3.1 Flow without blowing

The wake of the body with the additional rear device without open slits at the base is referred to as the reference case. It exhibits the well-known long-time bistable dynamics due to the $y$-instability as reported in Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a,Reference Grandemange, Gohlke and Cadotb). This behaviour is clearly observed by the random switches of the horizontal component of the pressure gradient $g_{y}$ in figure 4(a). They occur between two well-defined mirrored states denoted $P$ and $N$ in the corresponding probability density function (p.d.f.). The time spent in one state can be as large as a few orders of magnitudes of the convective time scale. The vertical component of the pressure gradient $g_{z}$ remains almost insensitive to the switching dynamics in figure 4(a). Its positive value is introduced by the presence of the body support and the ground. The wake topology corresponding to the $P$ state is obtained in figure 4(b) and reveals the static mode of the wake as reported in the literature. The $N$ state (not shown here) is a mirror symmetry ($y^{\ast }\rightarrow -y^{\ast }$) of the $P$ state.

Table 1.

Reference case (no slits). Mean values of force coefficients, base drag, recirculating bubble length, base pressure gradient modulus of the static mode with components for its $N$ and $P$ states, Strouhal numbers of the periodic modes.

The two periodic modes of the wake are identified in the PSD shown in figure 5 of the local velocity measurements in the wake. In accordance with the wake characterization of Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a), we find a low resonant frequency denoted $St_{y}=0.121$ (captured by the local velocity at $P_{L}$) associated with the global lateral motion and a high resonant frequency denoted $St_{z}=0.175$ (captured by the local velocity at $P_{T}$) associated with the global vertical motion. In addition, the random bistable dynamics in the horizontal direction can be identified in the PSD of $P_{L}$ with the $-1.7$ decay for low values of $St$.

Table 1 summarizes the main global flow characteristics of the reference case such as force coefficients, recirculating bubble length, base drag and mean pressure gradient. Recirculation region extension has been characterized as the maximum downstream location from the body base where $U_{x}^{\ast }\leqslant 0$. All these values are in accordance with previous measurements for similar square-back geometries. However, strict comparison with other published works is meaningless because none of the geometries are identical, with significant variations such as fore-body shape, support design, ground clearance and body dimensions. In this regard, the present drag coefficient $C_{x}^{Ref}=0.371$ obtained with a body length $l^{\ast }=4$ and a ground clearance $c^{\ast }=0.278$ is substantially larger than the $C_{x}=0.274$ obtained by Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a) with a body length $l^{\ast }=3.6$ and a ground clearance $c^{\ast }=0.174$. It is worth mentioning that for both cases the base drag is similar, $C_{B}=0.177$ in the present case and $C_{B}=0.185$ in Grandemange et al. (Reference Grandemange, Gohlke and Cadot2013a). The drag discrepancy can be reasonably explained by the additional length of the cylindrical supports together with the friction introduced by the body extension. We can see in table 1 that the floor and supports produce a negative lift coefficient $C_{z}=-0.055$; the bistable dynamics between the two asymmetric states is responsible for the large fluctuation in the side force coefficient $C_{y}^{\prime }=0.019$ due to the random switches of the $y$ component of pressure gradient between the mean values of both states $G_{y}^{N,Ref}$ and $G_{y}^{P,Ref}$. The mean pressure gradient modulus $G^{Ref}=0.133$ gives the strength of the permanent wake asymmetry.

Figure 6.

Base drag modification for all slit configurations as displayed in figure 2 without blowing versus slit area. The axisymmetric data are from García de la Cruz, Oxlade & Morrison (Reference García de la Cruz, Oxlade and Morrison2017).

