47
FLOW OF A FLUID IN AN AXIALLY ROTATING
By A. White*
This papcr presents the results of a preliminary experimental investigation into the flow
of a fluid in an axially rotating pipe. It has been shown that the flow is stabilized when
the pipe is rotated about its axis. At values of Reynolds number corresponding to turbulent
flow in a stationary pipe it has been shown that the flow resistance is considerably reduced
when the pipe is rotated, and at high rotational speeds reductions in pressure loss up to
as much as 40 per cent were observed. Flow visualization experiments have shown that
radial migration of fluid particles from the core of the fluid towards the pipe wall are
diminished, and thus the rate of energy dissipation is reduced.
INTRODUCTION
of a fluid along a revolving pipe does sometimes
occur in practice, but a search of the literature revealed
little information for this case. A number of investigations
into annular flow have been made with one or other of the
cylindrical walls rotating, following the classic work of
Taylor ( ~ >An
t .interesting review of work in this field is
included in a paper by Kaye and Elgar (2).
It has been found that rotation of the annular walls can,
depending on conditions, either stabilize the %owor make
the flow less stable. Rotation of the outer wall causes a
reduction in turbulence intensity near this wall, and rotation of the inner wall brings about an increase in turbulence and increased resistance to flow (2) (3).
Traugott (4) measured the turbulence intensity in a
flow along an annulus, using hot wire anemometers, and
observed a reduction in turbulence when the cylindrical
walls and the enclosed fluid were rotated as a solid body.
These results agree with a criterion for stability first
considered by Rayleigh (5) and later discussed by Prandtl
(6). This criterion indicates that the flow stability of a
fluid which is constrained to move in a curved path is
dependent on the tangential velocity gradient.
If the angular momentum increases with the radius,
then the flow tends to be stable. Any fluid particle from
an outer layer resists a tendency to being moved radially
inwards because its centrifugal
- force exceeds that on a
particle nearer the centre of curvature, and shows a
tendency to being thrown outwards. Similarly motion
outwards is impeded because the centripetal force acting
on an inner particle is smaller than that on a particle
THE FLOW
The M S . of this paper was first received at the Institution on 3rd
Y U 1963
~ and in its revised form, as accepted by the Council for
publication, on 10th October 1963.
* Graduate of the Institution.
t References are given in the Appendix.
further away from the centre. Consequently, radial motions
characteristic of turbulent flow are suppressed by centripetal body forces.
Conversely, for a flow where the angular momentum
decreases in a radially outwards direction, the forces on a
particle which deviates from its equilibrium position would
tend to move it still further away. In this case the centripetal body forces have a de-stabilizing influence, and can
lead to the formation of Taylor or Taylor-Gortler vortices. This stability criterion is not described in many
text books, but an account is given by Schlichting (7)
and Lin (S), together with additional references to work
on stability and transition to turbulence.
Notation
d
Pipe internal diameter.
f
Friction factor, T ~ & J V ~ .
I
Length.
d p Pressure loss.
Re Reynolds number, pVdIp.
V Mean velocity of flow.
p
Absolute viscosity of fluid.
p
Density of fluid.
T ~ , Wall shear stress.
w
Angular velocity.
Subscript 0 refers to conditions when
w
= 0.
FLOW ALONG A ROTATING PIPE
The case of flow along a rotating Pipe may now be considered in the light of the preceding arguments. Tangential
forces between the rotating pipe wall and the enclosed
fluid will cause the fluid to rotate with the pipe. If the fluid
rotates as a solid body with the pipe the tangential velocity
will increase in a radially outwards direction, and according to the Rayleigh criterion it would seem that the flow
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Vol6 No I I964
4s
A. WHITE
along a rotating tube would be more stable than the flow
along a stationary tube.
It is possible that, owing to momentum interchange,
the angular velocity of the core of the fluid could be
greater than that of the pipe. This would tend to reduce
the stabilizing effect, but it should be noted that even with
perfect momentum interchange instability cannot occur
since it is impossible for the angular momentum to increase towards the centre of the pipe.
