Leading Edge Vortex Circulation Developmenton Finite Aspect Ratio Pitch-Up Wings

Kyle Hord∗ and Yongsheng Lian†

University of Louisville, Louisville, Kentucky 40292

DOI: 10.2514/1.J053911

We study the leading edge vortex (LEV) development and LEV circulation of pitch-up wings at a low Reynolds

number of 500. Wings of different aspect ratios are linearly pitched up from 0 to 90 deg at two reduced pitch rates of

0.1 and 0.2. The flowfield is described by solving the unsteady three-dimensional incompressible Navier–Stokes

equations on composite overlapping grids. TheQ-criterion is used to isolate the LEV structure from shear layer. The

simulation shows that forces increase with the aspect ratio and the pitch rate. The stall angle increases with the pitch

rate but decreases with the aspect ratio. also It is shown that the LEV circulation depends on the wing aspect ratio:

increasingwing aspect ratio increases the rate ofLEVcirculation generationduring the pitch-upmotion. It also shows

that the reduced pitch rate could delay the LEV circulation development. Furthermore, the study shows that the LEV

circulation distribution better matches the spanwise force distribution for higher aspect ratio wings.

Nomenclature

b = span, mCD = coefficient of dragCL = coefficient of liftc = chord, mK = reduced pitch rateU = freestream velocity, m/sz = span locationα = angle of attack, deg_α = angular velocity�α = angular accelerationΓ = circulationω = vorticity

Subscript

CW = clockwise

I. Introduction

F LAPPING wings have the potential to provide superior liftgeneration, excellentmaneuverability, and robust gust resistance

for micro air vehicles (MAVs) operating at low Reynolds numbers[O�103–104�] [1–9]. The enhanced lift generation of a flapping wingis in part due to the existence of a leading edge vortex (LEV). TheLEV in close proximity to the wing enhances the lift [10–12]. Thishigh-lift generationmechanism is similar to the dynamic stall processfrequently encountered in helicopters, turbomachinery, windturbines, and nature fliers [13]. Dynamic stall refers to an unsteadyphenomenon in which the lift on a rapidly pitched wing continues toincrease even after the angle of attack passes the wing’s static stallangle. This temporary increase in lift is followed by a rapid decreasein lift due to vortex shedding.Because of its scientific merits and practical significance,

dynamic stall problems have been studied both numerically andexperimentally [3,14–18]. Most numerical studies have investigated

dynamic stall as a two-dimensional (2-D) problem due to theimmense computational resources required for a three-dimensional(3-D) study [4,13,19,20]. Studies with complicated flappingkinematics have been conducted at low Reynolds numbers; however,the focus is on force generation instead of the dynamic stallphenomenon [17,21]. For the dynamic stall problem, high Reynoldsnumber (Re> 106) 3-D experiments have been conducted mainlyfor helicopter application [22,23], and low Reynolds number(Re ≈ 103–104) studies have focused on MAVapplications [24–33].BecauseMAVs employ low aspect ratiowings, dynamic stall must beinvestigated as a 3-D problem at low Reynolds numbers in order forthe dynamic stall research to be applicable to MAVs.The impact of Reynolds number on the force generation and LEV

development has been investigated [34–37]. For example, Hord andLian [34] found that lift and drag on a pitch-up wing increased onaverage by 15% when the Reynolds number increased from 500 to20,000. Upon inspection of the flowfield, they found that the LEVbecame more compact in size with increased interior vorticitymagnitude as the Reynolds number increased. They concluded thatthe change in LEV circulation is responsible for the increase inaerodynamic forces. Other studies [35,37], however, showed thatforces are not sensitive to the Reynolds number.The study of finite aspect ratio wings at low Reynolds numbers is

