Oblique and Parallel Modes of Vortex Shedding in the Wake of a Circular Cylinder at Low Reynolds Numbers

TL;DR: In this article, it was shown that the Strouhal discontinuity is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode.

Abstract: Two fundamental characteristics of the low-Reynolds-number cylinder wake, which have involved considerable debate, are first the existence of discontinuities in the Strouhal-Reynolds number relationship, and secondly the phenomenon of oblique vortex shedding. The present paper shows that both of these characteristics of the wake are directly related to each other, and that both are influenced by the boundary conditions at the ends of the cylinder, even for spans of hundreds of diameters in length. It is found that a Strouhal discontinuity exists, which is not due to any of the previously proposed mechanisms, but instead is caused by a transition from one oblique shedding mode to another oblique mode. This transition is explained by a change from one mode where the central flow over the span matches the end boundary conditions to one where the central flow is unable to match the end conditions. In the latter case, quasi-periodic spectra of the velocity fluctuations appear; these are due to the presence of spanwise cells of different frequency. During periods when vortices in neighbouring cells move out of phase with each other, ‘vortex dislocations’ are observed, and are associated with rather complex vortex linking between the cells. However, by manipulating the end boundary conditions, parallel shedding can be induced, which then results in a completely continuous Strouhal curve. It is also universal in the sense that the oblique-shedding Strouhal data (S_θ) can be collapsed onto the parallel-shedding Strouhal curve (S_0) by the transformation, S_0 = S_θ/cosθ, where θ is the angle of oblique shedding. Close agreement between measurements in two distinctly different facilities confirms the continuous and universal nature of this Strouhal curve. It is believed that the case of parallel shedding represents truly two-dimensional shedding, and a comparison of Strouhal frequency data is made with several two-dimensional numerical simulations, yielding a large disparity which is not clearly understood. The oblique and parallel modes of vortex shedding are both intrinsic to the flow over a cylinder, and are simply solutions to different problems, because the boundary conditions are different in each case.

Summary

1. Introduction

• The problem of the wake formation behind bluff bodies has received a great deal of attention, both from an experimental standpoint and more recently from a theoretical/numerical standpoint.
• A further characteristic of the low-Reynolds-number flow around cylinders, which is not well understood, is the presence of oblique vortex shedding, whereby the axes of the vortices are shed at some angle to the cylinder axis.
• The results to be shown in the present study are in accordance with both of the above conclusions.
• The physical mechanisms for the oblique modes of shedding, and also for an induced parallel mode, are discussed in § 3.3.

2. Experimental details

• Measurements of velocity fluctuations were made with a miniature hot wire situated in the wake ofthree different cylinders (of diameters 0.051, 0.061, 0.104 em.) in a 6 in.
• The test cylinders (actually wires) were placed across a diameter at a position about 16 in.
• Each cylinder was passed through small holes in the test section, and clamped between felt pads to a tensioning structure outside of, and mechanically separate from, the test section.
• The outputs from the hot wires and the 'vibration detector' were fed into a HP-3582A two-channel spectrum analyser.
• Flow visualization and further frequency measurements were conducted in their new X-Y Towing Tank at the Graduate Aeronautical Laboratories.

3.1 Proof that the present Strouhal discontinuity is not caused by cylinder vibrations

• In the present measurements it was clear that, for the required accuracy, reliable temperature and pressure readings in the laboratory were necessary, but also the Pitot tube (used to measure free-stream velocity) had to be checked often.
• Therefore, endplates (circular disks) were placed at two points along the cylinder span.
• As a further demonstration that cylinder vibrations do not cause the discontinuity, it was decided to study simultaneously the wake and vibration spectra as Re was decreased through the discontinuity at Re = 64.
• These natural frequencies (at 84Hz and 168Hz) are shown by the vertical dashed lines in figure 2.
• Here the wake and vibration spectra are shown at Re = 70.1, above the diseontinuity , and also at Re = 62.3, below the discontinuity and 2(c), the vibration speetrum remains essentially unaltered and similar to that measured when there was no flow in the tunnel at all.

3.2 Measurements and observations of oblique vortex shedding modes

• The key to this breakdown was found by traversing a hot wire across the complete span of the cylinder wake.
• Re above the discontinuity, a single frequency fL was found across the whole span as shown schematically in figure 3 (a), with the exception of small regions near the ends of the cylinder where the lower-frequency cells are found.
• (a) 'Periodic' oblique shedding mode (for Re = 64-178), with a chevron-shaped oblique vortex pattern.
• Whole span is then directly influenced by the end conditions.

