2.1 Wind-tunnel facility and model geometry

The experiments are performed inside the working section of a subsonic wind tunnel of $2.4~\text{m}$ width and $2.6~\text{m}$ height. The free-stream velocity $U_{o}$ varies between $10~\text{m}~\text{s}^{-1}$ and $20~\text{m}~\text{s}^{-1}$ with a turbulence intensity of the order of $0.5\,\%$ and no measurable velocity gradients upstream of the bluff body. Figure 2(a) displays a scheme of the bluff body arrangement inside the working section. The front edges of the bluff body are rounded with a radius $R=0.085~\text{m}$ . The whole model with height $H=0.297~\text{m}$ , width $W=0.350~\text{m}$ and length $L=0.893~\text{m}$ is fixed at a geometrical ground clearance of $G=0.05~\text{m}$ above a flat plate floor. The distance between the elliptical leading edge of this false floor and the model is $2.42H$ . The ground clearance $G$ is about five times greater than the oncoming boundary layer, which measures $\unicode[STIX]{x1D6FF}_{99\,\%}\sim 0.034H$ with displacement thickness $\unicode[STIX]{x1D6FF}^{\star }\sim 0.004H$ and shape factor $\overline{H}=1.58$ . To compensate for the lift produced by the whole arrangement, a trailing edge flap inclined with angle $\unicode[STIX]{x1D6FC}_{flap}=5.7\,^{\circ }$ allows us to align the flow parallel to the false floor. The overall blockage ratio of the model in the cross-section above the flat plate is $2.2\,\%$ making blockage corrections unnecessary.

The Reynolds number based on the height of the model is defined as $Re_{H}=U_{o}H/\unicode[STIX]{x1D708}$ , where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the air at ambient temperature. Most of the results are conducted with $U_{o}=15~\text{m}~\text{s}^{-1}$ , corresponding to $Re_{H}=3\times 10^{5}$ . For reference purposes, we introduce a Cartesian coordinate system with $x$ the streamwise, $y$ the transverse or cross-stream and $z$ the spanwise direction. Its origin $O$ is arbitrarily set on the ground at the rear end of the bluff body. Unless otherwise specified, all physical quantities are normalized by $U_{o}$ , $H$ and by the dynamic pressure $q_{o}=0.5\unicode[STIX]{x1D70C}{U_{o}}^{2}$ , where $\unicode[STIX]{x1D70C}$ is the air density at ambient temperature and pressure.

Figure 2.

Experimental set-up. (a) Wind-tunnel flat plate dimensions and model positioning. The details of the model geometry are given by its 3-D view in the inserted picture. (b) Pressure taps locations used to evaluate the base pressure. (c) Particle image velocimetry (PIV) fields of view in the symmetry plane: a large window (A) to perform 2D2C PIV and a smaller window (B) for time-resolved PIV close to the upper shear layer.

2.2 Drag measurements

In order to quantify the effects of actuation on the drag, the model supports are directly fixed on an in-house one component aerodynamic balance with a high sensitive sensor (9217A Kistler piezoelectric, $1~\text{mN}$ upwards). The precision of the force measurements is $\pm 0.07~\text{N}$ . The measured drag force $F_{x}$ is normalized by the frontal area $S=WH$ of the bluff body as follows:

where $C_{x}$ is the drag coefficient.

The $C_{x}$ is calculated from six independent time averages which are all obtained during a single time interval of 10 s by an ETEP^® acquisition system. The acquisition frequency is set to $6.25~\text{kHz}$ and a low-pass filter is applied to obtain the mean drag coefficient. For example, at $Re_{H}=3.0\times 10^{5}$ the drag coefficient of the baseline flow equals $C_{x_{o}}=0.293\pm 0.005$ with $95\,\%$ confidence. Based on the standard deviation from measurements with several configurations, the overall error in the drag coefficient is less than 2 %. All drag measurements are corrected to take into account the thrust of the pulsed jets used for wake forcing. This thrust is evaluated by measuring the reaction force caused by the expansion of the pulsed jets into quiescent air. For the strongest jet blowing used in this study, this thrust achieves approximately 3–4 % of $F_{x}$ at $Re_{H}=3.0\times 10^{5}$ .

To quantify the drag variations with respect to the reference flow, we use the drag parameter $\unicode[STIX]{x1D6FE}_{d}=C_{x}/C_{x_{o}}$ which indicates an increase (respectively decrease) of drag for values greater (respectively smaller) than 1.

2.3 Base pressure coefficient

Pressure drag variations are measured by 17 pressure taps, which are distributed on the rear surface as illustrated in figure 2(b). As large eddy simulations of this bluff body configuration by Östh et al. (Reference Östh, Noack, Krajnović, Barros and Borée2014) demonstrated, both the number and disposition of the pressure taps are sufficient to determine the base pressure distribution along the rear side of the model. The differential offset pressure sensors (HCLA 02X5DB) operate in a range of $\pm 250\,\text{Pa}$ with the upstream static pressure $p_{o}$ as reference. The nominal response delay of these sensors is approximately $0.5\,\text{ms}$ , which enables us to perform unsteady pressure measurements according to Ruiz et al. (Reference Ruiz, Sicot, Brizzi, Laumonier, Borée and Gervais2009). The acquisition frequency of all pressure taps is set to $6.25~\text{kHz}$ .

By using the dynamic head $q_{o}$ , the pressure coefficient reads:

We define $\langle C_{p}\rangle$ and $\overline{C_{p}}$ as the spatial and the time-averaged pressure coefficients respectively taken over the rear surface of the model and during 60 s of acquisition. Although the wake may exhibit long time scales associated with intermittent reversals of the recirculating flow (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013), the spatially and time-averaged pressure coefficient $\langle \overline{C_{p}}\rangle$ should not be affected according to the recent measurements from Volpe, Devinant & Kourta (Reference Volpe, Devinant and Kourta2015).

Based on standard deviations from preliminary long-time acquisitions, the averaged base pressure for the unforced reference flow $\langle \overline{C_{p_{o}}}\rangle$ is measured with a precision of $\pm 0.004$ corresponding to less than $3\,\%$ of $\langle \overline{C_{p}}\rangle$ . Like in the case of drag measurements, we define a pressure parameter $\unicode[STIX]{x1D6FE}_{p}=\langle \overline{C_{p}}\rangle /\langle \overline{C_{p_{o}}}\rangle$ : since the $C_{p}$ values along the base are negative, $\unicode[STIX]{x1D6FE}_{p}\,<\,1$ (respectively $\unicode[STIX]{x1D6FE}_{p}\,>\,1$ ) is associated with an increase (respectively decrease) of pressure.

2.4 Velocity measurements in the wake

Velocity measurements are performed in the wake by the use of particle image velocimetry (PIV) and hot-wire anemometry (HWA). Two PIV fields of view are located in the symmetry plane ( $z=0$ ). The position of these fields is detailed in figure 2(c). A larger field (A) spans the whole wake containing entirely the recirculating flow domain. The second domain (B) corresponds to a zoom on the upper shear layer extending to $x/H\sim 0.4$ .

The largest field of view is used to capture the essential global modifications of the forced wake. Both the streamwise ( $u$ ) and the cross-stream or transverse ( $v$ ) velocity components of the flow are measured by two LaVision Imager pro X 4M cameras with resolution of $2000\times 2000$ pixels. A laser sheet is pulsed with time delays of $120\,\unicode[STIX]{x03BC}\text{s}$ (when $U_{o}=15~\text{m}~\text{s}^{-1}$ ) in the symmetry plane and image pairs are acquired at a sampling frequency of $3.5~\text{Hz}$ . Velocity vectors are processed with an interrogation window of $32\times 32$ pixels with a 50 % overlap. The resulting spatial resolution is approximately $1\,\%$ of the model’s height. Ensembles of 1000–2500 independent velocity fields are used to compute first- and second-order statistics.

The second PIV set-up enables a zoom close to the upper edge of the model with a high-speed PIV (HSPIV). One Photron^® SA-Z 1.1 camera with a resolution of $1024\times 1024$ pixels was used to acquire images at a rate of $10~\text{kHz}$ . The laser sheet is generated by a Quantronix MESA $532~\text{nm}$ system. Most of the tests are performed during $8~\text{s}$ with 80 000 snapshots. The images are processed with an interrogation window of $16\times 16$ pixels and an overlap of 50 %. The final spatial resolution is $0.003H$ .

Hot-wire measurements using a single wire probe (55P11) are acquired by a StreamlinePro Anemometer System (from Dantec Dynamics^®). Typically, they are performed to quantify the boundary layer characteristics and to analyse the spectral content in the wake downstream of the closure of the recirculating flow. The HWA also serves to calibrate the exit velocity of the pulsed jet system. A 55H21 support fixes the probe to a profiled displacement system installed on the roof of the wind tunnel. The velocity measurements are sampled at $6.25~\text{kHz}$ and the duration of each test varies between 60 and $120~\text{s}$ .

Figure 3.

