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 turbulence intensity of the upstream flow is of the order of $0.3\,\%$ at most operating conditions with flow homogeneity better than $0.5\,\%$. A sketch of the bluff body arrangement inside the working section is given in figure 1(a). The front of the model consists of curved edges rounded with a non-constant radius leading to a smooth curvature transition with the flat side surfaces of the model. This is aimed at minimizing the flow detachment just after the rounded front surface, limiting its impact on the downstream wake flow (Spohn & Gilliéron Reference Spohn and Gilliéron2002). The model with height $H = 0.3\,\text {m}$, width $W = 0.36\,\text {m}$ and length $L = 1\,\text {m}$ (with an aspect ratio $H/W = 0.83$ slightly higher than the original geometry of Ahmed et al. Reference Ahmed, Ramn and Faltin1984) is fixed on a raised floor with a ground clearance $G = 0.05\,\text {m}$, which corresponds to approximately five times the thickness of the turbulent boundary layer upstream of the model. The value of the aspect ratio falls into the ’interfering region’ defined by Grandemange, Gohlke & Cadot (Reference Grandemange, Gohlke and Cadot2013a) where both lateral and vertical bi-modality of the wake should be expected. As observed by Dalla Longa, Evstafyeva & Morgans (Reference Dalla Longa, Evstafyeva and Morgans2019), bi-modality can be observed in both directions when the body is out of ground proximity ($G/H = 0.59$). Nevertheless, in our experiments and other similar configurations (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013a; Bonnavion & Cadot Reference Bonnavion and Cadot2018), only bi-modality aligned on the direction of the larger dimension of the base (horizontal for our set-up) is observed. Only some experiments by Cabitza (Reference Cabitza2015) with also a higher ground clearance ($G/H = 0.56$) capture bi-modality in both directions. The closer ground proximity ($G/H = 0.167$) may therefore be responsible for the inhibition of the vertical bi-modality. Only one of the vertical asymmetric states can be locked, as shown by Barros et al. (Reference Barros, Borée, Cadot, Spohn and Noack2017). The influence of flow blockage above the raised floor was neglected due to a low blockage ratio of $2.2\,\%$. An inclinable flap fixed at an upwards angle of $\alpha = 1^\circ$ ends the raised floor in order to compensate for the lift and the streamwise pressure gradient generated by the whole set-up.

Figure 1. Experimental set-up. (a) Arrangement of the model in the test section. Field-of-view in the vertical plane of symmetry from the particle image velocimetry. (b) Locations of pressure taps on the base. (c) Disposition of the spanwise cylinders of diameter $d$ and length $W$ on top of (left) and under (right) the model to passively perturb the natural equilibrium of the wake.

All the experiments carried out in this work are performed at $U_0 = 25\ \textrm {m}\,\textrm {s}^{-1}$ or $Re_H = U_0 H / \nu = 5 \times 10^5$, where $\nu$ is the kinematic viscosity of the air at operating temperature. The boundary layer at separation at the rear edges of the model is fully turbulent with a characteristic momentum thickness $\theta _0 = 2.2\times 10^{-3}$ m measured by hot-wire anemometry for the top boundary layer and corresponding to $Re_\theta = U_0 \theta / \nu = 3670$. We use conventional notations in the Cartesian coordinate system with $x$, $y$ and $z$ for respectively the streamwise, spanwise and cross-stream or transverse directions (accordingly $\boldsymbol {u} = (u_x,u_y,u_z)$ for the velocity field), with the origin $O$ arbitrarily located on the floor in the vertical plane of symmetry of the model. Unless otherwise specified, all physical quantities are normalized by the appropriate combination of the physical parameters $H$, $U_0$ and $\rho$, the air density at operating conditions. In the remainder of the paper the Reynolds decomposition of a quantity $\chi$ into its time-averaged $\bar {\chi }$ and fluctuating $\chi '$ parts is introduced as $\chi = \bar {\chi } + \chi '$.

2.2. Passive perturbations

Similarly to the technique used by Barros et al. (Reference Barros, Borée, Cadot, Spohn and Noack2017) and Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a), passive perturbations are used to change the orientation of large-scale asymmetry of the wake as sketched in figure 1(c). These passive perturbations consist of cylinders placed along the whole span of the model either under or on top of the model at $x/H = -1$. The size of the cylinders used are $d = 0.053H$ on top and $d = 0.066H$ at the bottom to lock the natural horizontally bi-modal wake in a respectively top- and bottom-oriented asymmetric wake.

2.3. Forcing system

In order to force the wake of the model, a series of solenoid valves are used to generate pulsed jets. A 12 l pressurized air tank is contained inside the model (see figure 1a). By controlling the pressure $p_i$ inside the tank, the magnitude of the forcing, i.e. the exit velocity of the pulsed jets, can be changed, and by controlling the actuation parameters of the solenoid valves the frequency and duty cycle (the fraction of period during which the valve is opened) can be changed. Pulsed jets are issued through 26 slits of $h = 1\,\text {mm}$ thickness and $40\,\text {mm}$ width each separated by $4\,\text {mm}$ and localized at $0.5\,\text {mm}$ of the base edges. Additional curved surfaces of radius of curvature $r = 9h$ are placed flush to the slits in order to take advantage of an unsteady Coanda effect. A detailed sketch of the actuation system is given in figure 2(a) with flow visualization of the formation of small-scale vortical structures with a characteristic size of $h$. In this study high-frequency forcing is applied with a frequency of $f = 1050\,\text {Hz}$ and duty cycle of $0.5$, which corresponds to a Strouhal number based on $H$ of $St_H = fH/U_0 = 12.6$, or on the top boundary layer momentum thickness at separation $\theta$ of $St_{\theta } = f\theta /U_0 = 0.1$. The pair $(r,f)$ used here corresponds to an optimum in the unsteady Coanda effect occurring and the drag reduction achieved when forcing is applied along all the edges of the base (Haffner et al. Reference Haffner, Borée, Spohn and Castelain2020b). It results in an optimized cross-flow momentum near the edges of the base to both shape and reorient the wake. The solenoid valves are driven by four in-house electronic cards allowing us to drive independently each edge of the base. This allows the asymmetric forcing conditions considered in this work, which are displayed in figure 2(b). Each forcing distribution is named $F_X$, where $X$ designates the active edges of the base ($T, B, L$ and $R$ for respectively the top, bottom, left and right edges). As the focus is on permanent wake asymmetries in the vertical direction, only vertically asymmetric forcing distributions are considered here.

