The Behavior of 

Straight Open Channels 

With Movable Beds

GEOLOGICAL SURVEY PROFESSIONAL PAPER 622-A

The Behavior of 

Straight Open Channels 

With Movable Beds

By THOMAS HADDOCK, JR.

BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 622-A

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1969
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 - Price $1 (paper cover)
CONTENTS

Abstract _ ___------------_---_--__------

Introduction. ____________________________

Relation between sediment load and velocity 

Unigranular material _ _______________

Mid- velocity range... _ ____________

Low- velocity range. __ _ ___________

High- velocity range, ______________

Mixtures of sediment sizes. ____________

Mid-velocity range _ _ ____________

Low- velocity range __ _ ___________

High-velocity range. ______________

Extrapolation from flume studies ___ _ _ 

Two-dimensional flow_ ____________________

Independent versus dependent variables. ______

Constraints ________________________________

Discharge and sediment load as independent 

variables _ __________________________

Discharge and slope as independent 

variables __ _________________________

Variance of shear stress and friction 

factor. __________________________

Variance of velocity and friction 

factor. __________________________

Page

Al

1

3

5

6

7

12

14

15

18

19

22

25

25

28

30

30

33

33

35

Pnge

Two-dimensional flow Continued 

Slope Continued

Constraints Continued

Discharge and slope as independent 

variables Continued 

Variance of concentration and friction 

factor.__________________________

Behavior of flumes operated at constant 

depth-______________________________

Summary of slope as an element in stream 

behavior__ _____________________________

Three-dimensional flow._____________________________

Width as an independent variable_______________

Width as a dependent variable.__________________

Sediment size as dependent or independent 

variable _________________________________

Summary of width as a dependent variable. _ 

Stable channels.._______________________________

Gravel channels ____________________________

Sand channels._____________________________

"Regime" channels.________________________

Nonregime channels.________________________

Conclusion.________________________________________

References.___-____-____---_-_____-_-_-_____--._-_-

A 37 

39

48

50

50

52

54

56

56

57

60

61

66

67

69

ILLUSTRATIONS

Pj.ge 

FIGURE 1-6. Graphs showing relations between velocity and load:

1. In the Gilbert experiments, sand B__ ________________________--___-_--_--_---_---_--__------- A4

2. In the Gilbert experiments, sand E---_____-____-___-_-__---_------_----__------------------- 5

3. In the Gilbert experiments, sand G_______________________-_--___-_----_-----_---_-_----_-- 6

4. In the Williams experiments-.____-________________-________-__-_-_----__--_---------------- 7

5. In the Simons and Richardson experiment._______________-_-______-_--_-_-_-___-----_----_-- 8

6. In the Waterways Experiment Station studies and Gilbert experiments___________--__---_---__-- 9

7. Dimensionless plot of mean velocity, fall velocity, load, and discharge per foot of width in the mid-velocity

range.__----________-_________________________________-__-________---_-------_---_---__-- 10

8-12. Graphs showing:

8. Relations between velocity and load, unit discharge, and size of sediment for unigranular material in

the mid-velocity range_____--_-_-_-____------___-_---_---------_----------------------- H

9. Values for the coefficient ki in equations 4 and 5 with variations in the diameter and fall velocity of

sediment particles.______________________________________________-__--___-__----------- 12

10. Relations between velocity and effective load and size of sediment in the low-velocity range- _________ 13

11. Relations between load and discharge, slope, and sediment size, in the high-velocity range. ___________ 14

12. The velocity-load relation for unigranular material and a unimodal mixture, __________-_--_---_---_ 15

13. Dimensionless plot of velocity, fall velocity, and load for sands A through H, Gilbert experiments. ___________ 16

14-49. Graphs showing:

14. Relations between velocity and load, unit discharge, and mean grain size for sediment mixtures in the

mid-velocity range.__________.__.___________-_-_________--_--__---_------------------- 17

