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J. Fluid Mech. (2007), vol. 589, pp. 125–145. c

2007 Cambridge University Press

doi:10.1017/S0022112007007872 Printed in the United Kingdom

125

Geometry of unsteady ﬂuid transport during

ﬂuid–structure interactions

ELISA FRANCO

1, DAVID N. PEKAREK

2,

JIFENG PENG

3AND JOHN O. DABIRI

3,4

1Control and Dynamical Systems, California Institute of Technology, Pasadena, CA 91125, USA

2Mechanical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

3Bioengineering, California Institute of Technology, Pasadena, CA 91125, USA

4Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, CA 91125, USA

jodabiri@caltech.edu

(Received 7 November 2006 and in revised form 23 May 2007)

We describe the application of tools from dynamical systems to deﬁne and quantify

the unsteady ﬂuid transport that occurs during ﬂuid–structure interactions and in

unsteady recirculating ﬂows. The properties of Lagrangian coherent structures (LCS)

are used to enable analysis of ﬂows with arbitrary time-dependence, thereby extending

previous analytical results for steady and time-periodic ﬂows. The LCS kinematics are

used to formulate a unique, physically motivated deﬁnition for ﬂuid exchange surfaces

and transport lobes in the ﬂow. The methods are applied to numerical simulations

of two-dimensional ﬂow past a circular cylinder at a Reynolds number of 200; and

to measurements of a freely swimming organism, the Aurelia aurita jellyﬁsh. The

former ﬂow provides a canonical system in which to compare the present geometrical

analysis with classical, Eulerian (e.g. vortex shedding) perspectives of ﬂuid–structure

interactions. The latter ﬂow is used to deduce the physical coupling that exists

between mass and momentum transport during self-propulsion. In both cases, the

present methods reveal a well-deﬁned, unsteady recirculation zone that is not apparent

in the corresponding velocity or vorticity ﬁelds. Transport rates between the ambient

ﬂow and the recirculation zone are computed for both ﬂows. Comparison of ﬂuid

transport geometry for the cylinder crossﬂow and the self-propelled swimmer within

the context of existing theory for two-dimensional lobe dynamics enables qualitative

localization of ﬂow three-dimensionality based on the planar measurements. Beneﬁts

and limitations of the implemented methods are discussed, and some potential

applications for ﬂow control, unsteady propulsion, and biological ﬂuid dynamics

are proposed.

1. Introduction

It is often of interest in ﬂuid mechanics to quantify the exchange of mass,

momentum, and energy between diﬀerent regions of a ﬂow. In many cases these

mixing processes can be described in terms of speciﬁc kinematic boundaries in

the ﬂow, material surfaces that delineate ﬂuid particles with distinct behaviours.

These surfaces governing the exchange of ﬂuid between diﬀerent regions of the ﬂow

(hereafter referred to as exchange surfaces) can be identiﬁed in steady ﬂows from

inspection of streamlines derived from the Eulerian velocity ﬁeld. In cases of steady

ﬂow, the exchange surface commonly manifests itself as a closed recirculation bubble

that traps ﬂuid particles over long convective time scales. Examples include the

126 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

γ1

γ2

Figure 1. Schematic of manifolds for the recirculation bubble of a vortex pair. Fluid particle

trajectories (i.e. streamlines in steady ﬂow) that asymptote to the front stagnation point γ1as

time t→∞belong to the stable manifold of γ1, whereas trajectories that asymptote to γ1as

time t→−∞belong to the unstable manifold of γ1. The same considerations apply to γ2.

laminar separation bubble over an airfoil at low Reynolds number (e.g. O’Meara &

Mueller 1987) and the cardiovascular recirculation zone caused by an aneurysm

(e.g. Faturaee & Amini 2003).

Knowledge of the geometry and kinematics of the exchange surfaces in a ﬂow can

be used to monitor the performance of a given ﬂuid transport system or to improve its

performance via ﬂow control. Indeed, these ﬂow kinematics can be a useful surrogate

for the ﬂuid dynamics (i.e. forces and moments) when they are diﬃcult to evaluate

directly. However, since the majority of ﬂows of practical interest exhibit unsteadiness

(time-dependence), streamline representations are of limited use for capturing the

kinematics of the exchange surfaces.

The application of dynamical systems tools to ﬂuid mechanics has enabled precise

identiﬁcation of exchange surfaces in unsteady ﬂows that exhibit a well-deﬁned

temporal periodicity in the ﬂuid motion. The theory governing ﬂuid transport

in time-periodic ﬂows is now well developed and has been demonstrated in a

variety of canonical systems including the oscillating vortex pair, isolated and

leapfrogging vortex rings, and cylinder crossﬂow (e.g. Aref 1984; Rom-Kedar &

Wiggins 1990; Rom-Kedar, Leonard & Wiggins 1990; Shariﬀ, Leonard & Ferziger

1989, 2006; Shariﬀ, Pulliam & Ottino 1991; Duan & Wiggins 1997; see Wiggins 2005

for an excellent review). In each case, the analysis relies on the identiﬁcation of stable

and unstable manifolds, which are the collection of ﬂuid particle trajectories that

asymptote to a point in the ﬂow as time moves forward or backward, respectively.

Figure 1 illustrates this concept for the exchange surface that encloses the cores of a

vortex pair. The manifolds of interest for deﬁning the exchange surface are typically

those of the stagnation and/or separation points in the ﬂow, as shown in the ﬁgure.

A geometric deﬁnition of the governing ﬂuid exchange surface based on the

manifolds in the ﬂow is in general not unique; multiple deﬁnitions can be derived from

the same set of stable and unstable manifolds. For simple manifolds in time-periodic

ﬂows there is typically a single deﬁnition for the exchange surface that stands out

because of the relative simplicity of the ﬂow geometry that it suggests (Rom-Kedar

et al. 1990). In the case of the steady vortex pair in ﬁgure 1, the elliptical boundary

connecting γ1and γ2most appropriately deﬁnes the exchange surface.

In unsteady ﬂows with arbitrary time-dependence, however, it is often diﬃcult to

distinguish between the many possible deﬁnitions of the exchange surface that can be

Geometry of unsteady ﬂuid transport 127

constructed from the stable and unstable manifolds (whose deﬁnition is appropriately

modiﬁed to account for the lack of periodicity in the ﬂow). The level of diﬃculty in

applying a particular exchange surface deﬁnition to the ﬂow can vary substantially

from one deﬁnition to the next and even for the same deﬁnition evaluated at diﬀerent

times during the temporal evolution of the ﬂow (Malhotra & Wiggins 1998). Hence,

computing transport rates in aperiodic ﬂows currently relies on the implementation

of ad hoc transport deﬁnitions that are speciﬁc to the particular ﬂow being

investigated.

The goal of this paper is to propose and demonstrate an unambiguous, robust,

and physically motivated geometric deﬁnition of ﬂuid exchange surfaces that can

be easily applied to compute transport rates in arbitrary unsteady aperiodic ﬂows.

