Exp Fluids (2013) 54:1430 DOI 10.1007/s00348-012-1430-1

RESEARCH ARTICLE

Experimental detection of a periodically forced turbulent boundary layer separation Timothe´e Chabert • Julien Dandois Eric Garnier • Laurent Jacquin

•

Received: 14 May 2012 / Revised: 30 November 2012 / Accepted: 8 December 2012 / Published online: 29 January 2013 Ó Springer-Verlag Berlin Heidelberg 2013

Abstract Wind tunnel experiments are conducted to investigate the effect of periodic blowing over a non-slotted NACA-type flap equipped with seven pulsed slots. The tests are performed at Re ^ 3 9 106 for different deflection angles ranging from 2° to 35°. Wall-mounted hot-films are chordwise distributed on the flap. The purpose of the experiments is to assess the ability of wall shear stress signals to detect flow separation with and without periodic control, and to be used as reliable feedback information in a closed-loop control framework. Laser beam tomoscopy is first used to visualize flow structures with and without flow control. Two separation criteria based on higher-order statistical moments of hot-film signals are proposed to detect flow separation. They present the advantage of being independent of the free-stream velocity. The two criteria are finally practised to follow the steady-state response of the system and determine its static map local gradient.

1 Introduction Turbulent boundary layer separation over flaps is responsible for large performance losses during take-off and

This article is part of the collection Topics in Flow Control. Guest Editors J.P. Bonnet and L. Cattafesta. T. Chabert (&)  J. Dandois  E. Garnier Applied Aerodynamics Department, ONERA, The French Aerospace Lab, 92190 Meudon, France e-mail: [email protected] L. Jacquin Fundamental and Experimental Aerodynamics Department, ONERA, The French Aerospace Lab, 92190 Meudon, France

landing phases of an aircraft flight, including loss of lift and drag increase. Therefore, arrangements of slats and flaps are used to maintain the flow attached and to prevent flow separation. Since periodic excitation is able to maintain or reattach the flow on a non-slotted flap (see Nishri and Wygnanski 1998; Darabi and Wygnanski 2004a, b), this technique is studied in order to simplify the mechanical systems of high-lift configurations. To provide only the required amount of momentum for the flow to remain attached or to reattach it, the control device system has to be adapted, which means a feedback strategy is required. Closed-loop flow control has already been successfully implemented on airfoils with different kinds of feedback laws. Non-intrusive measurement techniques like wall pressure sensors, more convenient for applications, have already been largely employed. Using pulsed jets, Becker et al. (2007) succeeded in controlling flow separation over the single-slotted flap of a NACA 4412 high-lift configuration by maximizing the pressure difference between two chordwise distributed points thanks to an extremum-seeking algorithm. The optimization can also be done by maximizing the lift coefficient or by minimizing the drag coefficient. Both coefficients can also be optimized in the same time as proposed by Tian et al. (2006a) whose closed-loop algorithm minimizes the drag-to-lift ratio. Tian et al. (2006b) try to reduce the pressure fluctuations measured by unsteady pressure sensors integrated in a NACA 0025 airfoil. The feedback used by Benard et al. (2011) to control a NACA 0015 airfoil by means of plasma actuators is the wall static pressure measured by a single sensor located close to the leading edge. The pressure signal monitored by the sensor reveals that the onset of separation is accompanied by a large increase in pressure fluctuations. Thus, the root mean square value (referred to as rms value hereafter) of the pressure signal exhibits a sharp rise at

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incipient separation. This detection method based on the monitoring of rms pressure was first proposed by Pack et al. (2005) but on wall shear stress measurements, and further developed by Seifert and Pack-Melton (2006). Actually, the detection of a turbulent boundary layer separation location by means of wall shear stress measurements is not commonly used in experiments, but a new wave of interest for this technique has emerged with the closed-loop flow control framework expansion. Preliminary works showed the ability of non-calibrated hotfilm sensors to detect flow separation. The so-called phase reversal phenomenon is known to be a particularly wellsuited criterion to detect laminar boundary layer separation since its first description by Stack et al. (1988). In order to detect separation on an airfoil in transonic flow by means of hot-film array, Meijering and Schroder (2001) prefer the correlation coefficient computation of the time traces between two adjacent sensors to the phase reversal since the latter method is ineffective in turbulent regime in their case. To detect separation, the calculation of correlation coefficient was previously proposed by Nakayama et al. (1993). It is interesting to notice that transition location from laminar to turbulent regime is detected by computing rms and skewness values of hot-film signals in the work of Meijering and Shroder (2001). Those two methods will be tested to detect separation in the present work. Moreover, Cuvier et al. (2011) use the skewness direct calculation of hot-film signals to determine whether the flow over a ramp is reattached or not by means of fluidic vortex generators. Considering also the work of Seifert and Pack-Melton (2006) previously mentioned, it appears that flow separation on an airfoil can be detected by means of hot-film sensors which do not need to be calibrated. As a consequence, some authors already use this kind of measurements as an input of a closed-loop flow separation control scheme. Poggie et al. (2010) use hot-film sensors to follow the separation line which moves upstream towards the leading edge of a flap as its deflection angle increases. The closed-loop strategy is as follows. The plasma actuation is activated when the shear stress on the flap exceeds a predetermined threshold. Shear stress signal is not calibrated to physical values in their case. Hot-film signals are also used by Rethmel et al. (2011) on the chord of a NACA 0015 airfoil. The mean power dissipated by a sensor close to the leading edge is computed and triggers the plasma actuation when entering a predetermined dead-zone, namely a high and a low threshold are defined to prevent the closed-loop to oscillate around a single threshold. In the same way, the evolution of the rms value computed each 5 9 10-2 s of hot-film signal is used by Packard and Bons (2012) to evaluate whether the flow over an oscillating NACA 643 618 airfoil is separated or not. Since the rms value time history exhibits a dramatic increase at separation, the closed-loop strategy consists in triggering the

