3.1 The genetic algorithm evolution

Figure 5 shows the cost history through the GA optimization. There, $i$ (abscissa axis) indicates the numbering of each simulated actuation signal, ranked from the lowest to the highest drag value (within one single generation). Each solid line in figure 5 represents one generation, which evolves from the first (, green) to the last (, yellow). For clarity, a symbol every fifth actuation signal is placed along every generation line. The red dashed line (– – –, red) indicates the $C_{d}$ value of the unforced 2-D case, which corresponds well to classical results presented in the literature (Hoerner Reference Hoerner1965). The first noticeable achievement is the large drag reduction brought about by the multi-frequency control. For this 2-D case, a drag reduction at almost 50 % is observed, resembling the drag reduction achievable by an extreme rounding of the front corners (Hoerner Reference Hoerner1965). A deeper comparison of the flow topologies will be addressed in the following section. Looking at the behaviour of the GA process, one can observe that exploration and exploitation are preserved during the optimization. Specifically, one can notice the following: (i) The first generation (, green) already contains drastically lower values of drag compared to the baseline (– – –, red). (ii) After the $20$th generation, only small absolute drag reductions are observed, indicating a good exploitation of the minimum drag value. (iii) The mutation probability coefficient still guarantees a good level of exploration in the search space. The good level of exploration is in fact indicated by a large range of $C_{d}$ values within each and every generation. (iv) The overlapping of the last four lines (from generation 20 to generation 35) can be taken as a good convergence indicator. The case that gives the lowest $C_{d}$ value is therefore selected for further investigation and comparisons.

Figure 5. The evolution cost of the GA optimization procedure. Each line connects the individual $i$ of each generation from the lowest to the highest ranked. Dashed line (– – –, red) indicates the $C_{d}$ of the unactuated case; (, green) connects the individuals of the first generation; (, yellow) connects the individuals of the last generation. Sixty individuals were simulated for each of the 35 generations.

Considering the control laws generated by the GA process, one can analyse the trends of the control variables. By the use of multi-dimensional scaling (MDS) (Mardia, Kent & Bibby Reference Mardia, Kent and Bibby1979), one can visualize high-dimensional data in a low-dimensional feature space. The purpose is to define a distance matrix $C$ able to quantify the relative difference between control laws. If one considers $U_{afc,l}(t)=c_{l}(t)$ and $U_{afc,m}=c_{m}(t)$ as the control laws associated with the $l$th and $m$th individual, it is possible to calculate the difference $C(l,m)$ between two generic control laws as

(3.1)$$\begin{eqnarray}C(l,m)=\unicode[STIX]{x1D6FC}\sqrt{\langle |c_{l}(t)-c_{m}(t)|^{2}\rangle }+|J_{l}-J_{m}|.\end{eqnarray}$$

Here, the first term is the average difference between two control signals, while the second is a penalization based on the cost $J$. The parameter $\unicode[STIX]{x1D6FC}$ is chosen to impose the maximum variation of the first and second terms equal. The reader is referred to Duriez, Brunton & Noack (Reference Duriez, Brunton and Noack2017) for further details.

Figure 6 shows a proximity map of the control laws (for clarity, only the individuals of every fifth generation are shown). Each dot represents a control law, in a 2-D plane ($\unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}$) and the scatterplot is coloured by the cost, which in this case is the $C_{d}$ value. For clarity, the variables $\unicode[STIX]{x1D713}_{1}$ and $\unicode[STIX]{x1D713}_{2}$ are feature coordinates corresponding to the eigenvectors associated with the largest eigenvalues of the distance matrix $C$ populated by the generic terms $C(l,m)$. The individuals draw a parabolic curve with the exception of fewer control laws due to the continued exploration of the GA optimization. Control laws from A to E show the evolution of the control parameters along the drawn parabolic curve and the optimal control law is identified by individual E. Table 2 lists the variables defining each of the selected five control laws on the parabolic curve, and their respective $C_{d}$ values. The primary actuation frequency, $F_{1}^{+}$, evolves from high to low values and stabilizes around the second harmonic of the vortex shedding frequency, reducing progressively its related amplitude $A_{1}^{+}$. Cases C–E do not show significant changes in the $F_{1}^{+}$ and $A_{1}^{+}$ values, the main drag reduction effect being due to the secondary frequency $F_{2}^{+}$. In particular, considering cases C–E, where the primary frequency does not change significantly, one can still observe an important drag reduction related to the secondary frequency change. Comparing cases C and E, one can clearly observe the influence of $F_{2}^{+}$, where, even though $A_{2}^{+}$ remains small, the decrease of the secondary frequency brings a substantial drag reduction. In the optimal case E, $F_{2}^{+}$ is chosen as the third superharmonic of the vortex shedding frequency. A deeper analysis of the flow behaviour of the optimal compared to the non-actuated case is provided in the next section.

