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Flow Resistance in Sinuous

or Irregular ChannelsGEOLOGICAL SURVEY PROFESSIONAL PAPER 282-D

Flow Resistance in Sinuous or Irregular ChannelsBy 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-____-___-__-___

PageIV

111

111113114115117119119121

Part 2. A theoretical model of energy loss in curvedchannels, 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

123124126127129

130131132133

ILLUSTRATIONS

PLATE 3. Straight and sinuous channels molded in sandof laboratory flume______________ Facing 114

4. Spill phenomenon due to bank projection inBaldwin Creek, Wyo_________ Facing 122

FIGURE 68. Diagram showing postulated relation ofstress to flow velocity._______________ 114

69. Cross section of channels used in experi- ments.-------..-_____________________ 115

70. Diagrams of resisting stress as function ofsquare of Froude number...__________ 116

71. Diagrams of resisting stress as function ofsquare of Froude number_______________ 117

72. Experimental relation of T,-/T. to fc/r_,_______ 11973. Diagram showing postulated relation of s

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

PageFIGURE 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 sur- face in sinuous channels-.---.-..------- 120

76. Topographic map of water surface at Froudenumber above threshold-_______ -___. 121

77. Diagram of sinuous channel-___________ 12278. 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 of6/r_........ _-___ -- --. 127

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

TABLES

TABLE 1. Characteristics of channel patterns investi- gated.. ________________________________

2. Experimental values of Ti/r t in channels of various indices of curvature_____________

3. Experimental values of threshold Froude num- ber for channels of various indices of cur- vature. --___-_-________________--___---

Page

117

118

119

PageTABLE 4. Calculated fractions of channel cross section

within which energy loss due to spill is con- centrated______________ ______ __ 129

5. Frequency distribution of values of Froudenumber at bankf ull stage in rivers _________ 130

m

SYMBOLS

Units

a A measure of limiting depth near the bank at which local velocityno longer approximates mean velocity.

A Semiamplitude of repeating bend in channel- ____________________ LA Cross-sectional area; A' cross-sectional area within which spill

resistance causes energy dissipation __________________________ L2b Width of flowing water_______________________________________ Lc Resistance coefficient.C Chezy factor.c? Local depth of water __ ______________________________________ Ld Mean depth______________________________. __________________ LE Discrete resistance. / Friction factor.

at

F Froude number at a particular point,

"~F Mean Froude number for whole flow -7='

JgdFe Threshold value of mean Froude number. g Acceleration of gravity__-__-_-_____-________----_-____---__-_ LT~2H Flow energy level_________________________________._____-_____ Lh Local depth of water __ ______________________________________ LI A length along a conduit _ ___________________________________ LL Half the repeating distance analogous to half meander length in a

river _____________________________________________________ LQ Discharge___________________________________________________ L3^1q Discharge per unit width-_-_-___-____-__-___--_-_---__-------- L2T-1R Hydraulic mean depth__--___-_--_-____--__--------------__-_- Lrm Mean radius of curvature __ _ ________________________________ Ls Water surface slope.u Local flow velocity. __________________________________________ LT-1u Mean value of flow velocity._________________-_-_______-------- LT" 1s A vertical distance between a local superelevated water surface and

its accompanying depressed elevation_________ ________________ Lp Mass density of water __ _ _________________________-------_ ML~3T Shear on boundary, force per unit area_____-_------_-------- ML-1T~ 2Tt Shear due to eddies, secondary circulation, and internal distortion ML~ 1T~ 2 T, Shear due to skin friction___________________--------__---_- ML~1T~2Tg Shear due to energy dissipation by spill, or collapse of streamline

continuity. ___________________________________________ ML~1T~2f 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 resist- ance. 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 remark- able 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 resist- ances, E, being mostly of an indefina