Measurement of Unsteady Aerodynamic Characteristics of a Heaving Wing in a Low Reynolds Number Flow

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Measurement of Unsteady Aerodynamic Characteristics

of a Heaving Wing in a Low Reynolds Number Flow

*

MasatoOKAMOTO,† ShotaFUKATSU, and DaisukeSASAKI

Department of Aeronautics, Kanazawa Institute of Technology, Hakusan, Ishikawa 924–0838, Japan

The objective of this study is to determine the two-dimensional unsteady aerodynamic forces and moment acting on a heaving wing in a uniformflow using a wind tunnel. However, it is difficult to measure the aerodynamic forces acting on the heaving wing due to measuring device oscillation and the large inertial force of the wing model. In this study, a new type of wind tunnel test, named‘‘heaving wind tunnel,’’ was developed. Here, the wing model remains stationary as the wind tunnel oscillates with a heaving motion. The advantage of this experimental method is that the measurement results are unaffected by the large inertial force acting on the oscillating wing model. Therefore, the wing model can be used in the same way as in steady state experiments. The normal force, thrust and pitching moment coefficients of a heaving airfoil were measured using the heaving wind tunnel test developed in this study. Throughflow visualizations and pressure meas-urements, we found that the rapid drop in normal force coefficient after it reached its maximum value was due to a large growing leading-edge vortex.

Key Words: Unsteady Aerodynamics, Heaving Wing, Wind Tunnel Test

1. Introduction

Many insectsfly by flapping their wings and this method can achieve high performance in both forwardflight and hov-ering. During a wing beat, the blade element undergoes three types of motion: those are heaving (vertical motion to the wing chord), feathering (rotation about the aerodynamic cen-ter), and lead-lag (parallel motion along the wing chord). Many experimental and computational studies have been conducted to understand the unsteady aerodynamic charac-teristics of an insect wing beat at a low Reynolds number. In experimental studies, water tank tests are often used as the higher density of the water compared to air allows the un-steady aerodynamic forces at a low Reynolds number to be measured more easily. Dickinson et al.1)_{studied the}_fluid

dy-namics of some airfoils under translational and rotational motions by conducting a towing test, which will be discussed later. Sunada et al.2)measured thefluid dynamics of a wing model assuming hovering flight with sinusoidal plunging and pitching oscillations. Additionally, many experimental tests have been performed for three-dimensional wings with unsteady motion, considering insect wings such as the wings of the fruitfly and hawkmoth.3–10)_{Each study con}_{firmed that}

unsteady high lift generation is caused by leading-edge vor-tex reattachment and the vorvor-tex behavior related to the rota-tional motion at the end of a stroke.

Towing tests in water or oil tanks were often conducted for these experimental studies. The Reynolds number, deﬁned as Re ¼ Uc=, is about ten times larger in water than in air be-cause the kinematic viscosity coeﬃcient ¯ of water is an

or-der of magnitude smaller than that of air. At a low Reynolds number, the force acting on a model in water is easy to meas-ure due to the large dynamic pressmeas-ure (U2=2), which is a re-sult of water being 800 times denser than air.

Although water tank tests are useful for taking measure-ments at very-low Reynolds numbers, it will be difficult to measure a wing model close to an actual insect wing. An in-sect wing is quite different from the wing of a large aircraft in the section profile. The thin airfoils, such as thin flat plate, circular arc, and corrugated profile, are known to perform well in low Reynolds numberflows.11)The objective of the present study is to obtain the unsteady aerodynamic charac-teristics of a moving wing in a uniformflow using a wind tunnel and the aerodynamic characteristics of a steady wing. In the past, one of authors of this paper conducted a wind tunnel test to examine the sinusoidal heaving and feathering motions of those types of airfoils in uniform flow.12) How-ever, the wing models used in our experiment were limited by the inertial force of the mass of the wing model. The wing models, made of light balsa wood, were a little thicker than the thin airfoil that is effective for insect sized wings. The in-ertial force is particularly significant when the wing model is oscillating with heaving motion. Terefore, the unsteady aero-dynamic forces acting on an oscillating wing model are more difficult to measure than when the wing is in a steady state. In the present study, we developed a small wind tunnel that oscillates with sinusoidal heaving motion, which will be referred to as a “heaving wind tunnel” in this paper. In the test section of the wind tunnel, the airflow oscillates around the wing model, which is placed into the tunnel in a steady state. Under these conditions, the shape and the mass of the wing model barely affect the aerodynamic meas-urements of heaving motion, and the inertial force of the wing model can be ignored. We studied the unsteady

†_{Corresponding author, okmt}_{@neptune.kanazawa-it.ac.jp} Vol. 64, No. 3, pp. 147–155, 2021

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namic characteristics of a wing oscillating in a sinusoidal heaving motion using the heaving wind tunnel, and the e ﬀec-tiveness of this type of wind tunnel test was conﬁrmed.