an area that has not been well explored, even more so with thedynamic stall phenomena. Recent studies on static wings at lowReynolds numbers [O�104�] have concluded that the aspect ratio(AR) significantly affects the performance of the wing [38–40]. Forflappingwings, aspect ratio has been found to influence the flowfield.Granlund et al. [18] experimentally found that a finite aspect ratiopitch-up wing had a less defined LEV structure when comparedwith a 2-D case with the same operating conditions. Yilmaz andRockwell [28] experimentally studied a 3-D pitch-upwing and founda spanwise velocity induced by tip vortices altered the LEVdevelopment. This phenomenon is also seen in Coton and Galbraith[25] and Spentzos et al. [26]. Hord and Lian [34] came to a similarconclusion after performing an aspect ratio survey on a pitch-upwing. They found the LEV development is impeded to a less extentby tip vortices as the aspect ratio increased. They also noticed that thedynamic stall angle decreases with the increase of the aspect ratio.They suggested that tip vortices can attenuate aerodynamic forces onlow aspect ratio wings when there is no clear LEV, which is similarto the conclusions made by Shyy [41] for hovering wings andby Okamoto and Azuma [39] and Mizoguchi and Itoh [40] forstatic wings.The pitch rate of the dynamic stall motion has been known to

influence the maximum achievable lift and stall angle [11,19,20].Granlund et al. [27] experimentally surveyed several pitching rates ofa nominally 2-D airfoil undergoing a single pitch-up maneuver at aReynolds number of 20,000. They found that, even at a low pitch rate,

Received 9 September 2014; revision received 4 May 2015; accepted forpublication 8 May 2015; published online 5 July 2016. Copyright © 2015by the authors. Published by the American Institute of Aeronautics andAstronautics, Inc., with permission. Copies of this paper may be made forpersonal and internal use, on condition that the copier pay the per-copy fee tothe Copyright Clearance Center (CCC). All requests for copying andpermission to reprint should be submitted to CCC at www.copyright.com;employ the ISSN 0001-1452 (print) or 1533-385X (online) to initiate yourrequest.

*Ph.D., Mechanical Engineering Department.†Associate Professor, Mechanical Engineering Department. Senior

Member AIAA.

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lift would be higher than its corresponding static stall value.Interestingly, the slope of the lift curve changes little with the pitchrate except when the reduced pitch rate is greater than 0.2. Later,Granlund et al. [18] surveyed reduced pitch rates for a 3-D wing andfound that increased pitch rates causes tip vortices to develop athigher angles than LEVs. Hord and Lian [34] found similarphenomena in their investigation of a 3-D pitch-up wing. They alsofound that increasing the pitch rate produced a compact LEV withincreased vorticity magnitude and a reduction of tip vortexdevelopment.Early studies have shown that LEVs are affected by the Reynolds

number, wing aspect ratio, and pitch rate. To better our understandingof the aerodynamics associated with LEVs, it is important to make aquantitative analysis of the size and strength of LEVs. This in itselfcan be difficult, because LEVs cannot bemeasured directly like lift ordrag forces. In recent years, several experimental studies havetargeted the calculation of LEV circulation strength by measuringand filtering the flowfield about rotating, flapping, or pitching wings[12,42–46]. For example, Baik et al. [44] experimentally studied theeffect of reduced frequency on the development of the LEVcirculation and LEV size of a nominally 2-D pitch and plunge wing.They found that increasing the frequency could increase the LEVcirculation strength but decrease the growth rate of the LEV. BecauseLEV circulation is related to lift generation, calculating the LEVcirculation can provide a uniquely defined value to understand howwing kinematics and geometry affect circulation.In this paper, we will examine the effect of wing aspect ratio and

pitch rate on the LEV circulation by eliciting a dynamic stall processwith a single pitch-up motion. Specifically, we will show thedifferences in circulation development across three different finiteaspect ratio wings (1, 2, and 4) tested at two reduced pitching rates(0.1 and 0.2) at a lowReynolds number (Re � 500). The LEVwill beisolated using vortex identification techniques before the circulationof the LEV is calculated. The circulation across the span of the wingwill be evaluated throughout the entire pitching motion and com-pared with force distributions. To better understand the circulationdevelopment, this study will examine 1) the aerodynamic forcesgenerated, 2) the spanwise distribution of circulation, and 3) howcirculation compares with aerodynamic force distribution for eachwing and pitch rate combination throughout the entire pitchingmotion.

