InvestigationsonQuasi …downloads.hindawi.com/journals/ijae/2010/940528.pdfWind tunnel experiments were conducted on NACA-0015 airfoil model to investigate the eﬀect of the reduced

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2010, Article ID 940528, 11 pagesdoi:10.1155/2010/940528

Research Article

Investigations on Quasi-Steady Characteristics for an AirfoilOscillating at Low Reduced Frequencies

Deepakkumar M. Sharma and Kamal Poddar

Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh 208016, India

Correspondence should be addressed to Deepakkumar M. Sharma, [email protected]

Received 29 August 2009; Revised 10 May 2010; Accepted 20 September 2010

Academic Editor: Hui Hu

Copyright © 2010 D. M. Sharma and K. Poddar. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Wind tunnel experiments were conducted on NACA-0015 airfoil model to investigate the effect of the reduced frequency(κ = 0.0001 to 0.5) and that Reynolds number (Re = 0.2E06 to 0.7E06) on the aerodynamic characteristics and hysteresis behaviorassociated with the oscillating motion of the airfoil. Pressure measurements were conducted on the midspan of the airfoil forquantitative results. Although dynamic stall study has been conducted to greater extent of reduced frequencies, for current work,emphasis is made on investigating the low reduced frequency regimes up to which the steady conditions prevail under dynamicmode of operation. In present investigations of oscillating sinusoidal α = 10◦ + 15◦ sin(ωt) motion, the airfoil was allowed toexecute oscillating motion from static to almost steady (quasi-steady) conditions with the incremental increase in the reducedfrequency. The limiting conditions of κ and Re for quasi-steady state are carefully extracted and analyzed. Static and quasi-steadyconditions are critically assessed in context of the existence of laminar separation bubble within the opted Re regime. The flowseparation was found to be trailing edge flow separation with existence of LSB upstream towards the leading edge with no evidenceof unsteady flow reversals. Normal force defect and pitch damping factor are estimated for varied range of Re and κ within thequasi-steady conditions. Re effect is dominant at steady and quasi-steady regimes while the effect of reduced frequency is negligiblein this domain. Minor hysteresis effect associated with the stalling and reattachment point is due to existence of weak stall flutterwhich is further restrained by the oscillatory motion of the airfoil. Laminar separation bubble traced during steady condition isconserved in quasi-steady domain as well.

1. Introduction

Dynamic stall phenomena are the result of airfoil andwing undergoing ramp or oscillatory motion and havinga maximum angle of attack greater than the static stallangle. These unsteady flows are characterized by massiveseparation and formation of large-scale vortical structures.Numerous experimental and computational investigations[1–16] have shown that the unsteady flow can be separatingor reattaching over a large portion of the top surface ofthe airfoil. The predominant feature of dynamic stall is theformation, shedding, and convection over the upper surfaceof the airfoil of an energetic vortex-like disturbance fromthe leading edge of the airfoil. It induces a nonlinearityfluctuating pressure field and produces transient variationsin forces and moments that are fundamentally different fromtheir steady state counterparts.

As shown in Figure 1 [17], in steady flow, separation ofa boundary layer is said to occur where the boundary-layerflow velocity near the surface is reduced to zero and theflow detaches from the wall to form a wake that encloseda reversed-flow region. The separation and flow reversalpoints are coincident, and the external stream is alwaysdistorted. In unsteady flow as shown in Figure 2 [17], thelocations of the point of flow reversal and the point of flowseparation are two distinctly different points. Goldstein [18]in his theoretical studies of laminar flows has shown that theboundary layer equations have a singularity at the point offlow reversal in steady flow. But in unsteady flow as shown bySears and Telionis [19], the location of singularity associatedwith separation is distinctly different from the locationof flow reversal. Thus, there exists an unsteady boundarylayer that has a region of reversed flow near the surface,while not showing any strong variations in the boundary

2 International Journal of Aerospace Engineering

Typical velocity profileEdge of boundary layer

Separation pointReversed flow region

Wake region

Figure 1: Steady separation on an airfoil [17].

Edge of boundary layer

Typical velocity profile

Reversed flow boundary

Wake region

Figure 2: unsteady flow reversal on an oscillating airfoil [17].

layer displacement or momentum thickness and little or nodistortion of the external stream. This distinction betweenflow reversal and separation has also been demonstrated byNash et al. [20] for unsteady turbulent boundary layer flows.