Recently, there has been increased interest in micro air ve-hicles (MAVs) at low Reynolds numbers.13)_{For MAVs as}

small as an insect,flapping wings may be a more effective method of generating thrust than a small propeller. Further-more, researchers have proposed the use of flapping wings in unmanned aircraft for Mars exploration, which would per-form in a low Reynolds numberflow as the atmospheric den-sity on Mars is only 1% of Earth’s atmosphere.14)Therefore, a better understanding of the aerodynamics offlapping wings would be beneficial to the development of low Reynolds number aircraft.

2. Experimental Materials and Methods 2.1. Wind tunnel

Figure 1 shows the heaving wind tunnel, which is a small pusher-type wind tunnel, developed for this study. In the large fixed chamber, three small fans (9GV1212P1J01, Sanyo Denki Co., Ltd.) provide airflow to the wind tunnel chamber with little velocityfluctuation. The flow straighten-ing chamber consisted of 40 mesh nylon screens and an alu-minum honeycomb. It is connected to the test section via a contraction unit with a contraction ratio of 6. The size of the rectangular test section was 150 200 200 mm (H W L), and a 400 mm long diffusion unit was attached to the back of the test section. The wind tunnel system, which consists of theflow-straightening chamber and the test sec-tion, is placed onto slide rails, where it can be oscillated with sinusoidal heaving motion using a direct-current (DC) motor. The displacement during the heaving motion is detected us-ing a laser displacement sensor (IL-S100, KEYENCE Co., Ltd.).

The wind velocity U in the test section was regulated to be 1–5 m/s by changing the rotational speed of the fans that were driven by a pulse width modulation (PWM) system. The turbulence intensity of the wind velocity was less than

0.4% ( u2=U < 0:4%). The wind velocity was determined using a Pitot tube, and the very small diﬀerential pressure be-tween the total and static pressures was measured using an electric manometer (DMP200N, Okano Works, Ltd.).

The velocity measured by the Pitot tube was modiﬁed by counting the Karman vortex-street frequency, determined by the Strouhal number, using a hot-wire anemometer located downstream of the circular cylinder. The angle-of-attack on theﬁxed wing model was measured using a three-hole Pitot tube that was installed independently from the test section moving with the heaving motion.

The load measuring device, consisting of a three-component balance and a stepping motor, was set under the test section, and the wing model in the test section was attached to it. The three-component balance was fabricated from aluminum blocks, on which strain gauges were at-tached, thereby allowing the vertical and horizontal forces and pitching moment to be measured. The balance output signals were magniﬁed using a strain ampliﬁer (AS2503, Nippon Avionics Co., Ltd.), and the setting angle of the wing model could be varied using the driving mechanism of the stepping motor, which was controlled using a microcom-puter.

The displacement of the oscillating wind tunnel, the wind velocity, the air stream angle-of-attack, and the forces acting on the model wing were analyzed using a personal computer through a 16-bit analog-to-digital (A/D) converter (AIO-163202FX-USB, CONTEC Co., Ltd.). All of the output sig-nals werefiltered to reduce high-frequency noise using the same low-pass filter device (9B02, A&D Co., Ltd.), which did not affect the signal phase. A low-pass filter with a corner frequency three times larger than the frequency of the sinus-oidal wing oscillation was used for data processing, because aerodynamic oscillations with higher frequency could not be distinguished from mechanical noise.

The aerodynamic data was collected at a sampling fre-quency of 200 Hz through the 16-bit A/D converter. The aerodynamic data of the steady wing model, which could be reproduced with suﬃciently high repeatability, were ob-tained by averaging 300 datasets. The linearity of the output signals of the load measuring device was less than 0.1% for the forces and 0.5% for the moment in the measurement range studied. The minimum readable loads were 4:7 105_{N for the vertical force,}_{2:4 10}5_{N for the horizontal}

force and3:4 107Nm for the pitching moment. The inter-ference between the sensors was conﬁrmed to be small and the natural frequencies of the load cells were more than 70 Hz for the forces and 60 Hz for the pitching moment.

