345
Experiments on the
flow
past a circular cylinder at
very
high
Reynolds number
By
ANATOL
ROSHKO
Guggenheim Aeronautical Laboratory, California Institute of Technology,
Pasadena, California
(Received
15
November
1960)
Measurements on
a
large circular cylinder in
a
pressurized wind tunnel
at
Rey-
nolds numbers from
106
to
lo7
reveal
a
high Reynolds number transition in
which the drag coefficient increases from its low supercritical value to
a
value
0.7
at
R
=
3.5
x
lo6
and then becomes constant. Also, for
R
>
3.5
x
lo6,
definite
vortex shedding occurs, with Strouhal number
0.27.
1.
Introduction
Shortly after the closing and before the dismantling of the Southern California
Co-operative Wind Tunnel (CWT), some time was made available to us for
a
high Reynolds number experiment on a circular cylinder. In this large pressurized
wind tunnel (Millikan
1957)
it
was possible to reach
a
cylinder Reynolds number
R
of close to
107,
compared to about
2
x
lo6,
the highest value for which wind
tunnel measurements were previously reported in the literature. There are, in
addition, some measurements in natural wind, up to about
R
=
4
x
lo6
(Dryden
&
Hill
1930;
Pechstein
1940),
about which we comment more fully later.
The many experimental measurements of drag coefficient
C,
at subcritical
Reynolds numbers are in fairly good agreement as
to
the values of
C,(R),
but in
the supercritical range, i.e. after the transition to low values of
C,,
there is little
agreement, except that
C,
lies between values of
0.2
and
0.4.
It
is not clear
whether the relatively large discrepancies here are due to difficulties in measure-
ment
or
whether the flow here is more sensitive to the conditions of the experi-
ment. The measurements of Delany
&
Sorensen
(1953)
at Reynolds numbers up
to
2
x
lo4
exhibit
a
multivaluedness which they attribute (private communica-
tion) to changes in the flow from
a
symmetrical to an unsymmetrical type, higher
values of
C,
occurring in the unsymmetrical flow. Other authors claim to observe
no asymmetries.
On the question of vortex shedding
at
high Reynolds number there is little
information.
It
is
well known that vortex shedding occurs
at
Reynolds numbers
below the critical value, with
a
dimensionless frequency (Strouhal number
X)
of about
0.2.
At critical and supercritical Reynolds numbers there are, to our
knowledge, only two sets of measurements. The early ones by Relf
&
Simmons
(1924)
indicate that in the transition range there is
a
predominant frequency in
the wake, which they called ‘aperiodic’, compared to the ‘accurately periodic’
flow at
R
<
lo5.
Their measurements show that the Strouhal number of these
frequencies increases as the drag coefficient
C,
decreases.
346
Anatol
Roshko
More recently, Delany
&
Sorensen
(1953)
obtained measurements
at
still
higher Reynolds numbers
(106
to
2
x
lo6)
using
a
pressure pick-up in the wake
close behind the cylinder. The shedding frequencies, which were determined from
the predominant frequencies on an oscillograph record, show considerable
scatter, as do those of Relf
&
Simmons; the values of
S
are between
0.35
and
0.45.
From these two sets of measurements, it would appear that
S
rises rapidly in
the interval
R
=
2
x
lo6
to
lo6,
and that there may be
a
rapid decrease
at
about
2
x
106.
Our intention in the present experiments was to overlap the Reynolds number
range of the existing measurements, while extending them to values of
R
as
high as possible. The time available for preparing and performing the experi-
ments was
so
short
as
to preclude
a
thorough investigation of all aspects of the
flow, but it was hoped to obtain answers to
a
few obvious questions: Does the
drag coefficient continue to change! Is there vortex shedding?
Is
there an
asymptotic state, i.e. can we say anything about the ultimate form of the flow
as
R
-+
co,
and
at
what values of
R
are we approaching
it
Z
2.
Experimental arrangement
The experiments were performed in the subsonic test section of the CWT,
which had
a
height of
8.5
ft. and width
of
11
ft.
It
could be pressurized to
4
atm;
pressures of
1
and 2atm were also used. To avoid compressibility effects, the
flow
speed was limited to
a
Mach number of about
0.25.
Hot,
wire
Splirrer
position
5.i
in.2
FIGURE
1.
Arrangement
of
cylinder
in
the
wind
tunnel.
The cylinder was
a
seamless 'black steel' pipe which had been sandblasted
to remove its protective paint and scale. This resulted in a surface roughness of
about
200
pin. The cylinder had
a
diameter of
18
in. and was round to within
&in. (in diameter).
It
spanned the
Sift.
height of the test section. Pressure
orifices were located every
10"
(with additional ones at
6'
=
95"
and
loso),
over
half the circumference at the middle section. These were connected to
a
pressure
measuring system consisting of pressure transducers which read out to digital
indicators and recorders. The sensitivity of this system was set to give full output
at the highest dynamic pressure, consequently there was
a
deterioration of
accuracy
at
lower values of the dynamic pressure.
A
hot-wire anemometer was mounted at
a
fixed position,
7.3
diameters down-
stream of the cylinder axis and
0.7
diameters
off
the centreline, as shown in figure
1.
The output of the hot wire, in
a
standard circuit, was fed into
a
spectral analyser,
Flow
past a circular cylinder at
high
Reynolds number
347
which could scan the frequency spectrum and record it on
a
pen recorder. We
were looking for possible vortex shedding peaks in the spectrum. Whenever
one was indicated on the record, it was tuned by hand and its frequency accurately
determined by comparing it to an oscillator signal whose frequency was measured
by an electronic counter.
