J. Fluid Mech. (2009), vol. 632, pp. 245–271.
c
2009 Cambridge University Press
doi:10.1017/S0022112009007058 Printed in the United Kingdom
245
On vortex shedding from an airfoil in
low-Reynolds-number flows
SERHIY YARUSEVYCH
1
†, PIERRE E. SULLIVAN
2
AND JOHN G. KAWALL
3
1
Department of Mechanical & Mechatronics Engineering, University of Waterloo, Waterloo,
N2L 3G1, Canada
2
Department of Mechanical and Industrial Engineering, University of Toronto, Toronto,
M5S 3G8, Canada
3
Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, M5B 2K3, Canada
(Received 22 May 2008 and in revised form 23 February 2009)
Development of coherent structures in the separated shear layer and wake of an airfoil
in low-Reynolds-number flows was studied experimentally for a range of airfoil chord
Reynolds numbers, 55×10
3
6 Re
c
6 210×10
3
, and three angles of attack, α =0
◦
, 5
◦
and
10
◦
. To illustrate the effect of separated shear layer development on the characteristics
of coherent structures, experiments were conducted for two flow regimes common
to airfoil operation at low Reynolds numbers: (i) boundary layer separation without
reattachment and (ii) separation bubble formation. The results demonstrate that roll-
up vortices form in the separated shear layer due to the amplification of natural
disturbances, and these structures play a key role in flow transition to turbulence.
The final stage of transition in the separated shear layer, associated with the growth
of a sub-harmonic component of fundamental disturbances, is linked to the merging
of the roll-up vortices. Turbulent wake vortex shedding is shown to occur for both
flow regimes investigated. Each of the two flow regimes produces distinctly different
characteristics of the roll-up and wake vortices. The study focuses on frequency scaling
of the investigated coherent structures and the effect of flow regime on the frequency
scaling. Analysis of the results and available data from previous experiments shows
that the fundamental frequency of the shear layer vortices exhibits a power law
dependency on the Reynolds number for both flow regimes. In contrast, the wake
vortex shedding frequency is shown to vary linearly with the Reynolds number.
An alternative frequency scaling is proposed, which results in a good collapse of
experimental data across the investigated range of Reynolds numbers.
1. Introduction
In a growing number of new miniaturized mechanical systems, such as small-scale
wind turbines and unmanned aerial vehicles, lifting surfaces operate at relatively
low airfoil chord Reynolds numbers, i.e. Re
c
< 500 000. Airfoil operation at low
Reynolds numbers differs significantly from that typical for high-Reynolds-number
flows (e.g. Tani 1964; Carmichael 1981; Mueller & DeLaurier 2003). In particular, a
laminar boundary layer on the upper surface of the airfoil often separates and forms
a separated shear layer. The presence of laminar boundary layer separation has a
significant detrimental effect on airfoil performance, affecting airfoil lift and drag. The
† Email address for correspondence: syarus@uwaterloo.ca
246 S. Yarusevych, P. E. Sullivan and John G. Kawall
Laminar
separation
(a)
(b)
Laminar
separation
Separated
shear layer
Transition
Transition
Reattachment
Figure 1. Flow over an airfoil at low Reynolds numbers: (a) laminar separation without
reattachment; (b) separation bubble formation.
severity of this effect is mainly determined by the behaviour of the separated shear
layer. Figure 1 depicts two flow regimes common to airfoils operating at low Reynolds
numbers. As the inherently unstable separated shear layer undergoes laminar-to-
turbulent transition, it can reattach to the airfoil surface. At lower Reynolds numbers,
the separated shear layer fails to reattach, and a wide wake is formed (figure 1a). In
contrast, at higher Reynolds numbers, a turbulent separated shear layer may reattach,
resulting in a laminar separation bubble (figure 1b). A change between the two flow
regimes depicted in figure 1 is an unsteady phenomenon that occurs over a finite
range of Reynolds numbers for a given angle of attack (e.g. Carmichael 1981).
