Some Aspects of the Shape 

of River Meanders

GEOLOGICAL SURVEY PROFESSIONAL PAPER 282-E
Some Aspects of the Shape 

of River Meanders

By RALPH A. BAGNOLD

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 282-E

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

Reprinted 1962

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.C. - Price 20 cents (paper cover)
CONTENTS

Page 

Abstract------------------------------------------- 135

Energy dissipation in a curved channel________________ 135

Effect of breakaway and eddy formation_____________ 136

Conditions in a curved open channel_______-_-____-_-_ 139

Application to successive bends in constant channel

length------------------------------------------- 141

Comparison with experiment in open channels.--------- 141

Relation to river meanders_____-______-_______-__-_-_ 143

References._ __.._____-_-_-_____-_--_--.___--_-_-_-- 144

ILLUSTRATIONS

Page 

FIGURE 81. Transverse flow in a pipe bend.______________________________________________________________________ 135

82. Shearing flow in a pipe bend as alternative to transverse flow._____.___________-_-_-__-_____--_____-__--- 136

83. Theoretical relation of total resistance to curvature criteria in pipe bends._________________________________ 136

84. Effects of asymmetry of flow distribution in pipe bends of varying curvature.______________________________ 137

85. Theoretical resistance relation in pipe bends as modified by flow asymmetry and breakaway.________________ 138

86. Experimental relations in 90° pipe bends-_-__-_________-_____-_-____________---_--______--------_----- 140

87. Experimental relation between resistance and curvature criterion in open sinuous channels.__________________ 142

m
IV CONTENTS

	SYMBOLS

A cross-sectional area of flow

a a numerical coefficient

6 channel width

Cs a coefficient of wall friction

c a coefficient related to head loss

d diameter of pipe

e overall resistance coefficient

F force

F Froude number

g acceleration of gravity

H a drawdown in head in a pipe

j a numerical coefficient

k a numerical coefficient

n a numerical coefficient

rm mean radius of curvature of channel

R mean radius of curvature of pipe

s slope of water surface

u velocity

v tangential velocity

T distributed stress

p mass density of water

8 angle subtended by a curve, being the angle between radii at beginning and end of curve
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

SOME ASPECTS OF THE SHAPE OF RIVER MEANDERS

By RALPH A. BAGNOLD

ABSTRACT

From consideration of the probable nature of flow resistance 

in curved channels, a simple dynamical model is proposed to 

relate resistance to a criterion of bend curvature applicable both 

to closed pipes and to open channels. The theory indicates that 

resistance should fall to a minimum when the radius of channel 

curvature bears a certain critical ratio to the channel width; and 

this critical ratio should have approximately the same value for 

both closed and open channels, irrespectively of scale or of 

boundary roughness.

Resistances of pipe bends are known to fall to a sharply defined 

minimum when the curvature ratio, mean radius to diameter, is 

between 2 and 3. Recently measured flow resistances of sinuous 

open channels disclose that the same minimum occurs at approximately the same curvature ratio.

The theory therefore appears to go some way in explaining 

the mechanism tending to restrict the bends of rivers of all sizes 

to a curvature ratio between 2 and 3.

ENERGY DISSIPATION IN A CURVED CHANNEL

When water flows through a curved circular pipe of 

diameter d and mean radius of curvature R a transverse 

flow is created (fig. 81). Such a transverse flow is 

inevitable so long as the pipe walls exert a frictional 

drag on the tangential flow, for the radial acceleration 

gives rise to a calculable excess of fluid pressure on the 

outside of the bend and a corresponding pressure deficit 

in the inside. Throughout the main body of the flow 

the radial pressure gradient is balanced by the gradient 

of the centrifugal pressure. But it must remain unbalanced over the periphery, owing to the effect of 

boundary friction in reducing the flow velocity and 

therefore the peripheral pressure gradient in the radial 

direction. Hence water must flow radially inwards over 

the periphery, down the radial pressure gradient. And 

its place is taken by an outward flow across the centerline of the cross section.

The effect of the transverse flow is to reduce the 

internal shear due to the distortion of the flow round 

the bend. For if the transverse flow were prevented, 

say by the insertion of frictionless fins projecting inwards towards the central axis of the pipe, the maintenance of unaccelerated flow would require a uniform 

shear between the outside water and the inside water

FIGUKE 81.—Idealized diagram of transverse flow in a pipe bend. R Is mean 

radius of pipe curvature, and d is diameter of pipe.

as in figure 82. The rate of shear would be

R+f

where v is the tangential velocity, R is the radius of 

curvature, and d is the pipe diameter.