Figure 2. Pulsed jets system used for forcing. (a) Arrangement of one solenoid valve, tubing system and curved surface generating the pulsed jets. The inserted particle image velocimetry image visualizes small-scale vortical structures forming at the exit of the slit. (b) Different forcing distributions at the base used in the present study. (c) Phase-averaged velocity profile at the centre of the exit plane of each slit around the base for $p_i = 2.8$ bar. The black line denotes the average of all profiles and the error bar a ${\pm }5\,\%$ deviation from the average maximal pulsed jet velocity $\langle V_{j_{max}} \rangle$. (d) Evolution of $V_{j_{max}}$ with inlet pressure $p_i$ at the top centre slit. (e) Deviation of $V_{j_{max}}$ from the average maximal pulsed jet velocity $\langle V_{j_{max}} \rangle$ along each side of the base. Grey dashed lines denote a ${\pm }5\,\%$ bandwidth.

Hot-wire anemometry measurements were performed at the centre of the exit plane of the slits in order to characterize the forcing conditions. The evolution of the phase-averaged exit velocity $V_j$ at the exit of all slits at $f = 1050$ Hz and $p_i = 2.8$ bar is given in figure 2(c). The exit velocity profile is composed of a main peak followed by a trough. The peak occurs at approximately $t/T \sim 0.15$ and has a well-defined triangular shape. Its amplitude increases with increasing $p_i$, as shown through the evolution of the maximal velocity $V_{j_{max}}$ in figure 2(d). The trough, occurring around $t/T \sim 0.8$, is less pronounced and has more negligible variations in amplitude. Detailed measurements at the centre of all 26 slits have allowed us to quantify the homogeneity of the forcing, which is of primary importance to investigating the effect of forcing on wake asymmetries. The maximal velocity and the root-mean-square velocity at each slit are contained in a band of $\pm 5\,\%$ around the average value between all slits, as indicated in figure 2(e). More details on the forcing apparatus and its characterization can be found in Haffner (Reference Haffner2020). The forcing amplitude is defined by

(2.1)\begin{equation} C_{\mu} = d_c\frac{S_j V_{j_{{max}}}^2}{S U_0^2}, \end{equation}

where $S_j$ is the total section of the slits, $V_{j_{{max}}}$ the peak velocity of the pulsed jets and $S = HW$ the section of the model. Here $d_c$ is the effective duty cycle of the forcing based on the hot-wire anemometry measurements and defined as the relative period over which $V_j > 0$. The forcing apparatus allows for forcing amplitudes in the ranges $[2.5,9.5]\times 10^{-3}$ and $[1,3.6]\times 10^{-2}$ for respectively single-sided and all-sided forcings.

2.4. Pressure measurements

To perform surface pressure measurements on the model, a 64-channel ESP-DTC pressure scanner linked to 1 mm diameter pressure tappings around the model (35 taps on the base; see figure 1b) by $80\,\text {cm}$ long vinyl tubing was used and sampled at $200\,\text {Hz}$ with a range of $\pm 1\,\text {kPa}$. Measurement uncertainty of the system lies below $\pm 1.5$ Pa, which represents less than $2\,\%$ of the mean base pressure coefficient of the unforced flow. Pressure measurements are expressed in terms of the pressure coefficient $C_p$, defined as

(2.2)\begin{equation} C_p = \frac{p-p_0}{0.5\rho U_0^2}. \end{equation}

The reference pressure $p_0$ is taken at $x/H = -2$ above the model by a Pitot tube mounted on the ceiling of the test section. For each configuration studied, pressure measurements are performed over a time window of at least $t = 120\ \textrm {s}$, which corresponds to $10^4$ convective time units $H/U_0$ at $U_0 = 25\ \text {m}\,\text {s}^{-1}$. As the natural unforced flow behind the model presents a lateral bi-modal behaviour on long time scales of order $O(10^3 H/U_0)$ (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013b), this time window is not sufficient to obtain complete statistical convergence. Nevertheless, due to the important number of configurations studied involving all the forcing parameters in this work, this time window was chosen as a compromise to keep a reasonable experiment duration and considered satisfactory regarding the convergence of the mean base pressure based on comparison with longer experiments and on the standard deviation of repeated measurements (less than $2\,\%$ deviation from the mean value for the mean base pressure). Indeed, care was taken for the repeatability of the results by reproducing a couple of forcing experiments. For all the configurations where the flow was investigated in detail, the measurements were conducted for an approximate duration of $t = 350\ \textrm {s}$. To improve the accuracy and the repeatability of the measurements, a new acquisition of the unforced flow is performed before each forcing amplitude sweep for a given forcing distribution. This new unforced flow data is used as a reference for each amplitude of the sweep for the given forcing distribution.

The pressure drag of the model is quantified by the base pressure coefficient

(2.3)\begin{equation} C_{pb} = \frac{1}{N}\sum_{i = 1}^N C_p(y_i,z_i,t), \end{equation}

where $N$ denotes the number of pressure taps on the base of the model. To quantify the changes in base pressure, the base pressure parameter

(2.4)\begin{equation} \gamma_p = \frac{\overline{C_{pb}}}{\overline{C_{pb0}}} \end{equation}

is introduced. It represents the ratio between the forced and unforced time-averaged base pressure coefficients. The latter is always chosen as the corresponding baseline configuration depending on how the flow has been passively perturbed prior to forcing it.