Figure 2 shows the aerodynamic components acting on the wing model installed in heaving wind tunnel. In the test section of the wind tunnel, a wing model with a setting angle ª is exposed to a horizontal velocity U. When the test section is oscillating with a heaving displacement h, the air is oscil-lating at an inﬂow angle ¤, and ﬂows over the wing model at the angle-of-attack ¡. This is equivalent to a wing model moving at angleª and oscillating with heaving displacement h, while experiencing horizontal velocity U. The heaving dis-Honeycomb and screens

Heaving motion driving motor Laser displacement sensor

Wing model

Load measuring device

Slide rail Straightening chamber

Diffusion unit Test section

Fans

Seeding oil sheet production device

Large fixed chamber

Heaving motion U Top view Front view (Sectional drawing) Air Seeding particle Wing model Fans Straightening chamber

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placement h (positive downward) can be given by a sinusoi-dal heaving motion with heaving amplitude h1, as follows:

h ¼ h1cos !t ð1Þ

The ¡ and ¤ parameters are obtained using the following equation:

’ ¼ tan1ð _h=UÞ ¼ þ ’

)

ð2Þ The normal force n and thrust t components and the pitch-ing moment m actpitch-ing on the wpitch-ing model are acquired directly from the measuring device. The force components with re-spect to the inﬂow velocity, the lift l, and the drag d, can be obtained using: l d ! ¼ cos ’ sin ’ sin ’ cos ’ " # n t ! ð3Þ The non-dimensional aerodynamic coeﬃcients C_n, Ctand

Cm, and the pressure coeﬃcient Cpare given by the

follow-ing equations, which are normalized by the dynamic pressure U2=2: Cn¼ 2n U2S; Ct¼ 2t U2S; Cm¼ 2m U2Sc; Cp¼ 2p U2 ð4Þ where S is the model wing area and c is the wing chord.

In the testing setup, the wing model was attached to the measuring device and placed between the top and bottom walls of the test section. The clearance between the wingtip and the wall was less than 2.5% of the wing chord. Although the clearance was slightly large, compared to the experimen-tal results of other wind tunnels, it was confirmed that the ef-fect of this clearance was small enough to still enable accu-rate measurement of the steady aerodynamic characteristics of the wing at the Reynolds number studied. Using a hotwire anemometer, it was subsequently confirmed that there was a constantflow velocity distribution between the two walls of the test section, except within 2% of the wing width from the surface of the wall. In the test section oscillating with heav-ing motion, by movheav-ing the three-hole Pitot tube around the measurement part, dispersion in the distribution of theflow velocity in the test section was confirmed to be less than

1%, except in the range of the 5% section width near the wall. The pressure distribution of the ﬂat plate wing, which is shown in Fig. 4, was measured using 21 electric manometer sensors (KL17, NAGANO KEIKI Co., Ltd.) to simultane-ously measure the pressure at diﬀerent points along the wing. The minimum readable pressure value was 0.1 Pa and the re-sponse time was less than 20 ms. The output pressure data were sampled similarly to the force measurements using the A/D converter.

The flow around the wing was visualized in a sheet of seeding oil particles. The device to create this sheet was in-stalled in theflow generation chamber, as shown on the bot-tom left of Fig. 1. The device consists of many small fans, a honeycomb and screens, and the seeding particles made us-ing a seedus-ing generator (KANOMAX Co., Ltd.). The seed-ing particles were sprayed in front of the small fans, creatseed-ing a sheet about 10 mm in height thatflowed into the test section of the wind tunnel, as shown in Fig. 3. This seeding particle sheet moved with the wind tunnel as it oscillated with heav-ing motion. A laser light was used to irradiate the particle sheet, and theflow around the wing model was recorded at a frame rate of 300 frames/s using a high-speed movie camera (HLS-L2, DITECT Co., Ltd.) installed above the test section. The black regular lines, which are seen in theflow visualiza-tions that will be presented later, were caused by the rectify-ing honeycomb. In thisflow visualization, the vortex created

Lift, l Drag, d Normal force, n Thrust, t Angle-of-attack, α Setting angle, θ Horizontal velocity, U Pitching moment, m

Heaving velocity, h&

Inflow angle, φ

Inflow velocity, V

Wing model

Fig. 2. Aerodynamic components acting on the wing model in the heaving wind tunnel setup.

(a)ࠉAirfoils Thick flat plate (Airfoil 3)

Thin flat plate (Airfoil 1)

Thin circular arc airfoil (Airfoil 2) t=1.25%c t=0.75%c t=5%c camber=6%c c Tip plate Tip plate U Pressure holes (upper surface) Pressure hole (leading-edge) Pressure tubes Pressure tubes Pressure holes (lower surface) Wing section (Airfoil 3)

(b)ࠉPressure measurement model Fig. 4. Wing models tested in this study.

