2.1 Wind tunnel and experimental configuration

The experiments were conducted in a subsonic open-return wind tunnel located in the Hessert Laboratory at the University of Notre Dame. The wind tunnel has an inlet contraction ratio of 20 : 1 and contains a series of 12 turbulence management screens resulting in free-stream turbulence levels of less than 0.1 % for frequencies between 10 Hz and 10 kHz. Air is drawn through the inlet by an eight-bladed fan connected to an 18.6 kW AC induction motor. The test section is $0.610~\text{m}\times 0.610~\text{m}$ ( $2~\text{ft}\times 2~\text{ft}$ ) in cross-section and 1.83 m (6 ft) in length. One test section sidewall and the ceiling have large plate glass inserts in order to provide optical access to the flow field. The temperature is maintained constant during wind tunnel operation.

A schematic of the full model configuration is shown in figure 1(a). A fifth-order polynomial shaped contour, constructed from sheet metal, smoothly brings the flow from the floor of the wind tunnel inlet up to the level of a flat boundary layer development plate. At the termination of the contour, the boundary layer is tripped with a 50.8 mm (2 in.) wide strip of distributed sand grain roughness with an average roughness element size of $46~\unicode[STIX]{x03BC}\text{m}$ . The boundary layer development plate is 0.914 m (36 in.) long and spans the entire test section of 0.610 m (24 in.).

The boundary layer development plate is followed by a convex ramp of constant radius of curvature, $R=0.510$  m (20.08 in.). As shown in figure 1(b), the convex ramp geometry consists of multiple $5^{\circ }$ inserts, constructed using an epoxy mould, which smoothly connect together to create a circular arc that can be controlled in $5^{\circ }$ increments based on the number of inserts installed. At each ramp insert junction, great care was taken to ensure an aerodynamically smooth transition between the inserts. The results presented in this paper are for a fixed configuration with $\unicode[STIX]{x1D703}=25^{\circ }$ (as defined in figure 1 b). Following the curved section of the ramp, a 0.152 m (6 in.) flat section extends the ramp section to the tunnel floor. As shown in figure 1(b), $x$ denotes the streamwise spatial coordinate along the convex ramp surface while the total length of the ramp, consisting of circular arc and straight sections, was $L=0.375$  m. Measurements were performed over the region $0\leqslant x/L\leqslant 1$ . A fairing, constructed from Teflon sheeting with matched derivative end conditions, smoothly transitions the end of the ramp back to the floor of the wind tunnel test section, as shown schematically in figure 1(a). No measurements were performed on this fairing section.

A NACA 0015 airfoil model positioned in a nose-down post-stall configuration and equipped with leading edge DBD plasma flow control was located above the ramp section. Plasma flow control on the airfoil is not a focus of this experimental study but was used simply as a means to impose an unsteady APG on the ramp turbulent boundary layer below (without the use of any moving parts) while maintaining a constant approach flow. Those readers unfamiliar with DBD plasma actuators are referred to comprehensive reviews on the subject provided by Moreau (Reference Moreau2007), Corke, Post & Orlov (Reference Corke, Post and Orlov2009) and Corke, Enloe & Wilkinson (Reference Corke, Enloe and Wilkinson2010).

Figure 1.

(a) Schematic of the experimental apparatus. (b) Schematic of the convex ramp geometry and streamwise coordinate.

With the plasma off and the airfoil flow fully separated, the APG turbulent boundary layer on the convex ramp remains attached but near a state of incipient detachment. With airfoil leading edge plasma flow control on, the flow attaches to the airfoil suction surface. This has the effect of imposing a rapid increase in magnitude of the APG over the convex ramp section below, thereby precipitating turbulent boundary layer separation near the end of the ramp. Figure 2 compares visualization images of flow over the airfoil in the plasma-off and plasma-on cases. The influence of the leading edge plasma in reattaching the airfoil flow is apparent. Figure 3 presents corresponding images of the external flow over the convex ramp. The plasma-on case, which corresponds to a more severe APG, gives rise to flow separation on the aft section of the ramp. By temporally oscillating the leading edge airfoil plasma flow control, the ramp boundary layer is caused to undergo an unsteady separation and reattachment cycle. Since the region of oscillatory pressure gradient is spatially localized to a specific region of the ramp, the initial flow conditions at the beginning of the ramp are maintained constant in the experiment.

Figure 2.

Streakline visualization of the flow over the airfoil for (a) plasma-off (separated flow) and (b) plasma-on (attached flow) cases: chord Reynolds number $Re_{C}=1.2\times 10^{5}$ ( $U_{\infty }=10~\text{m}~\text{s}^{-1}$ ).

Figure 3.

Streakline visualization of the ramp flow for (a) plasma-off (ramp attached) and (b) plasma-on (ramp separated) cases: $U_{\infty }=10~\text{m}~\text{s}^{-1}$ .

The airfoil section was constructed from an epoxy mould with a 0.203 m (8 in.) chord. The airfoil had a span of 0.508 m (20 in.) and terminated in the spanwise direction in thin Plexiglas end plates that served to ensure a two-dimensional airfoil flow. The airfoil was mounted from the test section ceiling, and its mid-chord position ( $x_{a}$ , $y_{a}$ ) relative to the beginning of the ramp section was adjustable in both the horizontal (i.e. streamwise) and vertical directions. The airfoil angle of attack relative to the approach flow was also adjustable. For all of the results presented in this paper, the airfoil mid-chord was fixed at $x_{a}=216$  mm (8.5 in.) downstream of the convex ramp leading edge and $y_{a}=152$  mm (5.98 in.) above it. The airfoil geometric angle of attack with respect to the oncoming flow was fixed at $\unicode[STIX]{x1D6FC}=-12^{\circ }$ (nose-down configuration) with maximum blockage of less than 7 %, which posed no detrimental effects to the experimental arrangement.

2.2 Initial conditions for the ramp flow field

Measurements of the ZPG turbulent boundary layer that developed on the plate upstream of the ramp were conducted using constant-temperature hot-wire anemometry. A TSI IFA 100 anemometer and a conventional straight sensor probe (Auspex type AHWU-100) were used for the boundary layer surveys. The probe was traversed normal to the plate using a computer controlled traverse that had a resolution of $2.54~\unicode[STIX]{x03BC}\text{m}$ ( $10^{-4}$  in.) per step.

Figure 4.

Turbulent boundary layer mean velocity profile upstream of the convex ramp ( $x/L=-0.085$ ).

Table 1.

Approach ZPG turbulent boundary layer characteristics.

The boundary layer characteristics immediately upstream of the ramp section (i.e.  $x/L=-0.085$ ) were found to be identical, regardless of the imposed unsteady pressure gradient on the ramp. These initial boundary layer parameters are summarized in table 1. The parameters presented in this paper correspond to nominal upstream conditions of $U_{\infty }=10~\text{m}~\text{s}^{-1}$ and momentum thickness Reynolds number, $Re_{\unicode[STIX]{x1D703}}=1600$ just upstream of the convex ramp. The wall-shear stress was based on the Clauser method (and subsequently confirmed via oil film interferometry). Outer variable scaling of the measured mean velocity profile just upstream of the ramp showed very good agreement with the Klebanoff (Reference Klebanoff1955) ZPG profiles obtained at the considerably higher $Re_{\unicode[STIX]{x1D703}}=8000$ . Inner variable scaling of the measured mean velocity profile just upstream of the convex ramp is shown in figure 4 and, consistent with the approach Reynolds number, exhibits a limited wall-normal region in good agreement with the classic logarithmic law-of-the-wall profile.

It is important to note that the ratio of initial boundary layer thickness to convex ramp radius $\unicode[STIX]{x1D6FF}/R=0.049$ on the circular arc portion, so although there will be an influence of wall curvature on boundary layer development, it will be extremely small compared with the effect of the imposed unsteady APG (see Gibson, Verriopoulos & Vlachos Reference Gibson, Verriopoulos and Vlachos1984; Muck, Hoffman & Bradshaw Reference Muck, Hoffman and Bradshaw1985). As shown in figure 1(b), for $x/L>0.6$ the ramp is straight ( $R\rightarrow \infty$ ).

