations in the shear layer in a manner similar to what is observed in pressure-induced TSBs of different sizes (LeFloc'h et al. Reference LeFloc'h, Weiss, Mohammed-Taifour and Dufresne2020). The streamwise stresses appear to be reduced the most, with a decrease of approximately $50\,\%$ between the unforced case and $F_j=100\ \textrm {Hz}$. Instead of showing a strong peak in the upstream half of the unforced TSB, the streamwise variation of $\rho \overline {u'u'}$ is essentially flat when forcing is applied, even at $F_j=0\ \textrm {Hz}$. The wall-normal and shear stresses $\rho \overline {v'v'}$ and $-\rho \overline {u'v'}$ decrease only by approximately $20\,\%$ and keep their maxima at the same streamwise position in the downstream part of the TSB. This shows again that the shear layer is strongly affected near detachment but less so near reattachment. In fact, although the size of the average backflow region is strongly diminished with forcing, the downstream part of the shear layer appears to behave in a similar way with or without forcing.

The contour plots of average vorticity (figure 13a,e,i,m,q) confirm these findings by showing that periodic forcing redistributes vorticity close to the test surface in the detachment region. The separating shear layer, which is characterized by a region of increased (negative) vorticity in the wall-normal direction upstream of TD, is wider when forcing is applied because the distance between ID and TD increases (see figure also 12$a$). This spreading decreases the average shear and consequently reduces the streamwise stresses. In contrast, the vertical extent of the shear layer remains relatively constant in the downstream half of the TSB ($x>2\ \textrm {m}$), which limits the decrease in wall-normal and shear stresses. Furthermore, results from a triple decomposition of the velocity fields (not shown here), where the contribution of the coherent components at the forcing frequency and its harmonics are removed from the turbulent quantities (Hussain & Reynolds Reference Hussain and Reynolds1970), did not show any significant difference with the Reynolds decomposition of figure 13. This is a further indication that forcing with PJAs does not have a strong effect on the development of the shear layer but mostly affects the position of separation.

The p.s.d.s of wall-pressure fluctuations measured at $x=2.35\ \textrm {m}$ (near CR) are shown in figure 14. The shape of the p.s.d.s is not affected by forcing and the ‘natural’ shedding signature at 15–20 Hz can be observed irrespective of the forcing frequency. The narrow peak at the forcing frequency for $F_j=15\ \textrm {Hz}$ shows that large-scale vortices are slightly more coherent in this case but there does not appear to be a strong lock-on effect when forcing is applied close to the shedding frequency, as seen for example by Bhattacherjee et al. (Reference Bhattacherjee, Troutt and Scheelke1986) behind a backward-facing step or by Kotapati et al. (Reference Kotapati, Mittal, Marxen, Ham, You and Cattafesta2010) in a laminar separation bubble. For frequencies above 5 Hz, the amplitude of the forced p.s.d.s is moderately reduced compared to the unforced case, in accordance with the reduction in turbulent stresses. This confirms that forcing only slightly affects the dynamics of the shear layer wrapping the backflow region.

Figure 14. The p.s.d.s of wall-pressure fluctuations for the natural and forced flows, $x=2.35\ \textrm {m}$: $(a)$ standard representation and $(b)$ pre-multiplied representation.

Phase-averaged velocity fields obtained by PIV were analysed in an attempt to better understand the mechanism sustaining the reduction in TSB size. Figure 15 shows that periodic forcing generates a vortex system that is convected in the downstream direction close to the test surface. This vortex system is made visible by contours of the $\varGamma _2$ criterion introduced by Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001) and computed here after subtracting the mean velocity $\bar {U}(x,y)$ from the phase-averaged fields. In addition, the position of the vortices and their direction of rotation is highlighted by a dashed line that was manually placed based on the observation of the local velocity vectors. For $F_j<200\ \textrm {Hz}$ the signature of the vortex system in the two-dimensional PIV plane is a pair of contra-rotating vortices aligned in the streamwise direction, with a positive vortex (clockwise) upstream of a negative one. The combination of positive and negative vorticity generates a channel of positive vertical velocity that moves fluid away from the test surface at each forcing cycle. This is reminiscent of the ‘moving pump’ concept of Darabi & Wygnanski (Reference Darabi and Wygnanski2004a), although in their case the forcing through a loudspeaker acted on the shear layer by promoting the formation of a series of negative-vorticity structures. The present mechanism is different inasmuch as the contra-rotating vortices appear as the key elements that remove low-momentum fluid away from the wall.

