Post stall Control Using Flexible Fin onAir Foil

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47th AIAA Aerospace Sciences Meeting, 5-8, Jan 2009, Orlando, Florida

Post-Stall Flow Control Using Flexible Fin on AirfoilTianshu Liu 1, J. Montefort, 2 W. Liou, 3 S. Pantula, 4 Y. Yang 5

Western Michigan University, Kalamazoo, MI 29008Q. A. Shams 6

NASA Langley Research Center, Hampton, VA 23681

AbstractThis paper explores a new concept of post-stall flow control for airfoils by using a thin

flexible fin attached on the upper surface of an airfoil to passively manipulate flow structures infully separated flows for drag reduction and oscillation/flutter suppression. The relevantsimilarity parameters are given for aerodynamic and aeroelastic scaling of an airfoil with aflexible fin. Experiments are conducted on a NACA0012 airfoil section in a water tunnel. Forcemeasurements are made by using a balance and velocity fields are measured by using particleimage velocimetry. Drag reduction and oscillation suppression particularly for the natural low-frequency oscillation in deep stall are achieved by using a flexible fin attached at a suitablelocation on the airfoil. The analysis of the flow fields and the connection between the velocity

fields and the fin kinematics provides insight into the physical mechanisms of the low-frequencyoscillation and the post-stall flow control.

1. IntroductionFlow separation control is of immense importance to the performance of air vehicles and

other technologically important systems involving fluids. Generally, it is desired to postponeseparation such that the form drag is reduced, stall is delayed, lift is enhanced, and pressurerecovery is improved. Therefore, a great effort has been made over years for flow separationcontrol (or stall control) of airfoils and wings by using various techniques including synthetic jets(Glezer & Amitay 2002, Mittal et al. 2005), plasma actuators (Post & Corker 2004, Patel et al.2007), piezoelectric actuators (Seifert, et al. 1998, Mathew, et al. 2006), deflected flap

(Greenblatt & Wygnanski 2000), vortex generators (Gad-el-Hak 2000), passive and active blowing (Gad-el-Hak 2000), local suction (Atik et al. 2005), and oscillating camber (Munday &Jacob 2002). In general, separation control has been achieved by either injecting high-momentum fluid into separation regions or removing low-momentum fluid from the partiallyseparation flows for angle of attacks (AoA) that are smaller than or near the stall AoA. It has

been demonstrated that a partially separated flow can be almost re-attached by unsteady forcing,and the stall is delayed.

Post-stall flow control after the flow over an airfoil or wing is completely separated is morechallenging due to complicated separated flow structures. Post-stall flows contain not only

1 Associate Professor, Corresponding Author, Department of Mechanical and Aeronautical Engineering, G-220, ParkviewCampus, Western Michigan University, Kalamazoo, MI 49008, [email protected] , 269-276-3426, Senior Member AIAA

2. Assistant Scientist, Department of Mechanical and Aeronautical Engineering, Western Michigan University, Kalamazoo, MI490083. Professor, Department of Mechanical and Aeronautical Engineering, Western Michigan University, Kalamazoo, MI 49008,Associate Fellow, AIAA4. Graduate Research Assistant, Department of Mechanical and Aeronautical Engineering, Western Michigan University,Kalamazoo, MI 490085. Graduate Research Assistant, Department of Mechanical and Aeronautical Engineering, Western Michigan University,Kalamazoo, MI 490086. Research Scientist, NASA Langley Research Center, Hampton, VA 23681Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

AIAA Paper 2009-07361

47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-1106

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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vortex shedding due to the Kelvin-Helmholtz instability and random turbulence in shear layers, but also a natural, global low-frequency oscillation that could cause severe flutter in the marked regime in a diagram of AoA and Reynolds number (see Fig. 1). Zaman et al. (1989) havestudied in detail the low-frequency oscillation of a LRN(1)-1007 airfoil near stall and suggested that it might be due to switching between the stalled and un-stalled flows. Bragg et al. (1993,

1996) and Broeren & Bragg (1999, 2001) have conducted subsequent studies on unsteadystalling of a LRN(1)-1007 airfoil and provided experimental evidences on the combination of thin-airfoil stall and the trailing edge stall as the mechanism of the low-frequency oscillation.Large-eddy simulation for a LRN(1)-1007 airfoil has captured the some characteristics of thelow-frequency oscillation observed in the corresponding experiments (Mukai et al. 2006).

Figure 1 is a diagram of AoA and Reynolds number, indicating the possible regimes of flow patterns over thin airfoils, which is similar to that given by Wu et al. (1998). Data of thestall AoAs in are collected from Loftin & Smith (1945) and Selig et al.

(1995). Data of the stall AoAs in are from Laitone (1997), Kunz &Kroo (2001), Sunada et al. (2000) and Okamoto et al. (1996). The region marked by dashed lineindicates a possible regime in which the natural low-frequency oscillation may occur. Data fromthe experiments of Zaman et al. (1989) and Bragg et al. (1993, 1996) and Broeren & Bragg(1999, 2001) are marked in the diagram. The cases of the low-frequency oscillation investigated in this work are also marked in Fig. 1. It can be seen that the low-frequency oscillation occurs ina range of Reynolds numbers for micro-air vehicles and low-pressure turbine blades. The effectsof acoustic excitation on stalled flows over an airfoil have been studied by Zaman (1992). Stallflow control using periodic blowing and suction near the leading edge of an airfoil has beennumerically simulated by Wu et al. (1998). The excitation is typically based on the vortex-shedding modes. At this stage, however, there is still a lack of a systematical investigation of control of the low-frequency oscillation and vortex-shedding all together particularly at highAoAs.

7 4c 10103 Re −×=

43c 102102 Re ×−×=

This paper explores the post-stall flow control for airfoils by using a thin flexible finattached on the upper surface of an airfoil. Figure 2 illustrates a thin flexible fin attached to anairfoil at post-stall angles of attack. The flexible fin used in this study is a thin Mylar film for

passive control, which could be replaced by a polymer or composite sheet embedded withsensors and actuators for active control. To a certain degree, the idea of using a flexible fin isinspired by the fact that the major propulsive and control elements of birds, insect and aquaticanimals like wings, tails and fins are flexible. Nevertheless, the proposed flexible fin for dragreduction is not a replica of fish fins that are mainly used for locomotion and maneuver.Morphologically, a flexible fin attached to an airfoil is basically rectangular unlike fish fins. Inthis paper, the relevant similarity parameters are first discussed for aerodynamic and aeroelasticscaling of an airfoil with a flexible fin. Experiments conducted on a NACA0012 airfoil sectionin a water tunnel are described. The results obtained from force measurements by a balance and velocity field measurements by PIV are presented. It is shown that drag reduction and oscillationsuppression in stall can be achieved by using a flexible fin attached at a suitable location on the

NACA0012 airfoil. The spectral analysis of the flow fields and the correlation analysis betweenthe velocity fields and the fin kinematics are given. These results reveal the physicalmechanisms of the low-frequency oscillation and the post-stall flow control. This concept could

be useful in applications like gust alleviation for micro-air-vehicle and separation suppression of low-pressure turbine blades.





