J.
Fluid
Mech. (1978), vol. 87,
part
4,
pp.
641-672
Printed
in
Great
Britain
Structure and entrainment in the plane
of
symmetry
of
a turbulent spot
By
BRIAN
CANTWELL,
DONALD
COLES
AND
PAUL
DIMOTAKIS
Institute
of
Technology,
Pasadena,
California
(Received
31
May
1977
and
in
revised
form
3
October
1977}
641
Laser-Doppler velocity
measurements
in
water
are
reported
for
the
flow
in
the
plane
of
symmetry
of
a
turbulent
spot.
The
unsteady
mean
flow, defined as
an
ensemble
average, is
fitted
to
a conical
growth
law
by
using
data
at
three
streamwise
stations
to
determine
the
virtual
origin
in
x
and
t.
The
two-dimensional
unsteady
stream
function
is expressed as
1fr
=
U~
tg(£,
r/)
in
conical similarity co-ordinates g =
xfUoot
and
1J
= y
jU
00
t.
In
these co-ordinates,
the
equations
for
the
unsteady
particle displacements
reduce
to
an
autonomous
system.
This
system
is
integrated
graphically
to
obtain
particle
trajectories
in
invariant
form.
Strong
entrainment
is
found
to
occur along
the
outer
part
of
the
rear
interface
and
also
in
front
of
the
spot
near
the
wall.
The
outer
part
of
the
forward interface is passive.
In
terms
of
particle
trajectories
in
conical
co-ordinates,
the
main
vortex
in
the
spot
appears
as a
stable
focus
with
celerity
0·
77U
00
•
A second
stable
focus
with
celerity 0·64U
00
also
appears
near
the
wall
at
the
rear
of
the
spot.
Some results
obtained
by
flow visualization
with
a dense,
nearly
opaque
suspension
of
aluminium
flakes
are
also reported.
Photographs
of
the
sublayer
flow viewed
through
a glass wall show
the
expected
longitudinal streaks. These are
tentatively
interpreted
as longitudinal vortices caused
by
an
instability
of
Taylor-Gortler
type
in
the
sublayer.
1.
Introduction
During
the
past
few years, evidence
has
been
accumulating
which suggests
that
transport
in
turbulent shear flows
is
controlled by large coherent flow structures which,
although they differ from one type
of
flow
to
another, all represent typical
and
recognizable
concentrations
of
transverse mean vorticity at the largest scale
of
the flow.
In
this
context
a
turbulent
spot
in
a
laminar
boundary
layer
(Emmons
1951;
Schubauer
& Klebanoff
1955) provides
an
example
of
an
isolated
coherent
structure
which
may
have
important
properties
in
common
with
structures
in
a fully developed
turbulent
boundary
layer.
Coles &
Barker
(1975,
hereafter
referred
to
as
CB)
took
this
point
of
view when
making
some
exploratory
measurements
of
flow
in
a
turbulent
spot.
They
concluded,
on
the
basis
of
slender
and
partly
circumstantial
evidence,
that
a turbulent spot
in
a
laminar boundary layer at constant pressure
is
essentially a single large horseshoe vortex
structure,
that
the
vortex
centre
in
the
plane
of
symmetry
moves
at
about
0·83U
00
,
and
that
the
spot
grows
not
only
by
entraining
irrotational
fluid from
the
ambient
free
22
FLM
87
642
B.
Cantwell, D. Coles
and
P.
Dimotakis
stream
but
also
by
entraining
rotational
fluid
from
the
ambient
laminar
boundary
layer.
An
independent
and
more
extensive
study
by
Wygnanski,
Sokolov &
Friedman
( 1976,
hereafter
referred
to
as
WSF)
has
established
the
shape
of
the
turbulent
spot
in
three
dimensions
and
has
provided
data
on
all
three
components
of
the
mean
velocity
in
the
interior
of
the
spot.
In
the
plane
of
symmetry,
where
the
two
experiments
have
some
common
ground,
there
is
substantial
agreement
concerning
the
shape
and
mean-
velocity
field.