2.3 Airfoil leading edge plasma flow control

The time-dependent pressure gradient imposed on the convex ramp was created by alternately attaching and separating the airfoil flow. As described in § 2.1, this was accomplished by using a DBD plasma actuator positioned at the leading edge of the NACA 0015 airfoil which was cycled on and off at a prescribed frequency.

The dielectric material used for construction of the NACA 0015 airfoil leading edge plasma actuator consisted of two layers of 0.127 mm (0.005 in.) thick Kapton tape between two layers of 0.051 mm (0.002 in.) thick Teflon tape. The Teflon tape protects the Kapton tape from degradation due to the ozone produced by plasma formation (thereby significantly extending the lifetime of the actuator to several days of continuous operation). The leading edge of the airfoil was moulded with a recess of sufficient depth for the actuator to ensure an aerodynamically smooth leading edge surface. The electrodes were made from 0.127 mm (0.005 in.) thick copper foil tape. The covered electrode was 2.54 cm (1 in.) wide and had a span of 35.6 cm (14 in.), while the exposed electrode was 1.27 cm (0.5 in.) wide and had a span of 38.1 cm (15 in.) The electrodes had an overlap of approximately 1 mm (0.039 in) to ensure a uniform DBD plasma discharge throughout the span. Although the spanwise extent of the DBD plasma actuator only covered the central 75 % of the airfoil span, this was sufficient to fully attach the flow over the full extent of the airfoil suction surface.

Previous studies of active leading edge flow control with symmetric and cambered airfoils over a range of chord Reynolds numbers and Mach numbers have shown that the most effective forcing frequency, $f$ , occurs when the non-dimensional frequency $F^{\ast }\equiv fc/U_{\infty }\approx 1$ , where $c$ is the airfoil chord length and $U_{\infty }$ is the free-stream velocity (e.g. Seifert et al. Reference Seifert, Bachar, Koss, Shepshelovich and Wygnanski1993; Seifert, Darabi & Wygnanski Reference Seifert, Darabi and Wygnanski1996; Seifert & Pack Reference Seifert and Pack1999; Patel et al. Reference Patel, Ng, Vasudevan, Corke, Post, Mclaughlin and Suchomel2008). In this study, when the airfoil plasma actuator was on, it was operated at $F^{\ast }=1$ , which, for the conditions of the experiment, corresponded to a physical frequency of $f=1/T_{a}=50$  Hz.

Figure 5.

Schematic of quasi-steady and unsteady signals used for plasma actuation.

The DBD plasma a.c. carrier was a negative ramp function of frequency 3.5 kHz and peak-to-peak voltage of 15 kV. The negative ramp function has previously been shown to be optimum in plasma flow control applications (e.g. Thomas et al. Reference Thomas, Corke, Iqbal, Kozlov and Schatzmam2009). The actuator duty cycle was set at $T_{d}/T_{a}=10\,\%$ . Figure 5 contrasts ‘quasi-steady’ and ‘unsteady’ a.c. signals used for plasma actuation and also defines both the plasma excitation frequency and the duty cycle.

Figure 6 shows one cycle of the periodic signal sent to the airfoil leading edge plasma actuator (prior to amplification). The period of the time series $T=1$  s. The mid-point, $t/T=0.5$ , represents the stage in the unsteady actuation cycle just prior to plasma initiation, and so the airfoil flow is separated and the APG turbulent boundary layer on the convex ramp is attached. From $0.5\leqslant t/T\leqslant 0.75$ , the leading edge plasma actuator is operated in an unsteady mode at $F^{\ast }=1$ with an actuation duty cycle of 10 %. As shown in figure 6, the duty cycle for the unsteady separation time scale was fixed at 25 %. Figure 6 indicates the time interval after plasma initiation for the airfoil flow to fully reattach as well as the time interval after extinguishing the plasma for the airfoil flow to separate.

Figure 6.

One cycle of the signal sent to the airfoil leading edge plasma actuator (prior to amplification). Here, TBL stands for turbulent boundary layer.

As noted previously, an important and often-used parameter for characterizing unsteady boundary layer behaviour is the non-dimensional reduced frequency, $k$ , defined in (1.1). For example, the reduced frequency range relevant to dynamic stall in helicopter rotors is typically quoted in the range $k=0.05$ –0.15. In this experiment, the reduced frequency based upon the convex ramp length ( $\ell =L=37.54~\text{cm}=14.8~\text{in.}$ ) was $k\approx 0.12$ . This corresponded to a physical cyclic frequency of $f_{c}=1$  Hz.

The airfoil reattachment process was documented using flow visualization in conjunction with a Photron FASTCAM SA-1 675K-M1 high-speed camera operated at 2000 fps. High-speed imaging of the airfoil flow indicated that the actuator-induced reattachment process initiated at $t/T=0.50$ was complete by $t/T=0.56$ , as is denoted by the arrow in the voltage–time history shown figure 6. This imposes a more severe APG on the convex ramp flow field which leads to boundary layer separation. Figure 7 presents an interpretive sketch of the airfoil flow-field reattachment process based on the high-speed video imaging for the time interval $0.5\leqslant t/T\leqslant 0.57$ .

Figure 7.

Flow-visualization-based streakline sketches of the airfoil and ramp flow field for the time interval $0.5\leqslant t/T\leqslant 0.57$ over which the airfoil flow reattaches.

Figure 8.

Flow-visualization-based streakline sketches of the airfoil and ramp flow field for the time interval $0.75\leqslant t/T\leqslant 0.9$ . The dashed line delineates a region of separated flow on the convex ramp.

When plasma actuation is terminated at $t/T=0.75$ , the airfoil flow begins to undergo separation and high-speed video imaging shows that the flow over the airfoil is again completely separated by $t/T=0.88$ . This separation process is illustrated in the sequence of sketches contained in figure 8, which is also based on high-speed flow visualization. When the airfoil undergoes separation, the APG on the ramp is reduced and eventually leads to reattachment of the ramp turbulent boundary layer. As described in the experimental results section, this process is repeated in the experiment over a sufficiently large number of cycles to obtain suitable ensemble, phase and conditional averages of desired convex ramp boundary layer statistics.

The high-speed video camera system was also used to record flow visualization images of the ramp turbulent boundary layer, which was seeded with olive oil particles upstream of the wind tunnel inlet. A light sheet from an argon-ion laser was used to illuminate the particles in the centreline plane of the $25^{\circ }$ ramp section. The camera system was rotated by $25^{\circ }$ , and images with a resolution of $1024\times 512$ pixels were recorded along the 150 mm straight portion of the $25^{\circ }$ ramp section. This flow visualization technique allowed video imaging of the turbulent boundary layer separation process at $3000~\text{frames}~\text{s}^{-1}$ and a shutter speed of $1/4000$  s. This shutter speed is 14 times faster than the large-eddy characteristic time scale of the boundary layer, $\unicode[STIX]{x1D6FF}/U_{e}$ , measured at the centre of the field of view. The camera was synchronized to the unsteady plasma input signal.

Figure 9(a) compares images of the ramp boundary layer over the straight section of the ramp, $0.6\leqslant x/L\leqslant 1.0$ , at the instant of airfoil plasma initiation ( $t/T=0.5$ ) and after the airfoil suction surface flow has reattached ( $t/T=0.57$ ). The effect of the increase in boundary layer thickness associated with the more severe APG is apparent. In figure 9(b), the ramp boundary layer is shown over the same spatial extent at the instant the airfoil plasma is extinguished ( $t/T=0.75$ ). At this instant, the airfoil flow is still attached and the ramp boundary layer shows a region of separated flow near the aft end of the ramp as a consequence of exposure to the more severe APG. In the photo to the right ( $t/T=0.9$ ), the airfoil flow has now separated and the ramp boundary layer has thinned and undergone reattachment.

Figure 9.

Sample images of the ramp boundary layer flow for $0.6\leqslant x/L\leqslant 1.0$ during the time intervals for (a) airfoil flow reattachment and (b) airfoil flow separation.