Figure 6. Visualization of the 2-D proximity map of the control laws. Only every fifth generation is plotted for clarity. The red circles indicate the five representative cases A–E whose details are listed in table 2. Each dot represents a control law, and the distance between dots is a measure of the difference between two control laws. The colour scheme represents the $C_{d}$ value.

Table 2. Parameters and $C_{d}$ values of the five representative cases highlighted in figure 6.

3.2 The 2-D flow topology and the effect of the actuation

Looking at the average streamwise velocity field visualized in figure 7, one can observe a significant change in the distribution of streamlines, in the velocity field intensity and, most importantly, in the topology of the side recirculation and wake regions. The effect of the actuation has a strong global impact on the flow but it does not strongly affect the height and length of the side recirculation bubble. Previous research (Minelli et al. Reference Minelli, Krajnović, Basara and Noack2016, Reference Minelli, Krajnović and Basara2018), on a similar flow control case, focused (probably too intensively) on the suppression of the side recirculation. Even though drag was effectively reduced, the present strategy shows that a global approach to drag reduction results in a more effective control. In the present case, only the averaged $C_{d}$ is fed back in the control, discarding information on the flow topology (e.g. the size of the side bubble).

Therefore, two important aspects arise from this study. The first one is that the suppression of the side region is not the key factor to reach the highest drag reduction result. The second aspect is that a small disturbance (the control signal) introduced at the incipient pressure-induced separation point can significantly characterize the wake flow. The latter aspect is particularly important for the energy consumption introduced by the control device, as measured by the momentum coefficient $C_{\unicode[STIX]{x1D702}}$. In fact, the low turbulence level characterizing the attached BL upstream of the flow separation is easier to manipulate, in contrast to a highly turbulent flow (as the detaching shear layer from the trailing edge of the body), which requires a higher energy control to be manipulated. In this regard, one can notice that $C_{\unicode[STIX]{x1D702}}$ is from one to two orders of magnitude lower compared to flow control studies where the control device is directly placed at the base region of a bluff body. This is confirmed by both low- and high-$Re$ studies.

Starting from comparable $Re$ cases, Pastoor et al. (Reference Pastoor, Henning, Noack, King and Tadmor2008) showed a direct-wake control for a D-shaped body, achieving 15 % drag reduction using $C_{\unicode[STIX]{x1D702}}>0.005$ ($23\,000<Re<70\,000$). The same behaviour was confirmed by Parkin, Thompson & Sheridan (Reference Parkin, Thompson and Sheridan2013), Parkin, Sheridan & Thompson (Reference Parkin, Sheridan and Thompson2015). These authors state that ‘In any case, actuation at $C_{\unicode[STIX]{x1D702}}=0.004$ seems the appropriate level for peak drag reduction at highest efficiency’ (Parkin et al. Reference Parkin, Thompson and Sheridan2013). A study of a D-shaped cylinder wake was presented by Han et al. (Reference Han, Krajnović and Basara2013) and later by Gao et al. (Reference Gao, Li, Bai and Wu2016) using a $C_{\unicode[STIX]{x1D702}}$ of 3 % and 1 %, respectively, achieving 15 % drag reduction. In addition, Dalla Longa, Morgans & Dahan (Reference Dalla Longa, Morgans and Dahan2017) recently proposed a low-Reynolds-number ($Re=10\,000$) study of a D-shaped body, showing the use of a comparably low (but still higher than the present study and at a four times lower $Re$) momentum coefficient. In addition, in Dalla Longa et al. (Reference Dalla Longa, Morgans and Dahan2017) the momentum coefficient was set to low values only after a high momentum coefficient stabilized the wake. Considering higher-$Re$ applications, the momentum coefficient reaches values up to two order of magnitude larger than the one used in the present study (Barros et al. Reference Barros, Borée, Noack, Spohn and Ruiz2016; Lambert, Vukasinovic & Glezer Reference Lambert, Vukasinovic and Glezer2016; Li et al. Reference Li, Noack, Cordier, Borée and Harambat2017; Lambert, Vukasinovic & Glezer Reference Lambert, Vukasinovic and Glezer2018; Li et al. Reference Li, Cui, Jia, Li, Yang and Noack2019). In the literature, there also appear a series of bluff-body flow control studies at a much lower $Re=800{-}1200$ (Qu et al. Reference Qu, Wang, Sun, Feng, Pan, Gao and He2017; Ma & Feng Reference Ma and Feng2019). It is noteworthy to observe that those authors utilized a much larger $C_{\unicode[STIX]{x1D702}}=0.007{-}0.187$ to perform a direct-wake control, in line with the results listed above, and therefore encouraging for the use of a more efficient upstream control.