Figure 15. Illustration of vortical flow dynamics in the first PIV station with contour maps of $\varGamma _2 \leq -2/{\rm \pi}$ (counter-clockwise) and $\varGamma _2 \geq 2/{\rm \pi}$ (clockwise). Panels (a) to (d) show $F_j=15\ \textrm {Hz}$, $F_j=40\ \textrm {Hz}$, $F_j=100\ \textrm {Hz}$ and $F_j=200\ \textrm {Hz}$, respectively, with $C_\mu =1.4\,\%$ for all frequencies; $\varGamma _2$ was computed after subtracting the mean velocity field from the phase-averaged field.

A similar mechanism has already been described by Hecklau et al. (Reference Hecklau, Salazar and Nitsche2013), who investigated the active control of flow separation in a one-sided diffusor. They interpreted the contra-rotating vortices observed in the PIV plane as the two-dimensional signature of the three-dimensional starting vortex generated by a PJA slot. The starting vortex is created by the rapid ejection of fluid through the inclined slot in the test surface and its size depends, among other parameters, on the amount of ejected fluid (Steinfurth & Weiss Reference Steinfurth and Weiss2020). Because the duty cycle was kept constant in the present experiment, a higher forcing frequency results in a smaller amount of fluid ejected through the slots and, consequently, a smaller vortex diameter. While there is only one pair of relatively large vortices visible in the two-dimensional measurement station for $F_j=15\ \textrm {Hz}$ and $F_j=40\ \textrm {Hz}$, there are two pairs at $F_j=100\ \textrm {Hz}$ and, presumably, four at $F_j=200\ \textrm {Hz}$. At the latter frequency, the vortex system does not appear so clearly anymore because of the small amount of fluid ejected and the short time period between the pulses. It is also apparent that the high-frequency vortices tend to dissipate faster in the streamwise direction than the lower frequency, larger structures. This reduces the amount of fluid that is pumped away from the wall and may explain why forcing at $F_j=200\ \textrm {Hz}$ is less effective than at $F_j=100\ \textrm {Hz}$.

The effect of the vortex system on the separation front is illustrated in figures 16 and 17 for $F_j=15\ \textrm {Hz}$ and $F_j=100\ \textrm {Hz}$, respectively. Each figure illustrates one full cycle of forcing by showing snapshots extracted from supplementary movies 1 and 2 available at https://doi.org/10.1017/jfm.2021.77. In both figures the white line delimits the region of phase-averaged backflow. Here also, the position of the vortices and their direction of rotation is highlighted by dashed lines based on the direction of the phase-averaged velocity vectors.

Figure 16. Illustration of the flow dynamics for $F_j=15\ \textrm {Hz}$ and $C_\mu =1.4\,\%$ in PIV station 2 with contours of phase-averaged streamwise velocity (a,c,e,g) and $\varGamma _2$ (b,d,f,h). Time delay of 15.3 ms (approximately one quarter of a period) between the snapshots from (a,b) to (g,h). See also supplementary movie 1.

Figure 17. Illustration of the flow dynamics for $F_j=100\ \textrm {Hz}$ and $C_\mu =1.4\,\%$ in PIV station 3 with contours of phase-averaged streamwise velocity (a,c,e,g) and $\varGamma _2$ (b,d,f,h). Time delay of 2.5 ms (one quarter of a period) between the snapshots from (a,b) to (g,h). See also supplementary movie 2.

At $F_j=15\ \textrm {Hz}$ (figure 16) the sequence starts with a relatively large region of instantaneous backflow and a counter-clockwise vortex spanning almost the entire field of view. At the next time step the clockwise vortex enters the field and the instantaneous separation front is pushed downstream. Between the second and third time steps both the clockwise vortex and the separation front continue their downstream sweep. This results in a much smaller region of phase-averaged backflow compared to the start of the sequence. Finally, in the last time step the counter-clockwise vortex from the next period enters the field of view and the separated zone grows back to its original size. Supplementary movie 1 clearly shows that the process is highly unsteady and that the instantaneous recirculation zone grows and recedes at the period of forcing. Similar plots obtained further downstream in the reattachment region (not shown here) demonstrate a very similar phenomenon where the instantaneous reattachment point moves back and forth at $15\ \textrm {Hz}$. When averaged over time, this motion results in a smaller TSB compared to the unforced case, as can be seen in figures 9 and 13.