3.3 Physical mechanism of oblique and parallel shedding

• There are a number of advantages to visualizing the wake in the towing tank, one of which is the fact that the cylinder is started impulsively.
• This matching of the base pressure at the ends of the cylinder with that over the central span is a concept which was used by Ramberg (1983) in his study of the flow over yawed cylinders, and was also achieved by angling the endplates.
• Secondly, the results shown in figure 9(a) indicate that the following relationship holds well: i\.
• The present results have shown that it is the effects from the ends that cause an oblique angle of shedding to be generated near the ends of the span, and which then spreads inwards along the span from one shedding vortex to another.
• It is this successive vortex shedding in the near wake that is responsible for the gradual span wise shift of such disturbances from the ends, causing the oblique front that was discussed earlier.

4. Defining a universal and continuous S-Re relationship

• Judging by the results for a cylinder which sheds oblique vortices, it is not clear whether a continuous Strouhal curve, without discontinuities, actually exists.
• When the vortices are induced to be shed parallel to the cylinder, no mode transition occurs, because the vortices are shed parallel to the cylinder throughout the Reynolds-number range.
• The parallel-shedding data lie somewhat above the oblique-shedding data (the lines), and are thus consistent with the results of figure 1 , where the trend of increasing frequency for decreasing angle of shedding was observed.
• The results are shown in figure 12 by the crosses, and agree well with the other data, although the frequencies could not be measured to the same accuracy as in the wind tunnel.
• This good agreement provides strong evidence that the parallel-shedding Strouhal curve is indeed universal, as well as being completely continuous.

5. An explanation for the transition between oblique shedding modes

• Initially, the authors shall investigate the similarities between Tritton's Strouhal discontinuity and the present one, and see whether it is likely that his discontinuity is caused by the same oblique-mode transition.
• In the case of Tritton's data, one can see that both of his curves lie below the parallel-shedding curve, in a manner not unlike the present oblique-shedding data .
• This suggestion to explain the transition of oblique shedding modes is explored below.
• An interpretation of the present results can be given by making the analogous supposition that, in the cylinder wake, two-dimensional or parallel vortex shedding is the most unstable mode.
• The bend is not displaced spanwise a sufficient distance (say z*) per cycle to keep the vortices matched in phase.

6. Structure of 'vortex dislocations' between cells of different frequency

• Browand & Troutt (1985) have also found vortical structures to be generated naturally in the shear layer (downstream of a splitter plate) between spanwise cells of different frequency, or between cells of similar frequency but which are out of phase with each other.
• The present results are consistent with their suggestions, and it is found in this section that the generation of stream wise vorticity is clearly a fundamental characteristic of vortex dislocations.
• Physically, it is useful to know how many shedding cycles there are between one dislocation and the next.
• The lower data in this plot show the number of shedding cycles per beating period for the end-cell boundary (i.e. between cells of frequency fL andfe), while the upper data, at the lower Reynolds numbers, correspond with the boundary between the cells of frequency fu and fL.

7. Discussion

• There are a number of results (in the literature) to support the assertion that Strouhal-Reynolds number discontinuities can be caused by either flow nonuniformities, in the form of shear in the free stream (e.g. Gaster 1971) , or by cylinder vibration (e.g. VanAtta & Gharib 1987 ; see review by Bearman 1984) .
• There has nevertheless been little agreement as to the critical Reynolds number at which there is a change of shedding modes.
• The present experiments yielded a critical Reynolds number for all three cylinders of 64± 1 (despite their different length/ diameter ratios, and end plate diameter j cylinder diameter ratios).
• An investigation into such possibilities was not made here.
• One of the questions prior to the present work was whether the wake over large cylinder spans was truly unaffected by the presence of the end conditions , as had previously been supposed.

