3.1 The unactuated mixing layer

Analysing the temporal evolution of the different hot-wire signals at $x=200~\text{mm}$ provides indications on the naturally present frequencies at this location. Figure 3 shows the power spectral density (PSD) of each sensor signal (used in closed-loop control) for the two flow configurations. With the exception of the sensor signals at the centre of the mixing layer, $s_{10}$ and $s_{13}$ , all the spectra of the sensors show a distinct local maximum. In accordance with previous findings of Winant & Browand (Reference Winant and Browand1974) and Morris & Foss (Reference Morris and Foss2003), this ‘hump’ points to the existence of a preferred mode of the flow, which occupies a narrow band of frequencies. Following Ho & Huerre (Reference Ho and Huerre1984), the natural Strouhal number in a laminar mixing layer is $St_{{\it\theta}}=(f_{n}{\it\theta})/U_{c}=0.032$ , where ${\it\theta}$ is the local momentum thickness. The frequencies corresponding to this Strouhal number are estimated using the values of ${\it\theta}$ and $U_{c}$ given in the tables of figure 1. The estimation yields 12 and 25 Hz for the LS and HS configurations, respectively. As indicated in figure 3, these values are well inside the frequency range of the humps, visible in the spectra of the sensors outside the centre of the mixing layer. Hot-wire sensors $s_{10}$ and $s_{13}$ , close to the centre of the mixing layer are less sensitive to vortex shedding due to a vanishing mean value of the vertical velocity component. For comparison, the PSD measured near the trailing edge of the splitter plate at $x=50~\text{mm}$ (not shown here) feature much more clearly defined peaks around 90 Hz (LS) and 120 Hz (HS). These higher frequencies are related to the K–H instability caused by the boundary conditions imposed at the trailing edge of the splitter plate (Morris & Foss Reference Morris and Foss2003). The lower frequencies measured at $x=200~\text{mm}$ (shown in figure 3) correspond to the amplifier dynamics of the mixing layer in the self-similarity region, which is no longer influenced by the local formation conditions of the shear layer.

Figure 4.

Mapping of objective functions $J_{W}$ and $J_{K}$ , for a range of open-loop actuation frequencies $0~\text{Hz}<f_{a}<100~\text{Hz}$ , and four different duty cycles (15 %, 30 %, 50 %, 70 %), in (a,c) $Re_{{\it\theta}}=850$ (LS) and (b,d) $Re_{{\it\theta}}=1400$ (HS) configurations of the mixing layer (sensors located at $x=200~\text{mm}$ ). Red vertical line (colour online only) in each case represents the obtained frequency and the associated duty cycle of open-loop control for which the corresponding objective function is maximized. These values are noted as value pairs ( $f_{a},dc$ ) in the text, for each case.

3.2 Periodic forcing and closed-loop control

In order to set a reference for the closed-loop control, the mixing layer response to periodic forcing has been mapped for the entire range of available actuation frequencies ( $f_{a}$ ), up to 500 Hz and for several values of duty cycles ( $dc$ ). The mapping results for the LS and HS configurations are shown in further text up to $f_{a}=100~\text{Hz}$ , since this is the most interesting actuation range with respect to the defined objective functions.

From the open-loop mapping we find the optimal settings of $(f_{a},dc)_{W(LS)}=(21~\text{Hz},50\,\%)$ for maximization of the objective function $J_{W}$ in the LS configuration (figure 4 a). The maximum of $J_{K}$ is obtained for $(f_{a},dc)_{K(LS)}=(12~\text{Hz},70\,\%)$ (figure 4 c). The open-loop mapping experiment in the HS configuration of the mixing layer is shown in the right column of the figure 4. In the this configuration, open-loop mapping yields $(f,dc)_{W(HS)}=(35~\text{Hz},50\,\%)$ as the best periodic actuation parameter pair for the maximization of $J_{W}$ (figure 4 b), and $(f_{a},dc)_{K(HS)}=(24~\text{Hz},70\,\%)$ for the maximization of $J_{K}$ (figure 4 d).

We can conclude that the two objective functions are maximized for two different actuation frequencies. Also, the mapping experiments in LS and HS configurations show that the optimal actuation frequency depends on the free-stream velocities of the mixing layer. In both mixing layer configurations, the objective function $J_{K}$ is maximized when the actuation frequencies are around the estimated local natural frequency. Objective function $J_{W}$ reaches a maximum when the actuation is performed around 1.45–1.75 times the optimal frequency for $J_{K}$ .

The duty cycle of actuation apparently plays a significant role only around the actuation frequency where a given objective function is maximized. This means that by choosing an arbitrary objective function, we may not see the effect of the duty cycle outside of the actuation range corresponding to the optimal parameters for this objective function.

These results show that a maximization of an arbitrary objective function might require a frequency of actuation which is not necessarily the local natural frequency, nor its multiple. Such an actuation frequency can be easily imposed by an open-loop control, but the question is: can we build a closed-loop controller which will be able to find and use such frequency automatically, taking into account that such a hypothetical frequency may not be a dominant natural mode?

The GP search process has been performed for the maximization of the two objective functions $J_{W}$ and $J_{K}$ , yielding control laws $\mathscr{F}_{W(LS)}^{GP}$ and $\mathscr{F}_{K(LS)}^{GP}$ for the LS mixing layer configuration:

(3.1) $$\begin{eqnarray}\displaystyle \mathscr{F}_{W(LS)}^{GP}=\sin (\exp (s_{19}\times \exp (s_{1}^{2})-s_{16})-\cos (\exp (s_{13})\times s_{1}^{2}\times s_{19}\times s_{16})) & & \displaystyle\end{eqnarray}$$

and

(3.2) $$\begin{eqnarray}\displaystyle \mathscr{F}_{K(LS)}^{GP}=s_{19}+s_{19}+\sin (\sin (\sin (\sin (s_{16}-s_{1})))). & & \displaystyle\end{eqnarray}$$

The index $K$ or $W$ denotes that the control law $\mathscr{F}^{GP}$ is the optimal solution for the maximization of the respective objective function. Index (LS) or (HS) indicates in which flow configuration has the control law been trained by the GP.

