Flow Resistance in Sinuous 

or Irregular Channels

GEOLOGICAL SURVEY PROFESSIONAL PAPER 282-D
Flow Resistance in Sinuous 

or Irregular Channels

By LUNA B. LEOPOLD, RALPH A. BAGNOLD, M. GORDON WOLMAN, and 

LUCIEN M. BRUSH, JR.

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGIC SURVEY PROFESSIONAL PAPER 282-D

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

REPRINTED 1963

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

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

Symbols. 

Abstract.

Part 1. The problem and the experiments.

General statement-_________________________________

Flow in irregular open channels__________________.

Design of experiment_____________._...______

Experimental conditions. ______________________

Experimental results._________________________

Character of the flow; deformation of the free surface. _ _ 

Conditions below threshold____________________

Conditions at and above threshold-____-___-__-___

Page

IV

111

111

113

114

115

117

119

119

121

Part 2. A theoretical model of energy loss in curved

channels, by Ralph A. Bagnold. 

Fluid-dynamic considerations—— -__-_____-___--_-----

Magnitude of superelevation in a continuing channel 

bend. _-____________-___------__-----_---__----_-

Energy changes in zone of curvature reversal_____ ______

Energy dissipation._________________________________

Comparison with experiment____-_--____ — __________

Discussion._________________---___-______--___--__-

Part 3. Implication of the results. 

Application of experiments to natural channels._____--_

Summary.. _ ________________-________--____-------_

References cited.___________________________________

Appendix. Flume data for fixed grain channel-___--____

Pago

122

123

124

126

127

129

130

131

132

133

ILLUSTRATIONS

PLATE 3. Straight and sinuous channels molded in sand

of laboratory flume______________ Facing 114

4. Spill phenomenon due to bank projection in

Baldwin Creek, Wyo_________ Facing 122

FIGURE 68. Diagram showing postulated relation of

stress to flow velocity._______________ 114

69. Cross section of channels used in experiments.-------..-_____________________ 115

70. Diagrams of resisting stress as function of

square of Froude number...__________ 116

71. Diagrams of resisting stress as function of

square of Froude number_______________ 117

72. Experimental relation of T,-/T. to fc/r_,_______ 119

73. Diagram showing postulated relation of s

to F^........ __________________ 119

Page

FIGURE 74. A, Rate of increase of resistance as function 

of index of curvature; B, experimental 

relation of F*e to 6/rm_____.___ — _. 120

75. Sketches of flow features seen on water surface in sinuous channels-.---.-..------- 120

76. Topographic map of water surface at Froude

number above threshold-_______ — -___. 121

77. Diagram of sinuous channel-—___________ 122

78. Diagrams of deformed water surface in 

curved channels. A, Inclined channel 

banks; B, Vertical channel banks._______ 125

79. Theoretical curves of Fae as a function of

6/r«_—........ _-___ — — -- — --.— 127

80. Theoretical curves of z as a function of F*__ 128

TABLES

TABLE 1. Characteristics of channel patterns investigated.. ________________________________

2. Experimental values of Ti/rt in channels of 

various indices of curvature_____________

3. Experimental values of threshold Froude number for channels of various indices of curvature. --___-_-________________--___---

Page

117

118

119

Page

TABLE 4. Calculated fractions of channel cross section 

within which energy loss due to spill is concentrated______________ — ______ —— __ 129

5. Frequency distribution of values of Froude

number at bankf ull stage in rivers _________ 130

m
SYMBOLS

Units

a A measure of limiting depth near the bank at which local velocity

no longer approximates mean velocity. 

A Semiamplitude of repeating bend in channel- ____________________ L

A Cross-sectional area; A' cross-sectional area within which spill

resistance causes energy dissipation __________________________ L2

b Width of flowing water_______________________________________ L

c Resistance coefficient.

C Chezy factor.

c? Local depth of water __ ______________________________________ L

d Mean depth______________________________. __________________ L

E Discrete resistance. 

/ Friction factor.

at

F Froude number at a particular point,

"TM~

F Mean Froude number for whole flow -7='

•Jgd

Fe Threshold value of mean Froude number. 

g Acceleration of gravity__-__-_-_____-________----_-____---__-_ LT~2

H Flow energy level_________________________________._____-_____ L

h Local depth of water __ ______________________________________ L

I A length along a conduit _ ___________________________________ L

L Half the repeating distance analogous to half meander length in a

river _____________________________________________________ L

Q Discharge___________________________________________________ L3^1

q Discharge per unit width-_-_-___-____-__-___--_-_---__-------- L2T-1

R Hydraulic mean depth__--___-_--_-____--__--------------__-_- L

rm Mean radius of curvature __ _ ________________________________ L

s Water surface slope.

u Local flow velocity. __________________________________________ LT-1

u Mean value of flow velocity._________________-_-_______-------- LT" 1

s A vertical distance between a local superelevated water surface and

its accompanying depressed elevation_________ ________________ L

p Mass density of water __ _ _________________________-------_ ML~3

T Shear on boundary, force per unit area_____-_------_-------- ML-1T~ 2

Tt Shear due to eddies, secondary circulation, and internal distortion ML~1T~2 

T, Shear due to skin friction___________________--------__---_- ML~1T~2

Tg Shear due to energy dissipation by spill, or collapse of streamline

continuity. ___________________________________________ ML~1T~2

f Dimensionless factor proportional to flow resistance.

IV
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS

By LUNA B. LEOPOLD, RALPH A. BAGNOLD, M. GORDON WOLMAN, and LUCIEN M. BRUSH, JR.

ABSTRACT

The resistance to fully developed turbulent flow at constant 

depth in an open channel increases as the square of the mean 

velocity as long as the boundary conditions remain completely 

unchanged. The presence of the free water surface allows the 

possibility of departure from the relationship of resistance to the 

square of the velocity. Experimental evidence is given, which 

is in quantitative agreement with fluid dynamic theory, that 

such departure may be abrupt, with a marked increase of resistance. These departures are observed under conditions of 

boundary and flow which occur commonly in natural rivers.

It is shown that the condition under which this discontinuous 

increase in resistance occurs is definable by the mean Froude 

number for the whole flow which may be as small as 0.4. At 

this initial state, the rate of resistance increase with the square 

of the velocity may be more than double.

The phenomenon, which is absent in straight uniform channels, 

is associated with excessive deformations of the free surface due 

to transverse deflections of the whole or a part of the flow by 

changes along the channel in the curvature of the flow boundary.

In the simple cases examined the critical Froude number at 

which the sudden jump occurs depends mainly on the ratio of 

channel width to mean radius of channel curvature, though the 

inclination of the banks appears also to have a minor effect.

Over the range of values of the above ratio usually to be found 

in natural rivers, the critical Froude number ranges between 0.4 

and 0.55. The possible significance is discussed of the remarkable correspondence between this range of critical Froude 

number and the range of Froude number within which river 

flow at bankfull stage appears to be restricted.

PART 1. THE PROBLEM AND THE EXPERIMENTS

GENERAL STATEMENT

The resistance to flow in open channels is usually 

discussed by engineers in terms of a friction factor or 

coefficient. Though in American practice the factor 

most commonly employed is of the Manning type, this 

factor originates from the Chezy relation

where u is the mean velocity, R is the hydraulic mean 

depth, and s is the slope of the energy grade line.

The Chezy factor C provided a link between the 

impelling force and the unaccelerated velocity. This 

link was at the tune of its development perforce an 

empirical one. The nature of internal shear stresses 

within the flow, upon which the velocities of both 

turbulent and viscous flow depend, was unknown at 

that time. Thus the general relation

iPoc force

was used because it was believed consistent with the 

fundamental Newtonian concept that force equals the 

time-rate of change of fluid momentum.

The d'Arcy school employed the same concept by 

expressing all the various elements of resistance exerted

by closed pipes in terms of the velocity head, u2/2g. 

The total head could be conveniently written

where the friction factor, f, represented the uniformly 

distributed wall friction per unit length of straight pipe, 

I the pipe length under consideration, and ~2E the sum 

of the individual discrete resistances associated with 

various fittings.

For the case of open channels in steady uniform flow 

and considering only skin resistance, the discrete resistances, E, being mostly of an indefinable nature, we get

H u* -j-JK=zr- I

I 2g 

Since the drop in head per unit length is s,

which is the simple Chezy formula.

It is now known that for fully developed turbulent 

flow, the Newtonian concept covers shear resistance as 

well as direct fluid-dynamic force. Thus the Chezy

ill
112 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

formula becomes theoretically correct in that all resistances, distributed or discrete, continue to vary as u2 

provided there is no change in any part oj the -How 

boundary as velocity is varied.

The application of the d'Arcy term "friction factor" 

beyond the context within which it was developed has 

tended to encourage the tacit assumption that flow 

resistance in open water courses is due principally to 

boundary friction associated with distributed skin 

roughness.

This simplified and traditional view of open-channel 

resistance disregards two facts. First, the "squarelaw" resistance may be appreciably increased, as it is 

in pipes, by the distortion of the flow at discrete bends 

and other large-scale channel irregularities. Second, 

such internal distortion is accompanied, inevitably, by 

some deformation of the free water surface* Such 

deformation invalidates the required condition that the 

whole boundary remains fixed and unchanging.

Thus the possibility exists that appreciable departures 

from the square-law relation may occur. Such departures need not be associated with the general resonance 

expected when Froude number has a value at or near 

unity.

"Resistance in open channels, like that in closed pipes, 

is composed in reality of at least three resistance elements of different kinds.

In a straight uniform pipe the whole of the resistance 

to flow may be assumed uniformly distributed along the 

flow as a skin resistance. Assuming fully developed 

turbulence, this skin resistance, expressed as force per 

unit boundary area, depends, for any given shape and 

size of cross section, only on the square of the flow 

velocity and on the roughness of the boundary surface. 

"Roughness may be considered to be measured by the 

mean size of distributed boundary irregularities which 

are small in relation to the pipe size.

In a run of piping, however, other additional resistances are exerted at discrete points wherever any boundary feature—bend, or other fitting—deflects the flow 

or part of it away from its former direction. In general 

any such deflection creates energy dissipation by eddying, secondary circulation, and increased shear rate. 

It is usual in pipe work to estimate each of these flowdeflecting resistances separately and to add their total 

to the resistance of the straight piping. This type of 

resistance we will refer to as internal distortion resistance*

There is a third and important kind of resistance, 

also applied at discrete places. It occurs when a sudden reduction in velocity is forcibly imposed upon the 

flow, as at the sudden expansion just beyond a partially opened valve in a run of piping. The foot of 

a waterfall would be perhaps an extreme example in 

the case of open-channel flow.

