Post on 08-Mar-2023
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
1
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
2
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
3
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]
4
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.
5
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
6
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
7
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.
8
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
9
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
10
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
11
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
12
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
13
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-
14
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.
15
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,
16
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.
References
[1] Young J. and Lai J. C. S., 2007, ”Vortex Lock-In Phenoenon in the Wake of a Plunging
Airfoil”, AIAA Journal 45 (2), pp. 485-490.
[2] Liang, C., Ou, K., Premasuthan, S., Jameson, A. and Wang, Z. J., 2011, ”High-Order
Accurate Simulations of Unsteady Flow Past Plunging and Pitching Airfoils”, Computer
& Fluids 40, pp. 236-248.
[3] Ashraf, M. A., Young, J. and Lai J. C. S., 2012, ”Oscillation Frequency and Amplitude
Effects on Plunging Airfoil Propulsion and Flow Periodicity”, AIAA Journal 52 (11),
pp. 2308-2324
[4] McCroskey, W. J., 1981 ”The Phenomenon of Dynamic Stall”, NASA TM-81264.
[5] Fukushima, T. and Dadone, L. U, 1977, ”Comparison of Dynamic Stall Phenomena for
Pitching and Vertical Translation Motions”, NASA CR-2793.
[6] Carta, F. O, 1979, ”A Comparison of Pitching and Plunge Response of an Oscillating
Airfoil”, NASA CR-3172.
[7] Sarkar, S., 2010, ”Comparing Pure-Pitch and Pure-Plunge Kinematics for a Symmetric
Airfoil”, AIAA Journal 48 (12), pp. 2962-2969.
18
[8] McGowan, G. Z., 2008, ”Analytical and Computational Investigations of Airfoils Under-
going High-Frequency Sinusoidal Pitch and Plunge Motions at Low Reynolds Numbers”,
PhD Dissertation, North Carolina State University, USA.
[9] McGowan, G. Z., Granlund, K., Ol, M. V., Gopalarathnam, A. and Edwards, J. E., 2011,
”Investgations of Lift-Based Pitch-Plunge Equivalence for Airfoils at Low Reynolds
Numbers”, AIAA Journal 49 (7), pp. 1511-1524.
[10] Diana, R. G., Aider, J. L. and Wesfreid J. E, 2008, ”Transition in the Wake of a Flapping
Foil”, Physical Review E 77.
[11] Diana, R. G., Marais, C., Aider, J. L. and Wesfreid J. E, 2009, ”A Model for the
Symmetry Breaking of the Reverse Benard-von Karman Vortex Street Produced by a
Flapping Foil”, Journal of Fluid Mechanics 622, pp. 23-32.
[12] Cleaver, D. J., Wang, Z., and Gursul, I., 2012, ”Bifurcating Flows of Plunging Airfoils
at High Strouhal Numbers”, Journal of Fluid Mechanics 708, pp. 349-376.
[13] Zheng, Z. C. and Wei, Z., 2012, ”Study of Mechanisms and Factors that Influence the
Formation of Vortical Wake of a Heaving Airfoil”, Physics of Fluids 24.
[14] Jones, K. D., Dohring, C. M. and Platzer, M. F., 1998, ”Exeprimental and Computa-
tional Investigation of the Knoller-Betz Effect”, AIAA Journal 36 (7), pp. 1240-1246.
[15] Lai, J. C. S. and Platzer, M. F., 1999, ”Jet Characteristics of a Plunging Airfoil”, AIAA
Journal 37 (12), pp. 1529-1537.
[16] Guerrero, J., 2009, ”Numerical Simulation of the Unsteady Aerodynamics of Flapping
Flight”, PhD Dissertation, University of Genoa, Italy.
[17] Nayfeh, A. H. and Mook, D. T., 1995, ”Nonlinear Oscillations”, Wiley Classic Library
Edition.