We denote by $C_{B0}$ the base drag obtained for each slit configuration as displayed in figure 2 without injection, i.e. $C_{q}=0$. Whatever the configuration, a slight reduction of $C_{B0}$ is obtained from the reference case. This effect due to open slits at the base was also found by Bearman (Reference Bearman1967) for a two-dimensional blunt body and by García de la Cruz et al. (Reference García de la Cruz, Oxlade and Morrison2017) for an axisymmetric blunt body. The relative change of base drag coefficient $(C_{B0}-C_{B}^{Ref})/C_{B}^{Ref}$ is shown in figure 6 against the slit area normalized by the base area, $S_{b}/hw$. The decrease obtained by García de la Cruz et al. (Reference García de la Cruz, Oxlade and Morrison2017) with a perimetric slit on a circular base is also reported in figure 6. The rectangular perimetric slit (configuration 4S) is shown for comparison with the circular perimetric slit. A much smaller relative base drag decrease is observed with the wake of the rectangular base that is approximately $1.5\,\%$ compared to the $4\,\%$ of the circular base. This reduction is interpreted in García de la Cruz et al. (Reference García de la Cruz, Oxlade and Morrison2017) as an effect associated with a decrease of the intensity of the wake asymmetry and likely to be produced by the stabilization of the static mode. In order to check this result in our geometry, table 2 shows base drag and mean pressure gradients for all configurations. It appears that, as a general trend in our accuracy, the asymmetry strength given by $G_{0}$ is rather increased in comparison to the reference value of 0.133 with open slits. However, according to the precision of $\pm 0.005$ for the mean pressure gradient and $\pm 0.001$ for the base drag, these weak effects seem unfortunately at the limit of our measurements, which may lead to unreliable conclusions. A significant effect, however, is detectable on the $z$ component of the pressure gradient for B and T configurations pointing out a local pressure increase on the side of the open slit thus enhancing the vertical component in the T configuration and reducing it in the B configuration.

Table 2.

Passive effect values ($C_{q}=0$) for all configurations.

3.2 Blowing effect on the base drag

The base drag variation $\unicode[STIX]{x0394}C_{B}=C_{B}-C_{B0}$ is shown for all configurations as a function of $C_{q}$ in figure 7(a) and $C_{u}$ in figure 7(b). We recall that the L and R configurations are symmetrically equivalent, explaining why the corresponding curves are superimposed.

Figure 7.

Base drag coefficient variation $\unicode[STIX]{x0394}C_{B}=C_{B}-C_{B0}$ versus (a) the flow rate coefficient $C_{q}$ and (b) the blowing velocity ratio $C_{u}$.

Table 3.

Data extracted from figure 7. Optimal values of base bleed coefficients for maximum base drag reduction $BDR=-\unicode[STIX]{x0394}C_{B}^{min}/C_{B0}$ for all configurations. Parameters of the linear fit in the mass regime, $\unicode[STIX]{x0394}C_{B}=(\unicode[STIX]{x2202}C_{B}/\unicode[STIX]{x2202}C_{q})|_{0}C_{q}$, and in the momentum regime, $\unicode[STIX]{x0394}C_{B}=(\unicode[STIX]{x2202}C_{B}/\unicode[STIX]{x2202}C_{u})|_{\infty }C_{u}+b_{\infty }$.

The same trend is observed for all configurations, characterized by an initial base drag reduction with increasing values of the blowing intensity, followed by a constant-slope increase of base drag. This general behaviour is similar to that obtained by Wood (Reference Wood1964) for two-dimensional bodies implementing rear blowing. Two blowing regimes have to be distinguished: we will refer to these as the mass regime, the small blowing for which the base drag is decreased; and the momentum regime, the larger blowing for which the base drag is increased. Both regimes are fitted with linear regressions whose parameters are given in table 3.

The maximum of base drag reduction, $BDR$ (approximately from $1\,\%$ to $7\,\%$), and the corresponding optimal blowing coefficients, $C_{q}^{Opt}$ and $C_{u}^{Opt}$, summarized in table 3, show a substantial dependence on the blowing configuration. Maximal reductions involve the top or bottom slits whether the configuration is symmetric or not. The B configuration achieves the best reduction. There is no evidence of a simple relation between either the optimal $C_{q}$ or $C_{u}$ and the blowing configurations. This high and non-trivial sensitivity to the blowing location and intensity is investigated in the next sections by studying the interaction of the wake modes with the blowing.