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flow control by means of leading edge normal pulsed jets when the rms value goes beyond a predetermined threshold. In the following work, the objective is to define a suitable criterion to estimate experimentally the turbulent boundary layer separation location shifting on a gradually deflected flap. The experimental set-up is presented in Sect. 2. Particularly, on-surface hot-films are integrated into the flap. In Sect. 3, two criteria based on higher-order statistical moments are derived to detect flow separation and estimate the detachment length in the uncontrolled case. The periodic blowing effect is then studied in Sect. 4. It appears that it is necessary to determine whether the flow is reattached or fully separated and two ‘decision’ criteria are defined for that purpose. In our application, the implementation of extremum-seeking closed-loop strategy is foreseen, since it does not need any a priori modeling of the flow physics assuming the system can be represented by a static map which exhibits an extremum (see King et al. 2004). The so-called static map of a system, or steady-state map, is merely obtained as its stationary response to a given input. A static map can be further extended to multiple input variables (e.g. Becker et al. 2007). In Sect. 5, the two criteria previously defined are used to estimate the steady-state map of the system. Nevertheless, one key point of the extremum-seeking control is the local gradient estimation of the static map. To this end, a sinusoidal disturbance is superimposed on the system input and the demodulation between the system output and the above-mentioned disturbance provides finally the gradient (the detailed procedure can be found in Pamart et al. 2010). To be exhaustive, one must recall that Henning et al. (2008) propose two other methods to determine the gradient. The ability of the two criteria to follow the static behaviour of the system (i.e. to estimate the gradient) is investigated in the second part of Sect. 5. A slope-seeking approach is generally found more appropriate when flow control amplitude is precisely the control parameter (see Becker et al. 2007; Benard et al. 2010). The gradient estimation is the same as in the case of extremum-seeking.

2 Experimental set-up 2.1 Test model and measurements The experiments are performed at the ONERA ChalaisMeudon research center in an Eiffel-type wind tunnel whose test section is shown in Fig. 2. The model consists of a 867-mm-long flat plate and a non-slotted c = 220-mm chord long flap, based on a NACA 4412 airfoil shape, as shown in Fig. 1 (d defines the adjustable deflection angle in degrees). It is placed inside a cylindrical slightly divergent 1,750-mm-long test section. The model covers the 800 mm of the wind tunnel span. Carborundum is used at its leading

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Fig. 1 Sketch of the instrumented flap; the angle between the jets direction and the local tangent is 30°; x is the chordwise position

edge to trigger transition to a fully turbulent boundary layer. Surface oil flow visualizations have shown that flow was mostly two dimensional. The two-dimensional assumption has also been verified by spanwise pressure measurements on both flat plate and flap. The average free-stream velocity U0 is 34.5 m s-1, giving a Reynolds number Reflap = 6 9 105 based on the flap chord, and Remodel = 3 9 106 based on the total length of the model. The external turbulence level is 0.2 %. The average free-stream velocity is lowered to 20 m s-1 during the flow visualizations. Beam laser tomoscopy is installed to provide views of the smoke-seeded flow at different flap deflections with and without flow control. The laser plane is placed in the midplane of the test section in order to picture two-dimensional flow structures through a PhantomÒ high-speed camera. Sixteen SenflexÒ hot-film sensors are bonded chordwise on the flap and provide wall shear stress measurements. The sensor’s resistance is regulated with three Dantec StreamLines CTA, so that thirteen hot-film signals can be acquired simultaneously. The idea is to cover the entire flap chord. The hot-film signals are acquired for 40 s at a sampling rate of 10 kHz. This is performed by a National Instruments (referred to as NI hereafter) PXIe-8102 Real-Time controller equipped with a PXIe-6358 data acquisition card. To prevent aliasing, the signals are low-pass-filtered at 5 kHz by the StreamLines. Additionally, 23 static pressure measurements are available on the suction and pressure sides of the flap, both chordwise and spanwise distributed. 2.2 Actuators and control parameters Seven actuator segments are integrated into the flap along the span. Each of them consists of a fast switching two-