2. Similarity ParametersThe similarity parameters for a rigid airfoil with a thin flexible fin are considered. The

similarity parameter for a rigid airfoil in an incompressible flow is Reynolds number ν / cU Rec ∞= , where is the freestream velocity and c is the airfoil chord. The non-

dimensional time parameter is∞U

c / U τ ∞ , where τ is a characteristic time scale of a flexible fin.When τ is replaced by / 1 , this parameter is inversely proportional to the reduced frequency

∞U / c , where is a characteristic frequency of the flexible fin.For a flexible fin attached to the rigid airfoil, additional similarity parameters should be

obtained. The flexible fin is considered as a thin cantilever plate deforming under fluid-dynamical loading. The differential equation for the displacement w of a thin plate (e.g. fin) isgenerally expressed by (Meirovitch 1967)

)t ,P(F t

)t ,P(w)P( M )]t ,P(w[C

t )]t ,P(w[ L

2

2

=∂

∂+∂∂+ , (1)

where P denotes the coordinates x and y, L is a linear differential operator, C is a linear homogenous differential operator for damping, M is also a linear operator, and is an

external distributed force on the plate. In our case,

)t ,P(F

p)t ,P(F ∆= is the fluid-dynamical pressuredifference across the plate. For a homogenous plate, the operator is the bi-harmonic

operator, where is the plate flexural rigidity, and is the massdistribution of the plate. The rigidity depends on the plate thickness ( h), Young’s modulus( E ), and Possion ratio ( ν ). The operator C is a linear combination of the operator L and the massfunction M , , where and are constant coefficients. An analysis for therectangular fin deformation is given in Appendix A.

4 E D L ∇=

)1(12 / Eh D 22 E ν −= )P( M

E D

M a LaC 21+=

1a 2a

Let us introduce the following non-dimensional variables∞= q / F ' F , l / w' w = , l / x' x = , l / y' y = , τ / t ' t = , (2)

where is the dynamical pressure and l is the fin length. Thus, substituting Eq. (2)into Eq. (1), we have the non-dimensional form of Eq. (1)

2

U 5.0q ∞∞∞ = ρ

( ) ' F ' t

' wG' w

' t G)a(' w'

' t G

a' w' G 2

2

2224

114

1 =∂∂+

∂∂+∇

∂∂

⎟ ⎠ ⎞

⎜⎝ ⎛ +∇ τ

τ , (3)

where the two similarity parameters related to the rigidity and mass distribution are

∞

=ql

DG 3

E 1 and

∞

=q

l M G 22 τ

. (4)

Other similarity parameters related to the damping are τ / a 1 and τ 2a . Therefore, the functionalrelation for the force coefficient of an airfoil with a flexible fin can be generally expressed by

)a , / a ,G ,G , Re ,( f C 2121F τ τ α ∞= . (5)In this study, the effect of AoA is mainly considered at a fixed Reynolds number. Since a Mylar fin with the fixed length and thickness is used, the non-dimensional parameters related to the finare fixed. Clearly, a considerable amount of work is required in the future in order to understand the behavior of the flow control in the whole parametric space.





3. Experimental SetupExperiments were conducted in the water tunnel (The Rolling Hills Research Corporation

Model 1520) in the Fluid Mechanics Laboratory at Western Michigan University. The testsection is nominally 15 in wide, 20 in high and 60 in long. The tempered glass, 3/8 in thick on

the sidewalls and 1/2 in thick on the bottom, is mounted with silicon rubber, allowing good optical access from the top, bottom, both sides and rear for flow visualization and particle imagevelocimetry (PIV) measurements. The tunnel is operated as a continuous flow channel and thewater level in the test section is typically adjusted to be roughly 50 mm below the top of thewalls. The free water surface provides simple access to the model and easy setup of an externalforce balance. There is a 6:1 contraction section before the test section for turbulence reductionand avoidance of local separation and vorticity development. The test section flow velocity isvariable from 0 up to 0.3 m/s. In the test section, the turbulence intensity is less than 0.1%, and the velocity non-uniformity is less than 2%, and the mean flow angularity is less than 1% in boththe pitch and yaw angles.

A plastic NACA0012 airfoil section model built by a rapid prototype machine was tested.

The chord and span of the model were 10 in and 12 in, respectively. A clear Mylar (PET- polyester) film was used as a flexible fin. The Young’s modulus and Poisson ratio for Mylar were 2.8 GPa and 0.37, respectively. The Mylar film was attached to the upper surface of theairfoil by Scotch tape. Figure 3 shows a flexible Mylar fin attached on the NACA0012 model inwater in a typical PIV image. To reduce the three-dimensionality of flow, the model wasmounted on two 6.35 mm thick Plexiglas end plates (406 mm by 279 mm) that were supported

by an aluminum bar. The bar was directly connected to a home-made external force balance thatwas above the water surface. For measurements of very small drag force, the main balanceelement was a 275 mm long aluminum beam with a 30 mm thick and 12.75 mm wide cross-section. Two sections of the beam ware machined and considerably thinned, and the center locations of the two sections were at 6.4 and 11.3 mm from the corresponding end. Thethickness and length of the thinned sections is 2.75 mm and 35 mm, respectively. Two Omegastrain gauges were installed at the middle of the thinned sections. When the balance element(beam) was mounted perpendicularly to the incoming flow direction, the difference betweenvoltage outputs from the two strain gauges simply equaled to a product of the drag and thedistance between the two gauges. Balance calibrations verified this simple and directrelationship. The measurement uncertainty of drag was less than one gram. A similar beam wasused for lift measurement when this beam was parallel to the incoming flow direction.

The flow around the model was illuminated by a 2-mm thick laser sheet generated by a BigSky laser (CFR190) from the rear window of the test section. The laser frequency was set at 15Hz, which was sufficient for time-resolved measurements in this study since the dominantfrequencies in flows were lower than 0.5 Hz. The interval between two pulses was 2 ms. Themeasuring area was imaged by a PIV camera (TSI PIVCAM 10-30, Model 630046) with an 85-mm lens. PIV images were processed using the TSI Insight5 PIV software (32 ×32 pixelswindow size).