Figure 5.

Resulting $\langle {u_{i}^{\prime }}^{2}\rangle _{T}$ profiles (a) and pseudovisualizations (b) for maximization of $J_{W}$ and $J_{K}$ experiments in LS configuration of the mixing layer. The best open- and closed-loop control are compared. Actuator system input is marked using black and white stripes above each case (black stands for active). The pseudovisualizations are constructed from $u_{i}^{\prime }$ in a range of $-1.5$ to $1.5~(\text{m}~\text{s}^{-1})$ and are based on Taylor’s hypothesis of frozen turbulence.

The results of both open- and closed-loop control on the velocity fluctuation profiles can be seen in figure 5(a). When maximizing the cost function $J_{W}$ , the resulting fluctuation profiles are a near match for both controls. However, their shape is significantly different compared to the natural flow; we observe a larger spread of fluctuations across the mixing layer and a lower maximum amplitude around the middle. Maximization of $J_{K}$ leads to an increase in fluctuation amplitude across the entire profile, while the general shape is similar to the natural case.

Figure 5(b) shows the pseudovisualizations of the natural LS flow, followed by open- and closed-loop control cases for the two objective functions. The closed-loop control laws $\mathscr{F}_{W(LS)}^{GP}$ and $\mathscr{F}_{K(LS)}^{GP}$ feature almost perfectly periodic actuation, although time is not an explicit variable in the control laws. Clearly, in both cases, the feedback control has caused a resonance close to the specific frequency which was found the most effective for open-loop control; around 20 Hz for $J_{W}$ and around $12~\text{Hz}$ for $J_{K}$ . The actuation stripes in figure 5(b) indicate that the duty cycles are also similar to the open-loop reference case. Corresponding pseudovisualizations reveal that the mixing layer flow is well ordered on the scales of actuation. The cost function values of open- and closed-loop control are comparable to within 5 %.

The whole process of the GP search for the best control law was repeated five times for each of the objective functions, and in all cases the control laws produced identical frequency selection. The resulting values of the objective function were reproducible for any given control law. The standard deviation of the maximum values of both objective functions, measured over all experiments (typically more than 20 evaluations), lies in the range $0.08<J_{rms}<0.2$ for both open- and closed-loop control.

3.3 Robustness of closed-loop control

In this section, we test and compare the robustness of both open- and closed-loop control strategies with respect to changes of the mixing layer stream velocities. The best solutions presented in § 3.2 for the closed-loop control have been trained by the GP in the LS mixing layer configuration. Using the GP search, we find the best closed-loop control law optimized specifically for the HS configuration:

(3.3) $$\begin{eqnarray}\displaystyle \mathscr{F}_{W(HS)}^{GP}=s_{7}\times \exp (s_{4}-s_{1})+s_{4}. & & \displaystyle\end{eqnarray}$$

For this control law, as well as for the open-loop actuation $(f,dc)_{W(HS)}=$ ( $35~\text{Hz}$ , 50 %), the HS configuration of the mixing layer is the design point. To test their robustness, we apply both of them to off-design conditions. For this purpose, the LS configuration, studied in more detail in the previous section, is selected as the off-design condition. We emphasize that the HS design condition comprises a fully turbulent mixing layer, from the onset. The off-design LS configuration with a laminar origin of the mixing layer is therefore the most different flow configuration available in our experiments, without significantly changing stream velocity ratio $r$ . Other stream velocities between the LS and HS configurations represent comparatively minor modifications of the flow conditions.

Figure 6.

Comparative tests of the best open-loop actuation $(f_{a},dc)_{W(HS)}$ and the best closed-loop control law $\mathscr{F}_{W(HS)}^{GP}$ applied at the design point HS and the off-design point LS. In the upper row, the objective function $J_{W}$ and the corresponding mass flow rate coefficient $C_{q}$ values are represented. In the middle row, the PSD of every third hot-wire probe are shown in log–log scale, shifted by a decade for clarity. The PSD of the binary actuation signal is shown in a linear-log scale in the bottom of each window (line ‘b’). In the lower row, pseudovisualizations are shown on a time duration of $t=0.25~\text{s}$ . The associated grey-scale map depicts $u^{\prime }$ in a range of $-1.5$ to $1.5~(\text{m}~\text{s}^{-1})$ . Black and white stripes depict the associated actuation (black for active).

Figure 6 compares best open- and closed-loop control when applied to (a) design and (b) off-design flow conditions. The values of the objective function $J_{W}$ and the actuation cost $C_{q}$ are comparable for the two controls in the design case (bar diagrams in figure 6 a), with a slight advantage for the open loop. However, the corresponding diagrams in the LS off-design case (figure 6 b) show a drastic decrease in performance when the open-loop actuation for the HS configuration is applied, while the best closed-loop control for HS maintains its effectiveness.

