Experimental investigation of flow-induced vibration of a pitch–plunge NACA 0015 airfoil under deep dynamic stall

Journal of Fluids and Structures 67 (2016) 48–59

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Experimental investigation of flow-induced vibration of a pitch–plunge NACA 0015 airfoil under deep dynamic stall Petr Šidlof a,b,n, Václav Vlček b, Martin Štěpán a a Technical University of Liberec, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Studentská 2, 461 17 Liberec, Czech Republic b Academy of Sciences of the Czech Republic, Institute of Thermomechanics, Dolejškova 5, 182 00 Prague 8, Czech Republic

a r t i c l e in f o

abstract

Article history: Received 2 October 2015 Received in revised form 28 July 2016 Accepted 31 August 2016

The flow-induced vibration of a NACA 0015 airfoil model with pitch and plunge degrees of freedom is investigated in a high-speed wind tunnel using motion sensors, pressure sensors on the airfoil surface and synchronized high-speed Schlieren visualizations of the unsteady flow field at Reynolds numbers Re = 180 000–570 000 . Compared to other studies, the model has considerably smaller dimensions (with a chord length of 59.5 mm) and operates at higher flow velocities (from 37 to 125 m/s). With a relatively low pitch to plunge natural frequency ratio and zero initial incidence angle, the model is highly susceptible to deep dynamic stall instability with the peak-to-peak pitch amplitude reaching up to 90°. The limit cycle response of the system as a function of freestream flow velocity is reported in detail, together with synchronized Schlieren visualizations of the flow field revealing the boundary layer behavior during dynamic stall. With increasing inflow velocity, the plunge amplitude increases dramatically, accompanied also by slight rise in the frequency of oscillation and decrease of the phase lag between pitch and plunge. The pitch amplitude has a maximum at 62 m/s and further decreases with increasing flow velocities. & 2016 Elsevier Ltd. All rights reserved.

Keywords: NACA 0015 airfoil Aeroelasticity Stall flutter Dynamic stall Limit cycle oscillation Schlieren

1. Introduction Flow-induced vibration is a serious issue in the technical design of airplane wings and control surfaces, helicopter blades and rotating machinery such as propellers, turbine blades and compressors. The problem arising from the interaction of unsteady aerodynamic forces with an elastic structure has been studied for decades both experimentally and computationally. In the context of streamlined bodies, there are two main types of dynamic aeroelastic instabilities: classical (coupled-mode) flutter and stall flutter. In the former, the flow is attached at all times. The instability is related to the natural frequencies of the pitch and plunge modes and may be predicted by linearized computational estimations, such as the classical theory by Theodorsen (1935). From the linearized theories, classical flutter leads to unstable oscillations with exponentially increasing amplitudes. Due to structural or aerodynamic nonlinearities, the amplitudes may stabilize at limit cycle oscillations (LCOs). Stall flutter, on the other hand, occurs when the airfoil enters high angles of attack during parts of the oscillation period and when the boundary layer separates at the suction side (Dowell, 2015). In the case of light stall, as described by n Corresponding author at: Technical University of Liberec, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Studentská 2, 461 17 Liberec, Czech Republic. E-mail address: [email protected] (P. Šidlof).

http://dx.doi.org/10.1016/j.jfluidstructs.2016.08.011 0889-9746/& 2016 Elsevier Ltd. All rights reserved.

