J.
Fluid
Me&.
(1990),
VO~.
214,
pp.
347-394
Printed in Great Britain
347
An analytical study
of
transport, mixing and chaos
in an unsteady vortical
flow
By
V.
ROM-KEDAR',t A. LEONARD'
AND
S.
WIGGINS3
Applied Mathematics, California Institute of Technology, Pasadena, CA
91
125,
USA
*
Graduate Aeronautical Laboratories, California Institute of Technology,
Pasadena, CA
91125,
USA
Applied Mechanics, California Institute
of
Technology, Pasadena, CA
91
125,'
USA
(Received
2
May
1988)
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 strain-
rate 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 strain-
rate 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.
1.
Introduction
In most fluid flows of interest, transport and mixing are dominated by convective
processes
so
that the relative motions of fluid particles are all important.
Unfortunately particle motion is generally more complex than the underlying fluid
dynamics.
For
example, while the motion of three point vortices in an unbounded
domain
is
integrable, particle motion in this flow can be chaotic (Aref
1983)
and
-f
Present address: The James Franck Institute, The University of Chicago, Chicago,
IL
60637,
USA.
348
V.
Rom-Kedar,
A.
Leonard
and
S.
Wiggins
certain simple steady, spatially periodic solutions to the Euler equations in three
dimensions, known
as
ABC (Arnol’d, Beltrami and Childress) flows yield chaotic
particle motion (Dombre
et
al. 1986).
Of course, if the fluid dynamics is sufficiently simple then particle motions are
integrable and
a
direct analytical attack on the problem may be fruitful. An example
in this class is the analysis of a diffusion flame by Marble
(1985),
involving the rolling
up of an initially plane interface in the flow of
a
viscous line vortex in two
dimensions. At the other end of the scale, when the flow is turbulent, direct numerical
integration of the Navier-Stokes equations plus convective equations for passive
scalars (Kerr
1985;
Pope
1987)
is a computational approach to mixing problems,
whereas
a
theoretical approach might consist of constructing reasonable physical
models for mixing processes (Broadwell
1989
;
Dimotakis
1989
;
Kerstein
&
Ashurst
1984).
In this paper we consider an intermediate case, one in which the flow is
relatively simple but the particle motion is chaotic. We show that the recent rapid
development in the theory of nonlinear dynamical systems and chaotic phenomena
gives much hope for a rather extensive analysis of particle motion in such flows.
Indeed, the dynamical systems approach to the study of fluid flows is very similar in
spirit
to
the flow visualization techniques utilized in the experimental study
of
coherent structures in the sense that dynamical systems theory is concerned with the
global topology of the flow from
a
Lagrangian point of view. Since to
a
good
approximation temperature and mass move with the fluid velocity, understanding
the structures governing particle motion in fluid flows is necessary for interpretations
of flow visualizations (the visualization of motion of mass particles) and predictions
of
mass and heat transfer in technological applications.
The application of dynamical systems theory to the study of the global topology
of fluid particle motions is not new. The
first
work appears to be that of H6non
(1966)
who, acting on
a
suggestion of Amol’d
(1965),
numerically studied the fluid particle
motions in ABC flow. Hdnon showed that the flow contained KAM tori
as
well
as
chaotic motions of the Smale horseshoe type. This flow has recently been the subject
of more extensive study by Dombre
et
al.
(1986).
Chaotic particle motions in the ABC
flows also have relevance to the kinematic dynamo problem, see Arnol’d
&
Kortine
(1983),
Galloway
&
Frisch
(1986),
and Moffatt
&
Proctor
(1984).
Aref
(1985)
made
the first explicit connection between particle motions in two-dimensional in-
compressible flow and two-dimensional Hamiltonian dynamical systems.
Since the study of fluid particle motions involves only kinematical considerations,
the application, and hence, results of dynamical systems theory are independent of
Reynolds number. For example, Aref
&
Balachandar
(1986)
showed that unsteady
Stokes flow between eccentric rotating cylinders, in which the rotation rate is
modulated periodically in time, can exhibit chaotic particle motions of the Smale
horseshoe type, Thus this particular Stokes flow is effectively non-reversible. This
same flow has also been studied experimentally
as
well
as
theoretically by Chaiken
et
al.
(1986, 1987).
Ottino and coworkers (see Chien, Rising
&
Ottino
1986;
Khakar,
Rising
&
Ottino
1986;
and Ottino
et
al.
