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Volume 50, 1918
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Art. II.—The Resistance to the Flow of Water through Pipes.

[Read before the Technological Section of the Wellington Philosophical Society, 7th July, 1917; received by Editors, 31st December, 1917; issued separately, 24th May, 1928.]


In a previous contribution to this subject communicated to the Philosophical Society, and printed in the Transactions of the New Zealand Institute,* an attempt was made to determine the limits between which the resistance to the flow of water in a turbulent state is found to vary, first, for riveted steel pipes, and, secondly, for wood-stave pipes. This was done by plotting all the experimental determinations of loss of head which are on record and afterwards enveloping the observations as a whole by curves, the form of which was deduced from analogy with the ascertained law of resistance to flow through smooth pipes. In the present contribution an attempt is made to analyse the effect of different surfaces more in detail and to extend the study of the subject. The principle herein employed has been applied by the author to the observations upon the resistance to the flow of water in open channels, and the results communicated to the New Zealand Society of Civil Engineers.

It is well known that the flow of water or any fluid assumes two different modes, the one in which the flow is linear and known as streamline or viscous motion, and the other in which the flow is non-linear or sinuous, the flow being otherwise described as eddying or turbulent. The two terms “linear” and “sinuous” describe the two states very well, and are used herein in the sense defined. Between the two states there is an unstable region below which the flow is linear and above which it is sinuous.

In the linear stage the between relationship the elements affecting the resistance to motion is simple in character, and in consequence the nature of the relationship was discovered by experiment at an early date and subsequently rationalized, and is expressed as follows:—

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rs/v2 = a (v/vd) (1)

where s is the hydraulic gradient, r the hydraulic mean depth, d the diameter of the pipe, v the mean velocity, v the kinematic viscosity (i.e., the viscosity divided by the density of the fluid), and a a constant. Here the resistance is expressed as a loss of head per unit length of pipe, as is customary in engineering practice, whilst the customary notation has also been adopted—viz., r and s for the hydraulic mean depth and hydraulic gradient respectively.

In the sinuous or eddying stage, on the other hand, the relation between the elements of resistance is evidently complex, and as a

[Footnote] * E. Parry, Resistance to the Flow of Fluids through Pipes, Trans. N.Z. Inst., vol 48, pp. 481–89, 1916.

[Footnote] † E Parry, A Critical Discussion of the Subject of the Flow of Water in Pipes and Channels, with Special Reference to the Latter, Proc. N.Z. Soc. Civil Engineers, vol. 3, pp. 116–32, 1917.

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consequence the efforts of experimenters to discover the nature of the relationship has been fruitless, and, as little or nothing is known of the transformation of energy within fluids in sinuous motion, a precise mathematical solution was impossible. It may, however, be deduced from certain dynamical principles that the resistance is some function of (v/vd) provided that there is a proportionality between the dimensions of the eddies and of the cross-section of the pipe, leaving the form of the function to be determined by experiment.

The law of resistance, then, in its most general form, which applies to both states of motion, is expressed as follows:—

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rs/v2 = φ (v/vd) (2) where φ stands for “function of” and the other symbols have the same significance as in equation (1). As already explained, the relation between the quantities in the linear state is a simple one, the left-hand expression in equation (2) being a simple linear function of the right hand.

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As regards sinuous or turbulent flow, it was supposed at one time that the nature of the function was of the form rs/v2 = a (v vd)n (3) but it is now known that this form is defective, and that the range of observations upon which it was based was not wide enough to determine the true form; it was soon found that equation (3) did not fit the facts, and in consequence a modification of this was adopted in which v was treated as a constant, and independent indices given to v and d, yielding a formula of the form v = krxsy(4) where k, x are constants and r is the hydraulic mean depth numerically equal to d/4 for round pipe.

This formula is one of considerable flexibility, and of late the whole phenomenon of the flow of water in pipes has been analysed afresh and expressed in the form given in equation (4). Its adoption has not, however, contributed anything towards extending our knowledge of the subject, and it is much to be regretted that steps were not taken to extend the range of observations when equation (3) was found to be defective. This aspect of the question has been apparently overlooked.

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Such a series of observations extending over a wide range was recently conducted in the National Physical Laboratory by Stanton and Pannell* upon oil, air, and water in smooth brass pipes. The diameters of the pipes used varied from 0 142 in. to 5 in., and the mean speed from a fraction of a foot to 20 ft. per second. These combined with other observations upon the flow of water in smooth pipes when plotted with rs/v2 as ordinates and log. vd/v as abscissae were found to be sufficiently near to enable a curve to be drawn through the mean which was fairly representative of the whole, despite the fact that the condition of geometric similarity was not observed in respect to the surface of the

[Footnote] * T. E. Stanton and I. R. Pannell, Similarity of Motion in Relation to the Surface Friction of Fluids, Phil. Trans Roy. Soc., A, vol. 214, pp. 199–224, 1914.