Two quantities which summarize the degree of hysteresisassociated with dynamic stall are given by the normal forcedefect (ξ) and the pitch damping factor (ζ). ξ is obtainedfrom the close line integral around the cycle of the normalforce, and ζ is obtained from the closed line integral of thepitching moment; in both cases the variable of integration isangle of attack α [21]:

ξ =∮Cndα

∫ π/2−π/2 Cndα

, ζ = −∮

Cmdα/4α12, (1)

Cn = 1c

[∫ c

0

(Cpl − Cpu

)dx]

,

Ca = 1c

[∫ c

0

(

Cpudyudx

− Cpldyldx

)

dx

]

,

∴ CmLE = 1c2

[∫ c

0

(Cpl − Cpu

)x dx

],

∴ Cm = CmLE +Cn

4

(moment about quarter-chord

).

(2)

The normal force defect represents the loss in loadcarrying ability arising from the fact that the normal forceexperienced during the pitch-down is usually less than thatduring the pitch up. If there is no hysteresis, both portions ofthe cycle trace out an identical curve, and the defect is zero.The numerical value assigned to this hysteresis is thereforebased on that percentage of the integral contributed during

NACA0015 airfoil model

Oscillating gear

Figure 3: Experimental Set-up.

XX

x/c = 1

LE TE

NACA0015

Y

x/c = 0.25

s/c

Figure 4: Pressure port locations of NACA0015.

the upstroke which is lost during the downstroke. The pitchdamping term, which is more frequently reported, has adifferent interpretation. Since the pitching moment acts inthe direction of motion of the airfoil, the closed-line integralof the pitching moment with respect to α provides a measureof the net accumulation of energy from the flow. Hence, ifthe normal force and pitching moment trace out clockwisepatterns when viewed as a function of α, then the normal-force defect will be positive, and the pitch damping will benegative. Within quasi-steady conditions, values of ξ andζ are quantified at varied Re and κ. Cn − Cm correlationplots are formed by computing following nondimensionalparameters [17]:

χ = Cn − Cn0

Cnmax − Cn0

, γ = Cm − Cm0

Cmmax − Cm0

, (3)

where Cn0 and Cm0 are taken at α = −4◦. This nondimen-sionalization emphasizes the chronology of the significantevents occurring during fully developed dynamic stall. Itsignifies the strength of hysteresis and identifies traces ofweak or strong vortices or fluctuations generated during theoscillation cycle for a wide range of κ and Re, thus forms avaluable technique for correlating dynamic stall test results.

The objective of current work is to determine andinvestigate the conditions for quasi-steady state. It couldbe defined as an unsteady state up to which steady flowconditions and static stall characteristics prevail. Thus, forgiven quasi-steady flow conditions, although the flow isunsteady, the boundary layer separation should have aunique singularity with no flow reversals, and moment stalland lift stall are presumably the same point. Also, as stated byEricsson [22], both the frequency-induced camber (q-effect)

International Journal of Aerospace Engineering 3

Flow

Airfoil model

Scanner 1

Scanner 2

Lineinterfacesystem

CPUPXIunit

230 V AC

Labviewbased

applicationsoftware

Power supply. Power supply.±5 V DC, ±12 V DC

Figure 5: Measurement chain for surface pressure data acquisition.

T′

T′′

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

Samples

8000 200 400 600

LE

TE

LBL

LSL

TSL

Retarding flow

100 300 500 700

TBLR

S

Time series CP signal (fluctuating)

T

Uppersurface ports

s/c

Figure 6: Time series CP signals (fluctuating) at Re = 0.2E06 andα = 9.5◦.

and the frequency-induced plunging (α̇-effect) tend to zero.The set of very low-ordered reduced frequencies at varied Reare extracted and critically assessed to limit the quasi-steadyconditions.

2. Experimental Methodology

Figure 3 shows the designed experimental set-up whichgenerates simple harmonic oscillating motion for the givenairfoil. The preferred mechanism (Scotch-yoke) is robustand capable of functioning at predefined inertial and fluiddynamic loads by giving due considerations to all its oper-ating parameters. The airfoil model shaft passing throughquarterchord is mounted on an oscillating gear perform-ing sinusoidal motion. Limiting to the cross-sectional size

x/c

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

LSB

St.P

R

S

T′

T′′

CP distribution

CP

T

CP upperCP lower

(a)

0

0.1

0.2

0.3

0.4

x/c

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

α = 9.5U∞ = 10 m/s

T′ T′′

Std.