II. Pitch-Up Kinematics

Unlike most dynamic stall studies that use a sinusoidal pitchingmotion [11,13,20,47], a single pitch-up motion will be used in thisstudy to initiate the dynamic stall process. The selection of thepitching motion stems from recent work examining the perchingmaneuver [18,27,34,35] in which a simplified motion is used toapproximate the landing kinematics of birds. The simplified motioncouples a linear wing pitch-up with a simultaneous decrease instreamwise velocity. In this study, only the pitch-up motion will be

considered. The wing will initially have a neutral angle of attack

(0 deg) and quickly ramps up to 90 deg. In this work, the following

smoothing function (similar to Eldredge and Wang [48]) is used to

define the time-dependent angle of attack:

α�t� � K

a

�cosh�aU�t − t1�∕c�cosh�aU�t − t2�∕c�

�� aU�t2 − t1�

c(1)

in which the parameter a in Eq. (1) is a user-defined value that

controls the sharpness of the smoothing function, c is thewing chord,α is the angle of attack, andU is the freestream velocity. Parameter t1is the start time of the pitch-up motion and t2 is the end time of the

pitch-up. K is the dimensionless reduced pitching rate, defined as

K � c _α

2U� c

2U

α2 − α1t2 − t1

(2)

in which _α is the pitch rate, c is the chord length, U is the freestream

velocity, and α is the angle of attack in radians at the respective times.For this study, reduced pitch rates ofK � 0.1 andK � 0.2will be

used, and the wing will pitch about the quarter-chord. Figure 1

compares the variation of angle of attack with time using different

values of the smoothing parameter a forK � 0.1 and usingK � 0.2for comparison. As a increases, the pitch-up motion transits from a

smoothed ramp to a sharp ramp function. The effect of a appears

benign on the angle of attack profile; however, the effects are apparent

with angular velocities and accelerations. With the increase in a,angular velocity and acceleration magnitudes increase at the

beginning of the motions. This leads to higher noncirculatory forces

during the nonzero acceleration phase. Noncirculatory forces refer to

the resultant pressure gradient caused by the acceleration of fluid

around the wing due to the wing’s motion [37]. The noncirculatory

force is known to be proportional to the pitch rate [27] and can

contribute large portions to lift [45]. Granlund et al. [49] found that adoes not change the aerodynamic forces during the constant angular

velocity portion of the motion.Noncirculatory forces can influence the magnitude of the forces

measured, depending on the pitch-up profiled used. Figure 1

illustrates how K and the smoothing variable a impacted the

acceleration and angular velocity. Noncirculatory forces are a

function of these two quantities. As K increases, noncirculatory

forces also increase. The noncirculatory forces can quickly increase

the lift coefficient during the early stage of the pitch-up motion. The

effect is most visible in the CL plots between 0 and 5 deg where the

rotational acceleration is nonzero (shown later in Sec. V.A). If the

reader wishes to knowmore about how the noncirculatory forces can

be calculated for a pitch-upmotion, the reader is referred to Hord and

Lian [45].

Fig. 1 Impact of the smoothing parameter and K on the pitch-up kinematics: (left) change in angle of attack, (middle) angular velocity, and (right)acceleration of the wing.

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III. Numerical Method and Computational Setup

The flowfield is simulated by solving the 3-D unsteadyincompressible Navier–Stokes equations and the continuity equationpresented here in Cartesian coordinates:

∂u∂t

� u · ∇u � −∇pρ

� νΔu (3)

∇ · u � 0 (4)

in which ρ is the density, u is the velocity vector,p is the pressure, t istime, and ν is the kinematic viscosity. The incompressible Navier–Stokes equations are solved using the pressure-Poisson method [50].The equations are discretized in space with a second-order accuratecentral difference onto a set of overlapping grids [51]. In overlappingregions, the Lagrange interpolation is used to interpolate solutionsbetween different regions. Time stepping is accomplished using thesecond-order accurate implicit Adams–Bashforth–Moulton predic-tor-corrector method [52]. The PETSc‡ package is used to solve thesystem of discretized equations. Thewing motion is represented by aseries of composed rotation and translation matrices developed byLian and Henshaw [53]. This computational code has been validatedfor the lower Reynolds numbers for 2-D and 3-D applications inseveral papers, and the author refers the reader to these [5,45,54,55].The overlapping gridmethod employs a series of Cartesian grids as

the background grid and structured body-fitted grids around thewingsurface. The use of a Cartesian grid preserves its inherited highcomputational efficiency, and the use of body-fitted grids ensures anaccurate representation of thewing geometry and a good resolution ofthe velocity boundary layer profile. The use of overlapping grids alsoallows for the easy handling of wing motions. Figure 2 shows thesample of a 3-D overlapping grid used for the study.The rectangularwing has a cross-sectional profile of a 5% thick flat