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on the wing appears black in the thin particle sheet irradiated by the laser light, and the vortex is specifically emphasized due to the three-dimensional vortexflow. This flow visual-ization method wasfirst reported by Aoki et al.15)

2.2. Wing models

Figure 4(a) shows the airfoils examined in the present study. A thinﬂat plate (Airfoil 1) was chosen to be the refer-ence airfoil in this experiment because it is not governed by the Reynolds number in a steady state for Re > 3000. A thin circular arc airfoil with a 6% camber (Airfoil 2) was chosen to show the best performance at the Reynolds number stud-ied.16) _{Airfoil 1 and Airfoil 2 have thicknesses that were}

1.25% and 0.75% of their chord length, respectively. These models were made from a thin aluminum plate. The sections of the leading- and trailing-edges of both thinner wing mod-els were rectangular proﬁles made by cutting vertically. The planforms of these wing models are rectangular, with a chord of 40 mm and an aspect ratio of 3.75. The thick ﬂat plate (Airfoil 3) was fabricated with a rounded leading-edge and a thickness that was 5% of its chord length. This airfoil was designed so that the pressure distributions around the airfoil could be measured. As shown in Fig. 4(b), the third wing model had many small 0.5 mm diameter pressure holes, and it was made from an acrylic resin plastic that was formed using an optical shaping apparatus.

2.3. Theflow made by the heaving wind tunnel Using the same angle-of-attack, the airflow around a heav-ing wheav-ing in uniformflow and the airflow around a wing in the heaving wind tunnel were compared using the smoke-wire method offlow visualization. In the former flow case, the wing model and smoke-wire were oscillated together with heaving motion in a stationary wind tunnel. In the latter flow case where the wing model is stationary, the smoke-wire was placed in the test section of the wind tunnel oscil-lating with heaving motion. In both cases, the images of the flow streamlines, which were path lines in the oscillating wing, were almost the same. This indicates that the aerody-namic characteristics of a heaving wing in a uniform flow can be accurately studied using the heaving wind tunnel de-veloped in this study.

Next, the inflow angle in the heaving wind tunnel was measured using the three-hole Pitot tube. In Fig. 5, the dis-placement h of the heaving wind tunnel and the inflow angle ¤ as a function of time are shown. Although a sinusoidal dis-placement was observed, the behavior of the inflow angle cannot be perfectly described by Eq. (2). This indicates that there is a slight difference between the flow behavior of a sta-tionary wing in the heaving wind tunnel and a heaving wing in a uniformflow. In the heaving wind tunnel, there is a delay in the change inflow direction due to the air viscosity. How-ever, when expressed as a function of the angle-of-attack, the aerodynamic forces and moment measured in the heaving wind tunnel can be considered a valid approximation of the unsteady characteristics of a heaving wing. Therefore, in the present study, the angle-of-attack¡ was used, which was obtained using the three-hole Pitot tube.

3. Results and Discussion

3.1. Steady aerodynamic characteristics of airfoils Figure 6 shows the steady aerodynamic characteristics of the thinflat plate (Airfoil 1), the circular arc airfoil (Airfoil 2), and the thickflat plate with a rounded leading-edge (Air-foil 3) at Re ¼ 4000. In steady state, the lift coefficient Cl

and drag coefficient C_dare equivalent to the normal force co-efficient C_n and the negative thrust coefficient C_t, respec-tively.

As shown in the lift curve of Airfoil 1, Clincreases

propor-tionally with the increase in angle-of-attack¡, and then re-mains constant for > 7. For < 3, the lift curve slope is slightly less than the theoretical slope of 2, and it ap-proaches close to the theoretical value for > 3. At the ap-propriate angle-of-attack, the separated laminar ﬂow of the ﬂat plate reattaches and forms a leading-edge separation bub-ble. When this occurs, the lift slope increases. As the angle-of-attack increases, the location of reattachment moves to-wards the rear of the airfoil, which was determined previ-ously using the particle image velocimetry (PIV) method.17)

The Cdincreases as Clincreases. The pitching moment

coef-ﬁcient at 25% of the chord length Cm; 0:25chas a slightly

posi-tive slope for3< < 3, and decreases to about¹0.1 as the angle-of-attack increases. Although the leading-edge sep-aration bubble causes an increase in the nose-up moment, the moment changes to the nose-down direction as the size of the separation bubble increases.

For the circular arc airfoil (Airfoil 2), the non-linearity of its lift curve is more remarkable than that of the ﬂat plate. The lift slope is small for 1< < 9 and large for 1 _{< < 1} _or ₉_{< < 10}_{. The C}

l max is also large.