devi

atio

nin

CPu

Std. deviation in CPu

T

Re = 0.2E06

(b)

Figure 7: CP distribution and its standard deviation at Re = 0.2E06and α = 9.5◦.

for one of the existing wind tunnel test section (1.22 m× 0.305 m), available at low speed aerodynamics lab ofaerospace engineering department at IIT-Kanpur, the chordlength of the selected NACA0015 airfoil is kept 0.31 mspanning the tunnel width, keeping the geometric blockageup to 10% at maximum angle of attack of 25◦. Free streamvelocity (U∞) ranges from 10 to 35 m/s. Accordingly, Reranges from 0.2E06 to 0.7E06 with κ ranging from 0.001 to0.5 for oscillation frequency ( fo) varying from 0.01 to 5.4 Hz.The free-stream turbulence level in the test section is within0.15%.

2 ESP Scanners (each having 32 ports) are used tomeasure the surface pressure all along the 60 ports opted onthe mid-span of the airfoil model along the chord as shownin Figure 4. The multiplexing frequency of each scanner is20,000 Hz, and the accuracy is of 0.05% for the full rangeof ±5000 Pa. Figure 5 indicates the detailed measurementand instrumentation chain used for the surface pressuremeasurements. For both static and dynamic analysis, thetime series pressure data were acquired with a sampling


Table 1: Nomenclature.

Description Symbol Value Unit

Reduced frequency κ ωc/2U∞ —

Chord length c m

Circular frequency ω 2π fo rad/s

Oscillating frequency fo Hz

Free stream velocity U∞ m/s

Reynolds number Re U∞c/ν —

Kinematic viscosity ν 1.5E − 5 m2/s

Angle of incidence α deg.

Amplitude of oscillations α1 15◦ deg.

Time t s

Normal force defect ξ —

Pitch damping factor ζ —

Normal force coefficient Cn —

Axial force coefficient Ca —

Pitching Moment coefficient Cm —

Co-ordinates of airfoil profile x, y m

Coefficient of Pressure CP 2(p − p∞)/ρU2∞ —

Local static pressure p Pa

Free stream static pressure p∞ Pa

Upper and lower surface u, l —

Leading and Trailing edge LE, TE —

Distance measured along thesurface

s m

Coefficients of lift and pressuredrag

Cl, Cdp —

Laminar Boundary Layer LBL — —

Turbulent Boundary Layer TBL — —

Leading Edge Vortex LEV — —

Laminar Separation Bubble LSB — —

Laminar Separated Shear Layer LSL — —

Turbulent Separated Shear Layer TSL —

rate of 500 samples/sec/per port. The steady time-seriesdata taken for 10 seconds is averaged to get the mean CP

distribution at a given angle of attack. The standard deviationin the time seriesCP signals is obtained which helps in tracingthe evidence of LSB on the upper surface of the airfoil. Thedynamic time series Cp data is phaseaveraged for 50 cycles ofoscillation. Uncertainty in the calculation of surface pressurecoefficients (Cp) is about 0.1%.

3. Results and Discussions

3.1. Steady State Analysis

3.1.1. Tracing the Evidence of LSB. Figures 6 and 7 showsthe sample procedure for identifying the occurrence of

Zone of LSB

TSLLSL

Tran

siti

on

α = 9.5◦

St.P

LE 5 8 10 14 17

RS

T′ T′′T

Re = 0.2E06

Figure 8: Surface flow visualization.

LSB in the flow phenomena by analyzing the fluctuatingcomponents of the acquired time-series CP signals at themid-span of the airfoil. The mean and the Standard deviationof the CP signals are also shown in CP distribution andstandard deviation plots, with respect to chord length(x/c) for α = 9.5◦ and Re = 0.2E06. Attempt is made tocorrelate the larger magnitudes of disturbances observed intime series CP signals with CP distribution and standarddeviation plots. The stagnation point (St.P) is at the lowersurface (Port ID L3). From there along the surface, flowaccelerates and initiates the growth of laminar boundarylayer. Further it crosses LE and reaches CP min up to whichfavorable pressure gradient holds (suction peak). The flowwithin the boundary layer then initiates to retard due toadverse pressure gradient and just aft of the suction peakafter port ID U5, a sudden deviation is observed in CP

