spanwise variation in the unsteady stalling flow fields of two-dimensional airfoil models - broeren,...

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

A01-16676

AIAA-2001-0861SPANWISE VARIATION IN THEUNSTEADY STALLING FLOWFIELDSOF TWO-DIMENSIONAL AIRFOIL MODELSA.P. Broeren and M.B. BraggUniversity of Illinois at Urbana-ChampaignUrbana, Illinois

39th AIAA Aerospace SciencesMeeting & Exhibit

8-11 January 2001 / Reno, NVFoLpermission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191


AIAA-2001-0861

SPANWISE VARIATION IN THE UNSTEADY STALLING FLOWFIELDSOF TWO-DIMENSIONAL AIRFOIL MODELS

Andy P. Broeren* and Michael B. Bragg^

University of Illinois at Urbana-ChampaignUrbana,IL61801

Recent investigations of two-dimensional airfoilstalling characteristics have revealed low-frequency andhighly unsteady flow in some cases and large-scalethree-dimensional structures in other cases. The latterwere referred to as "stall cells" and may form on two-dimensional configurations where the ends of the airfoilmodel are flush with tunnel side walls or end plates.This paper presents results of detailed investigations ofthe stalling characteristics of several airfoils thatexhibited both low-frequency unsteadiness and large-scale three-dimensional structures. The airfoils werewind-tunnel tested in a two-dimensional configuration.The primary measurements were spanwise wakevelocity and mini-tuft flow visualization. The resultsshowed that airfoils with trailing-edge separations atand above maximum lift (static stall) exhibited stall-cellpatterns. Conversely, airfoils that had leading-edgeseparation bubbles that grew in size as the angle ofattack was increased into stall developed the low-frequency, highly unsteady flow. This unsteadinesswas found to be essentially two dimensional.Therefore, the development of either of thesephenomena appears to be determined by thecharacteristics of the boundary-layer separation leadingup to the stall.

Nomenclature

b Model spanc Airfoil chordC/ Mean lift coefficient, L/q^cbCiifnax Maximum lift, coincident with astanCltnn5 Root-mean-square of the fluctuating lift

coefficientE Mean hot-film wake-velocity voltageE'^ Root-mean-square of the fluctuating hot-film

wake-velocity voltage

/ Flow oscillation frequencyL Mean airfoil lift#00 Free-stream dynamic pressureRe Reynolds number based on chord,St Strouhal number, fcsma/U^U«, Free-stream velocityx Distance in streamwise directiony Distance in spanwise direction

a Angle of attackGLstaii Stalling angle of attack, coincident with C\>max<j) Phase angleji Absolute air viscosityp Air density

Introduction

The stalling of airfoils is a complex fluid dynamicphenomenon involving strong viscous-inviscidinteraction, boundary-layer separation and unsteadyflow. There are various reports of unsteadinessassociated with airfoil stall in the technical literature.One study in particular describes a low-frequency,large-scale unsteady flow. Zaman et al.1 performed adetailed investigation into this naturally-occurring,quasi-periodic phenomenon. The flow oscillationfrequencies measured in the airfoil wake were non-dimensionalized using the free-stream velocity and theairfoil projected height (csinoc). The resulting Strouhalnumbers were on the order of 0.02, approximately 10times lower than those associated with bluff-bodyshedding or a von Karman vortex street. This low-frequency oscillation occurred in the range of staticstall, or maximum lift, from a « 14.5 to 16.5 deg andinvolved a quasi-periodic switching of the flowbetween stalled and unstalled conditions. This resultedin large-amplitude force fluctuations, up to 50% of themean lift coefficient at 15 deg angle of attack.Curiously, this low-frequency oscillation completely

* Post-doctoral Research Associate, Dept. of Aeronautical and Astronautical Engineering, Member AIAAf Professor and Head, Dept. of Aeronautical and Astronautical Engineering, Associate Fellow, AIAA

Copyright © 2001 by Andy P. Broeren and Michael B. Bragg. Published by the American Institute of Aeronauticsand Astronautics, Inc., with permission.

1American Institute of Aeronautics and Astronautics


diminished as the angle of attack was increased, withbluff-body shedding frequencies (i.e., St « 0.2) beingmeasured at a = 18 deg.

Research into this low-frequency oscillation on theLRN-1007 airfoil was subsequently performed byothers2"7 and the features of the unsteady phenomenonare well known. The study of Zaman et al.1 wasconducted for Reynolds numbers less than 100,000 andBragg et al.5 extended this range up to 1,250,000 andmeasured the oscillation frequency at twelve angles ofattack from 14.4 to 16.6 deg. The Strouhal numbervaried from 0.017 to 0.30 and had very littledependence on Reynolds number, but had a very strongdependence on angle of attack. In the same paper,Bragg et al.5 also provided surface-oil flowvisualization results obtained for the LRN-1007 airfoilprior to the onset of the unsteady flow. There was aleading-edge separation bubble that grew in size on theupper surface as the angle of attack was increased. Thedata also showed that there was significant boundary-layer separation from a point downstream of theseparation bubble reattachment. The photographs ofthis flow visualization revealed that these features wereuniform across the span of the airfoil model.4

However, the oil-flow visualization results wereessentially time-averaged. The separation bubble wasfound to play a key role in the oscillation, as itselimination (with a boundary-layer trip) caused the low-frequency oscillation to vanish.4'5

The role of the separation bubble and turbulentboundary-layer (or trailing-edge) separation wasinvestigated in more detail by Broeren and Bragg.7

They performed LDV (laser-Doppler velocimeter)measurements on the LRN-1007 airfoil upper surfacefor a = 15 deg and Re = 300,000. The authors wereable to conditionally average the time-dependentvelocity data using the wake velocity as thesynchronization source because the naturally occurringflow oscillation was nearly periodic for this case. Theresult was a quantitative description of the uppersurface flowfield over an averaged oscillation cycle.The data showed the development and growth of aleading-edge separation bubble that merged with theturbulent boundary-layer separation causing acompletely separated or stalled condition. While thisstudy revealed key information about the low-frequencyoscillation, the LDV measurements were onlyperformed for a single two-dimensional plane at themodel midspan. Therefore, no information about thespanwise, or three-dimensional, character was obtained.This may be an important factor as other studies havedocumented large-scale three dimensional features inthe flowfields of stalled airfoils.