Similar to the thin ﬂat plate, the non-linear lift slope was

1 5 . 0 0 -20 -10 0 10 20 -20 -10 0 10 20 Time, s Disp laceme nt, h, mm In fl ow angle, φ, de g h φ

Fig. 5. Time histories of heaving displacement and inﬂow angle (f ¼ 3 Hz). -20 -10 0 10 20 30 -1 -0.5 0 0.5 1 1.5 -20 -10 0 10 20 30 0 0.4 0.8 -20 -10 0 10 20 30 -0.2 0 0.2 -20 -10 0 10 20 30 -5 0 5 10 Cl Cd Cm, 0. 25 c l/ d Angle-of-attack, α, deg Angle-of-attack, α, deg Theoretical lift slope 2π

Airfoil 1 Airfoil 2

Airfoil 3 Re=4000

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caused by the separation bubble generated at the thin lead-ing-edge. Although Cd increases as Cl increases, the rate

of increase decreases as the lift slope increases rapidly. At that time, the maximum lift-to-drag ratio ðl=dÞ_max which is larger than that of Airfoil 1 is provided.

The Clof the thickﬂat plate (Airfoil 3) is slightly smaller

than that of Airfoil 1. This is because the generation of the leading-edge separation bubble is not clear at the rounded leading-edge of Airfoil 3. As Cd is larger than that of Airfoil

1, theðl=dÞ_maxis smaller than that of Airfoil 1.

3.2. Unsteady aerodynamic characteristics of airfoils Theﬂow velocity was ﬁxed at Re ¼ 4000, and the wind tunnel oscillated along the vertical direction with an amplitude-to-chord-length ratio h1=c of 0.5 at oscillation

fre-quencies f of 1.5 and 3 Hz. These frefre-quencies correspond to the reduced frequencies, which are defined as k ¼ fc=U, of 0.1 and 0.25 respectively. The setting angle ª of the wing model was changed from 15 to15 at3 step intervals while the normal force coefficient C_n and the thrust coe ffi-cient Ct, which are perpendicular and parallel to the constant

ﬂow, respectively, and the pitching moment coeﬃcient at 25% of the chord length Cm; 0:25cwere measured.

Figure 7 shows the Cn, Ct, and Cm; 0:25cof Airfoil 1

oscil-lating during heaving motion at diﬀerent setting angles ª. As shown in the Cn–¡ curve, Cnchanges according to the steady

Cl curve. However, there is a large hysteresis between the

change in Cnand the angle-of-attack, and Cntraces a

clock-wise rotation at all the setting angles studied. There was a phase advance of the change in Cn relative to the change

in the angle-of-attack, which was caused by the large ﬂow separation of the wing that increased asª increased. As the reduced frequency was increased from 0.13 to 0.25, there

was increased hysteresis observed in the rotational curve of Cn.

As shown in the Ct–¡ curves, Ct is mostly negative at

every angle-of-attack. However, Ct at ¼ 0 maintained a

constant minimum value regardless of the angle-of-attack for the wide range of13< < 13, even at the higher re-duced frequency of 0.25. Asª increased, C_t traced an anti-clockwise rotation.

As shown in the Cm; 0:25c–¡ curves, there is a negative

slope and Cm; 0:25c traces an anti-clockwise rotation.

How-ever, the hysteresis of Cm; 0:25c at ¼ 0 was small.

oscil-lating during heaving motion at diﬀerent setting angles ª. As shown in the Cn–¡ curve, Cntraces a clockwise rotation for

everyª, and an increase in hysteresis as the angle-of-attack changed was observed as the angle-of-attack increased. For k ¼ 0:13, the peak value of Cn occurred at ¼ 12, and

for k ¼ 0:25, the Cnpeak value was observed at every setting

angle of 9. At this point, the peak value of Cnwas over

two times greater than Clfor the steady wing.

Although Ct is mostly negative at every ª studied, there

were some positive values for k ¼ 0:25 at the setting angle of9 0. The angle-of-attack at which positive val-ues of Ctwere observed corresponds to the¡ range where a

comparatively large lift-to-drag ratio l=d was obtained for the steady wing. Therefore, positive thrust was obtained for the heaving airfoil at the negative setting angles, and the airfoil needs to have a large l=d.

-10 0 10 20 30 -0.5 0 0.5 1 1.5 -10 0 10 20 30 -0.5 0 -10 0 10 20 30 -0.2 0 Angle-of-attack, α, deg Cn Ct Cm, 0.2 5c -1.5 _-10 ₀ ₁₀ ₂₀ ₃₀ -1 -0.5 0 0.5 1 1.5 2 -10 0 10 20 30 -1 -0.5 0 -10 0 10 20 30 -0.4 -0.2 0 0.2 Cn Ct Angle-of-attack, α, deg Cm, 0. 25 c 12 h1/c=0.5, f=1.5Hz, k=0.1 (Upper chart) h1/c=0.5, f=3Hz, k=0.25 (Right chart) Re=4000 (a)ࠉHeaving frequency f = 1.5 Hz (b)ࠉf = 3 Hz θ=6㼻 θ=12㼻 θ=18㼻 θ=0㼻 θ=6㼻 θ=0㼻 θ=12㼻 θ=18㼻

Fig. 7. Aerodynamic characteristics of Airfoil 1 during heaving motion.