plate with round leading and trailing edges. The flat plate wing isrepresented by three separated body-fitted structured grids (onewingbody grid and two wing tip grids). This composite wing grid is thenplaced inside a uniform Cartesian background grid, where itsboundaries remain at least 10 chords away from the plate. Typically,the background grid ismuch coarser than thewing grids. Tominimizeinterpolation error, multiple intermediate uniform grids are addedbetween the background grid and the wing grids. For the casesstudied in this paper, three intermediate grids are used for transitionfrom the finewing grids to the coarse background grid. All but one ofthe boundaries in the background grid is set to inflow boundaryconditions, and the remaining is set as outflow with zero gaugepressure. For this study, the Reynolds number is set to 500 and theflow is assumed to be laminar.

We implement a vortex identification scheme to decipher actualvortices from shear layer. If no detection scheme is used, definition ofa vortex from streamlines or vorticity plots is ambiguous at best. Forthis study theQ-criterion, originally developed by Hunt et al. [56], isselected. The Q-criterion is a Galilean-invariant vortex criterion thatis based off the velocity gradient tensor∇u of the flowfield, in whichu is the velocity vector. The Q-criterion is a straightforwardcalculation shown in Eq. (5), in which Ω and S are the rotation andstrain rate tensors of ∇u, respectively.

Q � 1

2�kΩk2 − kSk2� > 0 (5)

Avortex is positively identified when Q is greater than zero. Thisoccurs when the vorticity tensorΩ is greater than the strain rate tensorjSj. In addition to theQ > 0 requirement, the pressure at the center ofthe vortex must be less than the ambient pressure.

IV. Three-Dimensional Grid Sensitivity

Before we study the physics associated with the finite aspect ratiowings, a grid sensitivity analysis is performed. Only the results of theflat plate wing with an aspect ratio of 2 and the reduced pitch rate of0.2 are reported here. The simulation is conducted at a Reynoldsnumber of 500, and the flow is assumed to be laminar. Three griddensities, referred to as coarse, medium, and fine, are tested. The griddensity increased by 20% in each direction from coarse to mediumand then frommedium to fine, resulting in a grid size of 0.9, 1.7, and2.8 million grid points for the coarse, medium and fine grid system,respectively. Table 1 lists the three test cases with the number ofpoints used to define the airfoil and span cross sections, the initial cellheight, and the total grid points in the domain.Figure 3 shows the computed lift and drag histories for each case.

Lift and drag from the medium grid shows little variance from the finegrid, whereas the coarse grid shows significant underprediction. Thewakevelocities are also examined toverify convergence. The samplingof the wake velocities are taken at two locations: 1) at the midspan aquarter-chord behind thewing in the vertical direction and 2) along thespan direction of the wing, a quarter-chord behind the wing. Both setsofwakevelocities are reported inFig. 4. Little difference is found in thevelocity distributions, suggesting that the velocity field is adequatelyresolved by the medium grid.

Fig. 2 Overlapping grids. Left: body-fitted structured grids around the plate; right: background grids.

Table 1 Three grids used in the grid sensitivity analysis

Grid quality Airfoil Span Initial cell height Total grid pointsa

Coarse 72 80 6.65 × 10−3 0.9 MMedium 90 100 5.54 × 10−3 1.7 MFine 108 120 4.46 × 10−3 2.8 M

aM, million.‡http://www.mcs.anl.gov/petsc/petsc-as/ [retrieved Sept. 2013].

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Finally, due to circulation of the LEV being of primary interest inthe study, the circulation of the LEV is calculated for the three testedgrids. As will be discussed later, Eq. (6) is used in conjunction withthe Q-Criterion (Q > 0) to calculate the LEV vorticity shown inFig. 5.As it is seen, refining the grid has little benefit to the circulation

calculation. Thus the medium grid is selected, because it provides aresolved velocity field and little degradation to the aerodynamicforces compared to the Fine case.

V. Computational Results

A. Aerodynamic Forces

Figure 6 plots the lift, drag, and moment coefficient histories. The2-D numerical results [34] at the same Reynolds number arepresented for comparison purpose. The referencemoment coordinateis the same as the pitching location of x∕c � 0.25. In the followingsubsections, we discuss the effects of aspect ratio and pitch rate onaerodynamic forces.

