055

T H E O N S E T O F D Y N A M I C S TA L L :

A T I M E - R E S O LV E D A P P R O A C H

karen mulleners and markus raffel

German Aerospace Center (DLR) – Institute of Aerodynamics and Flow TechnologyBunsenstr. 10, 37073 Gottingen, Germany

Abstract The flow over an oscillating OA209 airfoil un-der dynamic stall conditions was investigated by meansof unsteady surface pressure measurements and time-resolved particle image velocimetry. The characteristicfeatures of the unsteady flow field were identified andanalysed utilising different coherent structure identifica-tion methods. An Eulerian and a Lagrangian procedurewere adopted to locate the axes of vortices and the edgesof Lagrangian coherent structures, respectively. Thecomplementary information obtained by these methodsprovided deeper insight into the spatiotemporal evo-lution of vortical structures within a single dynamicstall life cycle. In particular, the physical mechanismsheralding the formation of the primary dynamic stallvortex were identified as the emergence and subsequentroll up of a shear layer at the interface between the freestream and the reversing flow near the airfoils surface.Accordingly, the stall development was subdivided intotwo parts: a primary instability stage followed by asecondary instability or vortex formation stage. Thecharacteristic time scales associated with the first stagerevealed an overall decrease of the dynamic stall delaywith increasing unsteadiness. The delay attributed to thevortex formation stage was observed to be unaffectedby variations in the airfoils dynamics. The combinationof time-resolved imaging and extensive coherent struc-ture analysis was shown to provide a new approach todifferent aspects of dynamic stall.

1 introduction

The dynamic stall process of an airfoil comprises a se-ries of complex aerodynamic phenomena in responseto an unsteady change of the angle of attack. It is ac-companied by a lift overshoot and delayed massive flowseparation with respect to static stall. The salient featureof the unsteadily separating flow is the formation and

convection of a large-scale coherent structure referredto as the dynamic stall vortex. The most prominentexample can be observed on the retreating blades ofan helicopter rotor in forward flight. Although thedynamic stall delay and the related increase of the max-imum lift can be beneficial in some applications, thelarge excursions of the aerodynamic loads that emergeduring vortex break down induce strong vibrations andstructural loads, potentially fatal for a helicopter rotor.Hence, due to the incessant interest in improving themanoeuvrability and performance of rotary-wing air-craft and rapidly manoeuvring aircraft, dynamic stallhas been and still remains the subject of vivid interest[11, 21, 20].

A detailed analysis of the dynamic stall events onan oscillating airfoil was presented by Carr et al. [1].They revealed that the prominent features within a fullcycle of oscillation are consecutively: the emergence andspreading of flow reversal on the airfoil’s suction side,the formation and convection of a large-scale leadingedge vortex, massive flow separation, and finally flowreattachment. Analogously, Shih et al. [19] classified theunsteady flow development over an airfoil pitching upat constant rate into four successive stages: 1) a vortexformation stage, 2) a vortex convection stage, 3) stallonset, and 4) a stalled stage. Both descriptions showthat the flow over either a constantly pitching or oscil-lating airfoil is qualitatively characterised by the sameprominent features, being the initiation, growth andshedding of a leading edge vortex and the associatedlift overshoot. For both types of motion the process ofvortex formation and convection result in a delay ofmassive flow separation to angles of attack beyond thestatic stall angle. During this delay the lift continuesto increase with increasing angle of attack yielding thelift overshoot which is characteristic of dynamic stall.The inception of stall is generally accompanied by a lossof lift and an increase of the negative pitching moment

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and marks the beginning of the stalled stage. This stagecan be recognised by large-scale vortex shedding andassociated large fluctuations of the lift, drag and pitch-ing moment. Furthermore, when the airfoil motion isoscillatory, a large amount of load hysteresis is present.

Although a lot of effort – analytical as well as nu-merical and experimental – has been devoted in the pastto enhance the comprehension of the phenomenologyof dynamic stall, it is not yet fully understood and char-acterised. In particular, the process that leads to theformation of the primary stall vortex and the mecha-nism that causes the vortex to break contact with theairfoil are still controversial issues requiring further anddeeper examination.