The effect of the transverse flow may be compared to 

the effect of the lay in a rope which makes it flexible by 

greatly reducing the internal shear occasioned by 

bending it.

If the path of a peripheral water particle is regarded 

as a helix whose projected cross section is semicircular, 

the internal distortion over any given central angle

135
136 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

FIGUEE 82.— Idealized diagram of shearing motion between water filaments in a 

pipe bend if transverse flow were prevented.

subtended by the arc of the channel bend may be 

measured as the difference in path length between that 

of the helix and that of its core arc. For any given 

ratio R/d this difference has a minimum value for a 

certain pitch of the helix. That is, a certain optimum 

ratio exists between the transverse and the tangential 

velocities.

Let it now be assumed, on the general principle of 

least effort, that the transverse flow ultimately attains 

this optimum velocity ratio. An angular acceleration 

is required to create the transverse flow, and this may 

be supposed to occur within some finite initial length 

of the channel arc. So, let it also be assumed that the 

arc of the curved channel exceeds this initial length, as 

defined by a certain minimum subtended angle 00 .

Then if we neglect, as a first approximation, any 

residual energy dissipation due to the small continuing 

internal distortion after the steady state has obtained, 

the resistance to flow through the whole channel arc 

is measurable by the sum of: (a) the rate of energy 

expenditure required to create the transverse flow 

(assumed dissipated beyond the end of the channel 

bend); and (b) the rate of energy dissipation by wall 

friction.

The total resistance to flow, due to the channel bend, 

is thus the sum of two forces: (a) force Fa required to 

overcome the inertia-resistance to the creation of the 

transverse flow, and (b) a force F* required to overcome 

wall friction.

Fa should be proportional to AptfdfR where A is the 

cross-sectional area of the flow=7rd2/4. Fb is equal to 

TsirdR6=Cspv?4:AI&R6 where Cs=Tspv2 , the coefficient of 

wall friction.

The total resisting force F=Fa -\-Fb should therefore 

be

where a is some unknown constant.

The overall resistance coefficient e should therefore be 

e=od/R+4(7g0R/d (1)

Comparing channel bends which subtend any constant central angle 0 exceeding 00 , equation (1) can be 

written in the general form

where n2 is a constant (assuming as an approximation 

that Cs is constant, depending only on the wall roughness).

Equation (2) contains only two independent variables 

R/d arid n, and &' has a minimum value 2n at R/d = 1/71.

The family of curves obtained by giving n a range of 

arbitrary values is shown in figure 83. It should be 

noted that the geometry (fig. 84D) imposes an absolute 

cutoff at R/d=0.5, for the inner boundary then has 

a zero radius of curvature and becomes an abrupt angle. 

Hence the curves of figure 83 for R/d<C0.5 are 

imaginary only.

EFFECT OF BREAKAWAY AND EDDY FORMATION

Further, as R/d is reduced towards this limit the 

tacit assumption that the flow conforms to its solid

3 4 5 6 7 

CURVATURE CRITERION, R/d

FIGURE 83.—Theoretical relation between overall resistance coefficient «' and curvature criterion, radius of curvature divided by pipe diameter, for different arbitrary 

wall resistances.
SOME ASPECTS OF THE SHAPE OF THE RIVER MEANDERS 137

R/d=3.0

R/d=2 R/d=0.5

FIGURE 84.—Diagrammatic relation between local radius of curvature of water filaments and radius of curvature of pipe bend. 

A. Curvature without appieciable asymmetry of the flow.

B. Flow asymmetry creates a relatively stagnant but still stable zone, and a decrease in the effective value of R/d. 

C. Breakaway occurs, at which the stagnant i,one becomes unstable and dissipates energy in eddying. 

D. When the bend ultimately forms a right angle, the unstable zone is large and very much pronounced.

boundary must cease to be true at some critical value 

of R/d considerably greater than 0.5. For a stage must 

be reached at which the flow along the inner boundary 

becomes unstable and breaks away from the boundary, 

leaving an intervening space occupied by a zone of 

unstable and confused fluid motion, figures 84(7 and D.