The proposed deﬁnition has several distinguishing features. First, the evolution of

the deﬁned exchange surfaces qualitatively resembles the processes of entrainment

and detrainment that are observed in ﬂow visualizations using a passive ﬂow marker

(e.g. Sturtevant 1981; Yamada & Matsui 1978). This is not true of alternative

deﬁnitions. Second, in the limit of time-periodic ﬂow, the proposed deﬁnition is

identical to the deﬁnition traditionally selected on the basis of aesthetic merits in

previous studies (e.g. Rom-Kedar et al. 1990). Third, in the limit of steady ﬂow, the

proposed deﬁnition is identical to the exchange surface that would be identiﬁed in a

streamline plot of the ﬂow (e.g. Milne-Thompson 1968).

In the place of stable and unstable manifolds, which are a valid concept for strictly

time-periodic systems, we identify analogous Lagrangian coherent structures (LCS) in

the ﬂows to be investigated. The LCS share many of the properties of manifolds (see

the following sections for details), but can be computed based on a ﬁnite-time record

of the ﬂow, which need not be time-periodic (Haller 2000, 2001, 2002; Shadden,

Lekien & Marsden 2005; Shadden, Dabiri & Marsden 2006; Green, Rowley & Haller

2007). In addition, an important beneﬁt of LCS for ﬂow analysis is its objectivity, or

invariance under linear transformations of frame (Haller 2005). By constructing the

proposed exchange surface deﬁnition using LCS, it too is made objective.

We apply the proposed exchange surface analysis to study ﬂuid–structure

interactions. Whereas much of the classical study of mixing has focused on isolated

vortical structures and unbounded ﬂows, most practical ﬂows involve the presence

of solid structures that either bound the ﬂow or are immersed within it. The ﬂow

created by a freely swimming jellyﬁsh provides the main application in this paper.

The selection of this model system is motivated by the fact that it exhibits aperiodic

ﬂow despite the relative simplicity of its body shape and motion, as shown in ﬁgures 2

and 3. Muscle contraction reduces the volume of the subumbrellar cavity (i.e. the

region underneath its umbrella-shaped body), resulting in a net downward ﬂux of

ﬂuid. The motion of the lower margin of the bell generates vortex rings of opposite

rotational sense during the contraction and relaxation phases of the swimming cycle

(see ﬁgure 2). These vortices act to entrain ﬂuid from above the animal into the

subumbrellar cavity, where the feeding and sensory apparatus of the animal are

located. Despite the approximate periodicity of the swimming motion, inspection of

the ﬂow created by the animal indicates that it is indeed aperiodic in time. Further,

since the animal does not swim at constant velocity, a periodic ﬂow cannot be

constructed by any Galilean transformation of frame.

Instantaneous streamlines of the ﬂow ﬁeld measured by using digital particle

image velocimetry (DPIV) indicate local entrainment of ﬂuid from above the animal

into the subumbrellar cavity during the entire swimming cycle. Simultaneously, a

net downward momentum ﬂux propels the animal forward (ﬁgure 3). Although

128 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

Figure 2. Dye visualization of jellyﬁsh vortex wake (Dabiri et al. 2005). Time series shows

vortices of clockwise and anticlockwise rotational sense generated during the contraction and

relaxation phases of the swimming cycle, respectively. Bell diameter is 10 cm.

(a)(b)

Figure 3. Instantaneous streamlines of ﬂow around a jellyﬁsh as it swims vertically. (a)End

of relaxation phase of swimming cycle. (b) End of contraction phase of swimming cycle. Bell

diameter is 10 cm

the ﬂow features in ﬁgure 3 lead one to anticipate the existence of exchange

surfaces surrounding the animal, these surfaces are not apparent in the instantaneous

streamline plots. We will show that the present methods are suﬃcient to deﬁne

Geometry of unsteady ﬂuid transport 129

and quantify the exchange surfaces governing ﬂuid transport induced by the animal

swimming motions

Shadden et al. (2006) have previously computed LCS analogous to stable manifolds

for a free-swimming animal, the same species as studied here. However, in that work

the LCS was computed for the purpose of demonstrating that the LCS behave as

material lines as predicted by theory. There is no quantiﬁcation of the associated ﬂuid

transport therein or elsewhere. Indeed, there could be no discussion of ﬂuid transport

previously because (i) the LCS analogous to unstable manifolds have not previously

been computed for this ﬂow and (ii) as mentioned above, an empirical treatment of

LCS in the context of aperiodic exchange surfaces, the goal of this paper, has not

been addressed previously to our knowledge.

We note that the present study is restricted to planar sections of a three-dimensional

ﬂow. Limitations of the two-dimensional measurements are inferred in this paper

by comparing properties of the measured LCS evolution with previous theoretical

considerations of two-dimensional LCS kinematics. In addition, we apply the methods

of analysis to direct numerical simulations of two-dimensional ﬂow past a circular

cylinder at a Reynolds number of 200. This canonical ﬂow allows comparison between

classical perspectives on ﬂuid–structure interactions (e.g. vortex shedding) and the

geometric viewpoint taken in this paper. In addition, the two-dimensional ﬂow

enables validation of the inferences made in the jellyﬁsh study. Salman et al. (2007)

recently computed LCS for a more complex two-dimensional bluﬀ-body conﬁguration.

Although the mechanism of ﬂuid transport is described in that paper, quantitative

measurements of transport rates are not presented.

The paper is organized as follows: §2 presents the foundational dynamical systems

concepts, including a review of the mechanism of unsteady ﬂuid transport via

exchange surfaces. This is followed by a presentation of the proposed deﬁnition

of exchange surfaces in aperiodic ﬂow and examples of its implementation in a

simple vortex model. We prove that the proposed deﬁnition satisﬁes the classical

manifold-intersection ordering criterion governing time-periodic ﬂows. The utility of

LCS for computing the exchange surfaces in ﬂows with arbitrary time-dependence

is then presented. Finally, the methods used to extract LCS and exchange surfaces

from the jellyﬁsh ﬂow and cylinder crossﬂow are described in this section. Section 3

reports results obtained from the case study of the freely swimming animals showing

both the measured LCS evolution and the associated transport rates computed

using the proposed exchange surface deﬁnition. A sensitivity analysis is conducted to

determine the robustness of the ﬂuid transport measurements to perturbations away

from the speciﬁc exchange surface deﬁnition selected for study here. In addition, ﬂow

dimensionality inferred from the manifold kinematics is compared with divergence

calculations of the corresponding Eulerian velocity ﬁelds. These conclusions are

supported by the results of the numerical study of cylinder crossﬂow, which is also

presented in this section. The paper concludes with a discussion of the beneﬁts and

limitations of the developed methods and suggestions for potential applications in §4.