(b)ࠉHeaving frequency f = 3 Hz (a)ࠉHeaving frequency f = 1.5 Hz -30 -20 -10 0 10 20 30 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -30 -20 -10 0 10 20 30 -1 -0.5 0 0.5 -30 -20 -10 0 10 20 30 -0.6 -0.4 -0.2 0 0.2 0.4 Angle-of-attack, α, deg Cn Ct Angle-of-attack, α, deg Angle-of-attack, α, deg Cm, 0. 25 c θ=18㼻 θ=12㼻 θ=6㼻 θ=0㼻 θ=-18㼻 θ=-12㼻 θ=-6㼻 h1/c=0.5, f=3Hz, k=0.25, Re=4000 -30 -20 -10 0 10 20 30 -1 -0.5 0 0.5 1 1.5 2 -30 -20 -10 0 10 20 30 -0.5 0 -30 -20 -10 0 10 20 30 -0.4 -0.2 0 0.2 Angle-of-attack, α, deg Cn Ct Angle-of-attack, α, deg Angle-of-attack, α, deg Cm, 0.2 5c θ=18㼻 θ=12㼻 θ=6㼻 θ=0㼻 θ=-18㼻 θ=-12㼻 θ=-6㼻 h1/c=0.5, f=1.5Hz, k=0.13, Re=4000

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The Cm; 0:25c reached the large negative value of¹0.4 at

large setting angles, similar to the trend observed for Cn.

Although Cm; 0:25ctraced an anti-clockwise rotation for most

setting angles studied, it traced a clockwise rotation for a small angle-of-attack at ¼ 0.

The mean aerodynamic coeﬃcients at every ª for heaving motion are shown in Fig. 9. Compared with the steady Cl,

Cd and Cm; 0:25c, these mean values deviate from the

steady-state values at a higher angle-of-attack beyond the stall angle. The non-linear curves of Cl and Cm; 0:25c seen

under steady-state conditions are moderated in the mean Cn

and Cm; 0:25ccurves.

oscil-lating during heaving motion at diﬀerent setting angles ª. By comparing the Cn–¡ curves of Airfoil 3 and Airfoil 1, we can

see that the Cnpeak value of Airfoil 3 is slightly larger than

that of Airfoil 1. The same tendency is seen in the Cm–¡

curves. This seems to be caused by the large separationﬂow due to the airfoil thickness.

These results were compared in a previous study,12) in which the forces and moment acting on the wing with heav-ing motion were measured in a static wind tunnel. The trends of amplitudes in theﬁrst harmonics of C_n, Ct, and Cm; 0:25c

with the changes in angle-of-attack were comparatively sim-ilar, and the phase advances of these coeﬃcients were also observed in both experiments. However, the phase diﬀerence of the present study was larger than that of the previous study. It is necessary to conduct two types of experiments under the same conditions to elucidate the cause.

3.3. Flow visualization around Airfoil 3

Flow visualization around the thick flat plate (Airfoil 3) was performed by changing the inflow angle ª. Figure 11 shows one period of airflow around the heaving flat plate at ¼ 6, which represents the distinctive flows. A flow along the airfoil surface is seen at ¼ 2:4, and a small vortex can be seen on the upper surface of the airfoil’s lead-ing-edge at ¼ 4:8. This is where the leading-edge suction

peak is generated, which will be discussed further when looking at the pressure distributions in Section 3.4. The lead-ing-edge vortex, which is created by the leadlead-ing-edge suction peak, grows as the angle-of-attack increases. There are three leading-edge vortices: Two large leading-edge vortices with a small one in between them, which can be seen in the en-larged photo in Fig. 11. The second large vortex, which has a clockwise rotation, grows as the angle-of-attack in-creases, and it is called the dynamic stall vortex.18)

When a downward ﬂow, which is almost vertical to the wing surface, is seen at the back of this vortex, a large in-crease in the normal force coeﬃcient C_n can be observed in the Cncurve, and Cnreaches a maximum value at ¼ 16

as the vortex grows. As the angle-of-attack increases, the leading-edge vortex that was induced by the clockwise-rotat-ing second vortex eventually disappears, and the second vor-tex grows largely over the wing. A perfectly separatedﬂow is seen at ¼ 18:5, and Cndecreases signiﬁcantly. However,

as the angle-of-attack decreases, there is lessﬂow separation at the upper surface, and ﬂow separation is no longer ob-served at the minimum angle-of-attack.