That such a stage must be reached is a matter of 

common experience. A real fluid will not flow smoothly 

round a sharp projecting curve. But it is not possible 

to predict quantitatively just what that stage will be 

in the case of a real frictional fluid. Nor is it possible 

to predict what effect the change in the flow condition 

will have on the overall resistance to flow.

The factors most likely to determine the breakaway 

stage are the ratio R/d, the degree of turbulence in the 

entry flow, and the velocity distribution over the entry 

cross section. Under comparable conditions as regards

the two last factors, the breakaway stage is likely to be 

determined only by the ratio R/d.

But there are no grounds for assuming that the critical value of R/d at which breakaway occurs will coincide with the critical value at which the resistance 

becomes a minimum by the considerations leading to 

equation (2). So, it is to be expected that breakaway 

may occur at a larger or a smaller value of R/d.

The study of the effects of breakaway has been confined largely to conditions of diverging flow. As the 

divergence is increased, the flow is found to become 

suddenly unstable in the neighborhood of the diverging 

boundaries, with a consequent sudden increase in the 

rate of energy dissipation.

But in the case of a bend in a channel of uniform 

width there is no initial divergence anywhere. As the 

radius of bend curvature is reduced, the velocity distri-
138 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

bution becomes progressively more asymmetrical. 1 The 

flow tends to recede away from the inside region of 

greatest curvature and towards the outer boundary. 

Consequently, in the region near the inner boundary 

the shear rate is reduced, and with it both the shear 

stress in the fluid and the pressure gradient in the 

direction of flow.

A final abrupt change occurs when the pressure gradient along the inner boundary is so far reduced as to 

allow a backward leakage to take place. Flow in 

the inside region now becomes unstable. Local eddying results, accompanied by increased energy dissipation and increased flow resistance associated with 

breakaway.

But if we confine attention to conditions that exist 

just prior to this breakdown of the inside flow into 

large eddies, we have: (a) A reduction in boundary 

resistance at the inner side. This is, however, probably 

compensated by a corresponding increase at the outer 

side where the velocity gradient is increased, (b) A 

restriction of the flow proper to the outer side of the 

channel because of inertia effects, while as yet the 

inside flow is still stable and no dissipatory eddying 

has set in.

As the critical value of the ratio R/d of channel 

boundary curvature is closely approached, and the 

asymmetry of the velocity distribution increases, the 

still stable and nondissipatory body of relatively unsheared fluid on the inner side may be expected to 

increase in width, thereby narrowing the effective 

width of the flow proper from d to d', and increasing 2 

its effective radius from R to R', as sketched in figure 

845.

Consistent with this notion we have therefore immediately prior to the breakdown of the inside flow a 

stage in which frictional dissipation has not appreciably 

increased, but in which the flow proper has an effective 

curvature ratio R'/d' increasingly larger than the 

channel value R/d.

As a result, we should expect the value of the first 

term of equation (2), which represents the inertia re-

> The experimental evidence is somewhat confused as regards the changes in the 

velocity distribution which takes place within a pipe bend. There is general agreement that the distiibution within the bend is principally asymmetrical, the effective flow being confined to a proportion only of the channel cross section. But 

whether the zone of high velocity is found to hug the outer or the inner wall of the 

bend at any particular cross section of it appears to depend on a combination of 

factois, such as velocity distribution at entry, angular distance from entry, and the 

curvature ratio R/d.

The reasoning which follows is, however, very largely independent of whether the 

effective flow hugs the outer or the inner boundary, being based only on the generally 

accepted fact that the effective flow is constricted to occupy a less width than that of 

the channel. Hence the words "outer" and "inner" may be interchanged without 

affecting the general sense.

* Interchanging if necessary the words "inner" and "outer," the effective radius 

R'is decreased. But the change in radius is small compared to that in width. Hence 

the effect of a decrease in width, from either the inner or the outer boundary, from d 

to d' is always an increase in the effective curvature ratio from R/d to R'/d'. And 

the changes are approximately proportional, the error becoming mote appreciable as 

R/d approaches its limit at 0.5.