There has been a significant amount of researchfocussed on three-dimensional structures known as stallcells. These "mushroom" shaped patterns form fromstrong recirculating flows on stalled airfoil models andwings. Winklemann and Barlow8 noted that the stallcells formed on both two-dimensional models wherethe ends of the model are flush with tunnel side walls orsplitter plates and on plane rectangular wings of finiteaspect ratio. The stall cells began to form as the angleof attack was increased into maximum lift and existedon the surface several degrees above the stalling angleof attack. The authors sketched a tentative flowfieldmodel showing the general features of a leading-edgeseparation bubble and trailing-edge separation. Thesefeatures are qualitatively similar to that described abovefor the LRN-1007 airfoil operating near stall. Theauthors pointed out that this flowfield was probablyunsteady in nature, but their oil-flow visualizationmethod produced only time-averaged results.Winklemann9 measured the fluctuating velocity spectrain the wake of a rectangular wing model and his resultsdid not show any evidence of low-frequencycomponents.

The unsteady features of stall cells were lateraddressed by Yon and Katz,10 who used fine-thread tuftflow visualization and high-frequency responsepressure transducers for measurements on a NACA0015 airfoil model of variable aspect ratio. Thevariable aspect ratio model was equipped with endplates that effectively eliminated the tip flow resultingin essentially a two-dimensional configuration. Theauthors discovered that certain aspect ratios resulted innon-integer numbers of stall cells that werecharacteristically unsteady. The power spectra of theirunsteady pressure measurements showed evidence oflow frequencies on the same order as those measuredfor the LRN-1007 airfoil. However, the intensity ofthis unsteadiness was apparently not as severe. In thiscase, maximum lift (or stall) occurred at approximately16 deg, with the stall-cell patterns being visible in therange of 17 to 19 deg angle of attack.

The objective of the present paper is to show thatthere is a fundamental difference in the stallingcharacter of airfoils exhibiting three-dimensionalflowfield variations versus airfoils exhibiting low-frequency unsteady flow. That is, it will be shown thatthe low-frequency oscillation as described for the LRN-1007 airfoil is essentially two-dimensional. The role ofthe stall-cell structures in "steady-stall" cases is alsoaddressed. The unsteady flow described for the LRN-1007 airfoil is not an anomaly and is shown to occur forother airfoils. Several experimental methods wereemployed to accomplish these objectives. Spanwisevelocity measurements were carried out using a

American Institute of Aeronautics and Astronautics


traversable probe in the airfoil wake. Thesequantitative measurements were supplemented withmini-tuft surface flow visualization data. A novelmethod of conditionally averaging the mini-tuft datawas developed to facilitate analysis of the unsteady stallcases.

Experimental Methods

All experiments were carried out at the Universityof Illinois Subsonic Aerodynamics Laboratory using thelow-speed, low-turbulence wind tunnel. The generalexperimental arrangement is shown in Fig. 1. The 12-inch chord wind-tunnel models spanned the test-sectionvertically, a distance of 33.63-inches. The width of thetest-section was 48-inches, so this model orientationminimized the blockage and facilitated flowvisualization since photographs were taken from theside. All data were acquired at a Reynolds number of300,000 because this corresponded to previousmeasurements.6 A traversable hot-film probe was usedto measure the wake velocity at 15 spanwise stations—one at midspan and seven stations above and belowmidspan. The probe location in the wake alsocorresponded to the location used for the conditionallyaveraged LDV measurements.6'7 Note the use of theterms "above" and "below" are convenient to use giventhe vertical orientation of the model, but should not betaken literally, since this model orientation is arbitrary.Several pieces of information were gleaned from thewake-velocity data. The power spectra were obtainedusing a dynamic signal analyzer and the Strouhalnumbers were computed. The mean and root-mean-square of the fluctuating velocity voltage were alsocomputed.

A rigorous uncertainty analysis was carried outusing the methods of Kline and McClintock11 andColeman and Steele12 for 20:1 odds. The wake hot-filmprobe was not calibrated to output velocity, so only thevoltages are reported here. These voltages wereacquired using a 16-bit analog-to-digital conversionboard that had a rated accuracy of ±0.76 (iV. Since thewake voltages were on the order of unity, the relativeuncertainty was nearly 0%. The quantization error was0.153 mV, however, the mean and RMS voltage weresufficiently resolved through the acquisition of 30,000samples. The Strouhal number was computed from thefrequency spectrum, the angle of attack, the airfoilchord and the free-stream velocity. The absoluteuncertainty hi angle of attack was ±0.05 deg. This andthe uncertainties in the other quantities (/, U^) led to arelative uncertainty in the Strouhal number of ±2.5%.More details regarding the uncertainty analysis can befound in Broeren.13

Flow

^ Hot-filmprobe Airfoil

leading edgex/c = 0.0

y/b = +0.401

\

Midspany/b = 0.0

Range ofspanwisetraverse

0.50c~ —

Flow

y/b = -0.401

Aifoil chord = c= 12.00 inchesAifoil span = b = 33.63 inches

Fig. 1 Schematic drawing showing the experimentalarrangement for the wake-velocity measurements.

In addition to the wake-velocity measurements,flow visualization was performed using fluorescentmini-tufts.14 The mini-tufts consisted of 0.002-inchdiameter monofilament nylon that were dyedfluorescent. This caused the tufts to "fluoresce" underultra-violet (UV) illumination. The small size of thetufts limited their effect on the boundary-layer flow andprovided excellent frequency response for the unsteadycases. The tufts were able to capture the key features ofthe unsteady flowfields over stalled airfoils. Whiledetailed boundary-layer information was not obtainable,general patterns of separation and reattachment wererecognizable. The tufts were applied to the entiremodel surface so that spanwise variations in thesefeatures could also be ascertained. The mini-tuft flowvisualization data were processed in a rather uniqueway to yield information about the spanwise variationof the flowfield near the surface of the airfoil. The hot-film sensor, positioned at midspan, was used as asynchronization signal for the acquisition of mini-tuftphotography. That is, a computer algorithm sampled



the wake-velocity voltage signal and triggered theshutter of a camera at designated points during eachoscillation of the flow. This was not only possible, butquite effective, since the unsteadiness was very nearlyperiodic. The photographs were ordered and sorted intotime slots (or bins) based on their phase-lockedrelationship. The phase angle (cj)) was used to representthe time designation within each cycle. Each time slot(or bin) was approximately 15 to 20 deg of the 360 degcycle. The actual width of each slot depended upon thegrouping of the photographs.