1. The Effect of Aspect Ratio

The lift, drag, and moment coefficients show strong dependenceon the wing aspect ratio. It is clear that increasing the aspect ratio hasseveral impacts on the force histories. Encompassing all the changesobserved, the peak lift, drag, and moment coefficients and therespective slopes before stall increase with the aspect ratio andapproach their 2-D counterparts. Stall angle, however, decreases with

Fig. 3 Aerodynamic forces for the grid sensitivity analysis of a pitching wing of AR � 2, K � 0.2, and Re � 500.

a) Vertical velocity distribution in the y-direction at the mid-span

b) Spanwise velocity distribution in the z-direction at y/c=0

Fig. 4 U and V velocity distributions quarter-chord behind the airfoil at 45 deg.

Fig. 5 Calculated LEV circulation comparison for grid sensitivity.

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the aspect ratio. Table 2 summarizes some of these changes for easier

comparison.

Figure 7 shows the vorticity contours at the midspan of the three

tested wings at both K values. Because of the diffusive nature of low

Reynolds number flows, it is difficult to distinguish the boundaries of

the vortices produced. The Q-criterion has been overlaid on the

vorticity contours to define the edges of the vortices identified. At the

same pitch rate, an increased vorticity magnitude extends furtherdownstream above the surface of the wing as the aspect ratioincreases.With the assistance of theQ-criterion, it can be inferred thatthe LEV is growing in both size and strength with the aspect ratio.This, however, raises the question of the mechanism inducing thedifferences in vorticity structure observed in the plots. As discussedin a later section, it is found that the interaction between tip vorticesand spanwise vorticity attenuates LEV growth on the lower aspectratio wings.

2. The Effect of Pitch Rate

Increasing the reduced pitch rate K has the following effects: 1) itincreases the stall angle, 2) reduces the peak drag angle, and3) increases the force and moment coefficient magnitudes. Figure 8compares the vorticity contours at different angles between the tworeduced frequencies for the AR � 4 wing. At the same angle ofattack, the LEV is smaller and the LEV stays closer to the wing atK � 0.2 than at K � 0.1. The LEV strength also increases with thepitch rate. However, it does not mean the LEV generates more lift-producing circulation. To properly judge the influence of K on theLEV, we calculate and compare the circulation in the followingsections.

Fig. 6 Comparison of lift, drag, and moment coefficients.

Table 2 Effect of aspect ratio and K on thepitch-up motion

Aspect ratio Stall angle↓ Peak CL↑ Lift slope↑

K � 0.11 42.40 1.27 0.0292 38.96 1.58 0.0384 38.97 1.82 0.0482D 34.37 2.40 0.064

K � 0.21 43.55 1.58 0.0312 41.26 1.98 0.0394 38.97 2.38 0.0462D 41.26 2.93 0.062

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B. LEV Circulation at the Middle Span

To measure the relative strength of LEVs, we calculate the LEV

circulation by integrating the clockwise vorticity (ωz;cw) in the

computational domain as follows:

ΓCW �ZZ

Q−ωz dx dy (6)

To minimize the influence of shear layer, we use theQ-criterion [56]

to filter the domain. This is illustrated by referring to Fig. 7. The

clockwise vorticity identified by the Q-criterion is integrated to

calculate the circulation in the domain. Similar approaches have been

used by other authors [12,42–46] to calculate the circulation of

a wing.Before examining the LEV circulation variation across the wing

span, we first evaluate the midspan LEV circulation. The LEV

circulation is calculated using Eq. (6) with Q > 0. This provides asimplistic view of the development of the LEV circulation with angle

of attack as aspect ratio increases and approaches an infinite (2-D)

span. Figure 9 plots the calculated circulation values (ΓCW) at the

midspan of each aspect ratio.The LEV circulation changes with the aspect ratio and K. The

aspect ratio is seen to increase the slope ofΓCW, which is similar to the

lift and drag coefficients. Differences initially are seen in theK � 0.1case at angles of attack less than 35 deg in which the 3-D circulation

Fig. 7 Comparison of the LEV vorticity contours at midspan with theQ-criterion (Q < 0) defining the boundaries of identified vortices. Snapshots aretaken at the midspan and 45 deg.