1988)
studied chaotic fluid particle motions
in
a
variety of flows, both
at
small and large Reynolds numbers with particular
emphasis on using dynamical systems techniques as a theoretical basis for the
discussion of mixing processes. Broomhead
&
Ryrie
(1988)
studied fluid particle
motions in the velocity field of Taylor vortices close to the onset of the wavy
instability and demonstrated the chaotic transference of fluid between neighbouring
vortices. Feingold, Kadanoff
&
Piro
(1988)
studied models
for
particle motion in
Transport, mixing and chaos
in
an unsteady vortical
flow
349
three-dimensional time-dependent flows. Additional references on the application of
dynamical systems techniques to the study of fluid particle trajectories are Suresh
(1985)
and Arter
(1983).
In this paper we study fluid particle motion in the velocity field induced by two
counter-rotating point vortices of equal strength subject to a time-periodic strain
field. This is a fundamental type of flow which is relevant to
a
wide variety of
applications
as,
for
example, in the study of oscillatory flows in wavy-walled tubes
(see Ralph
1986;
Sobey
1985,
and Appendix
A),
in the study of trailing vortices, and
in the study
of
perturbed vortex rings (Shariff
1989).
The main difference between
our
analysis of the topology of a fluid flow via
dynamical systems techniques and previous analyses is that, rather than just using
the framework of dynamical systems theory to give a description of the topology and
indicate the presence
of
chaotic fluid particle trajectories, we use the framework in
order to calculate physically measurable quantities such as fluxes and the distribution
of volumes via residence times. We do this by
first
identifying the structures
in
the
flow responsible for these physical processes and then by using the dynamics of these
structures to predict these physical quantities. Thus in some sense we realize the goal
of
the study of coherent structures for our problem. Additionally, in this paper we
introduce two new concepts that play an important role in the study of mixing and
transport processes due to chaotic fluid particle motions.
The first is
Tangle Dynamics.
In
$3
we review how the study of particle motions
in two-dimensional incompressible time-periodic fluid flows can be reduced to the
study of a two-dimensional map.
It
is well known in the dynamical systems literature
that such maps may possess
resonance bands
consisting of alternating hyperbolic and
elliptic periodic points. This has fluid dynamical significance in the sense that the
stable and unstable manifolds of the hyperbolic points create partial barriers to
transport in the flow. Additionally, these stable and unstable manifolds may
intersect many times resulting in a complicated geometrical structure that
dramatically influences the stretching and deformation of fluid elements. We develop
analytical and computational techniques which we refer to as
tangle dynamics
that
allow us to compute the rate of transport of fluid between regions separated by these
partial barriers. From this information we can compute residence time distributions
and, more generally, determine the effect of a resonance band on
a
fluid element. We
develop these ideas in the context of the specific flow considered in this paper;
however, recently the methods have been generalized to apply to any two-
dimensional time-periodic fluid flow, see Rom-Kedar
&
Wiggins
(1989).
These
techniques are mathematically exact and represent a fundamental improvement
over the approximate phase-space transport models of MacKay, Meiss
&
Percival
(1984).
These models rely on the assumption of an infinite diffusion rate within the
chaotic regions. Such an assumption is not applicable to the flow that we are
studying.
The second new concept is
Finite
Time
Xtretch.
Ottino
(1988)
has shown the
relationship between the notion of a Liapunov exponent from dynamical systems
theory and the stretching of fluid elements. However, the Liapunov exponent is a
quantity computed
for
a
single fluid particle trajectory which is time averaged in an
asymptotic sense. Thus there is a practical limitation of this quantity in that, for
many open flows, most fluid particles spend only a finite time in the chaotic zone
rendering the classical theory of Liapunov exponents inappropriate. This is so
because the asymptotic time average
for
such trajectories would give
a
zero
350
V.
Rom-Kedar,
A.
Leonard
and
S.
Wiggins
exponent. However, using tangle dynamics and proof of the existence of chaotic
particle motions, we are able to determine which particles should experience
temporary exponential stretching and the finite time interval over which most of this
stretching will take place. We then quantify the stretching by considering the total
stretch suffered during this finite period of time.
This paper is organized as follows: In
$2
we derive the velocity field for the
oscillating vortex pair and in
$3
we begin
our
analysis of the velocity field by
introducing the PoincarB map. In
$4
we discuss three qualitatively distinct regions
which arise in our flow: the free flow region, the core, and the mixing region. We
discuss tangle dynamics and the associated mechanism for mass transport in the flow
in
$5
and we consider mass transport in detail and give precise definitions to the
concepts of entrainment and detrainment in terms of tangle dynamics in
$6,
along
with the results of numerical computations. In
$7
we discuss the concept of chaos and
show how
it
arises in our flow, and in
$8
we discuss mixing and the total stretch of
fluid elements
as
they pass through regions containing localized chaos. Summary and
conclusions are given in
$9,
2.