Experimental measurements of pressure loss, described
later in this paper, have shown that the flow in in fact
stabilized, for at flow Reynolds numbers corresponding to
turbulent flow conditions the resistance to flow is considerably reduced when the pipe is rotated.
Flow visualization experiments have shown that the
radial turbulent migration of fluid particles is greatly
diminished, and hence there is a reduction in the rate of
energy dissipation.
EXPERIMENTAL APPARATUS
In order to obtain measurements of pressure loss along a
rotating tube the apparatus shown diagrammatically in
Fig. 1 was constructed.
it is not easy to obtain reliable pressure measurements
from a rotating tube, particularly the small diameter used
for these experiments, and therefore it was decided to take
pressure measurements from stationary pipes at either end
of the rotating section. The pressure loss along a length of
rotating pipe could then be obtained by taking measurements with two or more different lengths of rotating section
and subtracting the results for the same conditions of flow
and rotation. This method is satisfactory provided that
the shortest section is still long enough to ensure a uniform
flow before discharging into the stationary pipe.
The pipe sections were manufactured from +in bore
drawn copper tubing. The rotating section was supported
in ball races at each end, with rubber seals to prevent
//OVLIILOW
TO
WEIGHING
TANK
c
WATER
FROM
TOWN
MAIN
u
Fig. 1. Apparatus
leakage, and ball race pedestals were provided at intervals
along the length to prevent the tube whirling.
The revolving pipe was driven by a variable speed d.c.
motor through a belt and cone pulleys and the speed of
rotation was measured with a stroboscope.
Water was supplied through a control valve either
directly from the town mains for the high flow rates, or
from a constant-head tank for the smaller flows. The rate
of flow was measured by collecting the discharge in a
weighing tank, although for some of the later tests a
calibrated flow-rotameter was installed in the supply
pipe. A standpipe was provided at the outlet to ensure
full flowing conditions throughout the tests.
Static pressure tappings were provided in the stationary
pipes at either end of the rotating section and were situated
sufficiently far from the junctions with the rotating section
to eliminate any effects due to swirling flow (9). The
upstream tapping was placed 10.5 in (28 diameters) from
the rotating section, and the downstream tapping was
situated 38 in (100 diameters) downstream of the rotating
section.
The pressure loss was measured with an inverted
inclined water manometer for the low flows and a mercury
manometer for the higher flow rates.
DESCRIPTION OF T E S T S A N D RESULTS
The overall pressure loss between the two pressure
tappings was measured for three different lengths of
rotating pipe (26 in, 41 in and 87 in), at various speeds of
rotation and flow rate. The temperature of the water
supply was noted and only varied between 13°C and 15°C
throughout the period of testing.
The results for the 87 in rotating length are shown
graphically in Fig. 2 and the results for the other two
sections were similar in nature. It can be seen that
rotation of the pipe causes a considerable reduction in the
overall pressure loss, except at very low flow rates. This
is in spite of possible increases in loss at the junctions. It
is also evident that the reduction in pressure loss is
dependent upon the rotational speed.
Fig. 3 shows the relationship between the overall
pressure drop and the length of the rotating section and
it is seen to be linear down to the shortest length tested.
The overall pressure drop may be considered to be
made up of two components:
(1) The pressure loss along that section of pipe where
the flow is rotating uniformly.
(2) The pressure drop on either side of this region,
i.e. the stationary pipes and the regions of relative swirl
in the rotating pipe.
The longitudinal pressure gradient along the rotating
pipe where the flow is uniform was obtained by subtracting the overall loss figures for two rotating section
lengths, at the same conditions of flow and rotational speed.
This is justifiable from the linearity shown in Fig. 3.
Fig. 4a and b were plotted by subtracting the overall
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Vo16 No I 1964
FLOW OF A FLUID IN AN AXIALLY ROTATING PIPE
loss figures for the 87 in and 41 in rotating section lengths,
and represent the axial pressure gradient along a rotating
tube with the ‘end effects’ eliminated.
It can be seen that rotation considerably reduces the
pressure loss in the turbulent flow regime, but at values of
Reynolds number corresponding to laminar flow in a
stationary pipe (i.e. < 2000) the pressure loss is increased.
