Aerodynamic Characteristics of Pitching and Plunging

Airfoils at Low Reynolds Number

Muhammad Saif Ullah Khalid, Imran Akhtar and Naveed Iqbal Durrani

Department of Mechanical Engineering

College of Electrical & Mechanical Engineering

National University of Sciences & Technology

Islamabad, 44000, Pakistan

March 14, 2014

Abstract

Numerical simulations for flow over pitching and plunging NACA0012 airfoil are performed

using a two-dimensional incompressible Navier-Stokes solver at 103 Reynolds number for

a range of Strouhal numbers. Identifying role of trailing-edge of airfoil like a bluff-body in

vortex-shedding phenomena, we present an equal St-criteria for comparing aerodynamic per-

formances of pitching and plunging airfoils. We observe comparable scales of aerodynamic

coefficients for both the kinematics in temporal as well as spectral domains. It performs

better than effective angle-of-attack criteria proposed in literature. Secondly, we compare

the contributions of various aerodynamic force-producing mechanisms like; vortex-shedding,

added-mass, wake-capture and interaction of leading and trailing edge vortices in case of

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pitching and plunging airfoils. Wake-deflection is one of these mechanisms. Nonlinear anal-

ysis of unsteady aerodynamic coefficients reveals that wake-deflection phenomena occurs

due to presence of quadratic nonlinearity in the aerodynamic response (measured in terms

of lift-coefficient) of oscillating airfoil.

1 Introduction

Wings and fins act as control surfaces in insects, birds and fishes, respectively, to produce

unsteady lift and thrust forces at Reynolds numbers varying from 103 to 105. These species

put effort through their muscles to perform certain kinematic combinations of plunging

(heaving), pitching, banking and stroke-reversals of their respective wings and fins. Most of

the studies in literature consider flapping as a combination of plunging and pitching motions

because of their commonality in most of the natural species. Considering each of these

kinematics independently, a rigid wing appears to be a forced oscillatory system excited by

single frequency. While performing these motions independently or in a combined form,

wing produces unsteady aerodynamic forces; oscillatory lift and thrust. Young et al. [1]

examined the existence of lock-in phenomena for flows over plunging airfoil at 1.2 × 104

Reynolds number (Re). This study encourages to examine flows over forced oscillatory wing

as nonlinear system. Some of very recent studies [2, 3] shed light on nonlinear characteristics

of these systems.

Although pitching and plunging represent different degrees-of-freedom of a wing yet they

produce similar periodic aerodynamic forces that may be termed as their response under exci-

tation frequency. Based upon similar nature of their responses, an equivalence may be estab-

lished between their respective forcing amplitudes and frequencies. Being highly nonlinear

systems, their responses should be analyzed rigorously. As angle-of-attack (α) varies sinu-

soidally with respect to time for both of these motions, McCroskey [4] discussed dynamic stall

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for vertical translation and pitching of wings about quarter-chord point (0.25c). In this study,

pitching-amplitude, equivalent to that of plunging, was calculated to be α(t) = tan−1(h/U∞)

where α(t), h and U∞ represent instantaneous pitching amplitude, instantaneous plunge-

velocity and free-stream fluid velocity respectively. He found similarity in lift-coefficient

(CL) versus α curves for both kinematics in case of deep stall for 0.15 reduced frequency

(k = ωc/U∞). For light stall conditions, this criteria of equivalence did not perform well

as described by Fukushima et al. [5] and Carta [6]. Following these investigations, recently

Sarkar [7] attempted to examine CL temporal profiles for pure-pitching and pure-plunging of

NACA0012 airfoil at Re = 104. In her study, mean angle-of-attack (αm) was taken as zero

and maximum plunge-equivalent pitch amplitude was calculated by α◦ = tan−1(h◦k) where

h◦ shows maximum amplitude of pure-plunge motion. Although responses of both forcing

types were periodic, yet pitching produced greater amplitudes for CL. Results of present

study also confirms these findings. Sarkar [7] attempted to consider velocity-profiles (as a

measure of thrust) in the wake of pitching and plunging airfoils, however, strong disagreement

was found. McGowan [8] and McGowan et al. [9] presented pitch-plunge equivalence for lift

force produced by oscillating SD7003 airfoil at Re = 104 using quasi-steady airfoil theory

and Theodorson’s formula for unsteady aerodynamic theory, two-dimensional Navier-Stokes

equations and experimental techniques. Various cases of high-frequency/low-amplitude and

low-frequency/high-amplitude with Strouhal numbers (St = fh◦/U∞) equal to 0.125 and

0.375 were considered in their study. They found agreement for pitch-equivalence to plunge

for low St cases and there was a strong disagreement for high St. They related this mismatch

to the vortex stretching and flow separation. For these kinematic conditions, they found lift

histories deviated from a sinusoidal profile.

