ArticlePDF Available

Geometry of unsteady fluid transport during fluid–structure interactions

Authors:
Article

Geometry of unsteady fluid transport during fluid–structure interactions

Abstract and Figures

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.
Content may be subject to copyright.
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
22
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
2,...,τ
n1
n
n+1,...},nZ. 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)<sf1
n(q(τn+1)),nZ
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 12lobe 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
n1(q(τn1))], which is the segment on the stable
manifold of γ2(τn) connecting q(τn)andfn1(q(τn1)) (in words, the latter
term represents the current location of the previous b.i.p.). Similarly, U+
n,
[q(τn),f
n1(q(τn1))], 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,[f1
n(q(τn+1)),q(τn)]
taking the segment on Ws(γ2(τn)), and U
n,[f1
n(q(τn+1)),q(τn)] on Wu(γ1(τn)). In
words, the term f1
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
f1
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)>sf1
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 x0Dis 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),xD.
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
n1
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.).
REFERENCES
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.
Dabiri, J. O., Colin, S. P., Costello, J. H. & Gharib, M. 2005 Flow patterns generated by oblate
medusan jellyfish: field measurements and laboratory analyses. J. Expl Biol. 208, 1257–1265.
Duan,J.&Wiggins,S.1997 Lagrangian transport and chaos in the near wake of the flow around an
obstacle: a numerical implementation of lobe dynamics. Nonlinear Proc. Geophys. 4, 125–136.
Fatouraee, N. & Amini, A. A. 2003 Regularization of flow streamlines in multislice phase-contrast
MR imaging. IEEE Trans. Medical Imaging 22, 699–709.
Green, M., Rowley, C. W. & Haller, G. 2007 Detection of Lagrangian coherent structures in
three-dimensional turbulence. J. Fluid Mech. 572, 111–120.
Guckenheimer, J. & Holmes, P. J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations
of Vector Fields. Springer.
Haller, G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica
D147, 352–370.
Haller, G. 2001 Lagrangian structures and the rate of strain in a partition of two-dimensional
turbulences. Phys. Fluids 13, 3368–3385.
Haller, G. 2002 Lagrangian coherent structures from approximate velocity data. Phys. Fluids 14,
1851–1861.
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.
Krueger, P. S. 2006 Measurement of propulsive power and evaluation of propulsive performance
from the wake of a self-propelled vehicle. Bioinsp. Biomim. 1, S49–S56.
Malhotra,N.&Wiggins,S.1998 Geometric structures, lobe dynamics, and lagrangian transport
in flows with aperiodic time-dependence with applications to Rossby wave flow. J. Nonlinear
Sci. 8, 401–456.
Milne-Thompson, M. 1968 Theoretical Hydrodynamics. Dover.
O’Meara, M. M. & Mueller, T. J. 1987 Laminar separation bubble characteristics on an airfoil at
low Reynolds numbers. AIAA J. 25, 1033–1041.
Peng, J. & Dabiri, J. O. 2007 A potential-flow, deformable-body model for fluid-structure
interactions with compact vorticity: application to animal swimming measurements. Exps.
Fluids DOI: 10.1007/S00348-007-0315-1
Rom-Kedar,V.,Leonard,A.&Wiggins,S.1990 An analytical study of transport, mixing and
chaosinanunsteadyvorticalflow.J. Fluid Mech. 214, 347–394.
Rom-Kedar, V. & Wiggins, S. 1990 Transport in two-dimensional maps. Arch. Rat. Mech. Anal.
109, 239–298.
Geometry of unsteady fluid transport 145
Salman, H., Hesthaven, J. S., Warburton, T. & Haller, G. 2007 Predicting transport by
Lagrangian coherent structures with a high-order method. Theor. Comput. Fluid Dyn. 21,
39–58.
Shadden, S. C., Dabiri, J. O. & Marsden, J. E. 2006 Lagrangian analysis of fluid transport in
empiricalvortexringflows.Phys. Fluids 18, 047105.
Shadden, S. C., Leki en, F. & Marsden, J. E. 2005 Definition and properties of Lagrangian coherent
structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica
D212, 271–304.
Shariff, K., Leonard, A. & Ferziger, J. H. 1989 Dynamics of a class of vortex rings. NAS A Tec h.
Mem. 102257.
