Integrative and Comparative Biology, volume 50, number 6, pp. 1155–1166
doi:10.1093/icb/icq020
SYMPOSIUM
Turbulence: Does Vorticity Affect the Structure and Shape of
Body and Fin Propulsors?
P. W. Webb1,* and A. J. Cotel†
*School of Natural Resources and the Environment, University of Michigan, Ann Arbor, MI 48109, USA;
†
Department of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI 48109, USA
From the symposium ‘‘Contemporary Approaches to the Study of the Evolution of Fish Body Plan and Fin Shape’’ presented
at the annual meeting of the Society for Integrative and Comparative Biology, January 3-7, 2010, Seattle, Washington.
1
E-mail: [email protected]
Synopsis Over the past century, many ideas have been developed on the relationships between water flow and the
structure and shape of the body and fins of fishes, largely during swimming in relatively steady flows. However, both
swimming by fishes and the habitats they occupy are associated with vorticity, typically concentrated as eddies characteristic of turbulent flow. Deployment of methods to examine flow in detail suggests that vorticity impacts the lives of
fishes. First, vorticity near the body and fins can increase thrust and smooth variations in thrust that are a consequence of
using oscillating and undulating propulsors to swim. Second, substantial mechanical energy is dissipated in eddies in the
wake and adaptations that minimize these losses would be anticipated. We suggest that such mechanisms may be found in
varying the length of the propulsive wave, stiffening propulsive surfaces, and shifting to using median and paired fins
when swimming at low speeds. Eddies in the flow encountered by fishes may be beneficial, but when eddy radii are of the
order of 0.25 of the fish’s total length, negative impacts occur due to greater difficulties in controlling stability. The
archetypal streamlined ‘‘fish’’ shape reduces destabilizing forces for fishes swimming into eddies.
Introduction
Flows are often visualized and described in terms of
streamlines. Vorticity is defined as the curl of the
velocity vector, resulting in curvature of those
streamlines. Streamlines may curve in both laminar
and turbulent flows, so both may contain vorticity.
However, in laminar water flow, it is common for
streamlines to remain parallel to each other. In contrast, in turbulent flows, streamlines follow complex
paths with many different orientations relative to
each other. Turbulent flows are not truly chaotic,
but rather contain general structure in the form of
eddies when high levels of vorticity are present.
Laminar and turbulent flows are distinguished on
the basis of Reynolds number, Re, the ratio of inertial to viscous forces. Numerically, Re ¼ Lu/,
where L ¼ a length scale, u ¼ velocity of the flow,
and ¼ kinematic viscosity of the fluid (Vogel
1994). Flow is laminar at small Re, becoming turbulent at larger Re, on the order of 103, usually coupled
with high vorticity throughout the flow.
During much of the 20th century, studies of fish
locomotion paid little direct attention to vorticity
although much debate focused on whether the
boundary layer of a swimming fish might be laminar
or turbulent (Webb 1975; Blake 1983; Videler 1993;
Anderson et al. 2001). There was also interest in the
potential for motions of the body and fins to create
or manipulate eddies and thereby reduce the amount
of energy expended for swimming (Rosen,
1959; Ahlborn et al. 1991; Drucker and Lauder 2002).
In practice, vorticity is ubiquitous around fishes
and in their habitats and would be expected to
have many effects on the structure and form of the
body and fins and on behavior. Flows are driven by
gravity, the Coriolis effect, wind-driven waves, and
anthropogenic sources (e.g. boat-driven waves).
Advanced Access publication April 21, 2010
ß The Author 2010. Published by Oxford University Press on behalf of the Society for Integrative and Comparative Biology. All rights reserved.
For permissions please email: [email protected].
1156
Because of the viscosity of water, velocity gradients
arise, primarily from interactions between the flow
and surfaces, ranging from the fish’s body and fins,
through objects embedded in flows (e.g. protuberances of coral, large woody debris, macrophytes,
and other ‘‘roughness’’ elements), to boundaries of
the bottom and along shorelines. Shear forces due to
viscosity in these velocity gradients are the major
sources of vorticity and of the eddies that are characteristic of turbulent flow.
There has been little consideration of the potential
for vorticity to affect fishes because conceptual and
physical tools have been lacking. This changed
toward the end of the past century, when methods
were deployed to show details of flow and the vorticity near fishes and in their habitats (Stamhuis and
Videler 1995; Pavlov et al. 2000; Müller et al. 2001;
Crowder and Diplas 2002; Drucker and Lauder 2002;
Lupandin 2005; Lauder and Tytell 2006; Tytell 2007;
Liao 2008; Webb et al. 2010). Results from such
studies suggested that some ideas about fish’s swimming and the form of the body and fins needed
revisiting. Traditional ideas that losses of mechanical
energy for a swimming fish occurred in the boundary
layer is challenged by the recognition of large losses
of energy in the wake (Schultz and Webb 2002;
Tytell 2004, 2007). The boundary layer is the thin
region of flow adjacent to any surface in which the
velocity of the flow changes from that of the surface
to which the water sticks to the velocity of the free
stream. Velocity gradients are high in the boundary
layer, so that viscous forces—a component of drag
resisting motion—are high. Eddies shed into the
wake carry substantial energy that is lost at the trailing edge—the most posterior edge—of propulsors,
but wakes shed by upstream fins may interact with
downstream fins to improve production of thrust. In
addition, fishes in their natural habitats are embedded in flows whose vorticity varies over many orders
of magnitude. Eddies not only provide opportunities
but also have negative effects on performance and on
the ability of a fish to maintain posture in the flow.