The actuation signal shown previously in figure 6 is supplied to two separate two-channel Crown XTi4000 power amplifiers. The amplified signals are passed through a series of two $1~\unicode[STIX]{x03A9}$ resistors to limit the current. The signal is then split and sent through two 1 : 360 transformers. The transformers produce two high-voltage a.c. output signals that are $180^{\circ }$ out of phase. One signal is supplied to the exposed electrode and the other to the covered electrode of the plasma actuator on the leading edge of the airfoil. This gives rise to a 15 kV peak-to-peak version of the signal shown in figure 6. This applied voltage was found to be sufficient to produce a body force capable of consistently reattaching the airfoil suction surface flow within 60 ms of plasma initiation.

2.4 Ramp time-mean and phase-averaged pressure distributions

In this paper, flow quantities $g(\boldsymbol{x},t)$ are expressed in terms of a triple decomposition,

(2.1) $$\begin{eqnarray}g(\boldsymbol{x},t)=\overline{g}(\boldsymbol{x})+\tilde{g}(\boldsymbol{x},t)+g^{\prime }(\boldsymbol{x},t).\end{eqnarray}$$

In the above relation, $\overline{g}(\boldsymbol{x})$ is the time-averaged component, $\tilde{g}(\boldsymbol{x},t)$ is the phase-averaged coherent fluctuating component associated with the imposed unsteady pressure gradient cycle and $g^{\prime }(\boldsymbol{x},t)$ is the phase incoherent random turbulent component.

The phase average is given by

(2.2) $$\begin{eqnarray}\langle g(\boldsymbol{x},t)\rangle \equiv \lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}g(\boldsymbol{x},t+nT),\end{eqnarray}$$

where $T$ is the period of the unsteady cycle shown previously in figure 6 and $N$ is the number of unsteady cycles used for the phase average. It follows that

(2.3) $$\begin{eqnarray}\displaystyle & \tilde{g}(\boldsymbol{x},t)=\langle g(\boldsymbol{x},t)\rangle -\overline{g}(\boldsymbol{x}), & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & g(\boldsymbol{x},t)=\langle g(\boldsymbol{x},t)\rangle +g^{\prime }(\boldsymbol{x}). & \displaystyle\end{eqnarray}$$

A total of 34 static surface pressure taps placed along the centreline of the convex ramp model were used to measure the imposed streamwise surface pressure distribution for the two conditions of fully attached (plasma on) and separated (plasma off) airfoil flow. In essence, these ramp surface pressure distributions describe the two extremes of the temporal ramp surface pressure variation. The pressure taps were connected via internal tubing to a 48-port Scanivalve SSS-48C9 system. The Scanivalve housed a differential pressure transducer (model number PDCR23D) and signal conditioner (Scanivalve Corp. SCSG2) which were used to measure the time-averaged surface pressure distribution along the ramp section. The surface pressure coefficient is defined as $C_{p}\equiv (P-P_{\infty })/(P_{0}-P_{\infty })$ , where $P$ is the static pressure measured at the wall pressure tap, and $P_{0}$ and $P_{\infty }$ are the free-stream total and static pressures respectively, as measured by a Pitot static probe suspended from the tunnel ceiling 32 mm upstream of the ramp section. The measured pressure coefficients have a $\pm 2.8\,\%$ relative uncertainty at 20 : 1 odds. Spanwise-oriented arrays of static pressure taps at selected streamwise locations confirmed the two-dimensionality of the imposed surface pressure distributions in the mean to within $\pm 3\%$ .

Figure 10 shows the time-averaged surface pressure coefficient distribution over the ramp with the airfoil plasma flow control both off and on. The $C_{p}$ distribution is presented as a function of $x/L$ , where $x$ is the streamwise spatial coordinate along the ramp surface and $L=37.54$  cm (14.78 in.) is the total length of the ramp (figure 1 b). Figure 11 shows the corresponding non-dimensional pressure gradient parameter, defined as $\unicode[STIX]{x1D6FD}\equiv (\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D70F}_{w})_{o}\,\text{d}P/\text{d}x$ , where the subscript $o$ denotes that the displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }$ , and wall-shear stress, $\unicode[STIX]{x1D70F}_{w}$ , are evaluated at the origin of the convex ramp section ( $x=0$ ). The values of $\unicode[STIX]{x1D6FD}_{o}$ were calculated from the $C_{p}$ data shown in figure 10. It is important to note that the $\unicode[STIX]{x1D6FD}_{o}$ values shown in figure 11 are based on initial values of $\unicode[STIX]{x1D6FF}^{\ast }$ and $\unicode[STIX]{x1D70F}_{w}$ and serve only to provide a comparison of the non-dimensional pressure gradients for the plasma-on and plasma-off cases. Values of $\unicode[STIX]{x1D6FD}$ on the ramp surface will be much larger under the influence of the APG since $\unicode[STIX]{x1D6FF}^{\ast }$ grows, $\unicode[STIX]{x1D70F}_{w}$ is diminished and $\unicode[STIX]{x1D6FD}(x/L)=\unicode[STIX]{x1D6FD}_{o}[(\unicode[STIX]{x1D6FF}^{\ast }(x/L)/\unicode[STIX]{x1D6FF}_{o}^{\ast })(\unicode[STIX]{x1D70F}_{Wo}/\unicode[STIX]{x1D70F}_{W})]$ . For example, at $x/L=0.72$ , $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FD}_{o}=3.5$ for the plasma-off case and $\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FD}_{o}=12$ for the plasma-on case.

Figure 10.

Measured mean convex ramp surface pressure distributions for the airfoil plasma-on and plasma-off cases.

Figure 11.

Non-dimensional streamwise pressure gradient, $\unicode[STIX]{x1D6FD}_{o}\equiv (\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D70F}_{w})_{o}\,\text{d}P/\text{d}x$ , on the convex ramp for airfoil plasma-on and plasma-off cases.

Figure 11 shows that in the absence of airfoil leading edge plasma flow control (i.e. with the airfoil flow separated), a favourable pressure gradient ( $\unicode[STIX]{x1D6FD}<0$ ) exists over approximately the first 40 % of the ramp section followed by an APG over the reminder of the ramp. The APG peaks near $x/L=0.65$ and is subsequently reduced over the last 35 % of the ramp section. Laser Doppler velocimetry (LDV) measurements (to be presented) will show that in this case, the APG turbulent boundary layer remains attached to the ramp surface, but near a state of incipient separation.

With the airfoil plasma control on, the ramp surface pressure results show that, after an initially favourable pressure gradient over the first 24 % of the ramp section, there is a larger APG over the remainder of the ramp, with the peak value occurring near $x/L=0.5$ . A plateau of constant $C_{p}$ values for $x/L>0.9$ shown in figure 10 is associated with turbulent boundary layer separation from the ramp surface.

In order to measure the effect of the airfoil plasma actuation on the unsteady ramp surface pressure, three high-frequency-response Endevco 8507C-1 piezoresistive pressure transducers were flush mounted on the ramp centreline at the representative streamwise locations of $x/L=0.67$ , 0.80 and 0.94. In this manner, two of the transducers were located upstream of the steady-state plasma-on separation location at $x/L=0.90$ , with the third located just downstream. Signals from the three pressure transducers were recorded at a sample frequency of 20 kHz and anti-alias filtered at a frequency of 10 kHz using an Endevco model 136 signal conditioner. Figure 12(a) shows a single representative sample trace of the instantaneous surface pressure coefficient for each of the three streamwise locations on the ramp surface during the unsteady cycle.