Keeping this in mind, figure 7(c) shows the difference between the unactuated and the best drag reduction actuated case, which, for simplicity, will be named BDR. Two main differences are clearly visible: BDR presents a more elongated wake (— $\boldsymbol{\cdot }$ —, green) with respect to the unforced flow (– – –, red), while a similar size of the side recirculation region is observed.

Figure 7. A comparison of the 2-D wake topology with and without actuation. The unactuated and actuated flow-averaged streamwise velocity is shown in (a) and (b), respectively. Panel (c) highlights the comparison of the averaged flow topology: (– – –, red) unactuated flow; (— $\boldsymbol{\cdot }$—, green) actuated flow.

When controlled, the wake extends by 40 % compared to the uncontrolled case, significantly increasing the base pressure recovery as visualized in figure 8. For clarity, the wake end is identified with the saddle point and measured from the base surface to the averaged streamwise velocity change of sign on the symmetry line.

Figure 8. A comparison of the coefficient of pressure $C_{p}$ along the base: (– – –, red) unactuated flow; (— $\boldsymbol{\cdot }$ —, green) actuated flow.

Next, the best control signal selected by the GA is investigated. The four variables in (2.1)–(2.2) are selected in BDR as follows: $F_{1}^{+}=0.42$, $F_{2}^{+}=0.63$, $A_{1}^{+}=0.45$ and $A_{2}^{+}=0.04$. What one can notice is that $A_{1}^{+}$ is one order of magnitude larger than $A_{2}^{+}$ and that $F_{1}^{+}$ and $F_{2}^{+}$ are the second and the third harmonics of the natural vortex shedding frequency, recorded in the unactuated case as $F_{s}^{+}=0.21$. The selection of multiple superharmonics represents a very interesting result, especially keeping in mind that the control is situated upstream and distant from the wake. However, having a feedback variable dominated by the wake dynamics ($C_{d}$) shows the potential of an upstream control, which is able to actively interact with the downstream flow. The wake is in fact deeply affected in terms of both its average topology and its dynamics. Before investigating the effect of frequencies in the sampled domain, it is interesting to look at the recursive movement of the wake induced by the control.

In order to do this, the steps to recreate figures 9 and 10 are described here. (i) A lossless proper orthogonal decomposition (POD) is performed on the velocity fields of 1000 snapshots of the sampled domain (visualized in figure 2b). (ii) A distance matrix $E$ is calculated by the summation of the Euclidean norm of POD temporal coefficients $b_{k}(t)$ as follows:

(3.2)$$\begin{eqnarray}E(t_{i},t_{j})=\sqrt{\mathop{\sum }_{k=1}^{N}\Vert b_{k}(t_{i})-b_{k}(t_{j})\Vert ^{2}},\end{eqnarray}$$

where $N$ is the number of modes, and $t_{i}$ and $t_{j}$ are two instants of the sampled flow field. The result is a square matrix with zeros on its diagonal. (iii) A proximity map, which visualizes the first two eigenvalues of the distance matrix $E$, is presented in figures 9(a) and 10(a). These plots represent the relative distance between each snapshot, and it is an indicator of how much the flow evolves from snapshot to snapshot. This visualization was previously used for bluff-body wake flows (Kaiser et al. Reference Kaiser, Noack, Cordier, Spohn, Segond, Abel, Daviller, Östh, Krajnović and Niven2014) and flow control studies (Duriez et al. Reference Duriez, Brunton and Noack2017; Wu et al. Reference Wu, Fan, Zhou, Li and Noack2018). The present plots are coloured using the corresponding drag $C_{d}$ and side force $C_{s}$ coefficients (respectively from left to right). Moreover, in the last (to the right) scatterplot of figures 9(a) and 10(a), the same colour is used for snapshots falling in the same cluster. The latter clustering procedure is explained in the following point. (iv) The proximity map is further clustered using a $K$-means clustering procedure (Macqueen Reference Macqueen1967; Lloyd Reference Lloyd1982), using $K=4$ clusters. An average of the snapshots falling in each of the four clusters is then represented in figures 9(b) and 10(b).

Comparing figures 9(a) and 10(a) one can notice how the unactuated flow presents a more spread scatterplot, indicating a less regular flow field. For the BDR case, the dots are closer to each other, highlighting the regularizing effect of the control. In BDR it is also possible to distinguish high- and low-drag states of the wake, while it is not so evident in the unactuated case. By looking at the two wakes, one can notice the presence of a more elongated wake core (the region containing the highest reverse velocity value) in figure 9(b) (clusters 2 and 4) for the unactuated case and in figure 10(b) (clusters 1 and 3) for BDR, associated with low-drag states. Conversely, a more compact and shorter wake core is associated to high-drag states. Therefore, what is interesting here is not only the visualization of the wake motion in the $y$ direction, but also the presence of a clear separation between low- and high-drag states, which regularly oscillate with the shedding frequency. Compared to the unactuated case, BDR mitigates this effect by regularizing the motion of the wake and the oscillation range of the $C_{d}$ value.