WSF
agree
with
CB
that
the
spot
is a large horseshoe
vortex,
but
disagree
about
certain
important
properties
of
the
vortex,
especially
its
characteristic
speed
of
propagation.
CB
deduced
a
vortex
velocity
of
0·83U"'
by
following a con-
spicuous
minimum
in
the
(ensemble)
mean
velocity
as
observed
by
a fixed
probe
in
the
outer
part
of
the
flow.
WSF
proposed
as
the
characteristic
spot
velocity
the
celerity
deduced
in
the
conventional
way
from
space-time
correlations
of
the
streamwise
velocity.
When
they
applied
this
correlation
technique
very
near
the
wall
in
the
plane
of
symmetry,
they
obtained
the
value
0·65U
00
•
Both
CB
and
WSF
carried
out
stream-function
calculations
in
moving
co-ordinates
for
the
mean
flow
in
the
plane
of
symmetry
of
the
spot.
A
major
issue
in
these
calcula-
tions
was
entrainment,
and
an
obvious
objection
is
that
the
mean
flow was
treated
as
two-dimensional.
WSF
quieted
this
objection
by
showing
experimentally
that
ow/
oz
= 0
near
the
plane
of
symmetry
and
also
by
showing close
agreement
between
measured
and
computed
values
of
the
normal
velocity v.
A
second
and
more
serious
objection
is
that
spot
growth
was
not
taken
into
account.
In
reality,
a
typical
spot
roughly
doubles in size
during
the
time
interval
between
the
passage
of
its
leading
and
trailing
interfaces
past
a fixed probe.
At
best,
the
stream
function
Jjf(y, t)
as
calculated
from
data
at
one
station
by
CB
and
by
WSF
can
only
suggest
the
probable
form
of
the
instantaneous
streamlines. Moreover,
the
calculated
flow
pattern
is
extremely
sensitive
to
the
speed chosen for
the
moving
co-ordinate
system,
and
it
is
precisely
this
speed
which
is
an
important
unresolved
element
in
the
experiments.
Our
primary
objective
in
the present research
is
to
remove this second objection
by
viewing
the process
of
spot growth
in
appropriate conical co-ordinates.
We
describe
the
experi-
mental
means
of
achieving
this
end,
including
an
unusual
flow-visualization
technique,
in
§§
2-4.
In
the
main
analysis,
in
§§5-8,
we
establish
the
conical
property
for
the
present
data,
define a
similarity
form
for Jjfjt as a
function
of
xjt
and
yjt,
calculate
proper
instantaneous
mean
streamlines,
and
obtain
mean
particle
paths
in
Galilean-
invariant
form
by
integrating
the
Lagrangian
displacement
equations
graphically.
Several
singularities
revealed
by
this
process
are
interpreted
in
terms
of
structure.
In
§ 9
the
results
for
particle
paths
and
for
the
spot
shape
are
combined
to
determine
quantitatively
the
local
rate
of
entrainment
around
the
spot
boundary
in
the
plane
of
symmetry.
Finally,
an
attempt
is
made
in
§ 10
to
assemble
visual
information
from
§ 3
and
structural
information
from§§
7-9
into
a
coherent
kinematic
description
of
the
turbulent
spot.
Structure and entrainment
of
a turbulent spot
Spot
Measuring stations
virtual
Disturbance
I
j
l
origin
generator
~
l
Plate leading edge
y, y
X
x
0
=
15
em
l4cm
u'l0-59
cm/s
59
em --1
f4----
89 em
-----1
119-cm
-----1
Y'
U
------'\_-Real
growth
c2
-.:'
244cm
FIGURE
1.
Sketch
of
flat-plate
model,
showing
important
dimensions
and
co-ordinate
systems.
2.
Model
and test conditions
643
The
experiment
was
carried
out
in
the
GALCITt
low-speed
water
channel, which
has
a
working
section 5·8 m long, 46
em
wide
and
61
em
deep.