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pipes. According to Professor Lees,* the mean curve can be expressed in the form rs/v2 = a (v/vd)n + b (5) the values, the coefficients, and the index being as follows:— a = 0.00801; b = 0.000028; n = 0.35 all the quantities being in foot-pound units.

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More recently Lander carried out an extensive series of experiments upon the flow of water and steam at speeds varying from 1.91 ft. per second to 11.55 ft. per second through ordinary commercial drawn-steel pipe of 0.423 in. diameter, and upon plotting the values of rs/v2 against log. vd/v he finds than an equation of the form (5) satisfies the relation between them. He, however, obtains different values of the coefficient and of the indices, the values being a = 0.0202; b = 0.0000622; n = 0.44 all values being in foot-pound units.

It is evident on contemplating the two sets of experiments that an equation of the form given in (5) correctly expresses the relation between the quantities near enough for all practical purposes, and it remains to be seen how far the principle is applicable to larger diameters and rougher surfaces, and it is the purpose of this paper to test its applicability to cast-iron, riveted steel, and wood-stave pipes of such sizes and characteristics as are in common use in the arts.

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Before proceeding further in the direction indicated it may be useful and interesting to compare the form of equation (5) with Chezy's formula, viz.:— v = crs (6) where c is a coefficient and r the hyraulic mean depth. It will be seen that c can be expressed in the form c = 1/√ a (v/vd)n + b (7) Comparing this with other well-known formulae for c, we have Prony's equation, viz.:— c = 1/√ a (1/v) + b (7) whilst Darcy and Bazm's formula may be expressed as follows:— c = 1/√a(1/d). + b Evidently the influence and value of v predominate in Prony's experiments, whilst the value of d predominated in Darcy's experiments; and

[Footnote] * C. H. Lees, On the Flow of Viscous Fluids through Smooth Circular Pipes, Proc. Roy. Soc., A, vol. 91, pp. 46–53, 1914.

[Footnote] † C. H. Lander, Surface Friction: Experiments with Steam and Water in Pipes, Proc. Roy. Soc., A, vol. 92, pp. 337–53, 1916.

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it does not seem to have occurred to any one to combine the two and thereby obtain an approximation to equation (7).

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Kutter's formula for c is too complicated for ready comparison, and, after all, what is required is not a formula for c, but a sufficient number of observations for each class of pipe to enable a curve to be drawn correlating rs/v2 to vd/v. The precise form of the equation expressing the relationship is really only of academic interest.

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Returning to equation (5), the results of the experiments on smooth pipe by Stanton and Pannell are plotted in figs. 1, 2, and 3, and indicated by the number 6, whilst the result of Lander's experiments on drawnsteel pipe is indicated by the number 10, the abscissae being values of log. vd/v and the ordinates values of rs/v2. In fig. 4 the same equations are plotted in terms of log. (rs/v2b) and log. vd/v.. Line 6 represents Stanton's experiments, and line 10 Lander's. These two lines converge at or near to a point 0, where the motion changes from linear to sinuous. Line 12 represents linear flow, and should be common to all pipes within limits. The convergence of these three lines indicates that the two pipes fulfil the condition as regards geometric similarity. The method of plotting adopted in fig. 4 affords a ready means of determining the characteristic of any description of surface, provided that the condition before mentioned is fulfilled, for it is only necessary to make one observation of the quantities involved and to join the point representing the observed value to the point 0 in order to determine the whole characteristic. There is one remarkable coincidence between Stanton's and Lander's results—viz., the ratio of a to b is the same in both; which suggests a possible relationship which would be most useful if it can be proved to have any dynamical significance, but no deduction can be made in the absence of such a proof.

In applying the principle involved in equation (2) to experiments upon large pipes we encounter several elements of uncertainty. One is that the temperature of the water has not, as a rule, been observed and recorded; but as the error involved in assuming a uniform temperature and applying it to all the experiments is considerably less than the error arising out of other disturbing factors, and probably less than the error of observation under the conditions prevailing during the experiments, the temperature error is of no great moment.