When this kind of resistance occurs at the sudden 

expansion of a closed pipe, it is often called impact 

resistance, the mathematical expression for the resulting energy dissipation being of the same form as that 

for the dissipation resulting from inelastic solid-body 

impact.

A sudden forced reduction in flow velocity may be 

visualized as resulting in an abrupt telescoping of the 

streamlines, accompanied inevitably by violent local 

vorticity. This telescoping of streamlines may also 

result in violent vorticity due to internal breakaway 

or mushrooming not directly associated with a fixed 

boundary. This abrupt, discontinuous expansion of 

the streamlines is forced to take place when the flow 

is rigidly confined, as in a closed pipe, and must do 

so regardless of the oncoming velocity. The closed 

pipe does not permit any volume adjustment, so in 

this case this resistance element obeys the square-law. 

But owing to the volume adjustment permitted the 

flow in an open channel, a discontinuous telescoping 

and expansion of the streamlines becomes possible 

only when the oncoming flow is supercritical; that is, 

when its velocity exceeds that at which an increase 

of fluid pressure can be propagated upstream. No 

prior volume adjustment is then possible.

To avoid any ambiguity, this kind of resistance in 

open channels will in the present context be called 

spill resistance.

In natural river channels skin and internal distortion 

resistance are common, and it is probable that spill 

resistance is not uncommon in parts of the_flow, even 

though the mean river flow is subcritical (F less than 

unity). Yet, for practical reasons, distortion and spill 

resistances are largely overlooked because large-scale 

and discrete boundary irregularities which cause them 

in natural channels are not easily described in quantitative terms; and therefore their resistance effects 

cannot easily be reduced to quantitative coefficients 

of useful form. Further, the ordinary laboratory 

flume in which much of the experimental work on 

open channels is conducted is straight and uniform 

and wholly devoid of the large-scale boundary irregularities which cause distortion and spill resistance.

As a consequence, the general resistance coefficient 

is often thought of as synonymous with the skin friction of pipes, even when the worker is dealing with 

natural channels. The large-scale irregularities of 

natural channels are conspicuous features. Indeed, 

few river channels are straight for distances exceeding 

10 times the channel width (Leopold and Wblman, 

1957, p. 53), and undulations and protuberances on 

the banks exist even in reaches of channel which in 

plan view appear straight.

Einstein and Barbarossa (1952) made an important
FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 113

advance when they broke down natural channel roughness into grain roughness and that caused by "sand 

bars, sand waves, and other irregularities of the channel" (p. 1125). These latter elements they considered 

analogous in a mathematical sense to a series of "uniformly sized and uniformly spaced protrusions of the 

stream bed, such as piles of single rocks" (p. 1125). 

In their analysis they postulated that the flow resistance 

created by these irregularities would be a function of 

sediment transport.

But relatively few experimental tests have been made 

under controlled conditions to assess the relative magnitude of resistance due to boundry irregularities of largescale as compared with small roughness elements. Nor 

have tests been made to verify the assumption that the 

overall resistance of natural open channels that include 

such large irregularities varies as the square of the velocity through a wide range of velocities.

In the present paper some preliminary tests in the 

laboratory are described. The results suggest that the 

prevailing approach to resistance in natural rivers may 

be seriously oversimplified. The tests indicate that 

large boundary irregularities may greatly increase the 

square-law resistance over and above that attributable 

to boundary roughness. And further, in combination 

with the presence of a free water surface, these irregularities, by the creation of a spill resistance, may under 

certain conditions give rise to an astonishingly large 

upward departure from the square-law at a threshold 

mean Froude number considerably less than unity.

The experiments were designed to explore the resistance effects of only one simple type of boundary irregularity that related to regular or repetitive sinuousity of 

a channel of uniform cross section. A mathematical 

model of flow under such conditions was not available 

when the experimental work was begun. A preliminary 

theoretical model was developed after the experimental 

results were complete, and for this reason it is the plan 

of the present paper to describe the experiments after 

a brief introductory discussion of flow in irregular channels. Following this is a presentation of a mathematical 

model which appears to explain many of the observed 

features. We conclude with a discussion of some possible implications of the theory in natural channels.

The sequence of development of the ideas will help 

explain why some phases of the experimental data are 

somewhat less complete than might be desired. The 

experiments should be viewed as preliminary, but they 

do imply that an extension of such investigations may 

yield interesting and informative results.

FLOW IN IRREGULAR OPEN CHANNELS

When flow in a shallow open channel is locally deflected 

around a bulge in the bank boundary, the local water

is subjected to radial accelerations. As a consequence 

of the requirement for constant pressure at the water 

surface, the radial forces are offset by an increase in 

the gravity head on the upstream and a decrease on the 

downstream side of the bulge. This creates a local fall 

in water-surface level, often large enough to be easily 

noticeable. An acceleration of local water along the 

direction of flow is also noticed, as the water flows 

through this area of local fall in water surface.

Provided the velocity of the accelerated water does 

not exceed T/gd, adjustments between velocity head 

and elevation head can take place with no energy loss 

specifically attributable to the deformation of the free 

surface.

But as the mean flow velocity is increased, a stage 

must be reached at which the accelerated local flow 

from high to low level begins to exceed the local propagation velocity, i/gd. At this stage the Bernoulli 

adjustment ceases to be possible; that is, elevation 

head and velocity head cannot be interchanged smoothly. "Rather, the local high velocity flow impinges 

violently on slower water downstream at the lower 

level, with the creation of spill resistance as previously 

defined, and with an inevitable dissipation of energy.

To describe the types of resistances in terms of 

shear stress, let ra represent the shear resulting from 

skin friction, and rt a distributed stress equivalent to 

the resistance from internal distortion of flow by bank 

irregularities or bulges. Both elements may be expected to increase as the square of the mean velocity, 

as in pipes. In terms of a mean distributed boundary 

stress, T—T3 -}-Ti and

where c is the overall resistance coefficient ( c=n=7^ \ Jo

which should remain constant for flow of constant 

cross section through a given channel.

When, however, the energy dissipation due to spill is 

introduced, an expression for an equivalent distributed 

boundary stress might be in the form T=r,+r<+T2 

where rg is the resistance due to energy dissipation in 

the spill process.

The total resistance, r, could no longer be expected 

to vary as the square of the mean velocity when rt is 

introduced, because, as mentioned earlier, spill 

resistance in an open channel cannot exist at low 

velocities but must start increasing from zero at some 

finite mean velocity at which parts of the flow become 

locally supercritical.

It is visualized that the extra resistance element 

rt is not directly related to wave phenomena, nor 

should it necessarily be connected with any resonance
114 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

effect within the main flow as Froude number approaches unity. Rather, the elevations and depressions of the water surface are the direct result of local 

radial accelerations, the positions of which are fixed by 

local curvatures of the bank. The positions do not 

depend on flow velocity and therefore cannot be regarded as standing waves. The velocity merely 

determines the amplitude of the local water surface 

dislevelment, and thus the magnitude of excess local 

velocities.

DESIGN OF EXPERIMENT

The preliminary object of the experiments was to 

gain some idea of the relative magnitudes of the 

resistance elements represented by TS due to skin 

friction and rt due to internal distortion of the flow. 

It was further desired to discover whether the expected 

sudden departure from the square-law due to the 

postulated element rz might be sufficiently pronounced 

to be detected. The experiments were therefore 

designed to contrive a quantitative separation of these 

elements from one another.

Each of these elements of stress represents equivalent 

distributed values of force per unit of boundary area. 

It was desired to determine the separate effects of 

these three resistances on the flow velocity in a channel 

having a uniform cross section but analogous in shape 

to that of a natural river.

A quantitative expression of the relation between 

mean velocity, u, and applied stress in a given channel 

cannot be derived analytically but only by experiment. 

Similarly, when the channel form is changed the new 

relation between u and applied stress can be determined 

only by experiment. But the two relations may be 

compared to determine the effect of the change made 

in the channel.

Skin resistance can be measured in a straight channel 

of uniform and definable roughness. The resistance 

due to internal distortion or eddies, ri} can be compared with skin resistance, TS, only if the changes made 

in the channel do not alter either the flow cross section 

or the boundary texture roughness. Moreover, if it 

is desired to vary the type of channel irregularity, the 

effects can only be compared one to another if flow 

cross section and boundary roughness are again made 

identical.

This necessity unfortunately precludes the introduction of abrupt baffles into a straight channel, because 

any experimental variation of the size or spacing of 

the baffles in relation to the channel cross-sectional 

dimensions at once makes the cross section indeterminate.

The simplest definable type of nonuniformity whose 

variation does not affect the cross section is a regular 

sinuosity of channel course. This was the type adopted.

Other types suggest themselves, all of which involve 

definable fluctuation of the cross section along the 

flow about a constant mean shape, and should be 

practicable provided the repetition distances downstream are long compared to the cross-sectional dimensions and provided the "bulges" are reasonably 

streamline. For example, the channel bed can be 

corrugated, the banks remaining straight; or the bank 

sinuosities can be brought into phase, to cause alternate 

expansions and contractions; or both bed and banks 

can be corrugated.

The following simple method of estimating the 

relative values of the three forms of flow resistance is 

applicable when of necessity the channels, straight and 

sinuous, are in other respects identical, and when the 

flow depth is maintained constant, the flow being 

varied by varying the slope s only.

The skin resistance TS is measurable in the straight 

channel by pgRs. In figure 68 a hypothetical relation 

is suggested between pgRs and pu2 . Velocity, hydraulic 

mean depth, and slope can be measured in the experimental setup. It is then assumed that TS represents 

the skin resistance of the sinuous channel when a small 

correction is made for the very minor extension in 

channel length. The postulated relation between TS 

and pu2 is represented by the straight line OA in figure 

68.

For the sinuous channel pgRs no longer gives the 

skin resistance only, but instead pgRs=Ts -\-Ti. In 

figure 68, a relation of rs+r* to pu2 would give the 

second straight line OB if the resistance rz were negligible. For any given value of pu2 , Ti can be read off

I

Spilt resistance 

plus ti + T5

Internal distortion 

resistance T/ 

plus TS

Skin resistance, T5

FIGUEE 68.—Diagram showing postulated relation of stress, pgPs, to the square of 

flow velocity. The postulated types of resistances are indicated by the lines of 

different slopes. The abscissa, pu*, is also the product of pgR and the square of the 

Froude number.
GEOLOGICAL, SURVEY PROFESSIONAL PAPER 282 PLATE 3

This channel was used to measure skin friction due to grain roughness for later comparison with resistance due to

channel sinuosity.

This channel has a repeating distance of 2 feet and a semiamplitude of 0.3 foot.

STRAIGHT AND SINUOUS CHANNELS MOLDED IN SAND IN FLUME

FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 115

the figure as BD—AD. If, as thought, TZ is appreciable 

beyond some finite value of pu2 , E in figure 68, it should 

appear as a steepening of the line, sketched as EC in 

the figure. At higher values of pu2 , rs can be read off 

the figure as CB.