3.4. Pressure distributions of Airfoil 3

The pressure distribution along Airfoil 3 was measured, and the aerodynamic coeﬃcients of C_land Cdfor the steady

state and Cnand Ctfor the unsteady state were obtained by

integrating the pressure components along the wing chord. They were then compared to the force measurement results. 3.4.1. Steady-state conditions

Figure 12(a) shows the pressure coeﬃcient C_pdistribution along the upper and lower surfaces of Airfoil 3 at Re ¼ 4000. The Cp of the leading-edge point (x=c ¼ 0) is

positive as this is a stagnation point. At ¼ 0, the Cp of

0 10 20 -0.5 0 0.5 1 0 10 20 -0.2 0 -20 -10 0 10 20 -1 -0.5 0 0.5 1 1.5 -20 -10 0 10 20 -0.2 0 0.2 Cn (st eady Cl ), Ct (stead y -Cd ) Cn (stead y Cl ), Ct (s teady -Cd ) Cm, 0. 25 c Cm, 0. 25 c Angle-of-attack, α, deg Angle-of-attack, α, deg Steady 3Hz 1.5Hz (a)ࠉAirfoil 1 (b)ࠉAirfoil 2

Fig. 9. Comparison between mean unsteady values and steady-state val-ues in aerodynamic coeﬃcients.

-10 0 10 20 30 -0.5 0 0.5 1 1.5 -10 0 10 20 30 -0.5 0 -10 0 10 20 30 -0.2 0 Angle-of-attack,α, deg Cn Ct Cm, 0.25 c -1.5 -10 0 10 20 30 -1 -0.5 0 0.5 1 1.5 2 -10 0 10 20 30 -1 -0.5 0 -10 0 10 20 30 -0.4 -0.2 0 0.2 Cn Ct Angle-of-attack,α, deg Cm, 0.25 c θ=18㼻 θ=12㼻 θ=6㼻 θ=0㼻 θ=18㼻 θ=12㼻 θ=6㼻 θ=0㼻 h1/c=0.5, f=1.5 Hz, k=0.1 (Upper chart) h1/c=0.5, f=3 Hz, k=0.25 (Right chart) Re=4000 (a)ࠉHeaving frequency f = 1.5 Hz (b)ࠉf =3 Hz

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the upper and lower surfaces near the leading-edge (except at the stagnation point) is its negative peak value caused by the rounded leading-edge and increases towards the trailing-edge. As the angle-of-attack increases, the maximum negative Cp near the leading-edge increases negatively on

the upper surface and increases positively on the lower sur-face. For 6, the Cpnear the trailing-edge increases

neg-atively. As the angle-of-attack increases further, the Cp

dis-tribution along the upper surface ﬂattens due to ﬂow separation on the upper surface of the wing. At every angle-of-attack, the positive Cpof the lower surface decreases

con-stantly towards the trailing-edge.

Figure 12(b) shows the aerodynamic coeﬃcients obtained

by integrating the pressure distributions. The Cl, Cd and

Cm; 0:25care similar to those obtained by force and moment

measurements, except for a slightly larger positive Cl and

negative Cm; 0:25c near ¼ 10.

3.4.2. Heaving motion

The pressure distributions of Airfoil 3 during heaving mo-tion were measured for0 15 at 3 deg intervals and Re ¼ 4000. The ﬁgures in this paper show the Cp

distribu-tions at ¼ 6, which are a good representation of the dis-tinctive pressure distributions. Figures 13(a) and (b) show the Cpdistributions obtained by changing the angle-of-attack

at ¼ 6and the loci of the aerodynamic coeﬃcients calcu-lated by integrating the pressure distributions, respectively. As the angle-of-attack increases from 6, large increases in Cnare observed. The negative Cpof the upper surface near

the leading-edge begins to increase negatively (see circles No. 1–4), and the maximum negative C_p value is observed near ¼ 8(see circle No. 4). More negative Cp values

be-gin to be observed at the back of the leading-edge and the po-sition where the largest negative Cp value occurs moves

backward (see circles No. 5–8). At the same time, the nega-tive Cpnear the leading-edge begins to decrease as the

angle-of-attack increases. As described in theﬂow visualizations, the large negative Cp represents the dynamic stall vortex,

and the negative Cp behind the vortex increases rapidly to

-10 0 10 20 -1 -0.5 0 0.5 1 0 25 50 75 100 -1 0 1

Chord-wise direction of flat plate, %

Cp

θ=0 deg 3 deg 6 deg 9 deg 15 deg

Angle-of-attack, α, deg Cl , C d ,Cm, 0.25 c Cm,0.25c Cd Cl Theoretical lift slope 2π

Pressure integral Direct measurement Solid line: upper surface

Broken line: lower surface

(a)ࠉPressure distributions (b)ࠉIntegral values of pressure

Fig. 12. Pressure distribution along Airfoil 3 (steady-state).