Fig. 8 Comparison of midspan LEV vorticity during the pitch-upmotion of theAR � 4wing forK � 0.1 and 0.2 with theQ-criterion (Q < 0) definingthe boundaries of identified vortices.

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results exceed the 2-D results. This is attributed to earlier flowseparation in the 2-D simulation.The effect of reduced frequencyK is seen to delay the development

of circulation for the 2-D andAR � 2, 4 wings. This is evident by therightward shift of theK � 0.2 lines comparedwith theK � 0.1 lines.Both the aspect ratios of 2 and 4 show diverging LEV lines with anincrease in K. At low angles of attack, ΓCW for both K values trend

together until approximately 25 deg. The K � 0.2 lines remainstagnant until 35 deg, which then begin to increase at a similar rate asK � 0.1. This suggests that the timing of circulation development is a

function of K.The effect of K on the AR � 1 wing is less clear. The LEV

circulation shows little variation with the reduced frequency K. Aspreviously shown in Fig. 7, the midspan LEV vorticity is smaller,

resulting in lower calculated ΓCW. Additionally the tip vortices thatform on the AR � 1wing occupy a significant portion of the suctionside surface. (This is shown in Sec. V.B.) This would result in

circulation that is not captured by the ωz component.

C. Spanwise Circulation

Figures 10–12 plot the resultant ΓCW contours. It is important tostress that these contours are not ΓCW distributions across the entiresurface of thewing, but rather they are the variation in ΓCW across the

normalized span (z∕b) through the entire pitchingmotion.Because ofwing symmetry, only a half wing is shown. Each contour plot showsK values of 0.1 and 0.2 alongside isosurface plots of Q � 5 at threeangles of attack.Beginning with Fig. 10, the AR � 1 wing, the Q isosurfaces of

both K cases show similar vortex structures. The starting vortex isstill visible in the K � 0.2 case, whereas it has already convected

away from the plate in the K � 0.1 case. When wings are pitched tothe same angle of attack, twice the amount of time has elapsed for theK � 0.1 case compared with the K � 0.2 case. The lower pitch ratewing allows the LEV to havemore time to develop and also allows tip

vortices to grow larger and interfere with the LEV. This phenomenonis also seen in the experiment of Granlund et al. [18]. However, it isdifficult to identify from the ΓCW contour any spanwise LEVcirculation for either K value, even though a LEV-like structureappears above thewing. Several authors have commented on how tipvortices can affect LEV development. Jones et al. [42] and Coton andGalbraith [25] found that the tip vortices impeded LEV developmenton low aspect ratio wings (AR < 2). At low aspect ratios, tip vorticesoccupy a large portion of thewing,which disrupts the development ofspanwise ωz-related circulation. The lack of LEV circulation wouldsuggest a lack of or significant decrease in lift production; however,as previously mentioned, the lift-inducing circulation may not betracked by ωz.An examination of the vorticity transport equation [Eq. (7)], the

first two terms on the right side indicate that vorticity can be stretchedor tilted. Thus any LEVor ωz vorticity can be tilted into other planesfrom interaction with tip vortices. ωz vorticity is being modified(tilted) by tip vortices in and out of the z plane used to identifyvortices. Because of this complex LEVand tip vortex interaction, lift-producing vorticitymay exist in several planes that is not bound to thestrength of the LEV. Thus, this enhances aerodynamic forces on theAR � 1 wing because lift-producing vorticity does not occur inthe ωz form that would shed with the LEVat stall.

DωDt

� �ω · ∇u� − ω�∇ · u� � υ∇2ω (7)

Examining Fig. 11 for the AR � 2 wing reveals the difference incirculation developmentwith an increase in aspect ratio. It shows thatthe reduced pitch rate has an effect on the development of circulation,which is not apparent for the AR � 1wing. Examining the K � 0.2contours shows similar valued circulation contours occur at higherangles of attack when compared with K � 0.1. This delay indevelopment is also seen in Fig. 9. Increasing K is also seen to delaythe formation of tip vortices. This is evident in the contour plots.

Fig. 9 LEV circulation calculated by integrating the clockwise LEV vorticity for each aspect ratio and K.

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Fig. 10 Contours: (left) spanwise contour ΓCW through entire pitching motion for the AR � 1 wing at both tested K values and (right) isosurfacecontours where Q � 5 for both K values at 30, 45, and 60 deg.