Sketch of the exit zone responsible for emission of the pulsed jets. It is located downstream of the plenum chamber which feeds the solenoid valves with pressurized air. The periodic flow passes through an elbowed conduit until arriving at the exit slit with thickness $h=1~\text{mm}$ . An additional curved surface of radius $r=9~\text{mm}$ can be installed to the set-up in order to produce a Coanda effect.

2.5 Pulsed jet system and forcing parameters

As stated in the Introduction, wake forcing is obtained by unsteady, periodic jet blowing along the edges of the model (Cattafesta & Sheplak Reference Cattafesta and Sheplak2011). The generation of pulsed jets with an exit velocity $V_{j}$ results from the pressure difference between the external flow and a pressurized air reservoir located inside the body. The volume of this cylindrical plenum chamber is 3 l and its internal pressure is called the input pressure $P_{i}$ .

The mass flow is driven periodically by 32 solenoid valves distributed homogeneously along the rear part of the model, upstream of the trailing edges, as illustrated in figure 3 by the geometry details of the jet exit zone. The pulsating frequency $F_{i}$ is selected to be a rectangular waveform with a duty cycle of 40 % and can be settled independently for each trailing edge, but all of them are connected to the same $P_{i}$ . Although the number of solenoid valves is finite, the exit slit with thickness $h=1\pm 0.1~\text{mm}$ is continuous and spans all the periphery of the four edges. The 32 outlets from the solenoid valves are followed by a circular pipe section (I in figure 3). All these pipes are equally spaced along the four trailing edges. In this conduit, the flow is convected along $x$ until arriving at the region (II), where the flow is free to diverge either along $y$ or $z$ . The region (II) plays an important role in obtaining a homogeneous outflow. Upstream of the exit slit, the flow passes through a sharp elbowed geometry. This might introduce some additional vorticity in the boundary layers of the conduit, but an extension of length $7h$ helps to guide the flow up to the exit section.

Figure 4.

Pulsed jet velocity and vorticity field. (a) Velocity signals $V_{j}$ for two configurations corresponding to an input pressure $P_{i}$ of $1.7\,\text{bars}$ and frequency $F_{i}$ of 20 and $610~\text{Hz}$ . (b) Vorticity ( $\unicode[STIX]{x1D714}_{z}$ ) field during the stroke phase of the pulsed jet in quiescent air at an actuation frequency of $F_{i}=20~\text{Hz}$ corresponding to $St_{H}=0.4$ and $\overline{V_{j}}=6.5~\text{m}~\text{s}^{-1}$ . The formation of counter-rotating vortical structures is indicated by the signs of circulation $\unicode[STIX]{x1D6E4}$ . The presence of the Coanda surface deviates the pulsed jets.

Additionally, a Coanda surface can be added just below the jet exit as displayed in the same figure. This surface is a quarter of a disk and is installed along the four trailing edges. In the present study, the radius of this geometry is fixed to $r=9~\text{mm}$ , corresponding to $3\,\%$ of the model’s height. When the Coanda surface is installed, $\langle \overline{C_{p_{o}}}\rangle$ increases $3\,\%$ at $Re_{H}=3.0\times 10^{5}$ . This is due to a slight passive flow deviation in view of the rounded surface (Littlewood & Passmore Reference Littlewood and Passmore2010) and the inherent cavity effect created by the added geometry (Evrard et al. Reference Evrard, Cadot, Herbert, Ricot, Vigneron and Délery2016).

The jet velocity is quantified by a single hot-wire probe $1~\text{mm}$ downstream and at the centreline of the exit slit. The effective jet velocity $V_{eff.}$ and the momentum coefficient $C_{\unicode[STIX]{x1D707}}$ of the jet read:

where $\overline{V_{j}}$ , ${V_{j}}_{Sdv.}$ and $s_{j}$ are the time-averaged jet velocity, its standard deviation and the jet slit cross-sectional area, respectively. Examples of velocity signals for two selected frequencies are reported in figure 4(a).

The effective jet velocity $V_{j_{eff.}}$ takes into account not only the steady component of the jet velocity but also its oscillatory dynamics. Although the exit slit is 2-D due to the designed continuous geometry, the jet velocity presents spanwise variations due to the complex upstream flow conditioning. Measurements of $V_{j_{eff.}}$ along the spanwise direction indicate variations of at most $\pm 10\,\%$ compared to the spanwise-averaged velocity.

The quantities $\overline{V_{j}}$ and $V_{j_{eff.}}$ are calculated from time averaging during a large number of pulsation cycles and taking into account the time when there is no discharge of fluid from the exit slit. Details about the pulsed jet formation are further characterized by the non-dimensional stroke length $L_{o}=1/h\int _{0}^{\unicode[STIX]{x1D70F}}V_{j}(t)\,\text{d}t$ as defined in Glezer & Amitay (Reference Glezer and Amitay2002). More specifically, it is possible to show that $L_{o}=\overline{V_{j}}T_{i}/h$ with the pulsating period $T_{i}={F_{i}}^{-1}$ (Smith & Glezer Reference Smith and Glezer1998). As an example, $L_{o}=289\pm 3$ and $L_{o}=14.1\pm 0.2$ for the velocity signals displayed in figure 4(a) respectively when $F_{i}=20~\text{Hz}$ and $F_{i}=610~\text{Hz}$ .

Finally, for spectral analysis, we consider the actuation frequencies in non-dimensional form by defining a Strouhal number based on the model’s geometry $St_{H}=F_{i}H/U_{o}$ . The pulsed jet vorticity field $\unicode[STIX]{x1D714}_{z}$ during the stroke phase is exemplified in figure 4(b) with and without the Coanda surface. The actuation parameters are $St_{H}=0.4$ and $\overline{V_{j}}=6.5~\text{m}~\text{s}^{-1}$ corresponding to $L_{o}=325\pm 3$ .

4.1 Global effects of actuation

Unless otherwise stated, base pressure changes ( $\unicode[STIX]{x1D6FE}_{p}$ ) are quantified by maintaining the same actuation frequency $St_{H}$ along all edges. The global impact of forcing on $\unicode[STIX]{x1D6FE}_{p}$ is summarized in figure 7(a) for several values of $C_{\unicode[STIX]{x1D707}}$ at a Reynolds number $Re_{H}=3\times 10^{5}$ .

For very low frequencies corresponding to $St_{H}\in [0,0.1]$ , the quasi-steady jet recovers up to $5\,\%$ of the base pressure, comparing well with past numerical (Roumeas, Gilliéron & Kourta Reference Roumeas, Gilliéron and Kourta2009; Wassen et al. Reference Wassen, Eichinger and Thiele2010) and experimental (Krentel et al. Reference Krentel, Muminovic, Brunn, Nitsche and King2010; Littlewood & Passmore Reference Littlewood and Passmore2012) results on steady blowing with similar exit jet velocities $V_{j}$ .

Figure 7.

Base pressure and drag evolution as a function of the actuation parameters. (a) Impact of $St_{H}$ and $C_{\unicode[STIX]{x1D707}}=\{2,3,4,5\}\times 10^{-3}$ on $\unicode[STIX]{x1D6FE}_{p}$ . Changes of the $\overline{C_{p}}$ on the rear surface are indicated for representative configurations. (b) Drag parameter $\unicode[STIX]{x1D6FE}_{d}$ changes for varying actuation frequencies. The drag measurements are performed at a fixed input pressure $P_{i}$ of $1.45\,\text{bar}$ . The colour bar indicates the corresponding values of momentum coefficient for each measured point of the curve. An inserted correlation plot between $\unicode[STIX]{x1D6FE}_{d}$ and $\unicode[STIX]{x1D6FE}_{p}$ indicates that the drag variations correspond to 68 % of the base pressure changes.

This tendency is inverted at higher driving frequencies for $St_{H}\in [0.1,3.7]$ , where the base pressure decreases instead. Two different mechanisms contribute to this rapid increase of drag. Firstly, an abrupt augmentation of $\unicode[STIX]{x1D6FE}_{p}$ at $St_{H}\sim 0.4$ is related to the synchronization of actuation vortices with the oscillatory wake mode $(f_{n_{o}})_{y}$ decreasing the base pressure by up to $25\,\%$ when $C_{\unicode[STIX]{x1D707}}=5\times 10^{-3}$ . At the same time, a significant shortening of the wake length is observed (Barros et al. Reference Barros, Borée, Noack and Spohn2016). Secondly, a range of actuation frequencies around $St_{H}\sim 0.8$ provoke an increase of $\unicode[STIX]{x1D6FE}_{p}$ leading to maximum amplitudes of 1.15 at $St_{H}\sim 0.8$ and $C_{\unicode[STIX]{x1D707}}=5\times 10^{-3}$ . Overall, an increase of the jet amplitude $C_{\unicode[STIX]{x1D707}}$ leads to a drop of base pressure close to $St_{H}\sim 0.8$ .

A further rise of $St_{H}$ beyond 4.0 enables a gradual base pressure recovery. The maximum 16 % raise of $\langle \overline{C_{p}}\rangle$ is achieved by the highest actuation frequency ( $St_{H}=12.1$ ). Interestingly, the numerical simulations from Dahan et al. (Reference Dahan, Morgans and Lardeau2012) illustrated an increase of the step base pressure at high-frequency forcing, in agreement with recent tests conducted on axisymmetric wakes (Oxlade et al. Reference Oxlade, Morrison, Qubain and Rigas2015).