34567 

CURVATURE CRITERION, R/d

FIGURE 85.—Theoretical relation between total pipe resistance and curvature criteria 

(equations 2 and 3) modified on the assumption that the effective values d'/R' of 

the ratio d/R becomes reduced to half the channel value d/R just before the breakaway in stability occurs.

sistance, to decrease from d/R to d'/R', while the second 

term representing the boundary friction, to remain more 

or less unchanged at R/d. On this view the overall 

resistance represented by &' in equation (2) should begin 

to depart from the values indicated by figure 83 at 

some value of R/d not far exceeding the critical value 

at which the breakdown of the inner fluid zone occurs. 

And the overall resistance should reach a local minimum 

value at the point of breakdown, followed when the 

breakdown has actually occurred by a rapid and progressive rise due to the well-known throttling effect of 

an unstable flow zone.

In figure 85 the curves of equation (2) as shown in 

figure 83 have been modified on the assumption that as 

the abscissa value, channel ratio, R/d is reduced, (a) the 

inner zone becomes unstable and breaks down at R/d = 

2; (b) as the abscissa ratio is reduced towards 2 the 

effective flow ratio R'/d' in the first term of equation 

(2) increases to a maximum value of 4 attained at the 

point of breakdown, while R'/d' in the second term 

remains unchanged at R/d; (c) for still smaller abscissa 

ratios (actual channel dimensions) the overall resistance 

increases rapidly.
SOME ASPECTS OF THE SHAPE OF THE RIVER MEANDERS 139

All the curves show a minimum resistance at the 

assumed breakdown ratio R/d=2. But those for small 

values of n (smooth channels) also retain the original 

minimum at the larger value of R/d.

Now compare figure 85 with figure 86, which reproduces the summary diagram given by Hofmann (1929, p. 

45) of the experimental data on the resistance of 90° 

pipe bends of varying roughness.

The experimental curves appear inconsistent among 

themselves, presumably because of differences in the 

experimental conditions, most probably in the conditions of entry flow.3

In spite of the apparent mutual inconsistencies 

disclosed in figure 86, the correspondence in general 

pattern between figure 86 and figure 85 is remarkable 

in view of the gross simplifying assumptions introduced 

into the reasoning on which figure 85 is based.

In particular, all the experimental curves show either 

a sharp minimum or else an abrupt upturn in the value 

of the resistance coefficient to occur within the same 

narrow range from 2 to 3 in the ratio R/d, irrespective 

of large variations in the boundary roughness and irrespective of the variations in other experimental 

conditions which are evident from the disparity in the 

shapes of the different curves.

It would seem difficult to account for this nearly 

constant critical value of R/d and for the pronounced 

changes in the resistance value associated with it by 

any other explanation than that of the phenomenon of 

breakaway.

Whatever the explanation offered, the experimental 

evidence on the resistance of closed 90° pipe bends 

indicates that with the possible exception of very 

smooth bends, the resistance undergoes a decrease to a 

pronounced minimum value when R/d is between 2 

and 3.

If the foregoing explanation is correct, this marked 

decrease in resistance should occur within the same 

range of R/d whatever the subtended angle, provided 

it exceeds some minimum angle, and whatever the 

boundary roughness, provided the roughness of the 

channel upstream of the bend is of the same order.

CONDITIONS IN A CURVED OPEN CHANNEL

The conditions of flow round a bend in an open 

channel can, as a first approximation, be regarded as 

those of the flow in the lower half of a closed pipe 

bend. The single transverse flow can be regarded as 

that of one only of the two equal and opposite circulations in the case of a closed pipe.

3 For the results of subsequent work, none of which discloses any radical disagreement with figure 86, see Yarnell and Woodward, 1936; Robertson, 1944; Mockmore, 

1944.

Differences exist, however, which may or may not 

have an influence on the occurrence and effects of 

breakaway. The pattern of the internal turbulence 

cannot be the same because the free surface of the open 

channel flow prevents any internal fluid motion taking 

place through it in the vertical direction. Again, in 

any natural open channel the cross section is far from 

semicircular, being greatly flattened. And, whereas, 

experiment shows that the transverse peripheral flow 

extends over the whole of each semiperiphery from the 

outside to the inside of a circular pipe, in the same 

type of experiment on flow in a shallow open channel 

(Leopold and others, 1960) the transverse peripheral 

flow inwards did not extend much beyond the centerline. Towards the inner bank the transverse motion 

appears to become confused by merging in a random 

way with the main flow. This suggests that the 

degree of turbulence increases towards the inner bank 

in a way it does not do towards the inner wall of a 

circular pipe bend.