15. Relations between velocity and load for sediment mixtures. Flume data-__________-------__------ 18

16. Relations between velocity and effective load and mean grain size for sediment mixtures in the lowvelocity range._______________________________________________-----_-_--------------- 20

17. Relations between velocity and effective load and mean grain size for sediment mixtures in the lowvelocity range.__________________.____________________-____-__----_------------------- 21

18. The behavior of the sediment load of the Colorado River at Yuma, Ariz____________-___--_------- 22

in
rv CONTENTS

FIGURES 14-49. Graphs showing Continued page

19. Relations between load and unit discharge, slope, and sediment size for sand mixtures in the highvelocity range ________________________________________________________________________ A23

20. Relation between sediment discharge and water discharge for Mountain Creek, N.C., and Vrest

Goose Creek, Miss____-_______-___________---__---_--___--_--__-_--___-_---___________ 24

21. Relation between discharge of coarser grades of sediment and water discharge of the Colorado River

at Yuma, Ariz_______-____________________________________________-_____-_-___________ 25

22. Relations between velocity and unit discharge, slope, and size of sediment for unigranular material

in the mid-velocity range_ ______________________________________________________________ 26

23. Relations between velocity and unit discharge, slope, and mean grain size of sediment mixture in the

low-velocity range__________________________-____-_---__--_____-----------_-_________ 27

24. Relations between velocity and unit discharge, slope, and mean grain size of sediment mixture in the

low-velocity range.____________________________________________________________________ 28

25. Values of sediment size parameters from equations 19 and 20_ ____________________________________ 32

26. Relations between velocity and unit discharge, slope, and mean grain size of sediment-.____________ 34

27. Relations between velocity and unit discharge, slope, and mean grain size of sediments for certain

canals and natural streams.________________-____----_--___-____--_--__------___________ 35

28. Relation between unit discharge and depth for Mountain Creek near Greenville, N.C______________ 36

29. Relation between unit discharge and depth for West Goose Creek near Oxford, Miss________________ 36

30. Relation of velocity to hydraulic radius for San Francisco riverside drain near Bernardo, N. Mex_____ 37

31. Relations between velocity and the product of unit discharge and slope for locations on the Colorado

River at Grand Canyon, Ariz., and below Davis Dam_____________________________________ 38

32. Relation between unit discharge and depth for Colorado River at Taylors Ferry, Ariz_____________ 39

33. Relation of width, depth, velocity, water-surface elevation, and stream-bed elevation to discharge,

Colorado River at Yuma, Ariz________________________________________________________ 39

34. Relations between velocity and unit discharge and slope.______________________________________ 40

35. Relations between velocity and slope when depth is approximately constant-. _ _____________________ 41

36. Relations between velocity and unit discharge, slope, and mean grain size of sediment, small reci^culating flumes_______________________________________________________________________ 42

37. Relations between velocity and the product of unit discharge and slope for 0.4 and 0.45 mm sard__ 43

38. Relations between velocity and slope for various depths of flow and different sizes of sand________ 44

39. Relation between velocity and slope for certain depths_________________________________________ 45

40. Relation between depth and slope at constant discharge______________________________________ 46

41. Relation between mean velocity and qS for uniform and mixed sediments of about 0.3 mm mean diameter in flume experiments..________________________________________________________ __ 47

42. Relation between width and slope, the Rio Grande near Vinton, Tex____________________________ 51

43. Variations of width and slope of the water surface with discharge and relation of slope of water surface

to size of bed material, Brandywine Creek, Pa_____________________________________________ 55

44. Variations of mean depth and mean velocity with discharge in the downstream direction at the bank

full stage, Brandywine Creek, Pa__________________--_--___--_-_-_----__-__--___---_____- 56

45. Relation between tractive force and sediment size with slope a dependent variable-_______________ 57

46. Recommended limiting tractive forces (or shear stresses) for canals,_____________________________ 58

47. Relation between tractive force and sediment size with slope an independent variable._____________ 59

48. Relation between width and discharge, San Luis Valley, Colo., canals.___________________________ 60

49. Relations between sediment size and discharge and slope, San Luis Valley, Colo., canals____________ 60

50. Aerial photograph of the Colorado River near the Nevada-California boundary showing systematic bed

forms under water.___________________________-_______-_______-_____-_--_--_-__--__------__-- 61