It is possible to obtain a solution to the Navier-Stokes
equations for laminar flow in a rotating tube (IO), where
the tangential velocity is proportional to the radius, i.e.
rhe enclosed fluid rotates bodily with the tube. This
49
solution shows that the axial velocity profile and pressure
loss do not differ from the values for laminar flow in a
stationary tube, but a fairly long ‘entry length’ along the
rotating tube is required for these conditions to become
established.
At first sight it would appear that the increased pressure
loss actually measured at low values of Reynolds number
is due to turbulence being generated at the transition
between the rotating pipe and the stationary pipe. However, it is likely that turbulence induced by the junction
in this way would die out along the tube, but there is no
30
L
c
CI
25
0)
L1
E
u
.-
I
m 20
v)
0,
LI!
n
2
15
v)
o
W
Q
n
20
40
60
ao loo
LENGTH OF ROIAT’NG S E C T I O N - I 1
a
_I
2
1c
n
W
>
0
5
I
~
0
6000
30
8000
10000
Re= p v d / p
a
280
i
I
I
LENGTH OF ROTATING S E C T I O N -
in
b
0
2 0 000
10 000
30 00
0
20
60
40
80
100
LENGTH O F ROTATING S E C T I O N - i n
ffe=p vd/ )r
b
C
Fig. 2. Overall pressure loss with 87-in long rotating
section
Fig, 3. Variation in overall pressure loss with length of
rotating section
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VoI 6 No I 1964
A. WHITE
50
evidence of turbulence decay along the rotating tube as
can be seen from the linearity of Fig. 3a.
Although great care was exercised to maintain the
rotating tube axially true, there was a slight eccentricity
of a few thousandths of an inch, and it is probable that this
eccentricity, aided by mechanical vibration, is the cause
3
1.00
I
sPEED.rev/min
I
2.
Y
W
2
m
9 2
C
._
I
0.501
0.1
Lu
LL
h 2.
.
, , , ,I
0.5
1.0
, I
I
I
I
I
50
1.0
w d/ V
8
Plotted from Fig. 4a and b.
I-
0
0
,
Fig. 5. Reduction in friction factor with pipe rotation
1.
L
w
a
m
Y)
2 1
W
n
3
ln
m
w o
a
a
0
0
10 000
20 000
30 000
Re=p vd/
b
Pipe diameter 3 in.
Fig. 4. Pressure loss along a rotating pipe: pressure loss
against Reynolds number for various speeds of rotation
of the increased turbulence and pressure loss at low
Reynolds numbers.
Fig. 5 represents the results in dimensionless form, and
shows the pressure loss along a rotating pipe expressed as
a fraction of the loss along a stationary pipe plotted as a
function of wdlV and Reynolds number. It appears that
at values of Reynolds number above about 8000 all the
experimental points lie on a unique line, but for Reynolds
numbers below this value the results deviate from this
line and the reduction in pressure loss caused by rotation is
not so great.
As stated previously, it is believed that some turbulence
is created in the rotating section by eccentricity and vibration, and at low values of Reynolds number the turbulence
intensity caused by the flow along the pipe may be small
in comparison. However, at high Reynolds numbers the
intensity of turbulence generated by the flow is probably
large compared with that due to eccentricity and vibration.
This is the probable explanation of the smaller reduction
in pressure loss at values of Reynolds number below about
8000, for although rotation of the pipe stabilizes the flow,
some turbulence is also created by pipe malalignment,
and it is only at the higher flow rates that this latter effect
becomes negligible compared with turbulence generated
by the flow.
Fig. 6 shows the results plotted in the form of a friction
factor chart, which was prepared using data from Fig. 5.
This chart clearly shows the large reductions in pressure
loss obtained when the tube is rotated, and also the
asymptotic nature of the reduction with increasing
rotational speed.
It is of interest to note that the foregoing results
provide a method by which the total pressure loss across a
pipe system with a rotating section may be estimated.