Now experiments on the conditions under which the 

flow breaks away from the boundary, in the case of 

rectilinear diverging flow, show that the occurrence of 

breakaway can be appreciably delayed by increasing 

the degree of turbulence near the boundary. This has 

the effect, by increasing the shear stress in the fluid 

in the downstream direction, of tearing away any 

incipient eddy as it begins to form. This follows 

from the fact that the greater the downstream shear, 

the greater the stability of the flow against the effects 

of a local upstream pressure gradient, or a local reversed 

gravity slope in the open channel case. That is, in 

terms of the inverse ration b/rm used in a previous 

paper (Leopold and others, 1960) on the results of 

relevant open channel experiments, the critical value 

of b/rm might be nearer 0.5 than 0.33, where b is the 

channel width and rm the mean radius of the channel 

centerline.

On this view it would not be unexpected that the 

breakaway stage should occur in a shallow open 

channel at a rather smaller value of R/d than in a 

circular pipe. The critical value might well be nearer 

2 than 3.

Apart from this possible change, no clear reason is 

apparent for any radical difference in the shape of the 

curves of resistance coefficient against R/d when the 

closed pipe bends are exchanged for open channel 

bends which subtend the same central angle. But the 

experimental curves are likely to have a like diversity 

among themselves, as is apparent in Hermann's curves 

for the closed bends, on account of differences in 

experimental entry conditions. In particular, some 

special peculiarities are to be expected in the case of 

a succession of alternately reversed bends, because the
140 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0 8 10 12 14 16 18 20

CURVATURE CRITERION, R/d 

FIGURE 86.—Experimental relation of resistance as a function of curvature criterion for 90° bends in pipes (after Hofmann, 1929).
SOME ASPECTS OF THE SHAPE OF THE RIVER MEANDERS 141

entry flow into each bend will be asymmetrical about 

the centerline of the channel.

APPLICATION TO SUCCESSIVE BENDS IN CONSTANT 

CHANNEL LENGTH

It must be remembered that the existence of the 

resistance minimum shown in figure 83 arises solely 

on account of the definition of the bend as subtending 

a constant central angle. The increase of resistance 

with increasing R/d towards the right of the figure is 

due to the increase in frictional resistance as the wall 

area increases with R.

The conditions become very different in the case of a 

succession of bends contained within a constant length 

of channel. The second term in equation (1) then becomes constant; and the equation takes the simple 

form

(3)

where es is constant and is measurable by the resistance of a straight channel of identical cross section 

and roughness. The curve of e against R/d is now 

that of a rectangular hyperbola.4 And if the theory 

is sound, an experimental plot of e against R/d should 

conform to this curve for all values of R/d except 

in the immediate neighborhood of the critical value 

at which breakaway occurs.

But by inference from the results of the pipe experiments, we should expect an abrupt departure 

from this curve in the order of R/d ==2 to 3, in the form 

of a marked fall in resistance that is due to conditions 

immediately preceding the occurrence of breakaway.

COMPARISON WITH EXPERIMENT IN OPEN CHANNELS

In the results of laboratory experiments on flow 

resistances in open sinuous channels (Leopold and 

others, 1960) the flow resistances derived from measurements made during three of the many runs appeared 

anomalous. The data respecting two of these three 

were omitted from the analyses in that paper on 

grounds of relatively inadequate coverage by individual 

measurements. The data were, however, published 

in the appendix of that paper (p. 134), and those data 

are included in figure 87 of the present paper. All 

three results previously thought to be anomalous 

now appear in the light of the present discussion to be 

wholly consistent.

* Eobertson (1944) having plotted In his figure 18 the measured resistance coefficients of bends in both pipes and rectangular open channels against R/d, suggested 

this hyperbolic relationship on empirical grounds. Unfortunately his experimental 

range of R/d was confined to small values. So the anomalous kink disclosed by the 

present figure 87 was not apparent.

Figure 87 shows all the resistance valueso btained 

(appendix Leopold and others, 1960) without omission, 

plotted 5 against the channel curvature ratio 

r»/6(=R/d).

The three previously anomalous resistance values are 

those lying together at a value of rm/b in the order of 1.75. 