4. Flow Control Using Flexible Fin4.1. Drag Reduction and Oscillation Suppression

A 0.25c long rectangular Mylar fin with a thickness of 0.1 mm was first used in preliminary tests, where c is the airfoil chord of 10 in (254 mm). To examine how a flexible fin





affects the drag of the airfoil, the fin was placed at different locations on the upper surface of the NACA0012 model. In addition, two tandem fins at different locations were tested. Theincoming flow velocity was 0.25 m/s, and the Reynolds number based on the chord was

. Figure 4(a) shows the time-averaged drag coefficient as function of AoA

for different arrangements of flexible Mylar fins. When AoA is larger than 12

4c 103.6 Re ×= DC

o, of the NACA0012 model with a flexible fin is smaller than that of the baseline model. For non-separated flow at smaller AoAs, it has been observed that a flexible fin was not naturallyattached on the surface in water due to its buoyancy, and thus it became an intrusive object to theattached boundary layer. In this case, as also shown in Fig. 4(a), of the NACA0012 modelwith a flexible fin is greater. This problem could be practically solved by somehow making theflexible fin attached on the surface for small AoAs before stall. This study focuses on the effectsof a flexible fin on separated flow after the airfoil stalls. Instead of completely removingseparation, a flexible fin passively alters flow structures to suppress separation and reduce thedrag. As a typical case for detailed investigation, a significant drag reduction at high AoAs by a0.25c fin located at 0.1c is shown in Fig. 4(b). The absolute measurement uncertainty of drag is

1 gram, and the relative uncertainty of is 1-3% in a range of AoAs from 12

DC

DC

DC o

to 20o

. TheCFD results of the drag coefficient for the same case are also given in Fig. 4(b), which areobtained by using a combined Navier-Stokes immersed boundary solver coupled with a finite-element structural code for laminar flows (Pantula 2008). The CFD prediction is roughlyconsistent with the measured results, particularly indicating a drag reduction when AoA is larger than 12 o, although the CFD drag values are considerably larger. As shown in Fig. 5, CFDcalculations indicate that the pressure drag is considerably reduced by the 0.25c fin at highAoAs. The skin friction drag is increased for small AoAs since the fin is detached from thesurface in calculations to simulate the situations in the water tunnel tests. Figure 6 shows the liftcoefficient as a function of AoA for the baseline model and the model with the 0.25c fin, wherethe correlation given by the McCormick’s formula for the effective wing aspect ratio of 4.42 is

shown as a reference (Liu et al. 2007). The lift of the model with the fin is considerablydecreased. For low AoAs, the smaller lift is caused by the detached fin from the surface. After stall at high AoAs, the decreased lift is probably caused by suppressing the lift-generatingorganized vortices by the fin. It will be pointed out that the suppression of the organized vorticesis also responsible to the drag reduction after stall. Therefore, the lift decrease after stall isconsidered as a side effect of this flow control method.

Actually, the drag of the NACA0012 model varies with time due to the highly unsteadynature of separation. The root-mean-squared (r.m.s) variation of the drag coefficient of the

baseline model is about 3% of the mean value. The power spectra of are shown in Fig. 7 for the baseline model and model with the flexible fin at AoAs of 14

DC

DC o and 18 o. The spectral peaks at

0.07 and 0.12 Hz are observed for the baseline model at AoA of 14 o, and these low-frequencycomponents have the Strouhal numbers based on the projected chord e LF U / )sin(c f St α = of 0.017 and 0.03, respectively. For AoA of 18 o, the spectral peaks at 0.04, 0.07 and 0.08 Hzcorrespond to = 0.012, 0.022 and 0.025, respectively. These low-frequency fluctuations of the drag are basically consistent with the natural low-frequency oscillation studied by Zaman etal. (1989). The similar low-frequency components are also found at AoAs of 16

LF St

o and 20 o. Asignificant backward-forward swing of the baseline model has been observed in a range of AoAsfrom 14 o to 20 o. Interestingly, as indicated in Fig. 7, the flexible fin attached at 0.1c





considerably dampens the low-frequency drag fluctuations and the model swing. The physicalmechanisms of the drag reduction and oscillation suppression will be explored based on time-revolved PIV measurements of flow fields.

4.2. Development of Flow Structures

4.2.1. Mean Flow PropertiesTo understand the physical mechanisms of the drag reduction by a flexible fin, PIVmeasurements of flow fields were conducted and velocity fields were obtained at 15 Hz. Figure8 shows the time-averaged streamlines and vorticity fields for the baseline model and the modelwith a 0.25c fin located at 0.1c at AoA of 18 o. The change of the velocity fields by the flexiblefin is appreciable. For the baseline model, a time-averaged large vortex or a re-circulating flowregion is observed on the upper surface near the trailing edge of the airfoil, which can be looselyconsidered as a separation bubble in a long-time average sense. Indeed, a time sequence of velocity and vorticity fields indicates that strong large-scale organized vortices are developed from the Kelvin-Helmholtz instability in the shear layer. Although the evolution process of thesevortices is highly unsteady, they occur at a higher probability in that region. Due to the presence

of the flexible fin, the time-averaged large vortex is largely destroyed.Figure 9 shows the profiles of the x-component of the mean velocity U at x/c = 0.4, 0.51,

0.66, 0.79 and 0.91 on the upper surface for AoAs of 18 o and 20 o, where x is the coordinate alongthe incoming flow direction and c is the projected chord onto the x-coordinate. Clearly, themomentum loss is reduced due to the presence of a flexible fin in the separated flow region,which corresponds to the drag reduction found in force measurements by the external balance.Further evidence is provided by the momentum thickness development along the x-direction, asshown in Fig. 10. Here, the momentum thickness is defined as

dyU

) y(U 1

U ) y(U

e y es

⎥⎦

⎤

⎢⎣

⎡−= ∫

∞ ∆∆θ , (6)

where )U min(U U −=∆ , U is the x-component of the mean velocity, is the externalvelocity, and is the y-coordinate at the surface. As indicated in Fig. 10, the development of the momentum thickness is significantly suppressed by a flexible fin.

eU s y

4.2.2. Unsteady Flow Properties

Figures 11 and 12 show the power spectra of the x-component of velocity U across theseparation region along the y-direction on the baseline model and the model with a 0.25c finlocated at 0.1c at five x locations for AoA of 18 o. As illustrated in Fig. 2, the x-coordinate is thecoordinate starting from the leading edge along the incoming flow and the y-coordinate is thenormal coordinate through the trailing edge. For the baseline model, the dominant spectralcomponent at is developed downstream. The Strouhal number based on the initial

shear layer momentum thickness is that is much smaller than the

most unstable mode ( ) in the shear layer, where

Hz07 .0 f =4

e0shear 104.8 U / f St −×== θ

032.0St shear = mm20 =θ is the estimated initialmomentum thickness. Therefore, such a low-frequency component cannot be the shear layer mode from the linear shear-layer instability, and it rather represents a natural, global low-frequency oscillation of the separation region. In fact, the Strouhal number based on the front-

projected chord )sin(c α is 022.0U / )sin(c f St e LF == α , which is consistent with the natural





low-frequency oscillation observed by Zaman et al. (1989). This low-frequency component of the velocity U is responsible to the drag fluctuation shown in Fig. 7. Figure 12 shows the effectof a flexible fin on the power spectra of U at the same locations, indicating that the dominantcomponent and other components of U are significantly suppressed. This is also consistent withthe drag measurements shown in Fig. 7.