Since the pioneering research into airfoil operation at low Reynolds numbers,
summarized and extended by Tani (1964) and Gaster (1967), a number of related
studies have been performed over the past several decades. For conciseness, the
following discussion of previous studies is focused on those most pertinent to the
development of coherent structures in the separated shear layer and airfoil wake.
As illustrated in figure 1, the laminar-to-turbulent transition in the separated shear
layer plays a key role in the overall flow field development over an airfoil operating at
low Reynolds numbers. Although most of the previous studies dealing with separated
shear layer development were performed for a separation bubble forming on a flat
plate in an adverse pressure gradient rather than on an airfoil surface, they provide
valuable insight into the transition process. It has been shown that, during the
initial stage of transition, small-amplitude disturbances centred at some fundamental
frequency experience nearly exponential growth in the separated shear layer (e.g.
Dovgal, Kozlov & Michalke 1994; Watmuff 1999; Boiko et al. 2002). Experimental
and numerical studies by Haggmark, Bakchinov & Alfredsson (2000), Lang, Rist &
Wagner (2004), Marxen, Rist & Wagner (2004) and Marxen & Rist (2005) suggest
On vortex shedding from an airfoil in low-Reynolds-number flows 247
that two-dimensional growth of the disturbances dominates the initial stage of the
transition. The final stage of transition, which results in rapid flow breakdown to
turbulence, is associated with nonlinear interactions between the disturbances. Some
studies show that coherent structures form during this stage of transition (e.g. Wilson
& Pauley 1998; Watmuff 1999; Lang et al. 2004; Marxen & Rist 2005, McAuliffe &
Yaras 2007). Specifically, Watmuff suggests that these structures are associated with
the Kelvin–Helmholtz instability and persist into the attached turbulent boundary
layer. In contrast, experimental results by Lang et al. and direct numerical simulations
by Marxen & Rist and McAuliffe & Yaras indicate that vortices forming in the
separated shear layer breakdown in the reattachment region.
Separated shear layer transition on an airfoil has not been as thoroughly examined
as that on a flat plate in an adverse pressure gradient. Nevertheless, the available
results obtained for various airfoil profiles indicate that the initial stage of transition
is similar to that observed on a flat plate (e.g. Brendel & Mueller 1988; Brendel &
Mueller 1990; Boiko et al. 2002; Yarusevych, Sullivan & Kawall 2006). Following the
initial linear growth of disturbances, in some studies, nonlinear effects were shown to
be associated with the growth of the sub-harmonic of the fundamental frequency wave
(e.g. Brendel & Mueller 1990; Dovgal et al. 1994; Boiko et al. 2002), but the underlying
mechanism has not been investigated. The numerical results of Lin & Pauley (1996),
supported by the experimental results and stability analysis of Yarusevych et al.
(2006), suggest that coherent structures can form in the separated shear layer and
are attributed to the Kelvin–Helmholtz instability (cf. Watmuff 1999). Several other
experimental studies on airfoils (e.g. Brendel & Mueller 1988; Hsiao, Liu & Tang
1989) found evidence of coherent structures forming during the transition process;
however, the behaviour and characteristics of these structures were not investigated in
detail. Recently, Burgmann, Brucker & Schroder (2006), Burgmann, Dannemann &
Schroder (2008) and Burgmann & Schroder (2008) performed detailed experimental
investigations of flow development within a separation bubble on an airfoil. The
results identified the roll-up vortices forming in the separated shear layer, which is
supported by the results of Zhang, Hain & Kahler (2008) on the same airfoil profile.
Agreeing with the numerical findings of McAuliffe & Yaras (2007), Burgmann et al.