51. Strip aerial photograph of the Rio Grande near Vinton, Tex., at low flow showing river reworking

systematic bed forms that are relics from higher discharges.______________________________________ 61

52. Graph showing relations in equation 70_____________________________________________________________ 66

TABLES

Page 

TABLE 1. Behavior of flumes of unit width and same sediment sizes with different independent and dependent variables_ _____ A29

2. Data for the Marala-Ravi Link Canal relating to behavior__________-_____________-__-__---__---_------ 64

3. Data for the Marala-Ravi Link Canal___________________________-_-__---_____-____--_--_-------_------ 65
SYMBOLS

Symbol Definition

a A constant.

& An exponent of Q or q relating discharge to width.

c A constant.

d The diameter of a sediment particle, in feet.

e The base of natural logarithms.

e An exponent of depth.

/ An exponent of Q or q relating discharge to depth.

g The acceleration of gravity, in feet per second per second.

h An exponent of concentration.

j An exponent of Q or g relating discharge to load.

k A constant.

ki A variable which is a function of sediment size and fall velocity.

m A variable in an equation by Velikanov (1949).

m An exponent of Q or g relating discharge to velocity.

q The unit discharge of water, in cubic feet per second per foot of width.

q° A constant.

t An exponent of slope relating slope to velocity.

t Time, in seconds.

u An exponent of slope relating slope to depth.

w Fall velocity of a sediment particle, in feet per second.

x An exponent of load relating load to depth.

z An exponent of Q or g relating discharge to slope.

A A constant.

B A constant.

Bf Bed form.

C The concentration of the sediment discharge, in parts per million.

D Mean depth of channel, in feet.

F A Froude number.

G Total sediment load.

J Head loss of a water sediment mixture moving in a pipe.

Je Head loss in a pipe with clear water of the same discharge as in /.

K A coefficient. i

L Rate of sediment movement, in pounds per second per foot of width.

L1 Load to the first power.

La A constant.

Le An effective load, in pounds per second per foot of width.

Q The total water discharge of the stream, in cubic feet per second.

Q, The total sediment discharge of a stream, in pounds per second.

S Slope or energy gradient.

S1 Slope to the first power.

/S° A constant.

V Mean velocity of the flowing stream, in feet per second.

Vb Velocity of water at the bed, in feet per second.

Vc A critical velocity at which sediment motion begins, in feet per second.

W Width of channel, in feet.

a An exponent of velocity relating velocity to load.

/8 An exponent of load relating load to slope.

 y The weight of water, in pounds per cubic feet.

M The dynamic viscosity.

v The kinematic viscosity.

* A function of.

TT 3.1416.

P Mass density of water, in mass per cubic foot.

ps The mass density of sediment, in mass per cubic foot.

<r log D The standard deviation of the logarithm of the depth.

<r log Q The standard deviation of the logarithm of the discharge.

a log V The standard deviation of the logarithm of the velocity.

ff log W The standard deviation of the logarithm of the width.

T A shear force.

AT The submerged weight of the sediment, in pounds per cubic feet.

r The friction factor.

BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

THE BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

By THOMAS MADDOCK, JR.

ABSTRACT

The chief variables related to the behavior of alluvial channels 

are mean velocity, depth,- width, slope, quantity of sediment 

transport, and sediment size composition. Except for a relation 

between mean velocity and quantity of uniform-sized sediment 

in motion, relations among these variables are to some degree 

indeterminate. Because changes in one variable result in mutual 

adjustment of all other variables, the actual results depend on 

the constraints that may be imposed, either natural of artificial.