As a result, the presence of vorticity could affect
habitat selection and behavior (Pavlov et al. 2000;
Galbraith et al. 2004; Cotel et al. 2006; Smith et al.
2006).
Thus, vorticity has implications for fish biology,
from the structure and shape of the body and fins,
through relationships between the structure and
function of propulsors and behaviors affecting ecology and evolution, to fisheries management and
designs of fishing gear and fishways (Castro-Santos
et al. 2009). Here we consider principal areas in
which vorticity appears to impact the lives of fishes.
P. W. Webb and A. J. Cotel
Vorticity near the body and fins
Discussions of the role of the form of the body and
fins on the flow around swimming fishes have a long
tradition. Historically, much attention was paid to
the boundary layer of fishes using the body and
caudal fin to swim (BCF swimming). An important
concept is streamlining. Human engineers working
with rigid bodies found fusiform shapes, with an elliptical shape along the longitudinal axis for the anterior third of the body length before tapering to the
trailing edge, and with ratios of length/width of
about 5–7 (Ahlborn et al. 2009) maintained laminar
flow, delayed transition to turbulent flow, and minimized or prevented boundary-layer separation, all
reducing drag. The fusiform shapes of the bodies
of pelagic fishes are streamlined similar to vehicles
engineered by humans, and the shape is assumed to
minimize losses of energy for such fishes (Webb
1975; Blake 1983; Videler 1993).
It is generally thought that boundary-layer flow
varies along the length of a fish, with flow more
likely to be laminar anteriorly, transitioning to attached turbulent flow posteriorly. A turbulent
boundary layer is also considered more likely for
swimmers moving like eels versus fishes such as
trout, jacks and tunas, and for fishes with roughen
skins (placoid scales of elasmobranchs and similar
factors) (Webb 1975; Aleyev 1977; Blake 1983).
Larger and faster fishes swim at high Re when the
boundary layer is more likely to be turbulent. Direct
observations using particle image velocimetry (PIV)
of the boundary-layer during swimming have been
made on only two species, the dogfish, Mustelus
canis, and scup, Stenotomus chrysops (Anderson
et al. 2001); data are desirable on vorticity in the
boundary layer for a broader range of species with
different shapes of their bodies and fins, and over a
range of sizes of fish.
Vorticity is also generated outside of the boundary
layer but close to the surfaces of the body and fins by
propulsive movements. It is known that the nature of
vorticity over the surface of insects’ wings can have
large effects on the creation of thrust (Lighthill 1975;
Lewin and Haj-Hariri 2003; Wang 2005). Similar
principles should be applied to the fins of fishes,
but data are lacking. PIV observations have been
made of flow near the body during BCF propulsion.
Eel-like fishes, with characteristic anguilliform motions, swim by passing a wave of bending backward
along the body. There is more than a whole wave
within the body length. Amplitude increases caudally,
but is relatively large over the whole length of the
body. Vorticity shed along the body appears to
1157
Vorticity and propulsors
Fig. 1 (A) Mechanical energy budgets based on ESBT (Lighthill 1975) and (B) an energy budget recognizing energy losses in vorticity of
the wake for a given common limiting input from the muscles and metabolism.
smooth variation in thrust that can waste energy due
to accelerations and decelerations of the body when
thrust varies from beat-to-beat of the propulsor
(Videler et al. 1999; Müller et al. 2001; Tytell et al.
2004, 2007). It has also been suggested that creation
and control of a vortex can reduce locomotor costs,
even reducing these toward zero (Rosen 1959;
Ahlborn et al. 1991). However, PIV evidence from
swimming fishes over the past two decades does not
support such conclusions.
Vorticity in wakes
Enhancing thrust
Wakes are shed from the sharp trailing edges of the
body and fins. Energy associated with vorticity of
eddies shed at the trailing edge of propulsors is
lost mechanical energy, but in BCF swimming,
eddies shed upstream can interact with propulsors
downstream and thereby enhance thrust (Rockwell
1998; Triantafyllou et al. 2002; Tytell 2006). The interaction is especially clearly explained by Drucker
and Lauder (2002) for vorticity shed from the
dorsal fin of bluegill, Lepomis macrochirus, which
combines with and adds to that of the caudal fin.