The asymmetry of the wake is characterized by the position of the base centre of pressure (CoP) $(y_b,z_b)$ relative to the centre of the base. The coordinates $y_b$ and $z_b$ are defined as

(2.5a,b)\begin{equation} y_b = \frac{\sum_{i = 1}^N y_i C_p(y_i,z_i,t)}{\sum_{i = 1}^N C_p(y_i,z_i,t)},\quad z_b = \frac{\sum_{i = 1}^N z_i C_p(y_i,z_i,t)}{\sum_{i = 1}^N C_p(y_i,z_i,t)}. \end{equation}

2.5. Aerodynamic force measurements

To quantify the effects of forcing on the drag, the model was directly mounted on a six-component aerodynamic balance (9129 AA Kistler piezoelectric sensors and 5080 A charge amplifier). The balance has been calibrated in-house using known masses and a system of pulleys applying pure forces, pure moments or a combination of both on the balance. A whole volume including the expected application point of the aerodynamics torsor of the model has been covered for calibration by using various lever arm lengths for moments. Total measurement uncertainty is less than $0.6\,\%$ of the full-scale range, which represents less than $1\,\%$ uncertainty on the mean drag force $F_x$ for instance. The pulsed jets system used for forcing induces a small thrust which is included in the drag force measurement. In order to evaluate the contribution of the pulsed jets thrust in the measured drag, each forcing configuration is also tested at quiescent free-stream conditions. At $U_0 = 25\ \textrm {m}\,\textrm {s}^{-1}$ for example, the thrust contribution to the total measured drag $F_x$ is less than $2\,\%$ (at maximum when forcing is applied at the highest velocities). Drag and lift measurements are expressed as non-dimensional aerodynamic coefficients

(2.6a,b)\begin{equation} C_D = \frac{F_x}{0.5\rho U_0^2HW},\quad C_L = \frac{F_z}{0.5\rho U_0^2HW}. \end{equation}

Drag measurements were performed simultaneously to the pressure measurements, leading to similar conclusions concerning their statistical convergence. Similarly to base pressure, the drag parameter

(2.7)\begin{equation} \gamma_D = \frac{\overline{C_D}}{\overline{C_{D0}}} \end{equation}

quantifies the drag changes under forcing.

2.6. Velocity measurements and pressure field reconstruction

Particle image velocimetry (PIV) is used to gain insight into the flow structure with a planar two-component set-up, as shown in figure 1(a). A large field of view covering the whole recirculation region in the wake in the vertical plane of symmetry of the model (plane $y = 0$) is imaged by a LaVision Imager LX $16\ \text {Mpx}$ equipped with a Zeiss Makro-Planar ZF 50 mm lens. A laser light sheet of 1 mm thickness is provided by a Quantel EverGreen $2\times 200\ \textrm {mJ}$ laser and the flow is seeded from downstream of the raised floor by atomization of mineral oil producing $1\ \mathrm {\mu }\textrm {m}$-diameter particles. For each configuration studied, a total of 1000 image pairs are acquired at a rate of ${\sim }4\ \text {Hz}$, which is satisfactory for convergence of second-order statistics. Image pairs are processed with Davis 8.4 with a final interrogation window of $16\times 16$ pixels and overlap of $50\,\%$ leading to a velocity vector each of $1.2\ \text {mm}$ or $0.004H$.

Pressure fields were calculated from the mean PIV velocity fields using a method similar to that used by Oxlade (Reference Oxlade2013) by explicit integration of the two-dimensional Reynolds-averaged momentum equations. Details on and validation of the method can be found in Haffner (Reference Haffner2020).

5.1. Conceptual picture of the main drag-reduction mechanisms

To introduce the flow analysis of the observed drag changes under asymmetric forcing, figure 6 presents a qualitative illustration of the wake flow mechanisms which can be triggered by the forcing for drag reduction. A clear distinction can be made between two different drag-reduction mechanisms: wake symmetrization as observed by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2014), De La Cruz et al. (Reference De La Cruz, Oxlade and Morrison2017b), Li et al. (Reference Li, Borée, Noack, Cordier and Harambat2019) and Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a) (among others) and wake shaping as described by Oxlade et al. (Reference Oxlade, Morrison, Qubain and Rigas2015), Barros et al. (Reference Barros, Borée, Noack, Spohn and Ruiz2016) and Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020b). On the one hand, the wake-shaping mechanism globally increases the pressure through the whole wake by reducing its transverse section with minimal reorganization of the recirculating flow. In this sense, it acts by raising the pressure $C_p$ through the whole near wake with a small influence on the pressure gradients inside the recirculating region. The shaping of the wake directly acts on the pressure gradients across the separatrix in the vicinity of separation. As shown by Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020b), the flow curvature in the vicinity of separation and its inversion have a strong influence on the drag decrease. On the other hand, the mean symmetrization of the wake acts by strongly reorganizing the mean recirculating wake and the pressure gradients inside governing the low-pressure imprint on the base. The symmetrization of the recirculation region allows us to reduce the imprint of the low-pressure region located farther downstream of the base. At the same time, it prevents the development of the interaction mechanism in asymmetric wakes leading to additional base drag, as shown by Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a). In the remainder of this section we will show how the drag changes observed in § 4 under asymmetric forcing can be explained by these two complementary mechanisms. Also, it will be shown how the combination of the baseline wake asymmetry and the forcing distribution strongly influences the drag changes.

Figure 6. Qualitative illustrations of different flow mechanisms leading to the reduction of pressure drag. I: wake shaping and change of flow curvature near the base (red rectangles denote the near separation region where flow curvature effects are preponderant according to Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020b)). II: symmetrization of the wake. In practice both mechanisms can be superimposed.

