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Streamline Flow through Curved PipesAuthor(s): C. M.
WhiteSource: Proceedings of the Royal Society of London. Series A,
Containing Papers of aMathematical and Physical Character, Vol.
123, No. 792 (Apr. 6, 1929), pp. 645-663Published by: The Royal
SocietyStable URL: http://www.jstor.org/stable/95217 .Accessed:
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645
Streamline Flow through Curved Pipes. By C. M. WHITE, King's
College, London.
(C ommunicated by GE. V. Appleton, F.iR.S.-Received, Fcbcuary 1,
1929.) NOTATION.
C, a niumerical coefficient, defined as Fd/8,uv, representing
the increase of resistance due to curvature. d, diameter of pipe.
l), mean diameter of coil. d/D, curvature ratio. F, intensity of
frictional drag on wall of pipe, i.e., shear stress at boundary. t,
gravitational acceleration. rn, hydraulic mean depth, =
area/perimeter. v, mean velocity of flow, - volume flowing per unit
time/cross sectional area of pipe. p, density of fluid. ,u,
viscosity of fluiid.
All the values given in the paper are dimensionless, except in
those cases in which sonie dimension is specifically stated.
NoTE.-All curves are plotted logarithmicallv.
The problem of the determination of the law of resistance for
flow throuagh smooth straight pipes has been the subject of many
exhaustive investigation The result is conclusive. It may be said
that the law of resistance is clearly defined over an extremely
wide range of those variables which influence flow through such
smooth straight pipes. Expressing the coefficient of resistance in
the form F/pv2, and the flow as pvdl/t, it has been shown that
F/pv2 is a function of pvd/ A. The general form of this function*
is given graphically in fig. 1, which is based primarily on the
experiments of Saph and Schoder, Stanton and Pannell, and
Schiller.t It will be seen that there is a very definite
.Oi8
'006
-004~~~~~~~~~~~~~, *002
*cx.s 88r _8
0 0 0 o 'l ?8 ?L_ 0*00; LOSN1VK 8 RE to Flow t s tIg
FiG#. I.-RDesistance to l?low through smnooth straight pipes. *
Davies and White, 'Engineering,' (in tae pre&s). t 'Trans. Am.
Soc. C. E.,' vol. 51, p. 253 (1903); 'Phil. Trans.,' A, vol. 94, p.
199 (1914);
' Z. angew. Math.,' vol. 3, p. 2 (1923). VOL. OXXITI.-A. 2 u
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646 C. M. White.
discontinuity in the curve when the Reynolds number reaches
2300, indicating the change from streamline to turbulent motion.
Below 2300 the tests agree very closely with the law F/pv2 -= 8
(pvd/l)-. This equation, apart from its corresponding forms for
non-circular pipes, appears to be the only solution of the
hydrodynamic equations which has as yet been obtainied for flow
through pipes. This solution represenlts streamline flow. The
second line in fig. 1, for flows greater than pvd/lt 2300,
indicates, of course, the existence of a second solution
representing turbulent flow. Above 2300, and up to at least
100,000, the equation F/p-V2 _0 04 (pvd/u)--which no doubt is an
approxi- mation to this second solution--represents the observed
results within the accuracy of the experimental errors.
With regard to the influence of curvature, however,
comparatively few records are available. In the present paper, the
author describes some tests recently carri.ed ouat in -the
Engineering Department at King's College, London. These tests, when
considered in relation to the results of other workers, appear to
define the influence of curvatuLre upon the law of resistance with
satisfactory accuracy for a range of pvd/p, up to about 9000.
While this range relates to flows far below those of chief
interest to the water engineer, it has, nevertheless, considerable
practical application. Coils of tube are largely used in
coinnection with heating and refrigeration in order to transfer
heat from one fluid to another. Unider such conditions, both the
pressure required to obtain the necessary circulation and the rate
of heat transfer for a given circulation will depend. upon the
magnitude of the resistance. The range is of particLular interest
also from the theoretical point of view, since it includes the
change from streamnline to turbulent flow, the cause of which still
remains unexplained, although the tests of Schiller* go far, in
that they show that the initial state of the fluid is a governinig
factor.
The present investigatioin had its origin in a re-examination,
by the author, of some earlier experimental work on curved pi.pes,
which it was hoped might, provide information concerning the
circumstances determining the limits within, which fllow must be
streamline in character. It was seen, however, that additional
experlinental data were necessary; the investigation actually
developed into a more general study of flow through curved pipes.
The experi- mental part of the work consisted of tests of three
pipes, Nos. I, II and III, wholse curvature ratios, (d/D, were
respectively 1115, 1150 and 1/2050. The pipes are nunmbered in the
order of their curvatures, but the order in which they were tested
was II, I and III. In each case, irregularities in the passage
* 'Z. an-ge-w. Alath,' vol. 1, p. 436 (1,921).