2. Analytical and experimental methods

2.1. Deﬁnition and analysis of exchange surfaces

As described by Malhotra & Wiggins (1998), the manifold geometry illustrated in

ﬁgure 1 is unique to a limited set of steady or quasi-steady ﬂows. In most situations of

practical relevance, the time-dependent hyperbolic trajectories γ1(t)andγ2(t) will be

perturbed, e.g. due to an external strain ﬁeld (Rom-Kedar et al. 1990) or ellipticity of

130 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

fn–1(q(τn+1))

fn–1(q(τn–1)

fn–1(q(τn–1)

γ2(τ)

Ws(γ2(τ))

Wu(γ1(τ))

γ1(τ)

q(τn)

Sn

–

Sn

–

Un

–

Sn

+

Un

+

Un

–

(a)

2

2′

3′

3

1

fn–1(q(τn+1))

γ2(τ)

Ws(γ2(τ))

Wu(γ1(τ))

γ1(τ)

q(τn)

Sn

–

Sn

+

Un

+

Un

–

(b)

2

2

1

2′2′

3′

1′

3

fn–1(q(τn–1)

fn–1(q(τn+1))

γ2(τ)

Ws(γ2(τ))

Wu(γ1(τ))

γ1(τ)

q(τn)Sn

+

Un

+

(c)

3

1

3′

1′

1′

Figure 4. (a) Schematic of perturbed recirculation bubble analogous to the unperturbed

steady case shown in ﬁgure 1. Symmetric right half of ﬂow omitted for clarity. Blue curve,

stable manifold of γ2(τ); red curve, unstable manifold of γ1(τ); ﬁlled circles, p.i.p.s.; open

circles, non-p.i.p.s.; ﬁlled diamond, b.i.p. Solid curves indicate the exchange surface derived

from the stable and unstable manifolds. Fluid is transported through consecutively numbered

lobes as the manifolds evolve in time. Unprimed indices indicate ﬂuid transport into the

recirculation region. Primed indices indicate ﬂuid transport out of the recirculation region.

Segment lengths S−

n,S+

n,U−

n,andU+

nare used to evaluate the b.i.p. criterion in equation (2.1)

and are deﬁned in the text. (b) Resulting ﬂow geometry using adjacent p.i.p. closer to γ1(t)as

the b.i.p. (c) Resulting ﬂow geometry using adjacent p.i.p. closer to γ2(t) as the b.i.p.

the vortex cores (Shariﬀ et al. 1989, 2006). In these cases, the heteroclinic trajectories

connecting γ1(t)andγ2(t) will break and exhibit spatial oscillations, as illustrated

in ﬁgure 4(a) for the left-hand side of the symmetric ﬂow. The stable and unstable

manifolds will then intersect, forming lobes. Formally, these lobes are deﬁned as areas

Geometry of unsteady ﬂuid transport 131

delimited by segments of the stable and unstable manifolds and by primary intersection

points (p.i.p.s) of the stable and unstable manifolds. Guckenheimer & Holmes (1983)

and Malhotra & Wiggins (1998) deﬁne p.i.p.s as follows:

Condition 2.1. At each time instant τ,p.i.p.s p(τ)are points such that:

p(τ)∈Wu(γ1(τ)) ∩Ws(γ2(τ)),

and [Wu(γ1(τ)),p(τ)] ∩[Ws(γ2(τ)),p(τ)] = p(τ),

where Ws(γ2(τ)) denotes the stable manifold of γ2(τ) at time τ,Wu(γ1(τ)) denotes

the unstable manifold of γ1(τ), and the bracketed expressions denote the segments

of these manifolds connecting the respective hyperbolic trajectory, γ1(τ)orγ2(τ), to

p(τ); see ﬁgure 4(a). The ﬁrst statement requires that the p.i.p. lies on both the stable

and the unstable manifold. The second statement requires that a p.i.p. is the only

intersection of the segments [Wu(γ1(τ)),p(τ)] and [Ws(γ2(τ)),p(τ)] that connect the

p.i.p. to γ1(τ)andγ2(τ), respectively. For example, the ﬁlled circles in ﬁgure 4(a)

indicate intersections that deﬁne p.i.p.s, whereas the open circles are not p.i.p.s.

The above deﬁnition implies that each lobe, deﬁned by the union of two

neighbouring p.i.p.s and the neighbouring segments of the stable and unstable

manifolds, is a region of trapped ﬂuid, because the manifolds are material lines

in the ﬂow. As a consequence, lobe areas of a two-dimensional incompressible ﬂow

must remain constant despite deformation and advection of the manifolds that occurs

due to the time-dependent nature of the ﬂow.

The fact that the manifolds in the ﬂow are material lines implies the following

rules regarding the temporal evolution of the ﬂow (Guckenheimer & Holmes 1983;

Malhotra & Wiggins 1998):

Rule 1: Maintenance of order under time evolution. Wu(γ1(τ)) and Ws(γ2(τ)) are one-

dimensional curves at any time τ. An ordering of points can therefore be deﬁned

on, e.g., Ws(γ2(τ)) as follows: for any two points p(τ),q(τ)∈Ws(γ2(τ)), p(τ)<sq(τ)

if p(τ) is closer to γ2(τ) along the arclength of the curve Ws(γ2(τ)). As the ﬂow

evolves temporally, p(τ+t)=fτ+t

τ(p(τ)) and q(τ+t)=fτ+t

τ(q(τ)) will still belong

to Ws(γ2(τ+t)), and p(τ+t)<sq(τ+t), where fis an orientation-preserving

diﬀeomorphism between two points in time. In the present context, fτ+t

τmaps the

ﬂow from time τto time τ+t.

Rule 2: Invariance of intersections. If at time τ,Wu(γ1(τ)) and Ws(γ2(τ)) intersect,

then they intersect at all times. This follows from the invariance properties of the

manifolds, i.e. the fact that the manifolds behave as material lines in the ﬂow.

The p.i.p.s travel along the stable manifold of γ2(t) as the ﬂow evolves.

Concomitantly, the lobes deﬁned by the p.i.p.s deform and stretch, transporting

the ﬂuid particles trapped in the lobe across a (still undeﬁned) exchange surface

formed by the intersection of the stable and unstable manifolds. Since our goal is to

quantify ﬂuid transport from empirical observations of lobe evolution, we must deﬁne

this time-varying exchange surface that will be computed along with the lobe areas.

To this end, the following criterion for the choice of an exchange surface is proposed,

respecting the theoretical p.i.p. ordering Rule 1 and Rule 2.

Following the work of Malhotra & Wiggins (1998, p. 415), let us ﬁrst consider

the evolution of Wu(γ1(τ)) and Ws(γ2(τ)) over a strictly increasing time sequence

T,{τ1,τ

2,...,τ

n−1,τ

n,τ

n+1,...},∀n∈Z. As previously noted, at each arbitrary

time τn,pointsp(τn) on the manifolds are mapped to new points p(τn+1)=fn(p(τn)),

132 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

where for notational simplicity the ﬂow map that advects ﬂuid particles forward in

time will henceforth be denoted as fn.