Cp 䐩䐪䐫䐬 0 25 50 75 100 -1 0 1 -10 0 10 20 -0.5 0 0.5 1 1.5 2 -10 0 10 20 -0.5 0 -10 0 10 20 -0.4 -0.2 0 0.2 䐟䐠䐡䐢䐣䐤䐥䐦䐨䐧䐩䐪䐫䐬䐥䐧䐠䐡䐦 Angle-of-attack, α, deg Angle-of-attack, α, deg Cn Ct Angle-of-attack, α, deg Cm, 0.25 c Pressure integral Direct measurement Solid line: upper surface

Broken line: lower surface 0 25 50 75 100 -2 -1 0 1 䐟䐠䐡䐢䐣

Cp 0 25 50 75 100 -2 -1 0 1

Cp 䐨䐤䐥䐦䐧

(a)ࠉPressure distributions (b)ࠉLoci of integral values

Fig. 13. Pressure distributions along Airfoil 3 during heaving motion (f ¼ 3 Hz, h1=c ¼ 0:5, k ¼ 0:25, ¼ 6).

Fig. 11. Flow visualizations around Airfoil 3 during heaving motion (f ¼ 3 Hz, h1=c ¼ 0:5, k ¼ 0:25, ¼ 6, Re ¼ 4000).

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near zero. As the vortex moves towards the trailing-edge, the Cn reaches a maximum at ¼ 16 (see circle No. 8). The

large Cn begins to decrease as the angle-of-attack increases

further to ¼ 19:5, and soon after this, there is a rapid de-crease in Cn(see circles No. 9–12). At this time, the Cp

dis-tribution along the upper surfaceﬂattens, and the positive C_p values of the lower surface retain large values, especially near the leading-edge. At the larger angle-of-attack, large negative and positive Cp values were observed

along the upper and lower surfaces, respectively. As the angle-of-attack decreased, the negative Cp values of the

upper surface increased, the positive Cpvalues of the lower

surface decreased (see circles No. 13 and 14), and a negative Cnwas observed.

As seen in Fig. 13(b), the loci of Cn, Ct and Cm; 0:25c

ob-tained by integrating the pressure distributions were compa-ratively similar to those of the forces and moment measure-ments, except for diﬀerences in maximum C_nand minimum Cm; 0:25c. In addition, the pressure distributions provided a

good simulation of theﬂow visualizations. That is, consistent trends were observed from both methods of calculating the aerodynamic coeﬃcients, which have large hysteresis at a large angle-of-attack, a large maximum Cn, and a rapid drop

in Cnafter reaching the maximum.

4. Conclusions

To obtain the two-dimensional aerodynamic characteris-tics of a wing oscillating with heaving motion, the heaving wind tunnel developed in this study was used at a low Rey-nolds number of 4000 and the following results were ob-tained.

1) By introducing the heaving motion, the maximum nor-mal force coeﬃcient acting on the wing increases remark-ably.

2) The forces acting on the oscillating wing have a large phase advance relative to changes in the angle-of-attack due toﬂow separation.

3) The phase advance increases as the angle-of-attack and the reduced frequency increase.

4) For a certain range of angle-of-attack, a positive thrust coeﬃcient was observed for the heaving thin circular arc air-foil, which also displayed a large lift-to-drag ratio under steady-state conditions.

5) A growing vortex generated at the leading-edge of the wing causes a rapid drop in the normal force coeﬃcient after it reaches its maximum.

The following advantages of the heaving wind tunnel were found in this study:

1) Wing models of diﬀerent masses and shapes can be used without aﬀecting the aerodynamic measurements of heaving motion.

2) The stationary wing model also allows pressure distri-bution measurement andﬂow visualization to be easily per-formed.

However, the following disadvantages were found: 1) Perfect sinusoidal change in the angle-of-attack is not

observed in the airflow created by the heaving wind tunnel. 2) It is difficult to change the angle-of-attack independ-ently because it is determined by the heaving andflow veloc-ities.

Our next study will investigate the effects of the airfoil and its planform on the unsteady aerodynamics of a moving wing in uniformflow as well as the aerodynamic characteristics of a steady-state wing. In addition, computationalfluid dynam-ics (CFD) solutions will be performed to compare the exper-imental results.

Acknowledgments

The authors are grateful to Dr. Akira Azuma (Professor Emeritus of the University of Tokyo) for his useful advice and warm encour-agement.

This work was supported by KAKENHI (19K04844).

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Shigeru Saito Associate Editor