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Streamline Flow through Curved Pipes. 647
leading to the pipe were provided, in order to produce a
turbulent state of flow at the entrance to the pipe. This
turbulence, however, would not influence the motion within the
testing length of the pipe, unless the conditions within the pipe
were such that the turbulence was self-sustaining, because, in
every case, a length of curved pipe at least equal to 180 diameters
was interposed as an entrant or stilling length between the
entrance and the first gauge point. It will be seen later that the
particular dimensions of each of the individual pipes were selected
in order to obtain information on some specific point which was in
doubt at the time. When the investigation was undertaken there
appeared to be only two sets
of relevant experiments on record: those of Grindley and
Gibson,* which were made to determine the viscosity of air; and
those of Eusticet who was directly interested in the effect of
curvature. The former tests, while they extended over a wide range
of Reynolds number, from 25 to 1400, were unfortunately confined to
one radius of curvature, namely, 112 times the pipe radius. On the
other hand, the experiments of Eustice-although they covered a wide
range of curvature, and provided much valuable information of a
qualitative nature-were open to certain criticisms. The pipes them-
selves were oval in cross section, and, moreover, no provision was
made to enable a uniform state of flow to become established
throughout the length of pipe under test. Taking this into account,
it was reasonable to question whether these results would be found
to be truly representative of flow through coiled pipes of circular
section, and further experimental evidence appeared to be necessary
before the point could be decided. The only coiled pipe for which
he gave the results in full was that with a cross sectional area of
0 * 0735 sq. cm. and a coil diameter of 2*59 cm. If the section had
been of circular form the curvature ratio would have been 1/8.5,
but, in view of the actual form, it appeared reasonable to assume
the curvature ratio to be 1/14.6. The tests of this pipe covered a
remarkably wide range of flows, the Reynolds numbers varying from
21 to 6000.
An investigation of the problem had been made by Dean,: who
showed mathematically that the product F/pv2. pvd/L,, i.e., Fd/uv
i~ a function of the criterion pvd/V. (d/D)1.? It should be
emphasized that, in obtaining this result,
* 4Roy. Soc. Proc.,' A, vol. 80, p. 114 (1908). t 'Roy. Soc.
Proc.,' A, vol. 84, p. 107 (1910). : 'Phil. Mag.,' vol. 5, p. 673
(1928), also vol. 4, p. 208 (1927). ? The actual notation used by
Dean was somewhat different. His result is
F,/=8 f [G2a7/p'2^R], where JF, = flux in curved pipe, F8 = flux
in straight pipe, G = pressure gradient, a = radius of pipe, R =
radius of coil, I-- coefficient of viscosity, v = kinematic
viscosity.
2 u 2
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648 C. M. White.
Dean assumed an essentially streamline type of motion.
Unfortunately the series he obtained in his solution was such that
a numerical result could be obtained for only extremely small
values of the criterion; nevertheless, the work was of the utmost
value in that it pointed to a basis of correlation for the existing
data.
It appeared to the author that, if it could be shown that the
coniditions of flow assumed by Dean did in fact represent the
actual motion of the fluid, then a single but extended series of
tests, with one pipe only, would provide all the information that
was necessary to determine the function
Fd /v = f ['pvd/lt. (dID)] As neither Grindley and Gibson's nor
Eustice's experiments provided sufficient data for the purpose, the
author arranged for the pipe No. II to be tested. It was evident
that the tests would have to extend over a range of flows at least
twice as great as that investigated by Eustice, and this
consideration led to the selection of a diameter of 0 63 cm. A
larger pipe would not give measurable pressure gradients with very
small flows, since, for reasons of manufacture, the length without
joints is limited to 20 metres, while a smaller pipe is undesirable
on account of irregularities in diameter. The curvature ratio,
1/50, intermediate between those of Grindley and Gibson and that of
Eustice, was selected in the hope that a comparison would show that
Dean's assumptions were justified. This hope was not realised,
since the test results with this pipe, when considered on the basis
of Dean's theory, definitely differed froom those of Eustice, and
were only in rough agreement with those of Grindley and Gibson.
Tests with a pipe of greater curvature were clearly necessary,
and pipe No. I with a curvature ratio of 1/15 was accordingly made
and tested. It should be noted thatu both pipe No. II and the pipe
used by Grindley and Gibson were very long, and there were
possibilities in both cases of small inaccuracies in the
determination of their diameters, while Eustice's pipe was not
circular in cross section. It was thus desirable that the cross
section of this pipe, No. I, should be as neagly circular as
possible, and that its dimensions should be known to a high degree
of accuracy. In view of this and the greater curvature to which
this pipe had to be coiled, it was necessary, from a purely
constructional point of view, to adopt a larger diameter than that
used for pipe No. II. This led to the use of lubricating oil in the
place of water for those tests with the smallest flows, since the
pressure gradients, using water, would have been too small for
accurate measurement with the available manometers.
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Streamline Flow through Curved Pipes. 649
The results of these tests were particularly satisfactory. The
agreement with pipe No. II was so good that it was reasonably
certain that the motion closely followed the type investigated by
Dean. And since by using oil, in addition to water, a range of
Reynolds numbers from 0 06 to 40,000 was explored, it could be said
that the results covered the whole practical range of pvd/li. .