There were fifteen chordwise rows of mini-tufts onthe model, placed at spanwise locations coincident withthe wake-velocity measurements described above.Each row in each photograph was analyzed todetermine the approximate locations of flowfieldfeatures such as separation bubble reattachment orboundary-layer separation. The estimated uncertaintyin determining boundary-layer separation features in themini-tuft patterns was ±10% chord, or less, and ±5%chord for bubble reattachment locations. Thechordwise locations of these features were tabulatedand then averaged with the data from the otherphotographs in each of their respective time slots. Thismethod produced information about the unsteadyflowfield averaged over an oscillation cycle.

A total of five airfoils were tested in this study andare shown in Fig. 2 along with their correspondingthickness and camber. The airfoils encompassed abroad range of stall behavior and were selected toillustrate the differences in the stalling behavior. Moredetails regarding the experimental methods, datareduction and uncertainty can be found in Broeren.13

THICKNESS CAMBER

18.6% 0.00%

14.0% 2.00%NACA2414

NACA64A010 10.0% 0.00%

LRN-1007

E374

Fig. 2 The airfoils tested.

7.3% 5.90%

10.9% 2.24%

Results

Review of Time-dependent Lift Data

Time-dependent lift data were acquired for severalairfoils having different stalling characteristics andthese results are presented and discussed by Broerenand Bragg.15'16 Five of these airfoils were selected formore detailed study and are considered here. The meanlift coefficient (C/) and root-mean-square (RMS) of thefluctuating lift (C'ltrms) data are presented in Fig. 3. Ofparticular interest here is the stalling behavior of theseairfoils. The heavy vertical lines near stall in each plotindicate the range for which the salient flowfieldfeatures were observed. For the Ultra-Sport and NACA2414 airfoils, stall-cell structures, similar to thosedescribed above were observed over the indicatedrange. For the NACA 64A010, E374 and LRN-1007airfoils stall cells were not observed, instead, low-frequency unsteady flow characterized the stall. Theremainder of this paper presents the major differencesin these flowfields and intends to show that they existexclusively, that is, they do not appear to coexist.

Before presenting these arguments in detail, somediscussion of the airfoil data in Fig. 3 is warranted.These five airfoils represent four different stall types.Time-averaged stalling characteristics can be dividedinto three fundamental types based upon the flowfielddevelopment leading up to the stall. It is common forairfoils to exhibit a combination of these features, thusresulting in more than the three basic types.McCullough and Gault17 performed systematic testingand formulated the present definitions andunderstanding of airfoil stall type. The Ultra-Sportairfoil (Fig. 3a) had a classic trailing-edge stall type.That is, the boundary-layer separation locationgradually moved forward on the airfoil as the angle ofattack was increased into stall. The NACA 2414 airfoilhad characteristics of the leading-edge stall type. Inthis case, the stall occurred due to abrupt flowseparation from the leading edge, without subsequentreattachment. The result was a discontinuous loss oflift as illustrated in Fig. 3b. For this stall type, a smalllaminar separation bubble formed near the leading edgeand the "abrupt" flow separation likely resulted fromthe "bursting" of this bubble. The third basic stall typeis thin-airfoil stall. This stall type is characterized byboundary-layer separation from the leading-edge withreattachment (a separation bubble) at a point that movesprogressively aft on the airfoil upper surface as theangle of attack is increased into stall. The effect of thislarge bubble is shown in Fig. 3c for the NACA 64A010airfoil. There was a distinct reduction in the lift curve



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Fig. 3 Mean and fluctuating lift coefficient variationwith angle of attack for airfoils with different stalltypes, after Broeren and Bragg 15,16

0.0 5.0 10.0 15.0 20.0 25.0a (deg)



slope near a = 4 deg associated with the growth of theleading-edge bubble. The gradual stall, at fairly low liftcoefficients, is also a hallmark of the thin-airfoil stalltype. The remaining two airfoils, the LRN-1007 andthe E374, represent a combination of both the thin-airfoil and trailing-edge stall types. That is, both airfoilflowfields exhibited a leading-edge separation bubblethat increased in size with angle of attack (characteristicof thin-airfoil stall) and trailing-edge separation thatmoved forward with increasing angle of attack(characteristic of trailing-edge stall). The relativemagnitudes of flow unsteadiness is also revealed in theRJVIS lift variation. The C'ltfna levels at maximum liftfor the Ultra-Sport and NACA 2414 (increasing a) arevery low and increase as the angle of attack increases.In contrast, the C'ltms for the other airfoils reaches apeak nearly coincident with maximum mean lift, thendecreases as the angle of attack increases. While low-frequency unsteady flow was observed for both thethin-airfoil and combination thin-airfoil and trailing-edge stall types, it was much more pronounced andperiodic for the latter combination stall type.

Spanwise Flowfleld Data

As described above, the Ultra-Sport airfoilexhibited characteristics of trailing-edge stall, where theturbulent boundary-layer separation point movedforward on the airfoil as the angle of attack wasincreased into stall. The extent of boundary-layerseparation is illustrated in Fig. 4 at a = 10 deg, thestalling angle of attack. The orientation of the tufts, inthe photograph, revealed that there was more boundary-layer separation above the midspan location asindicated by the tufts that failed to align themselves inthe streamwise direction. In contrast, there wereseveral rows below midspan that indicated separationlocations much closer to the trailing edge. The last fewtuft rows, near the bottom of the model showedincreased separation. Note that the flow direction isopposite of the standard convention, due to theorientation of the wind-tunnel facility. Photographstaken at a = 13 and 16 deg revealed that the extent ofseparated flow progressed toward the leading edge.Any flow unsteadiness in this angle of attack regionwas weak and broadband.

A quantitative analysis of the information from themini-tuft visualizations was performed by determiningthe boundary-layer separation location for each of the15 tuft rows. This could be estimated to within x/c = ±0.10 or better, since the tufts were applied in incrementsof 10% chord. The result of this process is also shownin Fig. 4. The vertical axis of the plot represents themodel span, with the midspan location at y/b = 0.0, the

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Fig. 4 Photograph at left shows mini-tuft flowvisualization patterns on the Ultra-Sport airfoil at a= 10 deg. The plot (at right) shows thecorresponding boundary-layer separation locationsdetermined from the flow visualization.

top of the model is at y/b = 0.5 and the bottom of themodel is at y/b = -0.5, so the vertical coordinate of eachdata point represents the spanwise location of the tuftrows. The horizontal axis is the chordwise location ofthe boundary-layer feature of interest, which is theboundary-layer separation location in this case. Notethat the ;c-axis is reversed from that shown in thephotograph as the leading edge at x/c = 0.0 is now in itsusual position on the left. This means that the flowdirection implied on the plot is from left to right,opposite that shown in the photograph. The vertical orspanwise direction remains the same. The decision toreverse the *-axis was not a deliberate attempt toconfuse the reader, but to standardize the plots. Theaspect ratio of the plot is identical to the wind-tunnelmodels, where b/c = 33.63/12.00 - 2.8.