Figure 9. The unactuated case, 2-D LES. (a) Proximity maps: left coloured by $C_{d}$ values, centre coloured by $C_{s}$ values, and right coloured by clusters. (b) Each phase is the average of the snapshots falling in each cluster. Phase 1 and 3 low-drag states, phase 2 and 4 high-drag states. The probability of each cluster is equally distributed.

Figure 10. The best drag reduction case, 2-D LES. (a) Proximity maps: left coloured by $C_{d}$ values, centre coloured by $C_{s}$ values, and right coloured by clusters. (b) Each phase is the average of the snapshots falling in each cluster. Phase 2 and 4 low-drag states, phase 1 and 3 high-drag states. The probability of each cluster is equally distributed.

In figure 11 the highest reverse velocity (global minimum velocity) for each snapshot is traced. In BDR all points fall relatively close to the centreline, while the unactuated case shows a wider distribution of the minimum velocity in the $y$ direction. Moreover, the reverse velocity is much higher, for the controlled case, explaining the highest pressure recovery visualized on the model base in figure 8. This aspect also indicates the beneficial effect of an increased lateral stability, where the spanwise motion of the wake is controlled by the actuation.

Figure 11. A scatterplot of the minimum reverse velocity in the wake: (a) unactuated case; and (b) actuated case. Each dot represents the position of the minimum reverse velocity for one single snapshot; 1000 snapshots are analysed for each case.

3.3 The 2-D versus 3-D flow topologies

So far, only 2-D cases obtained from the GA optimization procedure have been analysed. It is therefore necessary to verify the control behaviour in a fully 3-D case by comparing 2-D and 3-D results. The unactuated and BDR are simulated in 3-D using an extruded version of the 2-D domain (see figure 2c,d). The control is applied homogeneously along the $z$ direction using the same control signal applied to BDR (2-D). The authors are well aware of the fact that a 2-D simulation is far from representative of a turbulent flow, which by definition is 3-D (Pope Reference Pope2001). Nevertheless, the control case studied here should be treated slightly differently than a fully 3-D bluff-body flow.

Two main factors make this particular flow tend to a 2-D behaviour. The first one is the constant cross-section along the $z$ direction. This factor alone, however, is not sufficient for a two-dimensionalization of the flow. Many research studies have in fact found the presence of spanwise-travelling structures and modes in similar 2-D extruded geometries, from bluff plates (Roshko Reference Roshko1993), extruded circular cylinders (Bearman Reference Bearman1992; Lehmkuhl et al. Reference Lehmkuhl, Rodríguez, Borrell, Pérez-Segarra and Oliva2013; Radi et al. Reference Radi, Thompson, Rao, Hourigan and Sheridan2013; Rao et al. Reference Rao, Leontini, Thompson and Hourigan2013), and rectangular cylinders (Cimarelli, Leonforte & Angeli Reference Cimarelli, Leonforte and Angeli2018), to D-shaped bodies (Parkin et al. Reference Parkin, Thompson and Sheridan2013), the latter being more representative of a vehicle shape. Here is where the second factor plays a major role in the, herein called, two-dimensionalization of the flow. Having a control slot homogeneously spanning the extrusion direction and affecting the flow in its early separation stage is the main factor for this 3-D case to have a tendency towards 2-D features. Generally, as observed here, the control stabilizes the flow towards its 2-D nature, suppressing its aforementioned spanwise-travelling structures. In this section, two main parts are therefore described. The first one describes the 3-D nature of the unactuated flow, showing a remarkably different behaviour compared to the 2-D case. Conversely, the second part highlights the strong similarities between the 3-D actuated case and its 2-D counterpart. In this way, the choice of a 2-D optimization is justified and considered valuable for a 3-D application. In addition, a grid independence study of the 3-D LES is performed before going into the description of the flow features.

3.3.1 The 3-D LES grid independence study

Before considering any 3-D result, a grid independence study is performed on the flow simulated around the 3-D extruded geometry. Three meshes with different resolutions are considered for the non-actuated case. The details of the three grids are reported in table 3. A suitable LES resolution, as recommended by Piomelli & Chasnov (Reference Piomelli, Chasnov, Henningson, Hallbaeck, Alfreddson and Johansson1996), is kept for the streamwise $\unicode[STIX]{x0394}l^{+}$ and spanwise $\unicode[STIX]{x0394}s^{+}$ and wall-normal $n^{+}$ directions. In particular, the refining/coarsening procedure acts only on $\unicode[STIX]{x0394}s^{+}$. Plots of averaged velocity components $U$ and $V$ and Reynolds stress $\overline{u^{\prime }v^{\prime }}$ are visualized in figure 12. This shows that there is a very good agreement between the three meshes. For this reason, the medium resolution has been selected to study the flow behaviour in the next section.