Analysis of flume data indicates that no single simple equation 

describes the relation between mean velocity and load, the sediment discharge per foot of width of channel. Three separate 

equations for three ranges of velocity are developed. These ranges 

are called (1) the mid-velocity range, in which most natural 

streams and canals flow (hence, the most important); (2) the 

low-velocity range, associated with beginning of sediment motion; 

and (3) the high-velocity range, which is infrequently encountered in the field and is somewhat uncertain. Different equations 

for unigranular sediment and for sediment mixtures are proposed. 

These equations are based on flume data and are tested with 

data collected in the field.

Relations among mean velocity, depth, and slope are indeterminate because of the variability of the hydraulic roughness 

caused by changes in the characteristics of the bed forms. 

Consistent relations among the variables exist only when certain 

constraints are present and result from minimization processes. 

Velocity, depth, and slope may be expressed in terms of a unit 

discharge q:

V<Xqm, DdCqf, SdCq*.

Then the exponent of the independent variable is an index of 

the deviation of the dependent variable for any change in the 

independent variable. The variability of the dependent variable 

or of a function of the dependent variable is minimized when 

the variance of the variable or its function is minimized. If 

two different functions are jointly minimized, the sum of the 

variances of the two functions is minimized. For example, to 

minimize the variance of DS, which can be expressed as (/+2) 2, 

set the derivative with respect to / equal to zero from whence 

/= s. If the variance of the shear yDS and the friction factor 

qDS/V 2 are to be jointly minimized,

(/=22) 2 +(/+2-2m) 2 _» minimum.

Although there are a number of constraints, the principal one 

is the constraint of joint minimization of the variances of the 

shear stress and of the friction factor. The nature of the result 

is determined by the selection of the dependent and independent 

variables. If velocity, depth, and slope are all dependent variables, Voc(qS)°'*' If unit discharge and slope are independent 

variables, and load, velocity, and depth are dependent variables, 

Vcc(qS)°-*. If discharge is an independent variable and depth is 

not permitted to vary, then the variance of the friction facto" 

or the variance of the shear stress is minimized that is, eithe~ 

the friction factor or the shear stress is constant. Results of flum^ 

and field measurements are compared with the hypothesized 

values. I emphasize that these relations are not unique but onlj 

the most probable.

The effect of variation of width is considered. It is show"1 

that width and slope are related and that a three-dimension?,! 

analysis is associated with a minimization process. But, in contrast to using variances of the shear stress and the friction factor, 

the joint variance of certain ways streampower may be expressed 

is minimized. These ways are, power per unit length per unit 

discharge, or S; power per unit of area of channel bed, or qF: 

and power per unit length of channel, or QS. The Lacey equations 

are developed by using this approach, as are other "regime" 

equations for specific conditions found in streams with beds cf 

sand or of rarely moved coarse material.

Finally the problem of stable channels is considered. Recon?- 

mendations are made for equations to be used in design of stable 

channels and values of coefficients are expressed based on studies 

of flume and field data.

INTRODUCTION

The objective of workers in the field of movable-bed 

hydraulics has been to find unique or determinate 

relations among velocity, depth, slope, and the rate of 

sediment movement. The thesis of this report is that 

this objective is, to some degree, impossible to achieve. 

Although there are usable relations among discharge, 

mean velocity, and load for the transport of sediment 

of uniform size in a channel of symmetrically distributed 

velocities, it will be shown that these determinate 

relations are only partially applicable for mixtures cf 

sediment sizes and for odd-shaped channels. Further, 

I will show that any function containing slope as a 

parameter will be indeterminate within wide limits. 

A unique relation will be found only when constraints, 

natural or artificial, limit the possible combinations of 

depth, slope, velocity, and load.

This report will attempt to identify and describe 

some of the constraints that control the behavior of

Al
A2 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

alluvial channels. These will be the most obvious and, 

it is hoped, the most important.

Of all the elements that make up a plan of waterresources development, those concerning the movement of water in alluvial channels are the least understood. This is because the bed, where most of the 

stream energy is dissipated, constantly changes form. 

Thus, the flow in these channels is locally unsteady and 

nonuniform, and the departures from a steady or 

uniform state may vary widely.