Provision was made to install a 'splitter plate' on the centreline behind the
cylinder,
as
indicated in figure
1.
It
also spanned the
Sift.
height of the test
section, extended 4ft. along the centreline, and was 2in. thick, being made up
of two pieces of plywood bolted together.
3.
Wall
interference corrections
To
obtain the highest possible Reynolds number, the cylinder diameter chosen
was
a
little larger than might have been desirable from the point of view of the
wall interference. With
a
diameter
d
=
1-5ft. and tunnel breadth
b
=
11
ft.,
the blockage ratio was
dlh
=
0.136. To correct for the wall interference effects,
we made use of the formulas of Allen
&
Vincenti
(1944),
which give, for the cor-
rected values
of
velocity and drag coefficient
V
and
C,
in terms of the measured
values
V'
and
C&
V
_-
V,
-
l+&'i(~)+O-S2$2,
These formulas were obtained, following the earlier work of Lock, Glauert and
Goldstein, by using image doublets to represent the interference between wall
and cylinder, and image sources to represent the interference between wall and
wake; the two effects give the third and second terms, respectively, in the
formulas. Such an analysis does not take into account possible interference
effects on the separation mechanism and the structure
of
the wake close behind
the body; changes in these could have an important effect on the drag. This
would be especially important in regions where
C,
is changing rapidly with
R,
but probably less important where
C,
is nearly constant. The experimental
evidence relating to wall interference on cylinders is not entirely satisfactory.
We have used equations
(1)
and
(2)
as the best available, believing them to be
fairly accurate at
our
highest Reynolds numbers, where
C,
is nearly constant.
The maximum corrections to
V'
and
Ci
were about
4
and
10
%,
respectively.
The uncorrected drag coefficient
Cg
was obtained by integrating the measured
pressure distribution
Ck(0);
it was then corrected with equation
(2).
When
corrected values of pressure coefficient
C,
were needed (e.g. figures 3 and
4),
these were obtained
from
(C,-l)=
-
(Ck-1).
(";)"
(3)
Corrections
for
Reynolds number
R
and Strouhal number
X
are the same as
for the velocity.
348
Anatol
Roshko
(a) Drag coeficient
4.
Results
The corrected values
C,
are plotted against
R
in figure
2.
Also
shown, for com-
parison, is the well-known curve of
C,(R)
at
lower values of
R.
Different in-
vestigators do not agree exactly on this curve, but the differences are small,
except in the supercritical region. We have chosen Wieselsberger’s
(1921)
curve
as representative and have also plotted the results of Delany
&
Sorensen
(1953)
in the supercritical range (all the dashed lines). The latter extend to
a
higher
Reynolds number than any other wind tunnel experiments of which we are
aware, and they illustrate the difficulty in obtaining consistency in the super-
critical range.
It
had been hoped that
our
experiments would overlap the
R
FIGURE
2.
Drag
coefficient.
existing measurements more than they do, but we were unable to obtain sufficient
accuracy
at
our
lowest Reynolds numbers, since the dynamic pressures were
so
low that the resolution of the pressure-measuring system became inadequate,
and there was insufficient time to change it.
Our
results match the trend of those at lower Reynolds numbers,
as
well as
this can be determined from the multivalued behaviour, and
it
is
apparent that
C,
increases in the range
lo6
<
R
<
3.5
x
lo6,
from
a
value of about
0-3
to about
0.7,
and then levels off at the latter value.
It
will be noted presently that the
value
R
=
3.5
x
lo6
apparently marks the end of
a
transition range.
Also
shown on figure
2
is the value of
C,
obtained by Dryden
&
Hill
(1930)
in
some experiments that are apparently not well known. These measurements
were made on
a
smoke stack with
a
clear height
of
120ft., the values
of
C,
being
obtained from pressure distributions
at
a
section
41
ft. from the top, where the
Flow
past
a circular cylinder
at
high
Reynolds number
349
diameter was 11.Sft. The wind speeds were about 25-40m.p.h., which corre-
sponds to Reynolds numbers of 3-5
x
lo6.
Their mean value of
C,
for
a
group
of
observations is
0.67
&
0.04,
and it may be seen that this agrees quite well with
our results.
A
point of interest
is
that Dryden
&
Hill inferred the wind speed
from the measurements on the cylinder, making use of the fact that the maxi-
mum pressure is stagnation pressure, while the pressure
at
31" on either side of
that point must be nearly static pressure. For experiments in the natural wind,
this technique is clearly better than one using
a
velocity measurement
at
some
location removed from the cylinder.
R
FIGURE
3.
Base
pressure
coefficient.
In another set of experiments with
a
short stack
(I
=
30ft.,
d
=
loft.), Dryden
&
Hill obtained much lower values of
C,
(approximately 0.4)
at
the same values
of
R.
The lower value may be attributed to the small length-diameter ratio, as
may the similar results of Pechstein (1940) on
a
cylinder of length 10m and
diameter 2m in a natural wind.
(b)
Base pressure coeficient
Changes in
C,
are closely related to changes in
Cpa,
the pressure coefficient on the
back of the cylinder. This was determined from the average pressure over 20
or 30" on either side of the rearmost point. The results (corrected values) are
plotted in figure 3, together with
a
curve obtained by Flachsbart (1929) at lower
Reynolds numbers. The curve determined by our points is faired into Flachsbart's
curve. (We have not included other investigators' results on
this
plot
;
inclusion
of these would show
a
scatter or multivaluedness
at
supercritical Reynolds
numbers, like that on the
C,(R)
curve.)