Dividing both ordinate and abscissa by pgR, hydraulic mean depth being kept constant in the experiment, the diagram is unaffected and the abscissa is 

then expressed in terms of F2=u2fgR which is plotted 

againsts. The overall resistance coefficient c=fj2=g/C2 

is equal to the slope of the straight line OE in figure 68, 

which represents the linear relation between shear and 

the square of the velocity.

But the overall resistance coefficient will not be 

given by the changed slope of the curve beyond E. 

If such a resistance coefficient is required, it would at 

any given stage be proportional to the varying slope 

of a straight line OC. But since the elemental coefficients are no longer additive, they appear to serve no 

useful purpose in the present context.

EXPERIMENTAL CONDITIONS

A trapezoidal channel was moulded in noncohesive 

sand contained in a flume having a length of 52 feet 

and a width of 4 feet. The adjustable-slope flume in 

which a channel could be moulded, built by the Geological Survey in the hydraulics laboratory of the 

University of Maryland, is described elsewhere by 

Wolman and Brush 1 and is similar to the one used for 

previous experiments at the California Institute of 

Technology (Leopold and Wolman, 1957).

The sand in which the channel was formed had a 

median diameter of 0.0067 ft (2.00 mm) and was rather 

well sorted. The Trask sorting coefficient was 1.21.

In order to maintain a uniform cross section even 

around bends in the channel, the template used to mold 

the channel was not a thin plate but was turned on a 

wood lathe in a form resembling a round cake pan with 

sloping sides. All of the data discussed pertain to one 

cross-sectional shape made with the same template 

(see fig. 69).

i Wolman, M. Q., and Brush, Jr., L. M., Factors controlling the size and shape of 

stream channels in coarse noncohensive sands: U.S. Oeol. Survey Prof. Paper, in 

preparation.

In molding a straight channel the template was held 

rigidly in a carriage which rolled along the length of 

the flume. For a sinuous channel, the template was 

moved laterally by a mechanical linkage as it was 

pushed by hand along the length of the flume. Plate 

3 shows two of the channels thus formed

Because we desired to go to Froude numbers well 

above those observed in rivers and to maintain constant 

cross section of channel, it was necessary to prevent 

scour of grains. This was accomplished by spraying 

the molded channel with three thin coats of semigloss 

enamel paint. The grain roughness was still prominent 

and, in fact, the paint was so thin that it would not hold 

the grains indefinitely as the discharge was raised. The 

material used was fast-drying paint packaged in cans 

under pressure and equipped with a spray nozzle. It 

was found desirable to use a different color for each 

coat of paint, making it easy to see whether the new 

coat was of uniform thickness and covered the whole 

area.

The entrance consisted of a weir which discharged 

into a 2-foot approach channel having gradually 

converging metal sides terminating in a 4-foot reach of 

straight channel molded in the sand. The working 

length of channel was 46 feet. The discharge end of the 

flume was controlled by a vertically adjustable tailgate.

Water was delivered from a constant-head tank 

through, either a 2-inch or 1-inch pipe to the entrancebox of the flume. Discharge was measured by piezometer tubes inserted in a 90° pipe bend, which was rated. 

Head was measured by a manometer containing carbon 

tetrachloride.

Elevations were measured by a point gage mounted 

on an overhead travelling carriage. The vernier could 

be read to 0.001 foot.

Runs were made for each of two depths of water. A 

run consisted of setting the flume to a desired slope, 

adjusting the discharge until the water depth in the 

channel was equal to the chosen value. Profile of the 

water surface was then measured to determine whether 

the water surface slope was parallel to the channel bed. 

Tailgate setting and discharge where then adjusted 

until the depth was the desired value and the flow

0.735 ft Water surface width, full depth

.625ft Water surface width,"half" depth

Water Wetted Area Hydraulic

depth perimeter mean depth

(ft) (ft) (ft8) (ft)

0.090

133

0.685

.825

0.045

.075

0.066

.090

FinuBB 69.—Cross section of all the channels used in the experiments. The two values of depth shown were used in each channel pattern investigated.
116 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

uniform. The discharge was then determined from 

the manometer reading and the slope and depth 

recorded.

The experimental conditions investigated consisted 

of the following channel patterns, each one of which was

run at various discharges, and at each of the two chosen 

depths. The dimensions of the cross section at each 

depth are shown in figure 69 and the definitions of L 

and A can be seen on the sketches of sinuous channels 

in figures 70 and 71.

.008

o 

o 

u.

o: .006

ui

0.

.004

.002

STRAIGHT 

CHANNEL

y .

Depth (c 

(feet)

0.133

090

Channel 

Sinuous Straight

.2 .6 .8 1.0

0.010

.008

.006

.004

.002

7/

B

.2 .4 .6 1.0

0.010

.008

.006

- .004

.002

FIGURE 70.—Diagrams of channel slope, a, proportional to resisting stress, plotted against the square of the Proude number. Data for the two depths of flow used In the 

experiments are represented by different symbols. As a standard for comparison, mean lines for the straight channels are shown. A, Data for straight channel; 

B, sinuous channel having repeating length 4.00 ft, amplitude 0.30 ft; and C, sinuous channel having repeating length 3.87 ft, amplitude 0.64 ft
FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 

2.12ft

117

0.012

.010

.008

.006

.004

.002

0.012

.010

.008

.006

.004

.002

.2 .4 .6 .8 1.0

Depth (tf) Channel 

(feet) Sinuous Straight

.—Diagrams of channel slope, a, proportional to resisting stress, plotted against the square of the Froude number. Data for the two depths of flow used In the 

experiments are represented by different symbols. As a standard for comparison, mean lines for the straight channels are shown on all the graphs. A, Sinuous channel 

having repeating length 2.12 ft, amplitude 0.30 ft; B, sinuous channel having repeating length 2.30 ft, amplitude 0.54 ft.

TABLE 1.—Characteristics of channel patterns investigated, 

expressed as length and amplitude of repeating sinuosities l

Series No.

70A. ._.__._.

7QB _________ _

70C__________

71A. ________

715 _____ __

Channel form

Straight channel __ ____

Sinuous channel____ _ ___

-__-_do.___________ ___ _

-___-do.__________________

_-__.do.__________________

Repeating 

distance 

(2 L, in feet)

4.00

3.87

2. 12

2.30

Amplitude 

(2 A, 

in feet)

0.30

.54

.30

.54

1 The geometry of sinous channel patterns has previously been discussed (Leopold 

and Wolman, 1957), in terms of a meander "wavelength" and "amplitude," using 

these terms as they are usually used to describe a sine curve. Since the presence of 

a free water surface admits the possibility of true wave effects being found relevant, 

in future discussions it may be advisable to reserve the term "wavelength" accordingly. For meanders, the term "meander length" will, in the future, perhaps be 

better than "wavelength." Amplitude is synonymous with "width of meander 

belt." The molded curves of the present study are here described in terms of the 

ratio of repeating length to amplitude, where repeating length is analogous to 

"meander length," a downvalley straight line distance between symmetrically 

located points on successive channel curves. Amplitude is the total spread in height 

from wave crest to wave trough. L and A are respectively the semilength and 

semiamplitude.

The repeating distances shown above are average 

values for different curves along the length of the flume. 

Two basic lengths were molded, approximately 2 and 

4 feet, and the slight variation from these values occurred in the molding of the channel. The bulge

forms were to the eye quite uniform, however, as can 

be seen in plate 3B.

EXPERIMENTAL RESULTS

The quantitative results are shown graphically in 

figures 70 and 71 in the form s plotted against F2 . Values of the stress element T and of u 2 can be obtained 

by multiplying by pgR (see fig. 68). The original data 

are tabulated in the appendix.

Some scatter of the plotted points will be noted. 

This we found is not in excess of the scatter expected 

from random errors involved in reading the discharges 

from the rating curve.

In spite of the scatter it is clear from the plots that 

the simple square-law relation expected for turbulent 

flow was obeyed in the straight channel and also at 

low Froude numbers in the sinuous channels. The 

smallest Reynolds number for the channel at the reduced depth and at the lowest mean velocities measured 

was 3600. It is possible that at these low velocities 

turbulence was not fully developed in the shallows 

over the inclined banks. However, the data for the
118 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

straight channel as plotted in figure 70.A discloses no 

significant departure from the linear relation between 

mean stress and the square of the mean velocity. It 

seems unlikely, therefore, that Reynolds number effects 

enter significantly into the results obtained at higher 

velocities.

The square-law relation ceased to hold in the sinuous 

channels when a certain critical or threshold Froude 

number was exceeded. Beyond such a threshold the 

curves break sharply upward to a steeper slope, indicating the onset of additional energy loss.

At values of F 2 below the threshold—that is, in the 

square-law region—an idea of the relative magnitude 

of the extra square-law resistance T{ introduced by 

channel curvature can be gained by considering the 

experimental values of the ratio T</TS ; that is, the 

ratio of internal distortion resistance to skin-resistance 

of the straight channel. This ratio is that of AB to 

AD in figure 68, and can be read off the plots of figures 

70 and 71. It is given in column 8 of table 2.

The variation of T</TS with flow depth is peculiar. 

In the absence of any theoretical understanding of the 

factors on which the square-law resistance of "irregular" 

channels depend, we offer no explanation. The general 

effect of the presence of irregularities is evidently very 

appreciable.

It will be noticed that the ratio TJ/TS tends generally 

to increase with the increase in sharpness of bend. 

This sharpness depends not only on the radius of curvature but on the ratio of channel width to radius. This 

ratio, b/rm, is used throughout the present paper as an 

index of channel curvature. The quantity rm is the 

radius of a circle which passes through the centerline 

of the channel at the points of inflexion and maximum 

amplitude.

In figure 72 are plotted mean values of the ratio 

Ti/Tg for the two water depths. These mean values are 

plotted against mean values of the index of curvature, 

also computed as the mean value for the two water 

depths. The data appear in columns 9 and 10 of table 2.

TABLE 2.—Experimental values of ratio of internal distortion resistance, T», to skin resistance, 

Tt, in channels of various indices of curvature

1

Series of runs in 

figure No.