Shariff, K., Leonard, A. & Ferziger, J. H. 2006 Dynamical systems analysis of fluid transport in
time-periodic vortex ring flows. Phys. Fluids 18, 047104.
Shariff, K., Pulliam, T. H. & Ottino, J. M. 1991 A dynamical systems analysis of kinematics in
the time-periodic wake of a circular cylinder. Lectuces in Applied Mathematics,vol.28(ed.
E. L. Allgower, K. Georg & R. Miranda). pp. 613–646. Springer.
Sturtevant, B. 1981 Dynamics of turbulent vortex rings. AFOSR-TR-81-0400, available from
Defense Technical Information Service, Govt. Accession. No. AD-A098111, Fiche N81-24027.
Taira, K. & Colonius, T. 2007 The immersed boundary method: a projection approach. J. Comput.
Phys., submitted.
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in
fluid mixing. Phys. Rev. Lett. 88, 254501.
Wiggins, S. 2005 The dynamical systems approach to Langrangian transport in oceanic flows. Annu.
Rev. Fluid Mech. 37, 295–328.
Ya m a da , H . & M at s u i , T. 1978 Preliminary study of mutual slip-through of a pair of vortices. Phys.
Fluids 21 292–294.
... Fluid mass conservation requires that forward advection of fluid in the vicinity of a migrating jellyfish must be compensated by a rearward mass flux (Darwin 1953). While the Eulerian velocity and vorticity fields (Colin & Costello 2002;Park et al. 2014;Hoover et al. 2017) reveal instantaneous flow features, the dynamical-system-based approach that is adopted herein pertinently demarcates the surroundings from a Lagrangian perspective (Haller 2001(Haller , 2015Shadden et al. 2006;Franco et al. 2007;Katija & Dabiri 2009;) and is shown to be able to isolate physically distinct transport behaviours in a finite time. Accordingly, the implemented procedure displays its direct implication for the capture of motile prey and explains the animal's feeding mechanisms. ...
... subject to initial conditions below in (3.10) reveals the rate of extension of a particle's trajectory that is advected by a jellyfish's swimming motion. The Jacobian of φ is computed with respect to the prey particle locations at t 0 (Franco et al. 2007;Wilson et al. 2009). The resulting finite-time Cauchy-Green deformation tensor C(x) is defined as ...
... The motion of plankton (of tiny mass) in this case coincides with that of ideal Lagrangian tracer particles. To demonstrate the prey interception mechanism, the essential forward-time and backward-time FTLE fields (Shadden et al. 2006;Franco et al. 2007;Green, Rowley & Haller 2007) for a suspended layer of 4.9 × 10 6 infinitesimal Lagrangian prey particles are computed using (3.7)-(3.10) and the flow field that is created (see figure 4a and supplementary movie 2) by paddling A. aurita ...
Article
Three-dimensional simulations are performed to investigate swimming and prey capture by a paddling jellyfish. First, the three-dimensional vortex–vortex and vortex–body interactions are revealed, as the jellyfish swims forwards through several cycles of active muscle contraction followed by passive energy recapture via shape recovery. For varied transient paddling force and paddling frequency, we analyse the resultant changes of a jellyfish's swimming speed, interactive power, cost of transport and prey clearance rate. The pressure field around the periodically deformed elastic bell and the circulation generated by starting and stopping vortex rings are presented in greater detail to better understand the biophysical interactions that support swimming. Second, to reveal prey-specific interception and feeding behaviour, using a dynamical-system-based approach and modified Maxey–Riley equation, we compute the trajectories of the surrounding infinitesimal, inertial, opposite and normally escaping prey or plankton that hover around the medusa and are swept differently via the paddling-created velocity field. Accordingly, the diverse prey trajectories are obtained with varied paddling force, resonant driving of the elastic bell and for two different bell fineness ratios. These trajectories are then used to compute the finite-time Lyapunov exponent fields and identify particle Lagrangian coherent structures for various motile/strategically evasive prey, for five swimming cycles. The detected geometric separatrices unambiguously map and demarcate differently driven upstream fluid regions of a medusa and illustrate precisely from where an intercepted prey can be brought into the jellyfish bell, or safely stored in a capture region for ingestion, and from where a prey will surely escape. Hereby, for the first time, the prey-specific target regions, the physically well-defined three-dimensional capture surfaces and the generated cycle-to-cycle prey clearance rate are presented/analysed like never before, which provide a significantly advanced understanding on diverse predator–prey interactions and resultant success rate in prey capture. Several supplementary movies that show detailed fluid–structure interactions, transient entrainment of the floating prey and eventual prey confinement inside a secured capture surface are provided for two different jellyfish morphologies (fineness ratios 0.3 and 0.5) that help to better comprehend the natural prey encounter and hunting processes.