Accurate knowledge of the state of the fluid flowduring dynamic stall is strongly tied to a fundamentalunderstanding of the development and interaction ofcoherent structures. Due to the incessant technologicalprogress during the last decades yielding the develop-ment of state-of-the-art digital cameras, high repetitionrate lasers and sophisticated evaluation algorithms, theparticle image velocimetry (PIV) technique establisheditself as a valuable and indispensable tool in experi-mental fluid dynamics. However, the investigation ofvortices and vortex dynamics from experimental dataremains a challenging task due to the lack of a uni-versally accepted definition of vortical structures andthe difficulty to quantify them. Several specific defini-tions have been proposed hitherto, i. a. [10, 6]. Despitethe plethora of publications on the issue no consensushas been reached yet, reflecting the complexity of thesubject area. As a direct consequence, unambiguousvortex detection remains elusive and various Eulerianand Lagrangian criteria have been introduced over theyears with different validity depending on the specificproblem.

Within the scope of this study, the conspicuousfeatures of the experimentally investigated flow overa sinusoidally oscillating airfoil in a uniform flow areidentified and analysed utilising a combination of anEulerian vortex centre allocation procedure [13] and aLagrangian approach based on the Lyapunov exponent[7]. Whereas past experimental investigations generallyinvolve phase-locked measurements, the present studyprovides time-resolved recordings of the velocity field inaddition to unsteady airfoil’s surface pressure distribu-tions and allows for the examination of the chronologyof events leading to the onset of dynamic stall.

The onset of stall under dynamic conditions is gen-erally defined as the detachment of the primary stallvortex and was specified earlier in the present contextby Mulleners and Raffel [14] based on a characteristicmode of the proper orthogonal decomposition (POD)of the velocity field. Variations in the flow field topol-

ogy that accompany the stall onset were verified by aLagrangian coherent structure analysis. Furthermore, asubtle but significant change was observed in the ori-entation of the trajectories of the vortices that originateat the very leading edge shortly before and after stallonset. Due to the congruence with the Eulerian andLagrangian picture shown in Mulleners and Raffel [14],the POD-based assignment of the stall onset is deemedto be reliable and was utilised to specify the dynamicstall onset for the present parameter combinations.

The next logical step – and the main focus of thispaper – is to assess the associated time delay with re-spect to static stall. Furthermore, in order to modeland predict the dynamic stall onset and delay, specialemphasis is placed on the identification and characteri-sation of the physical mechanisms and parameters thatplay a key role in the initiation, growth, and subsequentdetachment of the dynamic stall vortex.

The paper is organised as follows. Prior to the de-scription of the applied coherent structure identificationprocedures, the experimental details will be specified.Subsequently, the experimental results are presentedand discussed. The discussion covers the examinationof the chronology of events heralding stall onset anda detailed analysis of the associated time-scales. Thediscussion is concluded by a short summary of the mostimportant findings and by suggesting further avenuesof investigation.

2 experimental set-up

Wind tunnel experiments were conducted to investi-gate dynamic stall on a constantly pitching airfoil in auniform flow at a free stream Reynolds number Re =

9.2× 105 based on the chord length c, with c = 0.3m(Mach number Ma = 0.14). A two-dimensional air-foil model with an OA209 profile was subjected to asinusoidal oscillating motion about its quarter chordaxis with a mean incidence α0, an amplitude α1, andan oscillation frequency fosc. The latter is preferablywritten in dimensionless form as the reduced frequencyk = π fosc c/U∞ , where U∞ is the free steam velocity.The mean incidence, amplitude and reduced frequencywere varied such that α0 ∈ {18°, 20°, 22°}, α1 ∈ {6°, 8°},and k ∈ {0.050, 0.075, 0.10}.

xy

z

U∞

Figure 1: Position of the PIV field of view.