plot with no significant change in pressure up to port IDU8. Port ID U5 is identified as Separation point S. Still theflow within the boundary layer is laminar as no deviationis observed in standard deviation plot. Separated laminarshear layer gets developed and extends up to port ID U10. Atport ID U8, the standard deviation in CP signal just beginsto grow and reaches the maximum at port ID 10. PortsID U8 and U10 are identified as onset T′ and extent T ofTransition. Laminar to Turbulent transition occurs at pointT . As the turbulent shear layer grows, it sparks the turbulentmixing and entrainment of fluid from the free-stream untilthe pressure reaches the value of turbulent boundary layeron the surface. T′′ indicates the point where entrainmentprocesses ends (port ID U14) and the separated turbulentshear energizes to reattach on the surface at port ID U17defined as reattachment point R. The entire chronology ofthe boundary layer events, viz, S-T′-T-T′′-R, defines thezone of LSB.

Figure 8 shows the surface oil-flow visualization imagewhich is clearly distinct and identifies boundary layer


LSB

5 8 10 14 17

9.5◦U∞ = 10 m/s

RS

LE

T′ T′′T

BL

Figure 9: Schematic diagram of mean flow with LSB.

05

1015

2025

1

0.8

0.6

0.4

0.2

0

Angle of attack (deg)

Trace of LSB

0.1

0.2

0.3

x/c

Std.

devi

atio

nin

CP

−5

Figure 10: 3D plot of stander. deviation in CP .

events. This technique has a great potential in identifyingdistinguished flow features on the surface. The skill lies inpreparing the mixture of TiO2 + oleic acid + transformer-oilwith definite proportions and tactfully applying it on thesurface of the airfoil. The chronology of events determinedbetween the separation and reattachment points quantifiesthe band identified as the zone of LSB. For the given caseat Re = 0.2E06 and α = 9.5◦, the length of LSB along thesurface (s/c) comes out to be =22.79% of the chord. Thevery existence of LSB classifies the flow separation to be atrailing edge turbulent boundary layer separation precededby a laminar separation bubble. The results obtained throughthis technique are rightly compared with the boundarylayer and laminar separation bubble parameters determinedand validated with certain boundary layer approximation

Separated flow

Bubblebursting

Abruptstalling

30

−100

1020

α

Pitch-up Pitch-down

90−90

−50

−25 0 25 50 75 270

125

150

175

200

225

250

TE

U01

U29

LSBLSB

LEV LEV

LE

Phase angle Phase angle

κ = 0.0001

RS

s/c

Re = 0.2E06

(a)

Separated flow

Bubblebursting

Abruptstalling

30

−100

1020

α

Pitch-up Pitch-down

κ = 0.002

TE

U01

U29

LSBLSB

LEV LEV

LE

RS

90−90

−50

−25 25 50 75 270

125

150

175

200

225

250


0

s/c

Re = 0.2E06

(b)

Figure 11: Time series CPu distribution for κ = 0.0001 and 0.002 atRe = 0.2E06.


−5

−4

−3

−2

−1

0

1

10 0.2 0.4 0.6 0.8

Abrupt stall

φ = 22.024α = 15.625

κ = 0.0001

Stalling point

x/c

CP

Re = 0.2E06

(a)

κ = 0.0001

10 0.2 0.4 0.6 0.8

x/c

Reattachment point

φ = 158.39α = 15.525

−5

−4

−3

−2

−1

0

1

CP

Re = 0.2E06

(b)

−4

−3

−2

−1

0

1

Abrupt stall

x/c

10 0.2 0.4 0.6 0.8

κ = 0.002

φ = 22.96α = 15.85

−5

CP lower

CP

CP upper

Re = 0.2E06

(c)

−4

−3

−2

−1

0

1

x/c

10 0.2 0.4 0.6 0.8

κ = 0.002

φ = 162.24α = 14.575

−5

CP

CP lowerCP upper

Re = 0.2E06

(d)

Figure 12: CP Distribution plot at stalling and reattachment points for κ = 0.0001 and 0.002 at constant Re = 0.2E06.

method adapted for the boundary layer solution of theNavier-Stokes equation [23].

Figure 9 represents the proportionate schematic diagramof the mean flow structure with the evidence of LSB andits distinct features. Location of the boundary layer eventsshown in % of chord is measured along the surface startingfrom the stagnation point.

Figure 10 shows the 3D plot for the standard deviationsin CP-Signals of the upper surface of the airfoil for α = −5◦

to 25◦ at Re = 0.2E06. With the increase in angle of attack,the growth of transition peak and hence the LSB are clearlydepicted from these plots.