Enhancement of thrust due to additive assimilation
of vorticity shed upstream by downstream fins in
BCF swimmers depends on appropriate phase relationships between the motions of fins. This, in turn,
depends on the spacing of upstream and downstream
fins and their kinematics. The anterior dorsal fin and
the caudal fin can interact to enhance thrust for
elasmobranchs that swim with anguilliform body
motions. The distance between the trailing edge of
the anterior fin and the caudal fin is approximately
50% of the total length (L), so that the distance between the fins can be considered relatively large. The
propulsive wavelength for anguilliform swimmers is
typically small relative to L (Lighthill, 1975; Webb
and Keyes 1982). In highly derived acanthopterygian
fishes, such as the bluegill, the distance separating the
interacting fins is smaller, approximately a third or
less of the body length, but the wavelength of their
swimming motions is longer compared to L
(Drucker and Lauder 2002; Standen and Lauder
2005; Tytell 2006).
Losses of energy in the wake
Deployment of PIV to visualize the wakes of swimming fishes, computational fluid dynamics (CFD),
and physical models show that substantial energy is
dissipated in the wake shed by propulsors (Videler
et al. 1999; Müller et al. 2001; Schultz and Webb,
2002; Tytell 2004, 2006, 2007; Borazjani and
Sotiropoulos 2010). This requires revisiting the mechanical energy budget, especially for BCF swimmers
(Fig. 1). It is not usually possible to separate sources
of momentum loss (drag) to momentum gain
(thrust) for such swimmers, which has led to
debate concerning magnitudes of drag and ultimately
of efficiency (Schultz and Webb 2002; Tytell 2004,
2006, 2007). In contrast, for fishes swimming using
median and paired fins (MPF propulsion), the flow
induced by the propulsor is largely outside of that of
the body, especially when these fins have a base that
is a small proportion of the length or width of the
fin (Blake 1983). MPF swimming facilitates separate
analysis of drag, thrust, and wake dynamics.
Most recent views of the magnitudes of
mechanical-energy components for BCF swimmers
derive from applications of elongated-slender-body
theory (ESBT) (Lighthill 1975). ESBT recognizes
that changes in momentum during swimming can
be simplified when the body is elongated and slender, with the width of the body and amplitudes of
lateral motions at the trailing edge being 0.2L
(Lighthill 1975). The total rate of working, Wtotal,
of the propulsive wave determined from ESBT is
consistent with the power that can be made available
to propulsors from the muscles and metabolism
1158
(Webb 1975; Blake 1983; Videler 1993). The model
has been very effective in evaluating many aspects of
how the shape of the body and the size and location
of median fins affect swimming and locomotory
behavior.
In order to consider how new observations on
vorticity shed into the wake at the trailing edge
affect the mechanical energy budget, imagine that
the total rate of work, Wtotal, can be partitioned
into discrete categories of energy uses. ESBT concludes that the largest portion, 80% of this mechanical energy, is useful energy, or thrust energy,
required to overcome drag, Drag energy, DE, from
now on (Fig. 1A). Approximately 20% of the mechanical energy is kinetic energy, KE, an energy loss
due to accelerating water replacing losses of momentum associated with drag. ESBT models predict that
DE KE, so that the Froude efficiency, defined as
¼ DE/(DE þ KE), is large, typically 480% (Lighthill
1975).
Analysis of wakes visualized with PIV and determined by modeling suggests that substantial energy
is carried as vorticity in the wake. This wake energy,
WE, is lost energy (Fig. 1B), with WE4DE, leading
to the Froude efficiency being on the order of 25%
(Schultz and Webb, 2002; Tytell 2004, 2006, 2007;
Borazjani and Sotiropoulos 2010). The high DE predicted by ESBT for BCF swimmers was postulated to
arise from swimming motions thinning the boundary
layer. This was presumed to increase shear and hence
DE by a factor of at least 3 to 5. Instead, high WE
implies that drag is smaller than expected by ESBT,
and apparently similar to that of equivalent
human-engineered rigid bodies (Andersen et al.
2001). Consequently, it appears that swimming motions and postulated thinning of the boundary layer
do not have a large effect on drag. There have been
some apparently successful attempts to test the idea
that high DE increases drag by manipulating the area
of the tail (Webb, 1975). However, with the wisdom
of hindsight, such methods also imposed destabilizing forces, and recent recognition of the high costs of
stability (Webb 1992, 2006) provides a more plausible explanation for the effects of these manipulations
on the mechanics and energetics of swimming.
P. W. Webb and A. J. Cotel
locomotion could increase the surplus, which
should translate into fitness benefits. Therefore,
given the magnitude of WE, mechanisms should be
anticipated for its reduction. Such mechanisms may
occur at a variety of organizational levels that affect
the form of the body and fins. Our current understanding of how fishes swim suggests adaptations are
especially likely that affect the orientation of the
trailing edge, including the shape of propulsive
waves.