19
[18] Yu, M. L., Hu, H. and Wang, Z. J., 2012, ”Experimental and Numerical Investigations
on the Asymmetric Wake Vortex Structures Around an Oscillating Airfoil”, Paper No.
AIAA-2012-0299, 50th AIAA Aerospace Sciences Meeting Including the New Horizons
Forum and Aerospace Exposition, Nashville, Tennessee, USA.
[19] Nayfeh, A. H., Owis, F. and Hajj, M. R., 2003, ”A model for the coupled lift and
drag on a circular cylinder”, Paper No. DETC2003–VIB48455, Proceedings of DETC03
ASME 2003 Design Engineering Technical Conferences and Computers and Information
in Engineering Conference.
[20] Marzouk, O. A., Nayfeh, A. H., Akhtar, I. and Arafat, H. N., 2007, ”Modeling steady-
state and transient forces on a cylinder”, Journal of Vibration and Control 13 (7), pp.
1065-1091.
[21] Akhtar, I., Marzouk, O. A., and Nayfeh, A. H., 2009, ”A van der Pol-Duffing oscillator
model of hydrodynamic forces on Canonical Structures”, Journal of Computational and
Nonlinear Dynamics 4 (4).
[22] Mantia, M. L. and Dabnichki, P., 2012, ”Added Mass Effect on Flapping Foil”, Engi-
neering Analysis with Boundary Elements 36, pp. 579-590.
[23] Wang, Z. J., Birch, J. M. and Dickinson, M. H., 2003, ”Unsteady Forces and Flows in
Low Reynolds Number Hovering Flight: Two Dimensional Computations vs Robotic
Wing Experiments”, Journal of Experimental Biology 207, pp. 449-460.
[24] Andro J. Y. and Jacquin, L., 2009, ”Frequency Effects on the Aerodynamic Mecha-
nisms of a Heaving Airfoil ina Forward Flight Configuration”, Aerospace Science and
Technology 13, pp. 71-80.
[25] ANSYS Fluent, ”ANSYS Fluent Userguide” ANSYS Inc.
20
[26] Koochesfahani, M. M., 1989, ”Vortical Patterns in the Wake of Oscillating Airfoil”,
AIAA Journal 27 (9), pp. 1200-1205.
[27] Liu, Y., Li, K., Zhang, J., Wang, H. and Liu, L., 2012, ”Numerical Bifurcation Analysis
of Static Stall of Airfoil and Dynamic Stall under Unsteady Perturbations”, Communi-
cations in Nonlinear Sciences & Numerical Simulations 17.
[28] McCroskey, W. J. and Fisher, R. K., 1972, ”Detailed Aerodynamic Measurements on a
Model rotor in the blade stall regime”, Journal of the Americal Helicopter Society 17,
pp. 20-30.
[29] Akhtar, I., Mittal, R., Lauder, G. V. and Drucker, E., 2007, ”Hydrodynamics of a Bio-
logically Inpired Tandem Flapping Foil Configuration”, Theoratical and Computational
Fluid Dynamics, DOI: 10.1007/s00162-007-0045-2
[30] von Ellenrieder, K. D., 2006, ”Dynamical Systems Analysis of Flapping Wing Propul-
sion”, Australian Fluid Mechanics Workshop, Melbourne University, Australia.
[31] von Ellenrieder, K. D. and Pothos, S., 2008, ”PIV Measurements of the Asymmetric
Wake of a Two-Dimensional Heaving Hydrofoil”, Exp. Fluids 44 (5), pp. 733-745.
[32] von Ellenrieder, K. D., Parker, K. and Soria, J., 2008, ”Fluid Mechanics of Flapping
Wings”, Experimental Thermal and Fluid Science 32, pp. 1578-1589.
[33] Mantia, M. L. and Dabnichki, P., 2013, ”Structural Response of Oscillating Foil in
Water”, Engineering Analysis with Boundary Elements 37, pp. 957-966.
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 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