4. Investigation and Determination ofQuasi-Steady State

The steady state analysis does identify and quantify the zoneof LSB existing for the given flow configurations. The flowseparation is classified under Trailing edge flow separation

with evidence of LSB on upstream. Further assessments arecarried out by varying κ and Re, and quasi-steady flowconditions are critically investigated and determined.

5. Effect of κ at Constant Re

The lowest possible reduced frequency of κ = 0.0001 is optedfor a given Re = 0.2E06. Further, step by step, sequentially allthe higher values of κ are critically analyzed until the frameof time-series CP distribution shows considerable changewhen compared with κ = 0.0001. It has been found that thetime-series CP distribution up to κ = 0.002 shows closuresimilarity with κ = 0.0001, thus taken for further analysisto determine the limit of quasi-steady state for a given Re.Figure 11 shows the time-series CPu distribution for κ =0.0001 and 0.002. The symmetric trend in CP plots of pitch-up and pitch-down assures the resemblance of flow to thesteady state for both the cases.


1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.2

α

25−5 0

0

5 10 15 20

α

25−5 0 5 10 15 20

LEVs

Cn

Cm

11

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.1 0 0.25 0.5 0.75χ

−0.02

−0.04

−0.06

−0.08

−0.1

κ κ

0.00010.002

0.00010.002

(a)

(b) (c)

Re = 0.2E06

Re = 0.2E06

Re = 0.2E06

Figure 13: (a) Cn versus α, (b) Cm versus α, (c) x − γ plot; for κ = 0.0001 and 0.002 at constant Re = 0.2E06.

For κ = 0.0001, the strength, onset, and extent of LSB inpitch-up motion are retained in pitch-down motion as wellwithout any phase lag. Similar trends are also observed forκ = 0.002. Effect of reduced frequency change is negligibleon LSB characteristics. The suction peak, stalling point,the separation zone, and the reattachment point by largeappear identical but need to be closely looked into for morediscriminations.

Once the airfoil is stalled, large disturbed wakes areformed since the movement of LEVs continued for a definitephase period (Δφ ≈ 23.3◦ and 21.18◦ for κ = 0.0001and 0.002, resp.). However, as evident from the time-seriesCPu plot, once the LEVs completely shed from the surface,the mean flow near the surface becomes virtually quiescent,leaving only random fluctuations due to local turbulence.Large wake is formed, and thereafter the flow remains fullyseparated up to Δφ ≈ 88.32◦ and 91.25◦ for κ = 0.0001and 0.002, respectively. During pitch-down motion, LEVsare reintroduced prior to reattachment and continued forconsiderable phase period (Δφ ≈ 25.29◦ and 26.03◦ forκ = 0.0001 and 0.002, resp.). The stalling and reattachmentpoints are identified for both the cases, and their chord-wise CP distribution plots are shown in Figure 12. For κ =0.0001 negligible phase lag is observed between stalling andreattachment points (Δφ ≈ 0.414◦) but for κ = 0.002

incremental phase shift is observed. The onset of stallingis delayed by Δφs ≈ 0.94◦, and reattachment point isdelayed by ΔφR ≈ 3.85◦ to sum up total phage lag ofΔφ ≈ 4.79◦ for the given change in κ. No change insuction peak (CP ≈ −3.4) is observed at stalling andreattachment points for both the cases of κ. The effect ofreduced frequency is also visible to cause incremental risein the fluctuations of pressure waves due to the inception,movement, and shedding of LEVs during pitch-up and pitch-down motion. The fluctuation pattern of CPu for κ = 0.002 asshown in Figures 12(c) and 12(d) indicates intensity and themovement of LEV towards downstream. This is also evidentfrom the Cn − Cm cross-plots as shown in Figure 13(c)where the progressive movement of cloud of repeating loopsindicates the presence, strength, and movement of LEVs. Thestrength of LEV shows incremental rise with increase in κ.The Cn and Cm plots as shown in Figures 13(a) and 13(b)are almost identical for both reduced frequencies. It showsminor hysteresis effect due to minimal phase lag associatedwith stall behavior as discussed earlier. Also this part of thestall loop is insensitive to the rate of change of angle of attackα̇ and remains even when α̇ tends to zero, for κ = 0.002,normal force defect ξ = 2.27%, and pitch damping factorζ = −0.0031. Thus stall flutter of very low amplitude andfrequency is incepted and is prolonged due to very low α̇ until


