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Drift,partialdriftandDarwin'sproposition
I.Eames,S.E.BelcherandJ.C.R.Hunt
JournalofFluidMechanics/Volume275/September1994,pp201223
DOI:10.1017/S0022112094002338,Publishedonline:26April2006
Linktothisarticle:http://journals.cambridge.org/abstract_S0022112094002338
Howtocitethisarticle:
I.Eames,S.E.BelcherandJ.C.R.Hunt(1994).Drift,partialdriftandDarwin'sproposition.Journal
ofFluidMechanics,275,pp201223doi:10.1017/S0022112094002338
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J.
Fluid
Mech.
(1994),
vol.
275,
pp.
201-223
Copyright
@
1994
Cambridge
University
Press
201
Drift, partial drift and Darwin’s proposition
By
I.
EAMES’,
S.E.
BELCHERlt
AND
J.C.R.
HUNT2
Department of Applied Mathematics and Theoretical Physics, University of Cambridge,
Silver Street, Cambridge CB3 9EW, UK
*
Meteorological Office, Bracknell, Berks RG12 2SZ,
UK
(Received 27 September 1993 and in revised form
8
April 1994)
A
body moves at uniform speed in an unbounded inviscid fluid. Initially, the body
is infinitely far upstream of an infinite plane of marked fluid; later, the body moves
through and distorts the plane and, finally, the body is infinitely far downstream of
the marked plane. Darwin
(1953)
suggested that the volume between the initial and
final positions of the surface of marked fluid (the drift volume) is equal to the volume
of fluid associated with the ‘added-mass’ of the body.
We re-examine Darwin’s
(1953)
concept of drift and, as an illustration, we study
flow around a sphere. Two lengthscales are introduced:
pmax,
the radius of a circular
plane of marked particles; and
XO,
the initial separation of the sphere and plane.
Numerical solutions and asymptotic expansions are derived for the horizontal La-
grangian displacement of fluid elements. These calculations show that depending on
its initial position, the Lagrangian displacement of a fluid element can be either pos-
itive
-
a Lagrangian drift
-
or
negative
-
a Lagrangian reflux.
By
contrast, previous
investigators have found only a positive horizontal Lagrangian displacement, because
they only considered the case of infinite
XO.
For finite
XO,
the volume between the
initial and final positions of the plane of marked fluid is defined to be the ‘partial
drift volume’, which is calculated using a combination of the numerical solutions
and the asymptotic expansions. Our analysis shows that in the limit corresponding
to Darwin’s study, namely that both
xo
and
pmax
become infinite, the partial drift
volume is not well-defined: the ordering of the limit processes is important. This
explains the difficulties Darwin and others noted in trying to prove his proposition
as a mathematical theorem and indicates practical, as well as theoretical, criteria that
must be satisfied for Darwin’s result to hold.
We generalize our results for a sphere by re-considering the general expressions
for Lagrangian displacement and partial drift volume. It is shown that there are
two contributions to the partial drift volume. The first contribution arises from a
reflux
of
fluid and is related to the momentum of the flow; this part is spread over a
large area. It is well-known that evaluating the momentum of an unbounded fluid
is
problematic since the integrals do not converge; it is this first term which prevented
Darwin from proving his proposition as a theorem. The second contribution to the
partial drift volume is related to the kinetic energy of the flow caused by the body:
this part is Darwin’s concept of drift and is localized near the centreline. Expressions
for partial drift volume are generalized for flow around arbitrary-shaped two- and
three-dimensional bodies. The partial drift volume is shown to depend on the solid
angles the body subtends with the initial and final positions of the plane of marked
fluid. This result explains why the proof of Darwin’s proposition depends on the ratio
t
Present address: Department of Meteorology, University
of
Reading, Reading, RG6 2AU,
UK
Pmax
1x0.
202
used to illustrate the differences between drift in bounded and unbounded flows.
I.
Eames,
S.
E.
Belcher and
J.
C.
R.
Hunt
An example of drift due to a sphere travelling at the centre of a square channel is
1.
Introduction
When a sphere moves in an unbounded inviscid fluid, one might intuitively expect,
as Darwin (1953) remarked, that there would be a net
flux
of fluid in the opposite
direction to the motion of the sphere; a reflux of fluid. Detailed calculations by
Darwin (1953) show that there is a volume of fluid that drifts in the same direction
as the sphere. This flux of fluid drifting with the sphere can be interpreted as a
‘potential-flow wake’ behind the body.
Darwin (1953) examined the motion of an arbitrarily shaped solid body in an
unbounded region of inviscid fluid. The body starts infinitely far from an infinite
plane
of
marked fluid, and travels at a constant speed towards the plane, which
is
then
distorted by the passage of the body. Darwin defined the
drft volume
to be the volume
between the initial and final positions of the surface of marked fluid and suggested
that drift volume is equal to the volume of fluid associated with the hydrodynamic
mass of the body (figure
1).