The boundary-layer separation locations shown inthe plot preserved the spanwise variation observed inthe corresponding photograph. The separationlocations were close to x/c = 0.45 on the upper half ofthe model, for 0.10 < y/b < 0.30. This is contrasted



with the lower half of the model on the interval -0.18 <y/b < -0.30, where the separation location was at ordownstream of x/c — 0.80.

The wake-velocity measurements provided morequantitative results to confirm the spanwise flowfieldvariations observed in the mini-tuft data. The mean andRMS wake-velocity voltages for the Ultra-Sport airfoilare plotted in Fig. 5 for a = 10, 13 and 16 deg.Considering the data for a = 10 deg, the mean velocityshowed a defect centered at approximately y/b = 0.20,which coincided with the region of largest boundary-layer separation (cf. Fig. 4). Conversely, the meanwake velocity was larger and more uniform for theregion of least boundary-layer separation, in the generalrange of -0.10 < y/b < -0.30. These results werecomplementary since increased boundary-layerseparation would lead to a larger wake and hence alarger velocity defect. The RMS velocities were alsoconsistent, as there was a minimum at the spanwiselocation corresponding to the region of the leastboundary-layer separation. The mean velocity data fora = 13 and 16 deg indicated increased spanwisevariation in the flowfield. The large velocity defects for

these two cases suggested that the wake became largeras the angle of attack was increased, consistent withboundary-layer separation moving forward on theairfoil. These large spanwise variations indicated thatstall-cell structures likely existed on the surface. This isrevisited again in the Discussion.

The leading-edge stall type airfoils wererepresented by the NACA 2414 as this airfoil clearlyexhibited the chief characteristics of this stall type. Themini-tuft flow visualization data, summarized in Fig. 6,were similar to the Ultra-Sport data. There was a largerextent of separated flow above midspan than below, fora = 15 and 16 deg. For a = 17 deg, the boundary layerwas completely separated from near the leading edgeand this involved a rapid transition from an unstalledcondition. Recall that the key characteristic of this stalltype was the leading-edge separation bubble that wasthought to have "burst," thus leading to thediscontinuous loss of lift (cf. Fig. 3b). The separationbubble was observed in previous surface oil-flowvisualizations13 but could not be observed in the mini-tufts because the reattachment location was upstream ofthe first row of tufts at x/c = 0.10. This "abrupt" stall

— a -10 deg— a =13 deg— a = 16 deg

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was observed while recording the present data. Thefree-stream velocity was set to achieve a Reynoldsnumber of 300,000 and the angle of attack was slowlyincreased from 0 to 17 deg. The flow on the uppersurface remained unstalled at a = 17 deg forapproximately 30 seconds, then became instantlyseparated and fully stalled. It should be noted here thatthe lift curve for the NACA 2414 airfoil exhibitsaerodynamic hysteresis about Clmax, and it is importantto distinguish between increasing angles of attack anddecreasing angles of attack. For the purposes of thispaper, any angle of attack mentioned in connection withthe NACA 2414 airfoil should be taken as oneincreased from a lesser value, unless stated otherwise.

The wake-velocity data showed spanwisevariations corresponding to those shown in the mini-tuftdata. While the data presented for the NACA 2414 andUltra-Sport airfoils are complementary, they deeplycontrast the following results for the remaining airfoils.

The thin-airfoil stall type category is representedby the NACA 64A010 airfoil. The data presented inthis case are slightly different from the data presentedpreviously in that the mini-tuft and spanwise velocityresults are not shown for analogous angles of attack,since this airfoil exhibited low-frequency unsteady flowfluctuations in the range of 8.7 < a < 10.2 deg, withmaximum lift occurring at about 10.1 deg. The mini-tuft images were difficult to interpret in this rangeowing to the unsteady flow, therefore these data are forangles of attack leading up the onset of theunsteadiness. On the other hand, the wake-velocitydata were acquired for angles of attack beginning atCi>max and continuing up to the onset of bluff-bodyshedding.

The key flowfield feature preceding the stall of theNACA 64A010 was the leading-edge separation bubblethat grew in size on the airfoil upper surface withincreasing angle of attack. The separation bubblereattachment locations determined from the mini-tuftdata are shown in Fig. 7. The data reveal how rapidlythe separation bubble grew in size, from a reattachmentat x/c » 0.15 for a = 7.4 deg to reattachment at x/c »0.50 for a = 8.4 deg. There was significant variation inthe reattachment location across the span at a = 8.4deg. Unsteady flow in the reattachment region impededinterpretation of the tuft orientations. The poorchordwise resolution of the tufts also added to thisdifficulty. In spite of this, the reattachment locations,when averaged across the span, compared very well toprevious surface-oil flow visualization results.13

Finally, the mini-tuft data confirmed previous surface-oil flow results that showed very little turbulentboundary-layer separation downstream of the bubblereattachment.

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Fig. 7 Spanwise variation in the leading-edgebubble reattachment location determined from themini-tuft flow visualization for the NACA 64AGIOairfoil.

Spanwise wake-velocity data were collected forangles of attack beginning near C/max at a = 10 deg andcontinued into the onset of bluff-body shedding at a =15 deg. The mean and RMS velocity voltages for thesecases are shown in Fig. 8. Unlike some of the previousresults, these data exhibited more spanwise uniformity.The mean velocities for a = 12 and 13 deg showedsome retarded flow above the midspan (that was notpresent below), but the variation was much less thanthat previously observed for the Ultra-Sport and NACA2414 airfoils. The bluff-body shedding case at a = 15deg, was very uniform in both the mean and RMSvelocity, consistent with previous data.13 The mini-tuftimages for a = 10 to 13 deg (not shown here) indicatedseparated flow from the leading edge, which wasexpected since the airfoil was stalled at these angles ofattack.