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McCroskey (1981), the flow near the leading edge remains attached, separates at a certain point downstream and can reattach creating a separation bubble. During the deep stall, the flow detaches right at the leading edge and forms a leadingedge vortex, which briefly increases the instantaneous lift. The vortex travels downstream, and as soon as it passes the trailing edge, the lift reduces dramatically to the stall state with full separation. The alteration between the attached and separated flow during the pitch angle oscillation is known as dynamic stall. The essential feature of stall flutter is thus highly nonlinear aerodynamic response to the airfoil vibration, which cannot be explained by simplified linear theories. Stall flutter often leads to LCO. Unlike the classical flutter, it is not directly driven by the ratio of the still-air natural frequencies of the pitch and plunge modes. Rainey (1956) has shown that dynamic stall can occur even for a purely pitching airfoil. Stall flutter is less common in airplane wings and tails, but may represent a severe problem in helicopter blades, propellers, turbine blades and compressors (Fung, 1955). The critical flow velocity for stall flutter is then often lower than that for the coupledmode flutter. The studies focused on energy harvesting using flapping foils (Young et al., 2014, Fenercioglu et al., 2015) show that low Reynolds number stall flutter at large foil angles is particularly effective in terms of efficiency of the energy extraction from the flow. There are also other types of dynamic aeroelastic instabilities such as whirl flutter (Čečrdle, 2015), buffeting and galloping (Fung, 1955; Blevins, 1990) and others; however, this study focuses mainly on the dynamic stall phenomenon. A comprehensive review paper covering the flutter and dynamic stall problems was published by McCroskey (1982), a practical guide for technical designers can be found in Ericsson and Reding (1988). The experimental studies concerning dynamic aeroelasticity using scaled airfoil models fall into two major groups: models with externally forced oscillation and investigations where the airfoil vibration is flow-induced. The former allows for precise adjustment and predefined time development of the pitching and plunging motion, and is less susceptible to model destruction due to unforeseen oscillation amplitudes. Raffel et al. (1995) measured the unsteady flow fields above a NACA 0012 airfoil with a single degree of freedom (DOF) under deep dynamic stall by means of particle image velocimetry (PIV), estimating also the vorticity fields. The pitch angle oscillated between 5° and 25° with a mean incidence angle of 15°. Lai and Platzer (1999) performed dye visualizations and LDV measurements on a NACA 0012 airfoil with forced sinusoidal oscillation in the plunge mode investigating the drag coefficient. Lee et al. (2000) and Lee and Gerontakos (2004) investigated the boundary layer transition and separation for the same airfoil oscillating in pitch for various mean incidence angles and oscillation amplitudes. The results were summarized for all the possible cases – static airfoil, deep-stall, light-stall and attached-flow oscillations. All the aforementioned models work in the Reynolds number range Re = (105)–(106). Ol et al. (2009) performed dye visualizations and PIV measurements using an electrically driven pitch–plunge foil in a water tunnel at Re = 60 000. A study by Prangemeier et al. (2010) reports on PIV measurements of non-sinusoidal quick-pitch motion of a SD7003 airfoil at lower Reynolds numbers Re = 30 000, motivated by the interest in animal flight and the development of unmanned aerial vehicles (UAVs). The flow structures in the wake of the same airfoil under forced pitching and plunging motion at Re = 1000–10 000 were investigated by Fenercioglu and Cetiner (2012) in a free-surface water tunnel. The interest into micro air vehicles (MAVs) motivated also the study of Bhat and Govardhan (2013), where flow fields around a forced NACA 0012 pitchoscillating airfoil and unsteady aerodynamic forces were measured using PIV and a strain-gauge-based load cell. Recently, Uruba (2015) analyzed the time-resolved PIV velocity fields around a NACA 0012 profile with forced pitch oscillations by means of the oscillation pattern decomposition method. The measurements using self-oscillating airfoil models are less frequent, since their design must ensure that the model enters aeroelastic instability in a desired inflow velocity range, yet the oscillation amplitudes do not exceed dangerous amplitudes. However, these models provide unique information on airflow which is truly coupled to the structural vibrations. O'Neil and Strganac (1998) studied the LCOs of a NACA 0012 airfoil with two degrees of freedom, supported by springs with a cubic hardening structural response which were realized by a tailored pair of cams with nonlinear shape. Their design partly inspired the self-oscillating model by Marsden and Price (2005), where an additional mechanism allowing for freeplay in the pitch restoring moment was introduced. The model also had an adjustable ratio of natural frequencies of the plunge and pitch mode, ranging between ωy /ωα = 0.287 and 0.562. The flutter boundaries and LCO amplitudes for various parameter values were summarized. Tinar and Cetiner (2006) reported on phase-resolved PIV measurements of a purely pitching NACA 0012 airfoil under stall flutter at Reynolds numbers Re = 60 000–150 000. Poirel et al. (2008) designed a twodegree-of-freedom NACA 0012 model with an adjustable structural stiffness and position of the elastic axis, and performed the dynamic and hot-wire anemometric measurements in the wake of the airfoil oscillating with the plunge mode blocked. The mechanisms leading to flutter of a 2-DOF NACA 0015 airfoil after gust impulse were studied by Schwartz et al. (2009) at Re = 80 000–120 000. Razak et al. (2011) used a flexibly supported NACA 0018 airfoil model with close pitch and plunge natural frequencies to investigate under which conditions classical or stall flutter occurs. The setup was equipped with pressure and acceleration sensors and a time-resolved PIV system. Yet another self-oscillating airfoil system was designed by Song et al. (2012) to provide data for a semi-experimental method for evaluation of the unsteady aerodynamic forces. In the current paper, an original design of a self-oscillating pitch–plunge NACA 0015 airfoil is presented. The model is capable of flow-induced vibrations at moderate Reynolds numbers ranging from Re¼180 000 to 570 000 and Mach numbers Ma = 0.1–0.4 , motivated by interest in the aeroelastic instability in UAVs, propellers, compressor and turbine blades. Compared to the airfoil models described above, where a typical chord length is c = 15–30 cm , the current model is considerably smaller, with c = 5.95 cm only. In addition to the pressure and motion sensors, the model mounted in the test section of a wind tunnel provides lateral optical access in order to perform interferometric flow field measurements, Schlieren visualizations and possibly also PIV. The dynamic response of the aeroelastic system with a zero initial incidence