For effective angle-of-attack criteria to compare aerodynamics of pure-pitching and pure-

plunging, amplitudes traversed by LE was kept equal for both the kinematics to establish

an equivalence of α(t), Young et al. [1] noticed the role of trailing-edge (TE) of NACA0012

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airfoil in the process of flow separation from its surface. They argued that trailing portion

of airfoil provided effective bluntness for associated vortex-shedding process. This bluntness

of airfoil’s TE varies with time during oscillation cycle.

Using this idea, we consider that TE of purely pitching and purely plunging airfoils should

undergo similar kinematic profiles to provided equivalent bluntness to on-coming flow. In

this context, we propose an equivalence between aerodynamic performance of pure-pitching

and pure-plunging airfoils by keeping St same for both the cases. Equivalent angle-of-attack

approach is fundamentally based upon dimensional analysis and equal-St criteria is proposed

here using kinematic similarity. Aerodynamic forces not only match in temporal domain but

also in their spectral compositions. The strength of presented criteria here is that it provides

good agreement for thrust force as well.

After finding similarity in overall magnitudes of aerodynamic forces from both pitching

and plunging airfoils, we analyze contributions from various mechanisms; vortex-shedding,

added-mass effect, wake-capturing and interaction of leading-edge vortices (LEV) with trailing-

edge vortices (TEV), to generation of aerodynamic force. To the best of our knowledge, no

published study is found in literature till date that presents a comparison of the relative ef-

fects of these mechanisms in aerodynamics of pitching and plunging airfoils. Hence, current

study also helps to highlight some fundamental differences in their aerodynamics.

We particularly focus on wake-deflection phenomena that occurs due to symmetry-

breaking bifurcation [10, 11, 12, 13]. It was observed by Jones et al. [14], Lai et al. [15],

Guerrero [16] and Zheng et al. [13] in their respective experimental and computational in-

vestigations. Due to this factor, unsteady response of CL gains non-zero time-averaged value

along with presence of even harmonics in its spectrum. Even harmonic causes deviation of

unsteady profile from a typical sinusoidal curve. Quadratic nonlinearity is responsible for

bringing asymmetry in the response of a nonlinear system [17]. We quantify the effect of

quadratic nonlinearity in wake-deflection phenomena. Liang et al. [2], Cleaver et al. [12]

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and Yu et al. [18] noted that an upward-deflecting wake causes greater negative pressure

on upper-surface of airfoil and time-averaged lift-coefficient, CL becomes positive. Similarly

negative CL is experienced by oscillating airfoil with wake deflecting downwards. Because of

symmetrical oscillation of airfoil about its mean position, vortices of equal strength are shed

in the wake per half oscillation cycle. Zheng et al. [13] examined the role of vortex-pairing in

wake-deflection phenomena. But they did not discuss its effect on aerodynamic coefficients

that is more important from practical point of view.

To study nonlinear characteristics of flows over oscillating airfoils, Young et al. [1] em-

ployed unsteady velocity-profiles in the wake. Similarly, Ashraf et al. [3] used CT to de-

termine periodicity, quasi-periodicity, and chaotic character of this flow. In present study,

we use CL to investigate the nonlinear character of these systems because vortex-shedding

frequency is related to it directly. Bluff-body aerodynamics studied in Ref. [19, 20, 21] shows

presence of fundamental harmonic (having vortex-shedding frequency) in CL-spectra along

with its odd harmonics. CT -spectra carry even harmonics as frequency of CT is twice that

of CL [2, 22].

The manuscript is organized as follows. Section 2 provides mathematical model and

strategy for numerical simulations. It also includes grid-convergence, time-step refinement

and validation studies. In section 3, we present results of our newly proposed equal-St based

criteria to set an equivalence between pure-pitching and pure-plunging of NACA0012 airfoil.