(2008) and Burgmann & Schroder (2008) suggest that these spanwise structures break
down and change orientation in the vicinity of the reattachment point. However, the
observed process was found to be different from that reported by Watmuff (1999)
and Lang et al. (2004) for a flat-plate bubble. Burgmann et al. (2008) concluded
that varying the adverse pressure gradient on an airfoil surface, e.g. by changing the
angle of attack, has a different effect on salient bubble characteristics compared to
the effect produced by varying the pressure gradient on a flat plate. Despite these
research efforts towards describing the development of coherent structures in the
separated shear layer on an airfoil, insight into the role of such structures in the
transition process, as well as the effect of other flow parameters, such as the Reynolds
number, on their characteristics, remains limited. Also, the effect of the flow regime
on the development of these structures is uncertain, as most of the previous studies
were concerned with the case of a separation bubble. For the case of flow separation
without reattachment, some insight into separated shear layer development can be
gained from studies on circular cylinders (e.g. Unal & Rockwell 1988; Prasad &
Williamson 1997). However, in view of the significant differences in geometry, these
results cannot be applied directly to airfoils operating at nominally pre-stall angles
of attack.
248 S. Yarusevych, P. E. Sullivan and John G. Kawall
The development of coherent structures in the airfoil wake at low Reynolds numbers
is of interest since it affects airfoil performance and governs flow around downstream
objects. For instance, these structures can result in undesirable structural vibrations
and noise generation. The wake of an airfoil at post-stall angles of attack can be
expected to behave similar to that of a bluff body (e.g. Huang et al. 2001). The
structure and characteristics of a two-dimensional bluff-body wake have been the
subject of active research over the past decades (e.g. Roshko 1993; Williamson 1996).
A comprehensive description of the vortex formation mechanism in the wake of a
circular cylinder is presented by Gerrard (1966). The wake vortex shedding frequency
is usually scaled with global parameters to form a Strouhal number St, with typical
values of 0.21 and 0.14 reported for the case of a circular cylinder and a flat plate,
respectively (e.g. Roshko 1954b). For a NACA 0012 airfoil in the range of chord
Reynolds numbers from about 25 × 10
3
to 120 × 10
3
at angles of attack above about
15
◦
, Huang & Lin (1995) measured a constant Strouhal number St
d
based on the
length of the airfoil projection on a cross-stream plane. In agreement with the results
of Roshko (1954b), an increase of angle of attack above about 15
◦
resulted in a
decrease of Strouhal number, with an St
d
value of 0.12 obtained at 90
◦
.
At lower angles of attack, even when separation occurs without reattachment, airfoil
wake characteristics have been shown to be quite different. Huang & Lin (1995) and
Huang & Lee (2000) detected vortex shedding in the airfoil wake at low Reynolds
numbers and identified several vortex shedding modes, with a wide distribution of
Strouhal numbers observed over the investigated Reynolds number range. These
modes were found to be closely related to separated shear layer behaviour, and
wake vortex shedding was observed only when laminar separation occurred without
subsequent reattachment or in the presence of turbulent boundary layer separation.
In contrast, Yarusevych et al. (2006) also detected organized structures in the airfoil
wake for the case when a separation bubble formed on the airfoil surface. Huang
et al. (2001) proposed empirical correlations for the dimensionless vortex shedding
frequency in the wake of an impulsively started wing for Re
c
< 2500. Several other
studies confirm the existence of the wake vortex shedding phenomenon for airfoils in
low-Reynolds-number flows but did not investigate it in detail (e.g. Williams-Stuber
& Gharib 1990; Gerontakos & Lee 2005). It should be noted that there are limited
results available for vortex shedding in airfoil wakes compared to those concerned
with bluff-body wakes. Moreover, evolution of coherent structures in the airfoil wake
and the effect of separated shear layer development on their characteristics have not
been investigated in detail.