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states solenoid valve (FestoÒ MHE2) which allows compressed air to blow through a 0.25-mm-wide and 90-mm-long slot covering 80 % of the total flap span. The distance between the slots is 7 mm that is 0.9 % of the span. This leaves 64 mm (8 % of the span) between the lateral walls and the extreme slots. The seven actuators are operated synchronously. The jets are downstream inclined at an angle of 30° with respect to the local flap tangent. Frequency and duty cycle of the valves are adjustable directly via the NI PXIe real-time controller previously presented. The jets flow rate is settled via a pressure regulator (also controlled with the PXIe) which controls the compressed air supply. The air supply flow rate is measured thanks to a sonic throat which is equipped with upstream and downstream pressure sensors and a thermocouple. The amount of flow control is quantified by the so-called momentum coefficient Cl, first introduced by PoissonQuinton (1948) (see also Poisson-Quinton and Lepage 1961) for steady blowing. It is similar to a thrust coefficient and is defined as qj Uj2 Sj qm Uj ¼ ; 1 2 2 2 q 0 U 0 S0 2 q0 U0 S0

Cl ¼ 1

ð1Þ

where the subscript j refers to the jet, and qm is the mass flow rate. Sj and S0 are the slots and the reference surfaces, respectively. The reference surface is defined as the flap surface. When periodic blowing is considered, the oscillatory momentum coefficient (see Greenblatt and Wygnanski 2000) is defined as qj hUj2 iSj hCl i ¼ 1 ; 2 2 q0 U0 S0

ð2Þ

where hi stands for the time averaging operator. In the case of a square excitation of the valve, Uj is  Umax if 0\t  aT; Uj ðtÞ ¼ 0 if aT\t  T; where a is the duty cycle and T the period. This yields 2 hUj2 i ¼ aUmax ; so hCl i ¼

2 qj aUmax Sj : 1 2 2 q0 U0 S0

Since hUj i ¼ aUmax ; the oscillatory momentum coefficient for a pulsed jet excited with a square signal is finally 2 1 qj hUj i Sj 1 hqm ihUj i ¼ : hCl i ¼ 1 a 2 q0 U02 S0 a 12 q0 U02 S0

ð3Þ

Consequently, if the duty cycle is 50 %, a = 1/2, the oscillatory momentum coefficient hCl i is twice the ‘steady’ momentum coefficient Cl (relation (1)) for the

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Fig. 2 a Front and b rear views of the model: wall shear stress sensors are integrated into the flap (blue zone)

same feeding flow rate and average ejection velocity hUj i: This can be appreciated in Fig. 3a which represents the ejection velocity for constant and periodic blowing with qm = 5 9 10-3 kg s-1 of feeding mass flow rate, and a 50 % duty cycle (DC). It is measured by a hot-wire located as close to the slot exit as possible. The ejection velocity in the case of periodic blowing is averaged over 10 periods. It appears that the jet velocity in the constant blowing case is half the jet velocity during the blowing phase of the periodic case. Since the average exhaust velocity is the same in both cases, the definition in relation (1) reflects no difference between constant and periodic actuation. On the contrary, relation (3) takes into account the additional amount of momentum in the periodic case. This formulation is then preferred to compute the momentum coefficient in the periodic blowing case thereafter. Additionally, the reduced frequency of actuation is defined as Fþ ¼

fc : U0

ð4Þ

The jet settling time, namely the transient duration before the jet has reached its permanent state in the periodic blowing case, can also be appreciated in Fig. 3a. The indicated settling time corresponds firstly to the switching time of the valve, which can be evaluated to 5–6 9 10-3 s (the manufacturer indicates a maximum switching time of 7 9 10-3 s, and secondly to the fluidic response time of the system, negligible with respect to the

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switching time). Furthermore, the jet stopping time appears to be slightly longer than the settling time. The valve switch-over from opened to closed, whose duration cannot be evaluated here, is possibly slower than the valve opening. In the following, the settling and stopping durations of the blowing jet are considered to be equal. Besides, Fig. 3b depicts a comparison between the ejection velocity calculated from the mass flow rate measurement and the ejection velocity measured by the hot-wire. Both ejection velocities increase linearly as mass flow rate increases, but with slightly different slopes. The average difference between them is around 10 m s-1. This discrepancy is not surprising since an integrated quantity is compared with a punctual measurement very sensitive to the hot-wire positioning with respect to a 0.25-mmwide slot. As a consequence, the ejection velocity is calculated from the mass flow rate measurement in this paper, in order to evaluate the momentum coefficient in the studied cases.

3 General features of the uncontrolled flow 3.1 Progressive separation of the flow The model was designed in order to have a progressive separation of the flow over the flap. This assumption is firstly checked with static pressure measurements. The distribution of pressure coefficient Cp along the flap chord is presented in Fig. 4a for increasing deflection angles.

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Fig. 3 a Excitation signal for the valves and exit velocity for constant and periodic blowing with qm = 5 9 10-3 kg s-1 of feeding mass flow rate, and 50 % of duty cycle; b comparison between the exit velocity calculated from the mass flow rate measurement and the exit

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velocity measured by the hot-wire against flow rate. Even though there is no upstream flow, U0 is assumed to be 34.5 m s-1 in the calculation of F?

