The flow around an unsteady hydrofoil, undergoing simultaneous heaving and pitching motions which mimic the tail motions of carangiform fish (such as tuna), was investigated using simultaneous particle image velocimetry (PIV) and force measurement. These investigations were performed at Reynolds numbers, based on the foil chord length, in the range of 150

Kelso, Richard M.

Lau, Timothy C. W.

The flow around an unsteady hydrofoil, undergoing simultane-ous heaving and pitching motions which mimic the tail motions of carangiform fish (such as tuna), was investigated using simul-taneous particle image velocimetry (PIV) and force measure-ment. These investigations were performed at Reynolds num-bers, based on the foil chord length, in the range of 1500&lt;Re&lt; 12500. The Strouhal numbers, based on the foil heave ampli-tude, were in the range of 0.1 &lt; Sth &lt; 0.95. Hydrogen bubble visualisation was also performed on the foil at identical Sth, but at 500&lt; Re&lt; 3500. Two force measurements were obtained. Firstly, instantaneous thrust and side forces were obtained by direct strain gauge mea-surements on the foil, which were then time-averaged. Sec-ondly, time-averaged thrust forces were estimated using a mo-mentum integral around the time-averaged two-dimensional PIV velocity field around the foil. Both force measurements, which show excellent agreement for all foil parameters, indi-cate that the foil is able to produce very large thrust coefficients, Ct, in excess of 10. These large thrust coefficients typically oc-cur when the non-dimensional foil heave amplitude is large, at conditions where hydrogen bubble visualisation indicates that large leading edge vortices are generated. This suggests that the unsteady foil may be able to use unsteady flow mechanisms to generate large forces

Timothy C. W. Lau

Richard M. Kelso

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The forces on a fish-inspired unsteady hydrofoil