70J5__. _______________

70C'__________________

7l-l___-___-_- _______

71J5__. .__.__.______._

2

Semilength 

(L, in feet)

2.00

2.00 

1. 93

1. 93 

1.06

. Do 

/ 1. 15

1 1. 15

3

Semiamplitude 

(A, in feet)

0 15

15

27

27 

15

15

27

27

4

Mean 

radius 

(rm, in feet)

3.34

3.34 

1.87

1.87 

1.00

1.00

.74

.74

5

Surface 

width 

(&, in feet)

0 735

bzo 

735

b2o 

735

oJo 

735

oJo

6

Hydraulic 

mean depth 

(R, in feet)

0 09

Uoo

09

066 

09

Obo 

09

Uoo

7 

b/rm

0.22

. 19 

.39

. oo

.73

. \)£ 

1. 19

.99

8

T.-/T,

0

0 

.50

.43 

1.78

.43 

3. 11

2.19

9

Mean

b/rm

j 0.2

| .36

| . 68

} 1.09

10

Mean

T.-/T,

0

.46

1. 10

2.63

The data indicate that channel curvature alone can 

account for energy loss of the same order as that due to 

skin friction, and in tight curves may be double that 

quantity.

At values of F2 higher than the threshold the plots 

indicate an abrupt upward trend resembling that 

sketched in figure 68, but the experimental points do 

not provide any certain estimate of the abruptness of 

the departure from the square-law. A small but systematic trend will be noticed which suggests that the 

relation beyond the threshold value, F%, is curvilinear 

as indicated in figure 73, in which case the maximum 

gradient would occur at the point of departure, F%. 

That is, an actual discontinuity may occur. The theory 

to be outlined later suggests that this should be so.

Since the resistance to flow no longer obeys the squarelaw, the most useful way in which to discuss its variation in this region is in terms of the magnitude of the

change of gradient. In other words, the magnitude of 

the change in the rate of increase of resistance as F2 is 

increased.

If, as seems probable, the plots in this region are 

really curved, the relation will be as indicated in figure 

73. As that figure indicates, the change of gradient 

at the actual threshold F2e__ would be measured by 

ds{d(F'2 )—dsld(F2 ),_where_F" refers to the Froude 

number just above Fc and F to that below.

The experimental values of the change of gradient 

are shown in column 7, and the relative values in column 

8 of table 3. For each channel pattern indicated by 

the series number, data are given for full water depth, 

5=0.09 feet, and reduced depth, 5=0.066 feet, if both 

are available.

In figure 74 the changes of gradient from column 7 

are shown plotted against the channel curvature criterion b/rm. It will be seen that the results for the two
FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 119

3.0

2.5

2.0

1.5

1.0

.5

70B

7IA

71B

.6 .8 1.0 1.2

WIDTH TO CURVATURE RATIO, -yFIOURE 72.—Experimental rrlation of ratio of internal distortion resistance, n, to 

skin resistance, r,. at various values of index of curvature.

steepest plots, series 7lB, which in table 3 have values 

of b/rm equal to 0.99 and 1.19, have too small ordinate 

values in figure 74A Estimates of gradients from 

plotted points are subject to rapidly increasing error as 

the general inclination steepens. In figure 7lB if the 

gradients shown in column 5 of table 3 had been determined from the initial slope of curves drawn through 

individual plotted points, the values in column 5 would 

have been appreciably larger, and in figure 74,4, the 

point 7lB marked with a circle would plot at an ordinate value of 0.038. With this correction there appears 

to be a reasonably good correlation in figure 74-4 between the change of gradient and the index of curvature, 

b/rm.

Figure 74B shows the experimental variation of the 

threshold Fc2 with b/rm. The trend is consistent with 

the requirement on general grounds that Fc2 should 

approach unity as b/rm approaches zero (straight 

channel).

FIGURE 73.—Diagram showing postulated relation of channel slope, *, proportional 

to flow resistance, and square of mean Froude number, the relation becoming curvilinear above a threshold value, f*.

CHARACTER OP THE FLOW; DEFORMATION OF THE 

FREE SURFACE

CONDITIONS BELOW THRESHOLD VALUE

A lee eddy occupied the upstream portion of each 

re-entrant in cases of relatively high mean velocity and 

strong curvature. The eddy is shown in the surface

TABLE 3.—Values of threshold Froude number, Fc, for channels having various indices of curvature, b/rm 

[Columns 7 and 8 show abrupt jump in rates of resistance increase, and the threshold Froude number, F, (column 10), at which they occur]

1

Series of runs in 

figure No.

70B-. __ ______-___-.

70C---___- ___________

71A... ____.____.__._.

71B... ...............

2

Hydraulic 

mean

depth (12, 

in feet)

0.09

. ODD 

.09

. Uoo 

.09

. UDD 

.09

. ODD

3

b/rm

0.22

.19 

.39

. oo 

.73

. O-6

1.19

.99

4

Semilength (L,

in feet)

2.00

2.00 

1.93

1.93 

1.06

. UD 

1.15

1.15

5

ds(d(F"*)

0. 0125

.010 

.023

.0195 

.056

.0375 

.072

.047

6

dsld(&)

0.0085

. UUo

.0135

.012 

.025

. Ui_i

.037

. U/yU

7

Difference, 

column 4

minus 

column 5

0.004

. UU—i

.0095

. UU75 

.031

.0255 

.035

. \J£l£l

8

Column 

7X100

dsld(F») 

(percent)

47. 1

£o. U 

70.4

D_k O

124.0

—fio

95

oo

9 

F»

0.37

. Oi

.31

. oJl 

.21

. _ii

. 16

. ID

10 

F.

0.61

KC

. oo 

.56

ec

. Oo

.46 jtf*

.40

.40
120

0.05

«T- -04 

"k

{3 03

1

*""s

"k .02

x_

<

*.o.

PHYSIOGRAPHIC AND H

1 '! 

7OA 

Series number 

o

Runs at depth 0.133ft 

Runs at depth .090ft

70B

K

——— ^

/" 7C 

®/

r

)C

/

/7IA 

f

/ , 7/B

o

7/B 

revised

7/B

A

.2 .4 .6 .8 1.0 1.2 

WIDTH TO CURVATURE RATIO, b/rm

1.4

0.5

.4

.3

70B

70C

7/4

7/B

B

.2 .4 .6 .8 1.0 

WIDTH TO CURVATURE RATIO,

1.2 1.4

FIGURE 74.—A. Experimental relation showing difference in rate of increase of 

resistance above and below threshold value of 7*, as a function of index of curvature. 

B. Experimental relation of square of threshold Froude number, ~f\ and index of 

channel curvature.

paths sketched in figure 75. The effect of this eddy is 

to restrict the flow to a narrower cross section and to 

create secondary circulations. These effects undoubtEXPLANATION OF FIGURE 75

FIGURE 75.—Sketches of flow features seen on water surface. The four diagrams are 

arranged with Froude number increasing from top to bottom. Serial numbers are 

listed, with accompanying data, in the appendix. 

A. Froude number below threshold value; no standing waves observed; eddies

are developed at concave bank near center of curvature.

B. Froude number near threshold. Standing waves developed just upstream of 

points of maximum amplitude. Eddies located between point of convex bank 

and position of maximum amplitude of concave bank.

C. Froude number slightly above threshold. Slanting waves developed. Eddies 

located downstream of point of inflection and upstream of position of maximum amplitude of concave bank.

D. Froude number well above threshold. Standing waves well developed. 

Prominently steepened water surface near convex bank just upstream of the 

nose of each point. Eddies just downstream of point and hugs convex bank.

Serial No. 45

I ft

F -- 0.33

u - .48fps

Fc -- .46

b/rm -- .62

Serial No. 68

I ft

Serial No. 51

I ft

In this sector water 

surface steepened 

near convex bank

F = 0.73 

u = l.24fps 

Fc = .43

Run No.89, data 

included in appendix

Explanation

Direction of surface streamline

X Standing wave on surface

Edge of eddy at surface

FIGURE 75
FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 121

edly are responsible for much of the increase in the 

square-law resistance. A quantitative statement of 

the limiting conditions at which the eddy first appears 

cannot be made from the available data.

In appearance, the effect of the eddy upon the flow 

was to reduce its real width and also to displace the 

phase of its sinuosity downstream with respect to the 

sinuosity of the channel banks, without however appreciably affecting the mean curvature.

The water surface was undisturbed by any travelling 

deformations other than a general fine texture of surface-tension ripples originating at the grain-roughness 

at the water's edge.

The inevitable difference of local pressures immediately upstream and downstream of each bank 

convexity, which are due to the flow deflection, gave 

rise to slight local surface elevation upstream and a 

corresponding depression downstream. But the local 

current appeared to pass each bank convexity without 

any discontinuity in the streamlines.

CONDITIONS AT AND ABOVE THRESHOLD VALUE

The critical stage Fc as defined by the plots of figures 

70 and 71 coincided with the formation of a system of 

small standing waves shown on figure 75B located just 

downstream of the point of inflection of the bank 

curvature and at or about the estimated point of

inflection of the mean flow path. The crests of these 

wavelets, starting nearly normally to the waterline, 

curved downstream as shown, fading out as they did so.

The critical stage also coincided with a marked 

change in the appearance of the flow past each bank 

convexity. This now assumed the nature of a plunge, 

or spill, from high level to low, with a pronounced 

local acceleration and a pronounced disturbance below 

where the descending water met the main stream.

As the flow velocity was further increased, the 

general appearance of the water surface when one 

looked upstream resembled that of a mountain torrent 

where jets of water deflected by boulders at the stream 

bank impinge inwards upon the main flow.

Owing to the steepening of the local water surface 

slope past each bank convexity, the mounds and 

hollows were much more readily noticed. Figure 76 

shows the configuration of the water surface in detail 

for conditions approximating_closely to those of figure 

71^4, and at a stage when F2 was approximately_29 

percent in excess of F* (13 percent in excess of Fc ). 

The surface contours have been drawn in planes parallel 

to that of the mean water surface; that is, having the 

same slope as the channel. It will be noticed that the 

surface pattern repeats itself in mirror image from one 

reach to the next.

0 =0.071 cfs

u = .947 fps

F* = .27

/T= .2.

5 = .0078 

2Z. =2.22 ft 

2X1 = .30ft

Contours on woter surface abqve datum

Drawn relative to sloping datum

of mean water surface

Contours on woter surface below datum

Drawn relative to sloping datum

of mean water surface

Observed elevotion, in feet obove datum

o 

Observed elevation, in feet below datum

---*• 

Path of surface floats

Small standing surface riffles

*A .73 FEET 

1.0 i-

.8

.6

.4

-2 -

O 1—

TIGURZ 76.—Topographic map of waier surface in sinuous channel at Froude number above threshold value. The datum Is a sloping plane parallel to and coinciding 

with the mean water surface elevation. The contour lines represent elevations above and depressions below that sloping plane.
122 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

By drawing the surface contour lines relative to the 

mean channel slope and then assuming the slope to be 

zero, the surface topography thus represented should 

be a reasonable approximation to that of an ideal 

frictionless liquid subjected to the radial accelerations 

imposed by the boundary curvature. The only implicit 

assumption made here is that resistances remain uniformly distributed.