To deﬁne the exchange surface, we want to identify a sequence of boundary

intersection points (b.i.p.s) q(τn). The b.i.p.s will in turn deﬁne the exchange surface

(a curve in two dimensions) B(τn), as the union of two segments: [Ws(γ2(τn)),q(τn)],

which is the arclength from the b.i.p. q(τn) to the hyperbolic trajectory γ2(τn) travelling

on the stable manifold; and [Wu(γ1(τn)),q(τn)], which is the arclength from q(τn)tothe

γ1(τn) along the unstable manifold. Therefore the desired exchange surface is deﬁned as

B(τn),[Ws(γ2(τn)),q(τn)] [Wu(γ1(τn)),q(τn)]. Flow crossing the exchange surface

deﬁned by this bounding curve from time τnto time τn+1 is identically the ﬂuid

transport that occurs due to the lobe dynamics.

The sequence of b.i.p.s used to deﬁne the exchange surface should satisfy the

following ordering (Malhotra & Wiggins 1998):

Condition 2.2. At each time instant τn, the chosen b.i.p. q(τn)must satisfy:

q(τn)<sf−1

n(q(τn+1)),∀n∈Z

where the notation <sindicates the ordering on the stable manifold of γ2(t), using

the arclength distance of the candidate points from γ2(t), as deﬁned by Rule 1.

By itself, this condition does not specify a unique b.i.p. among the multiplicity

of p.i.p.s; it merely constrains the direction of the sequence of b.i.p.s, such that the

location of the current b.i.p. should be closer to γ2(τn) than the current location of

the next b.i.p. This prevents the b.i.p.s from approaching γ2as n→∞. Since the b.i.p.

is not uniquely deﬁned by this criterion, the exchange surface is also not uniquely

deﬁned. For example, the b.i.p. selected in ﬁgure 4(a) (ﬁlled diamond) results in an

exchange surface given by the union of the solid red and blue curves. However, one

could also select the p.i.p above (ﬁgure 4b), or below (ﬁgure 4c) this intersection

point, and that new b.i.p could also satisfy Condition 2.2 while producing a diﬀerent

geometry for the exchange surface. For the exchange surface deﬁned in ﬁgure 4(b),

ﬂuid enters the recirculation region as lobe 1 evolves into lobe 2, instead of during

the 2 →3 lobe evolution as in ﬁgure 4(a). However, ﬂuid exits the recirculation

region during the same 1→2lobe evolution as in the exchange surface deﬁned in

ﬁgure 4(a). Conversely, the exchange surface in ﬁgure 4(c) diﬀers from ﬁgure 4(a)in

the process of ﬂuid detrainment from the recirculation region, but has an identical

entrainment process.

Current practice is to select the p.i.p. giving the exchange surface that most closely

resembles an equivalent unperturbed ﬂow (e.g. Rom-Kedar et al. 1990); in the present

case, comparison of ﬁgure 4(a) with ﬁgure 1 shows that the point denoted by the

ﬁlled diamond is most appropriate from this perspective. Yet, for the majority of

unsteady ﬂows, there is no unperturbed reference state with which one can compare

in order to determine an appropriate deﬁnition for the exchange surface (e.g. Salman

et al. 2007). This ambiguity limits comparisons of unsteady ﬂuid transport between

systems, or even in the same system examined at diﬀerent times during its temporal

evolution.

We propose the following criterion for the b.i.p. sequence:

Criterion for boundary intersection points. At each time instant τn, choose as a

boundary intersection point the intersection q(τn)for which

S+

n<U

+

n,

S−

n>U

−

n,(2.1)

Geometry of unsteady ﬂuid transport 133

where we deﬁne S+

n,[q(τn),f

n−1(q(τn−1))], which is the segment on the stable

manifold of γ2(τn) connecting q(τn)andfn−1(q(τn−1)) (in words, the latter

term represents the current location of the previous b.i.p.). Similarly, U+

n,

[q(τn),f

n−1(q(τn−1))], i.e. the segment with identical endpoints but on the unstable

manifold of γ1(τn). The deﬁnitions of S−

nand U−

nfollow as: S−

n,[f−1

n(q(τn+1)),q(τn)]

taking the segment on Ws(γ2(τn)), and U−

n,[f−1

n(q(τn+1)),q(τn)] on Wu(γ1(τn)). In

words, the term f−1

n(q(τn+1)) represents the current location of the next b.i.p.

Qualitatively speaking, this criterion identiﬁes the b.i.p. as the p.i.p. connecting

the segments of the stable and unstable manifolds with least deformation from an

equivalent unperturbed state. In other words, we deﬁne ﬂuid transport according to

evaluation of the length relationships in equation (2.1). An ancillary beneﬁt of the

exchange surface deﬁned by this choice of b.i.p. is that it presents the smallest temporal

shape oscillation. Since the stable manifold of γ2(τn) becomes increasingly deformed

as it approaches γ1(τn) and the unstable manifold of γ1(τn) becomes increasingly

deformed as it approaches γ2(τn), the b.i.p. will be located away from both γ1(τn)

and γ2(τn). In the case of the vortex model in ﬁgure 4(a), the b.i.p. deﬁned by the

present criterion is in fact equidistant from both hyperbolic trajectories. Furthermore,

this choice of b.i.p. coincides with the one that would be chosen in order to deﬁne

an exchange surface that most closely resembles the bounding streamline of the

analogous unperturbed steady ﬂow in ﬁgure 1 (cf. Rom-Kedar et al. 1990). The

beneﬁt of the proposed criterion is that it can be applied to ﬂows with arbitrary

unsteadiness where there does not exist an analogous steady ﬂow for comparison.

Using the vortex model in ﬁgure 4(a), let us consider the qualitative evolution of

the exchange surface deﬁned by the present b.i.p criterion. Although the criterion

is evaluated on the discrete time sequence τn, the real ﬂow is continuous in time.

Hence, for closely spaced time sequences, the choice of b.i.p may not change at each

τn. In this case, the b.i.p. will be advected along Ws(γ2(τn)) while maintaining its

identity over successive time instants τn, and the corresponding exchange surface (i.e.

the curve [Ws(γ2(τn)),q(τn)] [Wu(γ1(τn)),q(τn)]) will deform. This deformation will

continue until the current b.i.p. no longer satisﬁes the aforementioned criterion in

equation (2.1). At this time, a p.i.p. (i.e. a diﬀerent ﬂuid particle) closer to γ1(τn) will

become the new b.i.p. and the exchange surface will be redeﬁned accordingly. The

pictorial evolution suggested by the present b.i.p. criterion will be shown in detail in

the following section.

The described b.i.p. criterion is objective (i.e. frame-invariant) and has a practical

relevance: given an empirical set of lobes evolving in time, it facilitates the deﬁnition

of the exchange surface directly from observations of segment lengths along the

stable and unstable manifolds. It also guarantees that the aforementioned theoretical

requirements (i.e. Guckenheimer & Holmes 1983; Malhotra & Wiggins 1998) are fully

satisﬁed.