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Stereoscopic time-resolved particle image velocity(TR-PIV) measurements were conducted in the crosssectional plane at model mid-span (see figure 1). Thewidth of the field of view covered the entire chord forthe relevant angle of attack range. As maximally al-lowed by the hardware, time series of 6144 frames withfull camera resolution were recorded at 3000Hz, cor-responding to an acquisition rate of 1500Hz for thevelocity fields. After mapping the views of both cam-eras, the dimensions of the PIV measurement win-dow were 335mm × 165mm and the spatial resolu-tion of the recording was 5.0px/mm. The PIV imageswere processed using an interrogation window sizeof 32px × 32px and an overlap of approximately 80%yielding a grid spacing of 6px or 1.2mm which is lessthen 0.005 c. The interrogation window size was min-imised ensuring an acceptable signal-to-noise ratio. Thewindow overlap on the other hand was maximised toavoid artificial smoothing of velocity gradients [17]. Bydoing so the spatial resolution of the results of the vortexdetection algorithms was improved. Prior to the coher-ent structure analysis, the velocity fields were rotatedinto the airfoil reference system with the x-axis alongthe chord, the y-axis along the span and the z-axis up-ward perpendicular to the chord. The origin coincideswith the rotation axis, i. e. the airfoil’s quarter chord axis,at model mid-span. Simultaneously to the TR-PIV, thesurface pressure distribution at the model mid-span wasscanned at approximately 6kHz for about 15 s. The dataacquisition was synchronised with the recording of thePIV images allowing for straightforward assignment ofthe instantaneous pressure distributions to each of theacquired velocity fields.

3 coherent structure analysis

The common goal of coherent structure identificationmethods is to locate, extract, and visualise conspicuousflow structures characterised by various spatial and tem-poral scales. In the context of the present paper, twodifferent methods to analyse coherent structure wereadopted; one Eulerian and one Lagrangian procedure.

The Eulerian method utilises the dimensionlessscalar function Γ that was first introduced by Michardet al. [13] to locate the axis of individual vortices. Thefunction is derived directly from the two-dimensionalin-plane velocity field and is defined in discrete form as

Γ(xi) =1

M

∑

xj ∈Si

[(xj − xi)× (uj − ui)

]·n

|xj − xi| · |uj − ui|

=1

M

∑

xj ∈Si

sin(θij) ,(1)

with Si a two-dimensional area around xi, M the num-ber of grid points xj inside Si with j 6= i, n the unitnormal vector, uj the velocity at xj, ui the local mean ve-locity around xi, and θij the angle formed by the vectorsxj − xi and uj − ui. The local mean velocity is taken intoaccount in order for Γ to be Galilean invariant (cf. [4]).

According to its definition, Γ is a dimensionlessscalar function, with −1 6 Γ 6 1. The location of pos-sible vortex axes is indicated by the local extrema ofΓ and the sense of rotation is given by the sign of thelocal extremum. Besides the detection of the location ofthe vortex centres, their trajectories over a time seriesof flow fields have been traced. For this purpose, theconvection velocity of the individual identified vortexcentres within the reference frame are used to predicttheir future position and narrow the number of possiblefollow-up vortices.

The most popular Lagrangian approach, whichwas adopted in the present study, leverages finite-timeLyapunov exponents (FTLEs) and was introduced byHaller and Yuan [7]. The FTLE method reverts directlyto the fluid particle trajectories and is therefore inher-ently objective, includes information on the history ofthe flow, and has a clear physical interpretation. The par-ticle trajectories can be integrated in forward as well asin backward time yielding positive finite-time Lyapunovexponent (pFLTE) and negative finite-time Lyapunovexponent (nFTLE) fields. The ridges in the pFTLE fieldreveal material lines normal to which fluid particles arebeing stretched or repelled, consequentially they are re-ferred to as repelling material lines or stable manifolds.Vice versa, ridges in nFTLE fields visualise attractingmaterials or unstable manifolds, i. e. lines along whichfluid particles are being elongated, when integrating thetrajectories in backward time. The flow field aroundthe intersection of a repelling and an attracting materialline resembles that of a saddle point. Moreover, whenattached to a solid surface attracting material lines de-pict separation lines while attachment lines are repellingmaterial lines. This FTLE method thus yields candidatematerial lines and captures features of the flow that arefamiliar from flow visualisation experiments. Accord-ing to Shadden et al. [18] the ridges in the FTLE fieldsdelineate regions that exhibit qualitatively different dy-namical behaviour, hence indicated the boundaries ofLagrangian coherent structures (LCSs). For a compre-hensive discussion of the general properties and basicconcepts involved in the computation of the FTLE, thereader is referred to i. a. [5, 3].