5.2. Influence of global forcing on the symmetry of the wake

The effect of global forcing $F_{TBLR}$ on the symmetry of the three wakes is first briefly described. Due to the symmetry of global forcing, it is expected to impose only minor changes on the wake symmetry. However, this forcing type represents a baseline for the exclusive action of the wake-shaping mechanism. As shown in figure 7, the initial type of asymmetry remains unchanged under global forcing: permanent vertical symmetry-breaking states and lateral bi-modal dynamics are kept for the configurations $\boldsymbol {T}$, $\boldsymbol {B}$ and $\boldsymbol {N}$. For the bi-modal wake $\boldsymbol {N}$, even if the forcing distribution presents a fair degree of homogeneity, the bi-modal dynamics is still very sensitive to a small deviation from a perfectly homogeneous distribution. This leads to the wake exploring quantitatively more than one of the lateral symmetry-breaking states for certain $C_\mu$. The influence on the pressure drag is only weak as $\overline {C_{pb}}$ differs by less than 2 % between the perfectly bi-modal wake and the wake locked in a lateral symmetry-breaking state (Haffner et al. Reference Haffner, Borée, Spohn and Castelain2020a). The main difference is the increase of the mean asymmetry strength $\overline {R_b}$ with forcing amplitude $C_\mu$, which is defined as $R_b = \sqrt {y_b^2+z_b^2}$. There is a linear increase of the asymmetry strength with $C_\mu$ for all baseline configurations in the direction of the initial asymmetry. A similar trend was found by Oxlade (Reference Oxlade2013) on an axisymmetric bullet-shaped body with high-frequency forcing at the base. This increase of the asymmetry strength $R_b$ suggests that wake shaping leads to more pressure recovery on the high-pressure side of the wake than on the low-pressure side. It will be shown in further discussions in § 6 how it is linked to asymmetries in the wake.

Figure 7. Effect of global forcing $F_{TBLR}$ on the base pressure for $\boldsymbol {T}$, $\boldsymbol {N}$ and $\boldsymbol {B}$ configurations. (a) Sensitivity maps of the CoP position $y_b$ and $z_b$ to the forcing amplitude $C_\mu$. (b) Probability density function $\mathcal {P}(y_b,z_b)$ of the CoP at the maximal base pressure recovery ($C_\mu = 0.024$ for $\boldsymbol {N}$, $C_\mu = 0.034$ for $\boldsymbol {T}$ and $C_\mu = 0.031$ for $\boldsymbol {B}$). (c) Evolution of the mean wake asymmetry $\overline {R_b} = \sqrt {y_b^2+z_b^2}$ with $C_\mu$. Dashed lines indicate linear fits to the evolution.

5.3. Asymmetric forcing of a bi-modal wake

In this section we focus on asymmetric forcing of the natural unperturbed wake presenting lateral bi-modality with a slight mean vertical asymmetry.

Figure 8 shows the base pressure dynamics when the wake is subjected to asymmetric forcing distributions at various forcing amplitudes $C_{\mu }$. In figure 8(a) the evolution of the mean vertical position of the base CoP $\overline {z_b}$ with $C_{\mu }$ is shown for all types of asymmetric forcings considered together with the corresponding trends of $\gamma _p$ and $\gamma _D$ for $\boldsymbol {N}_{{BLR}}$. In this configuration, at minimal $\gamma _p$ and $\gamma _D$, we find the smallest vertical asymmetry (i.e. the smallest value of $|z_b|$). Moreover, from the evolution of $\overline {z_b}$ with the forcing amplitude, at $C_{\mu }$ values above 0.013, a plateau around $\overline {z_b} \sim -0.04$ can be observed, which stands for the locking of the wake in a vertical asymmetric state similar to the unforced wake $\boldsymbol {B}$. Furthermore, the transition from the perfectly vertical symmetric wake to a vertically asymmetric wake occurs very abruptly around $C_{\mu } = 0.01$, which means that finding the right amount of forcing momentum in order to optimally balance the wake depending on the initial degree of asymmetry is an intricate task. Interestingly, when keeping only the bottom edge for forcing $\boldsymbol {N}_{{B}}$, the wake reverses its vertical asymmetry even more quickly so that a minimum in drag or base pressure is not observed over the range of $C_{\mu }$ studied. We can speculate that single-sided forcing $\boldsymbol {N}_{{B}}$ has greater authority on the wake orientation and even lower forcing momentum would be needed to reorient the wake asymmetry. This was not possible with the present set-up since a minimal pressure is needed to completely close the solenoid valves and obtain the pulsed jet presented in figure 2(c), but it remains an interesting question. One could thence expect that an optimal drag reduction depending on $C_{\mu }$ similar to $\boldsymbol {N}_{{BLR}}$ forcing could be obtained with only $\boldsymbol {N}_{{B}}$ forcing at smaller forcing amplitudes. In contrast, when forcing is applied on the opposite side ($\boldsymbol {N}_{{TLR}}$ or $\boldsymbol {N}_{{T}}$), the slight vertical asymmetry already present in the unforced wake is directly enhanced. As a consequence, drag is only increased since the vertical asymmetry is only increased. Surprisingly, a lateral forcing $\boldsymbol {N}_{{LR}}$ leads to a reorientation of the asymmetry in the vertical direction, on the side of the initial mean vertical wake asymmetry, explaining the increase of drag evidenced in this case in the previous section. This observation might explain why $\boldsymbol {N}_{{BLR}}$ has a weaker effect on the change of symmetry of the wake than $\boldsymbol {N}_{{B}}$ and at the same time allows us to get stronger wake shaping for the $C_\mu$ for which mean vertical symmetry is reached. Although the lateral mirror symmetry of this configuration is respected by the forcing, the initial slight vertical mean asymmetry of the wake or the ground proximity could be reasons for such behaviour. This kind of reorientation was not witnessed by Lorite-Díez et al. (Reference Lorite-Díez, Jimenéz-González, Pastur, Martńez-Bazán and Cadot2020) using continuous blowing on an Ahmed body with higher ground clearance and presenting lateral bi-modal wake dynamics with vanishing vertical asymmetry.