The phase-averaged fields obtained at $F_j=100\ \textrm {Hz}$ (figure 17) illustrate a strikingly different situation. At this frequency, one period of forcing corresponds to the translation of two pairs of relatively small contra-rotating vortices within the field of view. The first (upstream) pair of vortices is visible on the contours of $\varGamma _2$ for the first three snapshots only. In the last snapshot, the threshold of $\left \lvert \varGamma _2\right \rvert >2/{\rm \pi}$ is not attained but a careful inspection of the velocity vectors reveals the likely presence of the two contra-rotating vortices at their expected positions. As for the second (downsteam) structure, only its counter-clockwise half is observed, presumably because the second half of the pair has shifted out of the field of view. Furthermore, this vortex appears to move slower than its upstream counterpart because the local velocity is smaller. Nevertheless, the mechanism controlling the separation front appears to be similar to the 15 Hz case, i.e. the passage of a vortex structure is associated with a downstream shift of the separation region. In the 100 Hz case, however, the persistent arrival of new vortices essentially maintains the phase-averaged separation front at one constant streamwise position. As can be observed in supplementary movie 2, this results in an almost steady recirculation region in a phase-averaged sense. The results obtained at $F_j=40\ \textrm {Hz}$ (not shown here) are essentially similar to the 100 Hz case inasmuch as only a very small back-and-forth motion of the separation front at 40 Hz can be detected. We conclude that periodic forcing is more effective when the forcing frequency is significantly higher than an inherent characteristic frequency of the separation front. When this occurs, the front does not have time to move back towards its original position between two incoming structures and the TSB appears steady when phase averaged. The similarity of the results obtained at $F_j=40\ \textrm {Hz}$ and $F_j=100\ \textrm {Hz}$ compared to the $F_j=15\ \textrm {Hz}$ case indicates that such a situation already occurs at $F_j=40\ \textrm {Hz}$, which explains why there is only a limited effect of forcing above the latter frequency. Interestingly, these results bear some resemblance with those of Amitay & Glezer (Reference Amitay and Glezer2002), who used synthetic jets to control the flow separation near the leading edge of an airfoil at both low and high forcing frequencies. Forcing at $F^+=O(1)$ generated strong vortices that led to unsteady reattachment while forcing at $F^+=O(10)$ led to an essentially steady reattachment process.

In summary, the results presented in this section indicate that the size of the average TSB can be artificially reduced by periodic forcing in the upstream boundary layer. In contrast to flow cases where the separation point is fixed by the surface geometry, forcing with PJAs pushes the mean separation (strictly speaking the TD point) downstream while the mean reattachment (TR) moves upstream, thus leading to a smaller TSB and a reduction of turbulent stresses in the shear layer. Increasing the forcing amplitude decreases the size of the average TSB while the most effective range of frequencies, which leads to the smallest TSB, is of the order of 40–100 Hz Nevertheless, one may note that the use of other types of actuators may have led to a different response of the shear layer. The present results also suggest the existence of a characteristic time scale during which the TSB tries to return to its original state between two forcing inputs. Forcing at frequencies significantly higher than the inverse of this time scale (i.e. 40–100 Hz) is more effective in reducing the size of the TSB than forcing at lower frequencies. Furthermore, the phased-averaged fields offer an interesting insight into the physics of flow separation control with PJAs. In accordance with the mechanism suggested by Hecklau et al. (Reference Hecklau, Salazar and Nitsche2013), our results indicate that PJAs generate strong starting vortices that, when convected within an APG, are associated with a downstream shift of the separation front.

5. Transient forcing

The results presented above suggest that the separation front responds to upstream disturbances with a characteristic time scale. To investigate this aspect in more detail, the transient response of the TSB subjected to an abrupt starting (Off/On) or stopping (On/Off) of the periodic forcing is explored in this section.