... Fluid mass conservation requires that forward advection of fluid in the vicinity of a migrating jellyfish must be compensated by a rearward mass flux (Darwin 1953). While the Eulerian velocity and vorticity fields (Colin & Costello 2002;Park et al. 2014;Hoover et al. 2017) reveal instantaneous flow features, the dynamical-system-based approach that is adopted herein pertinently demarcates the surroundings from a Lagrangian perspective (Haller 2001(Haller , 2015Shadden et al. 2006;Franco et al. 2007;Katija & Dabiri 2009;) and is shown to be able to isolate physically distinct transport behaviours in a finite time. Accordingly, the implemented procedure displays its direct implication for the capture of motile prey and explains the animal's feeding mechanisms. ...
... subject to initial conditions below in (3.10) reveals the rate of extension of a particle's trajectory that is advected by a jellyfish's swimming motion. The Jacobian of φ is computed with respect to the prey particle locations at t 0 (Franco et al. 2007;Wilson et al. 2009). The resulting finite-time Cauchy-Green deformation tensor C(x) is defined as ...
... The motion of plankton (of tiny mass) in this case coincides with that of ideal Lagrangian tracer particles. To demonstrate the prey interception mechanism, the essential forward-time and backward-time FTLE fields (Shadden et al. 2006;Franco et al. 2007;Green, Rowley & Haller 2007) for a suspended layer of 4.9 × 10 6 infinitesimal Lagrangian prey particles are computed using (3.7)-(3.10) and the flow field that is created (see figure 4a and supplementary movie 2) by paddling A. aurita ...
Article
Swimming and prey capture by prolate and oblate jellyfishes are numerically examined in two-dimensions using multi-relaxation-time lattice Boltzmann (MRT-LB) and immersed boundary (IB) methods. The near-field fluid structure interaction (FSI) and predator-prey interaction mechanisms are revealed via the simulated Eulerian flow characteristics, finite-time Lyapunov exponent (FTLE) field, and Lagrangian coherent structures (LCS). We implement appropriate periodic body force (Fb) distribution at nodal points of the elastic bell in radial direction to model the paddled swimming as well as complex feeding behavior. For a paddling jellyfish the evolved starting and stopping vortices, as move close to each other in near-wake, create the necessary vortex induced thrust that facilitate the forward body motion. The forced bell contraction in power stroke assists quicker propulsive swimming, whereas passive bell expansion in resting phase facilitates the continued vortex induced forward movement for larger duration, via the refilled fluid momentum. For feeding the detailed prey interception and precise capture areas for various jellyfish models are identified hereby via the computed prey tracks, forward and backward time FTLE fields, and LCS. Swimming performances are analyzed based on interactive thrust and drag forces, input power (energy rate required for bell contraction), output power (thrust multiplied by centroid velocity), and cost of transport (COT). At low Reynolds number (Re) the COT becomes higher for an oblate jellyfish than that of a prolate one; while with the increased Re the oblate species appears more economical. However, for amplified paddling force (Fb) or reduced flapping frequency of bell, the COT for an oblate jellyfish steadily decreases. Hereby impacts of the varied force duration, flapping amplitude, flapping frequency, and bell-elasticity on the swimming are analyzed in greater detail. Notably, the propulsion efficiency increased for higher flapping frequency. The adopted numerical model efficiently unfolds the prey capture mechanisms that are adopted by prolate and oblate medusae and quantifies their success rate (clearance rate, CR) in prey capture. Unlike in past studies, the FTLE fields and LCS that are computed here by tracking the transient motion of a large number of suspended Lagrangian prey particles reveal the realistic predator-prey interaction and precise prey capture surface, which are difficult to measure or analyze empirically.
... In the Lagrangian perspective, fluid particles are applied to examine the flow dynamical system rather than a continuum [19]. It is widely used to present high-quality visible details of time-dependent vortex structures, which may not be apparent in the classical Eulerian fields [20,21]. In addition, to capture the associated vortex structures, Q iso-surfaces are defined [22]. ...