Figure 8. Effect of localized forcing on the base pressure of the $\boldsymbol {N}$ configuration. (a) Top: mean vertical position $\overline {z_b}$ of the CoP for all different asymmetric forcings (colour and symbol scheme defined in figure 2b are recalled for clarity). Bottom: focus on the evolution of base pressure $\gamma _p$ and drag $\gamma _D$ parameters with forcing amplitude $C_{\mu }$ for $\boldsymbol {N}_{{BLR}}$ forcing (evolutions for other asymmetric forcings of the $\boldsymbol {N}$ wake can be found in figures 4 and 5). Vertical dashed line shows the optimal drag reduction for $\boldsymbol {N}_{{BLR}}$. (b) Sensitivity maps of the CoP position $y_b$ and $z_b$ to the forcing amplitude $C_\mu$ for $\boldsymbol {N}_{{BLR}}$, $\boldsymbol {N}_{{B}}$ and $\boldsymbol {N}_{{LR}}$. Vertical dashed line shows the optimal drag reduction for $\boldsymbol {N}_{{BLR}}$. (c) Evolution of the base CoP under $\boldsymbol {N}_{{BLR}}$ forcing for particular forcing amplitudes indicated by the roman numbers in (a).

The mean pressure fields $\overline {C_p}$ associated with $\boldsymbol {N}_{{BLR}}$ forcing at given $C_\mu$ are compared in figure 9. In particular, it can be seen how at the minimum of drag (case iii) the wake presents an increased floor-normal symmetry compared with the unforced case (case i). When the forcing amplitude becomes too strong, the initial vertical asymmetry is reversed and the wake qualitatively resembles the unforced bottom-perturbed wake. Moreover, the pressure in the mean low-pressure structure in the wake is clearly lowered, leaving a stronger imprint on the base. The transition is also apparent from the evolution of the position of the flow reattachment point on the base, which is the most vertically balanced at the minimum of drag and evolves to the upper edge as drag is increased. As will be discussed in § 6, the position of this point is informative for the degree of interaction of the recirculating flow with the opposite shear layer and leads to increased drag generation due to the asymmetry. As previously mentioned, after the minimum, $\gamma _p$ and $\gamma _D$ increase rapidly with the forcing amplitude not only because of the appearance of a strong vertical asymmetry, but also because we are now in a configuration analogous to $\boldsymbol {B}_{{BLR}}$ which has been shown to increase the drag of the model. This point is of particular interest as it shows that the importance of the asymmetric state of the wake is twofold. On the one hand, the degree and kind of asymmetry in the wake contributes to the generation of a fair amount of drag, as shown in previous sections. On the other hand, it also dictates the ability of the forcing to provide drag reduction by shaping the wake.

Figure 9. Mean pressure field $\overline {C_p}$ in the near wake superimposed with mean velocity streamlines under $\boldsymbol {N}_{{BLR}}$ forcing for forcing amplitudes indicated by the roman numbers in figure 8(a). Blue square symbols indicate the position of the mean reattachment point on the base.

5.4. Asymmetric forcing of a mean asymmetric wake

We focus in this section on the effect of asymmetric forcing on the two wake configurations $\boldsymbol {T}$ and $\boldsymbol {B}$, which are both locked in a static vertical asymmetry. We aim here at generalizing to a wake with stronger initial vertical asymmetry the relation between drag changes and wake symmetry changes observed previously for the bi-modal wake.

Figure 10 illustrates the evolution with forcing amplitude $C_{\mu }$ of the base pressure under asymmetric forcing. For conciseness, the figure only displays results for the $\boldsymbol {T}$ configuration since the results for the $\boldsymbol {B}$ configuration are very similar upon taking into account the top/bottom symmetry.

Figure 10. Effect of asymmetric forcing on the base pressure of the top-perturbed configuration $\boldsymbol {T}$ ($\boldsymbol {B}$ configurations have qualitatively similar mirror trends and are not shown for conciseness). (a) Evolution of the mean vertical CoP position $\overline {z_b}$ with $C_{\mu }$ under asymmetric forcing. The associated evolutions of $\gamma _p$ are recalled below. Coloured roman numbers indicate the cases presented in (c). A yellow (respectively blue) vertical dashed line indicates the maximal base pressure recovery for $\boldsymbol {T}_{B}$ (respectively $\boldsymbol {T}_{BLR}$). (b) Sensitivity maps of the CoP position $y_b$ and $z_b$ to the forcing amplitude $C_\mu$ for $\boldsymbol {T}_{{BLR}}$, $\boldsymbol {T}_{{TLR}}$, $\boldsymbol {T}_{{B}}$ and $\boldsymbol {T}_{{T}}$. Vertical dashed lines show the maximal base pressure recovery for $\boldsymbol {T}_{{BLR}}$ and $\boldsymbol {T}_{{B}}$. (c) Probability density function of the CoP position $\mathcal {P}(y_b,z_b)$ for specific $C_\mu$ of asymmetric forcings $\boldsymbol {T}_{{TLR}}$, $\boldsymbol {T}_{{BLR}}$ and $\boldsymbol {T}_{{B}}$. Horizontal arrows point towards increasing forcing amplitude $C_{\mu }$.

Forcing along the CoP side of the base ($\boldsymbol {T}_{{TLR}}$ and $\boldsymbol {T}_{{T}}$) results in a monotonic increase of the vertical asymmetry $|z_b|$ with $C_{\mu }$. For $\boldsymbol {T}_{{T}}$, this is consistent with the monotonic $\gamma _p$ and $\gamma _D$ increase observed based on the contribution of asymmetries to the drag. In contrast, for $\boldsymbol {T}_{{TLR}}$, the relation between vertical wake asymmetry strength, base pressure and drag is not monotonic anymore. Indeed, the evolution of base pressure and drag with $C_{\mu }$ exhibits a minimum at a low $C_{\mu } \sim 0.01$ even though the drag reduction is very moderate. This is interpreted as the result of two competing effects: the reinforcement of the wake asymmetry and a global pressure recovery in the wake due to a wake shaping occurring when forcing along three of the four base edges similarly to what was discussed in § 5.2.