Fig. 4 a Pressure coefficient Cp distribution along the chord flap for deflection angles from 5° to 25°; b visualization of the uncontrolled fully separated flow at d = 35°

A plateau of Cp appears over the suction side of the flap as the deflection increases and covers it almost completely for angles larger than d = 20°. This reveals that a recirculation zone is growing from the trailing edge towards the leading edge of the flap. The flow is fully attached up to d = 5°, then separates from the trailing edge before the separation location progressively goes up towards the leading edge as flap deflection increases. A visualization of the uncontrolled fully separated flow at d = 35° is proposed in Fig. 4b. Besides, it has been observed that the evolution of separation length against flap deflection is not subject to any hysteresis effect.

3.2 Estimation of the detachment length by means of hot-film signals Since the flow progressively separates from the trailing edge of the flap, the separation length can be estimated provided a suitable separation location tracer is available. Such a criterion has to be applied to the hot-films that are regularly distributed on the flap chord in order to follow the progression of the flow separation location. A first possible candidate is the ‘‘high-frequency standard deviation’’, proposed by Seifert and Pack-Melton (2006). This method proposes to monitor the standard deviation of the so-called

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high-frequency standard deviation which is the normalized and time-averaged high-frequency FFT coefficients summation. The FFT coefficients stem from a short-time fast Fourier transform (STFFT) of the hot-film voltage time history. This procedure has been tested here but results are not presented. Indeed, this method failed to detect the progressive separation of the flow since a sharp increase in fluctuations in the high-frequency band monitored by the method is observed on the whole flap at a specific angle of 15°. Besides, the FFT calculation is quite demanding in CPU resources. The other method proposed in this paper is based on the calculation of higher-order moments of the hot-film signal probability density function, that is, skewness and kurtosis, which are asymmetry and flatness measurements of a given distribution, respectively. For a stochastic signal x, skewness and kurtosis are defined as S¼

hðx  lÞ3 i ; r3

ð5Þ

K¼

hðx  lÞ4 i ; r4

ð6Þ

with l and r the mean value and the standard deviation of x, respectively. Since the flatness factor of a Gaussian distribution is 3, subtracting 3 to the present definition in relation (6) gives the so-called excess kurtosis. This subtraction is not performed in the following. Figure 5 depicts the contours of skewness and flatness factors on the entire flap for deflection angles varying from 2° to 35°. The transition from the attached to the separated zone is characterized by a sharp rise of skewness and kurtosis, as underlined by the dashed line in both representations. Actually, this line represents the defined thresholds for skewness and kurtosis: S = 0.5 and K = 3.5 for skewness and kurtosis, respectively. Both thresholds are defined only to track the sharp rise of both skewness and kurtosis at

incipient separation all along the flap. The advantage of this method is that the thresholds are the same for all hot-films and do not depend on free-stream velocity. Another method, not tested here, would consist in detecting the slope change at incipient separation in skewness or kurtosis evolution. Moreover, the same conclusion as in Sect. 3.1 can be drawn: as deflection angle d increases, a recirculation bubble is developing from trailing edge towards leading edge until it covers the entire instrumented flap above d = 20°. The separated zone is characterized by skewness and kurtosis factors higher than those of the attached zone, even though kurtosis tends to decrease towards 3 after the separation has occurred. Meijering and Schroder (2001) already proposed the skewness factor as a criterion to detect laminar-to-turbulent transition. As the boundary layer is fully turbulent on the flap, the steep change in both skewness and kurtosis factors is due to the flow separation. Consequently, skewness and kurtosis appear to be suitable criteria to be used a posteriori to follow the growth of the recirculation zone along the chord once an appropriate threshold is defined. In the present uncontrolled case, S = 0.5 and K = 3.5 thresholds have been chosen for skewness and kurtosis, respectively. Anyhow, the ability of those two criteria to resolve the flow state (separated or not) fast enough to match on-line detection requirements needs to be further investigated. The detection method proposed in this paper is relatively easy to implement since a simple recursive calculation procedure can be derived to estimate both skewness and kurtosis. Hence, no storage of large data arrays is needed. Besides, skewness or kurtosis calculations are less demanding in computational resources than a fast Fourier transform. The only issue remaining is the length of the time windows needed to obtain converged values. Actually, for a given relative uncertainty, the higher-order moments estimation needs more samples than the mean or mean square

Fig. 5 Contours of a skewness and b kurtosis for the entire flap and deflection angles from 2° to 35°. The deflection angle is denoted by d and given in degrees. The red dashed line underlines S = 0.5 and K = 3.5 contours for skewness and kurtosis factors, respectively

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estimations. For a given number of statistically independent samples, the error made on the skewness and kurtosis estimations can be computed through the following relations (see Bruun 1995): rffiffiffi 6 ErrorSkew ¼ za=2 ; ð7Þ n rffiffiffiffiffi 96 ; ð8Þ ErrorKurto ¼ za=2 n where n is the number of statistically independent samples and za /2 = 1.96 for a 95 % confidence interval (chosen in