The MEMS calorimetric sensor was used to measure the temporal evolution of the forward-flow fraction $\gamma (t)$ when forcing was started or stopped. The procedure is illustrated in figure 18 for a stopping sequence at a forcing frequency $F_j=100\ \textrm {Hz}$. The forcing signal ($a$) is modulated with a low-frequency square-wave function at 0.5 Hz. In the first half of the square-wave period ($t<1\ \textrm {s}$), the forcing signal at $F_j=100\ \textrm {Hz}$ is on but stops abruptly at $t=1\ \textrm {s}$. The output of the calorimetric sensor ($b$) indicates that the near-wall flow direction is mostly positive for $t<1\ \textrm {s}$ but oscillates more strongly between positive and negative values when the forcing is stopped ($t>1\ \textrm {s}$). This indicates a shift of the separation front on the test surface when forcing is turned off. By ensemble averaging this signal over a large number of square-wave periods, the forward-flow fraction is obtained as a function of time ($c$) and the variation of the TSB size when the forcing is stopped can be observed. The ensemble-averaged data are further smoothed by a moving average filter of 0.05 s in width. The same procedure is used for starting sequences (Off/On) by considering the square-wave period starting at $t=1\ \textrm {s}$.

Figure 18. Illustration of the methodology used for transient forcing: ($a$) forcing signal; ($b$) output of calorimetric sensor at $x=1.75\ \textrm {m}$; ($c$) forward-flow fraction computed by ensemble averaging the calorimetric sensor output over 90 square-wave periods.

The square-wave period of 2 s was chosen as a compromise that is sufficiently long to observe a complete return of the TSB to its unforced state when the forcing is started or stopped while also being sufficiently short to enable the acquisition of a large number of periods in a manageable testing time. All data reported herein were obtained by recording the calorimetric sensor signal during 180 s, which resulted in 90 square-wave periods used for ensemble averaging. The forward-flow fraction calculated at each streamwise position between $t=0.5\ \textrm {s}$ and $t=1.0\ \textrm {s}$ (forcing on) and between $t=1.5\ \textrm {s}$ and $t=2.0\ \textrm {s}$ (forcing off) is compared in figure 19 to the distributions obtained in the unforced and continuously forced cases, respectively. The values of $\gamma$ measured with the square-wave modulated forcing signal are very close to the reference distributions, which demonstrates the validity of the methodology.

Figure 19. Comparison of forward-flow fraction distribution between unforced flow, continuous forcing, and transient forcing ($F_j=100\ \textrm {Hz}$). Dashed line indicates position of forcing.

The transient forward-flow fraction $\gamma (t)$ obtained with this procedure at $x=1.75\ \textrm {m}$ (unforced TD position) is shown in figure 20 for forcing frequencies of 15 Hz ($a$), 40 Hz ($b$) and 100 Hz ($c$), respectively. For $F_j=15\ \textrm {Hz}$, the back-and-forth motion of the separation front, already observed in figure 16, is obvious when forcing is turned on since $\gamma (t)$ shows an oscillatory behaviour at 15 Hz. For higher forcing frequencies, these oscillations are not observed and $\gamma (t)$ reaches a constant value consistent with continuous forcing after approximately 300 ms. For all forcing frequencies, the curves of $\gamma (t)$ show an exponential increase or decrease after $t = 1\ \textrm {s}$ that is reminiscent of the step response of a first-order linear system. To determine the corresponding time constants, $\gamma (t)$ was least-squares fitted to the following simple laws for $t>1\ \textrm {s}$:

(5.1)\begin{equation} \left. \begin{gathered} \gamma_{Off\text{--}On}(t) = \gamma_{Off} + (\gamma_{On} - \gamma_{Off}) \textrm{e}^{{-}t/\tau_{Off\text{--}On}}, \\ \gamma_{On\text{--}Off}(t) = \gamma_{On} + (\gamma_{Off} - \gamma_{On}) \textrm{e}^{{-}t/\tau_{On\text{--}Off}}, \end{gathered} \right\} \end{equation}

where $\gamma _{Off}$ and $\gamma _{On}$ are the values of the forward-flow fraction in the unforced and continuously forced cases, respectively, and where $\tau _{Off\text {--}On}$ and $\tau _{On\text {--}Off}$ are the time constants obtained when forcing is started or stopped. The time constants are summarized in table 3, where it can be observed that $\tau _{Off\text {--}On}$ is almost constant for all forcing frequencies, whereas $\tau _{On\text {--}Off}$ is larger and increases with $F_j$. This indicates that the TSB takes more time to return to its unforced state when forcing is stopped than to react to a start of upstream forcing. Finally, figure 20($d$) shows the transient forward-flow fraction obtained with a forcing frequency of $F_j=100\ \textrm {Hz}$ and forcing amplitudes varying between $C_{\mu }=0.57\,\%$ and $C_{\mu }=1.79\,\%$. In order to highlight the potential effect of the forcing amplitude on the TSB time scale, the curves of $\gamma (t)$ were scaled to allow their superposition on the same graph. This representation clearly shows that the values of $\tau _{Off\text {--}On}$ and $\tau _{On\text {--}Off}$, and hence the transient behaviour of the TSB, do not depend on forcing amplitude.