In contrast, when forcing along the edge opposite to the CoP ($\boldsymbol {T}_{{BLR}}$ and $\boldsymbol {T}_{{B}}$), the drag of the model decreases notably. Here $\boldsymbol {T}_{{B}}$ results in a clear vertical symmetrization of the wake with increasing forcing amplitude $C_{\mu }$. The sign of $\overline {z_b}$ even changes for the highest forcing amplitudes. For even higher forcing amplitudes, a wake reversal is expected, leading to further drag increase. In all the cases it suggests that asymmetric forcing leads to the existence of an optimal forcing amplitude for which the wake is symmetrized and for which the drag is minimal. Figure 10(b,c) shows the clear change of large-scale base pressure dynamics, where the p.d.f. of the base CoP under single-edge forcing at the highest studied $C_{\mu }$ exhibits a vanishing vertical asymmetry and a restored lateral bi-modality. For the $\boldsymbol {T}$ configuration, the transition between the two lateral symmetry-breaking modes occurs through a perfectly symmetric state, whereas for the $\boldsymbol {B}$ configuration the forcing amplitude only enables transit through a bottom asymmetric state, which is consistent with the mean values of $z_b$ observed.

Concerning $\boldsymbol {T}_{{BLR}}$ (corresponding to the side opposite to the CoP), the evolution of the vertical asymmetry is not monotonic anymore with $C_{\mu }$. In both configurations $\boldsymbol {T}$ and $\boldsymbol {B}$, $|z_b|$ is first increasing with the forcing amplitude for values of $C_{\mu } < 0.015$ and only then a reduction in $|z_b|$ is observed with increasing $C_{\mu }$ as in the single-edge forcing configuration. This is shown on the base CoP p.d.f. and sensitivity maps in figure 10(b,c), where forcing on the side leading to the highest drag reduction leads gradually to the restoration of lateral bi-modal wake dynamics with first an unlocking of the vertical static mode. This initial asymmetry increase is qualitatively similar to what was observed for a global forcing $F_{{TBLR}}$ of the wake and is linked to the wake-shaping mechanism existing which is preponderant for three-sides forcing but less when forcing along a single edge. Interestingly, the maximal base drag reduction occurs around the transition between the vertical static asymmetric wake and the lateral bi-modal wake. As shown in Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a), the interaction between opposite shear layers is the key mechanism for generating drag in asymmetric wakes. The strength of the interaction is directly related to the distance between the shear layers and, thus, a vertical asymmetric wake with distance $H$ between the shear layers will generate more drag than a lateral bi-modal wake with distance $W > H$ between them. This means that a key indicator for the symmetrization of the wake is more related to the type of large-scale dynamics present in the wake rather than the single information of $\overline {z_b}$. This suggests once again the importance of the symmetrization in drag reduction. On the other hand, forcing on the opposite side ($\boldsymbol {T}_{{T}}$ and $\boldsymbol {T}_{{TLR}}$) leads to an increase of the vertical asymmetry and, even more interestingly, to a further stabilization of the base CoP position, which appears through a narrowing of the support of the p.d.f. A drag reduction hence does not directly relate to a reduction of the fluctuations of the base pressure. Here the drag reduction is linked to a reorientation of the asymmetry in the direction of the larger side of the base, which leads to increased fluctuations as the bi-modal dynamics appear.

Under global forcing, Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020b) observed that the main mechanism leading to drag reduction is the inward flow deviation at the edges and the changes in flow curvature along the dividing streamline. We observe indeed a narrowing of the wake resulting from inward flow deviation under global forcing $F_{{TBLR}}$ for both configurations in figure 11 where the separating streamlines in the median plane are compared. For each edge, two streamlines are taken to illustrate the separation. The first one is at the closest location to the model surface available in the PIV measurements (not impacted by near-wall light reflections) and corresponds to a distance to the model surface equivalent to $\theta$ at $x/H = 0$. The second one is at a location farther away from the model surface, in the upper part of the boundary layer corresponding to a distance to the model surface equivalent to $0.6\delta$ at $x/H = 0$, with $\delta$ the mean boundary layer thickness at separation. Nevertheless, we can also point out that there is a similar flow deviation and curvature for $\boldsymbol {T}_{{TLR}}$ and $\boldsymbol {B}_{{TLR}}$ at the top edge and for $\boldsymbol {T}_{{BLR}}$ and $\boldsymbol {B}_{{BLR}}$ at the bottom edge but with fundamentally different effects on the drag and base pressure. This underlines the importance of the inner structure of the recirculating region and fundamental differences brought by the asymmetry to the flow on the different sides of the wake. We thereby aim at showing how the reorganization of the mean inner recirculation in terms of symmetry can explain these different drag variations.

Figure 11. Influence of forcing conditions on separating streamlines on the top and bottom edges of the model. For each edge, two streamlines corresponding to wall-normal locations of ${\sim}\theta$ and ${\sim }0.6\delta$ at $x/H = 0$ illustrate the separation. Panel (a) shows $\boldsymbol {B}$ configurations and (b) shows $\boldsymbol {T}$ configurations. Colours are defined in figure 2(b). Unforced separating streamlines are given in black. Forcing amplitudes are $C_{\mu } = 0.02$ for asymmetric forcings $F_{{TLR}}$ and $F_{{BLR}}$ and $0.028$ for global forcing $F_{{TBLR}}$.