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50

Fig. 1. Schematic of the airfoil model in the test section of the wind tunnel – side and frontal view.

angle and behavior of the boundary layer identified from the Schlieren flow field visualizations is reported.

2. Experimental setup The experimental setup is an original design and builds on previous experience in measurements of the flow-induced airfoil vibration in the wind tunnel of the Institute of Thermomechanics in Nový Knín (Vlček and Kozánek, 2011; Kozánek et al., 2013). The current airfoil model is a redesigned and improved version of the model used in interferometric measurements of Šidlof et al. (2014). 2.1. Airfoil model design and wind tunnel setup A NACA 0015 section with a chord length of c = 59.5 mm and span of 76.6 mm is composed of five perfectly matching segments machined from steel and has spanwise holes and slots for the rotary encoder, bearings, wiring and pressure sensors. The vertical guides, supported by flat linear springs, are free to move vertically within linear bearings providing the plunge degree of freedom, and have a C-shaped reinforcement not to block the lateral optical access in the proximity of the airfoil (see Fig. 1). The airfoil rotates around miniature ball bearings located at 1/3 of the chord, with the restoring moment realized by a spiral torsion spring inside the profile. The mechanical parameters of the model are summarized in Table 1. The initial angle of attack can be adjusted, and is set to zero throughout current measurements. The airfoil is fixed in a 80 × 210 mm test section of a high-speed suction-type wind tunnel, capable of reaching flow velocities up to Ma = 1. The turbulence intensity at the inlet is below 1% . The free opening of the wind tunnel is 310 mm Table 1 Mechanical properties of the airfoil model. Mass (total, moving in plunge) Mass (airfoil only, moving in pitch) Moment of inertia

1.498 0.148 0.0000478

Chord length Elastic axis position (from leading edge) Center of gravity position (from leading edge) Stiffness in plunge Torsional stiffness Natural frequency of the plunge mode (still-air) Natural frequency of the pitch mode (still-air) Plunge damping coefficient (still-air) Pitch damping coefficient (still-air)

59.5 19.8 26.4 16 383 0.00753 16.8 14.5 5.6 0.00078

kg kg

kg m2 mm mm mm N/m N m/deg Hz Hz kg/s kg m2/s

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Fig. 2. Schematic of the Schlieren visualization setup.