Under design conditions the spectral analysis shown in figure 6(a) shows for both types of control the fluctuation energy concentrated around the actuation frequency. In figure 6(b) when the best HS closed-loop control law $\mathscr{F}_{W(HS)}^{GP}$ is applied under the LS off-design conditions, the closed-loop control selects the correct optimal actuation frequency and synchronizes the mixing layer in a similar way as in figure 5. Furthermore, when the best HS open-loop $(f_{a},dc)_{W(HS)}$ is applied in the off-design case (LS), we observe a significant reduction in the energy content at the actuation frequency. In the LS mixing layer, the frequency of $35~\text{Hz}$ no longer corresponds to the optimal frequency, and the pseudovisualization reveals only weakly defined structures inside the mixing layer. The closed-loop control adapted itself to a nearly periodic 21 Hz actuation regime, in agreement with the optimal open-loop control for this flow configuration. Although the spectra for this case show other frequencies present, the most energetic peak is at this optimal frequency, and appears to be as well amplified as in the design case. The corresponding imposed periodic structures inside the mixing layer, visible in the pseudovisualization are very similar to the optimum for the LS case (see figure 5 b).

For these off-design conditions, the comparison between open- and closed-loop control clearly favours the closed loop. The reference control for the LS configuration is the optimal open-loop actuation $(f_{a},dc)_{W(LS)}$ , which achieves $J_{W}=2.49$ (see figure 4 a). Using the optimal HS actuation $(f_{a},dc)_{W(HS)}$ in the LS off-design configuration results in $J_{W}=1.4$ . This is a reduction of $44\,\%$ with respect to the reference objective function value. The closed-loop control $\mathscr{F}_{W(HS)}^{GP}$ adapts to off-design conditions much more efficiently: objective function value of $J_{W}=1.91$ is obtained, yielding only 22 % of reduction in effectiveness compared with the reference value. A reverse robustness test reveals similar results, when optimal closed- and open-loop actuation signals for the LS case are applied to the HS configuration as the off-design condition.

A priori, closed-loop control can perform either better or worse than the open-loop control in design and off-design conditions. Here, the robustness property of the GP process can be partially attributed (see the survey of Beyer & Sendhoff (Reference Beyer and Sendhoff2007)) to the stochastic nature of evolutionary algorithms for which the optimal solution is obtained after many generations of evolution by operating on a large population of potential solutions. In § 5.1, we propose another explanation of this particular robustness result based on the anatomy of the best control laws found by GP in the HS configuration (§ 4.1) and on the frequency response to periodic forcing at the sensor location. It may be noted that like in $\mathscr{H}_{\infty }$ -control problems (Burl Reference Burl1999), GP can, in principle, take into account the robustness to external perturbations if the operating conditions are taken into consideration in the cost function during the learning phase. In addition, GP can combine periodic forcing and sensor feedback or even choose between open- and closed-loop control, depending on which performs better. These generalizations shall be part of future publications.

4.1 The anatomy of the best control laws

The probabilistic manner in which the GP constructs and manipulates the expressions of the control laws (see appendix B) means that there may be multiple control laws with different analytical expressions, but which perform the same action. For example, we can consider a simple hypothetical situation where two control laws $\mathscr{F}_{n}^{GP}=s_{19}$ and $\mathscr{F}_{m}^{GP}=s_{19}+s_{1}-s_{1}$ are produced during the GP process. At a first glance we can immediately acknowledge that these two control laws are identical, but the GP process treats them as different individuals. In addition to this, the actuator system design requires that the Heaviside function is applied on the control law (see § 2), which is another reason why control laws of different analytical expressions may produce a similar result $b(t)$ .

With this in mind, an efficient way to classify the solutions obtained by GP is to compare the spectra of the actuation signals $b(t)$ . If the actuation spectra of two control laws match, we consider that the two control laws are equivalent, regardless of their analytical expressions. The matching is computed as a linear correlation coefficient $r_{c}$ between spectral amplitudes and needs to be better than $r_{c}\geqslant 0.95$ . If we examine the analytical expressions of control laws matched in this way, we can obtain a statistical image of participation of different sensors in their expressions. Only a single occurrence of any given sensor is counted, regardless of how many times this sensor was repeated in the control law expression. A histogram is then computed with bins corresponding to the different sensor labels $s_{i}$ .

We consider the best HS control law $\mathscr{F}_{W(HS)}^{GP}$ (3.3) as reference for the matching of actuation spectra. The reference spectrum of the control law actuation signal PSD( $b$ ) is shown in figure 7(a). The search for equivalent control laws yields 1151 matches of the actuation spectrum within the whole GP experiment. The histogram of the participating sensors, in the control laws acting similarly as $\mathscr{F}_{W(HS)}^{GP}$ , is shown in figure 7(b). We can see that sensor $s_{7}$ is present $100\,\%$ of the time, while other sensors appear less often in relatively equal proportions. We can conjecture that this sensor is necessary to enable a nearly periodic actuation at 35 Hz. But do the remaining sensors play a role, or are they just artefacts of the GP optimization process?

Figure 7.

Analysis of the best control law $\mathscr{F}_{W(HS)}^{GP}$ : (a) power spectral density of the square wave actuation signal $b_{W(HS)}(t)$ , (b) histogram of sensor occurrence in all individuals matching the actuation spectra of $\mathscr{F}_{W(HS)}^{GP}$ , (c) recorded sensor signals $s_{i}$ , (d) sensor signals with linear regression weighting applied ( ${\it\alpha}_{i}s_{i}$ ) and (e) linear regression reconstruction corresponding to $\mathscr{F}_{W(HS)}^{lin}=-0.34\times s_{1}+1.266\times s_{4}+0.943\times s_{7}$ , plotted for comparison with $\mathscr{F}_{W(HS)}^{GP}$ and $b_{W(HS)}(t)$ for reference. Label #ind indicates the number of individuals matching the reference actuation spectrum.