Nonlinear analysis of aerodynamic forces is also explained in this section. We also show

comparison of our results to those of previously mentioned effective angle-of-attack criteria.

Discussion for the effects of unsteady force producing mechanisms; wake-capture, vortex-

shedding, added-mass, and interaction of vortices in the wake are also included. In section 4,

we investigate the effect of growing quadratic nonlinearity (measured in terms of amplitude

of first even harmonic in unsteady CL-profile) on wake deflection phenomena.

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2 Numerical Methodology

For present study, we simulate flow past oscillating NACA-0012 airfoil by solving two-

dimensional incompressible Navier-Stokes equations using ANSYS-Fluent [25]; a finite-volume

based commercial software. Navier-Stokes in their integral form are :

∫V

∂ρφ

∂tdV +

∮ρφ~v.d ~A−

∮Γφ∇φ.d ~A−

∫V

SφdV = 0 (1)

where ρ is the density of fluid, ~v represents velocity vector, ~A shows surface-area vector,

Γφ is diffusion term, ∇φ denotes gradient term and Sφ shows the source term. In our case,

because we use moving-mesh technique [25], source-term is zero.

To avoid effect of disturbances on boundaries, radius of O-type domain is kept at 25c.

Flow domain is meshed using unstructured triangular cells. We maintain a high grid reso-

lution near airfoil surface so as to resolve boundary layer and capture wake characteristics

in the downstream direction of airfoil. 400 nodes are present on airfoil surface. Dynamic

meshing capability of ANSYS Fluent [25] is employed in the vicinity of moving airfoil for grid

transition adjusting itself in accordance with instantaneous position of airfoil. It includes

spring analogy and remeshing techniques [25].

Numerical solutions of flow-fields are highly dependent on the suitability and accuracy

of boundary conditions. In ANSYS-Fluent [25], motion of an object may be defined by

a user-defined function (UDF ) which is a computer code written in C-language environ-

ment coupled with Fluent-Macros. Kinematically, it follows a sinusoidal profile, defined as

h(t) = h◦ cos(2πft + φh) and α(t) = α◦ sin(2πft) where φh denotes phase to vary initial

position for plunging. Here, h(t) and α(t) show instantaneous plunging and pitch amplitude

respectively, h◦ and α◦ are relevant maximum amplitudes, and f shows excitation frequency

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Figure 1: Schematic of Geometry and Fluid-Domain

for oscillating airfoil. Dirichlet conditions are employed on the inlet boundary and pressure

outlet condition is used for outflow boundary. At this boundary, static pressure is specified.

For incompressible flows, pressure on the boundary is determined by taking average of the

specified values on the cell faces and computed values of static pressure on the correspond-

ing cell-centers. All other flow variables are computed by extrapolation of the computed

values from the inner domain. Reynolds number Re = ρU∞c/µ is calculated based on chord

(c), free-stream velocity (U∞) and dynamic viscosity (µ). Reynolds number is set using

appropriate values of viscosity, keeping all other parameters equal to 1. Strouhal number

(St = 2fh◦/U∞) is varied by changing oscillation amplitude while incoming reference flow

velocity and oscillation frequency are kept fixed. For present study, numerical simulations

are initiated by inlet velocity boundary conditions that is unity and zero velocity components

in horizontal and vertical direction respectively.

For grid-independence study, we perform numerical simulations for flow past a plunging

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Table 1: Details of Grid Convergence Study for Plunging NACA-0012 Airfoil with St = 0.25at Re = 103

Grid Size (Cells) Nodes on Airfoil Surface Time-Averaged CT67890 250 0.129282482 300 0.112998754 350 0.0862118440 400 0.0865139442 450 0.0865

NACA-0012 airfoil at Re = 103 at St = 0.25. Mesh size is changed through introducing more

nodes on airfoil surface. Fig. 2 shows CL and CT for one complete oscillation cycle of plunging

airfoil. It can be seen that temporal solutions for aerodynamic forces become independent

of grid-resolution when number of cells in the domain exceeds 1.18440 × 105. The values

for time-averaged thrust coefficient become independent of grid size after having 400 mesh

nodes on airfoil surface as presented in Table. 1. All present simulations are performed using

2000 time-steps per oscillation cycle of airfoil.