Factors affecting the orientation of the trailing
edge of the caudal fin
Most studies pertaining to potential impacts of the
orientation of the caudal fin on wake energy, and
possible factors affecting that orientation, derive
from analyses of BCF swimming. Recent analyses of
wakes (Tytell 2004, 2007) and CFD experiments
(Borazjani and Sotiropoulos 2010) of BCF swimmers
show that WE for a given body form is affected by
the length of the propulsive wave, . The speed attainable and efficiency are also affected (McHenry
et al. 1995). Lighthill’s (1975) ESBT and the modification for large amplitudes forecast Wtotal at a given
swimming speed increases with increasing while
keeping the other kinematic variables constant
(Fig. 2).
Thus, swimming with a smaller appears to be
more economical than using large so that a smaller
should be advantageous by reducing total daily
energy costs (McHenry et al. 1995). However, in
Minimizing energy losses to the wake
The energy expended for swimming constitutes the
largest component of the daily energy budget of
fishes (Brett, 1995). Of the energy assimilated by
an animal, the surplus after meeting requirements
for all metabolic functions is available for growth
and reproduction. Reducing costs of energy for
Fig. 2 Relationships between the total rate of work of the
propulsive wave and the swimming speed for an idealized
sub-carangiform fish with various specific wavelengths normalized
by total body length. The total length, L, of the fish is 25 cm,
trailing edge span and amplitude 0.2L and tail-beat frequencies
are calculated at various swimming speeds using Bainbridge’s
equation (Bainbridge 1958).
1159
Vorticity and propulsors
order to swim, the speed of the propulsive wave, c
(the product of tail-beat frequency and ), must
exceed swimming speed, u. Thus, u/c must be less
than unity. Many independent studies have shown
that, u/c!1 as swimming speed increases. When is small, u/c!1 at lower swimming speeds. As a
result, Wtotal reaches a maximum at progressively
lower speeds as decreases and may become too
small to provide the power required that increases
with increasing swimming speed (Fig. 2). Thus, small
may reduce swimming performance (Webb 1988;
McHenry et al. 1995; Plaut 2000). Another benefit of
larger has also recently been shown applicable to
larger and faster fish, because the Froude efficiency is
improved at high Re (Borazjani and Sotiropoulos
2010).
The shape of the propulsive wave is determined by
muscle-driven deformations of the body working
against hydrodynamic forces, further tuned by stiffness of the body and fins (McHenry et al. 1995;
Summers and Long 2006; Zhu and Shoele 2008;).
The stress–strain curve for biological materials is notoriously J-shaped, with low stiffness at low imposed
load and strain, and increasing stiffness at highly
imposed load and strain. Bending stresses on the
vertebral column and the fin rays will increase with
tail-beat frequency and hence at greater swimming
speeds. Therefore, stiffness should increase at higher
swimming speeds (Summers and Long 2006) and
affect energy shed to the wake.
The stiffness of the fin web, and hence angles of
the trailing edge relative to the flow over the fin
could be affected by the structure of fins in many
ways; the shapes of fin rays affect the second
moment of area; connective tissue between fin rays
or the winding of the ceratotrichia in elasmobranchs
may vary in elastic properties; the development and
activation of muscles associated with the fin rays
could directly modulate stiffness (Geerlink and
Videler 1987; Lauder et al. 2006; Summers and
Long 2006; Taft et al. 2008; Flammang and Lauder,
2009).
The potential for independent control of the orientation of the tail has been explored using wing
theory for the fastest thunniform swimmers among
large elasmobranchs, teleosts, ichthyosaurs, and cetaceans (Lighthill 1975). In these vertebrates, a
lunate-shaped caudal fin is attached to the body by
a narrow caudal peduncle providing the opportunity
for substantial independence in the motions of the
tail. The production of thrust and the Froude efficiency are influenced by relationships between the
angle, or pitch, of the tail and the lateral motions
(vertical motions in cetaceans), or heave of the tail.
It seems probable that the anatomy of thunniform
vertebrates permits control of pitch and heave relationships that maximize thrust or efficiency depending on behavioral needs.
Clearly, more detailed information on the structure and properties of the fins of fishes, their detailed
kinematics (Jayne and Lauder 1996), experimental
manipulation of fins, and analysis using CFD
(Borazjani and Sotiropoulos 2010) all are desirable
to explore the potential to reduce WE and hence
reduce overall swimming costs.
Alternative propulsors
At a coarser scale, losses of energy associated with
WE in BCF swimming might be circumvented by
using other propulsors. High speeds and acceleration
rates depend on harnessing the large mass of myotomal muscle. Therefore, alternative propulsors
would be expected only at lower swimming speeds
that contribute most to the daily metabolic energy
budget (Brett 1995). Swimming modes are especially
diverse at lower speeds, involving median and paired
fins in MPF propulsive modes (Breder 1926). MPF
propulsors can be more economical than BCF propulsors. Therefore, overall metabolic costs may be
reduced by using MPF swimming modes
(Korsmeyer et al. 2002; Jones et al. 2007; Kendall
et al. 2007), especially rowing and under-water
flight using the pectoral fins in the labriform swimming modes, sculling with combined dorsal and ventral fins in the balistiform mode, and swimming with
combined pectoral, dorsal, and anal fins in the tetraodontiform mode (Breder 1926; Fulton and
Bellwood 2005; Johansen et al. 2007). Perhaps the
diversity of MPF swimming modes reflects, at least
in part, multiple solutions among species to reduce
high WE during routine activities.
