2.1. Water tunnel

Experiments were carried out in a hydrodynamic channel in which the flow is driven by gravity. The flow is stabilized by divergent and convergent sections separated by honeycombs. The quality of the main stream can be quantified in terms of flow uniformity and turbulence intensity. The standard deviation ${\it\sigma}$ is computed for the highest free-stream velocity featured in our experimental set-up. We obtain ${\it\sigma}=0.059\ \text{cm}\ \text{s}^{-1}$ which corresponds to turbulence levels of ${\it\sigma}/U_{\infty }=0.0023$ . For the present experiment the flow velocity is $U_{\infty }=2.1\ \text{cm}\ \text{s}^{-1}$ giving a Reynolds number based on step height $Re_{h}=U_{\infty }h/{\it\nu}=430$ , where ${\it\nu}$ is kinematic viscosity. Following the assumptions of Hervé et al. (Reference Hervé, Sipp, Schmid and Samuelides2012) the Reynolds number was chosen to ensure a sub-critical linear two-dimensional flow.

2.2. Backward-facing step geometry and upstream perturbation

The BFS geometry and the main geometric parameters are shown in figure 1. BFS height is $h=1.5\ \text{cm}$ . Channel height is $H=7\ \text{cm}$ for a channel width $w=15\ \text{cm}$ . The vertical expansion ratio is $A_{y}=H/(h+H)=0.82$ and the spanwise aspect ratio is $A_{z}=w/(h+H)=1.76$ . The injection slot is located $d/h=2$ upstream of the step edge.

Figure 1.

Sketch of the BFS geometry and definition of the main parameters.

The principle of the method described in Hervé et al. (Reference Hervé, Sipp, Schmid and Samuelides2012) is to devise an input–output model for the flow based on experimental data. This model is used to compute actuation aimed at negating incoming upstream noise, thereby preventing its amplification. Because our sensor is two-dimensional in the symmetry plane and our actuator can only deliver span-wise homogeneous actuation, a two-dimensional upstream perturbation is required for effective control. As shown in figure 1 a two-dimensional obstacle with a rounded leading edge of height $O_{h}=0.8\ \text{cm}$ has been placed at $d_{h}=15\ \text{cm}$ upstream from jet injection ( $12h$ from the step edge) to perturb the flow. Because of the low Reynolds number the flow stays two-dimensional.

2.3. Sensor: two-dimensional real-time velocity field computations

The sensors used as inputs for the closed-loop experiments are visual sensors, i.e. regions of the two-dimensional PIV (particle image velocimetry) velocity fields measured in the symmetry plane as shown in figure 4 below. The flow is seeded with $20\ {\rm\mu}\text{m}$ neutrally buoyant polyamid seeding particles. They are illuminated by a laser sheet created by a 2 W continuous laser beam operating at ${\it\lambda}=532\ \text{nm}$ . Images of the vertical symmetry plane are recorded using a Basler acA 2000–340 km 8 bit CMOS camera. Velocity field computations are run on a Gforce GTX 580 graphics card. The algorithm used to compute the velocity fields is based on a Lukas–Kanade optical flow algorithm called FOLKI developed by Besnerais & Champagnat (Reference Le Besnerais and Champagnat2005). Its offline and online accuracy has been demonstrated and detailed by Champagnat et al. (Reference Champagnat, Plyer, Le Besnerais, Leclaire, Davoust and Le Sant2011) and Gautier & Aider (Reference Gautier and Aider2014b ). Furthermore this acquisition method was successfully used in Davoust, Jacquin & Leclaire (Reference Davoust, Jacquin and Leclaire2012) and Gautier & Aider (Reference Gautier and Aider2013). The size of the velocity fields is $17.2\ \text{cm}\times 4.6\ \text{cm}$ . They are computed every ${\it\delta}_{t}=40\ \text{ms}$ , for a sampling frequency $F_{s}=25\ \text{Hz}$ .

2.4. Uncontrolled flow

The swirling strength criterion ${\it\lambda}_{Ci}$ is an effective way of detecting vortices in two-dimensional velocity fields, introduced and improved by Chong, Perry & Cantwell (Reference Chong, Perry and Cantwell1990) and Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999). For two-dimensional data the swirling criterion is defined as ${\it\lambda}_{Ci}=\sqrt{4\det (\boldsymbol{{\rm\nabla}}\boldsymbol{ u })-\text{tr}(\boldsymbol{{\rm\nabla}}\boldsymbol{ u })^{2}}/2$ (when this quantity is real).

Figure 2(a,b) shows the mean velocity amplitude fields for the perturbed (with obstacle) and unperturbed (without obstacle) flows. Figure 2(c,d) shows corresponding ${\it\lambda}_{Ci}$ snapshots, highlighting the perturbations caused by the upstream obstacle. Figure 2(d) shows the steady stream of vortices created by the obstacle interacting with the recirculation. Quantitatively, ${\it\lambda}_{Ci}$ in the perturbed flow is an order of magnitude higher than in the flow unperturbed flow. The boundary layer thicknesses at the step edge for the flow with and without obstacles are ${\it\delta}=1.34h$ and ${\it\delta}=1.73h$ respectively.

Figure 2.

Mean velocity magnitude contour fields for the unperturbed (a) and perturbed (b) flow. Instantaneous snapshots showing contours of ${\it\lambda}_{Ci}$ for the unperturbed (c) and perturbed (d) flow.