We perform validation study for our numerical strategy using computational investiga-

tions of Sarkar [7] and Guerrero [16]. Experimental investigations at lower Reynolds num-

bers are not usually found in literature. Studies like Ref. [26] were performed at relatively

higher Reynolds numbers. We show our results of unsteady CL for a pure-plunging airfoil

at Re = 104 with k = 7.86 and oscillation amplitude, ho = 0.05 taking Sarkar’s work [7] as

reference in Fig.3. Our results show good match with those from reference study. CT are

compared from the Guerrero’s study [16] in Fig.3d at Re = 1.1 × 103 for a range of St. In

this case too, our simulation methodology gives good confidence on the results.

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3 Pitch-Plunge Equivalence

Figure 4 shows pitching and plunging airfoils at their mean, maximum and minimum ampli-

tudes. For pitching motion of airfoil about its quarter-chord point, LE reaches its maximum

amplitude when TE is at its minimum position and vice versa. It signifies a phase-difference

in oscillations of LE and TE. This phase-difference is absent for a plunging airfoil. In present

study, St number is defined as St = 2fh◦/U∞ where f and h◦ shows oscillation frequency

and maximum displacement traversed by TE respectively. We calculate maximum pitching

angle by θ◦ = tan−1(h◦/0.75c) and mean pitching angle (θ◦) is zero to avoid giving an initial

bias to the flow.

To compare aerodynamic performance of pure-pitching and pure-plunging airfoils, we

perform numerical simulations for Re = 103 and a range of oscillation amplitudes (from 0.05c

to 0.50c) whereas excitation reduced frequency is kept constant at k = 3.142. Selection of this

reduced frequency is based upon observation of Andro et al. [24] where they claim this value

to be in mid-range of wing-oscillation frequencies for natural species. Temporal histories

and spectra for lift and drag coefficients; CL and CT are shown in Fig. 5 for St = 0.10.

Black solid lines in time-histories are for CL and CT of plunging airfoil, red solid lines

with square symbols show results for pitching from equal-St based criteria (α◦ = 7.5946◦)

and blue dotted lines with circles present aerodynamic coefficients of pitching airfoil whose

motion parameters (α◦ = 17.44◦) were calculated from effective angle-of-attack criteria. We

observe same orders of magnitude of these aerodynamic coefficients for both pitching and

plunging airfoils. Presently proposed criteria performs better than effective angle-of-attack

based equivalence. It is important to note that our criteria not only gives comparable CL

but also CT is similar for both the kinematics. This similarity is captured in their relevant

spectra as well that shows their nonlinear characteristics. Results of effective angle-of-attack

are found unmatched in spectral domain due to presence of sub- and super-harmonics of

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excitation frequency. These trends persist for other St values also. Another drawback of

previous criteria, mentioned in Ref. [7], is related to unmatched stream-wise velocity profiles

in the wake of pitching and plunging airfoils. These wake velocity profiles were presented

there as measure of thrust force produced by oscillating airfoils.

To highlight performance of our pitch-plunge equivalence criteria, we compare stream-

wise velocity profiles at a distance of 1.25c from TE of oscillating airfoil (St = 0.50) in its

wake at four positions of TE in Fig. 6. Solid lines represent horizontal velocity profile in

the wake of plunging airfoil and dashed lines are for those with pitching airfoil. It shows

good match of velocity profiles for two entirely different oscillatory degrees-of-freedom of this

nonlinear system.

Next using vorticity features of flow-field, we discuss reasons of having differences in

the amplitudes of CL and CT for these two kinematics. We relate the motion of leading-

and trailing-edges to convection of LEVs and TEVs. Vorticity contours for plunging and

its equivalent pitching are shown in Fig. 7. Although both motions generate aerodynamic

forces of similar order, yet vortices in case of plunging airfoil seem larger in size which

ultimately leads to production of slightly greater lift due to the fact that vortical regions

suggest low-pressure regions in the field.

For planar oscillations of a wing, three mechanisms contribute towards generation of

lift force [24], namely vortex-shedding, added-mass reaction, and wing-wake interaction.

McGowan et al. [9] considered circulatory and non-circulatory factors for this purpose.