8. Conclusions

• The present study of the laminar vortex shedding from a circular cylinder has shown that a discontinuity in the Strouhal~~Reynolds number relationship exists, which is caused by a transition from one mode of oblique vortex shedding to another oblique mode.
• There are, in essence, two forms of influence exerted by the end conditions over such wakes.
• It is believed that the parallel-shedding case represents truly two-dimensional vortex shedding, in the sense that there is an absence of axial flow in the vortices and the flow within one cross-section is representative of (and in-phase with) the flow at all span wise positions (with the exception of close to the ends).
• The Strouhal data for this two-dimensional shedding has been compared with results from two-dimensional numerical simulations, although the comparison yields a large disparity amongst the computations themselves, and also with the experimental results.
• Lastly, the author is grateful to F. Raichlen for the use of a small towing tank (in 1985) which resulted in the visualization of figure 23.

Full text

J. Fluid
Mech.
(1989),
vol.
206, pp. 579--627
Printed
in
Great
Britain
Oblique and parallel modes
of
vortex shedding
in the wake
of
a circular cylinder at
low Reynolds numbers
By C. H. K.
WILLIAMSONt
Graduate
Aeronautical
Laboratories,
California
Institue
of
Technology,
Pasadena,
CA 91125, USA
(Received 30
September
1988
and
in
revised form 19
February
1989)
579
Two
fundamental
characteristics
of
the
low-Reynolds-number cylinder wake, which
have
involved considerable
debate,
are
first
the
existence
of
discontinuities in
the
Strouhal-Reynolds
number
relationship,
and
secondly
the
phenomenon
of
oblique
vortex
shedding.
The
present
paper
shows
that
both
of
these characteristics
of
the
wake
are directly
related
to
each other,
and
that
both
are influenced
by
the
boundary
conditions
at
the
ends
of
the
cylinder, even for
spans
of
hundreds
of
diameters
in
length.
It
is found
that
a
Strouhal
discontinuity
exists, which is
not
due
to
any
of
the
previously proposed mechanisms,
but
instead
is caused
by
a
transition
from one
oblique
shedding
mode
to
another
oblique mode.
This
transition
is
explained
by
a
change from one mode where
the
central
flow
over
the
span
matches
the
end
boundary
conditions
to
one where
the
central
flow is unable
to
match
the
end
conditions.
In
the
latter
case, quasi-periodic
spectra
of
the
velocity fluctuations
appear;
these are due
to
the
presence
of
span
wise cells
of
different frequency.
During
periods when vortices
in
neighbouring cells move
out
of
phase
with
each other,
'vortex
dislocations' are observed,
and
are associated
with
rather
complex
vortex
linking between
the
cells. However,
by
manipulating
the
end
boundary
conditions,
parallel shedding
can
be induced, which
then
results in a completely continuous
Strouhal
curve.
It
is also universal in
the
sense
that
the
oblique-shedding
Strouhal
data
(S
11
)
can
be collapsed
onto
the
parallel-shedding
Strouhal
curve
(S
0
)
by
the
transformation,
S
0
= S
11
jcos
(),
where
()
is
the
angle
of
oblique shedding. Close
agreement
between
measurements
in two
distinctly
different facilities confirms
the
continuous
and
universal
nature
of
this
Strouhal
curve.
It
is believed
that
the
case
of
parallel shedding represents
truly
two-dimensional shedding,
and
a comparison
of
Strouhal
frequency
data
is
made
with
several two-dimensional numerical simu-
lations, yielding a large
disparity
which is
not
clearly understood.
The
oblique
and
parallel modes
of
vortex
shedding
are
both
intrinsic
to
the
flow
over
a cylinder,
and
are simply solutions
to
different problems, because
the
boundary
conditions are
different in each case.
1.
Introduction
The
problem
of
the
wake
formation
behind
bluff bodies
has
received a
great
deal
of
attention,
both
from
an
experimental
standpoint
and
more recently from a
theoretical/numerical
standpoint.
Nevertheless, even
at
low
Reynolds
numbers
t Address from December 1989:
Dept.
of
Mechanical
and
Aerospace Engineering, Cornell
University,
Ithaca,
NY
14853, USA.