3.3 Static mode

3.3.1 Asymmetric blowing configurations

The two configurations L and R have been investigated identically and the experimental data are, as expected, symmetrically equivalent. For brevity of the paper, we only present results for one configuration, and the results for the other can be deduced by the left to right reflectional symmetry.

Figure 8.

Blowing in L configuration. (a) Sensitivity maps of the Cartesian components, $g_{y}$ and $g_{z}$, of the base pressure gradient. The white dashed line marks the $C_{q}^{Opt}$ value for which the mean base drag $C_{B}$ is minimum. (b,c) Streamlines of the mean velocity field $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}^{\ast N}$ (top, left), $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}$ (top, right) and $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{z}}^{\ast }$ (bottom) superimposed on their corresponding velocity fluctuations $U_{x}^{\ast ^{\prime }N}$ or $U_{x}^{\ast ^{\prime }}$ for (b) $C_{q}^{Opt}=0.0043$ and (c) $C_{q}=0.0185$. The white bands in (b) and (c) indicate the blowing slits.

The sensitivity map of the Cartesian components of the base pressure gradient for the L configuration is displayed in figure 8(a). As introduced in Bonnavion & Cadot (Reference Bonnavion and Cadot2018), it represents the p.d.f. of the gradient component normalized by the most probable value computed from each value of the varying parameter, here the blowing flow rate. The vertical dashed line locates the optimal value of blowing that separates the mass and momentum regime as defined above. The positive (respectively negative) branch corresponds to the $P$ state (respectively $N$ state). In the mass regime, $C_{q}<C_{q}^{Opt}$, the dynamics is bistable, involving both $P$ and $N$ states. From the initial case at $C_{q}=0$ where both states are equiprobable, we see a significant change in their probability of presence as blowing is increased, promoting the $N$ state. The preferential state selection in the mass regime is then opposite to that of the momentum regime. It can be seen that the change of blowing regime is accompanied with a permanent wake reversal towards the $P$ state (figure 8c) that was predominantly in the $N$ state (figure 8b) in the mass regime. The momentum regime is then associated with a selected $P$ state in L configuration, where the blowing jet (clearly identified by its large localized fluctuation $U_{x}^{\ast ^{\prime }}$ in front of the slit) is fixing the steady circular recirculation of the bubble. It is also worth mentioning that there is almost no trace of the blowing jet for the largest blowing in the mass regime reached when $C_{q}=C_{q}^{Opt}$ in figure 8(b). However, a direct effect of the blowing can be identified through the additional linear increase of $g_{y}$ in figure 8(a). The blowing is likely to produce a positive pressure gradient proportional to $C_{q}$ that is not affected by the wake state whether it is $P$ or $N$.

Figure 9.

Blowing in R configuration. (a) Probability of occurrence of the $P$ state and (b) relative variation of base drag for $N$ and $P$ states versus the blowing coefficient. The dashed line marks the $C_{q}^{Opt}$ value for which $C_{B}$ is minimum. (c,d) Contours of the mean streamwise velocity difference $\unicode[STIX]{x0394}U_{x}^{\ast i}=U_{x}^{\ast i}-U_{x0}^{\ast i}$ of states $i=P,N$ superimposed on the mean flow streamlines of the field $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}^{\ast i}$ for (c) $C_{q}=0.0031$ and (d) $C_{q}=0.0055$. White bands in (c) and (d) indicate the blowing slits.

We now turn to the effect of the L or R blowing during the bistable dynamics. Actually, there should be a distinction whether the wake is in an $N$ or $P$ state. Depending on the wake state, the jet blows in either an opposite streamwise velocity region as depicted in figure 8(b, top), or in an adverse streamwise velocity region as depicted in figure 8(c, top). Conditional averaging is thus performed next to distinguish the two cases.