Table 3. Force values and resolution parameters for coarse, medium and fine meshes. Here $\unicode[STIX]{x0394}s_{max}^{+}$ is the maximum normalized resolution in the spanwise direction, $\unicode[STIX]{x0394}l_{max}^{+}$ is the maximum normalized resolution in the streamwise direction and $\unicode[STIX]{x0394}n^{+}$ is the normalized resolution in the wall-normal direction. VSF is the normalized vortex shedding frequency.

Figure 12. Grid independence study. Comparison of normalized mean velocity components $U$ and $V$ and Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$. (a) Three locations are selected for the comparison at $z=0~\text{m}$. (b) Coarse (), medium ($\boldsymbol{-}\!\!\!\!\boldsymbol{-}\!\!\!\!\boldsymbol{-}$, light blue) and fine () meshes.

3.3.2 Comparison between 2-D and 3-D cases: unactuated and actuated flows

Proper orthogonal decomposition and fast Fourier transform (FFT) analysis are used to compare the similarities and differences between the 2-D and 3-D flow simulations. First, the differences in the unactuated flow are described, and later the similarities between 2-D and 3-D cases of the actuated flow will be taken into consideration.

Figure 13 shows the first (most energetic) POD mode obtained in the 2-D and 3-D unactuated flow simulations. The vortex shedding is clearly represented in both cases by the same structure topology (figure 13a), but the dominant frequency changes between the two cases. Figure 13(b) visualizes the temporal coefficient FFT and its temporal evolution (orbit plot) of the first POD mode. The shedding motion peak is distinct in both cases, being $F^{+}=0.21$ for the 2-D case and $F^{+}=0.16$ for the 3-D case. For clarity, four phases of the evolution of the 3-D shedding structures are presented in figure 13(c). The source of this discrepancy can be found in the inherited three-dimensionality of the 3-D case. In fact, the presence of a lower energy mode, travelling along the $z$ direction, is observed during the POD analysis.

Figure 13. The shedding mode of the unactuated 2-D (left) and 3-D (right) cases, represented by POD. (a) Structure topology. (b) Analysis of the temporal POD coefficient. The power spectral density (PSD) and the orbit plot of the shedding mode are represented. (c) Four phases of the evolution of the Fourier mode taken at the 3-D shedding frequency $F^{+}=0.16$.

Figure 14(a) shows the FFT of the temporal coefficient and the structure representation of this transverse mode. From this figure, one can notice that the mode dominant frequency is very close to the 3-D shedding frequency, possibly driving the shedding towards a slower motion when compared to the 2-D case. Since the POD temporal coefficient of this mode shows extra non-dominant peaks, it would be clearer to extract the Fourier mode at $F^{+}=0.16$ and produce its relative 3-D isosurface plots. The pattern shown by figure 14(b) is clear: a 3-D and a front view perspective show the four phases that describe the evolution of this mode. Yellow and blue structures alternate recursively along the $z$ direction, showing an opposite direction of motion from one side to the other. While on the right side (positive $y$) the yellow structures move upwards (from phase 1 to phase 4), on the opposite side (negative $y$) the yellow structures move downwards (from phase 1 to phase 4), creating a wavy pattern impossible to simulate in 2-D cases.

Figure 14. The transverse mode found in the 3-D unactuated case represented by POD. (a) PSD of the POD temporal coefficient and POD structure topology. (b) Four phases of the evolution of the Fourier mode taken at the 3-D transverse mode frequency $F^{+}=0.165$, 3-D (top) and front (bottom) views.

The first behaviour one can notice in the actuated case is that the vortex shedding frequency is stabilized by the control into a 2-D-like vortex shedding. Figure 15(a,b) shows the first POD mode of the 2-D and 3-D controlled cases. The vortex shedding remains the most energetic mode even when the control is engaged. Moreover, the 2-D and 3-D sheddings now stabilize on the same frequency, being $F^{+}=0.21$ (the same frequency as observed for the 2-D unactuated case, figure 13b). In both cases the shedding is very regular and recursive. A clear peak is identified by the PSD of the POD temporal coefficient and the orbit plots evolve regularly and predictably with time, figure 15(b). A similar comparison can be done for the dominant control frequency $F_{1}^{+}$ (2.1), figure 15(c,d). Also here, the similarity between 2-D and 3-D is evident considering the structure topology and the POD temporal coefficient dominant frequency. This mode moves symmetrically on the $xy$ plane since an in-phase control signal is given both to the right and left actuators. In addition, the mode appears stronger close to the side recirculation bubble, fading out progressively downstream. It remains more challenging to distinctively find the secondary control frequency $F_{2}^{+}$ (2.1) by using the POD approach. The energy associated with this frequency is very low ($A_{1}^{+}$ is one order of magnitude larger than $A_{2}^{+}$ in (2.1)). Therefore the mode is indistinguishable utilizing an energy ranking-based approach. Nevertheless, the respective Fourier mode of $F_{2}^{+}$ can be easily extracted and plotted for comparison.