The combination of two vortex identification pro-cedures that are different in nature allows for differentaspects of the flow to be highlighted and delivers a moredetailed insight into the vortex dynamics.

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4 results and discusion

Analogous to the classification of Shih et al. [19] withregard to the flow over an airfoil pitching-up at constantrate, the unsteady flow development over an oscillationairfoil can be divided into five different stages [14]. Withthe starting point of a cycle taken at the minimum inci-dence angle, the flow will consecutively pass throughthe following stages within each individual cycle: 1) theattached flow stage; 2) the stall development stage; 3) stallonset; 4) the stalled stage; and 5) flow reattachment. Theonset of dynamic stall, defined as the detachment of theprimary stall vortex, was specified for the available timeresolved velocity field data based on a characteristicPOD mode as reported in a preceding publication (see[14]). The focus in the present paper lies on the analysisof the peculiarities of the dynamic stall development,including the succession of the physical mechanismsthat herald stall onset, the governing parameters, andthe relevant time-scales.

4.1 Triggering Mechanism

The dynamic stall process is deemed to be an interestingexample of unsteady separation. It is initiated by anadverse pressure gradient and involves a recirculationregion originating within the region of this local adversegradient. The observations presented in [14] revealeda prominent recirculation zone on the airfoil’s suctionside which grows in a direction normal to the surfacewhen the airfoil pitches up. This is analogous to thebasic mechanism heralding unsteady separation in thelow-Reynolds-number regime described by Obabko andCassel [15]. Furthermore, the shear layer that developsat the interface between this region of reversed flow andthe free stream flow seems to play a major role in thestall development (cf. Ho [8]).

Based on the size of the region of flow reversaland the level of interaction between the shear layervortices, the stall development stage can be subdividedinto two parts. A first part essentially covers the growthof the recirculation region and the corresponding initialdevelopment of the free shear layer. This includes theemergence of individual shear layer vortices as a resultof a primary instability. These small-scale vortices inter-act only weakly with each other while being convected

x

z zvi(t)

zw(xvi(t))

xvi(t)

Figure 2: Definition of the ingredients required for the com-putation of ∆z.

downstream by the external flow. Inspired by the stan-dard notions used to describe the evolution of a mixinglayer (cf. [2]), this part of the stall development phaseis termed the primary instability stage. Accordingly, thesecond part which is characterised by a secondary insta-bility eliciting the free shear layer to roll up and form alarge-scale dynamic stall vortex is termed the secondaryinstability stage or alternatively the vortex formation stage.

Prior to focussing on the two stages and their gov-erning parameters individually, the transition betweenthem is identified based on the temporal evolution ofthe average height of the shear layer, which is equiv-alent to the average height of the recirculation region.The latter is given by the average normal distance ofthe shear layer vortices with respect to the airfoil’supper surface. More general, the average height of allclockwise rotating vortical structures detected in aninstantaneous experimental velocity field denoted by ∆z

(a)

−0.20 0.00 0.20 0.40 0.600.00

0.05

0.10

0.15

0.20

0.25

(t− tss)/T

∆z/c

(b)

0.00

0.10

0.20

0 0.05 0.10 tds2 0.150.00

0.10

0.20

(t− tss)/T

∆z/c

∆t1

∆t2

Figure 3: Evolution of the average vertical distance of thevortex cores from the airfoil’s upper surface over one period(a) and in detail (b) with the corresponding linear fits. Thehorizontal and vertical, dotted lines indicate respectively thenoise level due to experimental limitations and the specifiedonset of dynamic stall (α0 = 20°, α1 = 8°, k = 0.05).