Furthermore, Figure 13 shows the development of the spectral components averaged acrossthe shear layer ( 37 .029.0c / y −= ) along the x-direction (incoming flow direction) for the baseline model and the model with a 0.25c fin located at 0.1c. For the baseline model, thedominant low-frequency component at Hz07 .0 f = ( 022.0St LF = ) increases in a nearly linear fashion in , and then it saturates and decays. As indicated in Fig. 13(b), thedevelopment of all the spectral components particularly the dominant component at

8 .02.0c / x −= Hz07 .0 f =

( ) is suppressed by the flexible fin. Accordingly, as shown in Fig. 14, the Reynolds

stress

022.0St LF =uv− in the whole separated flow region is considerably decreased due to the presence of

the flexible fin, where U U u −= and V V v −= are the velocity fluctuations in the x- and y-

coordinates, respectively.

5. Vortex Shedding and Low-Frequency OscillationThe above results indicate that a flexible fin dramatically dampens the dominant spectral

components particularly the low-frequency oscillation in the separated flow region over a NACA0012 airfoil. In order to understand the underlying mechanisms, it is necessary toexamine the development of vortices and the generation of the low-frequency oscillation on the

baseline NACA0012 model. Figure 15 shows the streamlines and vorticity fields, indicating thegrowth of a counterclockwise vortex shedding from the leading edge of the baseline model in aninterval of 1.0 s. The vorticity sheet in the shear layer rolls up into discrete vortices due to theKelvin-Helmholtz instability and shed from the leading edge. The vortex travels downstream asits size increases, while another vortex remains near the trailing edge in a probabilistic sense.Then, the two vortices tend to merge together to form a larger vortex. This dual-vortex structureis typically observed in our experiments, which is similar to that in the dynamic stall (Carr 1988).The vortex shedding is at about 1 Hz that corresponds to the bluff-body vortex shedding Strouhalnumber 31.0U / )sin(c f St eshed == α .

PIV measurements were conducted to focus on the region near the leading edge. Figure 16shows the power spectra of the velocity components U and V at 0.45c for the baseline model.Interestingly, the dominant spectral peak at 1 Hz that is the bluff-body vortex shedding mode

appears in the spectrum of the normal velocity V . In contrast, the dominantcomponent in the spectrum of U is at 0.07 Hz that corresponds to the natural low-frequencyoscillation at . The low-frequency component in the spectra of U is detected in thewhole separated flow region. In the spectra of V , the vortex shedding component is strong in0.3-0.5c and the low-frequency component become more evident near the trailing edge. Thisindicates that the low-frequency oscillation is a global phenomenon, and in contrast the vortexshedding is more active in the shear layer near the leading edge. These two salient modescoexist in the separated flow region over the NACA0012 model. Figure 17 shows the Strouhalnumbers based on the local momentum thickness for the low-frequency mode ( f = 0.07 Hz) and the vortex shedding mode ( f = 1.0 Hz) for the baseline NACA0012 model. Although the vortexshedding Strouhal number increases with x/c due to the increasing momentum thickness, the

31.0St shed =

022.0St LF =


7




median value of is about 0.032 that is the most unstable mode in the shear layer. This provides a connection between the vortex shedding and the shear layer instability. On the other hand, the Strouhal number for the low-frequency mode is so small compared to the mostunstable mode in the shear layer that any direct relationship between the low-frequency modeand the Kelvin-Helmholtz instability is excluded.

θ St

θ St

Zaman et al. (1989) have suggested that the low-frequency oscillation may be attributed from switching between stalled and un-stalled flows. Phase-averaged flow field measurements

by Broeren & Bragg (1999) have found that the generation and growth of a separation bubblefrom the leading edge and its merging to the trailing-edge separation could cause the low-frequency oscillation. However, due to the highly unsteady flows with a considerable degree of randomness in our PIV measurements, the scenario of a steadily growing bubble near the leadingedge described by Broeren & Bragg (1999) is not clearly observed. To gain insights into theorigin of the low-frequency oscillation, the center of the large-scale vortex in the time-averaged velocity field in Fig. 8(a) is selected as a reference point (the mean zero-crossing point of U ) toexamine the temporal signal of U . In addition, when a U -velocity profile across the reference

point is plotted, there is an instantaneous zero-crossing point across which the velocity U changes its sign. The zero-crossing point defines the instantaneous boundary of the reversed flow. Figure 18 shows the time traces of the velocity U and the zero-crossing point position atthe reference point. Although there are a considerable amount of higher-frequency components,the low-frequency oscillation of 0.1 Hz is visible, and the power spectra clearly show thedominant peak at about 0.1 Hz for both the velocity U and the zero-crossing point position. It isalso observed in Fig. 18 that the phase angle between the velocity U and the zero-crossing point

position is about 180 o, which is confirmed by a correlation analysis. According to the continuityequation, the valley of the zero-crossing point position should roughly correspond to the peak of U . Further, the magnitude squared coherence between the velocity U and the zero-crossing point

position is calculated, i.e.,cp y

) f (P) f (P

) f (PC

UU y y

2

U y

U ycpcp

cp

cp= , (7)

where is the cross power spectral density, and and are the power

spectral densities. is a function of frequency with values between 0 and 1 that indicates

how well corresponds to U at each frequency. Figure 19 shows the magnitude squared

coherence , indicating a high correlation between the velocity U and the zero-crossing

point position at 0.1 Hz and 0.4 Hz. The above results indicate that the low-frequency

oscillation of U is related to the fluctuation of the zero-crossing point position or the boundary of the reversed flow. In a certain sense, this phenomenon corresponds to a cycle of the growth and

bursting of a separation bubble described by Broeren & Bragg (1999).