Objective of their study was to investigate regimes where Theodorson’s formula might be

used to find contributions of pitch and plunge towards lift-generating capability of flapping

airfoils. Because it is based upon superposition principle applicable only to linear systems,

we consider earlier study [24] because they are related to unsteady characteristics of highly

nonlinear flow over oscillating airfoils. Quantification of these factors for aerodynamics of

oscillating airfoils is quite difficult task. We discuss mechanisms qualitatively for performance

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of pitching and plunging airfoils. We also discuss interactions of LEVs and TEVs that may

signify lift-producing capacity of airfoil at the end of stroke. For that matter, we need

to understand fundamental difference between these two types of motions. Plunging and

pitching of an airfoil are translational and rotational degree-of-freedom respectively.

3.1 Vortex-Shedding Phenomena

An impulsively starting airfoil motion may produce large-sized LEVs if its incidence is greater

than its stall-angle (αstall). For NACA0012 airfoil operating at Re = 103 attains static stall

when its angle-of-attack becomes 29◦ [27]. Andro et al. [24] predicted a plateau from two-

dimensional study. But highly unsteady flow features around static airfoil at this Reynolds

number is observed [24, 27]. Looking at Fig.7, large-sized hence stronger vortices are observed

in case of plunging airfoil. Similarly, pitching airfoil experiences dynamic stall due to forma-

tion and convection of LEVs [28]. Equivalent incidence for h◦ = 0.50 becomes 57.52◦ which

is greater than stall-angle of NACA0012. Presently proposed criteria gives pitch-amplitude

of 33.96◦. It clarifies the reason behind greater lift shown by plunge-equivalent pitching mo-

tion following effective angle-of-attack criteria. Unlike this criteria, smaller pitch-amplitude

comes out from equal-St based criteria. It concludes that vortex-shedding mechanism affects

the lift-producing capability from plunging motion of airfoil than its pitching.

3.2 Added-Mass Reaction

Added-mass is a term used to identify volume of the surrounding fluid displaced by airfoil

during its oscillation. Generally, it enhances inertia of the wing. We employ this term here

to understand its contribution towards production of aerodynamic forces by pitching and

plunging airfoils. As a result of force applied by accelerating wing on the fluid in its vicinity,

a reaction force is experienced by the wing-structure. Andro et al. [24] defined this reaction

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force as FA.M = −ρVfluidafoil where FA.M , ρ, Vfluid and afoil denote reaction force due to

added mass, density of fluid, volume of fluid displaced by the wing and acceleration of wing

respectively. It shows that added-mass reaction force is directly proportional to negative of

airfoil acceleration. Because density of fluid remains same for both kinds of kinematics, Vfluid

and afoil controls this factor to generate aerodynamic forces. Functional form of pitching and

plunging kinematics defines accelerations as α(t) = −α◦ω2sin(ωt) and h(t) = −h◦ω2sin(ωt)

where ω shows radial oscillation frequency (rad/sec). Because α is angular acceleration, it

is multiplied to radius r to attain linear acceleration. Using this relation, we can approxi-

mate linear acceleration for TE of pitching airfoil as aTE,p= −rα◦ω2sin(ωt). Equating the

maximum accelerations of plunging and pitching airfoils’ TEs, we get α◦ = h◦/r. Radius r,

depends upon the pivot-axis for pitching and consideration of LE (in case of effective angle-

of-attack criteria) or TE (in case of presently proposed equivalence). α◦ = h◦/r relation

provides pitch and plunge amplitudes to attain equal contribution of acceleration terms in

added-mass reaction force for both the cases at least for currently considered situation.

Third factor remains here is the volume of fluid swept by moving body. Contribution

of unit acceleration of a body along or about any axis to hydrodynamic forces is quantified

by added-mass tensor. Each component of added-mass tensor may be represented as ai,j,

where i denotes the direction of force on the body and j shows the direction of body-

acceleration. For 2D systems, the order of this tensor is 3 × 3 while it is 6 × 6 in case

of three-dimensional objects. Relative contributions of pitching and plunging motions of

airfoils towards production of aerodynamic forces can be understood by magnitudes of these

coefficients. A two-dimensional added-mass tensor for a NACA-0012 airfoil for pitching-

axis lying at 33% of chord has a11 = 0.10387145, a22 = 0.795293977, a23 = 0.135127881,

a32 = 0.135145331 and a33 = 0.047885882 while rest of the coefficients are zero [22, 33].