Corresponding drag measurements at different actuation frequencies and amplitudes confirm this trend. The corresponding evolution of $\unicode[STIX]{x1D6FE}_{d}$ is displayed in figure 7(b). For experimental simplicity, the input pressure is set to $P_{i}=1.45\,\text{bar}$ , thus leading to several levels of $C_{\unicode[STIX]{x1D707}}$ as a function of the actuation frequency. This should not invalidate our analysis since the impact of $C_{\unicode[STIX]{x1D707}}$ on $\unicode[STIX]{x1D6FE}_{p}$ appears secondary compared to the influence of the actuation frequency $St_{H}$ at fixed $Re_{H}$ value. By comparing figures 7(a) and 7(b), we remark a strong connection between the evolution of $\unicode[STIX]{x1D6FE}_{p}$ and $\unicode[STIX]{x1D6FE}_{d}$ . Drag alterations of $\pm 10\,\%$ are measured at this Reynolds number. The correlation between these quantities is confirmed by a linear fit displayed in the inserted picture, whose coefficient reveals that 68 % of base pressure changes are converted into drag modifications, in excellent agreement with the numerical findings from Krajnovic & Davidson (Reference Krajnovic and Davidson2003).

4.2 Scaling of drag changes

To gain further insight on the scaling of the drag changes, we present in figure 8(a) the effects of $F_{i}$ on the base pressure at three values of $Re_{H}=\{2,3,4\}\times 10^{5}$ , when actuation is applied along the four edges. Clearly, two domains can be distinguished depending on whether $\unicode[STIX]{x1D6FE}_{p}$ varies with both $F_{i}$ and $Re_{H}$ . First, an increase of $Re_{H}$ enlarges the low-frequency range in which the base pressure decreases ( $\unicode[STIX]{x1D6FE}_{p}>1$ ). However, at actuation frequencies higher than $400~\text{Hz}$ , the evolution of $\unicode[STIX]{x1D6FE}_{p}$ with forcing frequency remains similar for all Reynolds numbers, despite the different magnitudes of base pressure and forcing amplitude ( $C_{\unicode[STIX]{x1D707}}$ ).

Figure 8.

Scaling of drag changes. (a) Reynolds number effect on the base pressure changes for variable forcing frequency $F_{i}$ . (b) Scaling of the low-frequency forcing with constant jet amplitude $V_{j_{eff.}}\sim 7.3~\text{m}~\text{s}^{-1}$ for all slit ( $F_{T,B,R,L}$ ) and single-slit ( $F_{T}$ , $F_{B}$ and $F_{L}$ ) actuation. The base pressure distribution of $\overline{C_{p}}$ for $St_{H}\sim 0.8$ is displayed in the inserts. (c) Effects of $Re_{H}$ and jet amplitude on the base pressure recovery at high-frequency forcing $F_{i}=610~\text{Hz}$ along all slits. The studied Reynolds numbers $Re_{H}=\{2.0,2.5,3.0,3.5,4.0\}\times 10^{5}$ are indicated in the graph. The jet amplitude is described by the stroke length $L_{o}$ and the vertical dashed line indicates its optimum value $L_{o}\sim 12.8$ , corresponding to an effective jet velocity $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ for all upstream conditions.

In the low-frequency range, when actuating with a constant jet amplitude ( $V_{j_{eff.}}\sim 7.3~\text{m}~\text{s}^{-1}$ is displayed here), $\unicode[STIX]{x1D6FE}_{p}$ scales with $St_{H}$ , as shown by the collapsed data of figure 8(b). This scaling remains the same whether the forcing is applied along all edges ( $F_{T,B,R,L}$ ) or only along individual slits $F_{T}$ (top), $F_{B}$ (bottom) and $F_{L}$ (left).

In appendix A, we show that forcing at $St_{H}\sim 0.8$ leads to an increase of the fluctuating velocities along the full recirculating flow. One may therefore wonder if the increase of drag at this actuation frequency is associated with the amplification of instabilities along the shear layers from separation. Most amplified frequencies are usually scaled with the momentum thickness of the boundary layer, such that $St_{\unicode[STIX]{x1D703}}=F_{i}\unicode[STIX]{x1D703}/U_{o}$ , in the range of $St_{\unicode[STIX]{x1D703}}=0.022{-}0.0024$ for turbulent shear layers (Ho & Huerre Reference Ho and Huerre1984; Morris & Foss Reference Morris and Foss2003). Previous results, however, illustrate strong scatter related to such a critical $St_{\unicode[STIX]{x1D703}}$ for which the recirculating flow region appears significantly shortened and the drag is increased. For example, experiments conducted on step flows reported frequencies of approximately $St_{\unicode[STIX]{x1D703}}\sim 0.01$ ( $St_{H}\sim 0.27$ , based on the step height) (Chun & Sung Reference Chun and Sung1996; Dahan et al. Reference Dahan, Morgans and Lardeau2012; Sujar-Garrido et al. Reference Sujar-Garrido, Benard, Moreau and Bonnet2015) while actuation in wake flows present critical values of $St_{\unicode[STIX]{x1D703}}\sim 0.02$ ( $St_{H}\sim 1.8$ ) for axisymmetric geometries (Morrison & Qubain Reference Morrison and Qubain2009; Oxlade et al. Reference Oxlade, Morrison, Qubain and Rigas2015) and $St_{\unicode[STIX]{x1D703}}\sim 0.003{-}0.005$ ( $St_{H}\sim 0.12{-}0.8$ ) for nominally 2-D or 3-D square-back models (Chaligné Reference Chaligné2013; Chaligné et al. Reference Chaligné, Castelain, Michard and Juvé2013). In the present work, $St_{H}\sim 0.8$ scales with an increase of drag for varying $Re_{H}$ , but, when $St_{\unicode[STIX]{x1D703}}$ is calculated, we do not obtain a similar critical scaling for each upstream velocity $U_{o}$ , suggesting that $\unicode[STIX]{x1D703}$ is not the right length scale of the problem. Based on the seminal paper from Oster & Wygnanski (Reference Oster and Wygnanski1982), however, we could conjecture that the forcing time scales are related to the most amplified frequencies further downstream, and this aspect is discussed in appendix A.

An amplification of the vortices emitted by actuation along the shear layer is observed in all studies mentioned above. Consequently, the circulation introduced at each jet slit may induce higher entrainment rates and a change of the mean large-scale recirculating flow orientation, evidenced here by the $\overline{C_{p}}$ distribution for each configuration (inserted pictures). Additional information is provided and discussed in appendix A.

These results also underline the inadequacy of $C_{\unicode[STIX]{x1D707}}$ to describe the effect of the forcing amplitude in the low-frequency range. Indeed, despite $C_{\unicode[STIX]{x1D707}}\propto s_{j}{U_{o}}^{-2}$ at constant $V_{j_{eff.}}$ , variations of $s_{j}$ (by reducing the number of actuated slits from four to one) or $Re_{H}$ (by changing $U_{o}$ ) do not impact $\unicode[STIX]{x1D6FE}_{p}$ significantly. On the other hand, variation of $V_{j_{eff.}}$ at a constant $Re_{H}$ implies changes of $\unicode[STIX]{x1D6FE}_{p}$ (Barros et al. Reference Barros, Borée, Noack and Spohn2016), indicating that the injected pulsed jet vorticity governs the jet amplitude scaling, whose physical details are beyond the scope of the present paper and needs to be further clarified by systematic measurements.

The decrease of drag by high-frequency forcing critically depends on the jet amplitude. In figure 8(c), we display the impact of $C_{\unicode[STIX]{x1D707}}$ when actuation is performed at a forcing frequency $F_{i}=610~\text{Hz}$ ( $St_{H}=12.1$ at $Re_{H}=3.0\times 10^{5}$ ) for variable Reynolds number. Base pressure parameter $\unicode[STIX]{x1D6FE}_{p}$ drops monotonically by increasing $C_{\unicode[STIX]{x1D707}}$ and decreasing $Re_{H}$ up to an optimum. This optimality has been similarly observed in the recent axisymmetric wake measurements from Oxlade et al. (Reference Oxlade, Morrison, Qubain and Rigas2015) at a fixed $Re_{H}$ value. Our results show that the optimum decrease of drag does not scale only with $C_{\unicode[STIX]{x1D707}}$ . Rather, it is measured at a constant stroke length $L_{o}\sim 12.8$ corresponding to an effective jet velocity $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ for all Reynolds numbers $Re_{H}=\{2.0,2.5,3.0,3.5,4.0\}\times 10^{5}$ , indicating that the intrinsic jet dynamics governs the forcing amplitude together with $C_{\unicode[STIX]{x1D707}}$ at high-frequency forcing.