Figure 12(b) shows the corresponding phase-averaged wall pressure coefficient, $\langle c_{P}\rangle$ , computed from (2.2) with $N=250$ unsteady cycles, which was found to be sufficient to produce a fully converged phase average. Shortly after the onset of plasma actuation at $t/T=0.5$ , each of the phase-averaged pressure coefficients shows a rapid increase towards a peak value of $\langle c_{P}\rangle$ followed by a shallow local minimum. Inspection of figure 12(b) reveals that the $\langle c_{P}\rangle$ peak convects downstream at speed $U_{c}/U_{\infty }=0.95\pm 0.02$ . The sudden shift to higher values of $\langle c_{P}\rangle$ is associated with the plasma flow control reattaching the flow on the suspended airfoil, as depicted previously in figure 7. The peak and local minimum shown in the traces in figure 12(b) represent the transient airfoil wake passage over the ramp as the airfoil flow undergoes reattachment. Following this initial overshoot, the values of $\langle c_{P}\rangle$ gradually increase during the interval $0.6\leqslant t/T\leqslant 0.70$ and the boundary layer undergoes separation on the most downstream portion of the ramp section (as demonstrated later in § 3). The pressure signal at $x/L=0.67$ (the location farthest upstream of the separation) shows the largest change in magnitude throughout the unsteady cycle. The pressure signal for the transducer just downstream of the unsteady separation, $x/L=0.94$ , shows comparatively little change in the magnitude for the unsteady cycle.

Figure 12.

(a) Instantaneous pressure coefficients on the ramp for one unsteady cycle. (b) Phase-averaged pressure coefficient on the ramp for one unsteady cycle. (c) Phase-averaged turbulent root mean square (r.m.s.) pressure coefficient for one unsteady cycle.

The airfoil plasma is extinguished at $t/T=0.75$ , and this initiates a gradual reduction in values of $\langle c_{P}\rangle$ as the airfoil flow separation process commences. A local minimum followed by a peak in $\langle c_{P}\rangle$ shown in the traces for $t/T>0.8$ is a result of the flow separating from the airfoil section after the airfoil plasma flow control has been turned off during the unsteady cycle. The convection speed for this $\langle c_{P}\rangle$ peak is reduced to $U_{c}/U_{\infty }\approx 0.31$ due to the imposed APG. Figure 12(c) presents the phase-averaged r.m.s. pressure coefficient, $\langle c_{P\,rms}\rangle$ , for the unsteady cycle. After passage of the airfoil wake transient, $\langle c_{P\,rms}\rangle$ is reduced on the ramp. Significant increases are noted after the plasma is extinguished, and this is largest for $x/L=0.94$ and is associated with the unsteady reattachment of the separated ramp flow. Not only do the largest values of $\langle c_{P\,rms}\rangle$ occur for $x/L=0.94$ , but this location also requires the largest time interval to relax back to nominal attached flow values (which appears to be complete near $t/T=0.3$ ).

3.1 Ramp boundary layer mean velocity and turbulence intensity

Local wall-normal profiles of two-component mean and fluctuating velocity along the centreline of the convex ramp section were obtained non-intrusively using a Dantec Dynamics Fiber Flow LDV system. The fibre optic LDV system was operated in $180^{\circ }$ backscatter mode. The boundary layer flow was seeded with olive oil particles of nominally $1~\unicode[STIX]{x03BC}\text{m}$ diameter which were introduced upstream of the wind tunnel inlet with a liquid droplet seeding generator. The LDV probe volume was traversed locally normal to the convex ramp surface over a range of $x/L$ locations using a three-axis Aerotech traverse system and Unidex 11 controller. The probe measurement volume had a maximum wall-normal dimension of 0.35 mm (0.0138 in), which set the effective spatial resolution of the boundary layer measurements. The true $y$ location relative to the surface was established via a separate optics-based calibration procedure.

Prior to characterizing the unsteady aspects of the APG turbulent boundary layer, consideration is first given to the time-mean boundary layer behaviour for both the imposed plasma-on and plasma-off pressure gradients shown previously in § 2. Figure 13 documents the spatial evolution of streamwise mean velocity and streamwise-component turbulence intensity profiles along the ramp for both the less severe (airfoil plasma-off) and more severe (airfoil plasma-on) APG cases. Each profile is made non-dimensional by the local external velocity, $U_{e}(x/L)$ , and the wall-normal coordinate is scaled by the local boundary layer thickness, $\unicode[STIX]{x1D6FF}(x/L)$ . It is apparent from figure 13 that the ramp boundary layer flow is significantly influenced by the variation in the imposed APG.

The mean velocity profiles for the plasma-off case show that the boundary layer remains attached at all locations on the ramp section. This was also confirmed with surface oil flow visualization (not shown). In contrast, for the plasma-on case, the velocity defect is observed to be much larger, and time-averaged near-wall reversed flow occurs at downstream locations on the ramp section ( $x/L>0.9$ ). This aspect will be considered further in § 3.2.

The corresponding turbulence intensity profiles show that in both APG cases, the location of peak r.m.s. turbulence shifts away from the wall as the flow evolves in $x/L$ . However, both the magnitude of the peak turbulence intensity and its distance from the wall increase significantly with the severity of the imposed APG, as does the reduction in near-wall turbulence levels. In contrast to the ZPG turbulent boundary layer, the turbulence peak is broader, is of greater amplitude and occurs much farther away from the wall. The data in figure 13 are consistent with the results of Krogstad & Skåre (Reference Krogstad and Skåre1995), Shah et al. (Reference Shah, Laval and Stanislas2009) and Monty, Harun & Marusic (Reference Monty, Harun and Marusic2011), where it was also found that r.m.s. velocity fluctuation profiles from APG turbulent boundary layers showed a turbulence peak that moved away from the wall as the magnitude of the APG grew.

Figure 13.

The LDV measured (a) time-mean streamwise velocity and (b) turbulence intensity profiles along the convex ramp for the plasma-on and plasma-off cases.

3.2 Transient response of the ramp boundary layer

In order to initially characterize the transient response of the ramp turbulent boundary layer to the imposed unsteady pressure gradient, an array of six Auspex straight hot-wire probes was utilized. The sampling frequency was 13 kHz and the probes were separated in the wall-normal direction by $\unicode[STIX]{x0394}y=8$  mm ( $\unicode[STIX]{x0394}y/\unicode[STIX]{x1D6FF}=0.21$ ). As a representative example, figure 14(a) presents a contour plot of the phase-averaged velocity based on 250 unsteady pressure gradient cycles with the hot-wire rake located at a streamwise location of $x/L=0.80$ , which is upstream of any reverse flow that takes place during the unsteady cycle (and which obviously cannot be detected by the hot-wire array). The time interval associated with airfoil plasma flow control is indicated in the figure.

Figure 14.

(a) Temporal variation in phase-averaged streamwise velocity at $x/L=0.8$ (dotted horizontal lines indicate probe positions). (b) Temporal variation in phase-averaged turbulence intensity at $x/L=0.8$ (dotted horizontal lines indicate probe positions).

It is apparent from figure 14(a) that the ramp boundary layer momentum defect grows very rapidly in response to the increase in APG that commences at $t/T=0.5$ . It should be noted that for fixed $y/\unicode[STIX]{x1D6FF}$ , there is an initial increase in phase-averaged velocity near $t/T=0.53$ followed by a large reduction. This transient behaviour is associated with the passage of the airfoil wake as part of the airfoil flow attachment process (see figure 7). After the plasma is extinguished at $t/T=0.75$ , the phase-averaged velocity contours remain constant until approximately $t/T=0.8$ , and reference to figure 8 indicates that this delay is associated with the time interval required for the airfoil flow separation process to commence. Figure 14(a) shows a marked reduction in the velocity defect commencing near $t/T=0.8$ . The airfoil flow is fully separated by $t/T=0.9$ , and at the same time figure 14(a) shows a transient increase in velocity defect which appears to be associated with the ramp boundary layer reattachment process.

Figure 14(b) presents the corresponding temporal evolution of the phase-averaged streamwise-component turbulence intensity at $x/L=0.8$ . The sharp peak near $t/T=0.55$ is due to the velocity fluctuations in the outer flow associated with the passage of the transient airfoil wake and its jitter. Shortly after the wake passage, the boundary layer peak turbulence intensity rapidly shifts away from the wall and peak values are found near $y/\unicode[STIX]{x1D6FF}=0.3$ . This is consistent with the turbulence intensity profiles shown in figure 13. Conditional LDV measurements to be presented in the next section will show that this is associated with the development of an embedded shear layer within the unsteady APG boundary layer. During this portion of the unsteady cycle in the interval $0.6\leqslant t/T\leqslant 0.85$ , the turbulence intensity continually increases to reach a maximum value before onset of the reattachment process.