Figure 9 presents results obtained with configuration R. During the bistable dynamics that extend until $C_{q}=6.6\times 10^{-3}$ in figure 9(a), the mass regime preferentially selects the $P$ state with a probability larger than 0.5 while the momentum regime selects the $N$ state. These are actually the expected results for the R configuration, which can be deduced from the mirror symmetry of the above observations concerning the L configuration described in figure 8. The mean base drag and mean velocity field of each state are respectively shown in figure 9(b) and figure 9(c,d). There is no distinction in the base drag of the two states until the blowing coefficient reaches $C_{q}=C_{q}^{Opt}$. Bistable dynamics occur as far as the $P$ state is observable, which extends beyond $C_{q}^{Opt}$ until $C_{q}=6.6\times 10^{-3}$. Around this blowing limit, the velocity fields displayed by the streamlines of the two states in figure 9(d) are globally mirrored by each other, while the corresponding base drag reductions are significantly different. The $P$ state is likely to be in a mass regime with a base drag reduction that is still increasing linearly with the blowing coefficient, while the $N$ state produces a base drag increase characteristic of the momentum regime. We deduce that it is more favourable for base drag reduction to blow in the opposite velocity flow region than in the adverse velocity region. Since the blowing jet is hardly discernible in the mean flows, we superimposed on the streamlines in figure 9(c,d) the difference of the mean streamwise velocity with that of the flow at $C_{q}=0$, say $\unicode[STIX]{x0394}U_{x}^{\ast i}=U_{x}^{\ast i}-U_{x0}^{\ast i}$, where $i=P$ or $N$ denotes the state. The blowing jet is not discernible in the mass regime in figure 9(c) for both states at $C_{q}=0.0031$, indicating that the injected momentum is efficiently mixed (i.e. dissipated) in the asymmetric recirculating bubble independently of the injection location. On the other hand, the jet is clearly observable in figure 9(d) at $C_{q}=0.0055$. For both states it contributes to increase the momentum, obviously more when located in the opposite flow region in the $P$ state in figure 9($d$, bottom) than in the $N$ state in figure 9($d$, top).

Figure 10.

Blowing in L configuration with yaw. (a) Relative variation of base drag versus the injection coefficient for the body yawed at $\unicode[STIX]{x1D6FD}=-3.5^{\circ }$. The case $\unicode[STIX]{x1D6FD}=0^{\circ }$ is added for comparison. (b) Sensitivity map of the horizontal component of the base pressure gradient to blowing with the yaw $\unicode[STIX]{x1D6FD}=-3.5^{\circ }$. In (b) and (c), the dashed line marks $C_{q}^{Opt}=0.0115$ for which the mean base drag $C_{B}$ is minimum.

The favourable configuration (injection in the reversed flow region of the bubble) can be forced by applying a yaw angle $\unicode[STIX]{x1D6FD}$ to the body. It has been previously shown (Volpe, Devinant & Kourta Reference Volpe, Devinant and Kourta2015) that a yaw of $1^{\circ }$ is sufficient to permanently select a $P$ (respectively $N$) state with a positive (respectively negative) yaw without any modification of the base pressure gradient modulus (Bonnavion & Cadot Reference Bonnavion and Cadot2018). Figure 10(a) shows the base pressure variation where the body has been yawed to $\unicode[STIX]{x1D6FD}=-3.5^{\circ }$ to select an $N$ state. As expected, the mass regime is extended by a blowing in the L configuration compared to the unforced wake ($\unicode[STIX]{x1D6FD}=0^{\circ }$). It reaches a larger limit value $C_{q}^{Opt}=0.0115$ and an increased maximum base drag reduction of $7.5\,\%$. The sensitivity map of the horizontal component of the pressure gradient shows that, in the mass regime, the gradient magnitude of the forced $N$ state remains unchanged until bistable dynamics appear due to the compensation effect (Bonnavion & Cadot Reference Bonnavion and Cadot2018) of asymmetries between yaw and local momentum injection at the base.

Figure 11.

Blowing in T configuration. (a) Sensitivity maps of the Cartesian components, $g_{y}$ and $g_{z}$, of the base pressure gradient. The white dashed line marks the $C_{q}^{Opt}$ value for which the mean base drag $C_{B}$ is minimum. (b,c) Streamlines of the mean velocity field $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}^{\ast N}$ (top, left), $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}$ (top, right) and $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{z}}^{\ast }$ (bottom) superimposed on their corresponding velocity fluctuations$U_{x}^{\ast ^{\prime }N}$ or $U_{x}^{\ast ^{\prime }}$ for (b) $C_{q}^{Opt}=0.0102$ and (c) $C_{q}=0.0185$. The white bands in (b) and (c) indicate the blowing slit.