From Fig. 3 it can be seen that if the lines on each graph
are extrapolated they intersect at a common value on
the vertical axis, which is found to be nearly equal to the
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Val 6 No 1 I964
FLOW OF A FLUID IN A N AXIALLY ROTATING PIPE
Plats d
Tube speed = 0 rev/min.
Fig. 8. Laminar flow
Re = 2100; tube speed = 0 revlmin.
Fig. 9. Onset of turbulence
Re = 3520; tube speed = 0 rev/min.
Fig. 10
Re = 3520; tube speed = 1040 revlmin.
Fig. 11
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Vol6 No I 1964
A. WHITE
Plate 2
Re = 8800; tube speed = 0 revlmin.
Fig. 12
Re = 8800; tube speed = 1600 revlmin.
Fig. 13
Re = 8800; tube speed = 1840 revlmin.
Fig. 14
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FLOW OF A FLUID I N AN AXIALLY ROTATING PIPE
.
\I \ \
a
h 2
51
in BORE HYPODERMIC TUBE'
Fig. 7. Arrangement for dye injection into rotating tzibe
0
0
0 1 5
1000
L---I
L-L
5000
I
10000
I
I
I
I
50 000
Re=p v d / p
Fig. 6. Variation of friction factor with Reynolds
number and rotational speed
pressure loss along the stationary inlet and outlet sections
with zero rotation of the intermediate section. Thus it
seems that the pressure loss across the whole system with
a rotating section and stationary inlet and outlet lengths
may be calculated by adding two components of pressure
loss :
(1; The loss along the whole of the rotating length
based on the friction factor from Fig. 6 corresponding
to the speed of rotation and the flow rate.
(2) The loss along the stationary inlet and exit
lengths, based on the normal friction factor for a
stationary pipe.
Ir was found that this method produces values which
correspond very closely with experimental measurements
of overall pressure loss, and over the whole range of the
tests the maximum error was not more than about 10 per
cent.
FLOW VISUALIZATION EXPERIMENTS
The large reductions in flow resistance caused by rotation
of the pipe indicate that the turbulent fluctuations must
be considerably reduced, and it was considered that much
interesting information could be obtained from visual
flow experiments.
A transparent rotating section waa manufactured from
perspex tube of in bore and Q in wall thickness. The
upstream end was attached to a rotating swirl drum with
vanes in an axial-radial plane. This was used to ensuie
that the water entering the perspex tube was rotating at
the same angular speed, and thus avoided the necessity of
a very long tube.
+
J 0 11 R N A I
Dye was injected at a point 12 in from the entrance by
the arrangement shown in Fig. 7, from a pressurized
container, and its flow rate was adjusted with a screw
down type hose clip. The dye injection arrangement also
gave support to the perspex tube to prevent whirling.
The dye used consisted of Fluorescein dissolved in
water, although in order to take flow photographs it was
rendered more opaque by the addition of a little Indian ink.
Dye was injected at a controlled rate for various flows
and speeds of rotation and the resulting flow patterns are
shown in Figs 8-14, Plates 1 and 2.
An electronic flash gun was used with a special troughshaped reflector to illuminate the tube along its length as
uniformly as possible, and the duration of the flash was of
the order of 0.001 s, which was sufficiently brief to arrest
the motion of the rotating tube.
Figs 8,9 and 10, Plate 1, show the flow along a stationary
pipe and show the onset of turbulence with increasing
Reynolds number. These results are of course well known,
and were first observed many years ago during the classic
experiments of Osborne Reynolds (11).
Fig. 11, Plate 1, shows the flow pattern at the same
Reynolds number as Fig. 10 but in this case the tube was
rotating. It can be seen from Fig. 10 that turbulence
causes the dye quickly to diffuse radially outwards and to
fill the whole tube. The stabilizing influence of rotation is
clearly shown in Fig. 11, where radial migrations of fluid
particles are greatly diminished and the dye moves along
the central core of the tube without much radial diffusion.