5.1 Overview of the main effects

We proceed by investigating the transient response when actuation is set at $St_{H}=12.1$ and applied with jet amplitude $C_{\unicode[STIX]{x1D707}}=4.6\times 10^{-3}$ ( $L_{o}\sim 12.8$ and $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ ) at $Re_{H}=3\times 10^{5}$ . It is shown in figure 9(a) that the control suddenly increases $\langle C_{p}\rangle$ and the top pressure coefficient $C_{p_{T}}$ located at ( $y/H\!=\!1.05$ , $z\!=\!0$ ), as indicated by their filtered evolution represented by the red lines. A magnified view in the bottom figure details the pressure oscillations associated with the pulsed jet frequency. The pressure magnitude alternates between the unforced and higher values, leading to an $18\,\%$ increase of $\langle \overline{C_{p}}\rangle$ .

Figure 9(b) displays three snapshots of the velocity field during this transition. The unforced shear layer prior to actuation is depicted at the top. A circulatory flow is formed very close to the edge when starting control as shown in the middle picture. The injected circulation induces a notable deviation of the shear layer, quite similar to the flow vectoring effect reported in Smith & Glezer (Reference Smith and Glezer2002) using adjacent synthetic-jet forcing.

Figure 9.

Transient response of high-frequency control applied with $St_{H}=12.1$ ( $L_{o}\sim 12.8$ and $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ ) at $Re_{H}=3\times 10^{5}$ . (a) Time evolution of the base pressure $\langle C_{p}\rangle$ (top) and the pressure coefficient $C_{p_{T}}$ (middle and bottom). (b) Snapshots of the velocity field: unforced flow prior to actuation (top), during actuation stroke (middle) and shear-layer vectoring (bottom). The vectors are coloured by the streamwise velocity $u$ .

In figure 10, we quantify this process by computing the circulation and the cross-stream velocity in the vicinity of the jet slit. The integration of the vorticity field inside a rectangular region $\unicode[STIX]{x1D6FA}$ limited by $(x/H,y/H)=([0,0.03],[1.14,1.17])$ shows an increase of flow circulation $\langle \unicode[STIX]{x1D6E4}\rangle _{\unicode[STIX]{x1D6FA}}$ inside this domain (see figure 10 a). The accumulation of $\langle \unicode[STIX]{x1D6E4}\rangle _{\unicode[STIX]{x1D6FA}}$ induces a deviation of the shear layer, confirmed by the integrated cross-stream velocity $\langle v\rangle _{\unicode[STIX]{x1D6FA}}$ inside an extended domain $\unicode[STIX]{x1D6FA}$ limited by $(x/H,y/H)=([0,0.07],[1.15,1.19])$ displayed in figure 10(b). When the actuation is started, the downward cross-stream velocity significantly increases to values of approximately $8\,\%$ of $U_{o}$ . The fluctuations of both $\langle \unicode[STIX]{x1D6E4}\rangle _{\unicode[STIX]{x1D6FA}}$ and $\langle v\rangle _{\unicode[STIX]{x1D6FA}}$ indicate that the shear layer flaps due to the high-frequency injection of circulation leading to the periodicity of $C_{p_{T}}$ .

Due to the limited resolution of our PIV system, we are not able to identify the core of the pulsed vortex. Its length scale is of the same order as the actuation slit width ( ${\sim}1~\text{mm}$ ). Yet, flow visualizations acquired using this set-up shed some light on the phenomenology of this forcing illustrated in figure 11.

Two counter-rotating vortices emerge from the exit slit at the beginning of the actuation cycle. The corresponding jet structures closely resemble the Schlieren visualizations of synthetic jets during the blowing phase obtained by Smith & Glezer (Reference Smith and Glezer2005). Their averaged diameter scales with the slit width $h\sim 1~\text{mm}$ . Since the convection velocity adjacent to each of these structures is different (i.e. boundary layer at one side and recirculating flow just below the jet), the lower clockwise rotating vortex stays close to the slit outlet while the upper structure is convected with $V_{c}\sim 0.5U_{o}$ . Interestingly, at the beginning of the forthcoming actuation period ( $t+T_{i}$ ), the new pulsed jet interacts with the preceding clockwise vorticity, resulting in a merged structure which is further convected downstream.

Power spectral density ( $S_{v}$ ) of the cross-stream velocity at $(x/H,y/H)=(0.03,1.16)$ reveals vortex pairing by the presence of the first actuation subharmonic $F_{i}/2$ (see insert figure). Similarly, the next clockwise vortex still remains first close to the edge and is convected downstream towards the end of the actuation period. In the following cycle ( $t+2T_{i}$ ), however, there is no interaction between the structures. The entire process leads to a periodic accumulation of circulation close to the edges, promoting shear-layer deviation towards the centre of the separated flow.

Figure 10.

Effects of forcing on the near-edge circulation and the shear-layer vectoring at $Re_{H}=3\times 10^{5}$ . (a) Circulation $\langle \unicode[STIX]{x1D6E4}\rangle _{\unicode[STIX]{x1D6FA}}$ in the domain $\unicode[STIX]{x1D6FA}$ limited by $(x/H,y/H)=([0,0.03],[1.14,1.17])$ adjacent to the exit jet slit. (b) Time response of the integrated cross-stream velocity $\langle v\rangle _{\unicode[STIX]{x1D6FA}}$ over the rectangular domain $\unicode[STIX]{x1D6FA}$ limited by $(x/H,y/H)=([0,0.07],[1.15,1.19])$ .

Figure 11.

Cyclic evolution of the pulsed jet formation at high-frequency forcing. The picture shows the vortex formation and its interactions along two actuation cycles. The power spectral density $S_{v}$ of the cross-stream velocity at $(x/H,y/H)=(0.03,1.16)$ contains peaks at half the forcing frequency which indicates the existence of vortex merging also visible in the flow visualizations.

The resulting time-averaged shear layer is illustrated in figure 12(a) by the streamlines together with the mean cross-flow velocity $\overline{v}$ . An overall increase of the downward velocity magnitude is clearly noted by the flow streamlines. Interestingly, the region of upward flow adjacent to the body surface (indicated by red colour in the maps) is significantly reduced, suggesting reduction of local entrainment along the shear layers and large-scale engulfment at the end of the bubble region.

In figure 12(b), we compute the velocity angle $\unicode[STIX]{x1D6FD}$ along the separating streamlines originated at $(x/H,y/H)=(0,1.17)$ . The reference flow presents low-velocity deviation close to the edge and the streamlines are only slightly directed downwards. The minimum velocity angles of $-4^{\circ }$ are calculated at $x/H\sim 0.3$ . On the other hand, the forcing strongly deviates the flow at the very beginning of the shear layer with angles of $-10^{\circ }$ gradually decreasing along $x$ . We can deduce from this streamwise evolution that significant changes of the velocity field curvature take place, modifying the pressure gradients across the shear layer. As an analogy to the passive control discussed in § 1, this averaged scenario points to a fluidic boat-tailing effect where the pulsating jets deviate flow without geometric modifications, resulting in base pressure recovery and drag reduction. It is worth mentioning here that flow deviation has been also noted along the lower and lateral shear layers (Barros et al. Reference Barros, Ruiz, Borée and Noack2014), and we focused on this effect close to one edge of the model with the high-speed PIV apparatus.

Figure 12.

(a) Streamlines and distribution of time-averaged cross-stream velocity $\overline{v}$ at $Re_{H}=3\times 10^{5}$ . The actuation parameters are $St_{H}=12.1$ and $C_{\unicode[STIX]{x1D707}}=4.6\times 10^{-3}$ ( $L_{o}\sim 12.8$ and $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ ). (b) Velocity angle $\unicode[STIX]{x1D6FD}$ along the separating streamlines for both unforced and forced wakes.

5.2 Influence of the jet amplitude

The flow deviation depends on the introduced circulation close to the edge and is related to the jet strength. We compare the effects of three forcing amplitudes $C_{\unicode[STIX]{x1D707}}=1.7\times 10^{-3}$ ( $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ), $C_{\unicode[STIX]{x1D707}}=4.6\times 10^{-3}$ ( $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ ) and $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ ( $V_{j_{eff.}}\sim 11.6~\text{m}~\text{s}^{-1}$ ) at fixed actuation frequency ( $St_{H}=12.1$ ) and Reynolds number ( $Re_{H}=3\times 10^{5}$ ). Profiles of the $\overline{v}$ velocity, the vorticity $\unicode[STIX]{x1D714}_{z}$ and the cross-stream fluctuations $\overline{v^{\prime }v^{\prime }}$ are compared in figure 13 at two streamwise positions along the upper shear layer ( $x/H=\{0.1,\,0.3\}$ ).

Given the decrease of $\overline{v}$ at $x/H=0.1$ , a deflection of the shear layer is observed for all cases, but it is reduced for the smaller jet amplitude. On the other hand, an increase of $V_{j_{eff.}}$ to $11.6~\text{m}~\text{s}^{-1}$ promotes smaller deviation, in agreement to the optimal value of $V_{j_{eff.}}\sim 9.1~\text{m}~\text{s}^{-1}$ discussed in § 4.2. Further downstream at $x/H=0.3$ , the profiles look similar for these jet amplitudes.