The L or R blowing configuration studied above breaks the same symmetry as the static modes (i.e. left/right symmetry), leading in the momentum regime to the permanent selection of a unique state. Now, we investigate B or T blowing configurations that break the top/bottom symmetry and will show very different interaction with the wake. The sensitivity map of the base pressure gradient in the T configuration is shown in figure 11(a). A bistable dynamics is observed during the entire mass regime. At the maximum of drag reduction, the wake topology (figure 11b) remains very similar to that of the reference flow. The momentum regime is first marked by the disappearance of the $P$ state and then the suppression of the asymmetry in the $y$ direction since the horizontal component $g_{y}$ eventually fluctuates around a unique value equal to zero. This suppression is simultaneously accompanied by a decrease of the gradient component $g_{z}$ in the $z$ direction. It indicates that the blowing at the upper part of the base has rotated the two states to a single state with a vertical asymmetry leading to an $N$ state in the $z$ direction. This is clearly illustrated by the velocity fields in figure 11(c) for the largest blowing coefficient. The $N$ state in the $z$ direction that is forced by blowing in the T configuration actually reproduces the same situation as observed in the momentum regime (figure 8c) with the $P$ state in the $y$ direction selected by blowing in the L configuration. These two cases are deduced from each other by a simple $\unicode[STIX]{x03C0}/2$ rotation.

Figure 12.

Blowing in asymmetric configurations: (a) T, (b) B, (c) L and (d) R. Mean pressure gradient components of the wake states $(G_{y}^{i},G_{z}^{i})$ with $i=N,P$ in mass regime (empty symbols) and momentum regime (filled symbols). The solid line ellipse is the model of Bonnavion & Cadot (Reference Bonnavion and Cadot2018) (see text) and dashed ellipses stand for variations of $20\,\%$ of the gradient magnitude obtained with no blowing. The arrows indicate gradual increase of the blowing coefficient $C_{q}$.

The gradual rotation of the base pressure gradient as the blowing is increased is better quantified in figure 12, which summarizes the base pressure gradient evolution for all asymmetric blowing configurations. Each point of the plot represents the mean gradient of a corresponding wake state, either $P$ or $N$, obtained through conditional averaging. By definition, the $P$ branch develops on the right-hand side of the plot and the $N$ branch on the left-hand side. The continuous ellipse is modelling all the possible pressure gradients of the unstable wake (i.e. dominated by a static mode) that would have been passively selected by steady disturbances of pitch and yaw of the body in the absence of blowing (Bonnavion & Cadot Reference Bonnavion and Cadot2018). This view gives useful information about the wake asymmetry strength measured as the modulus of the gradient. Gradients located on the initial ellipse indicate no modification of the asymmetry strength compared to that naturally developed by the body geometry. However, when the blowing is activated, the wake strength is modified and gradients locate in different magnitude levels, as illustrated in figure 12 (see dashed ellipses), where the arrows represent the gradual increase of the blowing coefficient $C_{q}$. In particular, the mass regime (empty symbols) in the T and B configurations in figure 12(a,b) leads to a decrease of 20 % of the initial mean gradient modulus. However, the momentum regime (filled symbols) in which the rotation occurs shows a significant amplification of the asymmetry likely to be fed by the blowing jet. The same representation of the gradient of the wake states for the L and R blowing configuration (figure 12c,d) indicates as well an attenuation of the asymmetry during the mass regime (approximately 10 %) and an amplification in the momentum regime. The reduction of the asymmetry is mainly ascribed to a decrease of the horizontal component of the gradient for B or T configurations and of the vertical component for L or R configurations.