A more extensive set of phase-averaged measurements of the unsteady boundary layer were performed non-intrusively using LDV. At each spatial location, $10^{6}$ samples were acquired for each velocity component. Due to the LDV data rate dependence on the particle crossings (9–12 kHz), a bin averaging technique was used to determine the phase intervals of the LDV data (see Sonnenberger, Graichen & Erk Reference Sonnenberger, Graichen and Erk2000). In this method, the unsteady period, $T$ , is subdivided into 180 elementary intervals referred to as phase ‘bins’, of width $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=2^{\circ }$ , and the centre of each bin is denoted by a phase angle, $\unicode[STIX]{x1D719}$ , in the unsteady cycle. The estimate of the phase-averaged value from bin averaging the samples of a velocity component is given by

(3.1) $$\begin{eqnarray}\langle u(\unicode[STIX]{x1D719})\rangle =\frac{1}{N(\unicode[STIX]{x1D719})}\mathop{\sum }_{n=1}^{N(\unicode[STIX]{x1D719})}u_{n}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

where $N(\unicode[STIX]{x1D719})\geqslant 5000$ denotes the number of samples in each phase bin. The phase-averaged r.m.s. of the velocity component is estimated by

(3.2) $$\begin{eqnarray}\left\langle \sqrt{u^{\prime 2}(\unicode[STIX]{x1D719})}\right\rangle =\sqrt{\frac{1}{N(\unicode[STIX]{x1D719})-1}\mathop{\sum }_{n=1}^{N(\unicode[STIX]{x1D719})}(u_{n}(\unicode[STIX]{x1D719})-\langle u(\unicode[STIX]{x1D719})\rangle )^{2}}.\end{eqnarray}$$

Figure 15 presents contour plots illustrating the temporal evolution of the phase-averaged velocity, $\langle u/U_{e}\rangle$ , during the unsteady cycle for ramp locations of $x/L=0.86$ , 0.93 and 1.0. The time interval for plasma flow control on the airfoil is again denoted in the figure by vertical dashed lines. Key features regarding the response of the turbulent boundary layer to the imposed unsteady pressure gradient at locations both upstream and downstream of the separation on the aft portion of the ramp section are shown. In each case, the wall-normal coordinate is normalized by the local boundary layer thickness, $\unicode[STIX]{x1D6FF}(x/L)$ . After the airfoil flow has reattached near $t/T=0.55$ , the velocity contour levels at fixed $y/\unicode[STIX]{x1D6FF}$ drop significantly, which is a manifestation of the growth of the momentum defect in the normalized wall coordinate. It should be noted that at the location $x/L=1.0$ , a region of phase-averaged reverse flow is measured near the wall for $0.65\leqslant t/T\leqslant 0.85$ , as indicated by the white contour region. Upon termination of the plasma at $t/T=0.75$ , the velocity contour levels increase at fixed $y/\unicode[STIX]{x1D6FF}$ as the boundary layer momentum defect is reduced during the reattachment process from $t/T=0.85$ to $1.0$ . Careful inspection of figure 15 shows that the boundary layer phase-averaged velocity continues to recover from the previous unsteady separation cycle until approximately $t/T=0.2$ .

Figure 15.

Contour plots of the temporal evolution of ramp boundary layer phase-averaged streamwise velocity, $\langle u/U_{e}\rangle$ , based on LDV measurement: (a) $x/L=0.86$ , (b) $x/L=0.93$ , (c) $x/L=1.0$ .

Figure 16 presents the fraction of time the phase-averaged near-wall flow is in the streamwise direction, $\langle \unicode[STIX]{x1D6FE}_{+U}\rangle$ , for a representative range of $x/L$ locations along the ramp section. In each case, the LDV probe volume was placed as close to the wall as possible. The indicated $y/\unicode[STIX]{x1D6FF}$ locations in figure 16 differ due to the growth in the local boundary layer thickness, $\unicode[STIX]{x1D6FF}(x,t)$ . For the period $0\leqslant t/T\leqslant 0.5$ (airfoil plasma off), no phase-averaged reverse flow is detected for the locations $x/L=0.58$ , 0.72 and 0.86 since $\langle \unicode[STIX]{x1D6FE}_{+U}\rangle =1$ . During the same time period, the flow at $x/L=1$ shows $\langle \unicode[STIX]{x1D6FE}_{+U}\rangle =0.9$ , which indicates that the boundary layer at the farthest downstream ramp location is in a state of incipient detachment for the comparatively mild APG.

Upon application of the more severe APG for $0.5\leqslant t/T\leqslant 0.75$ , the most upstream location, $x/L=0.58$ , still indicates no phase-averaged reverse flow throughout the unsteady cycle. At $x/L=0.72$ , phase-averaged reverse flow for the same interval occurs not more than 8 % of the time. In contrast, the measurements at $x/L=0.86$ in the interval $0.65\leqslant t/T\leqslant 0.85$ show that phase-averaged reverse flow is detected approximately 20 % of the time ( $\langle \unicode[STIX]{x1D6FE}_{+u}\rangle \approx 0.80$ ), which is defined as a state of intermittent transitory detachment (see Simpson Reference Simpson1996). At $x/L=1.0$ , in the interval $0.62\leqslant t/T\leqslant 0.82$ , reverse flow is detected more than 50 % of the time, which is indicative of the existence of a separated boundary layer (Simpson Reference Simpson1996). It may be noted that the return to a state of incipient separation at $x/L=1.0$ takes approximately twice as long as the onset of flow separation.

Figure 16.

Fraction of time the phase-averaged near-wall flow moves in the streamwise direction, $\left\langle \unicode[STIX]{x1D6FE}_{+u}\right\rangle$ , for selected $x/L$ locations along the ramp.

Figure 17.

Variation of phase-averaged turbulent boundary layer displacement thickness, momentum thickness and shape factor with $t/T$ for locations of $x/L=0.86$ and 1.0.

The phase-averaged velocity profiles measured on the ramp section were used to calculate the temporal variation of global boundary layer quantities throughout the unsteady cycle. As an example, figure 17 shows the temporal variation of the boundary layer displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }$ , the momentum thickness, $\unicode[STIX]{x1D703}$ , and the shape factor, $H$ , with $t/T$ for representative locations of $x/L=0.86$ and $x/L=1.0$ . After plasma initiation, the displacement thickness shows a brief reduction near $t/T=0.52$ , which is associated with the convection of the increased wake region behind the airfoil as a result of the unsteady forcing that reattaches the leading edge (see figure 7). The displacement thickness subsequently increases to reach a maximum value near $t/T=0.82$ just before the boundary layer reattachment process begins on the ramp section. After application of the increased APG, the momentum thickness at $x/L=0.86$ increases by 26 % to reach a nearly constant value for the separated part of the cycle. The increase in $\unicode[STIX]{x1D703}$ at $x/L=1.0$ is observed to be much smaller. The shape factor for the $x/L=0.86$ location increases from an initial value near $H=1.8$ to a peak value of 2.53 near $t/T=0.8$ . At $x/L=1.0$ , the velocity profile shape factor shifts from a value of 2.35 during the attached portion of the cycle to a maximum value of 3.9 at $t/T=0.68$ , when the boundary layer profile shows the largest degree of reversed flow near the wall. As a basis for comparison, it should be recalled that for near-equilibrium flows consistent with Cole’s mean velocity profile, intermittent transitory detachment occurs near $H=2.7$ .

With the basic character of the unsteady ramp boundary layer documented, the next section is focused on the influence of the imposed unsteady pressure gradient on the nature of the coherent structure in the ramp turbulent boundary layer. This is explored via a series of conditional measurements.