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most cases). Unsurprisingly, for a given number n, the uncertainty on kurtosis is higher than the one on skewness. The number of statistically independent samples can be roughly estimated by calculating autocorrelation functions for signals at different flow configurations. For the first approach, which consists in determining a posteriori whether the flow is separated or not, the signals are 40 s long. The numbers of independent samples are estimated to be 50,000 for attached flow state and only 1,000 for separated flow state. So applying the previous relations gives errors of 0.15 and 0.6 for skewness and kurtosis, respectively, in the worst case when the flow is separated. When the flow is

Fig. 6 Flow visualizations when periodic blowing control is applied at F? = 0.17, 50 % of duty cycle and hCl i ¼ 5:6 % of amplitude. The flap deflection is 35°. Remodel = 1.6 106

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attached, the errors become 0.02 and 0.08, respectively. For the second approach, it follows that in-line detection of flow separation by means of skewness or kurtosis requires a time window large enough to collect a sufficient amount of independent samples. Therefore, the detection method is not appropriate if fast response is required.

4 Separation of the periodically forced turbulent boundary layer 4.1 Description of the alternate detachment and reattachment process The main features of the uncontrolled flow being evidenced, the effect of periodic blowing is now studied. First, for tomoscopy flow visualizations, the flap is deflected to d = 35°; the reduced actuation frequency and the duty cycle are set to F? = 0.17 and DC = 50 %, respectively. Figure 6 presents six selected pictures regularly spaced over one period of actuation. Since the free-stream velocity is lowered to U0 = 20 m s-1 for flow visualization purposes, the momentum coefficient is artificially high: hCl i ¼ 5:6 %: The Reynolds number is then Remodel = 1.6 9 106, based on the total length of the model. Such an amount of momentum is far from matching the general purpose of reducing flow control cost, but the idea of this section is to magnify control effects to determine whether the pulsed jets implanted in the flap are able to fully reattach the flow or not, even at large deflection angles. The flow is periodically attaching and detaching at the actuation frequency and has not enough time to detach completely since the transient recirculation zone never covers the flap entirely. The periodic blowing enables the flow to be partially reattached, that is, the flow is halfperiod/part-time reattached, from a naturally fully separated flow at d = 35° (the flow visualization presented in Fig. 4b shows that the flow is fully separated at d = 35°). In addition, one can observe that the flow remains attached longer than detached. This means the time for detachment is longer than the time for reattachment since the duty cycle is still 50 %. Besides, it appears that blowing at hCl i ¼ 5:6 % is sufficient to maintain the flow attached and to reattach the flow from an initial partially separated configuration at d = 35°, that is the maximum deflection angle of the experimental set-up.

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and a duty cycle of 50 % (the free-stream velocity is increased up to U0 = 34.5 m s-1). Figure 7 depicts the phase-averaged wall shear stress signals at x/c = 0.239 and x/c = 0.784 locations on the flap deflected to d = 20° and the related excitation signal applied to the actuators. In this case, the applied control is not able to entirely reattach the flap since there are no remaining traces of the alternative blowing on the x/c = 0.784 signal pictured in Fig. 7. On the contrary, near the flap leading edge (x/c = 0.239), the shear stress switches from one state to the other. The first state corresponding to the blowing phase is characterized by a mean value higher than the other one corresponding to the non-blowing phase. Those two phases are referred to as phase-on and phase-off thereafter. The flow, which is naturally separated in the uncontrolled case for d = 20° and x/c = 0.239, reattaches during phase-on and separates again during phase-off. Roughly speaking, this characterizes a part-time or half-period reattached flow. Moreover, during the transition from phase-off to phaseon (i.e. the reattachment at x/c = 0.239), the shear stress experiences a steep rise, whereas the transition from phaseon to phase-off (i.e. the separation) is smoother. The durations of those transitions are estimated to be 0.1*T and 0.45*T, respectively. It must be noted that nothing guarantees that the flow reattaches at x/c = 0.239 from a fully separated configuration during phase-on (consequently, no estimation of reattachment and separation durations is proposed). The difference between both transitions reveals an asymmetry between the two states which can be appreciated on the probability density function (referred to as pdf thereafter). The pdf estimations are provided by the calculation of histograms defined as

4.2 From forced reattachment to fully separated flow Periodic blowing effect is now investigated through more realistic actuation, characterized by a periodic momentum coefficient hCl i ¼ 0:8 %; a reduced frequency F? = 0.19

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Fig. 7 One period of hot-film signals (averaging is made on ten periods T of signal) at x/c = 0.239 and x/c = 0.784 locations; d ¼ 20 ; hCl i ¼ 0:8 %; F þ ¼ 0:19