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is calculated according to

∆z(t) =1

N(t)

N(t)∑

i=1

∣∣zvi(t) − zw(xvi(t))∣∣ , (2)

where N(t) is the total number of clockwise rotatingvortices detected in the instantaneous velocity field attime t using the Eulerian vortex detection algorithm,and where

∣∣zvi(t) − zw(xvi(t))∣∣ is the normal distance of

the individual vortices to the airfoil’s upper surface asillustrated in figure 2. The average vortex height is calcu-lated for all instantaneous velocity fields even when thevortices considered are not arranged in the shear layerand ∆z can not be interpreted as the average shear layerheight. A typical temporal evolution of ∆z, normalisedby c, for one oscillation period, T , is depicted in figure 3

for the dynamic stall case with α0 = 20°, α1 = 8°, andk = 0.05. Here, the values on the abscissa show the timelag with respect to the instant when the angle of attackexceeds the static stall angle, tss.

For small angles of attack, vorticity is a priori con-fined to the thin boundary layer and its transport isdominated by convection. Vorticity lumped into sepa-rate vortices is convected downstream along the airfoil’scontour preventing the boundary layer from growing

beyond a certain thickness. Hence, as long as the flowis fully attached the average height ∆z is expected to beof the order of the boundary layer thickness. However,due to experimental limitations valid velocity informa-tion was only available down to approximately 2mmabove the airfoil’s surface. Consequentially, minimumvalues of ∆z for low angles of attack are erroneous andmisleading. The horizontal, dotted line in figure 3(a)marks the minimum level of ∆z corresponding to a fullyattached flow.

With increasing angle of attack, the strongly accel-erated flow around the leading edge is accompanied bya large pressure gradient which leads to an increasedproduction of vorticity. Immediately downstream of theleading edge the flow decelerates, i. e. a local adversestreamwise pressure gradient emerges, as a result ofwhich a recirculation zone develops there. The emer-gence and subsequent growth of the region of flow re-versal can be readily understood from a vorticity pointof view.

The vorticity distribution is initially confined to asmall vortex sheet near the airfoil’s surface which can berepresented by a discrete train of two-dimensional pointvortices. According to the vortex image pair analysisof Reynolds and Carr [16], mirror vortices – included

(a) α = 25.5° ↗

0.0

0.2

0.4

z/c

|U∞ |

(b) α = 26.4° ↗

(c) α = 25.5° ↗

−0.2 0.0 0.2 0.4 0.6

0.0

0.2

0.4

x/c

z/c

(d) α = 26.4° ↗

−0.2 0.0 0.2 0.4 0.6x/c

Figure 4: Difference of the flow topology and vortex distribution between the primary instability and the vortex formation stage(α = 25.5° and 26.4° on the upstroke ↗, respectively) indicated by: (a)-(b) the velocity field and the centres of (•) clockwise and(•) anticlockwise rotating vortices and (c)-(d) the corresponding Lagrangian coherent structures indicated by the ridges in thepFTLE and nFTLE fields (α0 = 20°, α1 = 8°, k = 0.050).

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to model the airfoil’s surface – induce an upstream ve-locity in their objects. In the region where a persistentadverse pressure gradient is present and the potentialvelocity no longer prevails the upward swimming veloc-ity of the individual vortices, the vortex flow is able tomove upstream. Due to the velocity difference with theouter flow a shear layer develops which is subsequentlysubjected to the primary instability. As a result of thelatter the initial vorticity is redistributed in concentratedsmall-scale vortices.

Initially the shear layer vortices, all unidirectional-rotating, are more or less aligned and they interact onlyweakly with each other (figure 4(a)). The growth of thelayer is approximately linear (cf. [2]) and the same istrue for the increase of the mean height ∆z. The linear fitdescribing the evolution of the shear layer height withinthe primary instability stage is indicated by a dashed linein figure 3(b). This linear relationship is valid until, at alater stage, the small shear layer vortices are subjectedto the secondary instability and start to merge heraldingshear layer roll up and the formation of the primarystall vortex. Simultaneously, counter-rotating vorticesare induced near the airfoil profile as a result of strong