It is straightforward to prove the following Lemma:

Lemma 2.1. The criterion deﬁned above for the choice of b.i.p. sequence satisﬁes

Condition 2.2.

Proof. It is suﬃcient to ﬁrst notice that all points on Ws(γ2(τn)) Wu(γ1(τn)) at

time τn+1 will have moved closer to γ2(τn+1) in the arclength sense; intersection points

travelling toward γ2will therefore decrease their distance from the neighbouring

intersection points on the stable manifold. Conversely, lobe area preservation imposes

stretching of the corresponding arclength segments on the unstable manifold (i.e. a

ﬁlamentation process).

134 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

Now at time τn+1, we choose q(τn+1) such that S−

n+1 >U

−

n+1 and S+

n+1 <U

+

n+1,asthe

criterion requires. By deﬁnition, S+

n+1 =[q(τn+1),f

n(q(τn))] = fn(S−

n), and fn(S−

n)<S

−

n

since segments are shrinking on the stable manifold. Therefore we have that q(τn)<s

f−1

nq(τn+1).

It is worth noting that the behaviour of q(τn) given by the proposed criterion is

not compatible with any other ordering than the one imposed by Condition 2.2. Ab

absurdo, let us assume that our criterion is applied:

S−

n+1 =fn(S+

n)>f

n(U+

n)=U−

n+1,(2.2)

where now q(τn)>sf−1

n(q(τn+1)). Always by the chosen criterion, we have

S+

n<U

+

n,

fn(S+

n)<S

+

n,

fn(U+

n)>U

+

n,

⎫

⎪

⎬

⎪

⎭

(2.3)

and therefore fn(S+

n)<S

+

n<U

+

n<f

n(U+

n), which contradicts (2.2).

This Lemma could also be proved using the orientation-preserving property of

ﬂow maps. There exist other b.i.p. criteria that will satisfy Condition 2.2, producing

exchange surfaces such as the alternatives illustrated in ﬁgure 4(b, c). The present

criterion however, based on comparison of segment lengths, is intuitive and easily

applicable to experimentally determined manifolds with very irregular shapes, where

lobes are not always clearly discernible to the observer. Furthermore, the present b.i.p.

criterion can be implemented in the cases of ﬁnite or inﬁnitely many p.i.p.s, as long

as there exists a suﬃcient number of p.i.p.s to evaluate the b.i.p. criterion stated in

equation (2.1).

2.2. Deﬁnition and properties of Lagrangian coherent structures

Given the preceding developments, we are left with the task of extracting the stable

and unstable manifolds from measurements or computations of the ﬂow. In steady

and time-periodic ﬂows, it may suﬃce to examine streamlines or a Poincar ´

emap,

respectively, in order to determine the manifold geometry. However, these tools are of

limited use in ﬂows with arbitrary time-dependence, e.g. aperiodicity. Here, we make

use of the ﬁnite-time Lyapunov exponents (FTLE; also referred to as direct Lyapunov

exponents in the literature) of the velocity ﬁeld.

The Lyapunov exponent describes the rate of extension of a line element advected

in the ﬂow. The line elements that experience the most rapid extension are proposed

to straddle (i.e. possess endpoints on opposite sides of) a material line that acts as a

barrier to ﬂuid particle transport (Haller 2000, 2002; Shadden et al. 2005).

Restricting our attention to a two-dimensional domain D, consider the following

system that describes the ﬂow:

˙

x(t;x0,t

0)=u(x(t;x0,t

0),t),

x(t0;x0,t

0)=x0,(2.4)

where x0∈Dis the initial position and t0is the initial time of the ﬂuid particle

trajectory. We will assume that the Eulerian velocity ﬁeld u(x, t) is at least C0in

time and C2in space. The ﬂow map satisfying equation (2.4) will be denoted as

ft

t0(x0)=x(t;x0,t

0). This solution satisﬁes existence and uniqueness properties, and is

C1in time and C3in space.

The Cauchy–Green deformation tensor Cgenerated by the ﬂow map ft

t0(x0) can be

evaluated over a ﬁnite time interval T, giving a measure of how particles are advected

Geometry of unsteady ﬂuid transport 135

under the action of the ﬂow:

C,[∇ft0+T

t0(x)]∗∇ft0+T

t0(x),(2.5)

where Cdepends on x0,t

0and T;[]

∗denotes the adjoint (transpose) of [ ]. As shown

previously (Haller 2000, 2002; Shadden et al. 2005), denoting the largest eigenvalue

of Cas λmax (C), the FTLE is deﬁned as

σT

t0=1

|T|ln λmax (C).(2.6)

The aforementioned assumptions on the vector ﬁeld imply that the ﬁeld σT

t0is C1in

time and C2in space.

LCS can be deﬁned as a ridge line of the function σ. Intuitively, a ridge line is

a curve normal to which the topography is a local maximum. There are two precise

deﬁnitions of a ridge line introduced by Shadden et al. (2005); here we adopt the

second of these, called the second-derivative ridge.

Deﬁnition 1. A second-derivative ridge of σis a curve c(s) whose tangent vector

dc/dsis parallel to ∇σ((c(s)) and whose Hessian Σ(n, n)<0, where nis the unit

vector normal to c(s).

At every time t, the LCS is deﬁned as a second derivative ridge of σT

t0(x),x∈D.

When ﬂuid particle trajectories are integrated forward in time (i.e. T>0), repelling

LCS are revealed. These LCS are said to be repelling because as ﬂuid particles

approach the hyperbolic trajectory (e.g. γ2) along the repelling LCS, particles on

either side of the LCS are strongly repelled. Hence, repelling LCS can indicate the

geometry of stable manifolds. Conversely, backward-time integration of ﬂuid particle

trajectories (T<0) reveals attracting LCS, along which ﬂuid particles on either side

of the LCS are repelled as they move toward the hyperbolic trajectory (e.g. γ1)in

backward time. Attracting LCS can indicate the geometry of unstable manifolds.

Physically, both attracting and repelling LCS are material lines separating regions

of ﬂow that exhibit diﬀerent dynamics, such as the recirculation regions that are of

present interest.

2.3. Empirical evaluation of the exchange surface deﬁnition

To demonstrate the utility of the methods described in the previous section, the

unsteady, aperiodic ﬂow generated by a free-swimming Aurelia aurita jellyﬁsh was

analysed. The ﬂow map of the ﬂuid advection around the animal is clearly not

available in closed form, providing an opportunity to investigate the proposed

methods in a relatively simple geometry that exhibits complex, coupled ﬂuid–structure

interactions.