4.1 Application of the quadrant-splitting technique

Ejection and sweep events were quantified using the $u^{\prime }v^{\prime }$ quadrant-splitting (QS) technique initially developed and utilized by Wallace, Eckelmann & Brodkey (Reference Wallace, Eckelmann and Brodkey1972) and Willmarth & Lu (Reference Willmarth and Lu1972). Ejection events reside in the second quadrant ( $u^{\prime }(t)<0$ , $v^{\prime }(t)>0$ ), and are generally associated with both spanwise vortices that form the head of a vortical hairpin arch and streamwise-oriented near-wall vortices (see figure 13 of Robinson Reference Robinson1991). The fourth-quadrant ( $u^{\prime }(t)>0,v^{\prime }(t)<0$ ) motions are associated with the tilted neck elements of a vortical arch. Lu & Willmarth (Reference Lu and Willmarth1973) were the first to show that discrete second-quadrant (Q2) and fourth-quadrant (Q4) events are the primary contributors to the Reynolds stress production in a turbulent boundary layer.

In this study, the QS technique was implemented by using an identification (ID) function to determine those instants of time when $u^{\prime }(t)$ and $v^{\prime }(t)$ are in a given quadrant, and their product $u^{\prime }(t)v^{\prime }(t)$ exceeds a specified threshold level. The ID functions for quadrant 2 and quadrant 4 are respectively

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \text{ID}_{Q2}(t,\unicode[STIX]{x1D6FC}_{q})=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }u^{\prime }(t)<0\text{ and }u^{\prime }(t)v^{\prime }(t)<-\unicode[STIX]{x1D6FC}_{q}\sqrt{\overline{u^{\prime 2}}}\sqrt{\overline{v^{\prime 2}}},\\ 0\quad & \text{otherwise},\end{array}\right. & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \text{ID}_{Q4}(t,\unicode[STIX]{x1D6FC}_{q})=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }u^{\prime }(t)>0\text{ and }u^{\prime }(t)v^{\prime }(t)<-\unicode[STIX]{x1D6FC}_{q}\sqrt{\overline{u^{\prime 2}}}\sqrt{\overline{v^{\prime 2}}},\\ 0\quad & \text{otherwise},\end{array}\right. & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{q}$ determines the size of the so-called ‘hyperbolic hole’ defined by the product of the r.m.s. values of $u^{\prime }$ and $v^{\prime }$ . Narasimha et al. (Reference Narasimha, Kumar, Prabhu and Kailas2010) found that the r.m.s. value is more appropriate than the mean Reynolds stress value for calculating the threshold since the fluctuations of $u^{\prime }(t)v^{\prime }(t)$ can be large multiples of the mean as the flow approaches separation. Only a single event is counted when the ID function is satisfied for an unbroken sequence of temporal samples, and ID functions for Q1 and Q3 events were similarly defined.

The relative contribution to the Reynolds stress from the various quadrant motions was determined from

(4.3) $$\begin{eqnarray}\frac{\overline{u^{\prime }v_{i}^{\prime }}(\unicode[STIX]{x1D6FC}_{q})}{\overline{u^{\prime }v^{\prime }}}=\frac{1}{\overline{u^{\prime }v^{\prime }}}\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}u^{\prime }(t)v^{\prime }(t)\text{ID}_{Q_{i}}(t,\unicode[STIX]{x1D6FC}_{q})\,\text{d}t,\end{eqnarray}$$

where the subscript $i$ refers to the $i\text{th}$ quadrant. The contribution to $\overline{u^{\prime }v^{\prime }}$ from within the hyperbolic hole region was obtained from the relation

(4.4) $$\begin{eqnarray}\frac{\overline{u^{\prime }v^{\prime }}_{H}(\unicode[STIX]{x1D6FC}_{q})}{\overline{u^{\prime }v^{\prime }}}=\frac{1}{\overline{u^{\prime }v^{\prime }}}\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}u^{\prime }(t)v^{\prime }(t)\text{ID}_{H}(t,\unicode[STIX]{x1D6FC}_{q})\,\text{d}t,\end{eqnarray}$$

where the ID function for the hyperbolic hole is defined as

(4.5) $$\begin{eqnarray}\text{ID}_{H}(t,\unicode[STIX]{x1D6FC}_{q})=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }|u^{\prime }(t)v^{\prime }(t)|<\unicode[STIX]{x1D6FC}_{q}\sqrt{\overline{u^{2}}}\sqrt{\overline{v^{2}}},\\ 0\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$

The contributions from the four quadrants and the hole region are all functions of the threshold level, $\unicode[STIX]{x1D6FC}_{q}$ , and it follows that

(4.6) $$\begin{eqnarray}\mathop{\sum }_{i=1}^{4}\frac{\overline{u^{\prime }v_{i}^{\prime }}(\unicode[STIX]{x1D6FC}_{q})}{\overline{u^{\prime }v^{\prime }}}+\frac{\overline{u^{\prime }v_{H}^{\prime }}(\unicode[STIX]{x1D6FC}_{q})}{\overline{u^{\prime }v^{\prime }}}=1.\end{eqnarray}$$

For validation purposes and to form a basis for comparison with the APG convex ramp boundary layer, the QS technique was first applied to the ZPG turbulent boundary layer on the flat plate just upstream of the convex ramp section. The QS results are based on $10^{6}$ LDV velocity measurements obtained at each spatial location. Figure 18 shows the $u^{\prime }(t)$ and $v^{\prime }(t)$ velocity components measured by LDV on the boundary layer development plate at a wall-normal location of $y^{+}=105$ . The figure shows the quadrants in the $u^{\prime }v^{\prime }$ plane and also sketches sample hyperbolic holes defined for $\unicode[STIX]{x1D6FC}_{q}=1.5$ and $\unicode[STIX]{x1D6FC}_{q}=3$ . This figure clearly shows the preference for the quadrants 2 and 4 Reynolds stress-producing events, which is typical of a ZPG turbulent boundary layer.

Figure 19 presents inner variable scaled profiles of the fractional contribution of each quadrant to $\overline{u^{\prime }v^{\prime }}$ in the approach ZPG turbulent boundary layer with $\unicode[STIX]{x1D6FC}_{q}=0$ . These results are in very good agreement with similar measurements presented in figure 4 of Wallace et al. (Reference Wallace, Eckelmann and Brodkey1972). For example, at $y^{+}=36$ , the Q2 and Q4 events account for 81 % and 66 % respectively of the mean Reynolds stress (with the balance associated with Q1 and Q3 events), in good agreement with both Wallace et al. (Reference Wallace, Eckelmann and Brodkey1972) and Lu & Willmarth (Reference Lu and Willmarth1973).

Figure 18.

The $u^{\prime }(t)$ and $v^{\prime }(t)$ velocity components measured near the end of the flat boundary layer development plate ( $x/L=-0.085$ ) at the wall-normal location of $y^{+}=105$ . Also shown are the hyperbolic holes for $\unicode[STIX]{x1D6FC}_{q}=1.5$ and 3.0.

Figure 19.

Sample profiles of the fractional contribution to $-\overline{u^{\prime }v^{\prime }}$ from each quadrant ( $\unicode[STIX]{x1D6FC}_{q}=0$ ) in the ZPG turbulent boundary layer near the end of the flat boundary layer development plate ( $x/L=-0.085$ ).

Figure 20.

Fractional contribution of each quadrant to $\overline{u^{\prime }v^{\prime }}$ as a function of the hyperbolic hole size, $\unicode[STIX]{x1D6FC}_{q}\sqrt{\overline{u^{\prime 2}}}\sqrt{\overline{v^{\prime 2}}}$ , at a sample wall-normal location of $y^{+}=36$ in the ZPG boundary layer at $x/L=-0.085$ .

Figure 20 shows the fractional contribution of each quadrant as a function of the hole size, $\unicode[STIX]{x1D6FC}_{q}$ , at the same wall-normal location of $y^{+}=36$ for the ZPG boundary layer. For a hole size of $\unicode[STIX]{x1D6FC}_{q}=4$ , which corresponds to a threshold level of 12 times the measured Reynolds stress, the Q2 events still account for 16 % of the contributions to $-\overline{u^{\prime }v^{\prime }}$ . These results are consistent with the measurements presented in figure 11 of Lu & Willmarth (Reference Lu and Willmarth1973) and further confirm the importance of the strong Q2 ejection events in the ZPG turbulent boundary layer.