(d/D)-I, the criterion found by Dean. This pipe, then, gave all the
data necessary to express the relationship Fd/ltv = f[pvd/,.
(d/D)-] in the form of a curve. It was felt, however, that unless
Dean's criterion could be shown to apply to a pipe of very
different curvature ratio, the results could not be accepted with
complete confidence. A wide range of the criterion pvd/,u. (d/D)*
had certainly been explored, but only by varying pvd/lL. The range
of (d/D)l throughout had not exceeded O 1 to 0 * 26, and, since on
the one hand the resistance is not at all sensitive to changes of
d/D, while on the other hand a small error in pipe diameter causes
considerable effect, it was just possible that the agreement
between pipes Nos. I and II might be accidental. In order to
examine this point, it was desirable to extend the range of
curvature as distinct from the range of Reynolds number, the
quantity pvdl /t. (d/D)* of course still coming within the range
previously covered. It is impossible to extend the range of
curvature much in an upward direction, because certain difficulties
of manufacture arise; but even could these be overcome, there still
remains the ultimate limit, in that the curvature ratio of a pipe
cannot exceed unity. No such definite limitation, however, exists
in the other direction, although, as explained later, in an almost
straight pipe the effect of curvature upon streamline flow may not
be sufficient to be measurable before it is masked by turbulence. A
curvature ratio approxi- mately 1/135th of that of pipe No. I was
selected for pipe No. III in the hope that, on the one hand, before
turbulence set in, the resistance would show a sufficientlymarked
deviationfrom that of a straight pipe to enable Dean's theory to be
established beyond question, and that, on the other hand, the
increase due to curvature would be less than that usually observed
due to the beginning of turbulence in a straight pipe. It was hoped
that as a result this curved pipe would show the characteristic
sudden increase of resistance at the critical point-a point which
was by no means well defined in the curves obtained from the other
pipes. The small internal diameter of 0X3 cm. was adopted in order
that the exact value of the diameter might be determined by the
most precise method, namely, by observation of the resistance to
flow through the pipe before it was bent. This method was
permissible in the particular case of pipe No. III, since the
slight curvature desired could be obtained by elastic
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6.5 0 C. M. VWhite.
bending, and without appreciable distortion of the bore. The
tests of this pipe provided the desired data, and so completed the
experimental part of the investigation. There were then five sets
of results available, and their scope is indicated in the table
below.
Diameter Diameter C ture Range of flow, of pipe, of coil, ratio.
Reynolds Fluid. Remarks. cm. Cm. o. Numbers.
2-59 20to 5000 Water Oval pipe, area 0 0735 cm.2. Tested by
Eustice.
0*317 36 6 1/112 25 to 1400 Air Tested by Grindley and Gibson
0-630 31-7 1/50 16 to 13000 Water Author's pipe No. II. 1-032 15 62
1/15.15 0-058 to 41000 Oil Author's pipe No. I.
water 0 -2980 610 5 1/2050 220 to 4000 Water Author's pipe No.
III.
Pipe No. II, a thick walled lead pipe, 18 metres long and 0 630
cm. bore, was coiled on a drum 30 5 cm. in diameter, giving a mean
coil diameter of 31P7 cm. A length of some 400 pipe diameters was
allowed for stilling, and there were 300 diameters between the last
gauge point and the outlet. The measuring length was thus 2000
diameters. Alternative gauge points were provided, in order that
any possible periodic effect might be checked, but they were found
to be unnecessary. No attempt was made to jacket the apparatus
against temperature changes; but the temperature of the water was
measured both at the inlet and at the outlet of the pipe, and the
supply was maintained at room temperature.
The great length of this pipe, 18 metres, and the smallness of
its diameter rendered impossible a precise determination of its
diameter by direct means. Pieces cut from the ends before it was
coiled had an internal diameter of 0 * 630 cm. determined by plugs.
The coiling would, however, modify this by an unknown amount; but
in view of the tbickness of the walls, it was hoped that the
alteration would be small. That this was so is shown by the test
points for the smallest flows, which do not show any marked
tendency to lie away from the line F/pv2 = 8p4pvd. In all, some 170
tests were made (see fig. 2). It will be seen that a range of flows
of about 1000 to 1, with a corre- spondingly greater range of
pressure gradients, was covered by these tests of a single pipe.
Such a wide range necessitated frequent changes in the measuring
devices. Considerable overlapping was adopted as a practice,
whenever a change was made; and, under these conditions, some
scattering of test
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Streamltne F'low through Curved Pipe.s. 651
points is inevitable. Bearing this in mind the results are
satisfactorily consistent.
Fig. 2 shows that for flows less thain pvd/ t == 80, there is no
measurable *ZJ
- t - _ J< 1 ., -_.