) f (P U ycp) f (P

cpcp y y ) f (PUU

U ycpC

cp y

U ycpC

cp y

It is required to extract the flow fields that are responsible to the low-frequency oscillationfrom the instantaneous velocity fields also containing the active vortex shedding at a higher frequency. In order to smooth out the velocity components associated with the vortex sheddingin the instantaneous flow fields, the velocity fields are averaged over a relatively short timeinterval of 0.73 s that is the vortex shedding period at five selected phases in a low-frequencycycle of U . These phases are marked by open circles on the time trace of U at the reference point





in Fig. 18. The short-time-averaged velocity fields at the five phases are shown in Fig. 20, whichreveal the flow structures associated with the low-frequency oscillation. As shown in Fig. 20(a),at the valley (a) in U at the reference point, a strong re-circulating flow region is observed in asense of the short-time average, which is caused by the occurrence of a strong vortex or a groupof interacting vortices developed in the shear layer. For simplicity, this re-circulating flow

region is informally called a vortex due to the closed streamlines although the definition of avortex is an arguable topic (Jeong & Hussain 1995). Figure 20(a) indicates that a smaller vortexforms near the leading edge. As shown in and 20(b), the newly-formed vortex grows as theoriginal large vortex travels downstream. The dual-vortex structure is typical in the separated flow region and a saddle point between the two vortices is usually observed. As shown in Fig.20(c), at the peak (c) of U at the reference point the velocity of the reversed flow becomes muchsmaller since the strong organized vortices are gone, and the topological structure of flow isconsiderably different. Figures 20(d) and 20(e) indicate that the dual-vortex structure occurs and develops again. The low-frequency oscillation results from the periodic switching between thestrongly and weakly re-circulating flows in the separated flow region, which may be triggered bythe global instability of the dual-vortex structure.

A simplified analysis is given in Appendix B to shed insight into the instability of the dual-vortex structure. Figure 21(a) shows the flow field of two point vortices on a flat plate at AoA of 18o, which is generated by using the Joukowski transformation as a crude model of the dual-vortex structure observed in the experiments. The normalized strengths of the vortex #1 (near the leading edge) and vortex # 2 (near the trailing edge) are 7 .0cU / 1 =∞Γ and 1cU / 2 =∞Γ ,respectively. A dynamical system for the two point vortices on a flat plate is given in AppendixB. The equilibrium positions of the two vortices are calculated depending on the non-dimensional strengths of the two vortices and AoA. Calculations indicate that the equilibrium

positions of the vortices are unstable for a finite-amplitude disturbance. This implies that thedual-vortex structure on an airfoil cannot remain stationary. Figure 21(b) shows typicaltrajectories of the two vortices moving away from their initial positions above a plate. This

problem is similar to the stability problem of a single vortex on an airfoil in trapping of a freevortex for lift enhancement (Saffman and Sheffield 1977, Huang and Chow 1982, Chow et al.1985). A free vortex on an airfoil is generally unstable for a finite-amplitude disturbance withoutactive flow control. After the two vortices move away from the airfoil, the dual-vortex structureis re-generated from interacting organized vortices developed in the shear layer. Although thedual-vortex structure is initially induced by the organized vortices developed in the shear layer due to the Kelvin-Helmholtz instability, the motion of the dual-vortex structure is governed byits own dynamical system once it forms.

6. Interaction between Flexible Fin and Unsteady FlowAs pointed out before, a flexible fin attached to the upper surface of the NACA0012 airfoil

passively interacts with flow, suppresses separation and reduces the drag at high AoAs. Tounderstand the interaction between the flexible fin and its surrounding flow, it is necessary tomeasure the fin kinematics and fluid dynamics from PIV measurements near the fin. A typicalPIV image near the fin is shown in Fig. 3. The fin is clearly visualized in PIV images as anintersection between the fin and laser sheet. The high-contrast edges of the fin in images aredetected by using the Canny’s edge detector. The coordinates of the fin edges are fitted using thecharacteristic beam functions and the fin magnitude is determined as a function of time. InAppendix A, the dynamics of a rectangular fin is discussed. For the nearly 2D deformation of





the fin observed in our experiments, the first eigenfunction is sufficient to describe the findeformation. Therefore, the normalized displacement of a thin fin is expressed as

) x( X )t (l / ) x(w 3113 η = , (7)

where is the coordinate along the incoming flow from the root of the fin and 3 x )t (1η are thetime-dependent amplitude. The first characteristic beam function is

) xsinh x(sink xcosh xcos) x( X 1111131 γ γ γ γ −+−= , (8)

where )cosh /(cos)sinh(sink 11111 γ γ γ γ +−= , l / x x 3= is the coordinate normalized by the

fin length and 875.11=γ . By using Eq. (7) for least-squares fit to the detected fin edges, the

time-dependent amplitude )t (1η is determined. Figure 22 shows the time-dependent amplitude)t (1η and power spectrum of the 0.25c flexible fin located at 0.1c for AoA of 18 o. The

dominant spectral peak is at 1.3 Hz and the secondary peak is at 0.12 Hz. For AoAs of 12 o, 14 o,16o and 20 o, the most dominant spectral peaks are 0.1 Hz, 0.1 Hz, 1.3 Hz and 0.25 Hz,respectively. In general, the fin amplitude has a dominant component around 1.3 Hz and somelow-frequency components in a range of 0.07-0.25 Hz. This indicates that the flexible fin is

more responsive to the vortex shedding associated with the shear layer instability.The velocity fields near the flexible fin are obtained in PIV measurements. A localcoordinate system located at the leading edge is used for its convenience, as shown inFig. 3. The transformations between the coordinate systems and are

)" y ," x() y , x( )" y ," x( x" x = and

α sinc y" y −= . Figures 23 and 24 show the power spectra of the x-component and y-component of velocity across the fin along the y-direction on the NACA0012 model with a0.25c fin located at 0.1c at several locations of for AoA of 18

U V

c / " x o. The power spectra of U have significant peaks at 0.1 Hz, 0.5 Hz and 1.3 Hz on the upper side of the fin in

that is approximately within the fin. More spectral peaks occur downstreamat . In contrast, the spectra of V have the significant peaks are around 1.3 Hz thatcorresponds to the vortex shedding mode. To provide a direct connection between the finkinematics and the surrounding flow, the velocity is averaged over a zone around the fin( and

39.027 .0c / " x −=45.0c / " x =

4.0c / " x25.0 ≤≤ 05.0c / " y05.0 ≤≤− ). Figure 25 shows the time traces of the relativefin amplitude and the zone-averaged velocity components U and V , where the fin amplitude and velocities are normalized by the fin length and the freestream velocity, respectively. Thewaveform of the fin amplitude approximately corresponds to that of the zone-averaged velocitiesexcept that there is a phase shift between them. Further, the magnitude squared coherences

between the fin amplitude and the zone-averaged velocity components and are

calculated, where

U 1C η V 1

C η

)t (1η is the fin amplitude, and U and V are the zone-averaged velocity

components. The definition of isU 1C η ) f (P) f (P)P UU

2

U U 1111 η η η η = f (C , where

is the cross power spectral density, and and are the power spectral densities

( is similarly defined). Figure 26 shows the magnitude squared coherence between the fin

amplitude and the zone-averaged velocity components. For both U and V , the dominant peaksin the magnitude squared coherence are in a range of 1 Hz to 1.7 Hz, indicating a highcorrelation between the fin amplitude and velocities at the vortex shedding mode. The phase

) f (P U 1η

) f (P11η η ) f (PUU

V 1C η


10




differences between the fin magnitude )t (1η and the zone-averaged velocity components U and around the fin are 14V o and 149 o, respectively.