3.3.2 Symmetric blowing configurations

The two symmetric configurations, obtained with a simultaneous blowing in either the left and right slits (LR) or top and bottom slits (TB), give very similar results. The sensitivity maps of the base pressure gradient, respectively given in figure 13(a) and figure 14(a), indicate that bistable dynamics are conserved as expected by the blowing symmetry. The two $P$ and $N$ states remain also very similar whatever the blowing injection is, as can be seen in figures 13(b,c) and 14(b,c). However, the horizontal gradient component $g_{y}$ reaches a small value for the optimal blowing coefficient in both figures 13(a) and 14(a), showing that maximum base drag reduction is associated with a strength reduction of the asymmetry.

Figure 13.

Blowing in LR configuration. (a) Sensitivity maps of the Cartesian components, $g_{y}$ and $g_{z}$, of the base pressure gradient. The white dashed line marks the $C_{q}^{Opt}$ value for which the mean base drag $C_{B}$ is minimum. (b,c) Streamlines of the mean velocity field $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}^{\ast N}$ (top) and $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{z}}^{\ast }$ (bottom) superimposed on their corresponding velocity fluctuations $U_{x}^{\ast ^{\prime }N}$ or $U_{x}^{\ast ^{\prime }}$ for (b) $C_{q}^{Opt}=0.0054$ and (c) $C_{q}=0.0185$. The white bands in (b) and (c) indicate the blowing slits.

Figure 14.

Blowing in TB configuration. (a) Sensitivity maps of the Cartesian components, $g_{y}$ and $g_{z}$, of the base pressure gradient. The white dashed line marks the $C_{q}^{Opt}$ value for which the mean base drag $C_{B}$ is minimum. (b,c) Streamlines of the mean velocity field $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{y}}^{\ast N}$ (top) and $\boldsymbol{U}_{\boldsymbol{x}\boldsymbol{z}}^{\ast }$ (bottom) superimposed on their corresponding velocity fluctuations $U_{x}^{\ast ^{\prime }N}$ or $U_{x}^{\ast ^{\prime }}$ for (b) $C_{q}^{Opt}=0.0078$ and (c) $C_{q}=0.0185$. The white bands in (b) and (c) indicate the blowing slits.

Figure 15 recaps the base pressure gradient evolution for the two symmetric blowings. We observe an attenuation of the asymmetry strength given by the mean gradient modulus of approximately 10 % for the LR configuration mainly caused by a decrease of the vertical component and 20 % for the TB configuration due to a decrease of the horizontal component.

Figure 15.

Blowing in symmetric configurations: (a) LR and (b) TB. Mean pressure gradient components of the wake states $(G_{y}^{i},G_{z}^{i})$ with $i=N,P$ in mass regime (empty symbols) and momentum regime (filled symbols). The solid line ellipse is the model of Bonnavion & Cadot (Reference Bonnavion and Cadot2018) (see text commenting on figure 12) and dashed ellipses stand for variations of $20\,\%$ of the gradient magnitude obtained with no blowing. The arrows indicate the gradual increase of the blowing coefficient $C_{q}$.

Figure 16.

Effect of blowing on the wake periodic modes in (a) L and (b) T configurations. Contours of PSD of local velocity measured at $P_{L}$, $P_{R}$, $P_{T}$ and $P_{B}$ (see § 2.2) as depicted at the top of the figure and also in figures 8 and 11. The horizontal dashed lines are the values of the Strouhal numbers, $St_{y}=0.121$ and $St_{z}=0.175$, of the reference case. The vertical dashed lines mark the $C_{q}^{Opt}$ value for which the mean base drag $C_{B}$ is minimum.

3.4 Periodic modes

Here, the global periodic modes are studied for steady blowing using the L and T blowing configurations only, each by means of three local point velocity measurements in the wake. Their locations (see § 2.2) are displayed in figure 16 with a sketch view towards the base, but they are also displayed in the velocity fields in figure 8 (L blowing configuration) and figure 11 (T blowing configuration) to specify their relative position in the spatial distribution of the turbulent fluctuations. The PSDs are computed at each location varying the blowing coefficient and displayed as a colour map versus the non-dimensional frequency $St$ and $C_{q}$ in figure 16(a) for the L configuration and in figure 16(b) for the T configuration.