-. EV ... -1-C-- rf?f-. *5- 1 __ - _ r- - X
z 1- -- l t l+~ ~~~~~~~~~~~~~~~ _+ t_+ - - +F-~~~~~~~~~~~~~~~~0
1- -- -? 2~ - - - - - - - *51 -1- 2- - - ~-- -+ s-He___k F.- TO to-
-I-tA - -HI-t S W++H _ _ _
3-C9LE PIP-E Ne Tr41
FIG. 2.
deviation from the line representing streamline flow in straight
pipes. It is not until the Reynolds number exceeds 100, that this
curved pipe begins to show a greater resistance than that of a
straight pipe of the same diameter and length. It is convenient to
think of the curvature as causing an increase of resistance such as
that indicated by C in fig. 2. From 100 up to 6000 the effect of
the curvature becomes progressively more marked, until at 6000 the
resis- tance is 2 9 times that of a straight pipe in which the flow
is assumed to be streamline. It will be seen that the curve
definitely touches the line which represents turbulent flow in
smooth straight pipes, and that there appears to be a change of law
at this point. The conclusion. is later drawn that the flow becomes
turbulent at a Reynolds number of 6000. The investigation was not
intended to extend to turbulent flows, neither was the apparatus
suitable. The work was not therefore pursued further in this
direction.
Before considering the tests with pipe No. 1, which had the
greatest curvature ratio 1/15.15, it is desirable to emphasise that
accuracy of measurement of the size of the pipe was the foremost
consideration in the selection of the pipe for these tests. The
earlier work had thrown doubt either upon the strict accuracy of
certain assumptions which had to be made in order to compare
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652 C. M. White.
Eustice's results with the present experiments, or,
alternatively, upon the truth of the assumptions upon which Dean
based his theory. The con- ditions of the problem were such that
even. a small error in the size of the pipe would, it seemed, leave
the question of th-e effect of curvature in almost as doubtful a
position as before. It was felt that, from the point of view of
accuracy, a pipe diameter less than 1 cm. could not be used. After
several attempts at coiling, a curvature ratio of 1/15 was found to
be the greatest which would give reasonable freedom fromn
distortion. The following were tlle leading dimensions of this
pipe
Coiled pipe No. T. N-urmber of turns .................. 2 Mean
diameter of turn .............. l 15' 62 cm. Total length of tube
............... 4-85 * 1. cm. Coiled length of tube .............
466 1 cm. Internal volume .................. I 406 0 c.c. Diameter
(from volume) .......... 1.. I 032 cm. Entrant length (diameters)
........... 178 coiled; 188 5 total. Gauge length (diameters)
............ 190 1. Exit length (diameters) .............. 83
5.
The coiling naturally caused the section to become somewhat
oval, buLt an estimate, based on external measurements, showed that
the amount was unimportant. The largest diameter was approximately
5-6 per cent. greater than the smallest. Such a distortion,
however, is negligible in its effect upon the relative value of the
hydraulic mean depth, which is only 0 04 per ceit. less than that
of a circle of the same area. On the other hand, the coiling did
have considerable effect upon the cross sectional area, and the
coiled tube had an area which was roughly 4 per cent. less than
that of the pipe in its original straight state.
As stated earlier, some of the tests of this pipe, No, I, were
carried out wi.th oils having a greater viscosity than water. The
oil supply was maintained by gravity; and, as the available height
was only 10 feet, it became necessary to use oils with different
viscosities in order to cover the desired range of Reynolds number.
This procedure has the peculiar advantage that the pres- sure
gradients in the pipe can always, even with the widest range of
jReynolds number, be m.easured accurately with a simple manometer.
Additional com- plications are, of course, involved-the most
troublesome is the determnination of the viscosity of the oil.
simultaneously with the tests of the curved pipe-but
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Streamline F low through Curved Pipes. 653
the resulting gain, both in accuracy and in ease of control more
than comn- pensates for the disadvantages.
In its original state, the nmineral lubricating oil used had a
viscosity which was roughly 500 times that of water. With this
particular oil it was possible to explore a range of Reynolds
number from 0*45 downwards, buit from 0 45 upwards it was necessary
to use a fluid with a somewhat lower viscosity. Actually the
lubricating oil was diluted with lamp oil to reduce its viscosity,
and the range 0 -06 to 500 was explored by stages, each
necessitating a pro- gressively greater dilution. From 500 upwards,
water could be used.
.1 2 4 '-2 5 3 24 s 0c 7 21 * 2 4 .4 s 6 -7 5 6 ? 49 t0 1 2 'I
.iijj > _ __ _4_ _ __
-
654 C. M. White.
might have introduced errors of a systenmatic nature. A further
check against such errors is also provided by the test points with
the smallest flows. These lie accurately on the line F/pv2 8
/pvd.
The leading dimensions ofE pipe No. IIl were
Rladius of curvature ...................... 3 05 inetres. Total
length of pipe. 425 metres. Diameter (from flow when straigfht) .0
298 cm. Entrant length .......................... 650 diameters.
Gauge len-gth ........ 671 diameters. 'Exit length......... 106
diameters.