30

−100

1020

270

125

150

175

200

225

25090−90

−50

−25 0 25 50 75


αRαS

κ = 0.0002

Pitch-downPitch-up

LEVLEV

Separated flow

LSB LSB

LE

TE

U01

U29

Abrupt stall

αRe = 0.2E06

s/c

(a)

30

−100

1020 αR

κ = 0.0002

Pitch-downPitch-up

LEV

LEV

Separated flowLSBLSB

LE

TE

U01

U29

Abrupt stall

s/c

α= 0.7E06

Gradual stall

αS1αS2

270

125

150

175

200

225

25090−90

−50

−25 0 25 50 75


Re =

(b)

Figure 14: Time series CPu distribution for κ = 0.002 at Re = 0.2E06 and 0.7E06.

it get restrained and hence promotes the minor quasi-steadyhysteresis. The quantum of varied phase lags, strength ofLEVs, and stall flutter associated with the change in κ is verylow as compared to larger domain of unsteady characteristicsand hence does not seriously affect the stall characteristicsand are widely comparable to its steady counterpart. Thusκ = 0.002 could be considered as the limiting condition fordefining quasi-steady state at Re = 0.2E06.

Similarly, limiting reduce frequencies for quasi-steadystate are identified and validated for all the selected rangeof Re and are summarized in Figure 17. Effect of Re for agiven reduced frequency in the domain of quasi-steady stateis assessed further.

6. Effect of Re at Constant κ

The effect of Re of dominant and obvious in quasi-steadystate, and the characteristics attained are comparable to itsbaseline characteristics at varied Re. A very low reducedfrequency, κ = 0.0002, is opted for analysis which lies wellwithin the limiting value defining the quasi-steady state forthe given range of Re. Here the two extreme Re are takenfor comparison, which through baseline tests are catego-rized under different stall regimes. As shown in Figure 14,Re = 0.2E06 undergoes abrupt stalling (αS) during pitch-up motion, and also its reattachment (αR) is instantaneousduring pitch-down motion. While, for Re = 0.7E06, gradualstall (αS1) followed by abrupt stall (αS2), referred as double

stall, is evident during pitch-up motion but during pitch-down motion, the reattachment (αR) is instantaneous. Fora given Re, the strength, onset, and extent of LSB in pitch-upmotion are retained in pitch-down motion as well withoutany time lag but an increase in Re causes significant changesin the characteristics of LSB. With the increase in Re, theonset of LSB and transition path is confined but the strengthof LSB considerably reduces and LSB gets extended furtherupstream closure to the vicinity of LE. For Re = 0.2E06,LSB has moved up to port ID-U08 (s/c = 11.88% of chordmeasured from the LE) and got stuck there to trigger bubblebursting followed by abrupt stalling. For Re = 0.7E06, theweak LSB is evident up to port ID- U05 (s/c = 6.84%). At thislocation LSB bursts but the energy impact due to burstingis considerably low as compared to the entrainment energyfrom the free stream thus results in the extension of TBL anddrop in CP , promoting gradual stall. Later on, due to furtherrise in α, TBL separates to promote abrupt stalling.

Here one thing is significantly noted that during pitch-up motion, in abrupt stall case, LEV is incepted after abruptstalling and prolongs for a definite phase period (Δφ ≈19.77◦) followed by a separation zone of Δφ ≈ 96.39◦

and reintroduction of LEV for a phase period of Δφ ≈22.31◦ prior to reattachment. But in double stall case, LEVis incepted during gradual stalling, and weak evidence ofLEV is tracked, for a phase period of Δφ ≈ 24.89◦, oncethe airfoil is abruptly stalled followed by a full separationzone of Δφ ≈ 66.24◦. Weak LEVs are reintroduced for aphase period of Δφ ≈ 25.29◦ prior to reattachment. Thus


−5

−4

−3

−2

−1

0

1

CP

Stalling point

Abrupt Stall

φ = 22.644α = 15.775

κ = 0.0002

10 0.2 0.4 0.6 0.8

x/c

Re = 0.2E06

(a)

−5

−4

−3

−2

−1

0

1

CP

10 0.2 0.4 0.6 0.8

x/c

φ = 161.34α = 14.8

κ = 0.0002

Reattachment point

Re = 0.2E06

(b)