Fig. 11 Contours: (left) spanwise contour ΓCW through entire pitching motion for the AR � 2 wing at both tested K values and (right) isosurface

contours where Q � 5 for both K values at 30, 45, and 60 deg.

Fig. 12 Contours: (left) spanwise contour ΓCW through entire pitching motion for the AR � 4 wing at both tested K values and (right) isosurfacecontours where Q � 5 for both K values at 30, 45, and 60 deg.

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Toward the tips, circulation quickly drops in the lower K case. By

delaying the tip vortex formation, the LEVcirculation occupiedmore

of the span. This lead to a larger area of LEV circulation, covering

the span.

TheAR � 4 plots (Fig. 12) sharemany similaritieswith theAR � 2plots: circulation increases to higher magnitudes later in the pitching

motion and a reduction inLEVand tip vortex interactionwith increased

K. Examining the contour plots shows that the circulation develops

later in the pitching motion, and it is more uniform across the span

compared to the lower ARwings. Again the circulation is seen to drop

toward the edge of the spans. In theK � 0.1 case, tip vortices push thecirculation inward by approximately 5% and even less for K � 0.2.The AR � 2, K � 0.1 case showed deeper inward push of 10% of

circulation by tip vortices. Geometric scaling (increased span) of the

plate would explain the decrease of inward encroachment of the

circulation by the tip vortices.

Overall, the spanwise circulation is shown to be dependent on the

wing aspect ratio. The lowest aspect ratio wing (AR � 1) showed

little to no coherent circulation in the contour, whereas the isosurfaceplots showed a small LEV pushed off the surface by the tip vortices.For the AR � 2 wing, spanwise circulation developed but is notuniform. Finally, the circulation on the AR � 4wing is affected littleby tip vortices. For the AR � 2 case, the higher K value caused adelay in circulation development. Tip vortex interaction is also shownto affect less of the plate surface with an increase in K. This is mostevident on the AR � 2 and 4 wings, where circulation would extendfurther across the span when compared with the lower K.

D. Correlation of Force and Circulation

We further examine the correlation between aerodynamic forcesand circulation along the span. Comparing the normal forcecoefficient CN and the circulation across the wing span can revealwhether forces are closely correlated with circulation produced byωz. Figures 13 and 14 plot normal force and circulation at three anglesof attack. These angles are chosen as illustrative points of interest. Ifthe shapes of the two distributions are similar, it can be assumed that

Fig. 13 Wing span distribution plots ofCN andΓCW at a single angle of attack for all aspect ratios atK � 0.1. Solid lines and lineswith circles correspondto CN and ΓCW span distributions, respectively.

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circulation responsible for lift generation is accurately captured.Hord and Lian [45] found lift and circulation are closely correlated atlow angles of attack if the circulation is modified for 2-D cases.FromFig. 13,wecan see that at 30deg theprofiles ofCN andΓCW are

similar at all aspect ratios. This is expected because the LEV is close tothe surface of thewing, as illustrated in Fig. 8.However, the intersectionbetween the LEVand tip vortices creates a valley in ΓCW not shown inCN at jz∕bj ≈ 0.3. This intersection moves closer to the tip of the spanas the aspect ratio increases. In this region, out-of-plane circulationmaycontribute to the lift, but it is not included in our calculation.At angles of attack of 45 and 60 deg in Fig. 13, lift starts to decrease

due to shedding of the LEV (Fig. 6). Drop in force at the center span isevident. Force generation primarily originates from tips, as seen fromthe peaks inCN near the tips. However, ΓCW fails to capture this shiftin location of force generation because LEV circulation is stillincreasing at the midspan and decreases toward the edge. This againillustrates the 3-D nature of the flow at these high angles of attack.The disconnection between ΓCW and LEV circulation is also seen atK � 0.2 (Fig. 14).