The main features disclosed by this map are a pronounced mound of water upstream of each bank convexity, a pronounced hollow downstream of it, and an 

accelerated local flow from the mound towards the 

hollow.

A further feature which may be particularly noted is 

that the water surface remained free of any appreciable 

travelling distrubances. The deformations shown in 

the map were in fact static, located by the fixed curvatures of the banks, their locations being independent

of the mean velocity of the flow. The wholly static 

nature of the deformations appears to rule out 

wave propagation as an explanation of the energy 

dissipation which results in the extra resistance T».

After observing the superelevated and depressed areas 

of water surface associated with the bulge in the 

sinuous channel of the laboratory, we sought similar 

phenomena in the field. An example is shown in 

plate 4. The bulge was formed by a piece of sod collapsing into the channel. The superelevated water 

surface upstream from the obstruction and a depressed 

surface just downstream were noticeable in the field and 

can be discerned in the photograph.

The characteristics mentioned above also applied to 

the field example: accelerated local flow from mound to 

hollow, static deformation—that is, not travelling with 

the flow, but associated with the obstruction.

PART 2. 4 THEORETICAL MODEL OF ENERGY LOSS IN CURVED CHANNELS

By RALPH A. BAGNOLD

FLUID-DYNAMIC CONSIDERATIONS

In the conventional straight channel the body-force 

acceleration is g, which remains constant in both 

magnitude and direction. But when the whole or a 

part of a liquid flowing steadily in an open channel is 

subject to transverse acceleration due to some largescale irregularity in the bank configuration, the direction of the resultant body-force acceleration is no 

longer that of g, though its magnitude may for many 

practical purposes be taken as that of g. Changes 

in che direction of the resultant body-force acceleration 

along the channel give rise to certain special hydraulic 

conditions. Superelevation of the water surface along 

the concave bank is but one of the effects which are a 

matter of everyday observation. But these effects 

have not been investigated as extensively as their 

ubiquitous occurrence might warrant.

Transverse accelerations in the plane of the free 

surface occur whenever the local flow is deflected from a 

straight course by a boundary surface which has a 

curvature component in that plane. Such accelerations 

occur whenever one or both banks'of a stream has any 

kind of bulge. A single bulge, whether streamline or 

otherwise, deflects the streamlines in its neighborhood 

so as to create a reversal of their curvature. Thus 

the forces applied at the boundary which maintains 

the radial accelerations are reversed in sense. Because the pressures must be constant everywhere, the 

free water surface is successively elevated and depressed. As a consequence the local water surface 

in the direction of local flow tends to fall steeply from a 

high to a low level as the flow passes the bulge.

In the case of a single bulge neither the pattern of the 

streamline curvature nor that of the velocity distribution is readily definable. Because the precise pattern 

of the surface elevation is doubtful, the dynamical 

effects of the local fall in level are not easy to investigate 

quantitatively. But consider the simpler case where 

the whole flow in a channel of uniform cross section 

is deflected around an S-bend of definable curvature. 

In figure 77 let the velocity u at any radius r, within the

FIGURE 77.—A sinuous channel showing the center of curvature of a bend and symbols 

designating the radii of curvature
GEOLOGICAL SURVEY PROFESSIONAL PAPER 282 PLATE 4

EXAMPLE OF SPILL PHENOMENON IN A NATURAL RIVER AS A RESULT OF A RANDOM BANK PROJECTION

Note turbulent zone (a) just downstream from tip of sod projection (b), where 

water has spilled from the superelevated position (c) into a depression. Stream 

is Baldwin Creek near Lander, Wyo., flowing somewhat less tban bankfull. Discharge is 69 cfs. View is diagonally downstream. Projection is piece of sod bank 

which fell into channel. Its tip is about 3 feet from the average bank.

FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 123

width b=rt —r2 of the water surface, be defined as the 

tangential component at r. Further, let this velocity 

u be the mean of the velocity through the vertical 

depth at r. Thus defined, radial velocity components 

inward and outward due to internal motions cancel 

each other everywhere. It is further assumed that the 

flow is slow or subcritical; that is, u<j/gR or V</d.

And let it also be assumed that the flow as it approaches the section of curvature reversal has already 

reached a state of equilibrium under the constant 

applied forces due to the constant channel curvature. 

In this state the body-forces exerted on each unit mass 

of liquid at radius r consist of the acceleration g upwards, and the radial acceleration u 2jr horizontally 

inwards towards the center of channel curvature and 

therefore perpendicular to the local flow direction.

The resultant body-force is thus inclined to the fixed

vertical at an angle tan" 1 ( — j; a pendulum moving

with the local current is tilted through this angle, 

and the local horizontal plane is likewise tilted. But 

an observer in a closed vessel drifting with the local 

current would not notice any peculiarity because his 

axes of reference are also tilted.1

The gradient of the free surface at radius r, with 

respect to the fixed horizontal through the axis of channel curvature, is uz/gr. Thus the increase in the water 

surface level at radius r above that at the inner bank

1 CT u2 

radius rz is - I — 9Jr2 r

the whole width 6 is

dr. The total superelevation z over

,=! P1 !*!

9Jr~ r

dr

Beyond the section of curvature reversal the sense of 

the radial forces acting on the fluid from the banks is 

reversed. The high water level along what was the 

outer bank has to be lowered through the distance z; 

and the low level along the other bank has to be raised 

through the same distance.

The raising of level presents no problem. It is 

achieved progressively by an upward displacement of 

local water by the positive radial force of the inwardly 

advancing bank.

But the fall of level may cause a plunge or spill assuming the nature of a collapse, depending on the velocity 

u attained in the fall z, and on the local depth d. In the

i The radial acceleration causes a small but real increase in the magnitude of "g", 

from g to

So the period of the pendulum is actually reduced, and the wave velocity T/gd is 

actually increased. But these changes are too small to be relevant in the present 

context.

event of such a spill, energy will be dissipated by local 

impact. The word "spill" is used here in the context 

described on page 112.

MAGNITUDE OF THE SUPERELEVATION IN A 

CONTINUING CHANNEL BEND

The pattern of the internal flow in a continuing bend 

of a river or canal is imperfectly known. Particularly is 

observational evidence incomplete on the variation of u 

over a cross section for channel curves of different radii.

But let u be some arbitrary f unction/(r) or r. Then, 

making a number of alternative reasonable assumptions 

as to/(r), it appears that although the transverse profile 

of the water-surface elevation varies in shape from function to function, the overall bank-to-bank superelevation z is remarkably independent of the function assummed. For instance:

Case 1: Assume that the velocity u at various positions across the channel is constant and equal to 

the mean channel velocity u.

Then the equation for superelevation becomes

0 f'ldr . 

r=-2= —=ln

u Jr2 '

Because ri=rm-f-6/2 

and r2=rm—b/2

This case results in a logarithmic profile of water surface 

across the channel. 

Case 2: u varies as r.

Then

and

, u=— and rm=:

•m

£L=JL ri2~ 

u2 rm2 2

9 -- z~~

The profile is the arc of a parabola having a vertical 

axis through the center of channel curvature.

Case 3: u constant everywhere, and flow such that 

the effective radius is constant at rm.

This case might possibly be approximate to flow in a 

channel consisting of a succession of short alternatively 

reversed arcs. It might also approximate local flow 

round a single bulge in one bank with suitable local 

values given to 6 and rm.

zr _r1 —r2_b 9 u2~~ rm ~~rm

The profile is now a straight line.
124 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Case 4: Whole discharge concentrated within a 

narrow midchannel zone. The profile is now steplike. 

But again

0 6

Case 5: As a realistic modification of case 4, the 

probable velocity profile can be sketched for the 

conditions of a sinuous channel.

Making allowance for: a narrowing of the flow by an 

eddy against the inner bank such as is shown in figure 

755, a reduction in velocity towards both flow boundaries because of friction, and for a consequent velocity 

increase towards the now modified midchannel radius, 

a graphical integration of v?jr gives a value to z which 

approximates very closely to that given by

z_= b_ 

9 u2 rm

(b and rm referring to the whole material channel).

The profile in case 5 resembles that of case 4 with 

the abrupt step smoothed out to a flattened S. This 

profile bears a reasonable resemblance to that indicated 

by the contours of the experimental water surface 

shown hi figure 76 if a transverse section is taken 

through the summit of the mound.

The choice of the expression most likely to give the 

true superelevation therefore seems to lie between 

b/rm and the log expression of case 1. And it happens 

that over a wide practical range of b/rm the log expression 

approximates very closely to b/rm itself. Thus

6/r. 0.1 0.2 0.4 0.6 0.8 1.0

In ———= 0.100 0.201 0.406 0.62 0.84 1.098 2— or In

Finally, as experimental confirmation that the superelevation z is really given to a close approximation by 

v>2b/grm the evidence of figure 76 may be referred to. 

Under the experimental conditions u was 0.95 ft per 

sec, b was 0.735 ft, and rm was 1.0 ft. Whence uzb/grm = 

0<0206 ft. The measured superelevation, as given by 

the overall drop between the mound top and the 

bottom of the hollow indicated by the contours, is 

0.021 ft. In another experimental run at half depth 

(5=0.09 ft) not pictured here, v?b/grm =0.0135 ft and 

the measured superelevation 0.013 ft. The agreement 

is well within the limits of experimental error.

It is sometimes assumed that the existence of a 

secondary circulation may greatly reduce the superelevation. But this seems unlikely. For whatever 

may be the internal motion the whole stream momentum pub per unit depth must certainly be changing at 

the mean time rate u/rm. Hence the excess pressure

_O7

against the outer bank will be -—> and to oppose this a I'm

ti?b . superelevation — is required.

£frm

Data giving the surveyed water surface profiles across 

meander bends in rivers or across bends in canals are 

not published in detail. Leliavsky (1955, p. 124) states 

that leveling across the "Rhine confirmed a logarithmic 

variation analogous to case 1 above, but no data are 

given. In default of factual evidence, it will be assumed 

that the superelevation is given with fair accuracy by

z=u?b/grm

ENERGY CHANGES IN THE ZONE OF CURVATURE 

REVERSAL

Consider now the conditions in the transition zone 

in the neighborhood of the reversal of curvature, along 

that bank at which the water level falls.

There appear to be two rather different cases. Case 

A where this bank, locally, is inclined to the vertical; 

and case B where it is vertical.

Case A.—Banks inclined to the vertical. Let the 

following boundary conditions be assumed. The surface width of the stream is uniform and has a value 

equal to b, as previously supposed. But the cross 

section may vary in shape from section to section and 

may be asymmetrical to any reasonable extent. Defining a reach as the curved channel between successive 

reversals of curvature, all the conditions, both of crosssectional shape and of flow, are repeated from reach to 

reach, those in one reach being antisymmetrical, or in 

mirror image with respect to those in the next.