Oscillating vortex pair
We examine the flow governed by a vortex pair in the presence
of
an oscillating
external strain-rate field. The vortices have circulations
f
f
and are separated by a
nominal distance
2d
in the y-direction. The stream function for the flow in a frame
moving with the average velocity of the vortices is
where (x,(t),
f
y,(t))
are the vortex positions,
E
is the strain rate and
V,
is the average
velocity of the vortex pair. If
E
=
0
then
(x,,y,)
=
(0,d)
and
V,
=
r/47~d.
The
equations of particle motion are
We show, as an example, in Appendix
A
that this flow approximates the flow induced
by
a
vortex pair in
a
wavy-walled channel. We obtain dimensionless variables as
follows
:
=+
Y.
E
2nd
V,
r
+t,
-+€,
-
+vv>
=+Y)
r
X
Y
ft
-+x3
d
FY3
-
2nd2
w
r
Then
(2.1)
and
(2.2)
become
EX
-
v,
+-sin (tly),
(2.3~)
dx
(Y-Yv)
-
(Y
+
Yv)
dt
(x-xv)2+(Y-Yv)2
(x-Xv)2+(Y+Yv)2
IY
1
]-E?lsin(t/y).
(2.3b).
dt
(x--xv)2
+
(Y
-!/,I2
-
(X--5A2
+
(Y
+
YJ2
Y
Using the fact that a point vortex is convected with the flow but does not induce self-
velocity we obtain the following equations for the vortex position locations
:
-
dx,
- -
__-
1
dt
2Yv
Y
dt
Y
v,+-sin
EX,
.
(tly),
-
dYv
-
-
__
EYvsin (t/y).
(2.4a,
b)
Transport, mixing and
chaos
in
an
unsteady vortical
flow
35
1
FIGURE
1.
Streamlines
of
the unperturbed flow.
The resulting motion of the vortices is relatively simple. Equations
(2.4)
with the
initial conditions
s,(O)
=
0,
y,(O)
=
1,
are easily integrated to give
x,(t)
=
e-s(cos(t/Y)-l)
[1
-
2~
ee(COS(8)-1)
3
ds, y,(t)
=
ee(cos(t/y)-l).
(2.5a,
b)
The requirement that the mean velocity of the vortex pair be zero in the moving
frame yields
v,
=
e"/21,(~), where
lo
is
the modified Bessel function of order zero.
From
(2.5)
it
is clear that the vortices oscillate in orbits near the points
(0,
f
1).
Thus
we term the resulting flow given by
(2.3)
the oscillating vortex pair
(OVP)
flow.
Equations
(2.3)
together with
(2.5)
give the equations of particle motion as a
function of two dimensionless parameters
y
and
E,
proportional to vortex strength
and strain rate, respectively.
For
most of the analysis that follows
E
can take on
arbitrary values. However, for the perturbation calculations we shall assume that
E
is small and will require an expansion of the right-hand side of
(2.3)
in powers of
E.
This expansion yields equations
of
motion for fluid particles which are of the form of
a
periodically perturbed integrable Hamiltonian system
:
rv
(2.6~)
(2.6b)
The functions
fi,
gi
are given in Appendix
B.
For
e
=
0
the phase portrait of the integrable Hamiltonian system,
or
equivalently
the streamlines of the flow induced by
a
vortex pair in the frame moving with the
vortices, appears in figure
1.
Note that for this case, there are two hyperbolic
stagnation points
p-,
p,
connected by three limiting streamlines
Yu,
Yo
and
Yt
defined
by
Y(~,y)l~-~
=
0,
1x1
<
43,
with
y
>
0,
y
=
0,
and
y
<
0
respectively. Thus
a
fixed, closed volume of fluid
or
'bubble
'
is bounded by the limiting streamlines and
moves with the vortex pair for all times. As we shall see below, this picture changes
dramatically when
E
$:
0.
Note also that, for any
E,
the flow is symmetric about the
x-axis and thus we need only study the flow in the upper half-plane. Such symmetry
would be present in axisymmetric flows.
If
the strain-rate field is not aligned with the
(x, 9)-axes the straight line connecting the two vortices also rotates periodically, but
the qualitative behaviour of the particle motion is the same as that discussed in the
following but with the added complication of transport between the upper and lower
half-planes.