Additionally to the flow deviation, a significant decrease of the minimum time-averaged spanwise vorticity $\unicode[STIX]{x1D714}_{z}$ is measured at $x/H=0.1$ . This is accompanied by an enlargement of the vorticity thickness, being higher for the optimal jet velocity. Both effects may decrease the spatial growth rate of the shear-layer modes in the linear stability framework (Ho & Huerre Reference Ho and Huerre1984), as will be further illustrated by the streamwise evolution of the vorticity (see figure 15). Finally, the increase of $\overline{v^{\prime }v^{\prime }}$ at $x/H=0.1$ is explained by the pulsed jet dynamics close to the slit edge. An interesting feature, however, appears downstream ( $x/H=0.3$ ), where a recovery and even diminution of $\overline{v^{\prime }v^{\prime }}$ along the shear layer is noticed. Hence, the enhanced entrainment due to the formation process of the pulsed jet is followed by a relaxation of velocity fluctuations further downstream.

In figure 13, it is worth mentioning that these three forced wakes differ only $3\,\%$ in base pressure at $Re_{H}=3\times 10^{5}$ . Hence, the modifications of these quantities are not as significant as their changes when compared to the natural flow. Meanwhile, we markedly note a lower vectoring effect and a smaller decrease of peak vorticity for the lowest jet amplitude, both of them responsible for their base pressure variation.

Figure 13.

Influence of jet amplitude for high-frequency actuation ( $St_{H}=12.1$ ) at $Re_{H}=3\times 10^{5}$ . (a–c) (from left to right): profiles at $x/H=0.1$ of the time-averaged cross-stream velocity $\overline{v}$ , time-averaged vorticity $\unicode[STIX]{x1D714}_{z}$ and cross-stream velocity fluctuations $\overline{v^{\prime }v^{\prime }}$ when $V_{j_{eff.}}\sim \{5.6,9.1,11.6\}~\text{m}~\text{s}^{-1}$ corresponding to stroke lengths $L_{o}=\{8.2,12.8,15.1\}$ . (d–f) Similar effects for a downstream streamwise position of $x/H=0.3$ .

Part of the $\langle \unicode[STIX]{x1D6E4}\rangle _{\unicode[STIX]{x1D6FA}}$ circulation previously analysed comes from the pulsed jet circulation $\unicode[STIX]{x1D6E4}_{j}$ . The circulation flux issued from the exit slit may vary as $(\unicode[STIX]{x1D6E4}_{j})_{flux}\propto (V_{j_{eff.}})^{2}$ , further emphasizing that higher jet velocities are capable of inducing more deviated flow. However, the existence of an optimum jet amplitude on the $\unicode[STIX]{x1D6FE}_{p}$ evolution contradicts this tendency. The studies of Oxlade et al. (Reference Oxlade, Morrison, Qubain and Rigas2015) show that this optimality is correlated to a smaller peak vorticity of the jet structures shown in figure 11. It is natural to conjecture that the dynamics of the jet structures might have an additional influence in this process.

Figure 14.

Effects of upstream Reynolds number on the forced shear-layer deviation with $L_{o}\sim 8.2$ ( $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ) and high-frequency actuation. (a) Cross-stream velocity distribution $|\overline{v}|$ and streamlines for varying $Re_{H}$ . (b) Streamwise evolution of the velocity angle $\unicode[STIX]{x1D6FD}$ along the separating streamlines indicated in (a). (c) Cross-stream velocity profile at $x/H=0.02$ presenting higher flow deviation at lower Reynolds number.

5.3 Reynolds number effects and shear-layer growth

In this section, we investigate how actuated flow is affected by varying the upstream Reynolds number $Re_{H}=\{2,3,4\}\times 10^{5}$ : we maintain fixed both the forcing frequency ( $F_{i}=610~\text{Hz}$ ) and the pulsed jet amplitude ( $L_{o}=8.2$ and $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ). This configuration corresponds to a 11 % drag reduction at $Re_{H}=3\times 10^{5}$ .

As shown in figure 14(a), the distribution of $|\overline{v}|$ and the flow streamlines are highly influenced by $Re_{H}$ . The comparison between the unforced (full lines) and forced (dashed lines) streamlines reveals that flow deviation becomes more significant for lower $U_{o}$ . The streamwise evolution of the deviation angle $\unicode[STIX]{x1D6FD}$ shown in figure 14(b) indicates curvature changes of the mean streamlines, which might explain the higher base pressure recovery at lower upstream velocities. In figure 14(c), the increased downward velocity in the initial part of the shear layer is attested by velocity profiles ( $x/H=0.02$ ). In the region $y/H\!>\!1.17$ , the entire boundary layer flow is deviated and $\overline{v}$ is more than doubled when $Re_{H}$ is reduced to one-half. Higher flow deviations at low free-stream velocities were also found in the experiments of Smith & Glezer (Reference Smith and Glezer2002). However, in this former case a main jet was forced by adjacent synthetic jets, in contrast to the present results for a wake flow.

The flow deviation is accompanied by a change of vorticity spread and growth along the shear layer. Figure 15(a) shows the streamwise evolution of the peak vorticity $(\unicode[STIX]{x1D714}_{z})_{min}$ . The unforced mean flow vorticity decays as $\unicode[STIX]{x1D714}_{z}\propto x^{-0.72}$ , in agreement with turbulent separated boundary layer measurements from Morris & Foss (Reference Morris and Foss2003) exhibiting an exponent of $-0.79$ . When control is applied, there is an overall decrease of the peak vorticity confirmed by a profile at $x/H=0.02$ : $(\unicode[STIX]{x1D714}_{z})_{min}$ reduces by more than 30 % at this location, but further downstream ( $x/H>0.25$ ) both evolutions collapse.

A diminution of $(\unicode[STIX]{x1D714}_{z})_{min}$ along $x$ is associated with an enlargement of the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}_{z}}$ , defined as:

In figure 15(b), the streamwise evolution of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}_{z}}/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}_{o}}$ clearly shows a decrease of the shear-layer growth rate, despite the initial increase by forcing of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}_{o}}$ computed very close to the trailing edge at $x/H=0.02$ . The robustness of these results has been attested for all Reynolds numbers (Barros Reference Barros2015).

Figure 15.

Decay of spanwise vorticity along the streamwise direction at $Re_{H}=3\times 10^{5}$ . (a) Decay of the peak vorticity $(\unicode[STIX]{x1D714}_{z})_{min}$ and vorticity profile at $x/H=0.02$ close to the model edge for unforced and actuated flows. The forcing parameters are $St_{H}=12.1$ and $L_{o}\sim 8.2$ ( $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ). (b) Streamwise growth of the vorticity thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}_{z}}$ , where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}_{o}}$ is computed for the unforced and forced configurations separately at $x/H=0.02$ .

This reduction of the growth rate is linked to a damping of the turbulent kinetic energy $k=0.5\times (\overline{u^{\prime }u^{\prime }}+\overline{v^{\prime }v^{\prime }})$ – more precisely, the two contributions to $k$ measured with the PIV system – displayed in figure 16(a). The unforced, natural evolution of the shear layer is associated with a streamwise raise of $k$ . Forcing produces an increase of turbulent kinetic energy up to $x/H\sim 0.13$ , while more downstream $k$ is damped. The initial growth of $\overline{v^{\prime }v^{\prime }}$ is particularly visible in the vicinity of the edge close to the pulsed jet emission.

Figure 16.

Turbulent kinetic energy $k$ and production $\unicode[STIX]{x1D6F1}$ along the upper shear layer. (a) Contour maps of $k$ at $Re_{H}=3\times 10^{5}$ . The inserted picture indicates the cross-stream velocity fluctuations $\overline{v^{\prime }v^{\prime }}$ near the exit slit. (b) Streamwise evolution of $k_{max}$ for varying $Re_{H}$ . The vertical dashed lines are located at $x=0.07,0.13,0.25$ corresponding to the positions where both unforced and actuated flow present the same $k_{max}$ . (c) Cross-stream profiles of turbulent kinetic energy production $\unicode[STIX]{x1D6F1}$ at three streamwise locations for $Re_{H}=3\times 10^{5}$ . The jet parameters are $L_{o}\sim 8.2$ and $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ .

Figure 16(b) illustrates the increase of $k_{max}$ promoted by actuation and successive damping downstream of a critical position, where natural and forced configurations present equivalent values. This location is displaced downstream from $x/H=0.07$ at the lowest Reynolds number up to $x/H=0.25$ at the highest upstream velocity. Further downstream, $k_{max}$ is smaller compared to the unforced flow counterpart, indicating a stabilization of the cross-stream dynamics. In the time-averaged framework, the evolution of $k$ is dictated by the production of turbulent kinetic energy $\unicode[STIX]{x1D6F1}(x,y)$ , whose in-plane contribution is:

Profiles of $\unicode[STIX]{x1D6F1}$ at $Re_{H}=3\times 10^{5}$ are shown in figure 16(c). The selected streamwise positions correspond respectively to the beginning of the shear layer, where the actuation significantly increases $k$ ; the position at which $k_{max}$ is approximately the same for both unactuated and actuated flows; and finally a location where the damping of the velocity fluctuations is clearly visible at the end of the PIV domain. Close to the jet origin, the production term of the actuated flow has more than twice as high peak values compared to the reference case. Then, $\unicode[STIX]{x1D6F1}$ is reduced further downstream. The highest reduction of $\unicode[STIX]{x1D6F1}$ of about 30 % is observed for $Re_{H}=\{2,3\}\times 10^{5}$ while for $Re_{H}=4\times 10^{5}$ it is reduced to 15 %.