−4

−3

−2

−1

0

1

CP

10 0.2 0.4 0.6 0.8

x/c

Cp lowerCp upper

Cp lowerCp upper

Gradual stall

Abrupt stall

α = 18.35

φ = 33.83

φ = 22.96α = 15.85

κ = 0.0002

−5

= 0.7E06Re

(c)

−4

−3

−2

−1

0

1

CP

10 0.2 0.4 0.6 0.8

x/c

Cp lowerCp upper

κ = 0.0002

φ = 160α = 17.5

−5

= 0.7E06Re

(d)

Figure 15: Cp Distribution plot at stalling and reattachment points for Re = 0.2E06 and 0.7E06 at constant κ = 0.0002.

increase in Re shows substantial reduction in the intensity ofLEV without any considerable change in the phase. This canalso be observed in the Cn −Cm plots shown in Figure 16(c).Figure 15 shows the chord-wise CP distribution plots at theonset of stalling and reattachment points for both the optedRe cases at κ = 0.0002. As shown in Figures 15(a) and15(b), the phase lag between the stalling and reattachmentfor Re = 0.2E06 is Δφ = φS − φR ≈ 3.98◦ with no changein its suction peak (CP ≈ −3.4). For the double stall caseat Re = 0.7E06, as shown in Figures 15(c) and 15(d), thephase lag between the inceptions of gradual and abrupt stallis Δφ = φS2 − φS1 ≈ 10.97◦. The reattachment CP-plot isfound to be more comparable to the onset of abrupt stall CP-plot with no evitable change in suction peak (CP ≈ −4). Alsothe phase lag between the two is found to be significant Δφ =φS2−φR ≈ 13.83◦, but when compared to the onset of gradualstall the phase lag is reduced to Δφ = φS1 − φR ≈ 2.96◦ with

a drop in suction peak as well from CP ≈ −4.8 to CP ≈−4. Thus, individually for each Re cases, there is not anyconsiderable phase lag between the onset of stalling andreattachment point. Figure 16(a) shows incremental rise inCn are reintroduced for a phase period of Δφ ≈ 25.29◦

prior to reattachment. Thus increase in Re shows substantialreduction in the intensity of LEV without any considerablechange in the phase. This can also be observed in the Cn −Cm plots are shown in Figure 16(c). Figure 15 shows thechord-wise CP distribution plots at the onset of stalling andreattachment points for both opted Re cases at κ = 0.0002.As shown in Figures 15(a) and 15(b), the phase lag betweenthe stalling and reattachment for Re = 0.2E06 is Δφ =φS − φR ≈ 3.98◦ with no change in its suction peak (CP ≈−3.4). For the double stall case at Re = 0.7E06, as shown inFigures 15(c) and 15(d), the phase lag between the inceptionsof gradual and abrupt stall is Δφ = φS2 − φS1 ≈ 10.97◦. The


1

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.2

κ = 0.0002

Cn

Cm

0.2E060.3E060.4E06

0.5E060.6E060.7E06

κ = 0.0002

LEVs

11

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.1 0 0.25 0.5 0.75χ

−0.1

−0.075

−0.05

−0.025

0

κ = 0.0002

α

25−5 0 5 10 15 20

α

25−5 0 5 10 15 20

(a)

(b)

(c)

Re

Figure 16: (a) Cn versus α, (b) Cm versus α, (c) x − γ cross plot; at κ = 0.0002 for varied Re = 0.2E06 to 0.7E06.

reattachment CP-plot is found to be more comparable tothe onset of abrupt stall CP-plot with no evitable change insuction peak (CP ≈ −4). Also the phase lag between the twois found to be significant Δφ = φS2 − φR ≈ 13.83◦, but whencompared to the onset of gradual stall the phase lag is reducedto Δφ = φS1 − φR ≈ 2.96◦ with a drop in suction peak aswell from CP ≈ −4.8 to CP ≈ −4. Thus, individually foreach Re case, there is not any considerable phase lag betweenthe onset of stalling and reattachment point. Figure 16(a)shows incremental rise in Cn and stalling angle with theincrease in Re. The minor hysteresis observed at lower Re getsreduced with the increase in Re. As shown in Figures 16(b)and 16(c), the peaks and valleys formed in Cm-plots andCn − Cm plots in the prestall region indicate the dominanceof LSB which further becomes flatter and flatter with the risein Re, reducing the impact of LSB and promoting the extentof turbulent boundary layer separation. Thus, by increasingthe Re beyond 0.7E06 to a certain critical value, onset of LSBcould get completely extinct from the flow phenomena. Theextent of separated flow (full stall conditions) reduces withthe increase in Re thus transforms large wake at lower Re tocomparatively smaller wake.