Another factor which affects the harmony of the results arises from the fact that large-diameter pipe lengths are shorter than small-diameter pipes, and that in consequence the joints are more frequent; and, as the joint is a disturbing element, a large pipe and a small pipe of the same material and surface—such, for instance, as cast iron—are not strictly comparable on account of the increase in the number of joints, and often also because of the different nature of the joint. There is also the possibility that in two experiments on pipes of the same size and material the joint of the one may be better made than that of the other, and greater care taken in aligning the pipes.

In the case of riveted steel pipes we have still other disturbing factors. The longitudinal joints may be lapped or butted. There may be one, two, or three longitudinal joints in the circumference The circumferential joints may be alternately in and out, or taper; in neither case is the diameter of the pipe uniform. In one case we have a larger diameter alternating with a smaller diameter by twice the thickness of metal, with

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the plate at the joint alternately facing and not facing the stream; in the other case we have the plate or section tapering from a large end to a small end by twice the thickness of the plate, whilst none of the joints face the stream. There is further the disturbance arising out of the different thickness of plate used, and in comparing two riveted pipes of different diameter it will be realized that they are not similar in all respects unless the thickness of plate bears some proportion to the diameter. The same remarks apply generally to spirally riveted pipe.

The principle involved demands that for the same values of vd at the same temperature the same value of rs/v2 shall be obtained; but unless the frequency of jointing and the nature of the joints is the same, and unless there is a proportionality between diameter, thickness of plate, and size of rivet, it cannot be expected that the principle can be strictly applied, or that it can be proved to be applicable at all unless the characteristics mentioned are taken into account.

In spite of the vast array of experiments upon pipes of different kinds, it will be found that few of them are of much assistance in the present investigation. The characteristics of the pipe are not always precisely defined. The experiments on any one set are usually not numerous enough, or, if numerous, do not cover a sufficient range. Those experiments that are at all suitable have been used in the present paper, and a study of the diagrams will afford an indication as to the scope which should be given to further experiments.

In addition to the disturbing factors arising out of the nature of the surface, and frequency and nature of the joints, the thickness of plate in riveted pipe, and riveting, there is evidently another disturbing element arising out of the elastic compression of the water and from the acceleration and retardation of the flow. Most of the available observations on large pipe have been obtained under working conditions, and subject to disturbances arising out of change in velocity of flow due to the operation of valves and governors. When a change in velocity of flow is made, a wave of alternate compression and expansion is set up which takes some time to die down, especially if the pipe is a long one, and it is quite possible to obtain widely conflicting results on the same pipe and for the same average flow, due to the operation of the various impulses that may be set up. Another possible source of irregularity is the occlusion of air in larger or smaller quantities due to fluctuations of pressure. This might affect the flow considerably at a given head, whilst the proximity of the gauge to a bend or to a discharge-opening has been found to vitiate the results. That some disturbances of the kind mentioned are at work will be quite evident on contemplating the graphs showing the results, to which attention will now be drawn.

Cast-iron Pipe.

A very complete list of experiments on loss of head in cast-iron pipe will be found in Barnes's work, Hydraulic Flow Reviewed, including some particularly careful determinations by the said author himself, which fulfil all the requirements. The examples selected are taken from the publication mentioned. Four experiments by Darcy on clean, new, uncoated cast-iron pipes are represented by a, b, c, and k in fig. 1, the diameters varying from 0.2678 ft. to 1.6404 ft. The readings are erratic, and no conclusion can be drawn from them further than that the trend of the observations generally follows the curve for drawn-steel pipes. The remaining experiments are upon asphalted cast-iron pipes, either

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Fig.1. Transactions.

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Fig. 2.

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newly cleaned or new. The diameters vary from 3.333 ft. to 5.0938 ft. by four different observers and six sets of observations. All the readings are remarkably close, and form a most valuable groundwork for further investigation. There are in all forty-six observations, all within the limit of observational errors, which could be represented by a single curve. All that is required in regard to this class of surface is a number of observations between the values of log. vd/v = 6 and log. vd/v = 7. Even as they stand a curve could be drawn with a fair amount of probability as to its correctness, as the observations follow the curve for brass tube; but, as the determination of the function is essentially an experimental one, the completion of the curve should be left to experiment.

Wood-stave Pipe.

Among the available experiments on wood-stave pipe, the most complete are those by Moritz.* Two classes of pipe are used—viz., jointed and continuous. The frequency of joints in the former case is, however, not specified. The observations on 18 in., 14 in., 12 in., 8 in. jointed pipe, and on a 55 ¾ in. by Moritz, and a 31 in. continuous by Moore, are plotted in fig. 2. It will be seen that in spite of the care exercised the results obtained on some of the pipes are somewhat erratic, due, no doubt, to the effect of impulses travelling through the water. The results as a whole are not consistent, and they do not lie near enough together to enable them to be represented to a single line, as the underlying principle demands. Nevertheless, they do not disprove the applicability of the principle, as the results are not consistent, whilst the difference between the observations on the same pipe are greater than the differences between the different pipes.