The two shedding frequencies initially observed at $St_{z}\approx 0.121$ and $St_{y}\approx 0.175$ with no blowing are clearly identifiable in figure 16 when the blowing coefficient is increased. We can see that the only significant frequency modification is observed for the well-defined low frequency that increases from 0.121 to approximately 0.15 with wider energy distribution when blowing in the L configuration in figure 16(a). Apart from this effect, the major changes lie in the intensity of the resonance peaks that directly reflects the anisotropy of the fluctuations of the static mode together with its interaction with the local blowing presented in § 3.3. Changes in intensities coincide with the transition between the mass and momentum regime at the optimal blowing $C_{q}^{Opt}$. For the L configuration in figure 16(a), the $N$ state is predominantly present in the mass regime such that the probe $P_{L}$ is more often in a high level of fluctuation as shown by figure 8(b) than $P_{R}$. The situation inverts in the momentum regime, which permanently selects the $P$ state, producing high fluctuation at $P_{R}$ location and smaller at $P_{L}$ location, as can be seen in figure 8(c). For the T configuration, the $\unicode[STIX]{x03C0}/2$ rotation of the static mode from horizontal bistability between $P$ and $N$ states of the mass regime, to a vertical $N$ state at maximum blowing (clearly observable in figure 11), is responsible for the intensity variations in the spectra in figure 16(b). However, the relative locations of the probes with respect to the spatial envelope of the fluctuations do not allow for a direct deduction between figure 11 and figure 16(b), as was the case between figure 8 and figure 16(a) for the L configuration.

3.5 Leakage of the recirculating region

The experimental implementations do not allow for the tracking of the injected mass, but the injected momentum can be assessed by looking at the difference between the mean velocity modulus at a given blowing rate with that of $C_{q}=0$, say $\unicode[STIX]{x0394}U_{xy}^{\ast i}=U_{xy}^{\ast i}-U_{xy0}^{\ast i}$ in the $z=0$ plane and $\unicode[STIX]{x0394}U_{xz}^{\ast }=U_{xz}^{\ast }-U_{xz0}^{\ast }$ in the $y=0$ plane, where $i=P$ or $N$ denotes the state. Both cases will be referred to as mean momentum modifications. We superimpose on the mean flow streamlines in figure 17 the mean momentum modification when blowing in the LR configuration. We only display the optimal $C_{q}^{Opt}$ (figure 17a) and maximal injection $C_{q}=0.0185$ (figure 17b) corresponding to the extreme cases of respectively low and high base drag. The blowing jets are not distinguishable in the mass regime (figure 17a, top), but a momentum reduction is clearly visible outside the bubble. This is an indication that the bubble is sweating low momentum by leaking the continuous injection mass. The bubble size is slightly increased when compared to the dashed curve representing the separatrix obtained with no injection. For the maximum injection coefficient in figure 17(b, top), the bubble size is decreased, and the momentum is significantly increased at the exit of the slits inside the bubble. There are also indications of leakage shown by the momentum decrease just outside the bubble at the right-hand side injection and momentum increases after the bubble closure (also observable in the perpendicular plane in figure 17b, bottom). In both cases, it reveals that momentum from the bubble has mixed with the flow outside the bubble and then been naturally transported downstream: at the right outer edge with high-speed flow and in the wake with low-speed flow.

Figure 17.

Blowing in LR configuration. Mean momentum modification superimposed on the mean streamlines for (a) $C_{q}=C_{q}^{Opt}$ and (b) maximum injection $C_{q}=0.0185$. (a, top) $\unicode[STIX]{x0394}U_{xy}^{\ast i}=U_{xy}^{\ast i}-U_{xy0}^{\ast i}$, (b, bottom) $\unicode[STIX]{x0394}U_{xz}^{\ast }=U_{xz}^{\ast }-U_{xz0}^{\ast }$, $i=P$ or $N$ denotes the state. The dashed curve represents the separatrix of the recirculating bubble with no injection.