Figure 15. The shedding mode and $F_{1}^{+}$ mode of the actuated 2-D (left) and 3-D (right) cases, represented by POD. (a) Structure topology of the shedding mode. (b) Analysis of the temporal POD coefficient. The PSD and the orbit plot of the shedding mode are represented. (c) Structure topology of the $F_{1}^{+}$ mode. (d) Analysis of the temporal POD coefficient, the PSD and the orbit plot of the $F_{1}^{+}$ mode are represented.

A similar structure topology is again observed between the 2-D and 3-D case, figure 16. In this figure, the ellipses (– – –, red) and ($\boldsymbol{-}\,\boldsymbol{\cdot }\,\boldsymbol{-}$, green) are added as reference to ease the visualization of similar structures. In contrast to the previous $F_{1}^{+}$ mode, the $F_{2}^{+}$ mode visualized here presents an antisymmetric nature (with respect to the $y$ axis) and it has a stronger effect downstream in the wake of the model. One possible explanation for describing the difference of modes is based on their energy content. Since the $F_{1}^{+}$ mode is more energetic, it dominates the region near its generation (the actuator slot and the side of the model). On the contrary, the $F_{2}^{+}$ mode is transported downstream by the antisymmetric shedding mode (which might be the reason for its antisymmetric connotation) and excited in the wake, where it shows its highest energy content.

To reinforce this supposition, the spatial distribution of the energy level of the shedding frequency, and the primary ($F_{1}^{+}$) and secondary ($F_{2}^{+}$) control frequencies are presented in figure 17. In particular, figure 17(a) shows the 2-D distribution and 17(b) shows the 3-D distribution, averaged over the homogeneous direction $z$, and presented in a 2-D plot for clarity. The shedding frequency and $F_{1}^{+}$ have the same colour scale, while a dedicated colour scale is given to $F_{2}^{+}$ to account for the lower energy content in this frequency. In addition, to better individuate the region with the highest energy level, (– – –, yellow) describes the contour lines at half of the maximum energy level, for each frequency. The $F_{1}^{+}$ mode has a comparable energy level to the shedding mode, but the area of interest of the primary frequency is confined very close to the actuation and in the side recirculation bubble. On the contrary, the area of interest of the shedding frequency extends longer in the wake of the model, introducing a higher global energy level. Of particular interest is the distribution of the secondary frequency. In both 2-D and 3-D this distribution extends further down in the wake, having its energy peak in the shear layers of the wake. As mentioned before, this low energy frequency seems to be transported downstream by the shedding motion acting in an antisymmetric fashion and predominantly in the wake. Moreover, by looking at the global comparison of these frequency distributions between the 2-D and 3-D cases, a good agreement is observed, corroborating the two-dimensionalization speculation mentioned at the beginning of this section. In other words, the control stabilizes the flow towards a 2-D behaviour, justifying the 2-D optimization process. Therefore, in the next section, we will discuss the 3-D controlled case and the benefit that a multi-frequency (based on vortex shedding harmonics) actuation brings compared to single-frequency (based on higher frequencies) controls. Three 3-D cases will be analysed: the unactuated, the multi-frequency control found by the 2-D GA optimization and a single-frequency control suggested from previous literature (high frequency to suppress the side recirculation).

Figure 16. Fourier mode distribution of the $F_{2}^{+}$ mode. The 2-D (a) and 3-D (b) actuated cases are presented. Ellipses (– – –, red) and (— —, green) are added as reference to ease the visualization of similar structures.

Figure 17. The distribution of the energy of one single frequency over the sampled domain: (a) 2-D case; and (b) 3-D case. From left to right: shedding frequency, $F_{1}^{+}$ and $F_{2}^{+}$. For an easier comparison, the distributions are averaged over the homogeneity direction $z$. Dashed line (– – –, yellow) represents the contour lines at half of the maximum energy of each frequency over the sampled domain.