Of course a similar effect would be observed if the dye
had a density less than that of water, since the dye
particles would then be forced to the centre of the tube
under rotating conditions. However, the flow patterns in
the photographs cannot be ascribed to this cause, since
Indian ink particles are colloidal and therefore would not
be centrifuged out of suspension. This was verified by
injecting some of the dye into a test tube of water, and
rotating the whole for some time in a centrifuge. The dye
was seen to remain in suspension and the particles were
not forced either to the top or the bottom of the tube.
Figs 12-14, Plate 2, show the flow pattern at a higher
value of Reynolds number, well into the turbulent regime,
and again the stabilizing influence of rotation is clearly
seen.
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VoI h N o I 1964
A. WHITE
52
CONCLUSIONS
This investigation shows that the flow along a pipe is
stabilized when the pipe is rotated about its axis. At values
of Reynolds number corresponding to turbulent flow along
a stationary pipe it has been shown that the flow resistance
is considerably reduced when the pipe is rotated, and at
high rotational speeds reductions in pressure loss up to as
much as 40 per cent were observed.
This phenomenon is in agreement with a criterion of
stability enunciated by Rayleigh, which when related to
the flow conditions of this problem shows that radial
migration of fluid particles would be suppressed.
Flow visualization experiments have confirmed this
reasoning, and it has been clearly shown that radial
movements of fluid particles from the fast-moving central
core of the fluid towards the pipe wall are reduced, and
hence the rate of energy hssipation is reduced.
It should be noted that although the results have been
represented in dimensionless form, only one size of pipe
was used for the tests. Furthermore, it is likely that results
with a compressible fluid would differ from Fig. 6 since
a positive radial density gradient should increase stabilization.
Although this work has not been related to any particular
practical problem, the results may find application in
connection with certain items of chemical plant containing
rotating elements, and also with rotating-tube heat
exchangers.
ACKNOWLEDGEMENTS
This work was carried out at Kings College in the
University of Durham, now the University of Newcastle
upon Tyne, during the year 1961-62.
The author is indebted to Professor A. F. Burstall, D.Sc.,
Ph.D. (Member) for the provision of the experimental
facilities, and to Mr B. A. Sutton, M.A., for his helpful
comments and advice.
APPENDIX
REFERENCES
(I)
TAYLOR,
G. I. ‘Stability of a viscous liquid contained
between two rotating cyclinders’, Phil. Trans. 1923 223
(Series A), 289. ‘Fluid friction between rotating cylinders’,
Proc. roy. Soc. 1936 157 (Series A), 561.
(2) KAYE,J. and ELGAR,
E. C. ‘Modes of adiabatic and diabatic
fluid flow in an annulus with an inner rotating cylinder’,
Trans. Amer. SOL.mech. Engrs 1958 80,753.
(3) YAMADA,
Y . ‘Resistance of a flow through an annutus with
an inner rotating cylinder’, Bull. Jup. SOC.mech. Engrzg
1962 5 (no. 18), 302.
(4) TRAUGOTT,
S . C. ‘Influence of solid body rotation on screen
produced turbulence’, Tech. Notes nut. adv. Comm. Aero.,
Wash. no. 4135.
( 5 ) RAYLEIGH,
LORD. ‘On the dynamics of rotating fluids’,
Proc. roy. Sac. 1916 93 (Series A), 148.
(6) PRANDTL,
L. ‘Einfluss stabilisierender Krafte auf die
Turbulenz’, Vortrage aus dem Gebiete der Aerodynamik
und verwandte Gebiete (Aachen 1929), 1930 (Springer,
Berlin).
(7) SCHLICHTING,
H. Boundary layer theory fourth edition,
1960 chapter 17 (McGraw-Hill).
(8) LIN, C . C. The theory of hydrodynumic stability 1955 49
(Cambridge University Press),
(9) WHITE, A. ‘The flow of fluid in rotating pipes’, M.Sc.
Thesis, Kings College, University of Durham, hlarch
1963.
(10) TALBOT,
L. ‘Laminar swirling pipe flow’, J . appl. Mech.
1954 21 (no. l), 1.
(11) REYNOLDS,
0. ‘An experimental investigation of the circumstances which determine whether the motion of
water shall be direct or sinuous’, Phil. Trans. 1883, 935.
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