The test results from this pipe are particularly interesting in
connection with the change from laminar to turbulent moti-on. The
pipe was tested both in its original straight state, and also when
elastically bent to a curvature of 1/2050. Both sets of results are
shown in fig. 4. It will be seen that the
CUJIVFTJ PPiE N ni_ 1\ I
lL 215HhhI1 .Z2S3 s _ X _ - S t atAI M _ _ _ _ _ _ _ _ _ _ _ _ _
_ _ _ _ _ _ \ heru tNETH ~~GT C"
4_ v_ . E _i .>r '3 _ . _9 _ % _ ____ - ---------Nt **-t*
---- t +.S '3
. L \+ 4 t-L - b - _t-w- the --lwofr -- F1pV2- ----v FpV2 - -t
0
FIG.~~~~~~~~~~~ 4
increa,sed resistance due to curvature does not become appare:nt
unti,l pvd/p = 550, and that the increase at 2250 is only about
half the vertical displacemnent between the two laws of resistance
F/pr2 =8,i./pvd and F/pr2 =0 ~04 ( 4pvpd)*. One of the factors
which led to the selection of the particular curvature of this pipe
was the desire to know whether the value of the lower critical
velocity would
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Streamline Flow through Curved Pipes. 655
be altered appreciably by slight curvature of the pipe. Fig. 4
shows that there is a discontinuity at pvd/p. = 2250, and the form
of the rising curve leaves but little doubt that, in this pipe,
turbulent flow is established in the region 2250 to 3200. It would
be unwise, however, to draw any definite conclusion con-. cerning
the exact value of pvdl/ at which the rise takes place, for the
reason that the conditions of the test of this pipe departed
considerably from the ideal. The pipe was only 0 298 cm. in
diameter, and, as such a small pipe would be manufactured by
drawing down from a larger size, it is more than likely that its
interior was not of uniform diameter. Further, the holes for the
gauge connections were relatively large, about 0 15 cm. in
diameter. Yet another defect became apparent when the calculations
were completed. Sufficient damping had not been provided between
the manometer and the gauge holes. The effect of this is noticeable
in the two curves shown in fig. 4 anld marked A and B. The curve A
was obtained with a mercury and water manometer, while B was
obtained with a simple water manometer. In the latter case there
was greater surging of water to and fro between the manometer and
the pipe, and this would tend to cause an earlier rise of the
curve. In spite of these defects, the tests of this piipe do show
definitely that there is not any marked change in the lower
critical velocity due to a curvature ratio of 1/2050. At the
sametime, the effect of this curvature is by no means negligible in
its influence upon the resistance, since at pvd/,u _ 2250 the
curved pipe offers roughly 25 per cent. greater resistance.
In order to compare the results given in figs. 2, 3 and 4 with
those of Eustice and of Grindley and Gibson, it is necessary to
express all in the same manner. As originally published, Grindley
and Gibson's results were not reduced to a dimensionless form.
Fortunately the test readings were given in detail, and it has been
possible to recalculate the results and to express them in the con-
ventional manner-fig. 5. One small adjustment has been made to the
original data given by Grindley and Gibson. Their determination of
the diameter of the pipe gave the value 0 317 cm. If this value be
used then the points relating to flows less than pvd/, = 200 lie
below the laminar line for straight pipes. The most probable
explanation appears to be that actually the diameter of the pipe
was slightly more than 0* 317 cm. A matter of 1 per cent. in the
diameter is sufficient to account for the discrepancy. An
adjustment of this magnitude has been made, and, accordingly, fig.
5 is based on an assumed diameter of 0 * 322 cm. The heavy line
passing among the test points has been selected by inspection to
represent a fair mean of the results.
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656 CC. M. White.
Eustice tested a relatively short length of flexible hose
constructed of rubber reinforced with canvas. The total length of
the pipe was 97 8 cm. and the
*4 *5 6 7 *89 I
P 0 2 43 56
8 /0
21 i3 5 ._. - _T_ __ ___ -~,-----, .- CQILE? LEAD PIE 4 ) 89X -
- - - - - --- - - - v
AJB -6.L
! t
.2__ _____-
,~ i -l \ _ Q 1 _ __ -r- r6
E
-0r~ v _ s _
r_ _ _ 8
.7 J--w-t-- - - -
II~~~FG
X
X~ --1------- - - __
- 4 - - < t
-2 LiilIt _X _ _ 3 -_l. ^ -tr~~~~~---__>-- 1
presen t cma ison t is onl neesr to conidr oneset of tets
naely
considerble degee (seefig. 6) and rslte in a xecto in th are of
317
log pz'cZ/. FIG. 5.
diameter when straight 0h358 cm. This pipe was coiled to various
radii in order to study the progressive esect of curvatutre; but
for the purpose of the present comparison it is only necessary to
consider one set of tests, namely, that with the greatest
curvature. In this case the coil ha,d a mean dia.meter of 2s 59 cm.