Figure 12 shows the reconstructed mean pressure fields in the vertical plane of symmetry with superimposed streamlines for the same unforced and forced cases as in figure 11. It has to be acknowledged that the turbulent wake past this Ahmed body is of a three-dimensional nature. Nevertheless, our rather two-dimensional reasoning for those previous aspects remains consistent as the wake presents a mean pressure distribution on the base which can be stratified in the vertical direction – for the vertical asymmetric configurations – or vertically homogeneous for the lateral bi-modal configuration but, in any case, with a rather homogeneous horizontal distribution except near the very side edges of the model. Moreover, the two-dimensional approach in the symmetry plane to reconstruct the pressure field gives results very close to the base pressure measurements, further accounting for the consistency of our analysis. In a general manner, where forcing is applied a local zone of intense depression is created around the edge of the model, witnessing the acceleration of the flow around the corner which leads to a narrowing of the wake. Also generally, when forcing is applied, regardless of its localization, the strength of the pressure minima in the near wake is reduced compared with the unforced case. Nevertheless, this wake pressure recovery is not translated in the same way on the base. The main changes occur inside the recirculating wake, where the pressure field is reorganized in different ways depending on the forcing localization. When global $F_{{TBLR}}$ forcing is used, there is a global pressure recovery through the whole wake which is thinner. This pressure recovery is stronger than that observed with asymmetric forcing $F_{{TLR}}$ or $F_{{BLR}}$. However, asymmetric forcings $F_{{TLR}}$ and $F_{{BLR}}$ are responsible for a reorganization of the pressure field in the wake, consistent with the observed changes of base pressure dynamics and asymmetry detailed previously. This suggests using a combination of wake shaping and symmetrization mechanisms to modify the base drag. Forcing on the CoP side ($\boldsymbol {T}_{TLR}$ and $\boldsymbol {B}_{BLR}$) induces a global pressure recovery through the wake but reorganizes the inner recirculating region and its asymmetry. This contributes to giving the low-pressure region in the wake a stronger imprint on the base.

Figure 12. Effect of asymmetric forcing on the mean pressure $\overline {C_p}$ field and the velocity streamlines in the wake of the perturbed steady vertical asymmetric configurations. Panel (a) shows $\boldsymbol {T}$ configurations and (b) shows $\boldsymbol {B}$ configurations. Global forcing $F_{{TBLR}}$ at $C_{\mu } = 0.028$. Forcing amplitudes are $C_{\mu } = 0.02$ for asymmetric forcings $F_{{TLR}}$ and $F_{{BLR}}$. The reattachment point on the base is indicated by a blue square.

6.1. Mechanisms of drag changes

All the different drag changes observed under asymmetric forcing can be explained by extending the conceptual wake model drawn by Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a), which is recalled in figure 13. The main aspects of this asymmetric wake model consist of the recirculating flow $(a)$ issued from one shear layer feeding the recirculation region flow $(d)$ and interacting with the opposite shear layer $(b)$ to trigger its roll-up and the large-scale engulfment $(c)$ of fluid of high momentum. To document this conceptual model, we focus on the mean flow momentum variations in the wake between the forced and unforced wakes. The velocity modulus $|\bar {\boldsymbol {u}}|$ gives a fair quantification of the flow momentum per unit mass in the case of these subsonic flows and is shown in figure 14 for the two baseline configurations $\boldsymbol {T}$ and $\boldsymbol {B}$. Similarly, the difference in velocity modulus $\Delta \bar {U} = |\bar {\boldsymbol {u}}| - |\overline {\boldsymbol {u_0}}|$ between the forced and unforced wakes allows us to quantify the flow momentum changes under forcing. Depending on the asymmetric forcing distribution and the initial wake orientation, the flow momentum is redistributed in quite different ways. The drag changes of the asymmetric wake are very sensitive to the asymmetric forcing distribution for the main reason that forcing is applied in flow regions of rather different nature.

Figure 13. Conceptual model of the drag changes based on the model proposed by Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a). The different flows sketched are $(a)$ the recirculating flow formed by one shear layer in the asymmetry direction, $(b)$ the opposite highly fluctuating shear layer triggered by $(a),\ (c)$ the engulfment flow across this shear layer and $(d)$ the recirculating flow in the separated region. The drag decrease ii (respectively increase iii) occurring when asymmetric forcing is applied on the opposite side of the CoP (respectively on the side of the CoP) is explained through the different redistribution of the momentum injected and deviated by the forcing $(\,f)$.

Figure 14. Effect of asymmetric forcing on the mean velocity modulus difference $\Delta \bar {U}$ between forced and unforced flows in the wake of the perturbed steady vertical asymmetric configurations. Panels (a–c) show $\boldsymbol {T}$ configurations and (d–f) show $\boldsymbol {B}$ configurations. From (a,d) to (c, f): the mean velocity modulus $\bar {U}$ in the unforced flow superimposed with mean streamlines, the mean velocity modulus difference $\Delta \bar {U}$ to the unforced flow for $F_{{TLR}}$ and $F_{{BLR}}$ asymmetric forcings at $C_{\mu } = 0.02$ superimposed with mean streamlines of the forced case. Dashed lines indicate the separatrix of the unforced flows. Encircled numerals ① and ② indicate respectively the lower and higher flow momentum parts of the recirculation region, as discussed in the main text.

When the forcing $(\,f)$ is applied on the side opposite to the base CoP (case ii in figure 13), an important drag decrease is observed. The forcing has the ability to efficiently shape the wake as the free-stream flow (from outside of the separated region) it deviates competes with the cross-flow momentum of the recirculating flow $(a)$. Moreover, the deviation of high-momentum flow from the free stream outside the separated region counteracts the interaction mechanism between the recirculating flow $(a)$ of opposite momentum and the shear layer $(b)$. This results in the weakening of this flow mechanism which controls the asymmetry of the wake, as shown in figure 14, and the asymmetry of the wake is thus balanced. The maps of momentum difference in figure 14 ($F_{BLR}$ in (a–c) and $F_{TLR}$ in (d–f)) evidence the symmetrization of the wake by a transfer of momentum from region ② to region ①. In this sense, the weakened interaction mechanism leads to base drag reduction related also to the symmetrization of the wake. As the wake is symmetrized in the vertical direction, bi-modality in the lateral direction appears. Nevertheless, given the aspect ratio of the base $H/W < 1$, the interaction mechanism is less intense as the distance between opposite shear layers considered is increased. As a consequence, the drag of the bi-modal wake is less than that of the wake with an asymmetry in the vertical direction, as was observed by Bonnavion & Cadot (Reference Bonnavion and Cadot2018) and Haffner et al. (Reference Haffner, Borée, Spohn and Castelain2020a). As a complement, when the wake evolves from a vertical static asymmetry to a lateral bi-modality, it explores a transient symmetric state with reduced base drag (Grandemange et al. Reference Grandemange, Gohlke and Cadot2014; Li et al. Reference Li, Barros, Borée, Cadot, Noack and Cordier2016; Haffner et al. Reference Haffner, Borée, Spohn and Castelain2020a), which further contributes to decreasing the drag.