To better compare the profile history ofCN and ΓCW, we introduce

an error ratio e. The error ratio is defined as follows:

e � j �CN − �ΓCWj�CN

(8)

in which

�CN�z; α� �CN�z; α�CN�α�max

�ΓCW�z; α� � ΓCW�z; α�ΓCW�α�max

The error ratio contour is plotted in Fig. 15. These contours are a

representation of a difference ratio between the CN and ΓCW

distributions, which is effectively a simplified method to examine the

differences in span distributions of force and circulation. The error

ratio is calculated using Eq. (8), in which �CN and �ΓCW are normalized

between 0 and 1 of the maximum values that occur at that angle of

attack. As e approaches 0, this corresponds to a similar distribution of

Fig. 14 Wing span distribution plots ofCN andΓCW at a single angle of attack for all aspect ratios atK � 0.2. Solid lines and lineswith circles correspondto CN and ΓCW span distributions, respectively.

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CN and ΓCW at that angle and span location, such as the AR � 1,K � 0.2 distribution in Fig. 14. As e approaches 1, the distributionsshare little resemblance. The contours in Fig. 15 again use the same

split plot that is seen in Figs. 10–12. Each contour is split along the

z∕b � 0, with the left side plotting the K � 0.1 case and the rightplotting the 0.2 case.It can be seen that, as the angle of attack increases, themidspan error

and ΓCW begin to increase while the mid-span CN begins to decrease.

However, the effect of aspect ratio is less clear. Examining each aspect

ratio, it is easy to conclude that increasing aspect ratio increases the

midspan error; however, this only occurs once the LEV begins to

separate from the wing. As the LEV sheds from the wing, it continuesto feed off the flow for a finite period of time, increasing inboth size and

circulation. As the size increases, the low pressure center of the LEV

moves away from the wing, reducing the suction on the wing. Thus

ΓCW andCN can only be compared at low angles of attack or where the

flow is fully attached. This is due to the flow quickly becoming three

dimensional at higher angles of attack, and simply trackingωz does not

capture the diverse nature of the flow where lift can be produced by

vorticity in alternate planes of ω. At lower K, additional error isobserved near the tip and at the midspan. This is due to the LEV

shedding at a lower angle of attack with increased aspect ratio and

additional time for tip vortices to development.In summary, the lower lift slope on theAR � 1wing is mainly due

to theweakenedLEVby the tip vortices.As seen in Figs. 7 and 10, the

LEV showed weaker midspan vorticity and lower span circulation.Through the pitching motion, the circulation never developed tolevels seen at higher aspect ratios. As the aspect ratio increased, theLEV structure developed with less impedance from tip vortices andachieved higher circulation values across the span. This in turnallowed for the ωz circulation to develop, resulting in a higher liftslope. This phenomenon has been seen in static wing analysis. Torresand Mueller [38] studied static wings with different aspect ratios andplanforms. They found that, as the aspect ratio increased, the lift slopeapproached the theoretical thin airfoil prediction. Cosyn andVierendeels [57] also saw weak LEVs in their computational resultsof static flat platewings at various aspect ratios ranging from 0.5 to 2.They found that the strength of LEVs on wings of AR < 1.5 aredampened by tip vortices; however, as the aspect ratio increased, theinfluence of tip vortices decreased and the LEV became stronger.This is also seen in Figs. 10–12.Because of thewing becoming larger,more area is allotted for LEV development.

VI. Conclusions

In this study, we investigated the effect of thewing aspect ratio andpitch rate on force generation and LEV circulation by eliciting adynamic stall process with a single pitch-up motion. The LEV thatformed due to the dynamic stall process was isolated from the shearlayer using the Q-criterion, and the circulation of the LEV was

Fig. 15 Span distributions of error ratio e throughout entire pitch motion for each simulated aspect ratio and K.

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calculated by integrating the ωz vorticity across the span. Thecirculation across the span of the wing was evaluated throughout theentire pitching motion.Our study showed that aerodynamic forces increase with the aspect

ratio and the reduced pitch rate. The stall angle increases with the pitchrate but decreases with the aspect ratio. Comparison of circulation andnormal aerodynamic force distributions revealed that at low angles ofattack, this increase in circulationdue toaspect ratio corresponded to anincrease in aerodynamic forces.As the aspect ratio increased, spanwisecirculation increased and approached a near uniform distributionsimilar to the normal forces. At higher angles of attack, the circulationdid not correlate well with aerodynamic force distributions, possiblydue to the exclusion of out-of-plane circulation.

Acknowledgment

Thiswork is partially supported byNASAKentucky ExperimentalProgram to Stimulate Competitive Research (EPSCoR).

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