upstream of the airfoil leading edge. The gap between the airfoil and the test section lateral walls is kept to minimum (1.7 mm ) in order not to disturb the two-dimensionality of the flow field. The blockage factor is 4% at zero incidence, 20% for the maximum allowed pitch angle of 45°. 2.2. Optical measurements The test section of the wind tunnel and its special lateral optical glasses are designed for measurements of the flow field using the Mach–Zehnder interferometer and for Schlieren visualizations. The former method is sensitive to the air density, which has local variations due to compressibility effects, the latter to the density gradient (Shapiro, 1954). For medium and higher Mach-number flows where density variations become significant, Schlieren imaging of the flow field with the Schlieren knife edge aligned horizontally may be well used to detect the shear layer separated at the suction side of the airfoil at high angles of attack. The schematic of the current Schlieren setup is shown in Fig. 2. The collimated beam from a mercury lamp passes through the test section, where the light rays deflect by refraction due to refractive index gradients. Outside of the test section, part of the light is blocked by the Toepler knife at the focal point of the imaging lens. The resulting Schlieren images are recorded by a NanoSense Mk III high speed camera (resolution 1280 × 1024 pixels at 1000 frames per second), triggered and synchronized from the Dewetron measuring hardware. 2.3. Displacement and pressure sensors Due to the size of the airfoil (chord 59.5 mm, maximum thickness 9 mm), there is a very limited space for installation of the sensors inside the model. The airfoil is equipped with a miniature rotary magnetic encoder (RLS – Renishaw RM08) measuring the pitch angle, and four dynamic piezoresistive pressure sensors with built-in preamplifiers (Freescale Semiconductor MPXH6115) mounted flush with the airfoil surface. The pressure sensors are denoted p1U, p2U (upper surface) and p1L, p2L (lower surface) and are located 10.8 mm and 29.5 mm from the leading edge, respectively (see Fig. 3). The signals lead out of the vibrating airfoil by a flexible bundle of thin wires designed for high mechanical loads. Outside of the test section, plunge deflection is measured by a contactless magnetic linear incremental encoder LM13 (RLS, Renishaw). A secondary signal monitoring plunge is provided by a strain gauge bridge fixed on the flat spring supporting the vertical frame. Signals from the pressure sensors, strain gauge bridge, and rotary encoders are conditioned and digitized by a Dewetron amplifier and data acquisition system. Together with the digital signal from the linear encoder they are monitored online and stored by the DeweSoft 7.1.2 measuring software. The measurement software includes a special module for triggering the high-speed camera, providing the same time base for the recorded Schlieren or interferometric images and signals from all the sensors. For the measurement of the inflow velocity, a Pitot-static tube measuring the mean value of the static pressure p and total pressure p0 is mounted shortly upstream of the test section. The Pitot-static tube is connected to a pressure scanner wired to a PC, which registers the pressures and evaluates the inflow Mach number from equation

Ma =

1− κ ⎞ ⎛ κ ⎟ 2 ⎜⎛ p ⎞ − 1⎟ , ⎜ ⎜⎜ ⎟⎟ κ − 1 ⎜ ⎝ p0 ⎠ ⎟ ⎠ ⎝

(1)

where κ = cp/cv is the heat capacity ratio, for ideal diatomic gas equal to 7/5. The flow velocity can be then calculated according to

u = Ma κ R T

(2)

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Fig. 3. Schematic of the sensors and wiring configuration.

with R = 287.1 J kg−1 K−1 the specific gas constant for dry air and T the absolute local static temperature calculated from the ambient temperature T0 as

⎛ ⎞−1 κ−1 T = T0⎜ 1 + Ma2⎟ . ⎝ ⎠ 2

(3)

Within the Mach number range Ma = 0–0.5 where the current measurements were performed, the flow velocity can be also simply estimated from the speed of sound c0 = 343 m/s at standard conditions as u = Ma·c0 with an error less than 2% .

3. Results In this section, results of seven measurements at low flow velocity in the range from Ma ¼ 0 to Ma ¼ 0.14 leading to Table 2 Mach number Ma, flow velocity u, initial displacement in plunge y0, plunge and pitch amplitudes Ay and Aφ, phase difference Δψ (pitch behind plunge), LCO frequency f and reduced frequency k = πfc /u in measurements 2915-09–2915-44. #

Ma (1)

u (m/s)

y0 (mm)

Type

Ay (mm)

Aφ (deg)

Δψ

f (Hz)

k (1)

2915-09 2915-10 2915-11 2915-12 2915-13 2915-14 2915-31 2915-16 2915-32 2915-33 2915-34 2915-35 2915-36 2915-37 2915-39 2915-40 2915-41 2915-42 2915-43 2915-44

0 0.110 0.115 0.120 0.125 0.130 0.130 0.140 0.140 0.150 0.170 0.190 0.210 0.230 0.270 0.290 0.310 0.330 0.350 0.370

0 37.8 39.6 41.3 43.0 44.7 44.7 48.1 48.1 51.6 58.4 65.2 72.0 78.8 92.4 99.1 105.8 112.5 119.2 125.8

1.7 1.7 1.7 1.7 1.7 1.7 3.3 1.7 3.3 3.3 3.3 3.3 3.3 3.3 3.3 3.3 0 0 0 0

Damped (still-air) Damped Damped Damped Damped Damped LCO Damped LCO LCO LCO LCO LCO LCO LCO LCO LCO LCO LCO LCO