Article
Full-text available
This paper aims to further the understanding of the mixing process of in-line twin synthetic jets (SJs) and their impact in the near-wall region in a flat-plate laminar boundary layer. A numerical study has been carried out, in which colored fluid particles and the Q criterion are used to track the SJ-induced vortex structures at the early stage of the evolution. Interacting vortex structures at four selected phase differences are presented and analyzed. It is found that the fluid injected at the early stage of the blowing stroke mainly contributes to the formation of the hairpin legs, the fluid injected near the maximum blowing mainly contributes to the formation of the hairpin head, and the fluid injected at the late stage of the blowing stroke contributes very little to the formation of the hairpin vortex. It is also confirmed that, irrespective of the phase difference, the hairpin vortex issued from the upstream actuator is more capable of maintaining its coherence than its counterpart issued from the downstream actuator. The influence of the interacting vortex structures on the boundary layer is also studied through investigating excess wall shear stress. In all cases, a pair of streaks of high wall shear stress can be observed with similar size. Among them, the streaks have the strongest wall shear stress, with the largest gap at phase difference 0 when partially interacting vortex structures are produced. The findings can provide valuable guiding information for the applications of synthetic jets in heat transfer, mixing control, and flow control in a crossflow.
... Previous analyses of jellyfish have used FTLEs to qualitatively show fluid regions where fluid is being pushed or pulled towards or away from the bell during a contraction cycle (Wilson et al., 2009). Through such analysis, fluid transport patterns of particular interest are revealed, such as those used in feeding Sapsis et al., 2011) and locomotion (Franco et al., 2007;Zhang, 2008;Lipinski and Mohseni 2009;Wilson et al., 2009;Haller and Sapsis, 2011;Miles and Battista, 2019). Note that the aforementioned studies all involve actively swimming jellyfish, unlike the benthic upsidedown jellyfish. ...
Article
Full-text available
Upside-down jellyfish, Cassiopea , are prevalent in warm and shallow parts of the oceans throughout the world. They are unique among jellyfish in that they rest upside down against the substrate and extend their oral arms upwards. This configuration allows them to continually pull water along the substrate, through their oral arms, and up into the water column for feeding, nutrient and gas exchange, and waste removal. Although the hydrodynamics of the pulsation of jellyfish bells has been studied in many contexts, it is not clear how the presence or absence of the substrate alters the bulk flow patterns generated by Cassiopea medusae. In this paper, we use three-dimensional (3D) particle tracking velocimetry and 3D immersed boundary simulations to characterize the flow generated by upside-down jellyfish. In both cases, the oral arms are removed, which allows us to isolate the effect of the substrate. The experimental results are used to validate numerical simulations, and the numerical simulations show that the presence of the substrate enhances the generation of vortices, which in turn augments the upward velocities of the resulting jets. Furthermore, the presence of the substrate creates a flow pattern where the water volume within the bell is ejected with each pulse cycle. These results suggest that the positioning of the upside-down jellyfish such that its bell is pressed against the ocean floor is beneficial for augmenting vertical flow and increasing the volume of water sampled during each pulse.
... FTLE and LCS have also been used extensively to analyse ocean flows [59][60][61], for example to model the spread of pollution [62]. More broadly, FTLE has also been used to compute coherent structures for a wide range of other flows [63][64][65][66][67][68]. In this work, we will use FTLE fields generated from passive particles to investigate the trajectories of active mobile sensors, to understand how and when these sensors exploit structures in the flow field for energy-efficient transport. ...
Article
Intelligent mobile sensors, such as uninhabited aerial or underwater vehicles, are becoming prevalent in environmental sensing and monitoring applications. These active sensing platforms operate in unsteady fluid flows, including windy urban environments, hurricanes and ocean currents. Often constrained in their actuation capabilities, the dynamics of these mobile sensors depend strongly on the background flow, making their deployment and control particularly challenging. Therefore, efficient trajectory planning with partial knowledge about the background flow is essential for teams of mobile sensors to adaptively sense and monitor their environments. In this work, we investigate the use of finite-horizon model predictive control (MPC) for the energy-efficient trajectory planning of an active mobile sensor in an unsteady fluid flow field. We uncover connections between trajectories optimized over a finite-time horizon and finite-time Lyapunov exponents of the background flow, confirming that energy-efficient trajectories exploit invariant coherent structures in the flow. We demonstrate our findings on the unsteady double gyre vector field, which is a canonical model for chaotic mixing in the ocean. We present an exhaustive search through critical MPC parameters including the prediction horizon, maximum sensor actuation, and relative penalty on the accumulated state error and actuation effort. We find that even relatively short prediction horizons can often yield energy-efficient trajectories. We also explore these connections on a three-dimensional flow and ocean flow data from the Gulf of Mexico. These results are promising for the adaptive planning of energy-efficient trajectories for swarms of mobile sensors in distributed sensing and monitoring.