The asymmetry, and the variation, of the crosssectional shape will affect the velocity distribution; but

U2 b

as shown previously, the distance z=— through which

9rm

the side-water falls is unlikely to be affected to any 

appreciable extent.

Consider an element 2, shown shaded in figure 78A, 

of discharge of this side-water flowing over the inclined 

bank. The lower sketch shows the element of discharge 

as it approaches the section Y—Y' where the curvature 

reverses. It is reasonable to assume that g_ remains 

continuous and constant from a point upstream at 

which the curved channel flow has developed its full 

water-surface superelevation, as far downstream as the 

section Y—Y'. This superelevation is equal to z/2 

above the mean level. Moreover, within this restricted 

zone the curvature is small and may be neglected. 

Hence, it is legitimate to compare the action of this 

element 2 to flow in a straight channel over a local 

broad-crested weir.
PLOW RESISTANCE IN SINUOUS OB IRREGULAR CHANNELS 125

A

Position downstream 

of inflection point

Position upstream 

of inflection point

=^-17 "u

V

H -r

"I 

^ I

* dfi "•'

I "*

FIGURE 78.—A (upper). Cross sections above and below point of Inflection of channel curvature showing superelevated and depressed water surface along a sloping bank, 

(lower) Longitudinal profile near bank through the same reach Indicating energy grade line (upper dashed curve), water surface, and channel bed. B, Same as A above 

but In a channel having vertical banks.

For lack of sufficiently definite knowledge of the 

distribution of flow velocity across a curved channel, 

let it be assumed as an approximation that the velocities 

HI and u2 of g in the equilibrium zones upstream and 

downstream of the transition zone may both be represented by the mean velocity u. And assuming that 

the inclined bank extends to a sufficient depth, the flow 

depth d of g will be the same both upstream and downstream of the transition zone, figure 7SA. Hence the 

total energy level H will have fallen, through the transition zone, by the same distance z as the water level, 

though without ultimate change in either velocity or 

flow depth.

Now if we consider the case of straight-channel flow 

over a broad-crested weir, the flow energy,

(1)

can be reduced without accompanying energy dissipation by spill only by the amount represented by the difference between HI upstream from the weir, and He over 

the crest, as shown in the longitudinal profile sketched 

at the bottom of figure 78A. The quantity Hc is the

energy level at critical depth and the minimum value 

of the quantity ~—\"d for constant g.

An expression for this critical energy level for abroadcrested weir is as follows. 

Assuming a constant discharge g,

(D

which substituted in the Bernouilli expression, equation (1) above, gives

To obtain the -minimnTn value, He, of the above, differentiate and equate to zero.

d(d) 2g

and

9 9
126

Because

Then

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

(2)

This merely states that the energy grade line over the 

weir at critical depth is half as high again as the water 

level dc .

In the present case, because q=ud 

then

(3)

The maximum energy removal which can take place 

without accompanying dissipation is therefore given 

by Hi— Hc, or

(4)

This amount of energy is not lost by the stream as a 

whole, for it is transferable across to the opposite bank. 

It may be noted here that the critical velocityue 

which occurs directly over a broad-crested weir in the 

case of a straight channel should occur in the present 

conditions at the section Y-Y' of curvature reversal, 

the dynamics involved in the reversal having here 

taken the place of the static thrust of the real weir. 

Thus the fall in water level should begin upstream of 

this point. This is consistent with the experimental 

evidence. The contour lines of figure 76 show the fall 

in level from the equilibrium state achieved at the top 

of the mound to begin well upstream of the local point 

of inflection. Moreover figure 75 shows a system of 

standing wavelets indicative of the critical state situated 

just at the point of inflection and of curvature reversal.

ENERGY DISSIPATION

If z does not exceed Hi—Hc, the stream should experience no real energy loss arising specifically from 

the local deformation of the free surface; and for 

constant depth the overall resistance to flow should increase as the square of the mean velocity u.

But if z exceeds Hi—Hc, the excess energy removal 

z—(Hi—He) will be subject to dissipational loss. The 

descent of the side-water q will now assume the nature 

of a downward plunge or spill. How much of this 

excess energy removed from one side of the channel 

will be dissipated will depend on the conditions below 

where the plunging side-water meets the main stream. 

But it is relevant to express the whole of it in terms of 

measurable stream quantities.

U

3 /~9

U

Now

and

where

It*VU '

z—-

can be written d? F2

i/2

F2=— 

gd

Then if d be the mean flow depth of the stream, and 

letting

which is the excess energy to be removed, we have

(5)

It is now required to express the local d and F in terms 

of d and F. In this inclined bank case the local depth 

d is supposed independent of u. Let it also be assumed 

tentatively that d is independent of the channel 

curvature characteristic b/rm, being a function only of 

bank inclination and of its effect on the real frictional 

velocity gradient towards the water's edge. Then we 

can write d=ad where a is some numerical constant 

less than unity.

Thus

and

j?2 ~_.±_

d a

-=F2d

Thence

F2 __ (6a)

Case B. — Vertical banks. Because the local bank is 

vertical, figure 78B, in this case the equilibrium depth 

di upstream is not equal to d2 downstream as before, 

but d2=di—z. With the same approximation that 

U1 =u2=u, the required fall in the energy level is still 

z and equation 5 remains unchanged.

But d is now

and

F2=-

F2
FLOW RESISTANCE IN SINUOUS OR IRREGULAR CHANNELS 127

Equation 5 now becomes 

b

2/3 (6b)

In each case the theoretical threshold values Fl of F2, 

at which a departure from the square law is to be 

expected, is obtained by equating the dimensionless 

quantity f to zero and solving for F2 by a simple 

graphical method. In case A some arbitrary assumption must be made as to the value of a (see below).

COMPARISON WITH EXPERIMENT

The theoretical threshold values Fl of F2 are shown 

in figure 79, plotted against the channel characteristic

b/rm. The three continuous curves are obtained from 

equation 6a (inclined banks), for a=l, 0.8, and 0.5. 

The experimental values of Fl as read from the plots of 

figures are also shown. The precise experimental 

values of Fze for the longer channel radii are, however, 

difficult to determine from the plot; and these readings 

may be in appreciable error. The general trend of the 

theoretical FJ5 as the channel curvature is varied is 

nevertheless in remarkable agreement with experiment. 

The best fit for the particular cross-sectional shape of 

the experimental channel used appears to be that for 

a=0.8.

The broken curve of figure 79 is that obtained from 

equation 6b (rectangular channel). As will be seen, 

it approximates closely to the curve for equation 6a,

70A

Series number

o 

Runs at depth 0.133 ft

x 

Runs at depth .090ft

.4 .6 .8

WIDTH TO CURVATURE RATIO, b/rm

FIGURE 79.—Theoretical curves showing relation of square of threshold Froude number, JF«*, to index of channel curvature, for various assumed values of factor o. This 

factor is a measure of the limiting depth of water near a bank at which the local velocity ceases to approximate the mean flow velocity. Points plotted represent 

experimental values of F«* and b/rm for comparison with theory.
128 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

for a=l. These two curves both give F\=\ for the 

extreme case of a straight channel, 6/rm =0; whereas 

for a straight channel with inclined banks, Fl is smaller, 

at Fl=a in the general case where a<l. This is 

consistent with the simplifying assumption made that 

ul=u; that is, bank friction causes negligible reduction 

in the velocity with decreasing depth. So a can be 

regarded as defining the limiting reduced depth near 

the bank at which, in practice, the local velocity 

ceases to approximate to u, owing to bank friction.

The family of continuous curves in figure 80 show 

the values of f according to equation 6a as F2 is increased 

above the threshold Fc2. The two broken curves show 

the corresponding values according to equation 6b for 

the rectangular channel. It will be noted that in both 

sets of curves the gradient d f/d (jp*2) has a finite and 

maximum value at the threshold f=0 and F2=Fe2. 

This again is wholly consistent with the experimental 

results. The finite value indicates that the gradient 

of the overall resistance to flow changes discontinuously

Inclined bank 

(solid line)

Vertical bank 

(dashed line)

0.9

O.I 0.2 0.4 0.8

FIGURE 80.—Theoretical curves of the relation of flow resistance to square of Froude 

number at values of ?2 higher than the threshold, "f£. Full lines represent conditions with sloping channel bank, and dashed lines with vertical bank. Curves 

are developed for various values of curvature index, b[rm. In each case the zero 

ordinate value of f corresponds to the threshold value, FJ.

at F 2 as the experimental results show. And the subsequent decrease of gradient as jF2 is increased above 

~F<? can indeed be detected in the experimental plots 

by a close inspection of the individual plotted points, 

though in the absence of theoretical support, this tendency might well be taken as being due to systematic 

experimental error. The subsequent reversal of this 

tendency as shown by the plots to occur at still higher 

values of ~F2 may well be attributable to the onset of a 

general instability on the part of the whole flow as 

general resonance is approached towards F2=l.

The dimensionless quantity f bears the same relation 

to the specific and localized resisting force to which 

the local energy dissipation is equivalent as the dimensionless quantity s bears to the overall resisting force 

acting on the whole flow.

Over the length L containing one reversal of channel 

curvature the local energy dissipation is measurable by 

(pgd£ ) X (-4') where A' is the effective cross-sectional 

area of the side-flow g. Over the same length of channel 

the overall energy dissipation is pgLsXA. So the 

ratio of the overall resistance to the specific localized 

part of it which appears when F*>FC2 is

s L A

assuming of course that the whole of the excess energy 

Sf is in fact dissipated.

Of particular relevance to the behavior of natural 

rivers is the magnitude of the discontinuous jump in 

the rate of resistance increase with velocity increase 

which occurs at the threshold stage defined by Fc2 .

From the experimental data the additional resisting 

force applied to the length L of the whole flow when 

F2 exceeds Fc2 is measureable from the plots by (stt— sft) 

XpgAL (fig. 73). And in the limit when F2 is brought 

very close to F 2 the discontinuous change of gradient is

i d(F2)

where F1 refers to conditions immediately beyond the 

discontinuity and F to square-law conditions in which 

s/F2 is constant.

The corresponding theoretical change of gradient is

d(F2)

Hence theory may be compared with experiment by 

considering the values of A'/A in 

ds ds "" " L

^'_d(F' 2) d(F2) L 

A df 'd

(7)

d(F2)
FLOW RESISTANCE IN SINUOUS OB IRREGULAR CHANNELS 129

It would be difficult to predict the proportion of the 

excess energy frf (or £R if the hydraulic mean depth is 

preferred) which will be dissipated, because it cannot 

be assumed that the side-flow g is definably continuous 

below the critical depth de . Nor is it possible to calculate the cross section A' involved, for this would 

require a knowledge of the actual distribution of the 

flow velocity over the channel section. From an 

inspection of the water surface map, figure 76, the area 

A' involved might be guessed as being between onetenth and one-third of the whole area A. Thus if it is 

assumed that the whole of the excess energy frf is 

dissipated, then the expression on the right of equation 

7 when evaluated should give values between, say, 

0.1 and 0.3. And these values should be reasonably 

constant from one experimental channel to another.