Details of the experimental methods were similar to a recent study involving the

same species of animal (Shadden et al. 2006). DPIV measurements of the symmetry

plane of the animal were collected for several consecutive swimming cycles executed

in a large tank. The animal swam vertically in a rectilinear fashion away from the

tank walls; hence, all of the observed ﬂow phenomena were induced by the swimming

motions of the animal.

To support the jellyﬁsh studies, the ﬂuid transport analysis methods were also

applied to direct numerical simulations of two-dimensional ﬂow past a circular

cylinder at Reynolds number Re = 200 based on the free-stream velocity and cylinder

diameter (Taira & Colonius 2007). Unlike the jellyﬁsh ﬂow, the cylinder crossﬂow is

time-periodic. In addition, the well-known kinematics of that ﬂow ﬁeld (e.g. vortex

136 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

shedding) should provide a useful comparison with the present perspective based on

exchange surfaces.

The measured or computed time series of Eulerian velocity ﬁelds was input to an

in-house code (Peng & Dabiri 2007) in order to compute the LCS. The integration

duration Twas ±13 s for the jellyﬁsh ﬂows (a shorter backward-time duration was

required toward the beginning of the measurements due to limited backward-time data

initially) and ±1.5 vortex shedding cycles for the cylinder crossﬂow. A second in-house

code analysed the LCS curves in order to identify p.i.p.s, b.i.p.s, and the corresponding

exchange surfaces. Our approach toward these calculations is as follows:

(i) The repelling (i.e. forward time) and attracting (i.e. backward time) LCS are

both broken into several short, linear segments that approximate the LCS curves.

(ii) For each segment of the repelling LCS, a rectangular neighbourhood of interest

is deﬁned, centred at the midpoint of the segment and enclosing the segment.

(iii) Linear segments of the attracting LCS that possess an endpoint within the

neighbourhood of interest are isolated. The size of rectangular neighbourhood relative

to the length of each segment is suﬃciently large that it is impossible for any segment

of the attracting LCS without an endpoint inside the rectangular neighbourhood to

intersect the repelling LCS segment in question.

(iv) For each of the attracting LCS segments with an endpoint inside the rectangular

neighbourhood, the intersection point of the line containing it and the line containing

the repelling LCS segment in question is calculated. In the case of parallel segments,

this intersection point does not exist.

(v) A Boolean check is performed to determine if the coordinates of the intersection

point lie on both the repelling and attracting LCS segments. If so, this point is in fact

an intersection of the two LCS.

The algorithm was veriﬁed manually for the data presented in this paper and was

shown to function correctly. Upon proper identiﬁcation of the set of intersection points

between the attracting and repelling LCS, p.i.p.s and b.i.p.s were determined based

on their respective deﬁnitions given in the previous section. The lobe structure of the

jellyﬁsh ﬂow was such that for every time instant considered, every LCS intersection

point satisﬁed the p.i.p. criteria. This was not the case for the cylinder crossﬂow. In

both cases, the b.i.p. criterion in equation (2.1) was evaluated unambiguously at each

time step.

With the p.i.p.s and b.i.p.s recorded, the LCS arclengths that deﬁne the lobes and

exchange surface were isolated from the full set of LCS data. The area of the ith

lobe was calculated with the following formula for the area of a polygon, based on

Green’s Theorem in the plane:

A(Li)(t)=1

2

n−1

j=0

[xj(t)yj+1(t)−xj+1 (t)yj(t)],(2.7)

where xjand yjare the ﬁrst and second components respectively for the beginning

and ending points of the jth segment deﬁning lobe Li.

3. Results

Figure 5 plots the results of the transport analysis at four instants during the

jellyﬁsh swimming cycle. The LCS curves analogous stable and unstable manifolds

are shown in yellow and green respectively. The p.i.p.s are denoted by open red

circles, whereas the b.i.p.s are denoted by ﬁlled red circles. Based on these b.i.p.s,

Geometry of unsteady ﬂuid transport 137

(a)(b)

(c)(d)

Figu re 5 . Forward- and backward-time LCS surrounding a freely swimming jellyﬁsh at four

instants during a swimming cycle. (a)t=0, (b)t=1.07 s, (c)t=2.13 s, and (d)t=3.27 s;

yellow, forward-time LCS; green, backward-time LCS; open red circles, p.i.p.s; ﬁlled red

circles, b.i.p.s; segments of the stable and unstable manifolds that constitute the exchange

surface are indicated in solid lines, the remainder of the manifolds in dashed lines. Lobes are

numbered consecutively. Light blue, lobe inside recirculation region; dark blue, lobe outside

recirculation region.

the exchange surface is deﬁned as the union of the solid portions of the yellow

and green curves. The lobes formed by the p.i.p.s and adjacent segments of the

LCS are numbered sequentially from lobe 1 upstream of the animal to lobes 5

and 6 in the wake. Based on the exchange surface deﬁnition, lobe 4 (light blue) is

initially located inside the recirculation region, whereas lobe 5 (dark blue) is located

outside. The temporal evolution of the ﬂow illustrates the transport of ﬂuid across the

exchange surface by the lobes during the swimming cycle. We note that although the

existence of transversely intersecting LCS is suggested by previous theoretical work

(e.g. Guckenheimer & Holmes 1983; Malhotra & Wiggins 1998), these interesting

ﬂow kinematics are impossible to detect from inspection of the velocity ﬁeld.

Equally interesting are the exchange surfaces observed in the canonical cylinder

crossﬂow (ﬁgure 6). This ﬂow, previously studied by using Poincar´

e maps (Shariﬀ et al.

138 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

1

lobe 1

lobe 2

(a)

(c)

(e)

(b)

(d)

( f )

0

–1

1

0

–1

1

0

–1

–3 –2 –1 0

xx

y

y

y

1 2 –3 –2 –1 0 1 23

Figure 6. Forward- and backward-time LCS surrounding a circular cylinder at six instants

during a vortex shedding cycle. Red, forward-time LCS; blue, backward-time LCS; ﬁlled black

circles, b.i.p.s; segments of the stable and unstable manifolds that constitute the exchange

surface are shown in solid lines, the remainder of the manifolds in dashed lines. Fluid

inside two lobes is shown: green, lobe outside recirculation region; light blue, lobe inside the

recirculation region. The circular cylinder is shown in grey.

1991), consists of two long, narrow lobes deﬁned primarily by the repelling LCS (red

curve) that propagate from upstream toward the rear of the cylinder. The ﬂuid carried

by these lobes crosses the exchange surface determined by the b.i.p.s (solid red/blue

curves and ﬁlled black circles, respectively) and enters a well-deﬁned recirculation

region behind the cylinder. The lower lobe crosses the exchange surface ﬁrst, as

indicated by its colour change from green to light blue. This ﬂuid will eventually

cross the exchange surface again as it is detrained downstream via interaction with

the attracting LCS (blue curve). As in the jellyﬁsh ﬂow, inspection of the velocity or

vorticity ﬁeld would not reveal this mass transport geometry. However, one does get

a sense for the locations where vortex shedding occurs by examining the kinematics

of the attracting LCS, especially where the this curve folds back onto itself. This is

not by coincidence: passive scalars, such as a dye used to visualize the ﬂow, will tend

to align with the attracting LCS (Shariﬀ et al. 1989; Voth, Haller & Gollub 2002).