Conversely, when the forcing flow $(\,f)$ is applied on the base CoP side of the wake (case iii in figure 13b), we observe an important drag increase. The high-momentum flow from the free stream on the CoP side (outside the separated region) is deviated by the Coanda forcing $(\,f)$ and feeds directly the recirculating flow $(a)$, as evidenced in figure 14 (regions of high $\Delta \bar {U}$ in the top shear layer for figure 14(a–c) with $F_{TLR}$ forcing and in the bottom shear layer for figure 14(d–f) with $F_{BLR}$ forcing which extend further towards the recirculating flow). The inner deviation of the mean flow along the base CoP side not only lowers the base pressure by curvature effects but also enhances the recirculating feedback flow $(a)$ and, thus, strengthens the interaction mechanism with the opposite shear layer $(b)$. This leads to a greater flow engulfment $(c)$ and to both increased wake asymmetry and base drag.

The conceptual model completed here also provides a framework for wake asymmetries stemming from ground proximity, for which the interaction between opposite top and bottom shear layers is a key aspect of the wake (Castelain et al. Reference Castelain, Michard, Szmigiel, Chacaton and Juvé2018; Haffner et al. Reference Haffner, Borée, Spohn and Castelain2020a). We provide evidence of the pertinence of the present analysis and discussions on a low-ground-clearance case at $G/H = 0.06$ in the appendix.

The two drag-reducing mechanisms introduced in figure 6, wake shaping and wake symmetrization, thus provide a complementary means of reducing the drag of such bluff bodies. Drag reduction obtained through wake shaping will be significantly more important but will also require more energy input, whereas symmetrization, albeit leading to smaller drag reductions, can be obtained with a larger efficiency by a minimal single-edge forcing. One last aspect of the forcing concerns the global forcing, which is a combination of both situations illustrated in figure 13. As explained, the momentum injected by the forcing is redistributed in different ways depending on the side considered compared with the initial wake asymmetry direction. With global forcing conditions, the behaviour is similar. On the side opposite to the CoP, the forcing will be mostly involved in wake shaping, whereas on the CoP side it will lead to less wake shaping and to feeding of the recirculating flow $(a)$. This might explain why the wake asymmetry is observed to increase even under global forcing as both the side opposite to the CoP undergoes wake shaping and the recirculating flow is partially fed by the forcing. This aspect might explain the trade-off existing between the two drag-reducing mechanisms. In this sense it could give hints about the potential lowest base drag achievable for this kind of body. A consequence of this is that we observed an initially lateral bi-modal wake and an initially vertical asymmetric wake reaching the same optimal base drag under global forcing while keeping similar asymmetry characteristics.

6.2. Outcomes for drag reduction of blunt bodies

From the viewpoint of energetic efficiency of the control, asymmetric control provides a highly efficient drag-reducing mechanism by acting on the symmetry of the wake. This is the result of both localizing the control and the high sensitivity of the symmetry-breaking modes to small amounts of momentum injected by the forcing. Nevertheless, the outcomes in terms of drag changes are very sensitive to blowing location and initial wake asymmetry direction. The initial asymmetric state of the wake has been shown to be strongly sensitive to small geometric changes or changes in flow conditions. For road vehicles under realistic atmospheric conditions, it can impair the drag reduction targeted by asymmetric forcing. From a dynamical point of view, in real conditions, asymmetries can constantly vary as flow conditions do. For instance, for small cross-flow changes, the wake asymmetry can oscillate between lateral bi-modal dynamics and a static symmetry breaking in the lateral direction. This remark points to very interesting research paths involving any form of closed-loop control which would adapt the asymmetric location to the measured wake asymmetry in order to achieve all the drag-reduction potential of asymmetric forcing strategies as proposed in Haffner et al. (Reference Haffner, Mariette, Bideaux, Borée, Eberard, Castelain, Bribiesca-Argomedo, Spohn, Michard and Sesmat2020c) and Mariette et al. (Reference Mariette, Bideaux, Bribiesca-Argomedo, Ebérard, Sesmat, Haffner, Borée, Castelain and Michard2020). There is here an important potential to provide a rather simple adaptive active flow-control strategy which would succeed in efficient drag reduction in dynamical flow conditions.

Of course, the focus of this study was on vertical asymmetries stemming from the lock-in of a symmetry-breaking mode in a static position and on lateral bi-modality when the vertical asymmetry vanishes. Nevertheless, it is expected that these results on control and the associated discussions on drag change mechanisms are of general relevance concerning asymmetries in the wake of such blunt bodies. The lock-in of the symmetry-breaking mode in static asymmetric positions can also occur in the horizontal direction of the base when the model is put in small yaw for instance (Cadot et al. Reference Cadot, Evrard and Pastur2015; Bonnavion & Cadot Reference Bonnavion and Cadot2018; Li et al. Reference Li, Borée, Noack, Cordier and Harambat2019). Our conclusions should be generalizable to such horizontal asymmetries as it only moves the asymmetry in the horizontal direction and turns the problem explored in this work in the orthogonal plane. It should provide control research paths to mitigate these asymmetries as well, as De La Cruz, Brackston & Morrison (Reference De La Cruz, Brackston and Morrison2017a), Li et al. (Reference Li, Borée, Noack, Cordier and Harambat2019) and Lorite-Díez et al. (Reference Lorite-Díez, Jimenéz-González, Pastur, Martńez-Bazán and Cadot2020) explored for the Ahmed body in small yawing conditions.