– – – – – – 2.2 – 2.4 2.7 3.6 4.5 5.5 6.3 7.3 7.5 8.1 8.6 9.5 10.0

– – – – – – 45.7 – 46.5 48.1 48.4 47.3 46.4 45.5 43.1 42.2 40.5 39.3 38.5 37.3

– – – – – – 1.55 π – 1.47 π 1.35 π 1.29 π 1.21 π 1.20 π 1.14 π 1.18 π 1.12 π 1.14 π 1.13 π 1.14 π 1.15 π

– – – – – – 16.4 – 16.3 16.4 16.5 16.7 16.7 17.0 17.3 17.4 17.6 17.6 17.5 17.7

– – – – – – 0.0685 – 0.0635 0.0595 0.0529 0.0479 0.0435 0.0402 0.0349 0.0329 0.0310 0.0292 0.0274 0.0262

P. Šidlof et al. / Journal of Fluids and Structures 67 (2016) 48–59

2915- 09 (Ma = 0)

30

2915- 09 (Ma = 0)

150 2915- 10 (Ma = 0.110)

25

2915- 10 (Ma = 0.110)

2915- 11 (Ma = 0.115)

20

2915- 12 (Ma = 0.120)

15

2915- 13 (Ma = 0.125)

10

pitch [deg]

plunge [mm ]

53

2915- 11 (Ma = 0.115)

100

2915- 12 (Ma = 0.120) 2915- 13 (Ma = 0.125)

50

2915- 14 (Ma = 0.130)

5

2915- 16 (Ma = 0.140)

0 0.0

0.5

1.0

1.5 time [s]

2.0

2.5

3.0

2915- 14 (Ma = 0.130) 2915- 16 (Ma = 0.140)

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

time [s]

Fig. 4. Comparison of the transient system behavior for increasing flow velocities – damped oscillations. The plots are staggered vertically by 5 mm and 25°, respectively.

damped oscillation and 13 measurements of LCO for Mach numbers ranging from Ma ¼ 0.13 to Ma ¼ 0.37 are compared. In all cases, the initial incidence angle was set to zero, although post-measurement analysis revealed slight positive pitch offset of about 0.3°. The main measurement parameters are summarized in Table 2. The measurement procedure in all cases was as follows: First, the wind tunnel regulating element was adjusted so that the desired Mach number was achieved. The airflow was stopped by the shutoff valve, and the model was displaced and blocked with an initial deflection y0 in plunge. Then, the airflow was switched on, which (especially for higher flow velocities) induced certain small initial pitch angle due to minor asymmetries in the model installation. After releasing the mechanism, the model started either damped oscillation or vibrations with increasing amplitudes converging to LCOs. In certain cases, two values of the initial displacement in plunge y0 = 1.7 mm and y0 = 3.3 mm were tested to reveal the influence of initial conditions. For low flow velocities, the vibration is damped for any value of y0. In the case of high inlet velocities, identical LCOs are reached from any y0 and sometimes the self-oscillations are triggered even with y0 = 0 by disturbances in the flow only. In a narrow range of inlet flow velocities (approximately Ma ¼ 0.13–0.15), however, the character of the flow-induced vibration strongly depends on the initial displacement. For y0 = 1.7 mm the flow remains attached and damped vibration is observed. By applying a stronger initial impulse y0 = 3.3 mm at the same inflow velocity, the airfoil enters dynamic stall leading to LCOs. The dynamic response of the aeroelastic system for zero (still-air) and two low flow velocities is compared in Fig. 4. Due to off-centre location of the airfoil centroid (see Table 1), the acceleration after releasing from the initial plunge displacement triggers also the pitch mode reaching maximum amplitudes of about 10°. The spectra of the signals (not shown here) reveal two frequencies close to natural frequencies of the pitch and plunge modes, however, due to damping in the system they do not provide sufficient resolution to judge if the frequencies approach with increasing flow velocity, as would be expected for classical flutter behavior. For higher flow velocities, the airfoil is able to extract enough energy from the airflow to enter oscillations with increasing amplitudes leading to limit cycle oscillations. Two measurements are analyzed here in detail: a low and high flow velocity case 2915-33 and 2915-44, respectively (see Table 2). Fig. 5 shows the pitch and plunge signals and pressure signals p1U .. p2L (see Fig. 3) acquired during limit cycle oscillations at Ma = 0.15. Even at this low flow velocity, the airfoil oscillates

Fig. 5. Time history of pitch, plunge and pressure signals during LCO, measurement 2915-33 (low flow velocity).