The coiiang cautsed the pipe to become oval in cross section to a
considerabl.e degree (see fig. 6), and resulted in a reduction in
the area of 312 per cent. Eustice rather ingeniously compensated
for this effect by comparing the resistance of the pipe in its
coiled state with that; when compressed to an oval form betwveen
straight boards. He mnade the comparison when both the cross
sectional areas and the velocities were the same in the two cases.
This comparison, however, may be open to the objection that,
although the areas may be the same in the two cases, it does not
follow that the geometrical form of the cross section is also the
same. The method used here in comparing this oval pipe with the
other circular pipes requires some explanation. The form
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Streamline Flow through Curved Pipes. 657
of the cross section is assumed to be an ellipse with the ratio
of the axes- 2 94.* It is customary, when comparing the resistance
of non-circular pipes,
. __ > - - __ >_ . __ __ _ __ I_ -
\ . 'BSE}!LE -co L .o Pi sr/ __ __ \ _ _ * 4
N _t __ .TER_ -- - A.Z8 et cm.
~~~~~.rst\ 'ar. Nen XEDG
*8 .. j __ 1-_ 1AREA..
-t _ _ ~ 1 ..0 __L~. .-__ _I ~k... 4 T e. 0v
C~~~~~Rij TE/Ot
.~~~~~~~~~~~-~~~.
Z-0~~ 4 .8
.8~~~~~~~~~~~~~~~~~~~~~~~~~N
*2 8 2 *2 4. 6 8 3.0 * . -6 8
FJI. 6.
to make the comparison under conditions giving the same value of
the flow criterion 4pvmn/t. But this comparison is satisfactory
only in the case of turbulent flow. For streamline flow in a
circular pipe F/pv2 X 4pvm/p = 8 but in an elliptical pipe of the
form mentioned F/pv2 X 4pvm/t = 9 very closely, a discrepancy of
12, per cent. In order that the law of resistance may in both cases
be expressed as F/pv2 X flow criterion 8, it is necessary to define
the flow criterion, for the elliptical pipe, not as 4pvm/f but as 3
-56pvm/n,x. In comparing the oval pipe with round pipes, it is here
assumed that 3 56 pvm/[t is equivalent to the Reynolds number
pvd/l, for round pipes. This assumption may be open to criticism,
but it does at least provide a practical basis for comparison, and
one which is not without the virtue of simplicity. It may give rise
to a small discrepancy for turbulent flow, but the amount is not
likely to exceed 3 per cent. of FJpv2 for the particular section
under con sideration.
* This ratio is obtained by taking the perimeter to be the same
as that of the original unstrained pipe and the area to be 31 5 per
cent. less.
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-
658 C. M. White.
In fig. 6 two curves are seen, of which the upper relates to the
coiled state, while the lower shows the result obtained when the
pipe was squeezed between straight boards. It will be seen that the
lower curve lies above the theoretical line. This is probably due
to the fact that the pipe was not provided with an adequate
stilling length, although it might be due partly to a lack of
uniforrnity in the cross sectional area. It may be mentioned that,
while the curved pipe would be less influenced by the lack of
stilling length, yet, on the other hand, it would be somewhat more
sensitive to irregularity of cross sectional area. It is not
un-treasonable, therefore, to regard the curvature in this case as
causing dis- placements such'as that marked C in fig. 6.
The five sets of results shown in figs. 2 to 6 are miore easily
compared when they are replotted in a modified form. In fig. 7,
instead of the resistance
L1] c 4-? LL\-1-- L i I i T
A 20 30 5 p Jo 2-0 26 _5
FIG. 7. Fi'G. 8.
coefficient F/pV2, curves are given showing the increase of this
coefficient of resistance cauLsed by the curvature, plotted on the
Lsual base pvd/ t. The observed coefficient for thle curved pipe is
divided by the theoretical value of the coefficient for laminar
flow through a straight pipe in which the Reynolds number has the
same value, and the resultinlg quotient is then plotted logarith-
mically on a base of the flow expressed as a Reynolds numiber. The
ordinates of fig. 6 are equal, therefore, to the vertical
displacements such as C in figs. 2 to 6-an advantage of logarithmic
plotting.
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-
Streambine F?low through Curved Pipes. 659
The results are now in a form which enables them to be
considered in relation to Dean's theory. But, before doing so, it
is well to point out that the theory involves certain assumptions
which may or may not be in accordance with the actual conditions of
flow. In this respect, of course, it is like all other theories, in
that it demands experimental confirmation before it can be accepted
at all, anld a clear definition of the range over which it applies
before it can be used with confidence. Fig. 7 provides considerable
information on both these points. Dean showed mathematically that,
for a given pressure gradienit, the ratio of the mass flows through
two pipes of different curvatures is determined by the value of the
criterion pvd/t (d/D)k. It is more convenient to express this
result in an alternative form, by stating that the ratio C of the
resistance coefficient of a curved pipe to that of a straight pipe
(in which the flow number pvd/l, is the same) is a function of the
criterion pvd/,u (d/D)).