Similar stabilizing behaviour of shear layers under high-frequency forcing has been previously reported. With the help of linear stability analysis of the time-averaged streamwise velocity profiles, Dandois et al. (Reference Dandois, Garnier and Sagaut2007) found that high-frequency forcing produces a drop in the growth rates for the most amplified shear-layer modes. This is confirmed in the present study by the increased shear-layer thickness. On the other hand, Vukasinovic et al. (Reference Vukasinovic, Rusak and Glezer2010) showed the presence of high-dissipation zones in the vicinity of the actuator and therefore claimed that this zone drains energy from the mean flow leading to a stabilization, suggesting a dissipative small-scale actuation. Our results seem to confirm both analyses for a bluff body wake and indicate that such mechanisms, coupled to the fluidic boat-tailing effect, alter the global balance of the recirculating flow leading to pressure drag reduction.

5.4 Global wake modifications

Up to now, we have focused on the effects of high-frequency forcing on shear-layer deviation and its growth rate close to the edges of the model. A global change of the whole wake dynamics occurs due to the reduction of the shear growth, and an extension of our analysis to the entire recirculating flow appears necessary. We present in figure 17(a) the cross-stream velocity fluctuations $\overline{v^{\prime }v^{\prime }}$ and the turbulent kinetic energy integrated vertically across any selected region of the flow:

Figure 17.

Velocity fluctuations in the near wake and streamwise evolution of the entrainment measure. (a) Contour maps of $\overline{v^{\prime }v^{\prime }}$ for both the upper and the bottom shear layers. Streamwise evolution of $K$ . (b) Streamwise evolution of $\mathscr{V}_{+}$ and $\mathscr{E}$ inside the region of reverse flow up to the end of the recirculating region limited by $x=L_{r}\sim 1.5$ . The forcing conditions are $St_{H}=12.1$ and $L_{o}\sim 8.2$ ( $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ) for an upstream Reynolds number $Re_{H}=3\times 10^{5}$ .

Both quantities are shown in two regions around the top and the bottom shear layers. Not only the reduction of $K$ is observed along the upper part, as reported before using time-resolved PIV, but also along the bottom development of the wake. Similar trends are obtained whatever the Reynolds number, but a more pronounced decrease of velocity fluctuations is confirmed for the lowest upstream velocity corresponding to $Re_{H}=2\times 10^{5}$ .

The reduction of mixing along the shear layers should lead to lower flow entrainment inside the recirculating zone. A physical measure of the entrained reverse flow represents the integral of $|\overline{v}|$ and the total kinetic energy along $y$ inside the domain $\unicode[STIX]{x1D6FA}_{\overline{u}<0}$ :

These quantities are displayed in figure 17(b) at Reynolds number $Re_{H}=3\times 10^{5}$ . Actuation leads to a decrease of the cross-stream motion and the total kinetic energy of the reverse flow. Similar behaviour is measured at other Reynolds numbers. Close to the model, a decrease of $\mathscr{V}_{+}$ appears, highlighting a diminution of the upward flow adjacent to the rear surface. This points to a decrease of the entire recirculating flow intensity and thus of the whole kinetic energy inside the bubble. It is important to remark that the analysed quantities are computed in the symmetry plane of the wake which can only give an idea of the order of magnitude of the true entrained flow, since in reality it is three-dimensional.

Figure 18.

Global wake topology and effects of high-frequency actuation. (a) Streamlines of the in-plane velocity field ( $Re_{H}=3\times 10^{5}$ ). (b) Streamwise velocity contours $\overline{u}=\{-0.25,0.25,0.65\}$ indicating a narrowing of the wake at different Reynolds numbers. The actuation amplitude corresponds to $L_{o}\sim 8.2$ ( $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ).

We turn our attention now to the modifications of the wake topology. The streamlines of the natural and forced wakes at $Re_{H}=3\times 10^{5}$ are depicted in figure 18(a). The limit of the bubbles corresponds to the recirculation length $L_{r}$ . Despite of a base pressure increase of $15\,\%$ , only a very slight increase of the mean recirculation bubble length in the symmetry plane is measured for actuation at $St_{H}=12.1$ with $L_{o}\sim 8.2$ ( $V_{j_{eff.}}\sim 5.6~\text{m}~\text{s}^{-1}$ ). The recirculating bubble length is increased by at most $2\,\%{-}4\,\%$ considering all upstream velocities. In comparison with the low-frequency actuation analysed in the appendix A, the studied forcing at $St_{H}\sim 0.8$ leads to a decrease of $12\,\%$ on $\langle \overline{C_{p}}\rangle$ and $18\,\%$ on the bubble length. Surprisingly, we do not observe significant changes of the mean streamlines of the wake despite a drag decrease of $11\,\%$ .

A careful analysis of the streamwise velocity $\overline{u}$ , however, sheds light on the subtle changes of the wake geometry. In figure 18(b), the contour lines corresponding to iso-values of streamwise velocity $\overline{u}=\{-0.25,0.25,0.65\}$ are shown respectively for $Re_{H}=\{2,3,4\}\times 10^{5}$ . We observe a narrowing of the wake, which is more pronounced at $Re_{H}=2\times 10^{5}$ , in agreement with the highest flow deviations measured in the vicinity of the upper edge. The flow deviation along the bottom shear layer appears similar, suggesting an equal deviation by forcing along all slits, also reported with respect to the lateral shear layers (Barros et al. Reference Barros, Ruiz, Borée and Noack2014).

Finally, we study the forcing effects on the global wake mode measured downstream the end of the recirculating bubble. From hot-wire measurements at $x/H=2$ and $x/H=3$ in the symmetry plane ( $y/H=0.9$ ) and in a horizontal plane ( $y/H=0.6$ and $z/H=0.5$ ) parallel to the ground, we conclude that no significant modifications are observed in the shedding frequencies, differently to what has been previously found at lower actuation frequencies (Barros et al. Reference Barros, Borée, Noack and Spohn2016). However, a closer inspection reveals a slight decrease of the peak intensity in the spectrum with control. This indicates that these modes are modestly damped and their detection is only possible in the rear part of the recirculating flow, as observed by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013).

5.5 A conceptual scenario for high-frequency drag reduction

The high-frequency forcing impacts the wake by two mechanisms: an initial flow vectoring narrows the whole wake and a reduction of the shear-layer growth rate diminishes the turbulent kinetic energy and entrainment.

The analysis of flow deviation dynamics produced by actuation highlights the importance of circulation flux in the vicinity of the rear edges, responsible for the deviation of the shear layers. Figure 19(a) shows the trailing edge and its vicinity filled with a patch of fluid containing clockwise circulation during actuation. We assume the flow deviation depends on the enhanced entrainment up to a distance $l_{\unicode[STIX]{x1D6FA}}$ from separation. In this region $\unicode[STIX]{x1D6FA}$ , the injected circulating motion deviates the flow.

The circulation $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FA}}$ induces the cross-stream velocity $v_{ind.}$ causing the flow deviation $\unicode[STIX]{x1D6FD}_{ind.}=(v_{ind.}/U_{o})$ . By considering an analogy to the velocity induced by a potential vortex, we assume $\unicode[STIX]{x1D6FD}_{ind.}\propto \unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FA}}/(U_{o}d)$ , where $d$ is related to the slit centreline position with respect to the outer surface adjacent to the boundary layer.

In particular, $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FA}}$ crucially depends on the organized circulation from each pulsed vortex $\unicode[STIX]{x1D6E4}_{j}$ and its convection outside $\unicode[STIX]{x1D6FA}$ . Following Dabiri (Reference Dabiri2009), we may write $\unicode[STIX]{x1D6E4}_{j}\propto 0.5T_{f}{V_{j_{eff.}}}^{2}$ , where $T_{f}$ is the vortex formation time. The pulsed jet circulation will be convected with $U_{c}\propto \mathscr{F}(U_{\unicode[STIX]{x1D6FF}},U_{r},V_{j})$ dependent on the boundary layer ( $U_{\unicode[STIX]{x1D6FF}}$ ) and the recirculating flow ( $U_{r}$ ) characteristic velocities, both increasing with $U_{o}$ . During a period of actuation $T_{i}$ , the convection distance of vortices is approximately $l=U_{c}T_{i}$ . If $U_{o}$ increases or the actuation period increases, $l$ becomes larger than $l_{\unicode[STIX]{x1D6FA}}$ and the average $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6FA}}$ decreases, causing less flow deviation.

By coupling both the circulation and its convection, the dependence of $\unicode[STIX]{x1D6FD}_{ind.}$ can be expressed as:

Figure 19.