54

P. Šidlof et al. / Journal of Fluids and Structures 67 (2016) 48–59

Fig. 6. Time history of pitch, plunge and pressure signals during LCO, measurement 2915-44 (high flow velocity).

Fig. 7. Schlieren visualization of the flow field during one vibration period, measurement 2915-33. Shown every fourth recorded frame.

P. Šidlof et al. / Journal of Fluids and Structures 67 (2016) 48–59

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Fig. 8. Schlieren visualization of the flow field during one vibration period, measurement 2915-44. Shown every fourth recorded frame.

with very high pitch amplitudes of 48.1°, while the plunge amplitudes are relatively low (2.7 mm) . The character of the airflow, in particular of the boundary layer, may be qualitatively judged from the Schlieren images (Fig. 7), taken in 16 time instants within the first period of vibration plotted in Fig. 5. The flow visualization, sensitive to vertical density (and thus velocity) gradient, shows that the boundary layer is attached up to the trailing edge for low pitch angles, but separates and forms a free shear layer when the pitch angle φ surpasses about 30° (frame #2613). The airfoil is clearly under the deep stall condition, with airflow fully separated from a point very close to the leading edge. When the pitch angle drops below φ = 8° (frame #2634), the airflow reattaches and remains attached until φ = − 29° (#2642), when the shear layer separates from the lower airfoil surface and stays detached until φ = − 4° (#2666). The conclusions drawn from Schlieren visualizations are confirmed by dynamic pressure signals in Fig. 5. The static pressure during the airfoil vibration oscillates with a peak-to-peak amplitude of 5 kPa , with the waveform strongly disturbed when the airflow is separated. The ripples on the pressure waveform from the sensors p1U and p2U mounted in the upper airfoil surfaces start at t = 12.016, precisely at the time instant where the airflow detaches from the upper surface. The same holds for the p1L and p2L pressure signal perturbations starting at t = 12.043, caused by boundary layer detachment from the lower surface. The same behavior of the shear layer repeats throughout the LCOs, with low cycle-to-cycle variability. Fig. 5 reveals that flow separation on the upper and lower airfoil surfaces occur near the time instants when the plunge reaches minimum and

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2915- 31 Ma = 0.13

2915- 33 Ma = 0.15

-5

5 0 -5 -10

2915- 36 Ma = 0.21

0 -5 -10

5 0 -5

2915- 41 Ma = 0.31

-20 0 20 pitch [deg]

0 -5 -10

5 0 -5 -20 0 20 pitch [deg]

0 -5 -10

-20 0 20 pitch [deg]

-40

40

40

-20 0 20 pitch [deg]

40

2915- 44 Ma = 0.37 10

5 0 -5 -10

-40

40

5

2915- 43 Ma = 0.35

-10 -20 0 20 pitch [deg]

-5

10 plunge [mm]

plunge [mm]

5

-40

0

-40

10

40

2915- 40 Ma = 0.29

5

40

-20 0 20 pitch [deg]

10

2915- 42 Ma = 0.33

10

-40

40

-10 -40

40

0 -5

2915- 39 Ma = 0.27

-10 -20 0 20 pitch [deg]

-20 0 20 pitch [deg]

10 plunge [mm]

5

5

-10 -40

40

10 plunge [mm]

plunge [mm]

-20 0 20 pitch [deg] 2915- 37 Ma = 0.23

10

-40

-5 -10

-40

40

0

plunge [mm]

-20 0 20 pitch [deg]

5

plunge [mm]

-40

10 plunge [mm]

0

2915- 35 Ma = 0.19

10 plunge [mm]

5

-10

plunge [mm]

2915- 34 Ma = 0.17

10 plunge [mm]

plunge [mm]

10

5 0 -5 -10

-40

-20 0 20 pitch [deg]

40

-40

-20 0 20 pitch [deg]

40

Fig. 9. Phase plots of the airfoil LCOs for Mach numbers Ma = 0.13–0.37.