580
C.
H.
K. Williamson
when
the
shed vortices
remain
laminar,
there
are
some
rather
basic questions
that
are
not
understood.
Since
the
first
extensive
measurements
of
vortex
shedding
frequencies
by
Roshko
(1954),
there
has
been
remarkably
little
agreement
between
the
many
published
curves
of
Strouhal
number
(S) versus
Reynolds
numbers
(Re) for
the
laminar
shedding regime (between
Reynolds
numbers
of
49
to
178 in
the
present
study).
Indeed
published
results
have
been found
to
differ
by
almost
20%.
It
is clear
that,
even
though
the
measurement
of
the
wake
frequencies
is
a simple
matter,
the
frequency
itself
is highly sensitive
to
the
experimental
arrangement
and,
as will be
found here,
it
is
particularly
sensitive
to
three"dimensional effects. A
further
characteristic
of
the
low-Reynolds-number
flow
around
cylinders, which is
not
well
understood, is
the
presence
of
oblique
vortex
shedding,
whereby
the
axes
of
the
vortices
are
shed
at
some angle
to
the
cylinder axis.
Although
this
oblique shedding
phenomenon
has
been
noted
by
several
authors,
no
investigation
has
been carried
out
to
understand
its
origin.
In
the
present
paper,
both
of
the
above
features
of
the
flow
around
a circular cylinder are
investigated.
The
measurement
of
vortex
shedding frequency
has
been
the
subject
of
considerable
debate
since
the
observation
by
Tritton
(1959)
that
his
Strouhal
curve
(i.e. his
plot
of
Strouhal
number
versus
Reynolds
number)
was discontinuous.
He
found
two
Strouhal
curves, one
of
them
corresponding
to
a
'high-speed'
mode
above
Re
= 100,
with
a
jump
to
a
curve
corresponding
to
a
'low-speed'
mode below
Re
= 100.
Tritton
suggested
that
this
'Strouhal
discontinuity'
was caused
by
a
transition
from
an
instability
originating
in
the
wake
to
an
instability
originating in
the
immediate
vicinity
of
the
cylinder, as
the
Reynolds
numbers
are
increased.
His
discovery
of
a
discontinuity
in
the
Strouhal
curve
has
led
to
a
number
of
different
explanations
since
that
time,
and
to
much
discussion
over
whether
the
discontinuity
is
an
intrinsic,
'fluid-mechanic'
phenomenon,
irrespective
of
the
experimental
set-up.
A
further
explanation
was
put
forward
by
Gaster
(1969).
He
observed irregular
modulations
of
the
velocity signal in
the
wakes
of
slender cones, which were caused
by
the
presence
of
spanwise cells
of
different frequency.
Based
on
the
similarity
with
Tritton's
velocity signals,
Gaster
suggested
that
Tritton's
breakdown
of
regular
shedding could possibly be caused
by
the
existence
of
some
non-uniformity
of
the
flow.
Tritton
(1971)
then
repeated
his
experiments
in a different wind
tunnel
and
again observed a
Strouhal
discontinuity
(in
this
case
near
Re
::::::::
110). Still
further
experiments
by
Gaster
( 1971)
provided
stronger
support
for his original suggestion
that
the
Strouhal
discontinuity
was
caused
by
free-stream non-uniformities.
By
forcing
the
flow
to
be
non-uniform
across
the
cylinder
span,
he
induced
spanwise cells
of
different frequency
to
occur.
Measurements
of
shedding
frequency
versus
tunnel
speed
at
a single
point
demonstrated
a
discontinuity
under
these
conditions, because
the
frequency cells were
found
to
move
along
the
span
as
the
speed was varied.
However,
the
discontinuity
was
made
to
disappear
when small
end-plate
disks were
placed
70
diameters
apart
along
the
span,
and
it
was suggested
that
these
endplates
limited
the
span
wise
movement
of
the
frequency cells.
In
recent
support
of
Gaster's
suggestions, Mathis,
Provansal
&
Boyer
(1984) also
proposed
that
the
Strouhal
discontinuity
is
caused
by
flow
non-uniformity.
Berger
& Wille (1972),
on
the
other
hand,
believed
that
two
intrinsic modes
of
shedding exist, as
put
forward
by
Tritton,
and
that
the
choice
of
which mode occurs
at
a given
Reynolds
number
could be
dictated
by
the
level
of
turbulence
in
the
free
stream.
Further
discontinuities were
observed
by
Kohan
& Schwarz (1973)
and
Friehe
(1980), who found
Strouhal
discontinuities
within
the
range
of
Re
from 70
to
110 for several different cylinders. These
investigations
confirmed
that
a
Strouhal