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Fig. 8 Probability density function (pdf) evolution against flap deflection for a x/c = 0.239 and b x/c = 0.784. The pdf peaks are underlined by the dashed lines

histðU  Þ ¼

NU   100; Ntotal

ð9Þ

where NU  is the number of hot-film voltage samples which fall within [U*i ; U*i ? Width]. Figure 8 presents the pdf calculated at every flap deflection from 2° to 35° at x/c = 0.239 and x/c = 0.784 locations. As expected, the pdf of the shear stress signal at x/c = 0.784 is single-peaked, whereas the one at x/c = 0.239 is double-peaked for low deflection angles. The peak of greater mean value, which corresponds to phase-on, is higher than the one of smaller mean value. As the deflection angle increases, both mean values decrease before the two peaks merge at d = 27°. The mean value of the one-peaked pdf continues to decrease until d = 35°. The two peaks merging marks the transition between part-time reattached flow (or half-period reattached flow) and fully separated flow at x/c = 0.239. So when the two peaks merge, the periodic blowing cannot maintain the flow part-time attached any more. The flow is then fully separated. The main idea to detect this transition is again to use higher-order standardized moments. Apart from very specific cases, a double-peaked histogram (bi-modal distribution) is characterized by a skewness different from zero if the two peaks are asymmetric. The skewness sign is then supposed to depend on the ratio between the intensity of each peak. Thus, skewness appears to be a well-suited factor to determine whether the hot-film pdf is double-peaked or not, and, as a consequence, if the flow is part-time reattached or fully separated. In the same way, the kurtosis factor of a double-peaked pdf is different from 3. So flatness is likely to be another suitable criterion to detect the transition between both states. Figure 9 presents the evolution of skewness and flatness factors against flap deflection angle for x/c = 0.239, x/c = 0.420, x/c = 0.602 and x/c = 0.784 locations.

Since the pdf is first double-peaked for low deflection angles and the peak of greater mean value is higher than the one of smaller mean value, skewness factor is first negative, as presumed, before reaching positive values after the merging of the two peaks. The last remark could have been anticipated since it has been shown (refer to Sect. 3) that skewness factor is positive when the flow is detached. In the same way, kurtosis is lower than 3 when the flow is double-stated and becomes greater than 3 when the flow is fully detached. Subsequently, skewness and flatness factors appear to be well suited to detect the transition from an alternately detached flow to a fully separated flow. One can remark that the fully separated zone detected through the kurtosis criterion is delayed by a few degrees compared to the one determined through the skewness criterion (Fig. 9). The average skewness in the attached zone is around 0.2, depending on the flap location (see Fig. 5a). On the contrary, the average kurtosis in the attached zone is around 3 (see Fig. 5b). So there is, in the skewness case, a kind of dead-zone between the part-time reattached and fully separated configurations that is not detected in the kurtosis case and this explains the delay in the latter case. This delay amounts to a few degrees only (less than 2°) anyhow and has no consequences for the rest of the study. The difference between the detection criteria of Sect. 3 in the uncontrolled case and those just above-mentioned in the periodic blowing case has to be clarified. In the former case, the single-peaked pdf of a given hot-film signal in the uncontrolled flow changes when flow separation occurs. This change corresponds to sharp rises of both skewness and kurtosis which increase beyond critical values (S = 0.5 and K = 3.5) defined as thresholds for the purpose of detection. In the periodic case, the pdf of a given hot-film signal is double-peaked as long as the flow is part-time

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Fig. 9 a Skewness and b kurtosis evolution against flap deflection for x/c = 0.239, x/c = 0.420, x/c = 0.602 and x/c = 0.784 locations; hCl i ¼ 0:8 %; F þ ¼ 0:19; DC ¼ 50 %

reattached; then, the two peaks merge when the flow fully separates. Consequently, the rise of skewness and kurtosis marks the transition between the two flow configurations. In this case, the suitable thresholds for the transition detection are S = 0 and K = 3. To summarize, skewness and kurtosis are mere tools to detect pdf changes which are of different nature depending on whether the uncontrolled or the periodic blowing case is considered.

5 Open-loop control of the flow separation The previous part led to the definition of two criteria able to determine whether the flow under periodic actuation is attached, part-time reattached or fully detached. 5.1 Flow system static map An increasing ramp straight followed by a decreasing ramp of hCl i is used as input to study whether the skewness criterion is able to follow the dynamic behaviour of the system. The results concerning the kurtosis factor are not reported here for the sake of brevity but are in good agreement with those presented hereafter. The experimental procedure is as follows. The reduced frequency of the periodic excitation is set to F? = 0.19. The feeding flow rate starts from 0, rises and then decreases making the momentum coefficient varies continuously between 0 and 1.2 % as depicted in Fig. 10a. The skewness evolutions at x/c = 0.239 for two fixed flap positions, that is, d = 10° and d = 25°, are presented in the same figure for comparison. The 100-s long 10-kHz sampling signals are divided into 100 blocks with no overlap, which implies that