By inference from the pipe bend results, the low resistances here indicate the special conditions obtaining 

when breakaway is closely approached. The circles in 

figure 87 represent the results of runs at full depth, and 

the crosses, the results at reduced depth. The broken 

curve is that of the hyperbola

e=0.0624d/R + 0.0026 (4)

the numerical parameters having been chosen to give 

the best fit.

The only discrepancy from the theory is that in order 

to make the hyperbola given by equation (3) fit the 

plotted experimental curve, the asymptote es must be 

given the value 0.0026, whereas the values of es as 

measured for the straight channel were 0.0165 for full 

flow depth and 0.0160 for reduced flow depth. These 

measured values are in fact very close to those obtained 

in the experimental curved channel of least curvature 

(rm/b 4.55 and 5.25), the channel curvature being then 

still very appreciable.

It appears, on the face of it, that channels of less 

curvature (rm/&>5) offer no greater resistance to flow 

than a straight channel. This is just possible on the 

assumption that no transverse flow is set up, the internal 

shear involved in simple tangential flow being permitted 

without any increase of resistance by some change 

towards organization in the otherwise random internal 

turbulent motion.6

It is of interest that in figure 87 the plotted points 

marked by a cross (reduced depth) give by themselves

8 In this plot the ordinate is the overall square-law coefficient 

T —^ (which is equal to s/F2 because in the experiment the

hydraulic mean depth was kept constant) given by the constant 

slope of the lower portion of each of the graphs, figures 70 and 

71, of the above-mentioned paper. For comparison of the 

numerical values of the ordinates of the present figures 86 and 87, 

it should be noted that the functional definitions differ by a

factor of 2. The coefficient e of figure 87 can be written —=j=-=j-»

h being the hydraulic mean depth, whereas the coefficient c

AH of Hofmann's pipe bend diagram, figure 86, is defined as c= 2 . •

«The above discrepancy appears consistent with the notion that the resistance of 

a straight channel may, for some purposes, be usefully regarded as comprised of two 

separate elements: a skin element proper due to energy dissipation arising directly 

from and proceeding very close to the boundary and an indirect body element proceeding internally throughout the general body of the flow. On this view the hyperbola of equation (4) shown by the broken curve of figure 87 would be expected to be 

asymptotic not to the whole straight channel resistance but to the lesser skin resistance whose contribution to e is only 0.0026.
142 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

09

08

07

cel-o

1

EXPLANATION 

o

Full depth

X 

Reduced depth

'!*u 06

g

& 

$ 05

CO

8

QC 

O

03

x\

Hyperbola e =0.06244+ 0.0026

02

01

sight channel

10

CURVATURE CRITERION, •§- = y2-

.—Experimental relation between resistance and curvature criterion for sinuous open channels of constant cross section. Data from Leopold and others (1960).
SOME ASPECTS OF THE SHAPE OF THE RIVER MEANDERS 143

a curve which is systematically displaced downwards 

from the curve given by the full depth plot marked by 

circles. And the ratio of the reduction in e is 1 to 1.03 

approximately between rm/b 3 to 5. The corresponding 

ratio of the two straight-channel values of eg was almost 

identical with this, being 1 to 1.04.

It is also of interest to note that the magnitude of the 

reduction in the resistance coefficient e at R/d=1.75 in 

figure 87 is consistent with the view that this reduction 

is due to the increase in the effective curvature ratio 

R'/d' in equation (3), for the magnitude of the reduction 

found experimentally appears to be predictable from a 

reasonable assumption regarding the relative constriction of the effective flow width.

Let us assume that the stagnant but still stable and 

nondissipatory zone occupies approximately the same 

proportion jd of the whole channel width d as when at 

a slightly increased curvature it breaks down into an 

easily observable dissipatory eddy zone. The effective 

flow width d' is then d (1—j). And the effective flow 

radius R' is R(l+$.?d/R). Where

5.'

d/:

and

R_R

d~d . 1-j

__l + jjd/R

The first (inertia resistance) term in equations (1) 

and (3) will be reduced by a factor 1/k. From observations made during the sinuous channel experiments, the 

effective flow width was reduced by about 0.35d; that is 

j=0.35. Then, if breakaway occurs at R/d=1.75, the 

factor 1/A:=0.59. And the fall in resistance from what 

it could have been if the breakaway phenomenon had 

been absent will be 0.41 od/R. Taking a as 0.0624 

from equation (4), and d/R=0.57, the fall in the value 

of the resistance coefficient e should be 0.015 approximately.