Figure 20. Transient forward-flow fraction measured at ($a$) $x=1.75\ \textrm {m}$ for $F_j=15\ \textrm {Hz}$, ($b$) $F_j=40\ \textrm {Hz}$, ($c$) $F_j=100\ \textrm {Hz}$ (the blue and green curves are fitted to (5.1)) and ($d$) variation of forcing amplitude at $F_j=100\ \textrm {Hz}$.

Table 3. Time constants of ensemble-averaged $\gamma (t)$ at $x=1.75\ \textrm {m}$ according to (5.1).

Darabi & Wygnanski (Reference Darabi and Wygnanski2004a) experimentally controlled the forced reattachment of the flow over a trailing edge flap by using a loudspeaker at the hinge line. Contrary to the present results, the time required for the flow to attach after forcing was started depended on the forcing frequency and amplitude. The minimum reattachment time obtained for a flap deflected 6 degrees over its critical angle was found to be $tU_{\infty }/L_f = 16$, where $U_{\infty }$ was the free-stream velocity and $L_f$ the flap length. Using $L_f=L_b=0.42\ \textrm {m}$ and $U_{\infty }=U_{ref}=25\ \textrm {m}\,\textrm {s}^{-1}$, this corresponds to a time $t=269\ \textrm {ms}$ in our case. As can be seen in figure 20, this value is close to the time where the $\gamma (t)$ curves reach their asymptotes (approximately 300 ms). Darabi & Wygnanski (Reference Darabi and Wygnanski2004b) then investigated the transient separation process on the flap by abruptly stopping the loudspeaker forcing and found that the time required for separation to completely reappear was approximately 30 % larger than the time required to force the flow to attach. We conclude that the values of both time constants $\tau _{Off\text {--}On}$ and $\tau _{On\text {--}Off}$ are consistent with those of Darabi & Wygnanski (Reference Darabi and Wygnanski2004a,Reference Darabi and Wygnanskib) despite significant differences in flow configuration and forcing method. It should be noted, however, that the separation line in Darabi & Wygnanski's (Reference Darabi and Wygnanski2004a; Reference Darabi and Wygnanski2004b) experiment was essentially fixed at the flap corner.

Time traces of the transient forward-flow fraction $\gamma (t)$ were also measured at multiple streamwise positions in order to investigate the time required between the start or stop of upstream forcing and the first indication of the TSB reaction along its length. As illustrated in figure 21($a$), a time delay $\delta t$ was defined between $t=1\ \textrm {s}$, which corresponds to the start or stop of forcing, and the instant when the $\gamma (t)$ curve reacts to forcing. The values of $\delta t$ were found to be almost identical for starting or stopping sequences and are plotted in figure 21($b$) as a function of the streamwise position. The streamwise distribution of $\delta t$ is essentially linear and hints at a convection velocity of $0.27U_{ref}$, which is very close to the convection velocity $U_c \simeq 0.30U_{ref}$ of both the naturally occurring and the PJA vortices in the shear layer. This indicates that at each streamwise position, the unforced TSB starts its deformation with the passage of the first vortex system emanating from the PJAs. Thereafter, several vortices are required for the TSB to reach its asymptotic state. As indicated in figure 20, approximately 300 ms are required for all transient effects to have disappeared when forcing is started. This time delay corresponds to the passage of 5, 12 and 30 vortex pairs for $F_j$ = 15, 40 and 100 Hz, respectively. Similarly, the forced TSB starts to move back to its unforced position after the passage of the last vortex pair from the PJA when forcing is stopped but requires more time to complete its motion.

Figure 21. ($a$) Illustration of the definition of the time delay $\delta t$ between $t=1\ \textrm {s}$ and the vertical dashed-dotted line at $x=1.85\ \textrm {m}$ and $x=2.05\ \textrm {m}$, respectively; ($b$) variation of $\delta t$ as a function of the streamwise position.