Multiplying these coefficients with fluid-density, we can find added-mass effect by different

degrees-of-freedom on the aerodynamic forces. Magnitudes of a22 and a23 show that plunging

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motion (j = 2) has got more added-mass reaction force to produce lift (i = 2) as compared

to that from pitching (j = 3).

3.3 Vortex Formation and Interaction

When airfoil oscillates, an instability in its boundary layer causes formation of vortices at

the leading-edge. These vortices tend to go downstream along the airfoil. Depending on

oscillation-frequency of airfoil, these may get detached from the surface before they reach

trailing portion of airfoil. When airfoil comes back to undergo reverse-stroke, these vor-

tices may be recaptured by the surface. Recapturing of these vortices drops pressure on

the respective upper or lower surface of airfoil and resultantly change in lift is experienced.

Pitching airfoil seems having greater capability for capturing of LEVs produced in the previ-

ous stroke. The reason is the 180◦ phase in movements of LE and TE. As seen in Fig.7, when

LE goes for downstroke, an LEV is convected on upper surface of airfoil, upward motion

of TE does not allow this vortex to separate from the airfoil and it prolongs dynamic stall.

Consequently, pitching airfoil may attain more percentage of lift from this mechanism than

plunging airfoil.

Depending upon kinematics of airfoil, LEVs produced during a stroke may interact with

TEVs appearing at the same time. If LEVs are of larger size or their convection rate is high,

these can interact with TEVs. Time-duration available for their interaction depends on

oscillation-frequency. Akhtar et al. [29] discussed constructive and destructive interference

of vortices to produce more lift using tandem arrangement of flapping airfoil. In the present

scenario, constructive interference of LEVs and TEVs tends to occur. Vorticity fields in left

column of Fig.7 shows that LEV produced during downstroke interacts with TEV appearing

in successive upstroke. This interaction is weaker in case of pitching airfoil where TEV is

shed in the wake before LEV reaches there. For lower Strouhal numbers, it even dissipates

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before it arrives at the trailing-edge. This discussion concludes that possibility of LEV-TEV

interaction is greater in case of plunging airfoil.

Diana et al. [10, 11] and Zheng et al. [13] proposed a criteria based upon formation of

vortex-dipole and their pairing mechanism in the wake of oscillating airfoil. They reported

that wake-deflection happens due to these factors. This deflecting jet affects time-averaged

value and amplitudes of thrust force. We show stream-wise velocity profiles for St = 0.50

in Fig.6. We see that pitching and plunging gives similar results not only in profile-shapes

but also in their magnitudes. It means wake is similar in both cases even at such large St.

Despite having more contribution from vortex-shedding and LEV-TEV interaction mecha-

nisms in case of plunging airfoil, pitching motion somehow manages to achieve equal order

of amplitudes for aerodynamic forces. In Fig. 7, we observe a greater distance between

vortices in the wake of plunging airfoil as compared to those in case of pitching airfoil. Their

close proximity contributes in unsteady lift and thrust [10, 11, 13]. Despite shedding of

smaller-sized vortices in the wake of pitching airfoil, role of this vortex-pairing may be more

significant.

4 Quadratic Nonlinearity and Wake-Deflection

In Fig. 8, we present time-histories of lift-coefficient for low St 0.05 and high Strouhal number,

0.75. Here, settling time is found to be greater for larger amplitudes. After transient response

settles down, a periodic steady-state solution is achieved. For current simulations, airfoil

starts plunging from its bottom-most position unless mentioned otherwise.

Although these signals look more like sinusoids, but presence of higher harmonics in

these signals indicates nonlinearities as shown in Fig.9. Unlike bluff-body aerodynamics

[19, 20, 21], we observe non-zero time-averaged value (CL or CL◦) and even harmonics (CL2)

in CL-spectra. From time histories of aerodynamic coefficients and their spectral composi-

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tions, it can be deduced that period-1 response exist for the given range of flow and kinematic

parameters. Vortex-shedding phenomena is synchronized with excitation and we find fun-

damental harmonic at forcing frequency in CL-spectra. For lower St, power of 2nd harmonic

is relatively lesser (almost of the order 10−4) than that for higher St where its order is

comparable to that of CL as shown in Fig. 10.