maximum, respectively. The plunge velocity being low, the relative incidence angle is close to the pitch angle at these time instants. During this dynamic stall condition, the airflow remains attached even for surprisingly high relative incidence angles. For comparison, the same data are reported for a second measurement 2915-44 at much higher flow velocity, Ma = 0.37. The time history of the pitch and plunge signals in Fig. 6 shows vibration with a slightly lower pitch amplitude, but much higher plunge amplitude than 2915-33. The pitch waveform is nearly triangular. The peak-to-peak pressure amplitude of signals p1U and p2U is now almost 30 kPa . The pressure signals and the Schlieren visualizations (see Fig. 8) reveal that in this case, the boundary layer separates from upper surface at φ = 24° (frame # 2636) shortly before the plunge minimum reattaches at φ = 15° (frame # 2652), detaches from the lower surface at φ = − 20° (frame # 2663) and reattaches at φ = − 10° (frame # 2682). The aeroelastic response of the system was measured in a whole range of flow velocities. Figs. 9 and 10 visualize the limit cycle oscillations in the phase plane and as a sequence of airfoil contours in one period of oscillation for all the measured flow velocities. The phase trajectory rotates counterclockwise, with increasing plunge amplitude, decreasing pitch amplitude and decreasing phase difference between the plunge and pitch oscillation. These trends are also plotted in Fig. 11 together with the dependence of the LCO frequency on the Mach number.

4. Discussion and conclusions The dynamic stall events of a NACA 0015 airfoil with pitch and plunge degrees of freedom, equipped with motion and pressure sensors, have been measured at Re = 180 000–570 000 in a high-speed wind tunnel. The airfoil model and the wind tunnel are designed for optical access to the test section, providing possibility to perform Schlieren flow visualizations and interferometric measurements. Due to the relatively soft torsional spring realizing the pitch restoring moment in the current configuration, the model has a low ratio between the pitch and plunge natural frequencies fφ /fy = 0.86 compared to other self-oscillating airfoil systems such as that of Marsden and Price (2005) with fφ /fy = 2–5 or the model of Razak et al. (2011) where fφ /fy = 1.3. The airfoil is highly susceptible to deep dynamic stall instability within a wide range of inflow velocities, with the peak-to-peak pitch amplitude reaching more than 90°. The data available from follow-up measurements using a set of torsional springs (not published here) show that the pitch angle amplitude drops significantly when torsional stiffness is increased, while all other parameters are kept constant. An interesting aspect is the triggering mechanism for the dynamic stall of the airfoil with zero initial incidence angle. Theoretically, the dynamic stall governed by flow separation from the suction side of the airfoil could be set off by a classical flutter instability once the airfoil reaches sufficiently high pitch amplitudes. The aeroelastic response of the system at low airspeed, with damping decreasing with increasing inflow velocity, suggests that for certain velocity exponentially increasing oscillations (i.e., negative damping) might occur. However, in current model configuration this assumption was

P. Šidlof et al. / Journal of Fluids and Structures 67 (2016) 48–59 2915- 31 Ma = 0.13

2915- 33 Ma = 0.15

57

2915- 34 Ma = 0.17

2915- 35 Ma = 0.19

30

30

30

30

20

20

20

20

10

10

10

10

0

0

0

0

-10

-10

-10

-10

-20

-20

-20

-20

-30

-30

-30

-30

-20 -10

0

10

20

30

-20 -10

40

2915- 36 Ma = 0.21

0

10

20

30

40

-20 -10

2915- 37 Ma = 0.23

0

10

20

30

40

-20 -10

2915- 39 Ma = 0.27

30

30

30

30

20

20

20

20

10

10

10

10

0

0

0

0

-10

-10

-10

-10

-20

-20

-20

-20

-30

-30

-30

-30

-20 -10

0

10

20

30

40

-20 -10

2915- 41 Ma = 0.31

0

10

20

30

40

-20 -10

2915- 42 Ma = 0.33

0

10

20

30

40

-20 -10

2915- 43 Ma = 0.35

30

30

30

20

20

20

20

10

10

10

10

0

0

0

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-10

-10

-10

-10

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-20

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-30

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-20 -10

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2915- 44 Ma = 0.37

30

-20 -10

0

2915- 40 Ma = 0.29

-20 -10

40

0

10

20

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40

Fig. 10. Motion of the airfoil for Mach numbers Ma = 0.13–0.37: location of airfoil contours (in mm) in 55 time instants of one oscillation period.