... FTLE and LCS have also been used extensively to analyze ocean flows [56][57][58], for example to model the spread of pollution [59]. More broadly, FTLE has been used to coherent structures and mixing in a wide range of other flows [60][61][62][63][64][65]. In this work, we will use FTLE fields generated from passive particles to investigate the trajectories of active mobile sensors, to understand how and when these sensors exploit structures in the flow field for energy-efficient transport. ...
Preprint
Intelligent mobile sensors, such as uninhabited aerial or underwater vehicles, are becoming prevalent in environmental sensing and monitoring applications. These active sensing platforms operate in unsteady fluid flows, including windy urban environments, hurricanes, and ocean currents. Often constrained in their actuation capabilities, the dynamics of these mobile sensors depend strongly on the background flow, making their deployment and control particularly challenging. Therefore, efficient trajectory planning with partial knowledge about the background flow is essential for teams of mobile sensors to adaptively sense and monitor their environments. In this work, we investigate the use of finite-horizon model predictive control (MPC) for the energy-efficient trajectory planning of an active mobile sensor in an unsteady fluid flow field. We uncover connections between the finite-time optimal trajectories and finite-time Lyapunov exponents (FTLE) of the background flow, confirming that energy-efficient trajectories exploit invariant coherent structures in the flow. We demonstrate our findings on the unsteady double gyre vector field, which is a canonical model for chaotic mixing in the ocean. We present an exhaustive search through critical MPC parameters including the prediction horizon, maximum sensor actuation, and relative penalty on the accumulated state error and actuation effort. We find that even relatively short prediction horizons can often yield nearly energy-optimal trajectories. These results are promising for the adaptive planning of energy-efficient trajectories for swarms of mobile sensors in distributed sensing and monitoring.
... In a nutshell, using LCSs for time-dependent fluid flows provides a systematic way to uncover the flow's complex hidden dynamics through visualizations that can be qualitatively deciphered. Moreover in terms of jellyfish, they help reveal particle transport patterns that are of particular interest, such as feeding and prey-capture [77,124] and/or locomotion [65,67,[125][126][127][128][129]. FTLEs were computed in the open source software VisIt [117], where trajectories were computed using instantaneous snapshots of the two-dimensional Eulerian fluid velocity field across the entire computational domain using a forward/backward Dormand-Prince (Runge-Kutta) time-integrator with a relative tolerance of 10 −4 and absolute tolerance of 10 −5 , a maximum advection time of 0.02 s that equates to 1.6% of a contraction cycle, across a maximum number of steps of 250. ...
Article
Full-text available
Jellyfish are majestic, energy-efficient, and one of the oldest species that inhabit the oceans. It is perhaps the second item, their efficiency, that has captivated scientists for decades into investigating their locomotive behavior. Yet, no one has specifically explored the role that their tentacles and oral arms may have on their potential swimming performance. We perform comparative in silico experiments to study how tentacle/oral arm number, length, placement, and density affect forward swimming speeds, cost of transport, and fluid mixing. An open source implementation of the immersed boundary method was used (IB2d) to solve the fully coupled fluid–structure interaction problem of an idealized flexible jellyfish bell with poroelastic tentacles/oral arms in a viscous, incompressible fluid. Overall tentacles/oral arms inhibit forward swimming speeds, by appearing to suppress vortex formation. Nonlinear relationships between length and fluid scale (Reynolds Number) as well as tentacle/oral arm number, density, and placement are observed, illustrating that small changes in morphology could result in significant decreases in swimming speeds, in some cases by upwards of 80–90% between cases with or without tentacles/oral arms.