3.4 Superiority of a GA-driven optimization versus previously published achievements

In this last section we want to compare the present GA-driven approach to previously published results, and highlight the advantages of a global drag reduction (GDR) approach to flow control. The use of flow control for this type of bluff body can in fact be driven by a local or a global approach. The first one aims to reduce, or eventually suppress, specific recirculation regions, keeping the flow locally attached to the body surface. The second takes into consideration only global quantities, e.g. the effect of the control on drag reduction. A confined phenomenon, like the appearances of localized flow separations, is usually defined by a limited number of variables. Referring to the side recirculation bubble visualized in figure 3(a), previous research (Minelli et al. Reference Minelli, Krajnović, Basara and Noack2016, Reference Minelli, Adi Hartono, Chernoray, Hjelm, Krajnović and Basara2017b) has shown that a careful selection of the actuator position and frequency can lead to the complete suppression of this local phenomenon. Nevertheless, this does not necessarily optimize global quantities like the total drag. When multiple variables come into play, the use of a global optimization approach is therefore preferable. In particular, it is shown here that sacrificing the suppression of the side recirculation does lead to a lower drag and improved lateral stability. Therefore, two control cases are compared: the side-recirculation-suppression (SRS) case and the GDR case. The former uses a specific frequency indicated by previous literature (Minelli et al. Reference Minelli, Krajnović, Basara and Noack2016, Reference Minelli, Krajnović and Basara2018) to suppress the side bubble, while the latter case is introduced in § 3.2 and obtained from the GA optimization procedure.

Figure 18 shows the comparison of the averaged streamwise velocity and flow topology of the three considered cases, while table 4 lists the averaged drag and side force coefficients together with their respective root mean square (r.m.s.) values. The actuation placed at the front rounded corners significantly affects both the topology of the side and the wake regions. Using a single-frequency actuation signal, with $F^{+}=fD/U_{\infty }=3$, the side recirculation bubble is completely suppressed (SRS case), while the GDR case, controlled by a two-frequency signal, shows the persistence of this feature. The side recirculation region is therefore not directly connected to a lower drag value.

From this comparison, it does in fact seem rather beneficial to keep a separated flow at the side region in order to obtain a lower drag. The presence of this bubble surely influences the wake topology. Moreover, the presence of the side recirculation could be one of the factors improving the propagation of the actuation frequencies into the wake. The shear layer detaching at the leading edge (after the actuation slot) contains and may transport structures characterized by the actuation frequencies to the trailing edge. In SRS, on the other hand, the shear layer reattaches very early after the actuation slot, exhausting the energy of the actuation frequency on the wall of the model. Considering the two actuated cases and as previously found for a square-back D-shaped body (Barros et al. Reference Barros, Borée, Noack, Spohn and Ruiz2016; Dalla Longa et al. Reference Dalla Longa, Morgans and Dahan2017), a longer wake (in this case GDR) is associated with a lower drag. However, this is not the only factor that describes the drag–wake relationship. The combination of the wake length with its reverse velocity, width and its dynamics complete its description. In addition, the unactuated case presents a substantially different wake topology, in which the side separation merges with the wake and the flow does not reattach at the trailing edge as experienced by the controlled cases (and by the 2-D unactuated case). This is probably the reason behind the longer wake produced by the 3-D unactuated case.

Figure 18. A comparison of the wake topology in 3-D cases with and without actuation. The unactuated and actuated flow-averaged streamwise velocity is shown. The bottom right picture highlights the comparison of the averaged flow topology: (– – –, red) unactuated case; (— $\boldsymbol{\cdot }$ —, green) GDR case; ($\cdots \cdots$, yellow) SRS case.

By looking at the minimum reverse velocity extracted from every snapshot and visualized in figure 19, one can obtain a first indication of the wake core movement and its intensity. For clarity, the wake core is taken as the position with the highest reverse velocity value; therefore, each dot in figure 19 represents the location of the wake core and they are coloured by its intensity. In the unactuated case, the reverse velocity is high and the position of the wake core varies in both the $y$ and $x$ directions. The GDR case, figure 19(b), shows a much lower (absolute) value of the reverse velocity and a narrower area of interest of the wake core position. The SRS case (figure 19c) instead presents a much more confined region of the wake core, which mostly varies its location along $y$. This case also presents a higher reverse velocity compared to the GDR case. This last parameter seems to better relate to the drag value, having the highest reverse velocity for the unactuated case, and the lowest for the GDR case, which gives the lowest drag value.

Table 4. Values of $C_{d}$, $C_{s}$, their r.m.s. values and the shedding frequency $F_{s}^{+}$ of the unactuated, GDR and SRS cases.

Figure 19. A scatterplot of the highest reverse velocity in the wake: (a) unactuated case; (b) GDR case; and (c) SRS case. Each dot represents the position of the highest reverse velocity for one single snapshot, spatially averaged along the homogeneous direction $z$; 1000 snapshots are analysed for each case.