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there is one point calculated per second of signal. The results presented in Fig. 10 are smoothed by Savitzky– Golay filtering. When hCl i is 0 for t \ 10 s and t [ 60 s (no control applied), skewness values are less than 0.2 for d = 10°, whereas they are higher than 0.2 and more scattered for d = 25°. That implies the flow is attached in the former case and detached in the latter one (at x/c = 0.239), as already observed in previous sections. When hCl i increases, skewness value decreases below 0 and reaches a plateau around S = - 0.6 in both cases, so the flow is part-time reattached, before becoming positive again as soon as hCl i decreases. The decrease in skewness is delayed for d = 25° since the minimum hCl i needed to part-time reattach the flow at x/c = 0.239, referred to as hCl i;R hereafter, is obviously more important than the one needed for d = 10°. Figure 10b presents the skewness evolution in response to the ramp of hCl i represented in Fig. 10a, at x/c = 0.239 for deflection angles varying from 10° to 35°. Several zones corresponding to different flow configurations can be distinguished from this representation. First, the flow remains naturally attached until d = 15°, so the change in flap deflection has no effect on skewness evolution from 10° to 15°. It is worth being reminded that after d = 15°, the uncontrolled flow at x/c = 0.239 is naturally separated. As deflection increases, the range of hCl i corresponding to part-time reattachment reduces. Finally, for deflection angles higher than d = 31°, the flow is always separated. Since the ramp of hCl i applied as input of the system is slow, the system response given in Fig. 10b can be considered to be a steady-state map, as defined by King et al. (2004). This static map exhibits a global extremum

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Fig. 10 a Skewness temporal evolution at x/c = 0.239 for d = 10° and d = 25° when a hCl i ramp input is imposed; b contours of skewness versus time and flap deflection

(minimum), and, in the range of hCl i covered during the test, the gradient is negative. As a consequence, in an attempt to implement extremum-seeking control apart from the difficulties evoked in the following paragraph about skewness convergence, it should be interesting to seek for the minimum of skewness to provoke the reattachment of the flow. In the representation of Fig. 10a, skewness was computed every second. Here again, the use of skewness criterion enables to describe a posteriori the flow state (attached or separated) in response to a pulsed excitation. Is it possible to use it as a real-time detection criterion? The implementation of such a detection method in a closedloop framework implies the reduction in the time windows length from which the skewness factor is deduced. The related problem already exposed in the Sect. 3.2 concerns the large amount of samples needed to get converged skewness values. Figure 11 presents the skewness raw computation on a time windows of 0.1 s. In this case, the signal corresponds to the hot-film voltage at x/c = 0.239 for d = 10° and d = 25° (same signals as in Fig. 10a). The result is clearly ‘noisy’ but reveals the expected behaviour (as in Fig. 10a). To reduce the noise associated with the time windows reduction, a filter can be employed. In Fig. 11, the black line represents the skewness temporal evolution of the raw signal filtered through a simple moving-average filter on 50 points. The problem arising is the time delay induced by filtering. So the time that is saved by the time windows length reduction can be lost in the filtering process. Furthermore, the hysteresis effect is examined. The hysteresis effect when control is applied (the control amplitude needed to reattach the flow is greater than the minimum amplitude needed to maintain the flow attached)

Fig. 11 Skewness temporal evolution at x/c = 0.239 for d = 10° and d = 25° when a hCl i ramp input is imposed. A new skewness value is calculated every 0.1 s. The black lines are moving-average filtered signals

is a well-known phenomenon studied by several authors (e.g. Nishri and Wygnanski 1998; Benard et al. 2011). In our case, no hysteresis effect is observable in Fig. 10a. In order to present the results more distinctly, the value of hCl i;R for each flap deflection is deduced from the last representation in Fig. 10b and reported in Fig. 12. The value of hCl i;S is also represented; it corresponds to the value of hCl i when the flow separates again. This clearly shows that the detachment is not delayed when hCl i decreases in comparison with the reattachment when hCl i increases. In other words, hCl i;R values are equals to hCl i;S values. A possible explanation for this is that the detachment is not subject to any hysteresis effect in

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Fig. 12 hCl i;R and hCl i;S against flap deflection at x/c = 0.239. Subscripts ‘R’ and ‘S’ correspond to reattachment and separation, respectively

the uncontrolled case, as explained in Sect. 3 Indeed, the flow separation is not sudden, as for Nishri and Wygnanski (1998) and Benard et al. (2011), for example, but progressive in our case. Consequently, there is no hysteresis effect at a given flap deflection when control is applied. 5.2 Sinusoidal perturbation of the flow system As reminded in the introduction, one key point in extremum-seeking control approach is the estimation of the static map local gradient. The previous section has given an example of a static map exhibiting a global minimum where the control algorithm should converge. In the present section, the purpose is to determine the local gradient. Subsequently, the system response to a sinusoidal perturbation of momentum at a fixed actuation frequency is studied. The momentum coefficient hCl i input is presented on six periods in Fig. 13a. This input constitutes a sinusoidal perturbation of the system. The reduced frequency is F? = 0.19. The flap is deflected to d = 25°. As seen before, the flow is naturally separated at x/c = 0.239 without control. The critical value hCl i;R determined in the previous section for d = 25° is around 0.55 % (see Fig. 12) and is represented by the dashed line in Fig. 13a. Since the maximum value reached by the hCl i sinusoid (see Fig. 13a) is slightly over 0.9 %, the flow is expected to be part-time reattached on more than half a period (i.e. for all hCl i values over hCl i;R ). Figure 13b presents the spectrogram (i.e. short-time Fourier transform) of the wall shear stress signal at x/c = 0.239. This spectrogram has been calculated on a 10-kHz sampling signal of 100 s long divided into 100 blocks with 50 % overlap.