University of Queensland eSpace

16th Australasian Fluid Mechanics Conference
Crown Plaza, Gold Coast, Australia
2-7 December 2007
The forces on a fish-inspired unsteady hydrofoil
Timothy C. W. Lau and Richard M. Kelso
School of Mechanical Engineering, The University of Adelaide, North Terrace Campus, SA 5005, Australia
Abstract
The flow around an unsteady hydrofoil, undergoing simultane-
ous heaving and pitching motions which mimic the tail motions
of carangiform fish (such as tuna), was investigated using simul-
taneous particle image velocimetry (PIV) and force measure-
ment. These investigations were performed at Reynolds num-
bers, based on the foil chord length, in the range of 1500<Re<
12500. The Strouhal numbers, based on the foil heave ampli-
tude, were in the range of 0.1 < Sth < 0.95. Hydrogen bubble
visualisation was also performed on the foil at identical Sth, but
at 500< Re< 3500.
Two force measurements were obtained. Firstly, instantaneous
thrust and side forces were obtained by direct strain gauge mea-
surements on the foil, which were then time-averaged. Sec-
ondly, time-averaged thrust forces were estimated using a mo-
mentum integral around the time-averaged two-dimensional
PIV velocity field around the foil. Both force measurements,
which show excellent agreement for all foil parameters, indi-
cate that the foil is able to produce very large thrust coefficients,
Ct , in excess of 10. These large thrust coefficients typically oc-
cur when the non-dimensional foil heave amplitude is large, at
conditions where hydrogen bubble visualisation indicates that
large leading edge vortices are generated. This suggests that the
unsteady foil may be able to use unsteady flow mechanisms to
generate large forces.
Introduction
It has often been suggested that, due to evolutionary forces,
many aquatic animals are capable of producing large thrust co-
efficients, or producing thrust very efficiently [9]. This propo-
sition is particularly applicable for the carangiform class of fish
(such as tuna and dolphins), which is roughly defined as any
aquatic animal that uses the motion of its tail to generate propul-
sion, due to the fact that carangiform swimmers have been ob-
served to swim continuously for long periods and have been
observed to be capable of large accelerations [2, 7, 9].
Traditionally, research in the kinematics of carangiform swim-
mers has been achieved using unsteady hydrofoils which un-
dergo simultaneous heaving and pitching which mimic the tail
motions of carangiform fish. Previous studies on unsteady foils
have indicated that such a flow can be strongly characterised by
the Strouhal number, St, and to a lesser extent, the maximum
foil angle of attack achieved during an entire oscillating cycle,
αmax [1, 3, 5].
Propulsive efficiencies of over 80% have been recorded [1], oc-
curring roughly at 0.25< St < 0.35, which is approximately the
same Strouhal number range as observed for fish during steady
swimming [6, 8]. Large thrust coefficients, on the other hand,
are observed to be generated at higher St, where αmax tends to
be large [1, 3, 5]. It has been suggested that these large thrust
coefficients are developed due to the generation and control of
large leading edge vortices [1], possibly indicating that fish use
unsteady flow mechanisms during propulsion [2].
This study forms part of a larger investigation of the forces on,
and the flow around a heaving and pitching hydrofoil. In par-
ticular, we intend to examine a large number of flow cases and
foil parameters, using a combination of force measurements and
flow visualisation, with the ultimate aim of determining the ef-
fect of the foil parameters on the flow, and on the forces devel-
oped by the unsteady foil.
Experimental Setup
The motion of a carangiform tail is approximated using a rigid
NACA 0026 hydrofoil undergoing simultaneous heaving and
pitching motion. The foil has chord length, c = 50mm, and a
span, s= 200mm, giving an aspect ratio of 4.
Two stepper motors, which are phase locked at a flapping fre-
quency f , provide the heaving and pitching motions via a scotch
yoke and cam mechanism respectively, as previously described
in [4]. The heaving and pitching motions can then be described
mathematically as
h(t) = h0 · cos(2pi f t− pi2 ) (1)
θ(t) ≈ θ0 · cos(2pi f t) (2)
where t is time, and the selected heave and pitch amplitudes re-
spectively were h0c = 0.25,0.5,0.75 and θ0 = 0
o,15o,30o,45o.
We define the Strouhal number, based on the heave amplitude
as
Sth =
2 ·h0 · f
U∞
. (3)
where U∞ is the freestream velocity. Based on the foil dynam-
ics, the relationship between Sth and αmax is
Sth ≈ tan(αmax+θ0)pi (4)
The foil was placed in a recirculating water channel with a
working section of 2000mm × 500mm × 500mm such that the
centre of the foil is aligned to the centre of the water channel
working section. This is to ensure that the walls and the free
surface of the water channel remain distant to the unsteady foil.
The experiments on the unsteady foil consist of three parts : a)
Particle Image Velocimetry (PIV) of the flow in the wake of the
foil, b) instantaneous force measurements via strain gauges and
c) hydrogen bubble visualisation.
A dual cavity Quantel Brilliant B Twins Nd:YAG pulsed laser,
with a fixed pulse rate of 10Hz, was used as the source of illu-
mination for all PIV experiments. The laser pulse is converted
into a light sheet approximately 1mm thick using a series of
three cylindrical lenses, with focal lengths of 100mm, -50mm
and -12.7mm respectively. The resulting light sheet was gener-
ated on a plane coincident with the mid-span of the foil. The
flow was seeded with hollow glass spheres with a mean diame-