From (6a)

df _b I 1.5 a 

~rm 2+3 a

6 1 . 0.431 , n Q /<rt = —— -+ _ . for a=0.8 (8)

rm 2^(f 2)l v

The precise experimental values of ds/d(F'2) at the 

threshold stage Fe2 cannot be determined from the 

plots. And the alternative adoption of the mean 

gradient through all the plotted points may introduce 

appreciable error. With this understanding, the resulting values of A' I A, as given by equationj are shown in 

the following table. Values of df/d(F2) are wholly 

theoretical, having been calculated from equation 8 by 

giving Fe the appropriate theoretical values shown in 

figure 80.

TABLE 4.—Calculated fractions of cross section of flowing water within which energy loss occurs because of spill resistance

1

Series of runs in figure No.

70B_________-_.___ — ___________

70C_ _______ ____________________

71A....... .....................

71B__ _ -_-__.-.._ _ _____ _ _.

2

Hydraulic

mean

depth (R,

in feet)

0 09

066

09

U66

09

066

f 09

\ 066

3

b/rn

0 22

19

39

33

73

6_!

1 19

99

4

d«/d(F'»)

0 0125

010

023

195

056

Oo7o

072

047

5

ds/d(F2)

0. 0085

.008

.0135

.012

.025

.012

.037

.025

6

Difference,

column 4

minus

column 5

0.004

.002

.0095

.0075

.031

.0255

.035

.022

7

dr/d(F')

0. 60

. oo

1. 156

. oo/

1.515

. 345

2. 14

. oD

8

L/R

22.2

30.3

21.5

-Jo. O

11. 7

lo. 1

12.8

17.4

9

A.' I A.

0. 148

. 105

.176

.264

.242

. 60o

.209

.207

It will be seen from table 4 that the ratio A'fA does 

in fact lie within the range of values expected on the 

assumption that the greater part of the excess energy 

is dissipated in the impact of plunging side-flow g against 

the main stream. And on this assumption the extra 

resisting force predictable from fundamental principles 

is adequate to explain the facts.

DISCUSSION

The foregoing conceptual model, based on fundamental dynamic principles, appears capable of reproducing all the salient experimental features. It explains the reason for the abrupt jump in the rate of 

increase of the flow resistance. It predicts correctly 

the threshold conditions at which this jump occurs, the 

magnitude of the extra resistance, and the general 

trend of its increase. The model also predicts the 

pattern and the scale of the water surface deformation.

The special case chosen is a simple one. But the 

underlying principles should apply to the effects of any 

single and discrete obstacle to the flow, provided the 

obstacle is such as to deflect some part of the flow

which has a velocity comparable with the datum velocity u. The effect would not be expected to result 

from the existence of a small obstacle in a shallow 

depth close to the water's edge, for here the velocity 

of the deflected flow might well be considerably less 

than u.

The stage Fe2 at which a discontinuous jump occurs 

in the rate of increase of resistance appears to be determined by the shape of the obstacle as defined by the 

horizontal curvature of its boundary rather than by 

its size. For a small obstacle the relevant cross-sectional area A' will be small in relation to the whole 

area A of the stream. But a number of small deflecting obstacles in a given length of channel may well give 

rise to as large an extra resistance as a single bend in 

the whole channel.

Moreover, it follows from the theory that it is not 

necessary that the obstacle shall form a part of a bank 

boundary. It could as well take the form of an island. 

This leads to the interesting question: to what extent 

will the effect occur in the case of a submerged island, 

such as an isolated bank or dune? This again leads
130 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

to a consideration of the effect of a meandering thalweg 

in an otherwise straight reach of river.

There thus appears to be scope for a great deal more 

experiment with fixed channels before conditions are 

further complicated by the presence of moving sediment.

A further interesting problem arises from the localized nature of the effect. A discrete source of energy 

dissipation has been shown to occur, and this must inevitably be associated with an equal resisting force applied at the flow boundary. How far, and where, is 

this extra resisting force concentrated at discrete portions of the boundary? It seems probable that if it is 

concentrated in the immediate neighborhood of the 

deflecting obstacle, which at low values of F would tend 

to protect the bank boundaries elsewhere, the obstacle 

would tend to be eroded away if above-threshold values 

of F were to be attained. As a consequence, the 

channel would widen.

PART 3. IMPLICATION OF THE RESULTS

APPLICATION OF EXPERIMENTS TO 

NATURAL CHANNELS

The present experiment appears to have a bearing on 

some aspects of the river width problem from two points 

of view. First, it was found in the experiments that an 

additional resistance became appreciable at a threshold 

defined in terms of the hydraulic Froude number, and 

that the threshold tended to be restricted to limited 

range of values. In most natural rivers, observed 

values of Froude number tend to be less than this 

threshold range of values. This warrants comment and 

further investigation. Second, the observation that 

spill resistance was associated with a concentration of 

stress on some portions of the channel boundary suggests a mechanism which under certain conditions may 

contribute materially to adjustments in width of natural 

channels. These two inferences will now be discussed.

The mean Froude number of a river cross section 

expressed in terms of mean velocity and hydraulic 

radius is

_ u

F=

From the average change of velocity and depth (Leopold 

and Maddock, 1953) Jn a downstream direction, the 

downstream change of F with discharge can be expressed

as

With increasing discharge at a given stream cross 

section, on the other hand, Froude number increases as 

a power function of discharge, the exponent having an 

average value of 0.23. The value of F is thus greatest 

at high stage; that is, bankfull stage, and for headwater streams. But even for such conditions, long 

reaches of natural channel practically never achieve 

critical flow (F=1.0) except in some streams whose 

beds are composed predominantly of sand. On the 

contrary, in most rivers values of Froude number are 

much less than unity. A sample of values which occur 

in rivers can be obtained from velocity and depth data 

compiled from gaging station data.

For the computation of bankfull value of F, it was

necessary to restrict the examples to those gaging stations for which an independent determination could be 

made of what discharge or stage represented bankfull 

conditions. These cases (see table 5) included stations 

at which the authors had made field surveys relating 

gage datum to flood-plain level, or where a "flood 

stage" had been determined by actual flood experience, 

or where flood frequency curves could be used to check 

determinations based on a channel cross section. Even 

with these restrictions the determinations of bankfull 

stage for 62 gaging stations in the United States are 

of varying reliability. Once the bankfull stage or discharge has been chosen, data on velocity and mean 

depth are readily available. The reliability of the final 

values of F is limited principally in the choice of 

bankfull stage.

TABLE 5.—Frequency distribution of values of Froude number, F, 

for rivers at bankfull stage

Range off

Number

of cases

6

11

13

12

7

8

2

1 1

1

1 J

Percent

of cases

10

18

21

19

11

13

3

5

<0. 20

0. 20-. 25

. 26-. 30

. 31- 35

. 36-. 40

. 41-. 45

. 46-. 50

. 51-. 55

. 56-. 60

>. 60

62 100

Table 5 shows that there is no single value of Froude 

number around which a large percentage of the cases 

are clustered. The salient feature of the data is the 

fact that there is a sharp cutoff at a value of 0.45, 

with only 8 percent (5 stations) of the cases exceeding 

this value, and only one case exceeding 0.60.

Were the value of Froude number at bankfull stage 

a random consequence of various combinations of bank 

materials, river size, and other features representing the 

range of conditions included in the sample, an approximation of a statistically normal distribution curve 

would be expected. In contrast the data do not ap-
FLOW RESISTANCE IN SINUOUS OB IRREGULAR CHANNELS 131

proximate a normal distribution. The sharp diminution observed at a value of 0.45 suggests strongly that 

there exists some kind of threshold beyond which processes operating in a natural channel alter the hydraulic 

relations at channel cross sections in such a way that 

the velocity-to-depth ratio is reduced, and thus the 

Froude number is limited.

Now the range of threshold values, Fc , observed in 

the present experiments is from 0.40 to 0.61 (table 3). 

Beyond Fc spill resistance becomes an appreciable and 

sometimes a dominant part of the flow resistance. The 

coincidence of the range of Fc observed and the cutoff 

value of Froude number in natural rivers is remarkable. 

Some inferences may be drawn from the theory presented previously which suggest that this coincidence is not 

merely fortuitous.

From equation 6a the graphs of figure 79 were plotted showing the relation of the square of the threshold 

Froude number, F02, to the curvature criterion, b/rm, 

for various values of the parameter a. As explained, 

this parameter is defined as a=d/d, and represents the 

ratio of local depth, d, over an inclined bank to the 

mean depth of the whole channel, "d, beyond which the 

local near-bank velocity no longer is approximated by 

the mean flow velocity in the whole channel. Thus the 

parameter a depends on the relation between bank inclination and the effect of bank friction in reducing the 

local velocity over the inclined bank. For a given bank 

inclination, the smoother the bank surface the closer 

to the water's edge would the local velocity approximate the mean velocity of the channel; and thus the 

smaller would be the value of a. Looking at figure 79, 

one can see that for any value of 6/rm, the value of the 

threshold Froude number, Fc2, depends on the value of 

a, and by the above reasoning, depends then on the 

roughness of the bank boundary.

The experimental values of Fc2 plotted on figure 79 

suggest a value of a equal to 0.8 for the experimental 

channel. If in natural rivers the value of a were 

slightly smaller, the threshold Fe2 would be appreciably 

reduced, as can be seen in figure 79.

It seems possible that the wide range of values of 

Froude number shown for rivers in table 5 may be due 

to variations in bank roughness from river to river. 

Also differences in Reynolds number from river to 

river would probably cause some variation in a. High 

velocities would probably extend further toward the 

banks at high Reynolds number than at low, and to the 

extent that this were true, high Reynolds numbers 

would tend to be accompanied by lower values of a and 

Fe.

Thus, the theory suggests why the Froude numbers

for various rivers might vary through a range of values 

but are quite sharply limited by an upper value of 

about 0.45.

It is necessary to discuss the elements of a physical 

mechanism by which the onset of spill resistance would 

set the upper limit for Froude number in a river. The 

experiments suggest that this mechanism operates 

through the fact that spill resistance represents a localized and concentrated dissipation of energy near the 

banks. It might be expected, therefore, that this local 

dissipation affects bank erosion, or erosion of the bulge 

or projection causing the spill process. Further, the 

energy dissipation increases rapidly after the threshold 

value of Froude number is exceeded, and thus the 

erosion process would probably tend to increase in 

intensity with increase in F above Fe .