Conceptual sketch of the high-frequency actuation. (a) Deviation effect on the shear layer by injected circulation. (b) Global effects by wake manipulation: reduced entrainment and bubble elongation (I), reduced entrainment coupled to bubble shaping for high-frequency forcing (II) and enhanced entrainment decreasing the bubble length for low-frequency forcing (III).

Keeping all other quantities fixed, the increase of jet velocities up to the critical $V_{j_{eff.}}$ leads to an increase of $\unicode[STIX]{x1D6E4}_{j}$ . From our relation, this is directly linked to more deviation $\unicode[STIX]{x1D6FD}_{ind.}$ causing an amplified fluidic boat-tailing effect. On the other hand, flow is convected more rapidly for higher $U_{o}$ and the pulsed jets leave $\unicode[STIX]{x1D6FA}$ , reducing the total circulation in this region and consequently $\unicode[STIX]{x1D6FD}_{ind.}$ . This is confirmed by the smaller vectoring angles $\unicode[STIX]{x1D6FD}$ measured at higher upstream flow velocities.

From the vortex formation studies with piston–cylinder arrangements, Gharib, Rambod & Shariff (Reference Gharib, Rambod and Shariff1998) and Dabiri (Reference Dabiri2009) showed that $T_{f}\propto hT^{\star }/\overline{V_{j}}$ , where $T^{\star }$ is universal for each pulsating geometry. This implies also that $h$ is implicitly contained in (5.6).

An additional effect of increasing $V_{j}$ is the possibility to provoke vortex pinch-off, avoiding further transfer of circulation to $\unicode[STIX]{x1D6E4}_{j}$ . In this sense we could hypothesize the optimality of $L_{o}$ (or $V_{j_{eff.}}$ ) linked to vortex pinch-off as observed by Gharib et al. (Reference Gharib, Rambod and Shariff1998). Further measurements with systematic varying parameters should be performed to draw a more complete scenario. The recent experimental results from Oxlade et al. (Reference Oxlade, Morrison, Qubain and Rigas2015) at fixed upstream flow conditions support our hypothesis by showing the correlation between the vorticity contained in the main vortex and the optimum drag and jet velocity.

Describing how actuation affects the entire bubble topology is necessary to understand base pressure and drag changes. In general, a decrease of shear-layer growth is related to smaller wake entrainment rates, which elongates the recirculating flow as indicated by sketch (I) in figure 19(b). On the other hand, the initial shear-layer deviation alone would imply not only a narrowing of the wake but also a reduction of the recirculating length. The balance of both effects might explain the very similar wake lengths $L_{r}\sim L_{r_{o}}$ for high-frequency actuation (II), where the reduced velocity fluctuations act to elongate the bubble length. The forced bubble presents a higher aspect ratio $L_{r}/H^{\star }$ by the diminution of $H^{\star }$ from aerodynamic shaping, differently to the low actuation frequencies where an enhancement of the entrainment decreases $L_{r}$ shown by (III) (see appendix A and Barros et al. (Reference Barros, Borée, Noack and Spohn2016)).

Both scenarios are consistent with Roshko’s models of the wake equilibrium and base pressure (Roshko Reference Roshko1955, Reference Roshko1993a ,Reference Roshko b ). In an analogy to the simplified 2-D wake, let us recall the streamwise mean momentum balance along the closing separating line of the bubble revisited by Balachandar, Mittal & Najjar (Reference Balachandar, Mittal and Najjar1997):

where $\boldsymbol{x}$ and $\boldsymbol{y}$ are the normal vectors of the reference system, $\boldsymbol{n}$ the local normal vector to the differential $ds$ of the bubble boundary $\unicode[STIX]{x2202}\mathscr{B}$ . Fluxes of mean momentum across the wake boundary are assumed negligible.

As discussed in Balachandar et al. (Reference Balachandar, Mittal and Najjar1997), the normal and shear contributions of the right-hand side of the equation are both significant for the wake equilibrium, their magnitude varying more or less depending on the bluff body characteristics. When forcing is applied, a variation of base pressure would obey $\unicode[STIX]{x1D6FF}\langle \overline{C_{p_{b}}}\rangle \sim \unicode[STIX]{x1D6FF}(\overline{C_{p_{\unicode[STIX]{x2202}\mathscr{B}}}},\overline{u^{\prime }u^{\prime }},\overline{u^{\prime }v^{\prime }})$ . However, by computing the Reynolds stress terms along $\unicode[STIX]{x2202}\mathscr{B}$ , we note that their variation nearly compensates mutually, as $\overline{u^{\prime }u^{\prime }}$ and $\overline{u^{\prime }v^{\prime }}$ induce, respectively, positive and negative contributions to $\langle \overline{C_{p_{b}}}\rangle$ . Then, at first order, one obtains $\unicode[STIX]{x1D6FF}\langle \overline{C_{p_{b}}}\rangle \sim \unicode[STIX]{x1D6FF}(\overline{C_{p_{\unicode[STIX]{x2202}\mathscr{B}}}})$ .

The problem being reduced to determine $\unicode[STIX]{x1D6FF}(\overline{C_{p_{\unicode[STIX]{x2202}\mathscr{B}}}})$ , one would interpret the pressure along the boundary as a result of large pressure gradients across $\unicode[STIX]{x2202}\mathscr{B}$ , which depend on its local streamline curvature $R$ and the local variation of velocity fluctuations along $\boldsymbol{n}$ :

where $s$ is the coordinate tangential to the streamline and terms negligible compared to $\overline{u_{s}}^{2}$ are neglected (Bradshaw Reference Bradshaw1973). It is clear from this formalism that changes of the wake aspect ratio would imply favourable pressure gradients by modifying $R$ along $\unicode[STIX]{x2202}\mathscr{B}$ . Concomitantly, the damping of velocity fluctuations across $\boldsymbol{n}$ would similarly decrease $(\unicode[STIX]{x2202}\overline{p}/\unicode[STIX]{x2202}n)$ leading to a higher local bubble pressure $\unicode[STIX]{x1D6FF}(\overline{C_{p_{\unicode[STIX]{x2202}\mathscr{B}}}})$ .

6.1 Impact on the base pressure and wake flow

In figure 20(a), the effects of unsteady Coanda forcing are characterized by the evolution of $\unicode[STIX]{x1D6FE}_{p}$ as a function of $St_{H}$ at constant $Re_{H}=3.0\times 10^{5}$ . The Coanda blowing obeys a similar trend when compared to the standard case of straight pulsed jets. The discrete peak with minimal base pressure at $St_{H}\sim 0.4$ is preserved, as shown in the insert. Although the range of frequencies decreasing the base pressure remains comparable to the standard actuation, it is reduced to the interval $St_{H}\in [0.1,1.5]$ . Interestingly, base pressure recovery starts earlier from $St_{H}\sim 2.0$ and extends to $St_{H}\sim 12$ : for the tested jet amplitudes, the Coanda forcing outperforms the straight jets lowering the values of $\unicode[STIX]{x1D6FE}_{p}$ .

Figure 20.

Impact of unsteady Coanda blowing on the base pressure parameter $\unicode[STIX]{x1D6FE}_{p}$ . (a) Comparison between the standard actuation and the Coanda blowing by varying the actuation frequency $St_{H}$ at $Re_{H}=3.0\times 10^{5}$ . The measurements are taken with a fixed input pressure $P_{i}$ of 1.45 bar. (b) Effects of the momentum coefficient $C_{\unicode[STIX]{x1D707}}$ for the steady ( $St_{H}=0$ ) and high-frequency forcing ( $St_{H}=12.1$ ) on $\unicode[STIX]{x1D6FE}_{p}$ (open symbols) and $\unicode[STIX]{x1D6FE}_{d}$ (filled symbols) selected for some configurations.

To quantify the influence of the jet amplitude on $\unicode[STIX]{x1D6FE}_{p}$ , we compare with and without the Coanda effect the steady ( $St_{H}=0$ ) and the high-frequency ( $St_{H}=12.1$ ) blowing configurations with variable $C_{\unicode[STIX]{x1D707}}$ . In addition, figure 20(b) indicates for some specific $C_{\unicode[STIX]{x1D707}}$ the measured drag and $\unicode[STIX]{x1D6FE}_{d}$ . Without curved appendices, steady blowing increases base pressure by approximately $5\,\%$ , reducing the drag by at most $3\,\%$ in the range $C_{\unicode[STIX]{x1D707}}\in [0,0.015]$ , corresponding to blowing velocities $\overline{V_{j}}/U_{o}\in [0,1.1]$ . This performance agrees with drag reductions obtained by previous studies with similar steady jet amplitudes in axisymmetric or square-back models (Freund & Mungal Reference Freund and Mungal1994; Krentel et al. Reference Krentel, Muminovic, Brunn, Nitsche and King2010; Wassen et al. Reference Wassen, Eichinger and Thiele2010). In the case of steady blowing, while no difference is measured up to $C_{\unicode[STIX]{x1D707}}\sim 7.5\times 10^{-3}$ ( $\overline{V_{j}}/U_{o}\sim 0.55$ ), the Coanda effect improves base pressure recovery at higher blowing velocities. For example, with $C_{\unicode[STIX]{x1D707}}=13.1\times 10^{-3}$ corresponding to $\overline{V_{j}}/U_{o}\sim 1.0$ , base pressure increases by $14\,\%$ ( $\unicode[STIX]{x1D6FE}_{p}=0.86$ ) and drag decreases by $6\,\%$ ( $\unicode[STIX]{x1D6FE}_{d}=0.94$ ), which is similar to what has been found by Pfeiffer & King (Reference Pfeiffer and King2012). The curve indicates that further increase of base pressure might be achieved using higher blowing velocities, a tendency in agreement with the measurements of Englar (Reference Englar2001) and Pfeiffer & King (Reference Pfeiffer and King2012).