New opportunities: Minimizing drag versus
minimizing energy lost to the wake
Nearly a century of influence by ‘‘Gray’s Paradox’’
(the 1930s speculation based on the best knowledge
of the time, now known with the wisdom of hindsight to be incomplete) that suggested drag exceeded
the energy available from muscular activity and metabolism, spawned a plethora of ideas about
drag-reducing mechanisms. The current view that
WE4DE argues that these mechanisms reflect marginal savings in energy for a much smaller component of the mechanical energy budget than
previously thought. Therefore, mechanisms reducing
WE would seem much more probable. Furthermore,
mechanisms postulated to minimize the drag of the
1160
body during BCF swimming are largely based on
analogy with practices for human-engineered vehicles
(Blake 2004). In contrast, ideas on reducing WE are
amenable to direct experimental evaluation with current techniques, while methods such as CFD provide
unparalleled opportunities for evaluating the consequences of even small changes in the motions of
propulsors. These studies could be coupled with phylogenetic analyses to explore adaptations in a rigorous evolutionary framework.
Vorticity in the incident flow
Vorticity near swimming fishes and in their wakes
clearly affect form and function in fishes. Vorticity
is also characteristic of fishes’ habitat where turbulence would be expected to affect fishes through
mechanisms that differ from those in the flow near
the body and in the wake (Webb et al. 2010). In the
real world occupied by fishes, turbulent flow, with its
concentrations of vorticity in the form of eddies of
various shapes and sizes, is created by surface waves
and by currents. Eddies shed in natural flows depend
on hydraulic parameters such as the longitudinal and
cross-sectional shape of stream channels and the bathymetry of shorelines, by flow interacting with surface roughness and with objects such as macrophytes,
boulders, large woody debris, and by the wakes shed
by other organisms. Eddies grow larger over time as
they move downstream and entrain more water. The
maximum size of an eddy is determined by the dimensions of the channel and basin. Eddies also spin
off smaller eddies, the smallest of which in inertial
flows is called the Kolmogorov scale. These smallest
eddies disappear as viscous effects dissipate their kinetic energy as heat. Thus, there is a large range of
eddies with radii from 4100 s of kilometers in oceanic gyres to very few millimeters at the Kolmogorov
scale. Furthermore, the number of eddies decreases
exponentially with eddy size so that there are many
more small eddies than large eddies (Tritico and
Cotel 2010).
There are many models for eddies. Commonly,
eddies have a core, in which vorticity increases
with distance r from the center, followed by a
larger region where vorticity decreases asymptotically
with r toward free-stream values. The radius of an
eddy, R, is then defined with respect to some r where
vorticity is close to background levels, typically
within 1% of that in the free stream (Drucker and
Lauder 1999).
Absolute values for eddy sizes are less important
than their size relative to that of a body embedded in
the flow; only eddies of appropriate size relative to
P. W. Webb and A. J. Cotel
fish size affect fishes (Pavlov et al. 2000; Odeh et al.
2002; Nikora et al. 2003; Liao 2008; Webb et al.
2010). Eddies with R L have negligible impacts.
These eddies are common enough that their perturbations would be small and they will be distributed
over the entire length of the body resulting in no-net
displacement of an embedded body. However, small
eddies, sometimes called micro-turbulence (Bell and
Terhune 1970), may induce transition of the boundary layer to turbulent flow and hence have a small
effect on drag. At the other extreme are very large
eddies as large as oceanic gyres. The rotational flow
in such eddies with R L and with time scales 410s
of hours or more days is treated as rectilinear flow by
embedded organisms. Such flow may be important
in migration (Brett 1995).
Eddies in streams with R ranging from approximately 0.15L to several body lengths have been postulated to be used by fishes for holding station
and/or to facilitate transport (McLaughlin and
Noakes 1998; Hinch and Rand 1998, 2000). Objects
such as woody debris in a flow shed eddies in a
regular fashion, first from one side of the object,
and then the other, each eddy alternating in rotation,
forming a Kármán vortex street. Fish can draft between the eddies of such a Kármán vortex street
comprising eddies with R ¼ 0.15L–0.25L, using few
swimming motions and with reduced energy expenditure; this special swimming pattern is now known
as the Kármán gait (Liao et al. 2003). The range of
sizes of eddies relative to fish size over which fish can
use the Kármán gait is not yet known. Fishes are also
postulated to ride the component of eddy flow in the
direction that a fish wishes to go, thereby facilitating
migration. Although fishes are able to detect and
utilize eddies (Hinch and Rand 1998; McLaughlin
and Noakes 1998; Standen et al. 2004), explicit
demonstration that fishes choose paths in eddydominated turbulent flow in ways that reduce
energy costs or transport time is still lacking.
In contrast to positive impacts of eddies on fishes,
those with R of the order of 0.25L–0.5L can decrease
swimming speed, increase metabolic rate, and may
lose control in spills (Pavlov 2000; Odeh et al.