α = 26.35° ↗

α = 26.41° ↗

α = 26.46° ↗

Figure 5: The pFTLE and nFTLE fields corresponding to con-secutive time steps around the transition from the primaryinstability into the vortex formation stage (α0 = 20°, α1 = 8°,k = 0.05).

interactions between the shear layer vortices and thereversed flow near the surface (figure 4(b)). Duringthis stage the recirculation region grows swiftly and be-comes of the order of the airfoil’s thickness. The locationand the size of the recirculation region within the twodifferent stall development stages are indicated by theridges in the pFTLE and nFTLE fields in figure 4(c)-(d).Furthermore, the rate of increase of ∆z during the vortexformation stage is a measure for the rate of growth of thedynamic stall vortex.

The discontinuity in the temporal evolution of theshear layer height in figure 3(b) thus indicates a growinginstability and marks the transition of the flow from theprimary instability into the vortex formation stage. Froman Eulerian point of view the transition point is de-fined by the intersection of the two linear fitting curvesdescribing the rate of increase of ∆z in the individualregimes. The time instant and angle of attack at whichthe transition takes place is denoted by tds1 and αds1,respectively.

Additionally, from a Lagrangian point of view, thedevelopment of the secondary instability around t = tds1

is nicely observable in figure 5 representing a sequenceof instantaneous LCSs. The depicted LCSs, identified bythe ridges in the pFTLE and nFTLE fields, are extractedbased on instantaneous velocity fields for subsequenttime instants around t = tds1. The increasing bulgingsof the ridges in the pFTLE and nFTLE fields readily indi-cate the growing instability of the shear layer leading toshear layer roll up. Consequently, the Lagrangian coher-ent structures analysis supports the conclusion that thepassage from the first into the second stall developmentstage is triggered by a secondary instability of the shearlayer.

4.2 Primary Instability Stage

The question that remains is in what way the airfoildynamics affect the formation and growth of the insta-bility and therewith the dynamic stall vortex. Since thestall development stage has been divided into two partscharacterised by a distinctly different behaviour of theshear layer it seems usefull, even necessary, to threatboth stages individually.

The first part of the stall process is termed the pri-mary instability stage and has been identified as the partof the dynamic stall life cycle that covers the initial devel-opment of a recirculation region on the airfoil’s suctionside and the formation of small-scale positive vorticesat the interface as a result of a primary instability of theshear layer. The flow is conveniently considered quasi-static for angles of attack up to the static stall angle.Hence, the start of the primary instability stage is set att = tss, i. e. the moment the angle of attack is increased

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Table 1: Overview of the calculated dynamic stall quantities for different oscillation parameter combinations.

k α0 [°] α1 [°] αds1 [°] ∆t1 [s] × 10−2 αds2 [°] ∆t2 [s] × 10−3

0.050 18 6 23.97(5) 5.38(55) 24.04(1) 4(3)0.100 18 6 23.65(12) 3.86(20) 23.94(8) -

0.050 18 8 25.35(10) 4.37(19) 25.75(5) 8(3)0.100 18 8 25.94(6) 3.07(20) 25.81(6) 10(2)

0.050 20 6 24.97(9) 4.49(16) 25.53(11) 12(1)0.075 20 6 25.60(6) 3.84(11) 25.99(2) 11(2)0.100 20 6 25.83(10) 3.37(24) 25.89(6) 11(2)

0.050 20 8 25.87(20) 3.92(22) 26.59(8) 8(1)0.075 20 8 26.48(27) 3.03(23) 27.44(13) 9(3)0.100 20 8 27.20(14) 2.83(13) 27.95(4) 9(2)

0.050 22 6 25.47(21) 4.36(27) 26.19(9) 9(3)0.100 22 6 26.73(11) 3.04(9) 27.61(7) 9(1)

0.050 22 8 26.01(13) 3.63(12) 27.06(6) 9(2)0.100 22 8 27.94(19) 2.75(11) 29.03(16) 7(1)

The number in between the brackets is the standard deviation of the data.

beyond static stall. Its upper bound is given by tds1.The characteristic time delay attributed to the primaryinstability stage is denoted by ∆t1 and is calculated forall parameter combinations relevant to this study. Theassessed values are listed in table 1.