It turns out that the best control laws obtained by GP (3.1)–(3.3) can all be represented by a linear expression. We have used linear regression to reconstruct the actuation signal $b(t)$ by representing the control laws as a linear combination of the sensors $s_{i}$ , with associated weighting coefficients ${\it\alpha}_{i}$ . This is further illustrated in figure 7(c–e) for the HS configuration. The three hot-wire sensors used ( $s_{1}$ , $s_{4}$ and $s_{7}$ ) are all located on the lower part of the sensor rake, inside the low-velocity region of the mixing layer. As shown in figure 7(c), the closer the sensor is to the centre of the mixing layer, the larger are the recorded velocity fluctuations. All these data were obtained for the actuated state and all three sensors show fluctuations principally caused by actuation. However, significant differences exist between each of the signals; sensor $s_{7}$ recorded a much noisier signal than the other two. The transient events observed in this signal appear to be stochastic perturbations caused by the evolution of small-scale structures inside the core of the mixing layer. Nevertheless, the linear regression weighting shown in figure 7(d) further highlights that sensor $s_{7}$ represents the main sensor input, in accordance with its domination in the histogram 7(b). Here $s_{4}$ is scaled up to roughly the same amplitude level as $s_{7}$ but $s_{1}$ is significantly reduced. The corresponding linear approximation of the control law $\mathscr{F}_{W(HS)}^{lin}$ in figure 7(e) suggests that the dominant (but noisy) signal of $s_{7}$ is smoothed out by its combination with $s_{4}$ .

The apparently non-essential sensors contribute to the efficiency of a control law by a modification of the duty cycle. As shown in figure 4, the objective functions are highly sensitive to the values of the duty cycle around the optimal open-loop parameters. In our experiment, the duty cycle is the only way to modify the amplitude of actuation, taking into account that the actuator system works with a fixed level of pressure, and that the Heaviside function must be applied to binarize the control signal. If we use only a single weighted sensor signal ${\it\alpha}_{i}s_{i}$ at a time, for the linear approximation, we can estimate what duty cycle would each sensor produce. The results for $\mathscr{F}_{W(HS)}^{lin}$ are: $48\,\%$ using $s_{1}$ , $54\,\%$ with $s_{4}$ and $60\,\%$ with $s_{7}$ only. The measured duty cycle of this control law is $55\,\%$ .

This implies that combining the signals contributes to the optimization of the duty cycle. However, there is no reason why scaling the ‘dominant’ sensor signal with an appropriate constant generated randomly by GP could not perform the same task. Genetic programming can generate random constants in a range of $-1.00$ to 1.00 (see appendix B), but in our experience, the GP seems to be more efficient in using other variables as scaling instead of constants. Moreover, from the histogram of the sensor participation in this control law, it appears that all of the other ‘supporting’ sensors can be used equally often, and that there is no obvious pattern for their selection. Therefore, in the case of $\mathscr{F}_{W(HS)}^{GP}$ , the use of other sensors can be considered as an idiosyncrasy of genetic programming.

The control laws $\mathscr{F}_{W(LS)}^{GP}$ (3.1) and $\mathscr{F}_{K(LS)}^{GP}$ (3.2), which are trained to maximize the two objective functions in the LS flow configuration, similarly have a single sensor which is always present in the control law expression. In the case of $\mathscr{F}_{W(LS)}^{GP}$ , only the sensor $s_{16}$ appears all the time, and the flow is synchronized around $20~\text{Hz}$ in the LS configuration. For $\mathscr{F}_{K(LS)}^{GP}$ , the dominant sensor is $s_{19}$ which produces synchronization around $12~\text{Hz}$ . We will further discuss some aspects of the vertical location of the dominant sensor in § 5.3.

4.2 Diversity of the feedback frequency selection

The chosen objective functions $J_{K}$ and $J_{W}$ are arbitrary, but physics based. Each is maximized by causing a lock-on state of the mixing layer at a different optimal frequency. The GP successfully designed closed-loop control laws which are based on those frequencies (see § 3.3). If the feedback control can be designed to use these two frequencies, can we then expect that there are other frequencies, on which the feedback can be established?

The GP process, in its learning phase, creates a multitude of control law individuals which are discarded because they do not maximize the imposed objective function. This does not mean, however, that all of them are meaningless and that such control laws would not maximize some other arbitrary objective function. The results in § 3 suggest that the most efficient control laws produce near-periodic actuation. In these cases, the spectrum of the actuation signal features a well-defined energy peak around a single frequency. If the spectrum of actuation features a badly defined peak or a broadband energy distribution, such a control law usually performs badly. In the experiments performed, non-periodic or stochastic actuation does not appear to produce any remarkable effects within the observed portion of the mixing layer. We therefore assume that any existing alternative solution of the feedback control will feature a nearly periodic actuation signal based on a frequency, other than what was observed so far. The control laws deemed as suboptimal and discarded by the GP process, can provide a large sample base in which alternative synchronized states of the mixing layer might be found. Instead of designing many new objective functions which would enable us to find these alternative solutions, we can simply browse through the already existing, but apparently suboptimal solutions.

After a short search through all the individuals produced within one GP experiment in the HS mixing layer configuration, we can find at least five control laws which feature an actuation spectrum with a well-defined peak around a suboptimal frequency (with respect to $J_{W}$ ). The matching of the actuation spectra (as in the § 4.1) allows us to discover all of the individuals which produce these alternative actuation spectra, and to analyse the sensor participation histograms. Since these control laws do not maximize $J_{W}$ , we identify them using the number of the individual, in the first generation of the experiment, for which this type of actuation spectrum has been first observed.

Figure 8.

Examples of suboptimal control laws found in the GP experiment for the maximization of $W$ in HS mixing layer configuration. Shown on the left are power spectral densities of different actuation signals $b(t)$ used for the matching analysis. The corresponding histograms of occurrence of specific sensors are shown on the right. The numbering in the control law name denotes the number of the individual in the first generation of the experiment. Label #ind indicates the number of individuals matching the reference actuation spectrum.