2002; Enders et al. 2003; Lupandin 2005; Webb
et al. 2010; Tritico and Cotel 2010). The common
mechanism underlying these negative effects is postulated to be eddy vorticity that, in turn, perturbs the
sensory–motor system associated with the deployment of the body and/or fin systems that control
body posture and trajectories. These control systems
are eventually overwhelmed by high vorticity, leading
to sudden changes in posture or to the breakdown of
swimming trajectories (Webb et al. 2010). However,
Vorticity and propulsors
using only the size of eddies and their relative size to
that of a fish is inadequate to explain the challenge of
eddies to locomotor control.
The impacts of an eddy on numerous ecological
processes (e.g. erosion of substrate, mixing of gases
and other dissolved substances in the water column,
contagion rates of food items and consumers, in
damaging animals and plants, challenging and overwhelming control systems) all depend on vorticity
and on parameters derived from vorticity such as
circulation (the integral of vorticity over the
cross-sectional area of an eddy) and shear rates in
the velocity gradient through an eddy. Thus, the important physical characteristics of an eddy likely to
displace fishes are momentum due to the angular
velocity of flow in the eddy or the forces and
energy associated with eddy rotation. Eddies may
briefly interact with a fish, so that impulse may be
an appropriate measure, integrating force over a specified time. In each situation, the consequences of
the magnitude of a physical measure depend on its
value relative to that of the fish. Thus, failure of
control occurs when momentum of the impulse
and circulation of eddies in the flow reach the
same order of magnitude as those of a fish (Cotel
and Webb 2010). Therefore eddy/fish ratios of
1161
momentum, impulse, forces, and energy are more
appropriate measures of size of eddies.
It is not known how a fish and eddy interact to
challenge fishes’ control systems leading to negative
impacts on performance and energy expenditure.
Simultaneous PIV recording of flow and paths of
fishes entering and responding to displacements in
eddies is challenging and adequate data are lacking.
In the absence of direct observations, we explore
boundary conditions at which failure might occur,
using a simple physical approach to imagine morphological and functional features of fishes that
might affect fish/eddy interactions.
Fishes may not experience the impact of entire
eddies, but are more likely to encounter some part
of an eddy. A fish may enter an eddy at any point
and from any direction. The limits of control should
be exceeded when forces acting on a fish are largest,
which occurs for a fish penetrating along the eddy’s
diameter (Fig. 3) and we simplify discussion by considering this boundary condition.
To explore the magnitudes of forces acting on a
fish, consider an idealized neutrally buoyant fish, cylindrical in cross-section, 10 cm in total length,
10 cm3 in volume, penetrating an inviscid eddy
with length L, along the eddy’s diameter.
Fig. 3 A schematic representation of a hypothetical cylindrical fish swimming through an eddy. As the fish enters the eddy the
cumulative absolute force increases as more elements of the body experience vorticity, until the fish is fully immersed in the eddy.
Declining cumulative forces then parallel the rise as the fish passes out of the eddy. Cumulative forces are normalized with the
maximum value experienced when the fish is just immersed in the eddy. Penetration into the eddy is normalized by fish total length.
The fish is 10 cm total length, 10 cm3 in volume, with neutral buoyancy. The eddy, in inviscid flow has a radius of 5 cm and vorticity of
10 per second.
1162
Based on Tritico (2009), consider a reference eddy at
which fishes lose control of stability, with R of 5 cm
and vorticity equal to 10 per second. We further
simplify the situation by neglecting trajectory
changes. Assumptions of inviscid conditions and neglecting trajectory changes would affect magnitudes
of forces but do not change conclusions. Similarly,
we neglect the effect the embedded body would have
on the eddy. In part, normalizing results with reference cases minimizes potential error. The perturbation caused by a fish entering an eddy is likely to be
small compared with the impact of the eddy on the
fish.
We focus only on side forces. Torques would increase in the same way as the cumulative absolute
force summed along the body, but with larger magnitudes. The force on any element of the body at
any point in the eddy is taken as proportional to
dA ur2 where dA is the projected lateral area of
an element along the fish’s body and ur is the local
tangential velocity at a distance r from the center of
an eddy rotating at ! s1. Thus, as the idealized
cylindrical fish enters an eddy (Fig. 3), the total
side force increases from A to B as more of the
body enters the eddy until half the body is in the
eddy. The cumulative absolute force increases at a
lower rate as the fish pushes further into the eddy
because ur decreases toward the center of the eddy.
The cumulative absolute force on the fish increases
further from B to C as the body passes through the
center of the eddy, with the leading elements again
experiencing higher ur until the fish is fully immersed in the eddy. The cumulative absolute force
subsequently declines as the fish passes out of the
eddy from C to E in the same way as forces grew
as the fish entered the eddy.
