Streamline Flow Through Curved Pipes

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: 07/05/2014 14:29

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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

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-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


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 experimental 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).


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


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.



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 approximately 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


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


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 resistance 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


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 conditions 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 pressure 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


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 progressively 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



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.


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

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considerble degee (seefig. 6) and rslte in a xecto in th are of 317

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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



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,

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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.


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 displacements 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.


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 discrepancy 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


660 C. Al. -White.

-< ---, --* -=-f--=-r

--9 ---


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


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,


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


Article Contentsp. 645p. 646p. 647p. 648p. 649p. 650p. 651p. 652p. 653p. 654p. 655p. 656p. 657p. 658p. 659p. 660p. 661p. 662p. 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]

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Streamline Flow Through Curved Pipes

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