A key objective of this study was to determine ifthe low-frequency flow oscillation occurring on theLRN-1007 airfoil was essentially a two-dimensionalphenomenon, or if there was large spanwise variation inthe unsteady flowfield. Spanwise wake-velocity data



a = 10 dega = 11 dega=12dega= 13 dega = 15 deg

0.5-1 0.5-r

-0.52.0 2.2 2.4 2.6 2.8 3.0 0.0

E (volts)0.1 0.2 0.3

E' (volts)rnr: x '

Fig. 8 Spanwise variation in the mean andfluctuating wake-velocity voltage for the NACA64A010 airfoil.

indicated good uniformity in terms of integratedquantities. For example, Fig. 9 shows the variation inthe Strouhal number and power spectra amplitudeacross the wake for three angles of attack in the low-frequency oscillation range. The Strouhal number wasbased upon the frequency at the midpoint of the -3 dBbandwidth of the fundamental spectral peak, and thepeak amplitude corresponds to the -3 dB level. TheStrouhal number was essentially constant across thespan, indicating that the fundamental frequency of theoscillation did not vary in this direction. The increasingStrouhal number with angle of attack trend wasidentified in earlier studies (e.g., Ref. 5) and are inagreement with the present data. The peak amplitudedecreased significantly toward the ends of the model.The unsteady flow at each end of the model was likelyattenuated by the presence of the tunnel walls.However, the reduction in amplitude across the spanwas essentially symmetric from midspan.

Similar trends were observed in the wake-velocityvoltage and its root-mean-square value (see Fig. 10).The mean velocity voltage near the ends of the model

— a -15 deg

-0.50.00 0.01 0.02

St0.03 -15.0 -10.0 -5.0 0!0 5!0

Peak Amplitude (dB)

Fig. 9 Spanwise variation in the Strouhal numberand peak amplitude determined from the fluctuatingwake-velocity voltage for the LRN-1007 airfoil.

was slightly higher than values near the model midspan,likely indicating that the wake was slightly larger at thislocation (midspan). As expected, the RMS valuesbehaved analogously to the peak amplitude variationfrom Fig. 9. Since the RMS voltage was related to theintegrated power spectrum amplitude, the similarvariation indicated the large contribution of the low-frequency oscillation to the total RMS. These trends inthe wake-velocity data contrast with the spanwisevariation observed for the previous Ultra-Sport andNACA 2414 airfoils.

The conditionally averaged mini-tuft flowvisualization results were consistent with the wake-velocity measurements. Figure 11 shows the boundary-layer separation location at various times (or phaseangles, <|>) in the conditionally averaged oscillationcycle. The data shown are for a = 15 deg, since it hadthe largest amplitude (cf. Fig. 9) and confirmed that thelow-frequency oscillation was most intense near thisangle of attack. For each point in the cycle, theboundary-layer separation location was essentiallyuniform across the span. Furthermore, the separation



0.5-

0.4

0.3

0.2

0.1

-0.1

-0.2

-0.3

-0.4

-0.5

— a - 14 dega-15deg

— a -16 deg

0.5-

0.4-

0.3-

0.2-

0.1-

!fo.o.

-0.1-

-0.2-

-0.3-

-0.4-

2.0 2.2 2.4 2.6 2.8 3.0E (volts)

-0.5-

0.5-.

0.4-

0.3-

0.2-

0.1-

0.0-

-0.1-

-0.2-

-0.3-

-0.4-

0.0 0.1 0.2 0.3-0.5-

= 129.3 deg|) = 229.2 deg

0.0 0.2 0.4 0.6 0.8 1.0x/c

Fig. 10 Spanwise variation in the mean andfluctuating wake-velocity voltage for the LRN-1007airfoil.

location moved downstream from an average locationat x/c » 0.35 for $ = 40.8 deg to x/c « 0.80 for $ = 229.2deg. During the cycle, a separation bubble formed onthe upper surface and the reattachment location wasvisible in the mini-tuft patterns. These data aresummarized in Fig. 12. The earliest time in theaveraged cycle when a definite reattachment patternbecame visible was § = 149.0 deg, with the location atx/c « 0.05. It was assumed that the bubble separationlocation was very near the leading edge, which wasfairly sharp for the LRN-1007 airfoil. The data in Fig.12 show that the bubble reattachment locationprogressed downstream with time during the cycle,reaching x/c » 0.35 for ty = 229.2 deg. This case alsocorresponds to the farthest downstream location of theboundary-layer separation location in Fig. 11. Thiscomplex time-dependent flowfield behavior isconsidered in more detail in the Discussion. For now,the key conclusion is that the unsteady flow was two-dimensional in the conditionally averaged mean of thefundamental oscillation.

Fig. 11 Spanwise variation in the boundary-layerseparation location determined from theconditionally averaged mini-tuft flow visualizationfor the LRN-1007 airfoil.

Discussion

Three-Dimensional Structures

The spanwise variation observed for the stalledflowfields on the Ultra-Sport and NACA 2414 airfoilsbore strong similarities to stall-cell structures. Thework of Winklemann and Barlow8 showed that multiplestall cells can occur on three-dimensional models andthat the number of stall cells is proportional to themodel aspect ratio. In a later study, Yon and Katz10

used fine-thread-tuft flow visualization on models ofvariable aspect ratio to further investigate thisrelationship as well as the unsteady characteristics onthe stall-cell patterns. Their model configuration wasdifferent in that the ends of the model were fitted withend plates that effectively eliminated the tip flowresulting in more of a two-dimensional configuration.Again, the number of stall cells increased with aspectratio. However, this trend was offset from theWinklemann and Barlow8 data. Yon and Katz10



0.5-

0.4-

0.3-

0.2-

0.1-

^0.0-

-0.1-

-0.2-

-0.3-

-0.4-

-0.5

—————— A. ————

- <(>= 149.0 deg(j) - 166.3 deg

- <j) = 205.2 deg- <j)- 229.2 deg

0.0 0.2 0.4 0.6 0.8 1.0x/c

Fig. 12 Spanwise variation in the leading-edgebubble reattachment location determined from theconditionally averaged mini-tuft flow visualizationfor the LRN-1007 airfoil.

attributed the offset to the difference in model endconditions. The latter data are also noteworthy becausethey showed non-integer numbers of stall cells. For anaspect ratio of 3.0, the number of stall cells was about1.4. This aspect ratio was very similar to the value of2.8 for the airfoil models used in the present study.