Many real fishes are not cylindrical in crosssection. The magnitude of the cumulative absolute
force for a given penetration is directly proportional
to the ratio of depth/width for a fish of fixed volume
and length. A compressed fish would experience
greater destabilizing forces and torques than would
a cylindrical fish exposed to vertical eddies, while a
depressed fish would be less stressed by vertical
eddies. The opposite would apply for horizontal
eddies. This speculation is consistent with both laboratory and field observations (Webb et al. 2010).
The depth of the body and fins varies along the
length of real fishes (Fig. 4A). The entry shape of a
fusiform fish, with depth increasing with distance
from the nose, would reduce the growth of the cumulative force on the fish’s body entering an eddy
compared to a fish of constant depth (Fig. 4B). For a
given penetration into an eddy, the cumulative force
P. W. Webb and A. J. Cotel
Fig. 4 Postulated effects of body form on the cumulative absolute
force as a fish enters an eddy. (A) The depth distribution for the
fish, based on creek chub. (B) Cumulative absolute side forces
normalized by the maximum value for all options for various
combinations of body and fin, and for a constant reference depth.
The box shows the range of penetration anticipated for routine
swimming speeds from 1 to 3 L/s and a response latency of
100 ms. Penetration into the eddy is normalized by total length
of the fish. Parameters of the fish and eddy are the same as for
Fig. 3.
is lower for a real fish’s shape than for one of constant depth until the dorsal fin enters, after which the
cumulative force due to the successive median fins
remains higher. Thus, dorsal and anal fins add to the
destabilizing forces on a fish entering an eddy.
When the shedding of vortices is predictable,
fishes can use them to hold station, such as with
the Kármán gait. However, over much of a flow in
habitats occupied by fishes, eddies are created,
grown, and spin off smaller eddies as they move
downstream. Eddies may be destroyed as downstream structures absorb them and create their own
wakes, so that many eddies will be transient. As a
result, the frequency at which eddies are encountered, or the occurrence of an eddy with strength
high enough to challenge control of stability, is
often unpredictable. Then temporal scales, whereby
perturbing properties of eddies vary over time relative to the time it takes a fish to respond, affect the
magnitudes of destabilizing forces experienced by a
fish and are expected to affect eddy/fish interactions
(Webb et al. 2010; Cotel and Webb 2010).
Consequently, the magnitude of destabilizing forces
experienced by a fish will depend on the response
time between sensing perturbing flows and initiating
a response. The latency in initiating complex
1163
Vorticity and propulsors
maneuvers and responding to instabilities are on the
order of 100 ms (Webb 2004, 2006). At typical routine
cruising speeds of 1 to 3 L/s, a fish would penetrate the
modeled eddy to 0.1–0.3R before a motor response
might be expected (box in Fig. 4B). For this simple
model, the fusiform anterior of a fish reduces the
growth of such forces during the latency period compared to a reference fish of constant depth. It is generally considered that the fusiform shape typical of
fishes reduces coasting drag and recoil during BCF
swimming (Lighthill 1977; Webb 2006). This analysis
suggests that a fusiform shape would also reduce displacements due to incident flow vorticity, such that a
fish should have a greater opportunity for a controlled
response rather than uncontrolled loss of stability.
Furthermore, the median fins are important for
both self-correcting and powered stability during
swimming and for maneuvering, but could be
sources of instability when eddies with diameters
large enough to overwhelm control systems are present in the incident flow. Our simple analysis suggests that latencies are small enough that fish could
respond to perturbations before these fins entered
such an eddy.
At higher swimming speeds, fish might be expected to penetrate further into eddies and hence
experience larger forces and torques before responding. This would suggest that fishes swimming at
higher speeds would experience greater challenges
to their stability and be more prone to lose control.
In practice, two factors mitigate against this. First,
stability is affected by the relationships between fluctuations in velocity relative to the average velocity
(Webb 2006). Destabilizing perturbations in a turbulent flow are related to the deviations in velocity
from the mean, these being the turbulent components of the flow; compared to a fish in a steady
flow, disturbances arise because of the turbulent momentum, turbulent kinetic energy, and similar metrics associated with the fluctuations in turbulent
velocity. At the same time, the flow of water over a
fish’s control systems and the momentum of a swimming fish damp the growth of displacements. In habitats occupied by fishes, the variance in the turbulent
components of velocity increases more slowly than
does the average speed of flow (Webb 2006; Webb
et al. 2010). As a result, at higher speeds, the greater
momentum and energy of a fish are better able to
damp and correct displacements. Second, as speed
increases, the sizes of eddies experienced by a fish,
the perceived diameters of eddies, decrease (Adrian
et al. 2000), analogous to well-known Doppler effects
in the perception of sound. Therefore, as speed increases, the apparent sizes of eddies and the
perturbations they bring become smaller in the
fish’s frame of reference. Then, potentially destabilizing eddies become less common.
Conclusions
Technological limitations to the observation, analysis
and modeling of vorticity have militated against evaluating the role of vorticity in the lives of fishes.