The
flat-plate model,
sketched
in
figure 1,
was
the
Plexiglas
plate
used
by
CB. Several changes were
made
in
the
model
and
in
the
test
conditions
to
suit
the
purposes
of
the
present
research.
The
stream
speed
was
approximately
doubled,
to
about
59
cmjs,
in
order
to
obtain
greater
separation
between
the
sublayer
scale
and
overall scale
of
the
flow.
The
orifice for
the
jet
disturbance
was
moved
upstream
to
a
point
14
em
from
the
plate
leading edge.
The
disturbance
generator
was
a small three-roller
peristaltic
pump
connected
to
a
constant-
speed
motor
through
a
magnetic-particle
clutch.
By
energizing
this
clutch
for a
short
time,
the
pump
could
be
rotated
rapidly
through
a
third
of
a
revolution
on
command.
One
other
structural
change
made
in
the
model was
the
installation
of
several
transverse
cables
on
the
underside
of
the
plate
about
2
em
from
the
plate
surface. Tension
was
applied
to
these
cables
to
bow
the
top
or
working surface
into
a
slightly
convex shape.
After
this
change,
laser-Doppler
measurements
could
be
made
as
close
to
the
wall as
0·013cm.
The
leading
edge
of
the
model
was
set
appreciably
lower
than
the
trailing
edge
in
an
effort
to
prevent
premature
natural
transition
by
placing
the
forward
stagnation
line
above
the
centre
of
the
elliptical nose.
The
depth
of
the
model
below
the
free
surface
therefore
decreased
in
the
direction
of
flow while
the
measured
velocity
in
the
free
stream
increased,
as
shown
in
figure
2.
The
product
of
depth
and
velocity increased
slightly,
presumably
because
of
displacement
effects
from
boundary
layers
on
the
various
walls.
t
Graduate
Aeronautical
Laboratories,
California
Institute
of
Technology.
22-2
644
B.
Cantwell, D. Coles and
P.
Dimotakis
Measuring
stations
1·10
h
(x)
h
ref
~
~
R
:::::,
~
3
8
1·00
:::::,
~
1;
~
3
"""
,Q·90
777777777777777
10
20
30
40
50
60
70
80
90
100
110 110
Distance
from plate leading edge. x (em)
FIGURE 2.
Measured
variation
of
water
depth
hand
free-stream
velocity
Uoo
along
plate.
Reference
values
at
third
measuring
station:
hret
==
12·1
em;
U,
••
1
==
59·5
cmfs.
1·0
0·8
0·2
2 3 4
'l=y
(~')
2vx
FIGURE 3.
Measured
laminar
velocity
profiles
at
three
measuring
stations.
!::,.,
x =
59
em;
0,
x =
89
em;
e, x =
119
em.
Computed
profiles
are
from
Smith
( 1954) ;
j3
is
the
Falkner-
Skan
parameter.
Although
the
undisturbed
boundary
layer
in
the
plane
of
symmetry
remained
laminar
almost
to
the
trailing edge,
transverse
contamination
from
the
side walls
began
to
interfere
with
spot
growth
about
half
way
along
the
plate.
The
three
main
series
of
measurements
were therefore
made
at
stations
located 59,
89
and
119 em from
the
leading edge,
as
indicated
in
figure
1.
The
Reynolds
number
at
the
third
station,
Structure and entrainment
of
a turbulent spot 645
based
on
the
local
free-stream
velocity
of
59·5
cmjs
and
the
distance
from
the
leading
edge,
was
about
0·82 x 10
6
•
Velocity profiles
obtained
in
the
undisturbed
laminar
boundary
layer
are
shown
in
figure 3
in
Blasius
co-ordinates.
The
effect
of
free-stream
acceleration
in
thinning
the
laminar
layer
is
apparent.
If
the
early
part
ofthe
distribution
Uoo(x)
in
figure 2 is
fitted
by
a
power
law
Uoo
,.,.,
xm,
the
exponent
m is
found
to
be
about
0·14.