7. ConclusionsForce measurements indicate that the drag is reduced and the natural low-frequency

oscillation is suppressed in deep stall for AoAs from 12 o to 20 o at by using athin flexible polymer fin attached at a suitable location on the NACA0012 airfoil. Due to the

presence of a flexible fin, the velocity deficit and the momentum loss in the separated flowregion are reduced, and the time-averaged large vortex causing stronger reversed flow is largelydestroyed. As a side effect, however, the lift is reduced in stall since the organized lifting-generating vortices are suppressed by the fin. Based on the spectral analysis of velocity fields, itis found that the dominant low-frequency oscillation at

4c 103.6 Re ×=

022.0St LF = and other spectralcomponents including the vortex shedding mode are suppressed by the flexible fin. For further understanding, the mechanism of generating the natural low-frequency oscillation on the baseline

NACA0012 model is explored. It is found that the vortex shedding associated with the Kelvin-

Helmholtz instability coexists with the low-frequency oscillation. The vortex shedding mode ismainly detected in the shear layer near the leading edge, and in contrast the low-frequencyoscillation is a global phenomenon in the entire separated flow region. The short-time averaged velocity fields show that the low-frequency oscillation is related to the development of the dual-vortex structure that is induced by the occurrence of a strong vortex or a group of interactingorganized vortices developed in the shear layer. A simplified analysis indicates that the dual-vortex structure on an airfoil is intrinsically unstable for a finite-amplitude disturbance, and therefore the low-frequency oscillation may result from the global instability of the dual-vortexstructure. Interaction between the kinematics of the flexible fin and the flow around it is studied.The flexible fin is mainly responsive to the vortex shedding associated with the shear layer instability although it is also affected by the low-frequency oscillation. The flexible fin dampensthe development of the shear layer instability and the organized vortices, and as a result the low-frequency oscillation is largely eliminated and the drag is reduced at high AoAs.

AcknowledgementsThis work was supported by the Air Force Office of Scientific Research, USAF, under the grantnumber FA9550-06-1-0187.

References:Atik, H., Kim, C. Y., Van Dommelen, L. L., & Walker, J. D. A., Boundary-Layer Separation

Control on a Thin Airfoil Using Local Suction, J. Fluid Mech., Vol. 535, 2005, pp. 415-443Bragg, M. B., Heinrich, D. C., & Khodadoust, A., Low-Frequency Flow Oscillation over Airfoils

near Stall, AIAA J., Vol. 31, No. 7, 1993, pp. 1341-1343Bragg, M. B., Heinrich, D. C., Balow, F. A. & Zaman, K. B. M. Q., Flow Oscillation over an

Airfoil near Stall, AIAA J., Vol. 34, No. 1, 1996, pp. 199-201Broeren, A. P. & Bragg, M. B., Flowfield Measurements over an Airfoil During Natural Low-

Frequency Oscillations near Stall, AIAA J., Vol. 37, No. 1, 1999, pp. 130-132Broeren, A. P. & Bragg, M. B., Unsteady Stalling of Thin Airfoils, in Fixed and Flapping Wing

Aerodynamics for Micro Air Vehicle Applications, Ed: T. J. Mueller, AIAA Press, Reston,Virginia, 2001, Chapter 10





Saffman, P. G. & Sheffield, J. S., Flow over a Wing with an Attached Free Vortex, Studies inApplied Mathematics, Vol. 57, 1977, pp. 107-117

Seifert, A., Eliahu, S., Greenblatt, D. & Wygnanski, I., Use of Piezoelectric Actuators for AirfoilSeparation Control, AIAA J., Vol. 36, No. 8, 1998, pp. 1535-1537

Selig, M., Guglielmo, J., Broeren, A. P. & Giguere, P., Summary of Low-Speed Airfoil Data,

Volumes 1, 2 and 3, SoarTech Publications, Virginia Beach, Virginia, 1995Sunada, S., Ozaki, K., Tanaka, M., Yasuda, T., Yasuda, K., & Kawachi, K., AirfoilCharacteristics at a Low Reynolds number, Journal of Visualization and Image Processing,Vol. 7, 2000, pp. 207-215

Tikhonov, A. N. & Arsenin, V. Y., Solutions of Ill-Posed Problems, John Wiley & Sons, NewYork, 1977, Chapter II

Warburton, G. B., The Vibration of Rectangular Plates, Proceedings of the Institution of Mechanical Engineers, Vol. 168, No. 12 , 1954, pp. 371-383

Wu, J-Z, Lu, X-Y, Denny, A. G., Fan, M. & Wu, J-M, Post-Stall Flow Control on an Airfoil byLocal Unsteady Forcing, J. Fluid Mech., Vol. 371, 1998, pp. 21-58

Zaman, K. B. M. Q., McKinzie, D. J. & Rumsey, C. L., A Natural Low-Frequency Oscillation of

the Flow over an Airfoil near Stalling Conditions, J. Fluid Mech., Vol. 202, 1989, pp. 403-442Zaman, K. B. M. Q., Effects of Acoustic Excitation on Stalled Flows over an Airfoil, AIAA J.,

Vol. 30, No. 6, 1992, pp. 1492-1499

Appendix A. Flexible Fin DynamicsThe solution to Eq. (1) for a rectangular thin plate can be written as an expansion based on

the eigenfunctions (Meirovitch 1967))P(wr

∑∞

=

=1r

r r )t ()P(w)t ,P(w η , (A1)

where )t (r η are the time-dependent amplitudes. Assuming that the eigenfunctions areorthogonal and substituting Eq. (A1) into Eq. (1), one has a system of the ordinary differentialequations for )t (r η

)t ( N )t ()t (c)t ( r r 2r s

1srsr

=++ ∑∞

=η ω η η &&& , ( L ,2 ,1r = ) (A2)

where the damping coefficients are

∫= D

sr rs )P(dD)]P(w[C )P(wc , ( L ,2 ,1s ,r = ) (A3)

and the generalized force is

∫= Dr r )P(dD)t ,P(F )P(w)t ( N . ( L ,2 ,1r = ) (A4)

Note that appropriate dimensional constants are absorbed into Eqs. (A3) and (A4) to make Eq.(A2) dimensionally consistent.