The normal force defect and pitch damping factorcalculated for each case of varied Re and κ within the quasi-steady domain are plotted and summarized in Figure 17.

7. Conclusions

The quasi-steady conditions for the varied range of Re and κare determined and summarized in Figure 17.

The limiting value of reduced frequency for quasi-steadystate decreases with the increase in Re. The effect of Re isdominant for a given κ but the effect of κ for a given Reis negligible. However for any Re and κ within the quasi-steady domain, the phase lag interpreted with the onset ofstall and reattachment is very low and comparable to itssteady counterparts. Also the onset of LEVs, weather weakfor strong, is incepted after stalling promotes boundary layerseparation followed by the formation of large wake. Thus theboundary layer separation and reattachment is analogous tosteady flow conditions.

Normal force defect ξ ≤ 2.27% and Pitch damping factorζ < 0 hint at instability in the oscillating motion with theinception of stall flutter. This may form minor quasi-steadyhysteresis which reduces with the increase in Re. The flowcondition of pitch-up motion by and large gets retainedduring the pitch-down motion without any considerableaffect on the stall characteristics comparable to its steadycounterpart.

The onset and extent of LSB, if evident for the given rangeof Re, are unaffected by any change in κ. Effect of Re does


0.025

0.005

0.01

0.015

0.02

0

0.001

0.002

0.003

Quasi-steady

ζξ

κ

−1E−2

−8E−3

−6E−3

−4E−3

−2E−3

0E+0

8E+51E+5 2E+5 3E+5 4E+5 5E+5 6E+5 7E+5

Re

Figure 17: Summary of results obtained from quasi-steady tests.

reduce the strength of LSB and increases its extent towardupstream without any change in its onset phase. In short, thetransition path is conserved for any change in Re and κ. Thisimpact of LSB and the transition path has been assessed inthe quasi-steady domain but needs to be critically assessedfurther for unsteady flow conditions.

Acknowledgments

The present paper is financially supported by the Aeronauticsand Research Development Board (ARDB) of India. Tech-nical support and assistance provided by the staff of Lowspeed aerodynamics lab and National Wind Tunnel Facility(NWTF) at IIT-Kanpur are gratefully acknowledged.

References

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[9] K. Biber and G. W. Zumwalt, “Hysteresis effects on windtunnel measurements of a two-element airfoil,” AIAA Journal,vol. 31, no. 2, pp. 326–330, 1993.

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[13] S. Mittal and P. Saxena, “Hysteresis in flow past a NACA0012 airfoil,” Computer Methods in Applied Mechanics andEngineering, vol. 191, no. 19-20, pp. 2179–2189, 2002.

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[15] M. R. Soltani and A. R. Davari, “On self-similar behaviour ofthe hysteresis loops in pitching motions,” Aeronautical Journal,vol. 108, no. 1086, pp. 427–434, 2004.

[16] W. J. McCroskey, “Unsteady airfoils,” Annual Review of FluidMechanics, vol. 14, pp. 285–311, 1982.

[17] L. W. Carr, K. W. McAlister, and W. J. McCroskey, “Analysis ofthe development of dynamic stall based on oscillating airfoilexperiments,” Tech. Rep. NASA TN D-8382, 1977.

[18] S. Goldstein, “On laminar boundary-layer flow near a positionof separation,” Quarterly Journal of Mechanics and AppliedMathematics, vol. 1, no. 1, pp. 43–69, 1948.

[19] W. R. Sears and D. P. Telionis, “Boundary-layer separation inUnsteady Flow,” SIAM Journal on Applied Mathematics, vol.28, no. 1, pp. 215–235, 1975.

[20] J. F. Nash, L. W. Carr, and R. E. Singleton, “Unsteady turbulentboundary layers in two-dimensional incompressible flow,”AIAA Journal, vol. 13, no. 2, pp. 167–173, 1975.

[21] K. W. McAlister, L. W. Carr, and W. J. McCroskey, “Dynamicstall experiments on the NACA 0012 airfoil,” Tech. Rep. NASATP 1100, 1978.

[22] L. E. Ericsson, “Comments on unsteady airfoil stall,” Journal ofAircraft, vol. 4, no. 5, pp. 478–480, 1967.

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