Riveted Steel Pipe.

Of the numerous experiments on riveted steel pipe, but two or three are suitable for the purpose of this paper. As a rule, the range is short and the readings erratic, whilst the particulars of the pipe are not complete. One of the most complete and extensive sets of observations is that made by Marx, Wing, and Hoskins upon a pipe 6 04 ft. in diameter, the circular joints being butted, with a strap on the outside. The longitudinal joints are also butted, with a strap both inside and out. The length of pipe was 4,427 ft., with fourteen joints, and contained thirteen bends of 30 ft. radius and one of 40 ft. radius. The temperature of water is also recorded. The results are plotted in fig. 3 and marked a. On the same diagram are plotted experimental values by Herschell on a 48 in. pipe marked b and a 36 in. pipe marked c. In each case the plates are ¼ in. thick and asphalted, built with alternate large and small cross-sections. All three sets of results are erratic, giving widely different values of 1/c2 for the same value of vd/v, and the readings on the same pipe differ more than the difference between the pipes, so that no conclusion can be drawn as to the complete applicability of the principle involved. All that can be

[Footnote] * E. A. Moritz, Experiments on the Flow of Water in Wood stave Pipes, Trans. Am. Soc. Civ. Eng, vol. 74, pp. 411–51, 1911.

[Footnote] † C. D. Marx, C. B. Wing, and L. M. Hoskins, Experiments on the Flow of Water in the Six-foot Steel and Wood Pipe Line of the Pioneer Electric Power Company at Ogden, Utah, Second Series, Trans. Am. Soc. Civ. Eng., vol. 44, pp. 34–54, 1900.

[Footnote] ‡ One Hundred and Fifteen Experiments on the Carrying-capacity of Large Riveted Metal Conduits, John Wiley and Sons, N.Y.

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Fig. 3.

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Fig. 4.

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gathered from contemplating them is that their general trend indicates that the law of resistance can be expressed in the form of equation (5). The observations are, however, not consistent enough among themselves, and, if they were, they do not cover a sufficiently wide range to enable a curve expressing the relation between rs/v2 and vd/v to be drawn. There is evidently some disturbing factor at work which seems to have a greater effect at low values of the mean velocity and low friction heads.


Comparing the results as a whole as plotted in figs. 1, 2, and 3, it may be said that their general trend is such as to conform with the law expressed in equation (5), and that they do not disprove the wide application of the principle to comprehend both large and small diameters, provided that the surface characteristics are similar.

As regards cast iron with clean, asphalted surfaces, the results of the more recent experiments are remarkably consistent and afford strong evidence in support of the theory, and it only requires a few more experiments in the proper region of exploration in order to enable a curve to be drawn for this class of surface.

As regards wood-stave pipe, the available results are not consistent, and new observations are required throughout the range.

As regards riveted steel pipes, none of the existing data are of much assistance, because of the wide variations between the readings. It is evident that in all the experiments some disturbing factors were operating in such a way as to vitiate the results, these making their influence felt more at low than at high velocities. More experiments extending over a wider range are required.


The result of this investigation is not very conclusive. A beginning is, however, made in the direction of applying a principle which has been found to be applicable throughout a wide range of values of vd, and for widely different fluids, such as water, air, and steam, and extending it to large pipes in commercial use; and before further progress can be made more experiments are required on pipes of different diameters and different surface characteristics, selected with a view to extending the range of observations already obtained.

Whether or not this theory is applicable under all conditions, there is considerable advantage to be derived from plotting the results of observations against vd/v, as by this means one is able to exercise a far greater degree of judgment in selecting a probable value of c than by studying all the literature on the subject which exists, and the method is to be recommended on this account.

Readers are referred to a previous paper,* printed in the Transactions of the New Zealand Institute, for a diagram representing the coefficient of viscosity and the coefficient of kinematic viscosity of water at different temperatures, and also a diagram showing the relation between the values of log. vd/v and vd for water at temperatures 0, 10, 20, and 30 degrees centigrade, the use of which will facilitate the manipulation of the diagrams presented in this paper.

[Footnote] * E. Parry, Resistance to the Flow of Fluids through Pipes, Trans. N.Z. Inst., vol. 48, pp 487–88, 1916.