Using high-frequency Coanda blowing, we reached an even higher base pressure increase of 33 % ( $\unicode[STIX]{x1D6FE}_{p}=0.67$ ) at the largest $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ , and a total drag reduction of $18\,\%$ . Curiously, no saturation of $\unicode[STIX]{x1D6FE}_{p}$ in the high-frequency Coanda actuation exists: one may therefore hypothesize that the shear-layer deviation is mainly dictated by the new imposed boundary condition, modifying the relation to the pulsed jet circulation discussed previously.

The flow deviation imposed by the Coanda effect is illustrated by the contour maps of $\overline{v}$ in figure 21. The unforced flow, together with the steady blowing at $C_{\unicode[STIX]{x1D707}}=13\times 10^{-3}$ ( $\unicode[STIX]{x1D6FE}_{p}\sim 0.85$ ) and the high-frequency blowing at $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ ( $\unicode[STIX]{x1D6FE}_{p}\sim 0.67$ ), are compared. The superiority of the unsteady forcing for flow deviation is clearly visible from both the vorticity $\unicode[STIX]{x1D714}_{z}$ and $\overline{v}$ distributions. Moreover, we note a reduction of the upward velocity close to the rear surface of the model, similarly to the standard high-frequency configuration.

When comparing the steady blowing to the periodic actuation, the jet unsteadiness appears to be particularly important to flow attachment along the rounded surface. Although the unsteady forcing with frequency $St_{H}=12.1$ presents a smaller averaged $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ , the instantaneous jet velocity may reach about three times its mean value, as shown in figure 4(a). Hence, the time evolution of the momentum flux competing with the pressure gradient imposed by the rounded surface plays a significant role on the flow mechanisms leading to the unsteady Coanda effect (Jukes & Choi Reference Jukes and Choi2009). The use of steady blowing at velocities similar to the maximum jet amplitudes during the unsteady cycle would therefore be necessary in order to compare the effects of both actuation on flow attachment. Yet, this was not possible in the present study due to the needed input pressure $P_{i}$ , beyond the limits of the pulsed jet system apparatus. Moreover, generating a steady jet for such control would require a large amount of power, considerably affecting the energy balance. This aspect is commented on in what follows.

Figure 21.

Impact of Coanda effect on the time-averaged cross-stream velocity $|\overline{v}|$ and vorticity $|\unicode[STIX]{x1D714}_{z}|$ fields. Comparison between (a) the unforced flow, (b) the steady forcing $St_{H}=0$ ( $C_{\unicode[STIX]{x1D707}}=13\times 10^{-3}$ ) and (c) the high-frequency Coanda blowing $St_{H}=12.1$ ( $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ ).

Let us describe the transient response of the high-frequency Coanda blowing by the time evolution of the top pressure coefficient $C_{p_{T}}$ depicted in figure 22(a) as well as the corresponding velocity vectors coloured by the streamwise velocity $u$ . The total time window displayed is $0.1~\text{s}$ , which corresponds to 5 convective time units ( $t^{\star }=tU_{o}/H$ ) or 1 shedding cycle from $f_{n_{o}}\sim 0.2$ . The initial raise of $C_{p_{T}}$ takes approximately 6 actuation cycles ( ${\sim}0.01~\text{s}$ ) to be established, after which a decrease of pressure is noted due to the rapid stabilization of $P_{i}$ in the compressed air reservoir. (All the time-averaged measurements presented in this work are performed with at least  $5~\text{s}$ following this stabilization, whose duration is roughly $0.05~\text{s}$ .) This time interval is equivalent to 0.5 wake convective units, indicating that the increase of pressure scales with the rapid wake adjustment by delaying flow separation from the Coanda effect.

The velocity snapshots emphasize the correlation between the increase of base pressure and an abrupt change of the velocity field curvature, as indicated in the right insert of figure 22(a), where the flow starts to deviate following the curved surface. It strongly suggests that the rapid shear-layer deviation is crucial for the base pressure recovery. Considering the transition of $\langle C_{p}\rangle$ in figure 22(b) when control is turned off, we note a very rapid decrease of pressure followed by a slow recovery during approximately 5 convective time units ( $0.1~\text{s}$ ) up to the unforced level, suggesting a period of reestablishment of the initial wake conditions by the detachment of the Coanda surface flow.

Figure 22.

Transient analysis of high-frequency with Coanda effect at $Re_{H}=3.0\times 10^{5}$ . (a) Time response of the top pressure sensor when control is applied at $St_{H}=12.1$ ( $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ ). (b) Transient analysis when control is turned off revealing the time scales of the integrated base pressure $\langle C_{p}\rangle$ response. The horizontal dashed line represents the unforced pressure coefficient from longer time-averaged measurements.

As shown for high-frequency actuation, the smaller shear-layer growth stabilizes the wake velocity fluctuations. In figure 23(a), the integrated turbulent kinetic energy ( $K$ ) shows an overall decrease of velocity fluctuations along the wake with unsteady Coanda actuation. For example, $K$ decreases by more than $20\,\%$ at $x/H\in [0.8,1.5]$ . The total kinetic energy $\mathscr{E}$ in the reverse flow presents a similar behaviour (see figure 23 b), with reductions of up to $30\,\%$ . Finally, the reverse mass flow $\mathscr{M}$ inside the wake, obtained by integrating:

is plotted in figure 23(c). Part of the calculated decrease of $\mathscr{E}$ comes from the streamwise reverse flow intensity, which is also reduced for this flow.

In summary, the unsteady blowing in combination to the Coanda effect contributes to the base pressure not only by amplifying the shear-layer deviation but also by preserving the stabilizing effect of the shear layers from high-frequency forcing, resulting in a less-fluctuating wake with lower entrainment.

Figure 23.

Global measurements in the near wake. Integrated turbulent kinetic energy $K$ (a), total kinetic energy inside the reverse flow region $\mathscr{E}$ (b) and integrated reverse streamwise velocity $\mathscr{M}$ (c). The actuation parameters are $St_{H}=12.1$ and $C_{\unicode[STIX]{x1D707}}=7.5\times 10^{-3}$ with the Coanda effect.

6.2 An evaluation of the flow control energy input

Of course, a power is needed to generate the pulsed jets capable of reducing drag. A relation of this power to the recovered energy from drag reduction must be established for flow control purposes. To this end, a selection of forced wakes is chosen to evaluate the pulsed jet energy and drag variations. Following the energy analysis discussed in former studies (Freund & Mungal Reference Freund and Mungal1994; Choi et al. Reference Choi, Jeon and Kim2008; Pfeiffer & King Reference Pfeiffer and King2012), we may define the power ratio $\unicode[STIX]{x1D701}$ as:

It is worth mentioning that the power necessary to maintain the compressed air $P_{i}$ in the reservoir, as well as the electrical energy spent to actuate the valves, are not taken into account here. We are mainly concerned with the jet mechanical energy compared to the recovered drag power. Table 4 lists the quantities used to compute $\unicode[STIX]{x1D701}$ for several drag reducing configurations using steady or high-frequency (HF) Coanda blowing when $Re_{H}=3.0\times 10^{5}$ .

Table 4.

Power ratio parameter for drag reduction with and without the Coanda effect.

Except the high-momentum steady blowing, all calculated ratios $\unicode[STIX]{x1D701}$ are greater than unity, showing a recovery of energy invested in actuation by drag reduction. Generally, as expected from the base pressure measurements, the highest efficiency corresponds to the HF forcing configurations in the presence or not of the Coanda surface. The best compromise between the recovered and actuation power comes with forcing at low momentum coefficient $St_{H}=12.1$ and $C_{\unicode[STIX]{x1D707}}=1.7\times 10^{-3}$ : the values of $\unicode[STIX]{x1D701}$ are 34 and 38 respectively with the use or not of the Coanda effect. The high values of $\unicode[STIX]{x1D701}$ are linked to the low exit jet velocities in these configurations, which significantly impact $\unicode[STIX]{x1D701}$ due to the cubic dependence on $V_{j}$ . The unsteady Coanda blowing is the configuration with the highest drag reduction of 18 % ( $\unicode[STIX]{x1D6FE}_{d}=0.82$ ), corresponding to an energy saved five times greater than the pulsed jet energy. Hence, it appears to be a promising strategy for further optimization in view of future applications in bluff body drag reduction. Parametric variations of $r$ at different upstream conditions $U_{o}$ would bring more insights on the influence of the attachment point along the surface in order to improve drag and power savings.