The evacuation of the injected mass by the wake is also observable in figure 18 for the TB configuration. The optimal blowing is associated with an overall momentum reduction outside the bubble in figure 18(a) and the maximal blowing to momentum increase after the bubble in figure 18(b, top). The leakage pattern in the momentum regime is dominant in the plane $z=0$ and very different from the LR configuration observed in figure 17(b, top). The optimal blowing at $C_{q}=0.0078$ also produces a larger bubble size (figure 18a) and the maximal blowing a smaller bubble size (figure 18b).

Similar results are obtained for all other configurations, whose main differences lay only in the leakage pattern of the momentum regime. As a conclusion, the path of the injected flow is never detectable in the mean flow displayed with streamlines as ideally sketched in Bearman (Reference Bearman1967) in the theoretical case of a steady wake but mixed and then evacuated in the wake through the unsteady dynamics of the shear layers and bubble closure.

Figure 18.

Blowing in TB configuration. Mean momentum modification superimposed on the mean streamlines for (a) $C_{q}=C_{q}^{Opt}$ and (b) maximum injection $C_{q}=0.0185$. (a, top) $\unicode[STIX]{x0394}U_{xy}^{\ast i}=U_{xy}^{\ast i}-U_{xy0}^{\ast i}$, (b, bottom) $\unicode[STIX]{x0394}U_{xz}^{\ast }=U_{xz}^{\ast }-U_{xz0}^{\ast }$, $i=P$ or $N$ denotes the state. The dashed curve represents the separatrix of the recirculating bubble with no injection.

3.6 Drag and recirculation length

The base drag coefficient variation versus the drag coefficient variation in figure 19(a) deviates slightly from the linear law of slope one. This linear law is expected for a blunt-base geometry when pressure drag is the only source of drag. However, the slope of one is satisfactorily achieved for the mass regime which concerns all the data points from zero to the optimal values of lower drag (figure 19a, inset). A significant curvature is observed when the blowing is further increased above the optimal values (momentum regime) leading to drag increase. This effect is likely to be due to the nonlinear interaction of the blowing jet with the flow at the base, which cannot be assessed in the simple expression of thrust and slit head loss removals as used in (2.6). The plot also points out the good accuracy of the base drag estimation using the four pressure taps, indicating that it is an effective way to access the main characteristics of the base pressure distribution for this geometry.

The bubble recirculating length is correlated to the base drag as can be seen in figure 19(b). The larger the bubble, the smaller the base drag and vice versa. It also indicates that increasing blowing in the mass regime (respectively momentum regime) increases (respectively decreases) the recirculating length. This relationship is illustrated by the results shown in both figures 17 and 18 displaying the shape of the recirculating bubble for lowest (a) and highest (b) base drag of two blowing configurations. The relation between separation length and base drag is a classical result for blunt-base bluff bodies, either two-dimensional (Bearman Reference Bearman1967), axisymmetric (Mariotti, Buresti & Salvetti Reference Mariotti, Buresti and Salvetti2015) or three-dimensional (Grandemange et al. Reference Grandemange, Gohlke and Cadot2013b). The small variations compared to the non-blowing value produce the affine relationship as observed in figure 19(b):

(3.1)$$\begin{eqnarray}\displaystyle C_{B}=C_{B0}-0.185(L_{r}^{\ast }-L_{r0}^{\ast }). & & \displaystyle\end{eqnarray}$$

The aim of (3.1) is to highlight the relation between $C_{B}$ and $L_{r}^{\ast }$ when a steady perimetric blowing (of small momentum) is injected through the base of a three-dimensional blunt body. The correlation may depend on the body geometry and flow conditions. Therefore, we cannot ensure that such a dependence is universal, and a different value of the slope and power law may be found in different configurations.

Figure 19.

Base drag coefficient versus (a) drag coefficient and (b) recirculating bubble length. The subscript 0 denotes values obtain with $C_{q}=0$. In (a), the continuous line is $\unicode[STIX]{x0394}C_{B}=\unicode[STIX]{x0394}\tilde{C}_{x}$ and the inset is a close-up view of the plot showing data in the mass regime only. In (b), the continuous line is $\unicode[STIX]{x0394}C_{B}=-0.185\unicode[STIX]{x0394}L_{r}^{\ast }$.