Considering the behaviour of ∆z in function ofangle of attack, i. e. comparing αds1 for the prevailingparameter combinations, yields the conclusion that thetransition into the vortex formation stage is postponedto larger angles of attack with increasing frequency.However, comparing ∆t1 for the different degrees ofunsteadiness reveals that the phase transition is actuallypromoted rather than delayed with increasing fosc forconstant mean incidence and oscillation amplitude. Re-garding α0 and α1, their impact can not be investigatedseparately since they are interrelated parameters for theairfoil’s motion and its derivatives.

The rate of change of the angle of attack1, i. e. thefirst time derivative of α(t), is a measure for the instanta-neous unsteadiness and is influenced by the oscillationfrequency as well as by the mean incidence and the os-cillation amplitude. The prevailing airfoil motion beingsinusoidal, the rate of change of the angle of attack isinherently time-dependent, and moreover nonlinear.

Furthermore, the unsteady separation process com-prises a series of events governed by much smaller timescales compared to the oscillation period. Hence, inorder to outline the impact of the dynamic effects, aninstantaneous effective measure for the unsteadinessshould be considered rather than the reduced frequencywhich is only an overall measure. Since the role of the

dynamic effects is generally considered minor up to thepoint where the angle of attack is increased beyond thestatic stall angle, the relevant instantaneous effectiveunsteadiness is determined here by the rate of change ofα at t = tss, denoted by αss. In doing so the interrelatedparameters i. e. mean incidence, amplitude, and oscil-lation frequency are gathered into a single parameterallowing for a more general investigation of the relationbetween unsteadiness and stall delay.

Presenting ∆t1 in function of the normalised ef-fective unsteadiness αss c/U∞ in figure 6(a) reveals amonotonically decrease of the stall delay with increasingunsteadiness. Again a different picture emerges whenregarding the dependence of the angle of attack αds1

on the unsteadiness (figure 6(b)). Although the firstpart of the stall delay in terms αds1 increases more orless nonlinearly with αss c/U∞ as predicted by Johnsonand Ham [9], the incidence angle is considered not anadequate indicator for the unsteady processes and theactual time delay is preferred. Hence, it is suggestedhere that the unsteadiness speeds up the first part of thestall process rather than delaying it.

In figure 6 there seems to be one data point thatis out of line with the others, namely the parametercombination α0 = 18°, α1 = 6°, and k = 0.10 depictedby the open gray circular symbol ( ). This particularparameter combination yields a dynamic stall case thatis also referred to as a ’light stall’ case. A light stallcase can be recognised by the fact that the oscillationdirection is changed before the dynamic stall onsetangle is reached. Hence, massive flow separation is no

1 α(t) =dα(t)

dt= 2πfoscα1 cos(−

π

2+ 2πfosct).

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(a)

30

40

50

60

∆t 1[m

s]

(b)

0 0.005 0.01 0.015 0.02 0.025

24

26

28

αss c/U∞

αds

1[°]

Figure 6: Influence of the rate of change of the angle of attackat t = tss on (a) the stall delay ∆t1 and (b) the angle of inci-dence αds1 for different oscillation parameter combinationsindicated by different markers (cf. table 1). Error bars indicatethe standard deviations.

longer initiated on the upstroke but is forced to occuron top of the cycle when less circulation is present. Asa consequence, the height of the viscous zone enclosingseparated flow is of the order of the airfoil thickness,and not of the order of the airfoil chord as for a deepstall case (cf. [12]).

4.3 Vortex Formation Stage

The second part of the stall development stage imme-diately following the primary instability stage is termedthe secondary instability or the vortex formation stage andis characterised by the roll up of the shear layer into alarge-scale dynamic stall vortex. The process leadingto the formation of the primary stall vortex is identi-fied as an instability of the shear layer that confines thereversing flow near the airfoil’s surface.