Up to this point, the GDR flow control case is shown to contribute to a 20 % drag reduction, while the SRS case improves drag only by 8 %; see table 4. In terms of drag reduction, the GDR case is shown to be significantly more effective than the previously published control strategies; therefore it is also important to verify the lateral fluctuations induced by the two controls. Considering table 4 and figures 20–22, one can better understand the dynamics of the wake and its implications in terms of stability. The natural vortex shedding observed in the unactuated case has a strong influence on $C_{s}$, resulting in a high r.m.s.($C_{s}$) value. The same high value is recorded for the SRS case, while the GDR case shows a much lower fluctuation of $C_{s}$. This is also clarified by figure 20, which depicts the $C_{s}$ time history of the three cases. Having a low r.m.s.($C_{s}$) increases the lateral stability, avoiding unwanted forces created by the shedding motion.

Figure 20. The $C_{s}$ time history of the three cases: (a) unactuated case; (b) GDR case; and (c) SRS case.

For a deeper analysis of the wake motion, one can concentrate on figures 21 and 22. With the same approach as described in § 3.2, a proximity map and a clustering procedure on the 3-D flow field is presented here. The scatterplots are coloured by $C_{d}$, $C_{s}$ and the respective cluster number (from left to right in figures 21 and 22). Considering the GDR case (figure 21), one can clearly recognize the recursive side force variation and the periodic alternation of low- and high-drag states. Figure 21 shows in fact that clusters 2 and 4 are defined by high drag values, while 1 and 3 are characterized by low drag. In addition, clusters 2 and 4 show the highest side force values (of opposite sign). By a 3-D representation of velocity isosurfaces, one can better visualize the feature of each cluster. Figure 21(b,c) shows a 3-D and a top view of these isosurfaces, highlighting in yellow the wake core (the region containing the highest reverse velocity value). Clusters 1 and 3 (low drag) show a more elongated and distant-from-base wake cores, while clusters 2 and 4 (high drag) show a more contained and very close-to-base wake cores. Clusters 2 and 4 then not only individuate the oscillation of the side force component, but they are also associated with the highest drag force. This phenomenon is an indication of the presence of a breathing mode of the wake core, whose motion is strongly related to the drag oscillation. In other words, figure 21 shows the periodic elongation and translation of the wake core (yellow isosurface in figure 21b,c) from high- to low-drag states. In particular, looking at the same figure, the representation of clusters 2 and 4 shows a more circular vortex core, with its centre positioned between $x=1$ and $x=1.5$, while in clusters 1 and 3 the core centre is located at $x=2$ and presents a more elongated shape. The alternation of these two states gives rise to the breathing mode introduced earlier and characterizes the periodic motion in the $x$ direction of the wake.

Keeping figure 21 in mind, one can compare it with figure 22. In contrast to the GDR case, figure 22(a) does not show a clear distinction of low- and high-drag clusters, but it only shows a clear side force distinction between different clusters. This case is described by a lower r.m.s.($C_{d}$) value, which indicates damped $C_{d}$ oscillations, due to the reduction of the streamwise motion of the wake core. The latter, represented in figures 19(c) and 22(b,c), remains in fact close to the model base surface, and it only oscillates in the $y$ direction, strongly affecting the side force rather than the drag and its breathing mode. From these results, one can infer that the propagation of the control frequency downstream in the wake has a beneficial effect on the stability control of the wake motion.

Figure 21. The GDR case, 3-D LES. (a) Proximity maps: left coloured by $C_{d}$ values, centre coloured by $C_{s}$ values, and right coloured by clusters. (b) Each phase is the average of the snapshots falling in each cluster. The probability of each cluster is equally distributed.

Figure 22. The SRS case, 3-D LES. (a) Proximity maps: left coloured by $C_{d}$ values, centre coloured by $C_{s}$ values, and right coloured by clusters. (b) Each phase is the average of the snapshots falling in each cluster. The probability of each cluster is equally distributed.

This is corroborated by figure 23, which shows the propagation of the shedding and actuation frequency for the SRS case. The same figure also shows the control frequency Fourier mode visualized with 3-D structures. Comparing this visualization to figure 17(b) (GDR case), it is clear to observe a stronger penetration of the control frequency into the wake in favour of the GDR case. In the SRS case in fact, the actuation frequency remains very localized to the side of the model and no observable energy content is shown in the wake in comparison to the energy content of the shedding frequency. Therefore, an actuation that may effectively control the wake dynamics is preferable compared to a localized control, if the overall drag reduction represents the main deliverable of an optimization study.

Figure 23. The distribution of the energy of one single frequency over the sampled domain, SRS case. From left to right: shedding frequency, $F_{1}^{+}$ and a 3-D representation of $F_{1}^{+}$ (Fourier mode). (– – –, yellow) represents the contour line at half of the maximum energy of each frequency over the sampled domain.