The transient response of the TSB was also investigated with ensemble-averaged PIV data at a forcing frequency of $F_j=100\ \textrm {Hz}$ and an amplitude of $C_{\mu }=1.43\,\%$. For this, 30 PIV sequences of 1 s each were recorded at the most upstream PIV station. The camera recordings were synchronised to start exactly when the forcing started (Off/On sequence) or stopped (On/Off sequence). Contours of $\varGamma _2$ were calculated after subtracting the mean velocity $\bar {U}(x,y)$ of the corresponding continuous forcing case (i.e. either On or Off) from the ensemble-averaged fields.

Figure 22 illustrates a few snapshots extracted from supplementary movie 3, where the TSB reacts to a start of forcing at $t_0$. In the first snapshot, the separation front is in its unforced position. While a few spots of $\left \lvert \varGamma _2\right \rvert >2/{\rm \pi}$ are visible in the field of view, they do not appear to be the signatures of the vortex structures observed in figures 15 and 17. Rather, they are interpreted as residual noise from the phase-averaging procedure. In the next snapshots, the counter-rotating vortices emanating from the PJAs at $F_j=100\ \textrm {Hz}$ can be seen in the two-dimensional field of view near $x=1.65\ \textrm {m}$ and $x=1.75\ \textrm {m}$. The position of the vortices is constant in this representation since the time between snapshots was chosen to be exactly six forcing periods. Between $t_0+60\ \textrm {ms}$ and $t_0+180\ \textrm {ms}$, only the counter-clockwise vortex is seen in the most downstream position ($x=1.75\ \textrm {m}$), again presumably because the other half of the pair is located outside of the field of view like in figure 17. In the last snapshot, however, the complete pair is observed again because the separation front is now downstream of the field of view, which results in a smaller average vertical velocity and a situation similar to that observed in figure 15. The translation of the vortices through the field of view is obvious in supplementary movie 3: each arrival of a new vortex pair slightly pushes the separation front downstream and, after 240 ms, or 24 vortex passages, the region of phase-averaged backflow is shifted downstream of the field of view. These PIV results are consistent with figure 20($c$) since a downstream shift of the separation front implies an increase of forward-flow fraction at $x=1.75\ \textrm {m}$. Indeed, figures 19 and 20($c$) indicate that $\gamma (t)$ still increases slightly after 240 ms and that the TD point finally settles near $x \simeq 1.90\ \textrm {m}$ after the disappearance of transient effects at $t \simeq 300\ \textrm {ms}$.

Figure 22. Illustration of the flow dynamics in PIV station 1 for a start of forcing at $F_j=100\ \textrm {Hz}$ and $C_{\mu }=1.4\,\%$ with contours of phase-averaged streamwise velocity (a,c,e,g,i) and $\varGamma _2$ (b,d,f,h,j). Forcing is started at $t_0$. See also supplementary movie 3.

The stopping of upstream forcing is illustrated in figure 23 and supplementary movie 4 for the same parameters as above. The first snapshot at $t_0$ is taken exactly when forcing is stopped. At this instant the separation front (i.e. the TD point) is located downstream of the field of view and the contour of $\varGamma _2$ shows the typical pair of contra-rotating vortices observed when the flow is continuously forced at $F_j=100\ \textrm {Hz}$ (see figures 17 and 22). In the next snapshots these vortices disappear from the field of view since forcing is stopped. Rather, the ensemble-averaging procedure produces a random contour akin to noise, which would probably have been completely eliminated with a larger number of averaging periods. Crucially, the time between snapshots is much longer in figure 23 than in figure 22, which shows that the separation front takes approximately twice as much time to come back to its original position when forcing is stopped than to move to its controlled position when forcing is started. This is consistent with the time traces of $\gamma (t)$ obtained with the calorimetric sensor (figure 20) and with the values of the corresponding time constants summarized in table 3.

Figure 23. Illustration of the flow dynamics in PIV station 1 for a stop of forcing at $F_j=100\ \textrm {Hz}$ and $C_{\mu }=1.4\,\%$ with contours of phase-averaged streamwise velocity (a,c,e,g,i) and $\varGamma _2$ (b,d,f,h,j). Forcing is stopped at $t_0$. See also supplementary movie 4.

In summary, the results presented in this section demonstrate that the TSB responds to upstream forcing with a characteristic time scale that is much larger than the forcing period of the PJAs. When forcing is started the TSB takes approximately 300 ms to reach its continuously forced state but, depending on the forcing frequency, it can take twice as long to return to its unforced state. The time traces of the transient forward-flow fraction $\gamma (t)$ further suggest that the TSB responds to upstream perturbations like a first-order linear system with a characteristic time constant of the order of 0.1–0.2 s.