Because a large portion of the overall vorticity isconcentrated in the shear layer, the instability can also beseen as an instability of the vorticity distribution. Slight

displacement of small vorticity packets and the gener-ation of extra vorticity will alter the induced velocityfor existing vortices and vortex lines, and consequen-tially change the vorticity pattern and the interactionsbetween the viscous and inviscid parts of the flowfield.In particular, due to the altered process of production,convection and diffusion, the primarily small-scale shearlayer vortices deviate from their initial trajectories, ex-ert stronger mutual interactions and merge into largercoherent structures.

Furthermore, stronger interactions between theviscous-dominated shear layer and the reversed flowunderneath it yield the emergence of counter-rotatingvortices near the airfoil’s surface. The presence of thesenegative vortices is indicated by the Eulerian picture(e. g. figure 4(b)). These vortices will play an essentialrole in the process leading to the dynamic stall onset. asthey will be induced towards leading edge and lead tovortex induced separation

The dynamic stall onset angle of attack, definedpreviously from the experimental data based on a char-acteristic POD mode [14], is denoted by αds2 and thetime delay associated with the vortex formation stage isindicated by ∆t2. The assessed values of the onset an-gle of attack and the corresponding stall delay for theprevailing parameter combinations are also included intable 1.

The second contribution to the dynamic stall delayis found to be approximately constant for the presentedparameter combinations (see also figure 7). Apparently,this part of the stall development is no longer influencedby dynamic effects, and the vortex formation process issolely governed by viscous interactions. Notice that forso-called ’light stall’ case given by the parameter combi-nation α0 = 18°, α1 = 6°, and k = 0.10, there is no vortexformation stage, consequentially no ∆t2.

0.01 0.015 0.02 0.025

2

6

10

14

αss c/U∞

∆t 2[m

s]

Figure 7: Dynamic stall delay attributed to the vortex formationstage for different oscillation parameter combinations indi-cated by different markers (cf. table 1). Error bars indicate thestandard deviations.

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5 conclusion and future work

Time-resolved velocity field information and unsteadysurface pressure distributions were gathered duringwind tunnel experiments and elaborated by an extensivecoherent structure analysis. The characteristic featuresof the unsteady flow field were identified andanalysedutilising an Eulerian and a Lagrangian procedure thatallow for the localisation of the axes of vortices and theedges of Lagrangian coherent structures, respectively.The complementary information obtained by both meth-ods provided deeper insight into the spatiotemporalevolution of vortical structures within a single dynamicstall life cycle, and in particular within the stall develop-ment stage.

The classical hallmark of the dynamic stall phe-nomenon is the dynamic stall vortex. The physicalmechanisms leading to the formation of this primarystall vortex were identified as the emergence and sub-sequent roll up of a local shear layer. Accordingly, thestall development stage was subdivided into two partsdistinguished by their different dynamics: a primaryinstability stage followed by a secondary instability orvortex formation stage. During the former stage a re-circulation region emerges on the airfoil’s suction sideas the result of a persisting unsteady adverse pressuregradient. At the interface between the reversed flow andthe free stream flow small-scale shear layer vortices wereidentified by the Eulerian method. While observing onlyweak interactions during the first instability stage, thesecond stage was associated with the roll up of the shearlayer into a large-scale dynamic stall vortex. The tran-sition between the two stall development regimes wasdetermined by a discontinuity in the development ofthe shear layer height. Furthermore, it was revealed thatthe dynamic stall delay attributed to the first instabil-ity stage decreases with increasing unsteadiness. Thedelay attributed to the vortex formation stage on theother hand was observed to be virtually unaffected byvariations in the airfoil’s pitching motion.

Due to the inherent unsteady nature of the dy-namic stall process, the time-resolved approach has tobe regarded the preferential approach. However, inorder to further improve dynamic stall modelling andprediction, the parameter range should be extended, en-compassing Reynolds number variations and alternativeairfoil geometries.

Acknowledgements This work has been part of the DLR andONERA joint project: Advanced Simulation and Control ofDynamic Stall (SIMCOS). The authors thank H. Mai, T. Bute,J. Nuhn and A. Henning for their contribution to the windtunnel measurements.

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