SUMMARY

Little is known hi terms of quantitative mechanics 

about how river width is determined and maintained. 

Common observation indicates that among rivers of 

comparable discharge, those with cohesive banks tend 

to be narrow and deeper than those having sandy or 

less cohesive banks. Under certain conditions vegetation must exert an important control on river width 

by encroachment on bars or banks that then become 

stabilized by roots. The stems and exposed roots also 

locally reduce flow velocity and thus shield the bank 

materials from stress.

Even in the nearly ideal condition of cohesionless 

material, details of width-determining processes are but 

poorly known.

In general the experiments imply that an appreciable 

part of the whole flow resistance pgsR of an irregular 

channel is due to internal energy loss in eddies and 

vortices at local deflections. The thrusts are probably 

borne by the projecting portions of the banks. A considerable portion of this thrust will consist of components normal to the local boundary and therefore of a 

nonerosive nature.

As a result, the river bed on the whole must be 

relieved of a portion of the overall tractive stress pgsR. 

Certain small parts of the bed may experience an excessive stress, but these may be expected to relieve themselves of the tangential component of this excess by 

receding as local depressions.

It may well be, therefore, that a river which is deficient of bed material of large enough size to stabilize a 

straight channel may become stabilized in an irregular 

channel merely by creating random bank projections. 

In computing the bed stress necessary to dislodge bed 

material, the present experiments suggest that the stress
132 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

at initial movement may in reality be less than the 

distributed stress given by pgsR.

The same reasoning may have implication to river 

meanders when the basic hydrodynamic mechanism 

becomes better understood. The meander form may 

represent one of the channel patterns to which the above 

generalization applies, only the random bank projections 

are replaced by somewhat symmetrical channel curves.

Finally, the experiments show that a great deal more 

needs to be known about open flow in irregular fixed 

channels in the absence of sediment movement, and 

that research on this subject might well proceed hand 

in hand with further research on sediment movement 

per se.

REFERENCES CITED

Einstein, H. A., and Barbarossa, N. L., 1952, River channel 

roughness: Am. Soc. Civil Engineers, Trans., v. 117, p. 

1121-1146.

Leliavsky, Serge, 1955, Introduction to fluvial hydraulics: London, Constable.

Leopold, L. B., and Maddock, Thomas, Jr., 1953, The hydraulic 

geometry of stream channels and some physiographic 

implications: U.S. Geol. Survey Prof. Paper 252.

Leopold, L. B., and Wolman, M. G., 1957, River channel patterns—braided, meandering, and straight: U.S. Geol. Survey 

Prof. Paper 282-B.

Wolman, M. G., and Leopold, L. B., 1957, River flood plains—- 

some observations on their formation: U.S. Geol. Survey 

Prof. Paper 282-C.
FLOW RESISTANCE IN SINUOUS OB IRREGULAR CHANNELS 133

APPENDIX

Flume data for fixed-grain channel having various sinuosities

Serial 

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19 

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

Discharge 

(Q,in 

cfs)

0. 0255

.032

.038

.045

.052

.058

.062

.024

.0395

.055

.063

.082

. 103

.115

.017

.025

.031

.037

.042 

.047

.053

.059

.064

.0305

.043

.057

.068

.078

.087

.096

. 108

.115

.023

.030

.037

.042

.046

.051

.041

.055

.066

.077

.084

.092

.022

.028

.031

.033

.036

.0375

.04

.030

.038

.046

.051

.058

.061

.066

.019

.027

.029

.032

Centerline 

water 

depth 

(feet)

0.090

.090

.090

.090

.090

.090

.090

.133

.133

. 133

.133

.133

.133

.133

.090

.090

.090

.090

.090 

.090

.090

.090

.090

. 133

.133

.133

.133

. 133

.133

.133

. 133

. 133

.090

.090

.090

.090

.090

.090

.133

. 133

.133

.133

. 133

. 133

.090

.090

.090

.090

.090

.090

.090

. 133

. 133

. 133

. 133

.133

.133

.133

.090

.090

.090

.090

Hydraulic 

mean 

depth 

(R, in 

feet)

0.066

.066

.066

.066

.066

.066

.066

.090

.090

.090

.090

.090

.090

.090

.066

.066

.066

.066

.066 

.066

.066

.066

.066

.090

.090

.090

.090

.090

.090

.090

.090

.X)90

.066

.066

.066

.066

.066

.066

.090

.090

.090

.090

.090

.090

.066

.066

.066

.066

.066

.066

.066

.090

.090

.090

.090

.090

.090

.090

.066

.066

.066

.066

Water 

surface 

slope («)

0. 00125

. 00175

. 00295

. 00395

.00505

.0062

. 00725

.00033

.00097

. 00146

. 00216

. 00375

.0055

.00754

.00035

.00097

.00166

. 00246

. 00337 

. 00448

. 00559

. 00723

. 00853

.00036

.00097

.00156

. 00239

. 00319

. 00448

.00559

. 00723

. 00853

. 00124

. 00255

. 00370

. 00544

. 00686

. 00914

. 00124

. 00255

. 00370

. 00544

. 00686

. 00914

. 00112

. 00217

. 00297

. 00382

.0050

. 00654

.0080

. 00122

. 00224

. 00299

. 00396

. 00516

. 00658

.0081

.0022

.0042

.0058

.0072

Mean 

velocity 

(u,.in 

fps)

0.560

.703

.835

.989

1.14

1.27

1.36

.32

.53

.73

.84

1.09

1.37

1.53

.374

.549

.681

.813

.923 

1.03

1.16

1.30

1.41

.407

.573

.760

.907

1.04

1.16

1.28

1.44

1.53

.505

.659

.813

.923

1.01

1.12

.547

.733

.880

1.027

1.12

1. 227

.484

.615

.681

.725

.791

.824

.88

.400

.51

.61

.68

.77

.81

.88

.417

.593

.637

.703

w»

0.314

.494

.697

.978

1.30

1.61

1.85

.102

.28

.54

.71

1.20

1.89

2.34

. 140

.301

.464

.661

.852 

1.06

1.35

1.69

1.99

. 166

.328

.578

.823

1.08

1.35

1.64

2.07

2.34

.255

.434

.661

.852

1.02

1.25

.299

.537

.774

1.05

1.25

1.51

.234

.378

.464

.526

.626

.679

.774

. 16

.26

.38

.46

.60

.656

.774

.174

.352

.406

.494

*~gR &_«! 

0.148

.233

.329

.461

.613

.759

.873

.035

.096

. 185

.243

.413

.65

.81

.066

. 142

.219

.312

.402 

.500

.637

.797

.939

.057

.113

. 199

.284

.372

.466

.566

.714

.807

. 120

.205

.312

.402

.481

.590

.103

.185

.267

.362

.431

.521

.110

. 178

.219

.248

.295

.320

.365

.055

.088

. 130

. 159

.206

.226

.267

.082

.166

. 192

.233

Froude 

number 

(*)

0.38

.48

.57

.68

.78

.87

.93

.187

.31

.43

.49

.64

.81

.90

.26

.38

.47

.56

.63 

.71

.79

.89

.97

.24

.34

.45

.53

.61

.68

.75

.84

.90

.35

.45

.56

.63

.69

.77

.32

.43

.52

.60

.66

.72

.33

.42

.47

.50

.54

.56

.60

.23

.30

.36

.40

.46

.48

.52

.29

.41

.44

.48

Temperature (°F)

63

63

63

63

63

63

63

63

59

59

59

59

59

59

64

64

64

64

64 

64

64

64

64

64

64

64

64

64

64

64

64

64

66

66

66

66

66

66

66

66

66

66

66

66

60

60

60

60

60

60

60

60

60

60

60

60

60

60

64

64

64

64

Remarks

Data plotted in fig. 7QA. Straight

channel.

Data plotted in fig. 70B.

Repeating distance 2I/=4.00 ft.

Amplitude 2-4=0.30 ft.

Thalweg length =4.1 ft.

Sinuosity (thalweg length: valley 

length) = 1.024.

Data plotted in fig. 70C.

Repeating distance 2L=3.87 ft.

Amplitude 24=0.54 ft.

Thalweg length =4.09.

Sinuosity (thalweg length: valley

length) = 1.056.

Data plotted in fig. 71 A.

Repeating distance 2L=2.12 ft.

Amplitude 2-4=0.30 ft.

Thalweg length =2.22 ft.

Sinuosity (thalweg length: valley

length) = 1.048.

Data plotted in fig. 71B.

Repeating distance 2L=2.30 ft.

Amplitude 2-4=0.54 ft.

Thalweg length=2.60 ft.
134 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Flume data for fixed-grain channel having various sinuosities—Continued

Serial

No.

63

64

65

66

67

68

69

70

71

Discharge

«?,in

cfs)

0.034

.033

.041

.050

.054

.055

.058

.0615

.062

Centerline

water

depth

(feet)

0.090

. 133

. 133

. 133

. 133

. 133

. 133

. 133

. 133

Hydraulic

mean

depth

(R, in

feet)

0.066

.090

.090

.090

.090

.090

.090

.090

.090

Water

surface

slope (s)

0. 0086

.0022

.0039

.0057

.0072

.0078

.0088

.0102

.0118

Mean

velocity

(u, in

fps)

0.747

.440

. 547

.667

. 720

. 733

.773

.820

.827

u2

0.558

. 194

.299

.445

.518

.537

.598

.672

. 684

— ifi

F2 = J1

gR

0. 263

.067

. 103

. 153

. 179

. 185

.206

. 232

.236

Froude

number

(F)

0.51

.26

. 32

.39

.42

.43

.45

.48

.49

Temperature (°F)

64

64

64

64

64

64

64

64

64

Remarks

Sinuosity (thalweg length: valley

length) = 1.1 3.

The following data were not included in the figures or analyses in the present paper because the number of runs in each channel was 

insufficient to define the relation between slope and square of Froude number. The data are published here for the record.

72

73

74

75

76

77

78

0.033

.041

.048

.041

.060

.078

.093

0.090

.090

.090

. 133

. 133

. 133

. 133

0.066

.066

.066

.090

.090

.090

.090

0. 00311

. 00566

.0095

. 00119

. 00311

. 00566

.0095

0.725

.90

1.054

. 547

.80

1.04

1.24

0.526

. 81

1. Ill

. 299

.64

1.08

1. 54

0. 248

. 382

.525

. 103

. 221

.372

.53

0.50

.62

.72

.32

.47

.61

.73

71

71

71

71

71

71

71

Sinuous channel.

Repeating distance 2L=4.00 ft.

Amplitude 24 = 1.14 ft.

Thalweg length =4. 71 ft.

Sinuosity (thalweg length: valley

length) = 1.178

Radius of curvature, rm =1.14 ft.

U.S. GOVERNMENT PRINTING OFFICE : 1963 O—671152