If C is actually a single valued function of pvd/t (d/D)It then,
owing to logarithmic plotting, the various curves of fig. 7 should
all be of identical form, and the influence of the different
curvature ratios should be represented by a horizontal displacement
of each curve in relation to its neighbours. This displacement
should be equal to half the difference of the logarithms of the
curvature ratios of any two pipes under consideration. Fig. 7 shows
that, on the whole, the experimental results do conform to these
requirements. The shape of those parts of the curves indicated by
full lines is approximately the salmne, and the spacing appears to
be approximately proportional to log (d/D)k. The spacing, however,
varies somewhat for different values of C, and reference to fig. 8
is necessary in order to obtain conclusive evidence. In fig. 8,
which shows the Reynolds number for a given value of C plotted log-
arithmically against the appropriate value of D/d, the uniform
slope of 2-1 of the lines, confirms Dean's deduction conclusively.
The only set of points which systematically disagree are those
relating to Eustice's pipe. The dis- crepancy was not tnexpected
however, and does not weaken the general conclusion. On the
contrary, bearinig in mind the different entry conditions of these
tests, the form of this curve ill fig. 7 is sufficiently like those
of the round pipes to lend support to the view that the function c
=f [pvcl . (d/D)11. determined by experiments with round pipes, may
be found to apply also to other cases of flow in curved paths. In
plotting the points marked Eustice in fig. 8, the curvature ratio
of this pipe was taken as 0 178/2. 59 or 1/14 6. It is to be
expected that a pipe having a wide cross section would show less
curvature effect than a circular pipe-and this is confirmed. The
oval pipe gives approximately the same resistance as a circular
pipe of curvature
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-
660 C. Al. -White.
-< ---, --* -=-f--=-r
--9 ---
-
Streamline Flow through Curved Pipes. 661
ratio = 1/32. The ratio Effective Curvature/Nominal Curvature
for the oval pipe is roughly the same therefore as the
Breadth/Width ratio of its cross section.
The results given in fig. 7 are replotted in fg. 9 in a more
general form. Here, the ordinates are identical with those of fig.
7, but Dean's criterion, pvdl/. (d/D)l, is used for the abscisse.
All the experiments-with the exception of those of Eustice, which
are omitted from this figure-conform satisfactorily to a single
curve for values of the criterion up to 50. Above this value the
various curves progressively leave the main curve, and the points
of successive departure appear to depend upon the curvature ratios
of the particular pipes concerned. It seems clear that these
points, at which the individual curves leave the main curve,
coincide with a change in the type of motion, and it is perhaps
permissible to suggest that the change is from the kind of double
helical streamline motion described by Dean, to something very
closely akin to the ordinary turbulent motion associated with flow
through straight pipes. Dean, in his theory of the manner in which
curvature causes an increase in the resistance, shows that it is
due, in effect, to an internal circulation in the plane of the
cross section of the pipe, superimposed upon the normal streamline
velocity distribution. Fig. 10, a reproduction of one of Dean's ex-
planatory diagrams, shows the circulation effect produced by the
pressure gradients i_(______ which arise owing to centrifugal
action. If complete slipping at the boundary were to occur, and as
a result the fluid were to move through the tube much as a rubber
FiG. 10. cord m:ight be drawn through it, then the whole effect
would cease. In a straight pipe, the velocity distribution curve
for turbulent flow is considerably flatter than that for streamline
flow. In the former case the maximum velocity is only about 1*25
times the mean velocity, as compared with twice the mean velocity
in the latter. The effective radial pressure gradients for
turbulent flow in a curved pipe may thus be expected to be
considerably less, at a given Reynolds number, than if the flow
were streamline. Since the pressure gradients are less, and since,
apart fro:m this, eddy motion itself would tend to restrict an
ordered circulation, it follows that the effect of curvature is
likely to become less marked. A closing up of the
VOL. CXXIIl.-A. 2 x
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-
662 C. M. White.
horizontal spacing of the various curves in fig. 7 is to be
expected, therefore, when the flow becomes ti bulent. This closing
up, which it will be seen is well defined, corresponds to the
dispersion in fig. 9. This marks the limit of the range over which
Dean's theory applies. Bearing in mind that the motion assLued in
deriving the theory is essentially streamline, it is reasonable to
suggest that the points at which the respective c-irves leave the
main curve, are those at which ttirbulent motion begins.
It may be said, therefore, that the point, at which the
individual curves leave the main curve in fig. 9, in providing a
rough indication of the upper limit of Dean's theory, also provide
an indication of the beginning of turbulence. The information is
little more than qualitative, but nevertheless it is sufficient to
show that this limit does not depend upon the value of the Reynolds
number alone-neither does it depend upon Dean's criterion. On the
contrary, it appears that turbulence begins--under the con'ditio is
of the tests-when the resistance coefficient F/pv2 decreases to
0-0045, more or less irrespective of the curvature, of the pipe.
Dean's theory applies only when the value of Flpv2 is greater than
0 0045.