It has been recognized since Gilbert's (1914) time 

that a river system is the result of the movement of 

both water and debris. The mutual adjustment of 

certain variables has not been generally recognized 

despite the fact that Rubey (1952, p. 131) clearly 

stated: "Summarizing, if discharge, load, grain size, 

and sorting are considered the controlling factors, then 

velocity, slope, width, and depth of channel are dependent variables that are affected not only by the 

independent variables but also by one another." (Author's 

emphasis.)

Unfortunately, Rubey's paper, although completed 

in 1936, was not published until 1952. During that 

period, analysis of the flow of water in alluvial channels 

followed, in general, the same pattern as that for channels 

with rigid boundaries. It is but speculative to consider 

whether the early publication by Rubey might have 

prevented this type of analysis and its unfortunate 

results. For example, even today it is not fully recognized that when given a depth, a slope, and a size 

of bed material, the velocity is indeterminate. What is 

lacking is a measure of the hydraulic roughness associated with some characteristic of the deformation of 

the bed.

Deformation of the bed is accomplished by the 

movement, in certain specific ways, of many grains of 

sediment. Because the grains must move with the 

current in a general downstream direction, deformation 

of the bed results in the sediment load of streams. 

Under a particular set of conditions of discharge, slope, 

and size of material, a given bed form requires a specific 

amount of sediment in motion. Thus, the load of a 

stream becomes an important factor in the consideration 

of the hydraulics of alluvial streams.

Another characteristic of alluvial streams is an ability 

to adjust width, depth, velocity, and slope. The fact 

that all rivers have the same general features indicates 

that this adjustment is constrained by some phenomena 

to within rather specific limits. Given a discharge and 

a load, as Rubey (1952) noted, the width, depth, velocity, and slope of a stream must be the result of 

mutual adjustments within a set of limits not yet defined in an acceptable manner.

These conditions clearly differ from those found in

rigid-boundary hydraulics and must be treated differently if they are to be analyzed successfully. One of 

the possible methods of analysis is to corsider the 

problem as indeterminate and use statistical p ^ocedures 

as much as possible. Thus, the variation in behavior of 

experimental alluvial channels can be attributed to 

errors of measurement, which are normally distributed, 

and to systematic variation due to constraints placed 

on the flow system because of the size of the channels, 

the magnitude of the discharge, certain physical limits 

of stability, or the way in which the experiment was 

conducted.

This report will consist of three major par^s: (1) an 

evaluation of the role of sediment transport in fluvial 

hydraulics; (2) an analysis of two-dimensional flow 

width will not be considered as having a specific influence; and (3) a study of the effect of the introduction 

of width three-dimensional flow on the principal 

variables and a consideration of the elements of channel 

stability. In the second and third parts, variations in 

load, velocity, depth, and slope will be evaluated.

The initial step must be to determine the parameters 

controlling the movement of sediment, as this movement 

is the distinguishing feature of the flow of water in 

alluvial channels. I will show that the movement of 

sediment cannot be described by a simple function 

and that the behavior of a load of uniform-fized sediment is different from that of a load of a mixture of 

sand sizes. There is considerably more uncertainty 

about pertinent relations when mixtures of sand sizes 

are involved because of the difference between the 

sizes of the sediment making up the bed and the sizes 

of the sediment in motion.

The discussion on sediment movement will lead to 

the conclusion that the mean velocity of a stream is 

the best available index of the rate of sediment transport. This statement is reversible that is, the sediment transport is the best index of velocity. This immediately leads to a corollary that the shear velocity 

is not at all well related to sediment movement.

The second part of the report will conside1- how the 

slope of the energy gradient responds to changes in 

other hydraulic variables and the rate of sediment 

movement. First, the nature of this response will be 

shown to vary with different types of experimental procedures in the laboratory and with different sets of 

independent and dependent variables in the field. 