There were a number of other similarities betweenthe present experiments and those of Yon and Katz.10

First, for the latter case, the airfoil was a NACA 0015section and the Reynolds number was 620,000. Thestall-cell patterns were observed over an angle of attackrange from 17 to 19 deg, with a5to// = 1 6 deg. Theauthors noted that boundary-layer (trailing-edge)separation was evident approaching ct5to// prior to theformation of the stall cells. The symmetric NACA0015 airfoil is a 15% thick section and at a Reynoldsnumber of 620,000 it probably exhibited similarcharacteristics similar to the 14% thick (cambered)NACA 2414 airfoil and the 18% thick Ultra-Sportairfoil, tested at a Reynolds number of 300,000. Thismeans that the stall type was probably some

combination of trailing-edge and leading-edge stall.The lift data presented in their paper generally confirmsthis assertion. Another common feature was that all ofthese airfoils exhibited turbulent boundary-layer (ortrailing-edge) separation as the angle of attack wasincreased to maximum lift.

Based upon these comparisons, it is not surprisingthat the flowfields contained similar characteristics.This is illustrated in Fig. 13 which shows mini-tuftphotographs of the Ultra-Sport airfoil at a = 11 deg andthe NACA 2414 airfoil at a = 16 deg. These data werefor angles of attack that were about one degree higherthan a,staii. The separation lines sketched on the photoscompared favorably with the sketches shown in Yon18

for the NACA 0015 airfoil at a - 17 deg (e.g., Fig.2.10). All of the frames showed this non-integernumber (approximately 1.4-1.5) of stall-cell patterns.The only substantial difference between the presentairfoils and the NACA 0015 is that the boundary-layerseparation in the middle of the main stall cell did notextend to the leading edge for the present airfoils.

Ultra-Sport NACA 2414.T. 16 ,c!ea

Approximate boundary-layer separation location

Fig. 13 Mini-tuft flow visualization photographsshowing a comparison of boundary-layer separationpatterns, flow direction is from right to left.



Yon18 and Yon and Katz10 found the non-integernumber of stall cells to be "inherently dynamic." Thatis, the whole stall cell and partial stall cell wouldintermittently switch places. However, no suchbehavior was observed in the present tests. The steadystall-cell patterns observed in the present data wereconfirmed in the wake-velocity data. As noted earlier,the mean velocity was a minimum at the same spanwiselocation where the boundary-layer separation was amaximum. This was due to the local enlargement ofthe wake caused by the stall cell. The combined resultsof the present data, Yon and Katz, Winklemann andBarlow, etc. suggest that the stall-cell phenomenon maybe related to stall type. A common trend through all ofthese studies was that they all involved airfoils withtrailing-edge or leading-edge stall types or somecombination of the two. The present data indicate thatthe stall behavior for the thin-airfoil and combinationthin-airfoil/trailing-edge stall types was fundamentallydifferent.

Yon and Katz10 performed time-dependent pressuremeasurements on the airfoil upper surface using achordwise row of five high-frequency response pressuretransducers. In some cases they recorded frequencycomponents in the fluctuating pressure spectra thatconverted to Strouhal numbers on the order of 0.040 to0.060. These frequencies were only observed when thestall cells were present—for angles of attack greaterthan avail. Due to the low values of the Strouhalnumber, the authors compared their measurements tothe low-frequency oscillation of Zaman et al.1However, it is clear that these are two differentphenomena. The low-frequencies measured by Yonand Katz10 were very low in amplitude, not readilyobservable in the tuft movement and occurred forangles of attack above maximum lift. In contrast, thelow-frequency unsteadiness in the present data wasvery large in amplitude, clearly observable in the mini-tufts and occurred for angles of attack leading up to andincluding maximum lift, but generally not abovemaximum lift. Further, the phase-averaged mini-tuftdata and spanwise wake-velocity data showed that thelow-frequency flowfield oscillation was primarily two-dimensional in character, markedly different from thestall-cell phenomenon.

It is important to note that Yon and Katz10 did notsuggest that the unsteadiness observed in their stall-cellpatterns was identical to the low-frequency oscillationdocumented by Zaman et al.1 and the present data.Instead, they speculated that this frequency wasassociated with large-amplitude motions of theseparated shear layer, noted for other cases as shear-layer flapping (e.g., see Driver et al.19). It is quitepossible that their speculation was indeed correct.

Balow4 suggested that shear-layer flapping was relatedto the origin of the low-frequency oscillation on theLRN-1007 airfoil. Bragg et al.5 further speculated thatthe low-frequency oscillation was symptomatic of aresonance, completed with a "feedback loop" thatcaused the low-frequency flapping to "lock on" at thelow frequency—thus resulting in the quasi-periodic,large-amplitude, stalling and unstalling behavior.Given these observations, it is possible that the low-frequency measurements of Yon and Katz10 wererelated to a shear-layer flapping instability, but the stall-cell dominated the flowfield and provided no "feedbackloop" to complete or to "lock on" this low-frequencyfluctuation. Thus, no large-amplitude fluctuations wereobserved.

The combined results of these investigationssuggests that airfoils with trailing-edge separations atand above maximum lift contained these stall-cellpatterns that did not result in the low-frequencyoscillation. The present data indicate that these airfoilswere of the trailing-edge, leading-edge and theircombination stall type categories. The airfoils that hadleading-edge separation bubbles that grew in size on theupper surface as the angle of attack was increased intostall exhibited more two-dimensional flowfieldcharacteristics and low-frequency oscillations. In thecombination thin-airfoil/trailing-edge stall case, thelow-frequency oscillations were very well defined.

More on the Low-Frequency Oscillation

The low-frequency oscillation that occurs duringthe static stall of the LRN-1007 has been partiallyillustrated in Figs. 11 and 12. A more completedescription of this complex flowfield is provided inReferences 6 and 7. However, a brief synopsis of thesefindings is offered here so that the significance of Figs.11 and 12 can be fully exploited. The LDVmeasurements from References 6 and 7 provided aquantitative distribution of the streamwise velocity onthe LRN-1007 airfoil upper surface conditionallyaveraged over one oscillation cycle. A time-dependentsurface flowfield map is depicted in Fig. 14. Themapping shows the boundary-layer state over theduration of the conditionally-averaged cycle. Thephase angle (<|>) is used to represent the time during thecycle and is identical to the (|>'s given in Figs. 11 and12. The chordwise separation and reattachmentlocations were obtained from the velocity profiles byextrapolating them to the airfoil surface. Since the flowoscillation is essentially periodic, the starting point wasarbitrary and chosen to be § = 15 deg. At this point intime, the upper-surface boundary layer separates at x/c= 0.10. As time increases through the cycle, the