Vortex shedding
in
the wake
of
a circular cylinder
581
discontinuity (much like
Tritton's
result)
can
exist
in
other
experimental
set-ups,
although
its
cause
remained
a question.
An
alternative
view was
taken
by
Gerrard
(1978) who suggested
that
his
Strouhal
discontinuity
at
Re
~
100 was
in
some
way
related
to
the
end
of
a regime
of
Reynolds
number
in which diffusion
of
vorticity
plays
a
primary
role in
the
vortex
shedding.
More recently,
Sreenivasan
(1985)
interpreted
his several
Strouhal
discontinuities
as being involved in
the
'route
to
chaos'
in
the
cylinder wake.
Upon
investigating
velocity
spectra
as
Reynolds
numbers
were increased in
the
laminar
regime, he found
narrow ranges
of
Re
in which
'chaos'
was observed,
and
these were sandwiched
between regions
of
'order',
(with
both
of
these
states
characterised
by
the
velocity
spectra). These were
interpreted
as
the
initial
stages
in
the
transition
to
turbulence.
These results
are
further
briefly discussed,
in
the
light
of
the
present
work,
in
§
7.
Subsequently,
Van
Atta
&
Gharib
(1987) showed convincingly how some
discontinuities, which
might
be
observed in a
plot
of
Strouhal
number
versus
Reynolds
number,
can
be
related
to
vibrations
of
the
cylinder itself.
To
show
this
clearly
they
monitored
cylinder
vibrations
using a
photodetector,
and
also
demonstrated,
by
damping
the
cylinder
supports,
that
several
Strouhal
dis-
continuities (which occurred
at
multiples
of
a
fundamental
frequency) could be
smoothed
out.
They
suggested
that
'if
there
were absolutely no
vibration,
a
Strouhal-Reynolds
number
plot
would
have
absolutely no
discontinuities'.
A
particularly
significant
aspect
of
their
work
was
this
suggestion
that
indeed
there
might
exist a
continuous
S-Re
relationship.
In
a
further
paper,
VanAtta,
Gharib
&
Hammache
(1988)
investigated
the
wake
of
a
vibrating
wire,
and
found
that
the
wake
velocity
spectra
were influenced
by
the
interaction
between
the
vibrating
wire frequency (near
the
antinodes)
and
the
natural
shedding
frequency
(near
the
nodes). This
interaction
was found
to
cause
both
quasi-periodic
spectra,
and
also
spectra
with
a
'chaotic'
appearance,
dependent
on
the
spanwise position,
and
these
results
demonstrate
further
the
importance
of
body
vibration
in
determining
the
character
of
the
wake.
Following
the
result
of
Van
Atta
&
Gharib
(1987), some two-dimensional
numerical
simulations
by
Karniadakis
&
Triantafyllou
(1989) also
have
relevance
to
the
present
question.
They
demonstrated
that
their
Strouhal
number
varies
continuously
with
Reynolds
number
over
a
range
of
Re
from 40
to
250,
and
they
found no evidence
of
chaotic
behaviour
of
the
velocity fluctuations
in
the
wake
(as
had
been described
by
Sreenivasan).
In
this
respect,
it
was
supposed
that
their
simulations
supported
the
conclusions
of
Van
Atta
&
Gharib
that
the
asymptotic
state
in
unforced
laminar
wakes (for example,
the
cylinder
not
vibrating)
is
periodic.
Their conclusion seems reasonable
provided
that
the
flow is, in fact, two-dimensional.
However,
in
an
experiment
a cylinder
must
be
of
finite
length
and
therefore will
always
encounter
certain
end
conditions. As will be seen
later,
it
is for
this
reason
that
the
flow
can
be three-dimensional
even
in a uniform free
stream,
and
it
may
then
be possible for
the
laminar
wake
fluctuations
to
be quasi-periodic,
without
the
presence
of
external
forcing.
Out
of
the
several
explanations
for
the
existence
of
discontinuities,
the
suggestions
that
flow
non-uniformity
(or shear)
and
flow-induced
vibration
can
cause
jumps
in
the
frequency
curve
are
consistent
with
the
results
of
several
other
studies. Jfor
example,
the
work
ofMaull
&
Young
(1973)
demonstrated
the
presence
ofspanwise
cells
of
different frequency when a cylinder is
placed
in
a
shear
flow. Some spanwise
movement
of
these cell boundaries, as
the
overall flow speed is varied, would
result
in discontinuous changes in
measured
frequency
at
a
point.
Also,
it
is well established
that
cylinder
vibrations
can
'lock-in'
(synchronize)
with
the
shedding frequency (to