The hydrodynamic mass of a body is the mass of
fluid that must be added to the body when calculating the total kinetic energy; it
is
commonly referred to as ‘added mass’ (Batchelor 1967, p.
407).
Darwin’s proposition
is an appealing result and has been further investigated by Lighthill (1956), Yih (1985)
and Benjamin (1986).
Darwin’s proposition has been referred to several times as Darwin’s theorem. But
it has been pointed out by Darwin himself and by Benjamin (1986) that the drift
volume depends critically on the method of evaluation of certain integrals and
so
the proposition cannot be proved to be a mathematical theorem without delicate
qualifications. We therefore prefer the term ‘Darwin’s proposition’.
Darwin’s proposition has been used to interpret measurements of bubble motion
and other two-phase flows. For example, Bataille, Lance
&
Marie (1991) set up an
experiment with a lower layer of fluid that was dyed and slightly denser than an upper
layer
of
fluid. Bubbles were released beneath the interface, which was distorted by the
passage of a bubble. The volume of fluid that drifted with
a
bubble was measured and
found to be equal
to
Darwin’s calculation, i.e. half the bubble volume which is equal
to the added mass volume of the bubble for potential flow. Interestingly, Rivero’s
(1990) numerical calculations of the added-mass volume of accelerating rigid spheres
and bubbles in viscous flows
(Re
w
100) also agree with the volume calculated using
potential flow.
In this paper, Darwin’s proposition is investigated and generalized by considering
the deformation of a finite-sized plane of marked fluid that is initially placed a
finite distance from the body. When the body is subsequently infinitely far from
its initial position, the volume between the final and initial plane of marked fluid
elements is defined here to be the
partial drft volume
(figure
2a).
To
illustrate the
properties of partial drift volume, we consider flow around a sphere and examine
in detail the Lagrangian displacement of a circular plane
of
marked fluid,
of
radius
pmax,
that starts a distance
xo
from the sphere. In
$2,
trajectories of fluid particles are
calculated numerically and using asymptotic analysis, which extends the earlier results
of Lighthill (1956). The partial drift volume is calculated in $3, for various values of
xo
and
pmax,
using a combination of the asymptotic formulae and numerical results.
Drift, partial drft and Darwin’s proposition
Initial position
body at
t
=
-
Solid
body is infinitely far
from the plane of marked
fluid particles at
t
=
0
RGURE
1.
Sketch
of
the drift volume (as defined by Darwin).
203
--.)
plane
t
=
0
_.--
-
stratification
1
_________________-
---
*-------
-_________________
....................................................
FIGURE
2.
Definition sketch
of
the partial
drift
volume.
(a)
Notation,
(b)
to
show a typical problem.
204
-
-
I.
Eames.
S.
E. Belcher and
J.
C.
R.
Hunt
/
-
I
FIGURE
3.
Streamlines close
to
the
sphere.
It is shown that, in the configuration studied by Darwin
(1953),
Yih
(1985)
and
Benjamin
(1986)
when both
xo
and
pmax
are infinite, the problem is not well-posed,
because the drift volume is not well-defined. Yih’s procedure does in effect assume as
pmax/xo
-+
0
as xo
--+
00.
In
$4
a general expression for horizontal Lagrangian displacement is derived for
an arbitary potential fluid flow superimposed on a constant-mean-velocity flow. The
partial drift volume is shown to depend on the solid angles subtended by the body
and initial and final positions of the plane of marked fluid. This analysis is generalized
to arbitrarily shaped two- and three-dimensional bodies.
In
$5,
we use our results for drift due to a sphere in unbounded flow to indicate
how drift in bounded flow can be calculated. The sphere is assumed to travel along
the centreline of a square tube. This three-dimensional example can be applied to
practical problems (e.g. the experiments of Bataille
et
al.
1991;
Kowe
et al.
1988).
Finally it is worth noting that there are an increasing number of practical fluid
mechanics problems whose solution depends
on
having a good estimate of the drift
volume (e.g. Kowe
et
al.
1988).
For example, the partial drift volume is also of
importance when large particles are ejected into
a
cloud of smaller particles, or a
rising bubble formed in a temperature gradient (figure
2b).
2.
Horizontal Lagrangian displacement for
flow
around a sphere
Consider the inviscid flow caused by a fixed sphere of radius
a
in a uniform stream
of speed
-Ux,
where
U
is constant and
x
is the unit vector parallel to the x-axis. The
flow
is
axisymmetric and
so
it
is
sufficient to consider a single (x,y) plane, shown in
figure
3.
In spherical polar coordinates the velocity potential and streamfunction are
$=-U(r+$)cos~, tp=-U(g-;)sin’H,
(2.1~1,
b)