Figure 10 shows variation of time-averaged (CL)/zero-frequency component (CLo) and

even-harmonic amplitude (CL2) with respect to increasing heave-amplitude. For St < 0.30,

CLo of the order of 10−2 with negative sign is observed. As we increase heave-amplitude, it

goes to values of −0.1137 and −0.0621 for St = 0.50 and 0.55 respectively. Negative time-

averaged values clearly indicate wake-deflection in downward direction. When St equals

0.60, CLo starts getting positive values that identifies switching of wake to upward direction.

When St is increased from 0.70 to 0.75, CLo jumps from 0.7832 to 3.4177. This kind of jump

in response with a slight variation of control parameter (plunging amplitude in this case) is

pertinent to nonlinear systems. Order of CL2 is found as 10−4 for St < 0.15. It becomes 101

for positive time-averaged lift coefficients.

This non-zero CL (measure of wake-deflection) may be thought of due to numerics but it

can be justified using velocity measurements at side downstream of airfoil. To show deflection

of wake using velocity profiles, we follow the approach of von-Ellenrieder et al. [31] and

Zheng et al. [13]. In these references, maximum stream-wise velocities at different locations

in the wake were used to approximate the deflection angle. We show horizontal velocity

profiles in Fig. 11 at five different locations in the wake. These locations include 0.50c, 1.0c,

2.0c, 3.0c and 4.0c units distance from TE of airfoil. A black solid line connects maximum

velocity points in these five profiles. Quite flatter trend of horizontal velocity is observed for

x/c = 0.5. It is perhaps due to non-developed flow region near TE. Inclination of this line

gives an approximate deflection angle of 0.8198◦. After the wake switches its direction from

downward to upward at higher St, these angles go up to 11.42◦ for h◦ = 0.80c.

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Presence of higher harmonics in these spectra encourages to identify type of nonlinearity

that is responsible for wake-deflection. Analytically, behavioral changes in response of a

nonlinear system can be identified by temporal histories, Fourier-spectra and phases portraits

etc. In time history, symmetry in response about time-averaged value axis gets disturbed

along with a DC-shift of concerned signal. In Fourier spectra, it may be associated with

emergence of zero-frequency component and even harmonics. Emergence of even harmonics

in fourier-spectrum of a signal indicates quadratic nonlinearity in the system [17]. Although

airfoil oscillates symmetrically without giving a bias in the form of a non-zero mean angle-of-

attack to fluid-flow going downstream, deflected wake brings time-averaged CL in the system

which may also be termed as DC-shift of CL- signal or its zero-frequency component. Our

objective is to examine the relation of quadratic nonlinearity and wake-deflection phenomena.

In this section, we characterize quadratic nonlinearity by modulus of first even harmonic in

Fourier-spectra of CL, represented as CL2 . As reported in earlier studies, time-averaged CL,

represented as CL ,comes out to be approximately zero for symmetrically oscillating airfoils.

Deflection of wake brings a greater change in CL. Sign (positive or negative) of CL depends

upon direction of wake as deflected vortex street changes pressure-difference on surface of

airfoil. Figure 10 explains a direct relation of quadratic nonlinearity (measured as CL2) and

wake-deflection (quantified by CL◦). Irrespective of its sign, CL◦ becomes non-zero whenever

quadratic nonlinearity is observed in the system. It is completely in agreement to theory of

nonlinear systems.

Direction for wake-deflection is considered to be highly dependent on starting position

of oscillating airfoil. Few studies [12, 13] identified that wake got deflected to downward

direction for initial upward motion of airfoil and vice-versa. Contrary to that, others like

Guerrero [16] and Yu et al. [18] noticed reverse of this condition. Having a low reduced

frequency (k = 3.14) in current study and range of oscillation amplitude (from 0.05 to 0.80),

we observe both of these phenomenon. Again, this is highlighted in Fig. 10. For lower St,

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wake is deflected downward which is also confirmed by its velocity profiles in Fig. 11. But

at higher St, it goes upward quantified by positive CL.