ter of 20µm and a density of 1030kg/m3. Images of the seeded
flow field were captured using a Kodak Megaplus ES 1.0 10bit
CCD camera, with a sensor resolution of 1008 ×1018 pixels.
Two different lenses were used in conjunction with the camera;
a Nikkor AF 50mm f/1.4D lens and a Nikkor AF 70-300mm
f/4.5-5.6D lens. The size of the field of view was typically set
to ≈ 2.5c×2.5c.
For each flow condition, a minimum of 4000 image pairs were
collected. Velocity fields were obtained by cross correlating the
1003
image pairs using PIV-View 1.75 (command line interface). A
16×16 pixel interrogation window size was used, with an over-
lap of 50%. The resultant velocity field consists of 126 (stream-
wise) ×125 vectors (cross-streamwise) vectors. Velocity out-
liers were detected using the method suggested by [10]. Typical
velocity fields consist of less than 0.1% outliers. All outliers
were disabled. A time-averaged velocity field is then obtained
by taking the mean of all the valid vectors at each grid point.
Due to this time averaging process, the random errors in veloc-
ity are estimated to be very low, below 0.5%. No smoothing
was performed on the velocity field.
Based on the time-averaged velocity field around the foil, the
time-averaged thrust force on the foil can be estimated by
performing a momentum balance around the control volume,
ABCD, encompassing the foil (as shown in figure 1). If the
control volume is sufficiently large such that the x-component
of the velocity along the boundaries AB and CD, into the con-
trol surface AC is ≈U∞, and the pressure gradients across the
control volume are negligible, then the thrust force based on the
PIV velocity field can be approximated by
Ft,piv ≈ ρwater
∫ b
−b
U2(y) [U∞−U2(y)]dy (5)
which can be solved by numerically integrating along the
boundary of the PIV-derived time-averaged velocity fields.
Equation 5 also assumes that the flows through the sides of the
control volume ABCD+ (into the page) and ABCD- (out of the
page) are negligible, and the flow is steady. Nonetheless, we
expect Ft,piv to be a reasonable estimate to the true thrust forces
on the foil, due to the fact that the PIV-derived time-averaged
velocity fields are steady, encompass a large perimeter around
the foil, and are measured along the mid-span of the foil where
span-wise flow through the sides of the control volume are ex-
pected to be small.
Direct force measurements via strain gauges, performed simul-
taneously with the PIV experiments, were also conducted on
the unsteady foil. Force measurements were made on two or-
thogonal axes, F∗C and F
∗
N (see figure 1). Each force compo-
nent was measured using a set of four strain gauges set up in a
temperature-compensatedWheatstone bridge. Voltages are then
conditioned using an in-house variable gain strain gauge condi-
tioning apparatus, before being measured by a 14-bit data log-
ger (NI-USB 6009) in differential mode. Sampling frequency
was fixed at fsamp = 200Hz, and the typical sampling time was
in the order of 10 minutes, leading to approximately 120,000
data samples per force component. The voltage signal was fil-
tered using a digital 4-pole Butterworth filter with a cutoff fre-
quency of 5 Hz. The inertia of the foil was accounted for by
subtracting the force measurements performed on the foil when
there was no water in the water channel, leading to hydrody-
namic force components FC and FN . It was then a simple matter
to convert the normal and chord-wise foil force components to
thrust and side force (in the streamwise and cross-streamwise
directions respectively, as shown in figure 1) components, mea-
sured using the strain gauge apparatus,
Ft,sg = −
∫ T
0
(FC(t) · cosθ(t)+FN(t) · sinθ(t)) ·dt (6)
Fs,sg =
∫ 3T
4
T
4
(−FC(t) · sinθ(t)+FN(t) · cosθ(t)) ·dt (7)
where T = 1f is the foil oscillation period.
Finally we define the force coefficient as
Ft
Fs ,y
F (t)N
U
1
b
q(t)
F (t)C
h(t)
ControlVolume
U2(y)
x
A B
C D
Figure 1: Definition of foil kinematics
C =
F
1
2ρwaterU2∞ · c · s
(8)
where s= 1 and s= 0.2 for the PIV and strain gauge thrust mea-
surements respectively. Furthermore, we also define a dimen-
sionless term analogous to the lift:drag ratio, which we name
the “productivity”, η
η=
Ct
Cs
×100% (9)
Hydrogen bubble visualisation was also performed on the un-
steady foil for a smaller number of foil heave and pitch combi-
nations. The hydrogen bubble wire (50µm tungsten wire) was
placed spanning the entire length of the foil on both sides, corre-
sponding to the location of the maximum thickness of the foil.
Images of the flow, viewed from the foil spanwise direction,
were captured using a Canon EOS 20D digital still camera with
a Canon EF 50mm f/1.4 USM lens, at a fixed rate of 8 frames
per second.
Results and Discussion
The experiments consisted of force acquisition and PIV mea-
surements across a large number of foil parameters and flow
conditions at 1500 < Rec < 12500. In this paper, we limit our
discussion to two foil heave/pitch combinations, Group I and
Group II. In Group I, the foil undergoes moderate heave and
pitch amplitudes, h0c = 0.5 and θ0 = 30
o, which are represen-
tative of the majority of the investigated flow conditions. In
Group II, the foil undergoes purely heaving motions, h0c = 0.5,
θ0 = 0o, which are representative of heave-dominated flow con-
ditions.
The PIV-derived thrust coefficients and the direct strain gauge
thrust and side force coefficients for both groups I and II are
shown in figure 2, for a range of Strouhal numbers. The agree-
ment Ct,piv and Ct,sg for both cases (figures 2a and b) is ex-
cellent, indicating that the force measurements are, most likely,
accurate.
Interestingly, in all measured cases, we do not observe a sudden