2π

6 4 2

46 44 42 40 38

0.20

0.25 0.30 Mach [1]

0.35

17.4 17.2 17.0 16.8 16.6

3π 2 π π 2

16.4

0 0.15

17.6

Phase Difference [rad]

8

frequency of vibration [Hz]

48 pitch amplitude [deg]

plunge amplitude [mm]

10

0.15

0.20

0.25 0.30 Mach [1]

0.35

0.15 0.20 0.25 0.30 0.35 Mach [1]

0

0.15 0.20 0.25 0.30 0.35 Mach [1]

Fig. 11. LCO plunge amplitude, pitch amplitude, frequency and phase difference of pitch behind plunge as a function of flow velocity.

never confirmed. In most measurements, the flow visualizations were acquired during the LCOs only. In a small number of cases, nevertheless, the measurement was set up such that the high-speed camera data are available also for the transient regime. In all these measurements, the flow visualizations show that there are no attached-flow gradually increasing oscillations. After the initial trigger induced either by releasing the airfoil from an initial displacement, or (for higher flow velocities) even by a small disturbance in the incoming flow, the pitch oscillations start abruptly with flow separation and dynamic stall occurring immediately in the first period of oscillations. The airfoil model does not allow setting initial displacement in pitch in a well-defined way, but an initial impulse in pitch may be induced indirectly by releasing the model from an initial displacement in plunge due to exocentric mass center location with respect to the elastic axis. Consequently, in certain flow velocity range the aeroelastic response of the system is highly sensitive to the initial conditions, leading, for

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the same inflow velocity, to damped oscillations for a low initial deflection (when the pitch angle remains small and flow attached), and to violent dynamic stall for slightly higher initial plunge displacement which triggers sufficiently high pitch angle leading to flow separation. The limit cycle response of the system as a function of freestream flow velocity was reported in detail, together with synchronized Schlieren visualizations of the flow field revealing the boundary layer behavior during dynamic stall. With increasing inflow velocity, the plunge amplitude increases dramatically, accompanied also by slight rise in the frequency of oscillation and decrease of phase lag between pitch and plunge. The behavior of the pitch amplitude is very similar to that reported by Poirel et al. (2008): increasing up to Ma = 0.18 and then significantly decreasing with higher flow velocities. The upper limit in flow velocity for stable LCOs is not clear. Above Ma = 0.37 (u = 125.8 m/s) the oscillation amplitudes reached the safety margins of the model design and thus the flow velocity could not have been further increased. For an airfoil under deep dynamic stall conditions under a similar Reynolds number regime, Lee and Gerontakos (2004) showed that at certain angles of attack the laminar boundary layer separates from the leading edge leading to stall, with significant hysteresis in the angles of attack between the flow separation and reattachment. The current measurements confirmed the same behavior of the boundary layer and agree quantitatively in terms of critical angles of attack, although the flow visualizations do not allow us to study in detail the leading edge vortex development and convection, turbulent breakdown, reattachment and return of the boundary layer to the laminar state. In addition to Schlieren visualizations, the current setup allows measurements of the flow field using Mach–Zehnder interferometer. In this context, the relatively small dimensions of the airfoil model and high flow velocities compared to other studies are beneficial for the spatial resolution (number of fringes) in the interferograms. The interferometric measurements have been already performed with the current model, but will be analyzed and reported later. Recently, a set of replaceable torsion springs with higher stiffness than that used in the current measurements have been prototyped and tested, proving significant pitch angle amplitude reduction. Also, measurements with non-zero initial angle of incidence are envisaged in near future.

Acknowledgments The research has been supported by the Czech Science Foundation, project 13-10527S “Subsonic flutter analysis of elastically supported airfoils using interferometry and CFD”. The measurements were performed in the wind tunnel of the Institute of Thermomechanics, Academy of Sciences of the Czech Republic in Nový Knín, in cooperation with M. Luxa, D. Šimurda, J. Horáček and I. Zolotarev. The natural frequencies and damping ratios of the mechanical model were measured by Š. Chládek.

Appendix A. Supplementary material Supplementary data associated with this article (video sequences of the Schlieren visualizations in measurements 291533 and 2915-44, see Table 2) can be found in the online version at http://dx.doi.org/10.1016/j.jfluidstructs.2016.08.011.

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