582
C.
H.
K.
Williamson
one
or
more
of
its
natural
frequencies)
and
cause
jump
changes in
the
shedding
frequency as
the
flow speed is varied, (see for
example
the
review
by
Bearman
1984).
However,
an
important
question
that
remains
is
whether
a
discontinuity
in
the
frequency curve
can
still occur even
if
cylinder
vibrations
or flow non-uniformity are
not
involved.
This
is one
of
the
central
questions
in
the
present
paper.
A
further
feature
of
laminar
vortex
shedding
that
is
relevant
to
the
question of
Strouhal
discontinuities is
the
observation
that
vortices
can
shed
at
oblique angles
to
the
cylinder. No
systematic
measurements
of
oblique shedding angles have
previously been made,
although
typical
angles
of
10°
to
20°
have
been quoted.
From
a theoretical
standpoint,
linear
stability
analysis
of
wake-type
profiles (with in viscid
parallel flow)
can
be
used
to
show
that
two-dimensional
disturbances
have a
greater
temporal
growth
rate
than
oblique disturbances.
This
can
be
demonstrated
in a
simple
manner
using
Squire's
transformation
(Squire 1933). Also, a
recent
study
by
Monkewitz (1988)
of
the
'absolute'
instability
of
a
limited
number
of
wake-type
profiles has shown,
in
all
the
investigated
cases,
that
the
two-dimensional
disturbances were
the
most
unstable,
i.e.
that
we should
expect
to
see two-
dimensional
or
parallel shedding,
'yielding
no clue as
to
why
oblique
vortex
shedding
is
often
observed'.
In
the
case
of
experiment,
there
are some conflicting results for
the
oblique angles
at
which vortices are shed.
The
observation
of
parallel shedding, in a towing
tank,
by
Hama
(1957)
contrasts
with
observations
of
oblique shedding,
made
in
a wind
tunnel,
by
Berger (1964). Berger & Wille (1972)
later
suggested
that
the
low
turbulence level
in
a towing
tank
enables parallel shedding
to
occur, whereas
the
higher turbulence levels
expected
in
a wind
tunnel
somehow causes
the
oblique
shedding.
They
also
pointed
out
that
end
effects could be
important.
With
relevance
to
this
question, a
detailed
investigation
by
Ramberg
(1983) on
the
wake
of
a
yawed
cylinder (a cylinder whose axis is
not
perpendicular
to
the
free
stream),
indicated
that
the
flow was sensitive
to
the
end
conditions.
He
showed
that
changing
the
angle
of
the
endplates
of
his
yawed
cylinder could influence
the
shedding angle
and
base pressure
near
the
ends.
He
further
observed
an
inter-
dependence between
the
shedding frequency
and
the
shedding angle. The results
to
be shown
in
the
present
study
are
in accordance
with
both
of
the
above
conclusions.
It
was found in
further
studies
by
Gerich &
Eckelmann
(1983)
and
Gerich (1986),
using a wind
tunnel,
that
cells
of
low frequency
exist
near
the
ends
of
a cylinder,
extending
over
spans
of
around
10 diameters.
For
large
length-to-diameter
ratios
(L/D),
the
flow
in
the
central
region
of
the
span
(outside
of
the
end
cells) was assumed
to
be
'unaffected'
by
the
ends,
but
when L / D was reduced
to
around
30,
the
end
cells
merged, causing a single low frequency
of
shedding
over
the
whole span. An earlier
investigation in a towing
tank,
by
Slaouti
&
Gerrard
(1981), showed
that
the
wake
structure
could be influenced
by
the
end
conditions,
and
they
concluded
that
slantwise (oblique) shedding was only observed when conditions
at
one end were
more
'dominant'
than
the
other.
However,
their
results were from observations
using a cylinder
with
LjD
~
25-30, which is
within
the
range
where Gerich &
Eckelmann
showed
that
the
end
cells cover
the
whole span.
The
question therefore
remains as
to
whether
the
flow outside
of
the
end
cells, for larger
LjD,
is indeed
'unaffected'
by
the
end
conditions.
It
is clear
that
there
are differences in
the
measurements
of
frequency
and
discontinuities between one
experiment
and
another,
and
it
is also
evident
that
there
are differences in
the
shedding angles between experiments.
It
might
then
be