Nevertheless, studies to date suggest that vorticity
over a wide range of scales, and arising near the
fish and from global phenomena, affect the anatomy
and morphology of fishes in ways that provide new
approaches to some traditional debates and lead to
new questions. Because considering vorticity in the
lives of fishes is in its infancy, the discussion above is
inevitably speculative. Nevertheless, it is already apparent that vorticity has large impacts upon the form
of the body and fins, and hence on evolutionary
patterns and on the ecological distribution of fishes.
Funding
Michigan Sea Grant R/GLF-53 and National Science
Foundation Career 0447427.
Glossary
Boundary layer: A thin region of flow adjacent to any
surface in which viscosity acts to create a velocity from that of the surface, where there is no
slip between the water and the surface, to the
velocity of the free
H stream. H !
Circulation, D: ¼ A ! dA ¼ V dl, where ! is
vorticity, A is the area of the region of interest,
V is velocity, and l the boundary enclosing the
region of interest for the line integral.
Circulation is the surface integral of vorticity
(first term in equation) or the line integral of
velocity (second term in equation) over a specified region. For a region containing an eddy, or
the region of an eddy containing a fish, this
provides a measure of strength of the eddy.
The momentum and forces due to an eddy,
and the lift on a wing, are all related to
circulation.
Computational fluid dynamics (CFD): A method
using computers to model flow, computer flow
patterns and derived parameters such as circulation, momentum, forces, kinematic energy in
the flow and of objects embedded in the flow.
Eddy: A rotating region of the flow with finite vorticity, especially common in turbulent flows.
Elongated slender body theory, ESBT: Analytical
theory for swimming in an inviscid fluid (i.e.
neglecting viscosity) by long and narrow
1164
animals, applied to fishes where the width of the
body and amplitudes of the motion at the trailing edge are 0.2L, where L is the body length.
Froude efficiency, : The ratio of total mechanical
work done by a propulsor to the useful component of that work to overcome drag.
Impulse: the integral of force over a specified time
period.
Kármán vortex street: A wake comprising a series of
eddies shed by a body at regular intervals, first
from one side of the object with rotation in one
direction, and then from the other side with
rotation in the opposite direction. The definition is most commonly used for bluff bodies
such as cylinders.
Kolmogorov scale: These smallest eddies in flows
typical of habitats occupied by post-larval
fishes, at a size at which the eddies disappear
as viscous effects dissipate their kinetic energy
as heat.
Laminar flow: Flow in which fluid particles move in
an orderly unidirectional fashion along streamlines that are parallel to each other.
Particle image velocimetry (PIV): A method to determine flow patterns by visualizing small markers
in the water, and recoding their locations at
known intervals from which velocities are calculated across the observation area or volume.
Radius of an eddy, R: The distance from the center
of rotation of an eddy to a location where vorticity is close to background levels, typically
within 1% of that in the free stream.
Reynolds number, Re: The ratio of inertial to viscous
forces. Re ¼ Lu/, where L ¼ a length scale,
u ¼ characteristic velocity of the flow, and ¼ kinematic viscosity of the fluid.
Second moment of area: A measure of the distribution of material over the cross-section of a
beam, such as a fin ray, that affects bending
and the deflection (strain) of the beam under
a load (stress).
Shear force: A force such as imposed by a velocity
gradient deforming a material by slippage along
planes parallel to the imposed load.
Strain: Deformation of an object, such as a fin ray,
under load (stress).
Streamlines: Lines in the flow that are instantaneously tangent to the velocity vectors.
Stress: Load or force causing deformation of an
object such as a fin ray.
Torque: The moment of a force about a point tending to cause rotation.
Trailing edge: The furthest downstream region on
the body or a fin.
P. W. Webb and A. J. Cotel
Turbulence intensity, TI: TI ¼ s/ū, defined as the
ratio of the standard deviation, s, of the instantaneous velocity, ui, to the mean velocity, ū.
Turbulent flow: There is no commonly accepted
definition of turbulence. Turbulent flows are
characterized by streamlines that follow complex
paths. In habitats occupied by fishes and in the
flow around fishes, turbulent flows are not truly
chaotic as coarse-scale coherent structures are
present as eddies.
Turbulent
kinetic
energy,
TKE:
TKE ¼ 0.5
(ux0 2 þ uy0 2 þ uz0 2), where ux 0 ; uy 0 and uz 0 are the
turbulent velocity components in the x, y and z
directions, respectively. Hence TKE represents
the additional kinetic energy due to fluctuations
in velocity that characterize turbulent flow.
Turbulent velocity component, u0 : u0 ¼ ui – ū, where
ui is the instantaneous velocity at a point as a
function of time and ū is the average velocity
over time. Hence, u0 captures the fluctuations in
velocity that characterize turbulent flow.
Visco-elasticity: A material with both viscous and
elastic properties when being deformed.
Deformation of a viscous material under a
load increases linearly with time, whereas that
of an elastic material is immediate.
Viscosity: A measure of the property of a fluid to
resist deformation.
Vorticity, !: !¼ 5x V, where 5x is a vector operator
(also called the curl or cross product) that determines the rate of rotation of a vector field for
velocity, V.
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