The experimental fall in e, along the ordinate R/d= 

1.75, from the hyperbola curve to the lowest plotted 

points is 0.016 approximately, which is in astonishingly 

good agreement.

It should also be noted that the relative amount of 

the fall in resistance, which takes place just before the 

final rise in the pipe results, figure 86, is of just the same 

order as that in the open channel results, figure 87, 

taking the different critical values of R/d into consideration in the above calculation.

A pronounced fall in the resistance to flow, confined 

within the same narrow range of boundary curvature, 

is thus found to occur in the same degree in both closedpipe bends and open-channel bends. Hence, evidently 

it cannot be associated with the instability of the free

surface of an open channel as defined by the Froude 

number criterion. There can be little doubt that this 

fall in resistance must be associated directly with the 

breaking away of the flow from a boundary of the bend. 

If so, since breakaway always occurs at some stage as 

the channel curvature is increased, whatever the shape 

of the channel cross section and whatever the degree of 

turbulence, the fall in resistance appears to be a very 

general property of the flow of real fluids, a property in 

which neither Froude number nor Reynolds number is 

essentially involved.

The precise channel curvature, as defined by the 

ratio R/d or b/rm, at which the resistance minimum 

occurs is likely, however, to be influenced by the flow 

conditions immediately upstream of the channel curvature. But provided these conditions are similar, for 

example, if in comparable cases the channel consists of 

a succession of alternately reversed bends as in a meandering river, the critical value of b/rm is likely to remain 

of the same order.

Thus the critical value found in the small-scale 

sinuous open channel experiments seern likely to be of 

the same order as that to be found for the largest 

meandering river flowing at a low Froude number and 

for a small stream flowing at a very high Froude 

number in a meandering melt channel cut in the steep 

ice surface of a glacier.

RELATION TO RIVER MEANDERS

Leopold and Wolman (1960) have compiled considerable evidence that when other conditions—as yet 

unknown—cause a stream of any size which flows in a 

deformable channel to develop a meander pattern, the 

ratio rm/b of the meander bends does in fact tend to a 

constant common value between 2 and 3. One might 

be tempted to conclude that a satisfactory explanation 

has now been found.

But such a conclusion would gloss over several important steps in the reasoning. Strong evidence has 

been found that the resistance to flow in a channel of 

uniform cross section falls to a sharply defined minimum within the narrow range of the curvature ratio 

rm/b between 2 and 3 approximately. But we have yet 

to show that the same minimum of resistance does in 

fact occur in a natural meandering channel whose 

cross section is not uniform.

If the same resistance minimum is found to occur in 

a natural channel, we have still to explain why the 

configuration of the natural channel should tend towards 

that giving minimum resistance. The reason is far 

from being self-evident.

Any explanation which may be put forward must 

remain speculative until the nature of the general 

dynamic mechanism is understood whereby flow in a
144 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

deformable channel tends to mould its channel to 

a certain preferred configuration.

REFERENCES

Hofmann, Albert, 1929, Loss in 90-degree pipe bends of constant 

circular cross-section: Hydraulic Inst. of Munich Tech. 

Univ. Bull. 3, published in 1935 by Am. Soc. Mech. Engineers, New York. (English translation of the original.)

Leopold, L. B., Bagnold, R. A., Wolman, M. G., and Brush, 

L. M., Jr., 1960, Flow resistance in sinuous and irregular 

channels: U.S. Geol. Surv. Prof. Paper 282-D, in press.

Leopold, L. B., and Wolman, M. G., 1960, River meanders:

Geol. Soc. America Bull. 

Mockmore, C. A., 1944, Flow around bends in stable channels:

Am. Soc. Civil Engineers Trans. 109, v. 71, p. 593-628. 

Robertson, J. M., 1944, Discussion of Mockmore, C. A., 1944,

Flow around bends in stable channels: Am. Soc. Civil

Engineers Trans. 109, v. 71, p. 619-628. 

Yarnell, D. L., and Woodward, S. M., 1936, Flow of water

around 180-degree bends: U.S. Dept. Agriculture Tech.

Bull. 526, 64 p.

* U.S. GOVERNMENT PRINTING OFFICE : 1961 O—621829