Turbulent kinetic energy (TKE) is defined as $k(x,y,t)=(u^{\prime }(x,y,t)^{2}+v^{\prime }(x,y,t)^{2})/2$ , where $u^{\prime }$ , $v^{\prime }$ are longitudinal and vertical velocity fluctuations. Figure 3(a) shows the time-averaged TKE field $\langle k(x,y)\rangle _{t}$ downstream of the step for the unperturbed case, while figure 3(b) shows the perturbed case. The field exhibits two regions of high TKE. The lower region corresponds to the perturbation-induced recirculation bubble. The upper region corresponding to perturbations induced by vortices shed by the upstream obstacle.

Figure 3.

TKE field $\langle k(x,y)\rangle _{t}$ downstream of the BFS: unperturbed (a) perturbed (b).

Figure 4.

Schematic description of the main terms used in the ARMAX model.

2.5. Actuation

In Hervé et al. (Reference Hervé, Sipp, Schmid and Samuelides2012) actuation is a Gaussian flow sink/source placed above the step, which is not physically feasible. In our case, actuation is provided by a flush slot jet, 0.1 cm long and 9 cm wide. This actuation has been chose to obtain a perturbation as homogeneous along the spanwise direction as possible. The jet angle to the wall is 45°. The slot is located 3 cm ( $2h$ ) upstream of the step edge (figure 1). Jet flow is induced using water from a pressurized tank. It enters a plenum and goes through a volume of glass beads designed to homogenize the incoming flow. Jet amplitude is controlled by changing tank pressure. Because channel pressure is higher than atmospheric pressure this allows us to provide both blowing and suction. The convection time from jet injection to measurement area is 2 s ( ${<}0.5\ \text{Hz}$ ). The maximum actuation frequency $f_{a}$ is approximately 1 Hz which is sufficient for these experiments.

The control law output is velocities, and these are converted into pressure commands using a pre-calibrated transfer function.

3.2. Model computation

Figure 6 shows a small segment of the calibration time series. The forcing law $u(t)$ used in these series is one of pseudo-random pulses. Pulses are made to occur at random intervals, long enough for the effects of the previous pulse to have subsided before the next pulse. During these intervals the only input to the system is $s(t)$ . This allows the effects of actuation and upstream perturbations to be computed using a single time series. Impulse amplitude for actuation $u(t)$ should be chosen such that it is high enough to affect the output $m(t)$ but low enough to avoid perturbations of the upstream output $s(t)$ . The mean value was removed for $m(t)$ because the focus is on suppressing fluctuations in the output. Calibration data were acquired over 25 min. Figure 7 shows the auto-correlation function for $m(t)$ . A quasi-oscillatory behaviour can be observed. It can be used to choose $n_{a}$ which is such that $n_{a}{\it\delta}_{t}$ equals half the oscillatory period, as recommended by Hervé et al. (Reference Hervé, Sipp, Schmid and Samuelides2012).

Figure 7.

Auto-correlation function for $m(t)$ .

Figure 8 shows the response to an impulse, which can be used to evaluate the coefficients $n_{du},n_{bu},n_{ds},n_{bs}$ . The time delay $t_{du}=2.5\ \text{s}$ between the beginning of the actuation and the response gives $t_{du}=n_{du}{\it\delta}_{t}$ . The upstream sensor is located 3.5 cm upstream of the jet injection. Assuming perturbations travel at channel velocity, this implies a time delay of $t_{ds}\approx 1.7\ \text{s}$ for an upstream disturbance to affect the output, thus $n_{ds}{\it\delta}_{t}=t_{du}+t_{ds}$ .

Figure 8.

Output impulse response.

Let $t_{bu}$ be the time during which an impulse in $u$ affects the output, as shown in figure 8, then $n_{bu}{\it\delta}_{t}=t_{bu}$ . Because the response to an impulse in $s$ is more difficult to distinguish we assume $n_{bs}=n_{bu}$ . Finally $n_{c}$ is chosen after the other coefficients have been fixed in order to get the best possible agreement between model and real outputs. Table 1 summarizes the final coefficients used in the computation of the ARMAX model using the Matlab armax function (Ljung Reference Ljung1999); it also shows the corresponding time delays and averages in seconds.

Figure 9.

(a) Calibration data set, model performance (dotted grey) compared to experimental results (in black). (b) Validation data set, model performance (dotted grey) compared to experimental results (in black).

Figure 9(a) compares ARMAX output to the source signal for the calibration series. Agreement is good at 96 %. Figure 9(b) compares ARMAX output to the source signal for the validation series; agreement is slightly lower at 94 %. Here $s,u,m$ are initialized at zero, which explains the poorer agreement at the start.

Table 1.

ARMAX coefficients.

3.3. Linearity

A major underlying assumption of this approach is the linearity of the system. In our set-up this was checked by imposing periodic pulsed forcing, for two different amplitudes. For both amplitudes, the response in $m(t)$ was averaged over each impulse. This phase-averaged response is shown in figure 10. Impulse amplitude ratio is also given for comparison. The signals were shifted in time to facilitate comparison. Figure 10 shows how a change in impulse amplitude leads to a proportional change in response amplitudes, confirming the linear behaviour of the flow. Linearity was also checked when varying the size of the window where TKE is computed. Averaging over smaller windows, closer to the step, where nonlinearities are weaker, did not improve system linearity.

Figure 10.

Time evolutions of $m(t)$ in response to two short impulses (square signals) of different amplitudes (thick and thin lines). The signals have been shifted in time to better highlight the linear nature of the response.