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The reduced excitation frequency is precisely F? = 0.19. Followed by its harmonics at F? = 0.38 and F? = 0.57, this frequency periodically appears in the spectrogram when the momentum of the applied control is sufficient to part-time reattach the flow at position x/c = 0.239. The periodic part-time reattachment of the flow is detectable when monitoring the skewness and kurtosis evolution as presented in Fig. 13c. Actually, the use of the previously defined separation criteria (S = 0 and K = 3), as underlined by the dashed line in Fig. 13c, enables to follow the flow state evolution by monitoring skewness or kurtosis. Once again, one can notice that the kurtosis evolution is slightly delayed if compared to the skewness when the flow separates, and slightly ahead of the skewness when the flow reattaches. Nevertheless, this tiny difference between the two criteria does not prevent either of them from being employed for flow separation detection. As a consequence, the skewness follows the periodic part-time reattachment of the flow as well as the kurtosis. More precisely, an increase in hCl i; corresponding to the rising part of the sinusoid, implies a reattachment of the flow and a decrease in skewness or kurtosis. On the contrary, a decrease in hCl i is followed by a rise of skewness or kurtosis. Thus, both skewness and kurtosis ‘signals’ and sinusoidal perturbation input are logically in opposite phase. As already underlined, the static map gradient estimation is of primary importance in an attempt to implement extremum-seeking algorithm. This estimation is in general (e.g. Becker et al. 2007) provided by the multiplication of the sinusoidal input perturbation by the corresponding sinusoidal response of the system. As illustrated by Pamart et al. (2010), the sinusoidal response is in opposite phase with respect to the sinusoidal perturbation if the local static map gradient is negative. On the contrary, the sinusoidal response is in phase with the sinusoidal perturbation if the local gradient is positive. In our case, the static map presented in Fig. 10 (at x/c = 0.239) exhibits, as already noticed, a negative gradient, so skewness and kurtosis ‘signals’ of Fig. 13c are logically in opposite phase against the sinusoidal perturbation input of Fig. 13a. Therefore, monitoring the skewness or the kurtosis evolutions in response to a sinusoidal input gives the sign of the static map local gradient. New values of skewness and kurtosis are computed at every second, that is each moment is calculated on intervals containing 10,000 samples not necessarily independent. As already discussed in the previous section, the relatively large number of independent samples needed to compute skewness or kurtosis remains the main limitation of the method. It is worth mentioning that the number of independent samples contained in a given time window is related to the flow

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Page 13 of 14 b Fig. 13 a Sinusoidal perturbation of momentum coefficient applied

at d = 25° of flap deflection; F? = 0.19; T* is the period and equals to 12.5 s; the hCl i;R at d = 25° and x/c = 0.239 is represented by the dashed line; b spectrogram (PSD stands for power spectral density) for the corresponding case; c skewness and kurtosis evolution when sinusoidal input presented in a is applied; the dashed line represents both separation criteria (S = 0 and K = 3). The time windows length for skewness and kurtosis calculation is 0.08 T*

6 Conclusion In the present work, experimental detection of flow separation over a non-slotted flap by means of wall-mounted hot-films has been studied. In the uncontrolled case, it has been shown that the computed third- and fourth-order statistical moments, skewness and kurtosis, of hot-film signals exhibit a sharp rise when the flow separates. This leads to the definition of two thresholds, S = 0.5 and K = 3.5 for skewness and kurtosis, respectively, in order to detect flow separation. Secondly, two other criteria have been defined to detect flow separation when periodic blowing at F? = 0.19 is applied. They are also based on higher-order moments of the hot-film signals. Both of them allow to detect the transition from a state where the flow is half-period attached to a fully separated situation. This transition is marked by a change in sign of skewness and excess kurtosis (i.e. kurtosis becomes greater than 3). Beyond that, the ability of skewness and kurtosis criteria to form a suitable feedback in a closed-loop control approach has been studied. It has been shown that skewness and kurtosis criteria are able to follow the flow configuration changes in response to a ramp or sinusoidal input of momentum coefficient hCl i; and thus to estimate the system static map local gradient in the case where extremum-seeking closed-loop strategy is adopted. The main limitation remains the necessary length of the time windows the moments are calculated on. So the number of statistically independent samples contained in the chosen time windows has to be taken into account before considering the use of such a detection method as a feedback for closed-loop algorithm. Acknowledgments The authors would like to acknowledge Michel Alaphilippe and Nicolas Severac for their constant and precious support, and all members of the ‘DAFE PIV team’ for their help during the tomoscopy experiment.

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