5 Conclusions

In this study, we present resemblance of flow-fields and aerodynamic forces produced by

flows over airfoils, performing pure-pitching and pure-plunging. To identify this equivalence,

we consider motion of trailing-edge in both the cases that actually provides bluff-body effect

to incoming flow. We compute plunge-equivalent pitch-amplitude by matching distances

traversed by trailing-edges in both the cases. Performance of this criteria seems better not

only in case of lift but also for thrust. Their velocity profiles in the wake are also found similar

in profiles and comparable in their magnitudes. We consider time-averaged lift-coefficient as a

result of wake-deflection. Choice of CL-signal for this analysis is justified here due to presence

of vortex-shedding frequency in it. Secondly, its non-zero time-averaged value characterize

the wake-deflection while its sign (positive or negative) gives information about direction

of deflection. We also show that direction for wake-deflection is not necessarily dependent

on starting position only. At lower Strouhal numbers, wake gets deflected in downward

direction for starting upstroke and vice-versa. But at higher Strouhal numbers, it changes

its direction. In nonlinear systems, appearance of quadratic nonlinearity is responsible for

symmetry-breaking as also seen in phase maps of aerodynamic states here. We quantify the

strength of this nonlinearity using first even harmonic of lift-coefficient. We prove quadratic

nonlinearity as governing element for wake-deflection.

17

Acknowledgments

This work is part of doctoral research of the first author. He is thankful to National University

of Sciences & Technology and Higher Education Commission, Government of Pakistan for

providing scholarship under Mega S&T scheme.

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21

t/T

CL

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

82482 Cells98754 Cells118440 Cells139442 Cells

(a)

t/T

CT

0 0.2 0.4 0.6 0.8

­0.4

­0.2

0

0.2

0.4

82482 Cells

98754 Cells

118440 Cells

139442 Cells

(b)

Figure 2: Results of Grid-Convergence Study

22

Time Period

CL

0 0.5 1

-6

-4

-2

0

2

4

6

8

(a)

Sarkar [7]

Present Study

StM

ean

CT

0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

(b)

Guerrero [16]

Present Study

Figure 3: Results of Validation Studies

Figure 4: Kinematic Profiles of Pitching and Plunging Airfoils for Equal St-based Criteria

23

f (Hz)

|FF

T| L

0 1 2 3 4 510-4

10-2

100

(c)

f (Hz)

|FF

T| T

0 1 2 3 4 510-4

10-2

100 (d)

t (sec)

CL

12 16 20

-2

0

2

(a)

t (sec)

CT

12 16 20

-0.3

0

0.3

(b

Figure 5: (a) & (b) show time-histories and (c) & (d) present spectra.

24

Figure 6: Comparison of Velocity Profiles at St=0.50

25

Figure 7: Vorticity plots (contour levels varying from -10 to 10) in the flow-fields due topure-plunge (left column) and pure-pitching (right column) motion of airfoil at St = 0.50.1st row: mean position of TE during upstroke, 2nd row: top-most position of TE, 3rd row:mean position of TE during downstroke and 4th row: bottom-most position of TE

Time

CL

0 4 8 12-1

-0.5

0

0.5

1

1.5

Transient Steady-State

(a)

Time

CL

0 20 40-30

-15

0

15

30(b)

Figure 8: Unsteady CL for Strouhal number (a). 0.05 and (b). 0.75

26

f (Hz)

|FF

T| L

0 1 2 3 4 510-4

10-2

100

CL1

CL2

CL3

(a)

f (Hz)|F

FT

| L

0 1 2 3 4 510-4

10-2

100

CL3

CL1

CL2

(b)

Figure 9: CL-spectra for Strouhal number (a). 0.05 and (b). 0.75

St

CL

o

0 0.2 0.4 0.6 0.8 1-1

0

1

2

3

4 (a)

St

CL

2

0 0.2 0.4 0.6 0.8 1-1

0

1

2

3

4(b)

Figure 10: Variation of mean CL (CL◦) and amplitude of even-harmonic CL2 with respect toSt

27

u-velocity

y/c

0.8 1.6 2.4

-1

0

1

x/c = 4.0

x/c = 0.50x/c = 1.0x/c = 2.0

Figure 11: horizontal velocity profiles at five different locations in the wake at St = 0.50

28