When the operator C is a linear combination of the operator L and the mass function M ,, where and are constant coefficients, the damping term in Eq. (A2) is not

coupled such that M a LaC 21

+=1a 2a

rsr r rs 2c δ ζ = . Thus, Eq. (A2) becomes





)t ( N )t ()t (2)t ( r r 2r r r r r

=++ η ω η ω ζ η &&& , ( L ,2 ,1r = ) (A5)where r ζ are the damping factors and r are the natural frequencies. The solution to Eq. (A5)is

)0(t sin)t exp(1

)0(t sin)1(

t cos)t exp(

d )t (sin)]t (exp[)( N 1

)t (

r rd r r rd

r rd 2 / 12r

r rd r r

t

0

rd r r r

rd

r

η ω ω ζ ω

η ω ζ

ζ ω ω ζ

τ τ ω τ ω ζ τ

ω

η

&⎥⎦

⎤

⎢⎣

⎡−+

⎥

⎦

⎤

⎢

⎣

⎡

−+−+

−−−= ∫, (A6)

where can be regarded as a natural frequency associated with the dampingsystem. The second and third terms are the starting transients. For an impulse force

2 / 12r r rd )1( ζ ω ω −=

)PP()t ()t ,P(F im−= δ δ , when 0)0(r

=η , the main behavior of )t (r η is described by

t sin)t exp( rd r r ζ − .

The approximation for is given by)P(wr

) y(Y ) x( X ) y , x(w mnr = , (A7)

where and are the characteristic beam functions (Warburton 1965, Reddy 1999).For the edge clamped and the other edge

) x( X n ) y(Y m0 x = a x = free, the characteristic beam function

is) x( X n) xsinh x(sink xcosh xcos) x( X nnnnnn γ γ γ γ −+−= , ( L ,3 ,2 ,1n = ) (A8)

where a / x x = is the normalized coordinate and )cosh /(cos)sinh(sink nnnnn γ γ γ γ +−= .

The values of nγ are given by solving the nonlinear equation 01coshcos nn=+γ γ . The first

three values of nγ are 875.11 =γ , 694.42 =γ and 8547 .7 3 =γ . For the edges and 0 y = b y = free, the characteristic beam function is) y(Y m

1) y(Y m = , ( )0m = y21) y(Y m −= , ( 1m = )

)2 / 1 y(cosh)2 / 1 y(cos) y(Y mmmm−+−= β , ( L ,6 ,4 ,2m = )

)2 / 1 y(sinh)2 / 1 y(sin) y(Y mmmm−+−= λ α λ , ( L ,7 ,5 ,3m = ), (A9)

where b / y y = is the normalized coordinate, )2 / sinh( / )2 / sin( mmm β −= , and

)2 / sinh( / )2 / sin( mmm λ λ α = . The parameters m and mλ are given by solving the equations

0)2 / tanh()2 / tan( mm=+ and 0)2 / tanh()2 / tan( mm

=− λ λ . The typical values of m

and mλ are 73.42 = , 99.104 = , 853.7 3 =λ , and 137 .145 =λ .

The combinations of and are) x( X n ) y(Y m) y(Y ) x( X ) y , x(w 011

= , ) y(Y ) x( X ) y , x(w 022= , ) y(Y ) x( X ) y , x(w 033

= ,

) y(Y ) x( X ) y , x(w 114= , ) y(Y ) x( X ) y , x(w 125

= , ) y(Y ) x( X ) y , x(w 136 = ,

) y(Y ) x( X ) y , x(w 217 = , ) y(Y ) x( X ) y , x(w 228

= , ) y(Y ) x( X ) y , x(w 239= (A10)





For a plate with any combination of fixed and free edges, the non-dimensional naturalfrequencies can be theoretically determined (Warburton 1954). Nevertheless, from anexperimental point of view, optical deformation measurements provide the time-dependentdeformation data at a number of locations. Given the eigenfunctions

( ) and measured for a number of points (

)t ,P(w )P(w ir

M ,2 ,1r L= )t ,P(w i iP N ,2 ,1i L= ) on the plate, a

system of equations for the coefficients )t (r η ( M ,2 ,1r L= ) at a given instant is

∑=

= M

1r

r ir i )t ()P(w)t ,P(w η . ( M ,2 ,1r L= , N ,2 ,1i L= ) (A11)

The coefficients )t (r η can be determined using least-squares method. Furthermore, todetermine the damping factors r ζ and the natural frequencies r , the impulse transient solution,

t sin)t exp()t ( rd r r r ζ η −∝ can be utilized in experiments when an impulse force is applied to

the plate. The use of spectral analysis for )t (r η gives and measurement of

the exponential decaying rate gives

2 / 12r r rd )1( ζ ω ω −=

r r ζ . In general, once r ζ and r are known, the

generalized force can be estimated from the measured magnitudes)t ( N r )t (r η by using Eq.(A5).

An inverse problem is how to recover the fluid-mechanical force from thegeneralized force . According to Eq. (A4), is a projected component of on the eigenfunction and therefore Fredholm integral equations should be solved for

. In general, this inverse problem can be solved by using the regularization method (Tikhonov and Arsenin 1977).

)t ,P(F

)t ( N r )t ( N r )t ,P(F

)P(wr

)t ,P(F

Appendix B. Two Vortices on a Flat Plate

In order to simulate the observed dual-vortex structure associated with the natural low-

frequency oscillation, the potential flow of two point vortices on a flat plate airfoil is considered as a preliminary model. In the ζ -plane, according the Thomson’s circle theorem, the complex

potential for an incoming uniform flow over a circle at the origin and two vortices outside thecircle is

ς π

Γ Γ Γ

ς ς

ς ς π

Γ

ς ς

ς ς π

Γ ς

ς ς α

α

ln2

i

/ aln

2i

/ aln

2i

eaeU )(w

210

22

22

12

11i22

i

+++

−−+

−−+

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ += −

∞

, (B1)

where α is the angle of attack, is the radius of a circle,a 1Γ and 2Γ are the strengths of thevortices #1 and #2, respectively,

1ς and

2ς are the strengths of the vortices #1 and #2,

respectively, and is the strength of a vortex located at the origin to meet the Kutta condition.The complex velocity is

0Γ


15




ς π Γ Γ Γ

ς ς ς ς π Γ

ς ς ς ς π Γ

ς ς ς α

α

12

i / a

112

i

/ a

112

iea

1eU d

)(dw

210

22

2

2

12

1

12

i22i

+++⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

−−

+

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

−−

+⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −= −

∞

. (B2)

By using the Joukowski transformation , the circle is transformed to a flat platewith a chord of in the z-plane. The inverse Joukowski transformation is

ς ς / a z 2+=a4c =

22 )4 / c()2 / z(2 / z −±=ς , where the positive and negative signs are taken for and , respectively. The complex velocity in the z-plane is

0) z Re( ≥0) z Re( <

ς ς ς ς

d / dz1

d )(dw

dz)] z([dw = . (B3)

At the trailing edge of the plate, the Kutta condition must be satisfied at or 2 / c z = a=ς suchthat

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−−−⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−−−++−= ∞

222

111210 / a1

1a / 1

1 / a1

1a / 1

1sinU a4)( ς ς Γ ς ς Γ α π Γ Γ Γ . (B4)

Figure 21(a) shows a typical flow field over a flat plate with two vortices. A question is whether or not this dual-vortex structure is stable.