Vortex shedding
in
the
wake
of
a circular cylinder
583
suggested
that
these characteristics
of
the
flow
are
related
to
each
other.
The
present
paper
will show
that
there
is a
direct
relation
between
the
frequency
measurements
and
the
phenomenon
of
oblique
vortex
shedding,
and
also
that
both
are
influenced
by
the
conditions
at
the
ends
of
the
cylinder. One
of
the
fundamental
results
of
the
present
work is
that,
even for large spans,
the
end
boundary
conditions (or
end
cells)
are able
to
affect
the
flow over
the
whole span, even outside
of
the
end cells.
It
is also
found
that
the
presence
of
oblique shedding does
not
require a difference in
the
two
end
conditions.
The
present
paper
originated from some
other
.work for which a simple
and
rapid
measurement
of
the
Strouhal
frequencies in
the
laminar
regime
was
anticipated.
It
was
very
soon found
that
the
flow
around
the
cylinder was sensitive
to
a
number
of
features
of
the
experimental
arrangement.
A
great
deal
of
care was
taken
to
dampen
(and monitor)
any
cylinder vibration,
and
to
check
the
flow
uniformity
and
turbulence
level
of
a small wind
tunnel
in which
the
cylinder was placed. Despite
these efforts,
it
became
apparent
that
a
Strouhal
discontinuity
near
Re
= 64
remained.
Not
only was
this
discontinuity
repeatable
on several occasions (even
after
an
earthquake
had
broken
the
original cylinder
and
hot
wire),
but
three
cylinders
of
different
diameters
all
produced
a similar
discontinuity
at
the
same
Reynolds
number.
It
was
because
of
these observations
that
the
present
study
was carried
out.
In
a
recent
letter
(Williamson 1988a), some preliminary results from
the
present
study
were outlined.
In
particular,
a link between
the
Strouhal
discontinuity
and
a
transition
between two oblique
vortex
shedding modes was made.
It
was
further
shown
that,
by
manipulating
the
end
conditions
to
cause parallel shedding, a single
continuous
Strouhal
curve could be found.
In
the
present
paper,
the
preliminary
results mentioned above
are
included
with
many
further
results in detail,
to
present
a comprehensive
picture
of
the
laminar
vortex
shedding modes behind a cylinder
at
low
Reynolds
numbers.
During
the
course
of
the
present
research,
it
was learned
that
H.
Eisenlohr
and
H.
Eckelmann
(1988,
private
communication) were
undertaking
a similar line
of
research, which has since
appeared
(Eisenlohr &
Eckelman
1989).
They
recognized,
as found in
the
present
paper,
that
the
phenomenon
of
oblique shedding was
influenced
by
the
end
conditions for long cylinders.
They
also showed
that
parallel
shedding could be induced
by
a suitable change
of
end
conditions.
Their
results
are
referred
to
further
in
this
paper,
and
it
is found
that
there
is
substantial
agreement
between
their
results
and
some
of
the
present
work
(and
also
with
Williamson
1988a).
In
§ 3.1 evidence is
presented
to
show
that
the
Strouhal
discontinuity
is
not
caused
by
cylinder vibrations.
Further
investigation shows
that
the
discontinuity
is
not
due
to
any
of
the
many
previously proposed mechanisms.
Instead,
measurements
and
observations discussed in §3.2 show
the
cause
of
the
discontinuity
as being a
transition
from one mode
of
oblique
vortex
shedding
to
another
oblique mode. One
of
the
modes
(at
the
lower
Reynolds
numbers) corresponds
with
the
presence
of
spanwise cells
of
different frequency. These cells should
not
be confused
with
the
small low-frequency cells found close
to
the
ends
of
the
cylinder
that
were
investigated
by
Gerich & Eckelmann,
and
which were also found in
the
present
study
at
all values
of
the
Reynolds
number
where
vortex
shedding occurred.
The
physical
mechanisms for
the
oblique modes
of
shedding,
and
also for
an
induced parallel
mode,
are
discussed in § 3.3.
Each
mode is caused
by
a
'matching'
of
the
flow
over
the
whole
span
with
the
end
boundary
conditions,
and
it
is
by
adjusting
the
end
conditions
that
parallel shedding
can
be induced
to
occur.
In
§4 a new parallel-