The
corresponding
Falkner-Skan
parameter
f3
=
2mj(m
+
1)
is
about
0·25.
The
profile for
this
value
of
f3
(Smith
1954) is
included
in
figure 3
and
is
reasonably
close
to
the
measurements
at
the
first
two
stations.
The
measurements
at
the
third
station
are
well
represented
by
the
Blasius
profile
ifthe
boundary-layer
thickness
is
taken
to
be
about
20%
less
than
the
thickness
calculated
for
dpjdx
=
0.
3.
Flow
visualization
Two
different flow-visualization
techniques
were
used
to
support
the
measurements
reported
here.
The
first
technique
was
dye
visualization
like
that
applied
to
the
spot
problem
by
Elder
(1960)
and
by
Meyer &
Kline
(1961
).
In
the
present
instance,
a
thin
layer
of
dyed
fluid
was
laid
down
very
near
the
plate
surface
through
a slot
about
10
em
from
the
leading
edge. CB
used
potassium
permanganate
dye
in
this
way,
but
did
not
take
photographs.
Coles
later
obtained
a
motion-picture
record
of
both
the
spot
and
the
synthetic
boundary
layer.
A
similar
record
was
made
during
the
present
experi-
ments,
except
that
the
dye
was
vegetable
food colouring.
Two
frames
from
these
motion
pictures
showing
the
spot
shape
in
plan
view
are
reproduced
in
figures 4(a)
and
(b)
(plate
1). Motion
pictures
of
a
turbulent
spot
in
air
have
also
been
made
by
M.
R.
Head
at
Cambridge, using
smoke
in
the
laminar
boundary
layer
for visualization.
By
kind
permission
of
Dr
Head,
two
frames
from
his
motion
pictures
are
reproduced
in
figures
5(a)
and
(b) (plate 2).
As
far
as
we
are
aware,
none
of
these
observers
of
the
turbulent
spot
has
reported
seeing
anything
like
the
splitting
which
is
known
to
occur
under
certain
conditions
for
the
puff
in
pipe
flow.
Each
spot
seems
to
remain
intact
and
recognizable for
as
long
as
it
can
be
seen.t
Dye
(or smoke) visualization is
particularly
effective
in
showing linear
growth
because
the
spot
entrains
dyed
fluid from
the
wall region, picking
it
up
like a
vacuum
cleaner.
This
dyed
fluid is
rapidly
broken
up
and
dispersed
throughout
the
body
of
the
spot
before
there
is
appreciable
dilution.
If
the
plate
surface is
initially
completely
covered
with
dye,
entrainment
is seen
to
occur
along
the
full
span
of
the
spot.
To
the
rear,
the
spot
leaves
behind
a wedge-shaped
wake
of
nearly
clear fluid.
This
wake
is
slowly
erased
by
downstream
motion
of
the
adjacent
dye.
When
a second
spot
follows
closely
behind
the
first one,
entrainment
is visible
only
at
the
lateral
extremities
of
the
spot
(cf. figure 4b,
plate
1
).
Dye
which
is
present
near
the
plane
of
symmetry
of
the
second
spot
must
therefore
have
been
picked
up
at
a
much
earlier
time.
This
dye
tends
to
be
more
concentrated
near
the
leading edge
of
the
spot
as
the
spot
moves down-
stream.
Another
obserYation, also
illustrated
by
figure
4(b),
is
that
a region
of
trans-
t
\Vygnanski,
Haritonidis
&
Kaplan
( 1978)
have
recently
reported
some
remarkable
observa.
tions
of
regular
wave
packets
at
the
outer
edges
of
the
laminar
wake
of
a
spot.
These
wave
packets
eventually
degenerate
into
new
turbulence.
vVe
have
found
no
evidence
of
this
pheno-
menon
in
our
motion
pictures,
perhaps
because
our
Reynolds
numbers
were
not
sufficiently
large.