By introducing the non-dimensional variables , , and

, a dynamical system for the two point vortices on a flat plate in the z-plane is

c / z z* = c / tU t * ∞= a / * ς ς =

∞= aU / * Γ Γ

) , ,; z , z(V dt

dz *2

*1

*2

*11*

*1 α Γ Γ = ,

) , ,; z , z(V dt dz *2*1*2*12*

*

2 α Γ Γ = , (B5)

where the complex velocities of the two vortices are

⎪⎭

⎪⎬

⎫+++⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

−−

+

⎪⎩

⎪⎨

⎧

−−⎟

⎟

⎠

⎞⎜

⎜

⎝

⎛ −

−= −

*1

*2

*1

*0

*2

*1

*2

*1

*2

*1

*1

*1

2*1

i2i

2*1

*2

*11

12

i / 1

112

i

/ 1

12

ie

1e / 11

1) z , z(V

ς π Γ Γ Γ

ς ς ς ς π Γ

ς ς π Γ

ς ς

α α

⎪⎭

⎪⎬

⎫+++⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

−−

+

⎪⎩

⎪

⎨

⎧

−−⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

−= −

*2

*2

*1

*0

*1

*2

*1

*2

*1

*2

*2

*2

2*2

i2i

2*2

*

2

*

12

12

i / 1

112

i

/ 1

1

2i

e

1e / 11

1

) z , z(V

ς π Γ Γ Γ

ς ς ς ς π Γ

ς ς π

Γ

ς ς

α α

.

The non-dimensional strength of the vortex added to meet the Kutta condition is





⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

−−

−⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −

−−

−++−=*2

*2

*2*

1*1

*1

*2

*1

*0

/ 11

1

1

1

/ 11

1

1

1sin4)(

ς ς Γ

ς ς Γ α π Γ Γ Γ

,

and the non-dimensional inverse Joukowski transformation is 1 z4 z22*** −±=ς . The

equilibrium positions of the two vortices satisfy0) , ,; z , z(V *

2*

1*2

*11 =α Γ Γ

0) , ,; z , z(V *2

*1

*2

*12 =α Γ Γ . (B6)

The equilibrium positions are obtained by solving Eq. (B6), which depend on the three parameters , and *

1Γ *2Γ α . However, calculations indicate that the equilibrium positions are

unstable for a finite-amplitude disturbance. This means that the dual-vortex structure on anairfoil cannot remain stationary.





Figure 1. Diagram of stall regimes

Figure 2. Schematics of a thin flexible fin attached to an airfoil for post-stall flow control

Figure 3. A flexible Mylar fin attached on a NACA0012 airfoil model illustrated in a typical PIVimage





(a) (b)

Figure 4. The drag coefficient as a function of AoA for (a) different fin arrangements and (b) a 0.25c finlocated at 0.1c

Figure 5. Pressure and skin friction drag coefficientsobtained by CFD

Figure 6. Measured lift coefficient as a function of AoA





(a) (b)Figure 7. Power spectra of the drag coefficient due to unsteady separation for the baseline NACA0012model and model with a 0.25c fin located at 0.1c at (a) 14 o and (b) 18 o

(a)

(b)

Baseline, AoA 18 o

Fin, AoA 18 o

Figure 8. Mean velocity vectors and vorticity distributions for (a) the baseline NACA0012 model and (b)model with a 0.25c fin located at 0.1c at AoA of 18 o


20




(a) (b)

Figure 9. Profiles of the mean velocity component U at different locations on the upper surface for (a)AoA 18 o and (b) AoA 20 o. The solid line: baseline NACA0012 model; the dash line: NACA0012 modelwith a 0.25c fin located at 0.1c

Figure 10. Effect of the flexible fin on the development of the momentum thickness in the separated region





Figure 11. Power spectra of the velocity component U across the separated region for different x locationsfor the baseline NACA0012 model at AoA of 18 o





Figure 12. Power spectra of the velocity component U across the separated region for different x locationsfor the NACA0012 model with a 0.25c flexible fin located at 0.1c at AoA of 18 o





(a)(b)

Figure 13. Development of spectral components of the velocity component U across the shear layer in the x direction at AoA of 18 o for (a) the baseline NACA0012 model and (b) model with a 0.25c flexible finlocated at 0.1c

(a)(b)

Figure 14. Reynolds stress in the separation region at AoA of 18 o for (a) the baseline NACA0012 modeland (b) model with a 0.25c flexible fin located at 0.1c





(a)

(b)

(c)

(d)

(e)

(f)

Figure 15. Snap-shot velocity fields of vortices shedding from the leading edge at (a) 0 s, (b) 0.2 s, (c) 0.4s, (d) 0.6 s, (e) 0.8 s, and (f) 1.0 s





(a) (b)

Figure 16. Power spectra of the velocity components U and V on the baseline NACA0012 model

Figure 17. The Strouhal numbers based on the local momentum thickness for the low-frequency mode (f = 0.07 Hz) and the vortex shedding mode (f = 1.0 Hz) for the baseline NACA0012 model





Figure 18. The time traces of the velocity U and the zero-crossing point position at the reference point,where the circles mark the phases for examination of the short-time-averaged velocity fields

Figure 19. The magnitude squared coherence between the velocity U and the zero-crossing point positionat the reference point





(a)

(b)

(c)

(d)

(e)

Figure 20. Velocity fields averaged over a short time of 0.73 s at the five phases indicated in Fig. 18 in alow-frequency cycle of the reference velocity U (about 12 s), (a) the first valley, (b) the first middle point

between the peak and valley, (c) peak, (d) the second middle point, and (e) the second valley





(a) (b)

Figure 21. (a) Two vortices on a flat plate at AoA of 18o

, and (b) typical trajectories of the two vortices

(a) (b)

Figure 22. Fin deformation, (a) time trace, and (b) power spectrum of the fin amplitude





Figure 23. Power spectra of the velocity component U across the fin for different locations for the NACA0012 model with a 0.25c fin located at 0.1c at AoA of 18 o





Figure 24. Power spectra of the velocity component V across the fin for different locations for the NACA0012 model with a 0.25c fin located at 0.1c at AoA of 18 o





Figure 25. Time traces of the relative fin amplitude and zone-averaged velocity components around thefin, where the fin amplitude and velocities are normalized by the fin length and the freestream velocity,respectively

Figure 26. Magnitude squared coherence between the fin amplitude and zone-averaged velocitycomponents

Post stall Control Using Flexible Fin onAir Foil

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