The actuation spectra of these alternative solutions are shown in figure 8, with their associated histograms of sensor participation. Of particular interest is the case (figure 8 a,b) where the control law $\mathscr{F}_{191(HS)}^{GP}$ provokes a well-defined actuation at $45~\text{Hz}$ . From its histogram, it appears that the hot-wire $s_{7}$ (same as in $\mathscr{F}_{W(HS)}^{GP}$ discussed earlier in § 4.1) is the only essential sensor. It follows that the selection of a dominant sensor is not the only condition to obtain a given feedback frequency. The scaling in the control law or the combination with another sensor(s) also plays an important part in determining the behaviour of the solution. The control law $\mathscr{F}_{203(HS)}^{GP}$ (figure 8 c,d) selects 23 Hz as the dominant actuation frequency, which is very close to the open-loop frequency which maximizes $J_{K}$ in the HS configuration (see figure 4 d). We have not performed a dedicated GP search to maximize $J_{K}$ in the HS flow, but this individual appears as a candidate to fulfil this objective. The other solutions found include well-defined peaks at actuation frequencies very close, yet distinctly different from the optimal actuation (35 Hz) for maximizing $W$ : $\mathscr{F}_{231(HS)}^{GP}$ 37 Hz (figure 8 e,f) and $\mathscr{F}_{238(HS)}^{GP}$ 32 Hz (figure 8 g,h). This confirms that an automatic selection of the sensor and an optimized scaling of the signal(s) within the control law can tune a mixing layer synchronisation frequency with high precision.

Finally, in spite of the MISO design of the control system in this experiment, and the ability to impose only a single actuation frequency in the open-loop case, the closed-loop control $\mathscr{F}_{230(HS)}^{GP}$ might even be able to produce some kind of frequency cross-talk effect, as evidenced in the spectra in figure 8(i,j), where two well-defined peaks can be observed. Here, the first peak is a low-frequency modulation of the actuation frequency based around the second peak. This case, however, does not appear in sufficient number of samples to provide a solid base for further discussion of the dynamics involved. Finally, all of these examples represent control laws either created randomly in the first GP generation or created by modification of later individuals by random mutations. This means that they have not undergone the optimization process and might still be improved if an objective function would target them.

4.3 The periodicity of feedback control

The linear approximation of the GP control law expression has been performed using the data from the actuated flow, i.e. where the controlled state of the mixing layer is fully established. Hence, the synchronization mechanism found by GP greatly simplifies the dynamics of the actuated mixing layer in comparison to the natural flow, presented in § 3.1. For both open- and closed-loop control, the input frequency of the actuation system was recovered by the sensor system, as shown in the spectra in figure 6. It seems therefore that this phenomenon called the lock-on flow in the literature (Baek & Sung Reference Baek and Sung2000) is a crucial lead for understanding the potential capabilities of the frequency selection by the feedback control. In our experiments, the amplitude of actuation is large and the energy of perturbations inserted into the mixing layer is of the same order of magnitude as the energy of the unactuated flow ( $U_{jet}\approx U_{c}$ ). In this case it is expected (Oster & Wygnanski Reference Oster and Wygnanski1982; Fiedler & Mensing Reference Fiedler and Mensing1985) that the mixing layer will be well organized at the frequency of actuation in the streamwise region where these perturbations are amplified (see also figure 9).

For closed-loop control this means that once the first perturbation created by the actuators reaches the sensor array, every consequent cycle of actuation might be triggered by the perturbations created by one of the previous cycles. This is also the reason, why the observed closed-loop actuation becomes so periodic; establishing a controlled state means triggering a perturbation of specific properties. The controller then just acts on the perturbation and the cycle is self-repeating. Therefore, the basic requirement is that the perturbation created by actuation is amplified sufficiently to be captured by the sensor array. Here, the well-known mechanism of wave amplification by a shear flow (Michalke Reference Michalke1965; Ho & Huerre Reference Ho and Huerre1984) contributes to the determination of which types of perturbations can be used for feedback synchronization at a given location of the sensor array.

Nonlinear mechanisms present in the turbulent mixing layer can sometimes affect a cycle in unpredicted ways. This is why the closed-loop control can never be as periodic as the open-loop, i.e. the next cycle depends on how the previous cycle was registered by the sensors. Clearly, the control law must be designed in such a way as to minimize the negative effects of random flow perturbations on the repeated creation of an optimal actuation pulse.

5.1 Frequency response to periodic forcing

The optimal closed-loop control laws presented in § 3 (see figures 5 and 6), demonstrate a significant ordering of the flow, which corresponds to the dominant frequency of the actuation signal. This is confirmed by the spectra of the sensor signals in figure 6. In the case of the closed-loop control, these frequencies are not fixed; they are continuously established by the control law as a function of the signals from the selected sensors. Yet, in both open- and closed-loop controlled flow, the achieved lock-on states appear very similar when we observe the flow pseudovisualizations. Therefore, we consider that a synchronized state of the flow, obtained either by open- or closed-loop, is identical as long as the dominant frequencies measured in the flow are the same. For this reason, open-loop experiments are used to analyse the actuated flow dynamics for a range of discrete forcing frequencies. The open-loop experiments provide a more complete sample base where we can track the physical phenomena, not only on the few known closed-loop solutions, but rather on many different open-loop actuation frequencies in between.

Figure 9.