boundary-layer separation point progresses toward thetrailing edge to about x/c = 0.80 at <j> = 165 deg. At <j> =165 deg, a separation bubble is observed and itsseparation and reattachment locations are indicated inFig. 14 (x/c ~ 0.05 and 0.12, respectively). The bubblegrows in size from (j) = 165 to 255 degrees as the bubbleseparation point moves slightly forward on the airfoiland the reattachment point moves downstream.Meanwhile, the turbulent boundary-layer separationpoint continues to progress downstream until <j) = 225deg, where it reverses direction, moving upstream andultimately merging with the separation bubblereattachment. Whence this occurs, the entire uppersurface boundary layer is separated aft of x/c « 0.05.The coalescence of the separation bubble reattachmentand the trailing-edge separation produces a large regionof separated flow on the upper surface from fy = 255 to360 deg. The boundary-layer separation point remainsfixed at x/c « 0.05. As the boundary-layer separationpoint begins to move downstream, the oscillationbegins again.

Boundary-Layer Separation PointLeading-Edge Bubble Separation PointLeading-Edge Bubble Reattachment Point

Attached Boundary LayerSeparated Boundary LayerSeparation Bubble

30 60 90 120 150 180 210 240 270 300 330 360Phase angle, (() (deg)

Fig. 14 Variation in the LRN-1007 airfoil uppersurface flowfield as a function of phase over theconditionally averaged cycle, after Broeren andBragg.7

While the previous LDV measurements providedexcellent details about the unsteady flow, the velocityfield was only measured in a single plane at modelmidspan. The significance of Figs. 11 and 12 is thatthey show identical trends to the LDV results andfurther reveal that the unsteady flow is essentiallyuniform over the entire span of the model. Not only dothe mini-tuft flow visualization results indicate theproper trends, but the absolute values comparefavorably as well. For the following comparisons, thephase-averaged bubble reattachment and boundary-layer separation locations were averaged over themiddle one-third of the span, i.e., -0.167 <y/b < 0.167,.This interval included data from the midspan mini-tuftrow and the first two rows above and below midspan.

The average bubble reattachment and boundary-layer separation locations were plotted as a function ofphase angle in Fig. 15. This plot is similar to Fig. 14,without the shaded regions. For a = 15 deg, theselocations corresponded fairly well with the LDV resultstaken from Fig. 14, further validating the phase-averaged mini-tuft method. The plot also shows thatthe amount of boundary-layer separation increased withincreasing angle of attack, particularly between a = 14and 15 deg. For a = 16 deg, the boundary-layerseparation data were corrupted by the point at x/c =0.62 (<j> » 115 deg), which was obviously spurious andprobably should have been closer to x/c = 0.50. Theseparation bubble was larger for the a = 15 deg caseover a = 14 deg, but both formed at about the sametime at <() = 140 deg. The lack of meaningful mini-tuftdata later than about <|> = 240 deg meant that the flowwas completely separated. As described above, thisoccurred at about (j) = 255 deg (for a = 15 deg), whenthe bubble reattachment point merged with theboundary-layer separation point. A key to this scenariowas that the boundary-layer separation locationgradually moved downstream until <|) = 225 deg, thenreversed direction and began moving forward,ultimately merging with the downstream movingbubble reattachment. Similar behavior is shown in Fig.15 for a = 14 deg, as the boundary-layer separationreached a maximum downstream location of about x/c= 0 .88at< | )«215 deg, then decreased as the bubblecontinued to grow in size. The data for a = 16 degshows that the bubble was first observed earlier in theoscillation cycle at about <|> = 90 deg. However, thebubble did not grow to be as large as in the other casesbefore the mini-tuft data became uninterpretable. Thissuggests that the flowfield completely separated earlierin the phase-averaged cycle than at the lower angles ofattack.



Bubble Reattachment, a = 14 degBubble Reattachment, a = 15 degBubble Reattachment, a = 15 deg, LDV Ref. 7Bubble Reattachment, a = 16 degBoundary-Layer Separation, a = 14 degBoundary-Layer Separation, a = 15 degBoundary-Layer Separation, a = 15 deg, LDV Ref. 7Boundary-Layer Separation, a = 16 deg

1.0-.

0.9-

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0.7-

0.6-

^0.5-

0.4-

0.3-

0.2-

0.1-

0.00 30 60 90 120 150 180 210 240 270 300 330 360

Phase angle, § (deg)

Fig. 15 Variation in the LRN-1007 airfoil uppersurface flowfield as a function of phase over theconditionally averaged cycle, present datadetermined from mini-tuft flow visualization.

The unsteady behavior of all of the combinationthin-airfoil/trailing-edge stall airfoils tested was foundto be very similar. In particular, the E374 airfoil wassingled out as having nearly identical unsteadycharacteristics to the LRN-1007 airfoil. The combinedresults of the phase-averaged mini-tuft data and thespanwise wake-velocity measurements showed that thelow-frequency oscillation was essentially two-dimensional in character on the model surface.Although not explicitly shown here, nearly the samelevel of spanwise uniformity shown for the LRN-1007airfoil (Figs. 9-12) was also exhibited by the E374airfoil and these data are given by Broeren.13

The complementary spanwise-average data for theE374 airfoil, Fig. 16, also showed analogous trends.For this airfoil, the separation bubbles were about thesame size for both angles of attack. In contrast, therewas substantially more turbulent boundary-layerseparation at a = 13 deg versus a = 12 deg, until $ =225 deg. The latter case also showed that the maximumdownstream boundary-layer separation location

Bubble Reattachment, a = 12 degBubble Reattachment, a = 13 degBoundary-Layer Separation, a = 12 degBoundary-Layer Separation, a = 13 deg

l.O-i

0.9-

0.8-

0.7-

0.6-

0.4-

0.3-

0.2-

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0.00 30 60 90 120 150 180 210 240 270 300 330 360

Phase angle, $ (deg)

Fig. 16 Variation in the E374 airfoil upper surfaceflowfield as a function of phase over theconditionally averaged cycle, determined from mini-tuft flow visualization.

occurred at x/c = 0.90 (cj) = 155 deg). This location wasobserved to be coincident with the first appearance ofthe bubble. This was different from the LRN-1007airfoil at a = 14 deg, where the maximum downstreamboundary-layer separation location occurred well afterthe first appearance of the bubble. The data in Fig. 16show that the separation bubble was visible on thesurface from <|> = 155 to 225 deg, which was about 28%of the 360 deg cycle. These data provide even moreconclusive evidence that the unsteady flowfields forthese two different airfoils were fundamentallyidentical. This further implies that this low-frequencyunsteadiness may be a general phenomenon that occursfor airfoils classified as having a combination thin-airfoil/trailing-edge stall. Even more importantly, it iskey flowfield features preceding stall, such as thegrowing leading-edge separation bubble in tandem withthe turbulent boundary-layer separation, that apparentlylead to the unsteady flow. Therefore, it is likely thatany airfoil exhibiting these features will becharacterized by the unsteady stall.