drop-off in the thrust coefficient as Sth and αmax increases, even
beyond the steady-state stall angle (which is estimated to be
≈ 7◦−10◦ at Re≈ 10000). In fact, the foil still produces thrust
even beyond αmax > 65o (for example, αmax = 66o at Sth =
0.714 for Group I) without any sign of stall. Since Ct and Cs
show a strong relationship to St2h (which was also noted by [3]),
this implies that Ct on the foil can be increased to arbitrarily
large values simply by increasing Sth.
Based on these results, it is not surprising that in most flow
cases, as represented by group I, the thrust coefficients achieved
1004
LeadingEdge
Vortex
(a) Group I, h0c = 0.5, θ0 = 30
o, Sth = 0.36, αmax ≈ 18.5o
Leading Edge
Vortex
(b) Group II, h0c = 0.5, θ0 = 0
o, Sth = 0.36, αmax ≈ 48.5o
Figure 3: Hydrogen bubble visualisation of unsteady foil at its
minimum heave position
are moderately high, on order of Ct ≈ 3− 4 at large Sth ≈ 0.7
(figure 2a). Furthermore, when the heave amplitude is large, the
thrust coefficients developed by the foil are very large, exceed-
ing Ct > 10 at Sth ≈ 0.7 (figure 2b), where the maximum angle
of attack is also very large (αmax ≈ 65o).
The results of the hydrogen bubble visualisation indicate that
flow around the foil at these high heave amplitudes is domi-
nated by the presence of a strong leading edge vortex (as shown
in figure 3b), whereas for most other foil parameters, the lead-
ing edge vortex is either small or non-existent (figure 3a). This
provides strong evidence that the large Ct values produced by
the heave-dominated foil are linked to the generation of a large
leading edge vortex, as previously suggested by [1].
The productivities for both groups asymptote to η ≈ 40−50%
as Sth increases, suggesting that the large values ofCt developed
at high Sth coincide with similar increases in Cs. Denoting the
mean foil velocity magnitude in the cross streamwise direction
as
∣∣h′(t)∣∣, and assuming that the moments on the foil are small,
we can estimate the propulsive (Froude) efficiency as
ηF ≈ Ft ·U∞
Fs · |h′(t)|
∝
η
Sth
(10)
since it can be shown from equation 1 that U∞|h′ (t)| ∝
1
Sth . This
implies that the propulsive efficiencies are low at large Sth.
Of additional interest is the x-intercept in figures 2a and 2b,
which is the Sth at which the foil is neither producing nett thrust
nor drag. This value, St0 ≈ 0.15− 0.3 in most flow cases, is
consistent with the conclusions of our previous study [4]. Thus,
if we assume that the forebodies of fish have very low drag, this
explains the fact that steadily swimming carangiform fish swim
have been observed to swim at St ≈ 0.25 [6, 8].
Based on these results, we propose two different modes of
propulsion :
1. “Cruise Mode” : Ct is low to moderate, and propulsive
(Froude) efficiencies are, most likely, relatively high. This
occurs at moderate heave and pitch amplitudes, and at
Sth ≈ 0.15−0.3.
2. “Burst Mode” : Ct is very large, and propulsive (Froude)
efficiencies are relatively low. The non-dimensional heave
amplitude and Sth should be large.
Conclusion
The force measurements around the heaving and pitching hy-
drofoil indicate that the foil can generate large thrust coeffi-
cients, in excess of Ct > 10 . These large thrust coefficients
occur when the foil heave amplitude is very large, where strong
leading edge vortices are observed. In all other flow cases,
where the heave amplitude is not very large, the thrust coeffi-
cients are moderate, and the leading edge vortices on the foil
are either small or non-existent. We provide evidence that the
generation of large thrust coefficients by the foil is inextricably
linked to the generation of a strong leading edge vortex on the
foil, confirming the proposition made in [1].
References
[1] Anderson, J. M., Streitlien, K., Barrett, D. S. and Tri-
antafyllou, M. S., Oscillating foils of high propulsive effi-
ciency, Journal of Fluid Mechanics, 360, 1998, 41–72.
[2] Fish, F. E. and Lauder, G. V., Passive and active flow con-
trol by swimming fishes and mammals, Annual review of
fluid mechanics, 38, 2006, 193 – 224.
[3] Hover, F. S., Haugsdal, O. and Triantafyllou, M. S., Ef-
fect of angle of attack profiles in flapping foil propulsion,
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alisation of a pitching and heaving hydrofoil, in Proceed-
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1005
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−1
0
1
2
3
4
5
Heave-based Strouhal Number, Sth
T
h
ru
st
C
o
effi
ci
en
t,
C
t
PIV Strain Gauge
(a) Group I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−2
0
2
4
6
8
10
12
Heave-based Strouhal Number, Sth
T
h
ru
st
C
o
effi
ci
en
t,
C
t
PIV Strain Gauge
(b) Group II
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
1
2
3
4
5
6
7
Heave-based Strouhal Number, Sth
S
id
e
F
o
rc
e
C
o
effi
ci
en
t,
C
s
(c) Group I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
5
10
15
20
25
Heave-based Strouhal Number, Sth
S
id
e
F
o
rc
e
C
o
effi
ci
en
t,
C
s
(d) Group II
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−120
−100
−80
−60
−40
−20
0
20
40
60
80
Heave-based Strouhal Number, Sth
P
ro
d
u
ct
iv
it
y,
η
(%
)
(e) Group I
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
−20
−10
0
10
20
30
40
50
60
Heave-based Strouhal Number, Sth
P
ro
d
u
ct
iv
it
y,
η
(%
)
(f) Group II
Figure 2: Thrust and side force coefficients, Ct and Cs, and productivities η (%) for the unsteady foil for a,c,e) group I, h0c = 0.5,
θ0 = 30o and b,d,f) group II, h0c = 0.5, θ0 = 0
o
1006


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The forces on a fish-inspired unsteady hydrofoil

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