Frequency response to periodic forcing at $f_{a}$ with respect to individual hot-wire probes at $x=200~\text{mm}$ , for the (a) $Re_{{\it\theta}}=850$ (LS) and (b) $Re_{{\it\theta}}=1400$ (HS) flow configurations. Duty cycle used is $50\,\%$ . Measurements are taken from the spectra of the flow actuated by open-loop in the range of $1~\text{Hz}<f_{a}<100~\text{Hz}$ . The horizontal axis denotes the range of actuation frequencies employed. The vertical axis represents the normalized amplitude of power spectral density at the frequency $f_{a}$ . Here $\text{PSD}(s_{i};f_{a})[f]$ denotes the amplitude of the power spectral density of sensor signal $s_{i}$ measured at the frequency of actuation $f_{a}$ and $P(s_{i};f_{a})$ represents the power of the signal. Values of the vertical axis are shifted by $m$ for clarity (where $m=0,1,2,3,4,5,6$ ). The symbols (see legend) mark the frequencies and the sensors which are selected by the different closed-loop control laws.

Figure 9 shows amplitudes of power spectral density of the mixing layer flow as a function of the frequency of open-loop actuation $\text{PSD}(s_{i};f_{a})$ , for the LS and HS configurations. The amplitude is normalized by the power of the signal. This serves as an estimation of the amplitude of the fluctuations appearing at the frequency of actuation, with respect to the combined amplitudes of fluctuations on all other resolved frequencies.

In figure 9(b), the hot-wires $s_{1}$ , $s_{4}$ and $s_{7}$ are used by the control law $\mathscr{F}_{W(HS)}^{GP}$ (3.3). From figure 7 we can see that probe $s_{7}$ is dominant in building the actuation signal. In figure 9(b) we note that the most amplified signal that this probe can register, corresponds to the actuation frequency of $35~\text{Hz}$ (see legend); exactly what the closed-loop control law selects as the dominant feedback frequency. Moreover, in figure 9(a), we observe that the PSD of the sensor $s_{7}$ is maximized for an actuation frequency approximately equal to $21~\text{Hz}$ , precisely the frequency found by using the GP control law $\mathscr{F}_{W(HS)}^{GP}$ (trained in the HS configuration) in the off-design LS configuration (figure 6 b). This observation suggests that the robust behaviour presented in § 3.3 may result from the dominant role of $s_{7}$ , and from the perturbation amplification properties of the actuated mixing layer that makes $s_{7}$ sensitive to $35~\text{Hz}$ in HS and $21~\text{Hz}$ in LS configurations, respectively. Hot-wire probes $s_{10}$ to $s_{19}$ were not used in the optimal HS control law, but are shown here to illustrate what kind of actuation frequencies they detect. For example, if the control law could be based on $s_{10}$ , one might expect a feedback on a frequency from 10 to $30~\text{Hz}$ , since these actuation frequencies appear most amplified at this probe’s location (figure 9).

The symbols in figure 9 denote the selected feedback frequencies and the dominant sensors for the closed-loop control laws presented in figures 5, 6 and 8. We can conclude that the frequency selected by using the dominant sensor in the control law is always well amplified. The control law $\mathscr{F}_{K(LS)}^{GP}$ (3.2), for example, has sensor $s_{19}$ as the dominant sensor and this control always selects a frequency around $12~\text{Hz}$ . In figure 9(a), we see that the signature of the open-loop actuation at $12~\text{Hz}$ is the most amplified for the sensor $s_{19}$ (filled circle). We can make similar statements for the rest of the closed-loop control laws marked on the diagram. In some cases the selected frequency was the most amplified for the sensor in question, but in other cases it was not a maximum. An example is the control law $\mathscr{F}_{191(HS)}^{GP}$ (empty circle) in figure 9(b). The frequency selected by this feedback is amplified, but it is not the maximum amplification for this sensor. We can therefore conclude that not only the most amplified, but also sufficiently amplified perturbations can be periodically returned by the feedback control.

The amplification envelopes shown in figure 9 resemble the universal amplification curve, proposed by Fiedler & Mensing (Reference Fiedler and Mensing1985), which follows a nonlinear growth until saturation, then a linear decay. In the case of sensor $s_{10}$ , the decay is interrupted by a sudden extinction of amplitude much earlier than for other sensors. This occurs for both LS and HS configurations, albeit for different frequency ranges. As we will discuss in the following § 5.2, this is a consequence of the mean flow modifications induced by the forcing.

We can conclude that the feedback control created by the search process of GP can potentially be based on any frequency $f_{a}$ , if the growth rate of the created perturbation estimated for the lock-on condition ( $f=f_{a}$ ) is strictly positive and significant up to and including the location of the feedback sensors.

5.2 Vortex interactions

Figure 10.

Mean velocity variance profiles (at $x=200~\text{mm}$ ) in (a)  $Re_{{\it\theta}}=850$ (LS) and (b) $Re_{{\it\theta}}=1400$ (HS) mixing layer configurations, for different open-loop actuation frequencies ( $0~\text{Hz}<f_{a}<100~\text{Hz}$ , $dc=50\,\%$ ). Fluctuation magnitudes are scaled and shifted by the value of actuation frequency for clarity. Black dots denote the maximum value of $\langle {u_{i}^{\prime }}^{2}(t)\rangle _{T}$ . Variance profiles at open-loop frequencies 12 and 20 Hz, for the LS case, and 35 Hz in the HS case are highlighted in red (colour online only), as being the closest to the best closed-loop solutions presented in § 3.