... Within fluid flows, LCSs help reveal particle transport patterns that are of particular importance in biology. They can be used to understand various processes for jellyfish, including feeding and prey-capture [77,124] and locomotion [65,67,[125][126][127][128][129]. FTLEs were computed using VisIt [117], where trajectories were computed using instantaneous snapshots of the 2D fluid velocity vector field on the entire computational domain using a forward/backward Dormand-Prince (Runge-Kutta) time-integrator with a relative and absolute tolerance of 10 −4 and 10 −5 , respectively, and a maximum advection time of 0.02s (1.6% of a contraction cycle) with a maximum number of steps of 250. ...
Preprint
Full-text available
Jellyfish - majestic, energy efficient, and one of the oldest species that inhabits the oceans. It is perhaps the second item, their efficiency, that has captivated scientists for decades into investigating their locomotive behavior. Yet, no one has specifically explored the role that their tentacles and oral arms may have on their potential swimming performance, arguably the very features that give jellyfish their beauty while instilling fear into their prey (and beach-goers). We perform comparative in silico experiments to study how tentacle/oral arm number, length, placement, and density affect forward swimming speeds, cost of transport, and fluid mixing. An open source implementation of the immersed boundary method was used (IB2d) to solve the fully coupled fluid-structure interaction problem of an idealized flexible jellyfish bell with poroelastic tentacles/oral arms in a viscous, incompressible fluid. Overall tentacles/oral arms inhibit forward swimming speeds, by appearing to suppress vortex formation. Non-linear relationships between length and fluid scale (Reynolds Number) as well as tentacle/oral arm number, density, and placement are observed, illustrating that small changes in morphology could result in significant decreases in swimming speeds, in some cases by downwards of 400% between cases with to without tentacles/oral arms.
... Within fluid flows, LCSs help reveal particle transport patterns that are of particular importance in biology. They can be used to understand various processes for jellyfish, including feeding and prey-capture [69,77] and locomotion [32,92,88,60,44,82]. FTLEs were computed using VisIt [19], where trajectories were computed using instantaneous snapshots of the 2D fluid velocity vector field on the entire computational domain using a forward/backward Dormand-Prince (Runge-Kutta) time-integrator with a relative and absolute tolerance of 10 −4 and 10 −5 , respectively, and a maximum advection time of 0.02s with a maximum number of steps of 250. ...
Preprint
Full-text available
Jellyfish have been called one of the most energy-efficient animals in the world due to the ease in which they move through their fluid environment, by product of their morphological, muscular, and material properties. We investigated jellyfish locomotion by conducting \textit{in silico} comparative studies and explored swimming performance in different fluids (e.g., changing viscosities) at different fluid scales (e.g., Reynolds Number), bell contraction frequencies, contraction phase kinematics, as well as bell morphologies and contraction amplitude. To study these relationships, an open source implementation of the immersed boundary method was used (\textit{IB2d}) to solve the fully coupled fluid-structure interaction problem of a flexible jellyfish bell in a viscous fluid. Complex relationships between scale, morphology, and frequency lead to optimized forward swimming speeds for a particular bell composition, stemming from intricate vortex wake topology, interactions, and fluid mixing. Lastly, we offer an open source computational jellyfish locomotion model to the science community that can be used as a starting place for future numerical experimentation.
... These ridges are adopted to describe the fluid dynamics phenomenon in a fully developed turbulence flow. Franco et al. 18 analyzed the structure of flow field around a cylinder under low Reynolds number. The results showed that LCS had a better performance in capturing the unsteady vortex region compared to the Eulerian system. ...
Article
Full-text available
In this study, a modified partially averaged Navier–Stokes (MPANS) model is applied to investigate the flow instability characteristics in a low specific centrifugal pump. In MPANS model, the unresolved-to-total ratio of kinetic energy f k is determined according to the local grid size and turbulence length scale. The numerical results by MPANS model are compared with that simulated by SST k-ω model and the available experimental data. It is noted that MPANS model shows better performance for investigating the unstable flow in the current pump under part-load operation conditions. The time-averaged internal flow and flow incidence in the pump impeller depicts that with the decreasing flow coefficient, flow separation develops in the impeller. Owing to the strong separation flow as well as vortex evolution, incidence angle is large and varies remarkably at the entrance of blade-to-blade passage in the pump impeller. The evolution dynamics of rotating stall is further discussed in detail based on vorticity transport equation. During the evolution of rotating stall, the vortex stretching term has an important effect on vorticity transport under the part-load conditions. The analysis of the pressure fluctuations excited by periodic evolution of rotating stall shows that the rotating stall cell propagates along the rotational direction, and identifies the rotating stall frequency (f stall ), which is much lower than the rotational frequency of the impeller, f n (f stall = 8.76% f n ). Finally, two-dimensional Lagrangian coherent structure (LCS) is used to reveal the separation flow in blade-to-blade passages of the pump by monitoring the trajectory of the particles. Both LCS and vortex structure by λ 2 can clearly demonstrates the passage blockage and flow separation under the part-load operation conditions, depicting that the separation flow occurs at blade suction side and develops from the leading edge to the main passage in the impeller.