4.2 Effect of APG on ejection and sweep events

It is known that the imposition of a streamwise pressure gradient influences the dynamic process of turbulence production in the boundary layer. For example, Krogstad & Skåre (Reference Krogstad and Skåre1995) showed that the APG turbulent boundary layer is dominated by strong Q4 sweep events, with Q2 ejection events nearly absent in the near-wall region. In stark contrast, Bourassa & Thomas (Reference Bourassa and Thomas2009) showed that for a favourable pressure gradient large enough to cause relaminarization, Q4 sweep events are largely eliminated, while infrequent but larger-amplitude near-wall Q2 ejection events still occur.

Figure 21.

Schematic showing quadrant event duration, $\unicode[STIX]{x0394}T_{i}$ , and time interval between events, $T_{i}$ .

Prior to examination of the unsteady APG turbulent boundary layer, a representative comparison is first made between QS measurements for the ZPG approach boundary layer flow at $x/L=-0.085$ and the APG flow on the $25^{\circ }$ ramp at $x/L=0.93$ with the plasma off and the ramp turbulent boundary layer attached. In particular, the average time between events occurring in the $i\text{th}$ quadrant, $\overline{T_{i}^{\ast }}$ , which is defined as the time between consecutive upward or downward crossings of the $u^{\prime }v^{\prime }$ time series past the specified threshold level, and the duration of those events, $\overline{\unicode[STIX]{x0394}T_{i}^{\ast }}$ , defined as the mean time between consecutive upward and downward crossings of the threshold level, are examined. These time intervals are defined in the schematic shown in figure 21. In order to remove the effect of convective time-scale differences between the ZPG and APG flow fields, following Antonia & Krogstad (Reference Antonia and Krogstad1992), these characteristic mean times are presented and compared in scaled form $\overline{T_{i}^{\ast }}\equiv \bar{T}_{i}\,U_{e}/\unicode[STIX]{x1D6FF}$ and $\overline{\unicode[STIX]{x0394}T_{i}^{\ast }}\equiv \overline{\unicode[STIX]{x0394}T_{i}}\,U_{e}/\unicode[STIX]{x1D6FF}$ . Also presented is the time-scale ratio, $\overline{\unicode[STIX]{x0394}T_{i}}/\overline{T_{i}}$ (an approach used by Krogstad & Skåre (Reference Krogstad and Skåre1995)), which is independent of convective speed. Large values of this ratio are associated with frequent and/or long-duration events that are expected to be the most dynamically significant. Conversely, small values of $\overline{\unicode[STIX]{x0394}T_{i}}/\overline{T_{i}}$ correspond to comparatively infrequent and/or short duration events, which would not be as dynamically important.

Figure 22 compares profiles of $\unicode[STIX]{x0394}\overline{T}_{2}^{\ast },\overline{T}_{2}^{\ast },\unicode[STIX]{x0394}\overline{T}_{4}^{\ast }$ and $\overline{T}_{4}^{\ast }$ as well as the ratios $\unicode[STIX]{x0394}\overline{T}_{2}/\overline{T}_{2}$ and $\unicode[STIX]{x0394}\overline{T}_{4}/\overline{T}_{4}$ for the ZPG and APG turbulent boundary layers at a threshold level of $\unicode[STIX]{x1D6FC}_{q}=4$ where the QS technique captures only the strongest events.

Figure 22(a) shows that for both ZPG and APG cases, strong Q2 events are of longer duration than strong Q4 events over the entire boundary layer. Furthermore, ZPG Q2 and Q4 events are of longer duration that the corresponding events in the APG flow with the exception of very near the wall where they become comparable.

Figure 22(b) shows that in the ZPG boundary layer, Q2 events are more frequent than Q4 events at all locations within the boundary layer. In contrast, for the APG case, Q4 events are more frequent than Q2 events for $y/\unicode[STIX]{x1D6FF}<0.12$ , with the disparity growing near the wall. For $y/\unicode[STIX]{x1D6FF}>0.12$ in the APG boundary layer, Q2 events become more frequent than Q4, with the disparity growing with increased distance from the wall.

Figure 22.

Comparison of (a) mean duration $\unicode[STIX]{x0394}T_{i}^{\ast }$ and (b) mean time $T_{i}^{\ast }$ between Q2 and Q4 events, and (c) their ratio $\unicode[STIX]{x0394}T_{i}/T_{i}$ for ZPG and APG boundary layers with $\unicode[STIX]{x1D6FC}_{q}=4$ .

Figure 23.

Comparison of fractional contribution of quadrant events to $\overline{u^{\prime }v^{\prime }}$ for ZPG and APG cases for strong events: (a) ZPG with $\unicode[STIX]{x1D6FC}_{q}=4$ ; (b) APG with $\unicode[STIX]{x1D6FC}_{q}=4$ .

Figure 22(c) compares the ratios $\unicode[STIX]{x0394}\overline{T}_{2}/\overline{T}_{2}$ and $\unicode[STIX]{x0394}\overline{T}_{4}/\overline{T}_{4}$ for $\unicode[STIX]{x1D6FC}_{q}=4$ . For the ZPG boundary layer case, $\unicode[STIX]{x0394}\overline{T}_{2}/\overline{T}_{2}$ exceeds $\unicode[STIX]{x0394}\overline{T}_{4}/\overline{T}_{4}$ for all $y/\unicode[STIX]{x1D6FF}$ , and so the Q2 events are more dynamically significant than the Q4 events throughout the boundary layer. In contrast, for the APG boundary layer, $\unicode[STIX]{x0394}\overline{T}_{4}/\overline{T}_{4}$ exceeds $\unicode[STIX]{x0394}\overline{T}_{2}/\overline{T}_{2}$ for $y/\unicode[STIX]{x1D6FF}<0.12$ , indicating that strong Q4 motions dominate in the near-wall region. For $y/\unicode[STIX]{x1D6FF}>0.12$ , strong Q2 motions dominate, with a fairly localized peak in $\unicode[STIX]{x0394}\overline{T}_{2}/\overline{T}_{2}$ occurring near $y/\unicode[STIX]{x1D6FF}\approx 0.28$ . The implications of this regarding an embedded shear layer instability mechanism will be discussed in § 6.

Figure 23 compares the fractional contributions of the Q2 and Q4 events to the Reynolds stress, $\overline{u^{\prime }v_{1}^{\prime }}/\overline{u^{\prime }v^{\prime }}$ , for the ZPG and APG cases for strong events ( $\unicode[STIX]{x1D6FC}_{q}=4$ ). In the ZPG boundary layer, the contribution from strong ejection events (Q2) exceeds that from sweeps (Q4) over the entire lower half of the boundary layer. In contrast, for the APG boundary layer, strong Q4 sweep events become the dominant contributor in the near-wall region, accounting for nearly 25 % of the mean Reynolds stress near the wall, while the Q2 events account for less than 5 %. The Reynolds stress contributions from Q2 and Q4 events become comparable near $y/\unicode[STIX]{x1D6FF}=0.12$ , with ejection events becoming the dominant contributors farther from the wall. At $y/\unicode[STIX]{x1D6FF}=0.32$ , strong Q2 ejection events account for approximately 47 % of the mean Reynolds stress.