In this section, we analyse the influence of the actuation frequency $f_{a}$ on the structure organization of the forced mixing layer and more precisely on the vortex interactions between the vortices created by the actuator system and those issued from the natural mixing layer. We first turn our attention to the evolution of the velocity variance profiles with respect to a range of open-loop actuation frequencies. The velocity variance profiles for actuation frequencies from 0 to $100~\text{Hz}$ are shown in figure 10, for the LS (a) and the HS (b) flow configurations. In both flow configurations, the respective natural profiles feature a single maximum, located between sensors $s_{10}$ and $s_{13}$ . As the actuation is applied with a progressively higher frequency, the vertical position of the maximum of variance is shifted vertically downwards. The single maximum of fluctuations achieves the lowest vertical position for relatively low frequencies of actuation: 8–10 Hz for LS and around 15 Hz for HS. The downward shift of the maximum of fluctuations indicates that the entrainment of the mixing layer is increased and the mixing layer is vectored towards the low-speed side. If the actuation frequency is still increased past this range, we observe a return of the maximum upwards, and a first appearance of a second, smaller velocity variance maximum. This second maximum starts to be visible from 10 to 12 Hz for the LS case in figure 10(a), and fully emerges as an equal to the first maximum for actuation at approximately 20 Hz. This significant discontinuity in the shape of the profiles appears around the actuation frequencies which maximize objective function $W$ . For both flow configurations, this change in the profile shape also corresponds to the abrupt extinction of the normalised PSD amplitude for the sensor $s_{10}$ , clearly visible in figure 9(a,b). The variance profiles featuring two maxima of velocity variance can be interpreted as representing the vortex pairing in the mixing layer (Winant & Browand Reference Winant and Browand1974; Ho & Huang Reference Ho and Huang1982). The sensor $s_{10}$ is the closest to the middle of the mixing layer and can find itself between two vortices if vortex pairing occurs (one vortex on top of the other, with the hot-wire vertically in between). For the HS case (figure 10 b) this discontinuity is much less pronounced, but is still quite discernible.

Figure 11.

Receding location (indicated by the arrow) of vortex pairing for various open-loop frequencies (from unactuated to 65 Hz) in the HS flow configuration. The approximate streamwise location of the sensor rake is indicated with white crosses. Flow snapshots correspond to: unactuated (a), $f_{a}=15$  Hz (b), 5 Hz (c), 30 Hz (d), 10 Hz (e), 65 Hz (f).

It follows that by actuating in a given frequency range, we can promote vortex pairing to occur at the sensor location, while for other actuation frequencies no pairing is observed by the sensors. Smoke visualizations of the flow for several actuation frequencies are shown in figure 11. The selected snapshots all show a typical location of vortex pairing and the associated actuation frequency. We can clearly observe that this location is changing with the actuation frequency and for a certain case it simply coincides with the sensor location. The objective function $W$ is designed to select a thick velocity variance profile, and so the optimal control laws aim to cause the vortex pairing which dominates the sensors.

The observed phenomena are in general accordance with the criterion of Moore & Saffman (Reference Moore and Saffman1975) which guarantees a stable vortex arrangement in a turbulent mixing layer. The criterion is $\ell \gtrsim 3.5\times {\it\delta}_{{\it\omega}}$ , where $\ell$ is the spacing between two vortices, for a single vortex to be stable. In the case of actuated mixing layer, the distance between two vortices is approximated as $\ell =U_{c}/f_{opt}$ , where $f_{opt}$ is the dominant frequency in the flow resulting from optimal open-loop actuation and the best GP closed-loop control laws. Resulting values are given for reference in table 1 for closed-loop control laws $\mathscr{F}_{W(LS)}^{GP}$ (3.1), $\mathscr{F}_{K(LS)}^{GP}$ (3.2) and $\mathscr{F}_{W(HS)}^{GP}$ (3.3). In the case of maximization of $K$ for the LS configuration, the control law produced coherent vortices at a frequency of approximately 12 Hz. In the velocity variance profile ( $b_{K(LS)}$ in figure 5 a) no sign of vortex pairing is evident, i.e. a single maximum is observed. The corresponding values of $\ell$ and ${\it\delta}_{{\it\omega}}$ in table 1 show that the criterion for a stable single vortex has been achieved. In contrast, the values of vortex spacing for control laws which maximize $W$ are just below the local vorticity thickness criterion. Such values confirm that the vortices have begun to deteriorate due to interaction with the neighbouring vortices, as one would expect when vortex pairing commences. Therefore, the two objective functions are maximized when the controller is triggered either by a large coherent vortex (in the case of $K$ ) or by a pair of vortices close together (in the case of $W$ ).

Figure 12.

Example of the streamwise evolution of the LS mixing layer actuated with $f_{a}=10~\text{Hz}$ . The profiles of averaged streamwise velocity fluctuations $\langle {u^{\prime }}^{2}\rangle _{T}$ obtained by particle image velocimetry (PIV) are superimposed over an instantaneous snapshot of smoke visualization of the flow. The profiles which exhibit two peaks correspond to streamwise location where vortex pairing is clearly evident. Upstream of this location, the resurgence of the natural instability (a) can be observed between two cycles of actuation. Vortex (b) is another K–H vortex, already starting to be influenced by the vortex created by actuation (c). The vortex pairing observed at the right-hand side of the frame appears to involve a large vortex (e), created by the previous actuation cycle, as well as several older K–H vortices (d) coalescing around it. The PIV results were obtained using a commercial Dantec system, with a spatial resolution of 3.45 mm between vectors.

Table 1.

Comparison of vortex spacing $\ell$ and vorticity thickness ${\it\delta}_{{\it\omega}}$ values for the best GP control laws in LS and HS mixing layer configurations. These values are used for estimation of the criterion of stability of a vortex in a turbulent mixing layer $\ell \gtrsim 3.5\times {\it\delta}_{{\it\omega}}$ , as given by Moore & Saffman (Reference Moore and Saffman1975).