The present work is motivated by the need for additional insight into coherent
structures forming in low-Reynolds-number flows over airfoils. The main goal is to
investigate the formation, evolution, and characteristics of the coherent structures
and their role in the overall flow field development for both flow regimes typical of
airfoil operation at low Reynolds numbers. In the following sections, the experimental
approach is discussed first. Then, a brief overview of the flow development is provided
for the cases investigated. The main results detailing formation, characteristics and
frequency scaling of coherent structures are discussed in two sections: (i) coherent
structures in the separated shear layer and (ii) coherent structures in the airfoil wake.
2. Experimental setup
All experiments were performed in a low-turbulence recirculating wind tunnel. The
5-m-long test section of this tunnel has a spanwise extent of 0.91 m and a height of
On vortex shedding from an airfoil in low-Reynolds-number flows 249
1.22 m. The free-stream turbulence intensity level in the test section is less than 0.1 %,
and there is no significant frequency-centred activity associated with the oncoming
flow. During the experiments, the free-stream velocity U
0
was monitored by a Pitot
tube, with an uncertainty estimated to be less than 2.5 %.
An aluminium NACA 0025 airfoil with a chord length c of 0.3 m was examined. This
NACA profile was selected because it allowed investigating both strongly separated
flows and separation bubbles at nominally pre-stall angles of attack and the Reynolds
numbers of interest. The airfoil was mounted horizontally in the test section 0.4 m
downstream of the wind-tunnel contraction, spanning the entire width of the test
section. With this arrangement, it was verified experimentally that end effects did not
influence flow development over at least 50 % of the airfoil span within the domain
of interest. The angle of attack was set by a digital protractor, with an uncertainty
of 0.1
◦
To enable surface pressure measurements, the airfoil was equipped with 65 pressure
taps, 0.8 mm in diameter, which were positioned at the midspan symmetrically
on the upper and lower surfaces. Surface pressure distributions were measured
with a pressure transducer connected to the taps through a 64-channel Scanivalve
module. The uncertainty associated with the surface pressure measurements was less
than 2 %.
Flow velocity data were obtained with Dantec constant temperature anemometers.
A normal hot-wire probe, a cross-wire probe and a rake of three cross-wire probes were
used in separate measurements. The probes were attached to a holder mounted on a
remotely controlled traversing mechanism. The mechanism allowed probe motion in
the vertical y and streamwise x directions with a resolution of 0.01 mm and 0.25 mm,
respectively. For boundary layer measurements, the probe holder could be manually
adjusted to change the angle between the probe and the airfoil surface. To minimize
possible probe interference, this angle was kept below 7
◦
± 0.1
◦
, as recommended
by Brendel & Mueller (1988). All hot-wire measurements were carried out in the
vertical midspan plane of the tunnel. Based on the results of Kawall, Shokr & Keffer
(1983), the maximum hot-wire measurement error was evaluated to be less than 5 %.
The origin of the streamwise coordinate x was located at the leading edge of the
airfoil. For boundary layer measurements, the vertical coordinate y was referenced
to the airfoil surface; whereas, for wake measurements, the vertical coordinate was
referenced to the trailing edge. It should be noted that a conventional single hot-
wire probe is incapable of determining flow direction and, therefore, cannot resolve
the velocity direction of the reverse flow that occurs near the airfoil surface in the
separated flow region. However, hot-wire measurements in the separated shear layer,
which is of particular interest in the present study, can be analysed without any
restrictions.
Spectral analysis of velocity signals was performed to uncover organized flow
structures and determine their characteristics. Autospectra of the velocity signals
were determined by means of the fast Fourier transform algorithm applied to the
experimental data. To allow adequate comparison of the velocity spectra, the velocity
data at each streamwise location were collected at a y/c position that corresponds
to the maximum r.m.s velocity. Such a position approximately corresponds to the
location of half the boundary layer edge velocity U
e
in the separated shear layer and
half the maximum velocity deficit in the wake. Each spectrum was normalized by the
variance of the sampled signal, so that the area under the spectral curve was unity.
The uncertainty of the spectral analysis was approximately 4.5 %, with a frequency
resolution bandwidth of 1.2 Hz.