It should be noted that, although the state of the fluid before
it entered the pipe may have differed somewhat in the various
tests, yet in every case there were irregularities in the passages
just before the curved pipes. These irregularities-a thermometer
pocket, stop cock, and sharp bend--would presumably tend to augment
any disturbance originally present in the fluid and would thus tend
to build up complete turbulence. It is not unreasonable, therefore,
to expect a considerable degree of turbulence in the fluid as it
was fed to the pipe, particualarly in the case of those tests in
which the Reynolds number exceeded 2000. In spite of this, it
appears that a Reynolds -number not less than 9000 is necessary in
order that the turbulence shall persist throughout the length of
the pipe when the c-urvature is 1/15. Even with the relatively
small curvature of 1150, streamline flow is maintained up to 6000,
a figure more than twice that whicb would be obtained were the pipe
straight. -There is an indication, therefore, that for large
disturbances, flow in curved pipes is more stable than flow in
straight pipes. This is directly opposed to the opinion sometimes
expressed that curvature tends to cause instability.
In conclusion, the foregoing results show that the loss of head
h in a length I of coiled pipe of diameter d, in which the mean
velocity of flow is v, may be .expressed by the equation
h C . 814pvd . 41/gLd,
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-
Streamline Flow throuqh Curved Pipes. 663
where [./p is the kinematic viscosity of the fluid, and wvhere
the numerical coefficient C depends upon both the flow and the
curvature.
The value. of C is given by the empirical equation
C-1 = i - [1 - (11*6jK)X]1IX,
where c = pvd/ . (d/D)1 and x = 0 45. This equation represents
the experimental results for values of K greater
than 11 6 and up to at least 2000. Below K = 11.6, C - 1.
Typical value. of C, taken from the curve in fig. 9, are given in
the table.
pvd/(l. (d/D)& Fd/8tv i pvd/l . (d/D)I. Fd/81tv
K. C. K. |.
0 1 60 1-309 11-6 1 100 1.503 13 1-014 200 1.897 17 -028 400
2*48 20 1-045 600 2*85 25 1 079 1000 3*61 40 1 189 2000 4.93
These values are probably true for a wide range of fluids and
pipes, but there is the definite limitation that they do not apply
under conditions in which the product C-. 8, 1pvd is less than 0
0045.
The author expresses his indebtedness to Mr. W. A. de Silva, a
research student, both for the entire experimental work in
connection with pipe No. II and for the calculations necessary to
reduce Grindley and Gibson's results to dimensionless form. He
would also like to thank Profs. Gilbert Cook and Alex. H. Jameson,
of King's College, for their support.
2x 2
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Article Contentsp. 645p. 646p. 647p. 648p. 649p. 650p. 651p.
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663
Issue Table of ContentsProceedings of the Royal Society of
London. Series A, Containing Papers of a Mathematical and Physical
Character, Vol. 123, No. 792 (Apr. 6, 1929), pp.
373-736+i-viiVolume Information [pp. 734-736]Discussion on the
Structure of Atomic Nuclei [pp. 373-390]The Total Reflexion of
Electric Waves at the Interface between Two Media [pp. 391-400]The
Measurement of Flame Temperatures [pp. 401-421]The Arc Spectrum of
Silicon [pp. 422-439]On the Vortex Theory of Screw Propellers [pp.
440-465]The Spectrum of H2. The Bands Analogous to the Parhelium
Line Spectrum. Part II [pp. 466-488]The Wave Equation in Five
Dimensions [pp. 489-493]The Structure of the Benzene Ring in C6
(CH3)6 [pp. 494-515]The Ionisation of Potassium Vapour [pp.
516-536]The Determination of Parameters in Crystal Structures by
means of Fourier Series [pp. 537-559]A New Band System of Carbon
Monoxide (3 1S 2 1P), with Remarks on the ngstrm Band System[pp.
560-574]A New Integrating Photometer for X-Ray Crystal Reflections,
etc. [pp. 575-602]The Adsorption of Hydrogen on the Surface of an
Electrodeless Discharge Tube [pp. 603-613]On the Design and Use of
a Double Camera for Photographing Artificial Disintegrations [pp.
613-629]The Absorption Band Spectrum of Chlorine [pp.
629-644]Streamline Flow through Curved Pipes [pp. 645-663]On the
Measurement of the Dielectric Constants of Liquids, with a
Determination of the Dielectric Constant of Benzene [pp.
664-685]The Molecular Dimensions of Organic Compounds. Part I.
General Considerations [pp. 686-691]The Molecular Dimensions of
Organic Compounds. Part II. The Viscosity of Vapours: Benzene,
Toluene and Cyclohexane [pp. 692-704]The Molecular Dimensions of
Organic Compounds. Part III. The Viscosity of Vapours: Thiophen and
-Methylthiophen, Pyridine and Thiazole [pp. 704-713]Quantum
Mechanics of Many-Electron Systems [pp. 714-733]Back Matter [pp.
i-vii]