Article
Full-text available
▪ Abstract Chaotic advection and, more generally, ideas from dynamical systems, have been fruitfully applied to a diverse, and varied, collection of mixing and transport problems arising in engineering applications over the past 20 years. Indeed, the “dynamical systems approach” was developed, and tested, to the point where it can now be considered a standard tool for understanding mixing and transport issues in many disciplines. This success for engineering-type flows motivated an effort to apply this approach to transport and mixing problems in geophysical flows. However, there are fundamental difficulties arising in this endeavor that must be properly understood and overcome. Central to this approach is that the starting point for analysis is a velocity field (i.e., the “dynamical system”). In many engineering applications this can be obtained sufficiently accurately, either analytically or computationally, so that it describes particle trajectories for the actual flow. However, in geophysical flows (a...
Article
Full-text available
We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikov's technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.
Article
This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence. The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.
Article
We derive analytic criteria for the existence of hyperbolic (attracting or repelling), elliptic, and parabolic material lines in two-dimensional turbulence. The criteria use a frame-independent Eulerian partition of the physical space that is based on the sign definiteness of the strain acceleration tensor over directions of zero strain. For Navier-Stokes flows, our hyperbolicity criterion can be reformulated in terms of strain, vorticity, pressure, viscous and body forces. The special material lines we identify allow us to locate different kinds of material structures that enhance or suppress finite-time turbulent mixing: stretching and folding lines, Lagrangian vortex cores, and shear jets. We illustrate the use of our criteria on simulations of two-dimensional barotropic turbulence.
Article
An investigation was conducted to demonstrate experimentally the phenomenon of mutual slip-through for a pair of vortices, which has been described in many textbooks, although Maxworthy (1972) had been unable to reproduce it in the laboratory. The investigation was concerned with the generation of vortex rings at larger Reynolds numbers. Air was employed as a working fluid and an orifice of 8 cm diameter was tested. Cigarette smoke and a smoke wire, stretched across the diameter of the orifice, were used to visualize the vortex rings. A streak camera and a 16 mm cine-camera were employed to take pictures of the successive motions of the rings. Pictures are presented showing five typical stages of the mutual slip-through of a pair of vortices. The pictures show that the slip-through phenomenon as described by Batchelor (1967) could be successfully reproduced in the experiment.
Article
This paper examines whether hyperbolic Lagrangian structures-such as stable and unstable manifolds-found in model velocity data represent reliable predictions for mixing in the true fluid velocity field. The error between the model and the true velocity field may result from velocity interpolation, extrapolation, measurement imprecisions, or any other deterministic source. We find that even large velocity errors lead to reliable predictions on Lagrangian coherent structures, as long as the errors remain small in a special time-weighted norm. More specifically, we show how model predictions from the Okubo-Weiss criterion or from finite-time Lyapunov exponents can be validated. We also estimate how close the true Lagrangian coherent structures are to those predicted by models.
Article
We use direct Lyapunov exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow. These results are compared with commonly used Eulerian criteria for coherent vortices. We find that despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold. As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow. We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures.
Article
The most widely used definitions of a vortex are not objective: they identify different structures as vortices in frames that rotate relative to each other. Yet a frame-independent vortex definition is essential for rotating flows and for flows with interacting vortices. Here we define a vortex as a set of fluid trajectories along which the strain acceleration tensor is indefinite over directions of zero strain. Physically, this objective criterion identifies vortices as material tubes in which material elements do not align with directions suggested by the strain eigenvectors. We show using examples how this vortex criterion outperforms earlier frame-dependent criteria. As a side result, we also obtain an objective criterion for hyperbolic Lagrangian structures.