Previous analytical studies of time-periodic ﬂows have demonstrated that the rate

of ﬂuid transport into the region bounded by the exchange surface is directly

proportional to the area of the lobes (in two dimensions; Rom-Kedar et al. 1990

and Shariﬀ et al. 2006). Figure 7 plots the temporal evolution of the area of each

of the lobes identiﬁed in ﬁgure 5. In addition to the direct area measurement, we

also present calculations of an equivalent lobe volume based on an assumption

of axisymmetry in the ﬂow. These calculations were made by assigning an axis of

Geometry of unsteady ﬂuid transport 139

1.6

(a)

(b)

1.4

Lobe 3

Lobe 5

Lobe 6

Lobe 4

Lobe 2

Lobe 1

Normalized lobe area volumeNormalized lobe area volume

1.2

1.0

0.8

0.6 0123

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4 012

Time (s)

3

Figu re 7 . Temporal evolution of normalized lobe area (solid lines) and lobe volume (dashed

lines) during a cycle of jellyﬁsh swimming. Normalized values represent the instantaneous lobe

area (volume) divided by the time-averaged area (volume) of that lobe over the swimming

cycle. (a) Lobes 1–3. (b) Lobes 4–6.

symmetry that coincides with the symmetry axis of the animal. To compare these

calculations with the lobe area measurements, we plot both quantities normalized by

the average value of that quantity over the swimming cycle. The average value is

taken separately for each lobe. Diﬀerences between the behaviour of the lobe area

and lobe volume are small for lobes above the animal and become more severe for

lobes near the lower margin. This is a direct consequence of the radial lobe motion

that occurs near the lower margin, which appears in the additional O(r) dependence

of the volume calculation relative to the area calculation.

Since the number of lobes that can be extracted from the ﬂow is dependent on the

integration time Tused to compute the LCS (i.e. more of the manifold is revealed as

the integration is carried out for longer times; cf. Haller 2000, 2002; Shadden et al.

2005), it is useful to consider the average lobe area as opposed to the total area of all

of the lobes in order to study ﬂuid transport. In a two-dimensional ﬂow, lobe area

preservation requires that the ratio of the total area of all of the lobes to the area of

any individual lobe is exactly equal to the number of lobes in the ﬂow (Rom-Kedar

et al. 1990; Malhotra & Wiggins 1998). Hence, in this case the behaviour of the

average lobe area is suﬃcient to characterize all of the lobe dynamics. Figure 8(a)

plots the ratio of the average lobe area (volume) to the area (volume) of the circulation

region. The average lobe area is approximately 2 % of the recirculation region area;

the average lobe volume is approximately 13% of the recirculation region volume.

One of these lobe volumes (in three dimensions) or two of these lobe areas (one per

140 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

1.2

(a)

(b)

1.0

0.8

Normalized area volume

Normalized area

0.6

0.4

0.2

0123

1.2

1.0

0.8

0.6

0.4

0.2

00.25 0.50

Time (t/TC)

Time (s)

0.75 1.00

Figure 8. (a) Temporal evolution of the area (volume) of the average lobe and the recirculation

region for the jellyﬁsh ﬂow. Normalized values represent the instantaneous area (volume)

divided by the time-averaged area (volume) of the recirculation region over the swimming

cycle. Solid lines, area; dashed lines, volume; lines with dots, average lobe; lines without dots,

recirculation region. (b) Temporal evolution of the area of each lobe and the recirculation

region for the cylinder crossﬂow. Normalized values represent the instantaneous area divided

by the time-averaged area of the recirculation region over a vortex shedding cycle of duration

TC. Filled circles, recirculation region; ﬁlled squares, lobe 1; ﬁlled triangles, lobe 2.

side of the animal in two dimensions) is added and removed from the recirculation

region per swimming cycle. Since the total recirculation region does not change

appreciably in size over time (ﬁgure 8a), the lobe dynamics give an indication of the

ﬂuid turnover rate within the recirculation region. For the jellyﬁsh ﬂow this turnover

rate is on the order of 10% per swimming cycle. A similar analysis can be performed

for the cylinder crossﬂow, as shown in ﬁgure 8(b). In this case, two lobes (one above

and one below the cylinder centreline) are added and removed from the recirculation

region during each vortex shedding cycle of duration TC. The corresponding ﬂuid

turnover rate is approximately 14% per vortex shedding cycle.

We now examine the sensitivity of the results to the choice of the b.i.p. that deﬁnes

the exchange surface and associated recirculation region for the jellyﬁsh ﬂow. Figure 9

compares the temporal evolution of the area bounded by the current exchange surface

deﬁnition to the corresponding areas enclosed by modiﬁed exchange surfaces. These

modiﬁed surfaces are deﬁned using the adjacent p.i.p.s that are either directly above

or below the current b.i.p. (cf. ﬁgure 4b, c). In cases where the current b.i.p. has no

adjacent p.i.p. below it (e.g. ﬁgure 5a), the current b.i.p. is used in the comparison.

The data are shown for the four time instants in ﬁgure 5. The results indicate

Geometry of unsteady ﬂuid transport 141

1.3

1.2

1.1

Recirculation area ratio

1.0

0.9

0.8

0.7

012

Time (s)

3

Figu re 9 . Ratio of the area bounded by modiﬁed exchange surfaces deﬁned using adjacent

p.i.p. above (ﬁlled triangles) or below (ﬁlled squares) the actual b.i.p., respectively, to the area

bounded by the original exchange surface. Data points correspond to the images in ﬁgure 5.

that the measurements are relatively robust to changes of the b.i.p. to its nearest

neighbour p.i.p. The use of the adjacent p.i.p. below the current b.i.p. results in a

slight underestimate of the area enclosed by the exchange surface. Conversely the

use of the adjacent p.i.p. above the current b.i.p. results in a slight overestimate. This

relative insensitivity suggests that despite the discontinuous shifts in b.i.p. that occur

as the b.i.p. criterion is evaluated on the temporally evolving LCS curves, physically

consistent results can be obtained and used for quantitative comparison of ﬂuid

transport systems.

The temporal variation of the lobe areas in ﬁgure 7 is in violation of the known

behaviour of two-dimensional lobes in incompressible ﬂow. A major source of this

spurious result is the three-dimensionality of the ﬂow, which is not captured by the

two-dimensional DPIV measurements. In principle, the amount of time-dependence

exhibited by each lobe area can therefore be used as a measure of the local ﬂow

three-dimensionality. For example, it can be inferred from the relatively constant

area of lobes 1 and 2 that the ﬂow in their vicinity (upstream of the animal) is

nearly two-dimensional. By contrast, the ﬂow near lobes 5 and 6 (in the vortical

wake) exhibits three-dimensionality that appears in the transient behaviour of the

corresponding lobe areas. As would be expected, the spatial transition between two-

and three-dimensionality is gradual, given the even spatial distribution of lobes in the

streamwise direction from lobe 1 to lobes 5 and 6.