In addition to vortex pairing, the actuated state can also feature very complex interactions of actuation and natural Kelvin–Helmholtz vortices, as illustrated in figure 12. The visualization shown here, along with the statistically converged data from PIV acquisitions, allow us to shift the analysis from a fixed framework of the hot-wire sensors and explore the spatial development of an actuated mixing layer, albeit for only one frequency of actuation. This example is a typical case of what can be expected to occur when the LS mixing layer is actuated at a low frequency ( $f_{a}=10~\text{Hz}$ ). The wavelength of actuation is so large that it allows a resurgence of the natural instability between two cycles of actuation. In figure 12, several K–H vortices (a) are observed appearing in the wake of the last known actuation vortex (c). Vortex (b) directly in the trail of the actuation vortex (c) is another K–H vortex already starting to be destabilized by the influence of the bigger, turbulent actuation vortex. Further downstream, we can observe older K–H vortices (d) being entrained by a large vortex (e) which is a product of the previous cycle of actuation. The corresponding profiles of $\langle u^{\prime 2}\rangle _{T}$ show a typical shape associated with vortex pairing: two peaks of comparable amplitudes. In this case, the pairing occurs around $350~\text{mm}<x<450~\text{mm}$ in the streamwise direction. However, the process visible in this example corresponds to the ‘collective interaction’ (Ho & Nosseir Reference Ho and Nosseir1981), where the K–H vortices are entrained around a much larger vortex created by a large amplitude actuation.

5.3 The impact of convection speed on the frequency selection

Since the perturbations produced by actuation dominate the flow, one might expect that the frequency of their repetition by feedback would be locked to a frequency based on the convective time. This does appear to be true for the closed-loop control which maximizes objective function $K$ . For the LS configuration, the frequency based on the convective time would be $f_{c(LS)}\approx 12~\text{Hz}$ , with respect to $U_{c}$ given in the table of figure 1. In § 3.2, the frequency of the closed-loop control law $\mathscr{F}_{K(LS)}^{GP}$ is found equal to $12~\text{Hz}$ , in agreement with the convective frequency. In the HS case, the corresponding frequency would be $f_{c(HS)}\approx 24~\text{Hz}$ . Although the GP was not applied to maximize $K$ in the HS configuration, the open-loop mapping in figure 4(d) confirms a maximum of $J_{K}$ at $(f_{a},dc)_{K(HS)}=(24~\text{Hz},70\,\%)$ , which corresponds to the aforementioned frequency.

However, the speed at which the velocity fluctuations of different length scales are convected is not uniform across the vertical extent of a turbulent mixing layer. Studies by Wills (Reference Wills1964) and most recently by Buxton, de Kat & Ganapathisubramani (Reference Buxton, de Kat and Ganapathisubramani2013) show that a ‘global’ convection velocity, containing all the length scales present in the flow, is similar to the mean velocity profile $U(y)$ of the mixing layer, and that fluctuations on the low-speed side convect on average at speeds greater than the mean and fluctuations on the high-speed side are observed to convect at speeds less than the mean. If the local convection characteristics are the only parameters which determine the frequency of feedback, then the vertical location of the sensor might be directly linked with the frequency that the closed-loop control based on that sensor will select.

Figure 13.

Participation of hot-wire sensors in various control laws and the resulting lock-on frequencies. The filled circles correspond to the sensors appearing in all the individuals matching a particular lock-on frequency. Sensors depicted with empty circles are not dominant, but appear in the control law from which the PSD of the actuation signal has been presented (see figure 8), or whose expression is given in the article (see § 3). The vertical dashed line differentiates between results obtained by tests in the LS configuration, from the results obtained in the HS configuration. Note that the control law $\mathscr{F}_{W(HS)}^{GP}$ appears two times, since it was tested in both LS and HS configurations yielding a different selected frequency (see figure 6).

In order to check this hypothesis, we can plot a summary of all of the closed-loop control laws in figure 13. Here we plot the participating sensors in each control law presented so far, versus the lock-on frequency they produce (see § 4). The sensors appearing in all of the matching individuals are referred to as the ‘dominant’ sensors (filled circles). The other sensors which appear in the reference control law are called the ‘supporting’ sensors (empty circles), and they do not appear in all the individuals which match the feedback frequency of the reference control law.

We can immediately recall that the control laws $\mathscr{F}_{W(HS)}^{GP}$ and $\mathscr{F}_{191(HS)}^{GP}$ share the same dominant sensor $s_{7}$ (see § 4.2), but produce different lock-on states (35 and 45 Hz). This already shows that the vertical location of that sensor does not impose a unique feedback frequency, based on the local convection speed.

Furthermore, there appears not to be a link between selecting a sensor on the high-speed side of the mixing layer and obtaining a high-frequency feedback. An obvious example is the control law $\mathscr{F}_{238(HS)}^{GP}$ which achieves a lock-on frequency of 32 Hz using only the high-speed side sensor $s_{16}$ , whereas $\mathscr{F}_{W(HS)}^{GP}$ achieves 35 Hz using always the low-speed side sensor $s_{7}$ .

We can conclude that some lock-on states produced by closed-loop control may be related to the convection speed, but certainly not all of them. In addition, the outermost sensors ( $s_{1}$ and $s_{19}$ ) are effectively in the far field of the mixing layer for both LS and HS configurations. In their case, the local convection speed observations from Buxton et al. (Reference Buxton, de Kat and Ganapathisubramani2013) probably do not apply. From these results, there is no clear and direct link to be made between the vertical location of the sensors and the selected frequency of feedback. If the feedback control was directly proportional to only one sensor, perhaps the vertical location would be the only parameter. This type of configuration is worth examining in future studies.

In the reported experiment, the transitional stage from the unactuated flow to the fully established actuated lock-on state is not elucidated. It is, however, this transitory period which must be studied in order to fully understand the selected lock-on frequency. It is also worth noting that the selection of the lock-on frequency is very robust. Every time the closed-loop control is activated using the same control law, the resulting lock-on frequency is the same, provided that there is no sensor drift between attempts. In other words, the lock-on state is independent of the unsteady conditions of the flow at the moment the control is activated.