Summary and Conclusions

Several airfoils having different stallingcharacteristics were studied to better understand theunsteady and three-dimensional flowfield variationsgoverning the stall. The airfoils were tested in a two-dimensional configuration where the ends of the modelswere flush with the wind-tunnel side walls. Wake-velocity measurements and surface mini-tuft flowvisualization were carried out at a Reynolds number of300,000. The results showed that the stall of airfoilshaving trailing-edge separation leading up to the stallwas characterized by large-scale spanwise structuressimilar to stall cells. The stall cells were observed onthe airfoil upper surface for angles of attack at or abovestall (maximum lift) and were generally steady.Conversely, the stall of airfoils having a thin-airfoil orcombination thin-airfoil and trailing-edge stall type,where a leading-edge separation bubble grows in sizeleading up to the stall, was governed by low-frequencyunsteady flow. This unsteady flow generally occurredprior to and including the mean maximum lift region.The flowfield was found to be two-dimensional in theconditionally averaged mean of the oscillation, with noevidence of stall cell formation. Therefore, theseappear to be exclusive phenomena that are determinedby the boundary-layer separation characteristics leadingup to the stall.

Acknowledgements

This work was funded, in part, through a NASAGraduate Student Researchers Program Fellowship.The authors wish to acknowledge K.B.M.Q. Zaman ofthe NASA Glenn Research Center for his contributionsto this research.

References

1. Zaman, K.B.M.Q., McKinzie, D.J., and Rumsey,C.L., "A Natural Low-Frequency Oscillation OverAirfoils Near Stalling Conditions," Journal of FluidMechanics, Vol. 202, 1989, pp. 403-442.

2. Bragg, M.B., Heinrich, D.C., and Khodadoust, A.,"Low-Frequency Flow Oscillation Flow Oscillationover Airfoils near Stall," AIAA Journal, Vol. 31, No. 7,July 1993, pp. 1341-1343.

3. Heinrich, D.C., "An Experimental Investigation of aLow Frequency Flow Oscillation Over a Low ReynoldsNumber Airfoil Near Stall," M.S. Thesis, Dept. ofAeronautical and Astronautical Engineering, Univ. ofIllinois, Urbana, IL, 1994.

4. Balow, F.A., "Effect of an Unsteady LaminarSeparation Bubble on the Flowfield Over an AirfoilNear Stall," M.S. Thesis, M.S. Thesis, Dept. ofAeronautical and Astronautical Engineering, Univ. ofIllinois, Urbana, IL, 1994.

5. Bragg, M.B., Heinrich, D.C., Balow, F.A., andZaman, K.B.M.Q., "Flow Oscillation over an AirfoilNear Stall," AIAA Journal, Vol. 34, No. 1, Jan. 1996,pp. 199-201.

6. Broeren, A.P., "Phase-Averaged LDV FlowfieldMeasurements About an Airfoil in Unsteady Stall,"M.S. Thesis, Dept. of Mechanical and IndustrialEngineering, Univ. of Illinois, Urbana, IL, 1996.

7. Broeren, A.P., and Bragg, M.B., "FlowfieldMeasurements over an Airfoil During Natural Low-Frequency Oscillations near Stall, AIAA Journal, Vol.37, No. 1, Jan. 1999, pp. 130-132.

8. Winklemann, A.E., and Barlow, J.B., "FlowfieldModel for a Rectangular Planform Wing beyond Stall,"AIAA Journal, Vol. 18, No. 8, Aug. 1980, pp. 1006-1008.

9. Winklemann, A.E. "Flow Field Studies Behind aWing at Low-Reynolds Numbers," AIAA Paper 90-1471, June 1990.

10. Yon, S.A., and Katz, J., "Study of the UnsteadyFlow Features on a Stalled Wing," AIAA Journal, No.Vol. 36, No. 3, Mar. 1998, pp. 305-312.

11. Kline, S.J., and McClintock, F.A., "DescribingUncertainties in Single-Sample Experiments,"Mechanical Engineering, Vol. 75, Jan. 1953, pp. 3-8.



12. Coleman, H.W., and Steele, W.G., Experimentationand Uncertainty Analysis for Engineers, John Wileyand Sons, New York, 1989.

13. Broeren, A.P., "An Experimental Study ofUnsteady Flow Over Airfoils Near Stall," Ph.D.Dissertation, Dept. of Mechanical and IndustrialEngineering, Univ. of Illinois, Urbana, IL, 2000.

14. Crowder, J.P., "Flow Visualization," TheHandbook of Fluid Dynamics, edited by R.W. Johnson,CRC Press, Boca Raton, FL, 1998.

15. Broeren, A.P. and Bragg, M.B., "Low-FrequencyFlowfield Unsteadiness During Airfoil Stall and theInfluence of Stall Type," AIAA Paper 98-2517-CP,Proceedings of the 16th Applied AerodynamicsConference. Albuquerque, NM, June 1998, pp. 196-209.

16. Broeren, A.P., and Bragg, M.B., "UnsteadyStalling Characteristics of Thin Airfoils at Low-Reynolds Number," Proceedings of the Conference onFixed, Flapping and Rotary Vehicles at Very LowReynolds Number, Ed. by T.J. Mueller, Notre Dame,IN, June 2000, pp. 396-420.

17. McCullough, G.B., and Gault, D.E., "Examples ofThree Representative Types of Airfoil-Section Stall atLow-Speed," NACA TN 2502, Sept. 1951.

18. Yon, S.A., "Coherent Structures in the Wake of aStalled Rectangular Wing," Ph.D. Dissertation,University of California, San Diego, San Diego StateUniversity, 1995.

19. Driver, D.M., Seegmiller, H.L., and Marvin, J.,"Time-Dependent Behavior of Reattaching ShearLayers," AIAA Journal, Vol. 25, No. 7, July 1987, pp.914-919.


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