Quantitatively, ﬁgure 10 shows that the standard deviation of the normalized lobe

areas plotted in ﬁgure 7 increases from approximately 10% upstream of the animal to

over 30% in the wake. The locations of two- and three-dimensionality in the ﬂow, i.e.

upstream and in the wake, respectively, match physical intuition for the ﬂow around

a self-propelled animal in quiescent surroundings. These locations are also consistent

with previous dye visualizations of ﬂow generated by the same animal species (Dabiri

et al. (2005)).

Some of the area variation is due to error in determining the position of the

LCS from the FTLE ﬁeld. The numerical simulations of cylinder crossﬂow enable

us to quantify this error, since that ﬂow is two-dimensional by deﬁnition. As seen

in ﬁgures 8(b) and 10, the lobes in this purely two-dimensional ﬂow exhibit an area

variation of approximately 4% over a vortex shedding cycle. This area variation

is wholly attributable to error in LCS identiﬁcation since no three-dimensionality

142 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

0.4

0.3

0.2

0.1

0

1234

Lobe number

2D cylinder crossflow

Normalized lobe stand. dev.

5

Figure 10. Standard deviation of normalized lobe area as a function of lobe number. Solid

line, lobe area; dashed line, lobe volume. Standard deviation of two-dimensional cylinder

crossﬂow lobes is included for reference.

exists. These results suggest that the ﬂow upstream of the animal may be closer to

two-dimensional than the magnitude of the standard deviation in lobe area implies.

Finally, ﬁgure 11 plots the divergence of the velocity ﬁeld shown in ﬁgure 5(a).

Deviations from zero divergence suggest three-dimensionality in the ﬂow. The

locations of highest velocity ﬁeld divergence (i.e. in the downstream wake) are

consistent with the regions of maximum lobe area variation, supporting the notion

that lobe area evolution can be examined as a metric for ﬂow three-dimensionality.

The divergence calculation has the added beneﬁt of point-wise evaluation of ﬂow

dimensionality, whereas the lobe examination only gives information regarding

average behaviour within a lobe. However, in cases where lobe evolution has already

been computed for transport measurements, the area variation can complement

existing metrics with no added computational eﬀort.

4. Discussion

The geometry of ﬂow elucidated by these methods shows some striking diﬀerences

from conventional Eulerian perspectives such as velocity and vorticity ﬁelds. Indeed,

much of the ﬂuid transport geometry is hidden when one examines the ﬂows studied

in this paper using those metrics. As suggested by Shariﬀ et al. (1989), the attracting

(backward-time) LCS shows qualitative similarity to what would be observed in the

ﬂow using a passive scalar such as dye to mark the ﬂuid particles. However, there

appears to be no analogous visualization to produce the repelling (forward-time) LCS

structure. The repelling LCS structure revealed in the cylinder crossﬂow provides an

interesting and unconventional perspective on that canonical ﬂow. The transport

role of the upstream lobes may be a useful target for manipulation in various ﬂow

control applications. Notably, despite the fact that the Reynolds number (Re = 200)

is in the regime of periodic vortex shedding, there exists a well-deﬁned, time-varying

recirculation region at the base of the cylinder that is revealed by the present methods.

The role of such regions in external ﬂows of this kind should receive greater attention

in the future. We hypothesize that similar structures will exist in other bluﬀ body

ﬂows as well as in structures of importance in aero- and hydrodynamics. The recent

results of Salman et al. (2007) support this conclusion.

The case study of jellyﬁsh swimming provided a test of the methods in a ﬂow with

more complex ﬂuid–structure interactions and temporal aperiodicity. Despite the

Geometry of unsteady ﬂuid transport 143

–2.0 –1.6 –1.2 –0.8 –0.4 0 0.4 0.8 1.2 1.6 2.0

Figure 11. Divergence of DPIV velocity ﬁeld surrounding a freely swimming jellyﬁsh at

t= 0, cf. ﬁgure 5(a). Locations of non-zero divergence indicate three-dimensional ﬂow. Animal

position is shown in dashed outline.

increased complexity, the lobe dynamics predicted in the simple model vortex system

appeared in this ﬂow as well. By extracting the geometry of ﬂuid transport, it was

possible to quantify the mass transport that occurs concomitantly with momentum

transport during self-propulsion. This mass transport has biological signiﬁcance

because local environmental sensing is a vital capability for many self-propelled

organisms like the jellyﬁsh. By increasing the rate of ﬂuid turnover within the

recirculation region, the animal is able to more eﬀectively deduce the properties

(e.g. chemical cues) of the surrounding ﬂuid.

The ﬂuid turnover metric can be similarly useful for identifying the precise eﬀect

of various ﬂow control strategies (e.g. blowing and suction) on the surrounding

ﬂuid. For example, the upstream lobes observed in ﬁgure 6 indicate exactly which

ﬂuid incident on the body in the ﬂow is actually aﬀected by the control strategies.

Moreover, the magnitude of the ﬂuid turnover rate can indicate the duration of the

interaction between that ﬂuid and the body, when coupled with statistical methods

(e.g. Rom-Kedar et al. 1990; Shariﬀ et al. 1991; Duan & Wiggins 1997).

Measurements of unsteady propulsion can beneﬁt from the present methods, as they

suggest the possibility of computing unsteady mass ﬂux induced by the propulsor. The

144 E. Franco, D. N. Pekarek, J. Peng and J. O. Dabiri

present tools can be especially useful where the propulsors generate an external ﬂow

(e.g. ﬂapping foils or undulating bodies), in which case the mass ﬂux can be diﬃcult

to estimate. Krueger (2006) has shown that measurements of this unsteady mass ﬂux

combined with an estimate of the mechanical power expended by a system can be

used to compute propulsive eﬃciency without making an assumption of quasi-steady

ﬂow, as must be done to compute an equivalent Froude eﬃciency. Hence, the concepts

described here can enable comparison of propulsive performance in swimming and

ﬂying organisms or in engineered propulsion systems that are unsteady.

Finally, although three-dimensional LCS and lobe structure is beyond the scope of

the present work, three-dimensional ﬂows are fully amenable to the present treatment.

In those cases, the intersection points are replaced by intersection lines, and lobe areas

become lobe volumes. The resulting exchange surfaces will then become truly three-

dimensional. That development will provide valuable new insight into many of the

ﬂows under investigation in ﬂuid dynamics.

The authors acknowledge insightful discussions with S. C. Shadden, J. E. Marsden,

K. Shariﬀ, very helpful suggestions from the manuscript referees, and funding from

the NSF Biological Oceanography Program (OCE-0623475 to J.O.D.).

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