4.3 Quadrant-splitting measurements in the unsteady APG turbulent boundary layer

The QS technique described previously was modified in order to analyse the unsteady APG boundary layer. Bin (i.e. phase) averaging was used to determine the number of quadrant events that take place during the unsteady boundary layer cycle. At each measurement location, the QS technique was applied for $10^{6}$ unsteady boundary layer samples in the interval where the phase resolution was $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=2^{\circ }$ , resulting in phase estimates for 180 different phase bins. The numbers of Q2 and Q4 events detected for a particular phase value are given by

(4.7) $$\begin{eqnarray}\displaystyle \#\text{Q}2(\unicode[STIX]{x1D719})=\mathop{\sum }_{n=1}^{N(\unicode[STIX]{x1D719})}\text{ID}_{Q2}(\unicode[STIX]{x1D719},n), & & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle \#\text{Q}4(\unicode[STIX]{x1D719})=\mathop{\sum }_{n=1}^{N(\unicode[STIX]{x1D719})}\text{ID}_{Q4}(\unicode[STIX]{x1D719},n), & & \displaystyle\end{eqnarray}$$

where the ID functions are now defined as

(4.9) $$\begin{eqnarray}\displaystyle \text{ID}_{Q2}(\unicode[STIX]{x1D719},n)=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }u_{n}^{\prime }(\unicode[STIX]{x1D719})<0\text{ and }u_{n}^{\prime }(\unicode[STIX]{x1D719})v_{n}^{\prime }(\unicode[STIX]{x1D719})<-\unicode[STIX]{x1D6FC}_{q}\left\langle \sqrt{u_{B}^{\prime 2}(\unicode[STIX]{x1D719})}\right\rangle \left\langle \sqrt{v_{B}^{\prime 2}(\unicode[STIX]{x1D719})}\right\rangle ,\\ 0\quad & \text{otherwise},\end{array}\right. & & \displaystyle\end{eqnarray}$$

(4.10) $$\begin{eqnarray}\displaystyle \text{ID}_{Q4}(\unicode[STIX]{x1D719},n)=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }u_{n}^{\prime }(\unicode[STIX]{x1D719})>0\text{ and }u_{n}^{\prime }(\unicode[STIX]{x1D719})v_{n}^{\prime }(\unicode[STIX]{x1D719})<-\unicode[STIX]{x1D6FC}_{q}\left\langle \sqrt{u_{B}^{\prime 2}(\unicode[STIX]{x1D719})}\right\rangle \left\langle \sqrt{v_{B}^{\prime 2}(\unicode[STIX]{x1D719})}\right\rangle ,\\ 0\quad & \text{otherwise}.\end{array}\right. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It should be noted that the threshold level for the unsteady QS technique is referenced against the phase-averaged r.m.s. values of $u_{B}^{\prime }$ and $v_{B}^{\prime }$ , which denote velocity components for the given bin value. The remaining two quadrants are similarly defined. The hyperbolic hole with a value of $\unicode[STIX]{x1D6FC}_{q}=3$ was used to emphasize relatively strong quadrant events. However, conditional measurements using other threshold values (not shown here) showed that the event detection statistics were not overly sensitive to the hyperbolic hole size for a range of $\unicode[STIX]{x1D6FC}_{q}$ values.

The dynamic behaviour of the Q2 and Q4 event contributions to the Reynolds stress for a threshold of $\unicode[STIX]{x1D6FC}_{q}=3$ is shown as a function of $t/T$ in the phase-averaged contour plots in figure 24. Results are shown for three representative locations on the ramp, $x/L=0.86$ , 0.93 and 1.0. During the plasma-off portion of the unsteady boundary layer cycle corresponding to $0\leqslant t/T\leqslant 0.5$ , the Q4 contribution to Reynolds stress is clearly dominant in the near-wall region. Peak values of the contribution to the Reynolds stress from Q2 events are relatively constant and lie above the Q4 contours at each streamwise location.

Figure 24.

Contour plots of the temporal evolution of phase-averaged contributions to the scaled Reynolds stress $(\times 10^{-4})$ from Q2 and Q4 events at three locations along the ramp.

During the plasma-on portion of the cycle ( $0.5<t/T\leqslant 0.75$ ) for which the APG is more severe, the peak value of $\langle -u^{\prime }v_{Q2}^{\prime }/U_{e}^{2}\rangle$ increases in magnitude and also rather suddenly shifts farther away from the wall, and there is very little contribution to the Reynolds stress near the wall from Q2 ejection events. The peak values of $\langle -u^{\prime }v_{Q4}^{\prime }/U_{e}^{2}\rangle$ show an increase in magnitude and are spread throughout the lower portion of the boundary layer below the Q2 contours.

It is clear from this series of figures that the Q4 events are the dominant contributors to the Reynolds stress near the wall for the unsteady APG turbulent boundary layer. Perhaps most notable from figure 24 is that, regardless of position, $x/L$ , or phase, $t/T$ , contours of peak $\langle -u^{\prime }v_{Q2}^{\prime }/U_{e}^{2}\rangle$ always lie at $y/\unicode[STIX]{x1D6FF}$ locations just above contours of peak $\langle -u^{\prime }v_{Q4}^{\prime }/U_{e}^{2}\rangle$ ; the Q2 and Q4 contributions to the Reynolds stress preserve this spatial relationship throughout the unsteady pressure gradient cycle. This time-dependent behaviour is similar to that documented in figure 23(b) for the time-mean APG turbulent boundary layer at $x/L=0.93$ with plasma off.

Figure 25.

Phase-averaged velocity, turbulence intensity and Q2 and Q4 Reynolds stress profiles at $x/L=0.86$ for $\unicode[STIX]{x1D6FC}_{q}=3$ and indicated $t/T$ value.

Figure 25 presents phase-averaged profiles of the Q2 and Q4 contributions to the Reynolds stress along with the corresponding phase-averaged turbulence intensity and velocity profiles as obtained at $x/L=0.86$ on the ramp for several representative times $t/T$ during the unsteady cycle. It should be recalled from figure 16 that this $x/L$ location exhibits intermittent transitory detachment ( $\langle \unicode[STIX]{x1D6FE}_{+u}\rangle \approx 0.80$ ) during a portion of the plasma-on part of the cycle. In figure 25, the $y/\unicode[STIX]{x1D6FF}$ locations of peak turbulence intensity, as indicated by the dashed lines, are shown to coincide with the inflection point of the local phase-averaged velocity profile. Furthermore, at this $y/\unicode[STIX]{x1D6FF}$ location, the Q2 and Q4 contributions to the Reynolds stresses have nearly identical values. For time $t/T>0.5$ with the plasma on and the severity of the APG increased, the location of the inflection point, the peak turbulence intensity and the $y/\unicode[STIX]{x1D6FF}$ location associated with matching contributions of Q2 and Q4 Reynolds stress all shift away from the wall in concert. For every $t/T$ value shown, peak Q2 Reynolds stress is located on the high-speed side of the velocity profile inflection point, above the peak turbulence intensity location. In contrast, the peak Reynolds stress due to Q4 sweep events is located on the low-speed side of the inflection point, below the peak turbulence intensity location. In fact, figure 25 shows that the wall-normal variation in the position of contours of peak values of $\langle -u^{\prime }v_{Q2}^{\prime }/U_{e}^{2}\rangle$ and $\langle -u^{\prime }v_{Q4}^{\prime }/U_{e}^{2}\rangle$ during the unsteady cycle shown previously in figure 24 actually reflects the wall-normal motion of the outer inflection point of the phase-averaged velocity profile. It should be noted that this behaviour was also found to be true for the other $x/L$ locations along the ramp.

Figure 26.

Schematic showing how coherent spanwise vorticity associated with an inflectional instability in the embedded shear layer gives rise to Q2 and Q4 peak Reynolds stress events.

Taken together, the results shown in figure 25 are consistent with the presence of an embedded shear layer within the APG boundary layer which contains spanwise-oriented coherent vorticity with clockwise rotation that arises from a Kelvin–Helmholtz inflectional instability. This is shown schematically in figure 26. In this view, the coherent vortical structures associated with the embedded shear layer are responsible for the localized peaks in Q4 Reynolds stress on the low-speed side of the shear layer as vortices transfer comparatively high-velocity fluid above the shear layer to the low-speed side $(u^{\prime }>0,v^{\prime }<0)$ . Similarly, the Q2 Reynolds stress peak on the high-speed side of the shear layer is the result of comparatively low-speed near-wall fluid being transferred across the shear layer to the high-speed side $(u^{\prime }<0,v^{\prime }>0)$ .

The results of the conditional measurements presented in this section suggest that the presence of an embedded shear layer inflectional instability could play a dominant role in the flow physics of the APG turbulent boundary layer. That being the case, consideration is next given to scaling the boundary layer based on embedded shear layer parameters.