Next, the relations between the independent and dependent variables will be determined by the method 

of minimizing variance as proposed by Langbein 

(1964, 1965). The theoretical result will be compared 

with observations made under specific sets of conditions in the laboratory and in the field. In this part 

the movement of water and sediment will be considered
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

as two-dimensional flow that is, discharge of both 

water and sediment will be evaluated in terms of a 

unit width.

The final section of the report will consider the behavior of the whole stream width will be treated as 

an independent or dependent variable. Using width as 

a dependent variable, a "regime" relation will be defined; "regime equations," such as the Lacey and Blench 

equations, will be developed as will "regime equations" 

for other particular sets of conditions. Finally, the 

problem of stable channels will be discussed in terms 

of movable and relatively immovable beds and the 

pertinent factors involved.

Throughout this report no attempt will be made to 

divide the hydraulic radius. Hydraulic radii of the bed 

and wall have not been computed, and data from all 

channels having width-depth ratios of less than 4.5 

have not been used in determining any of the hydraulic 

relations. Furthermore, hydraulic radius and mean 

depth are considered equivalent. This may introduce 

a systematic error, but it also eliminates one value, 

hydraulic radius, that is hard to compute.

Acknowledgments. This report has been in preparation for a considerable period of time. An initial draft 

was submitted to a number of colleagues for review 

late in 1965. The author is greatly appreciative of the 

comments received despite the additional work they 

caused. It is hoped that the report reflects some of the 

many constructive suggestions that were received.

The author is very pleased to acknowledge the counsel 

and criticism of W. B. Langbein, whose previous work 

outlined the premises upon which most of the report 

is based. The able assistance of M. E. Moss is 

acknowledged in the work of assembling and analyzing 

the mass of data used in the preparation of this report.

RELATION BETWEEN SEDIMENT LOAD AND 

VELOCITY

The relation between sediment load and velocity will 

be considered first, using data from experiments in 

flumes. The subject of sediment load was considered by 

Rubey (1938), and the following paragraph is a paraphrase of his statements.

Sediment loads must be observed by time averages; 

hence, the area over which equilibrium is to be studied 

also must be time averaged. If the velocity of flow near 

the bed is Vb , and t is a unit of time, a characteristic 

area would have the dimensions of Vj,t and a unit 

width. The force on this characteristic area would be 

pV/Vtt or pVbzt, where p is the mass density of the fluid, 

and the rate of expenditure of energy on this area 

would be pV^t. This expenditure is held in equilibrium 

by a force-velocity product that is equal to the product 

of load per unit width in pounds per second, the time,

329-510 O 09 2

t, and a characteristic velocity that is related to tl e 

settling velocity of the sediment, w, or together, Ltw. 

Hence, when ps is the mass density of the sediment,

Eliminating t,

or

which can be expressed in dimensionless terms as

w

<r OC

O1

(1)

(2)

The problem then resolves itself into determining 

the relation between the bed velocity, F6 , of Rubey 

(1938) and the mean velocity, V.

Support for the Rubey relations may be found in 

equations for the flow of sediment in pipes. Durand's 

(1953) equation for flow of sediment in pipes is

where J is the head loss for a sediment-water mixture, 

Je is the head loss for the same discharge of clear 

water, C, is the sediment concentration, D is the pipe 

diameter, g is the acceleration of gravity, and AY = 

g(ps p). Rearranging,

Multiplying both sides by irpVD2 to get the total load 

in the pipe, G,

Because L, the load per unit of circumference, is G/TrD, 

when both sides of the equation are divided by irpD,

or

T/4 V

T7Vcc

Lw\ (

P )

A

kJe

The above equations illustrate some of the problems 

introduced through the use of dimensionless parameters 

An examination of the terms in the bracket of the 

right-hand member of the last equation indicates that 

the greater the specific gravity of the sediment the
A4 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

greater the load for a given velocity. Actually, if A? 

increases, w must increase. Hence, in the Durand 

equation, the specific gravity of the sediment has no 

influence on the relation between velocity and load for 

sediment sizes larger than about 1 mm. In this respect, 

the Durand relation differs from the Rubey relation 

because an increase in the specific gravity in the 

Rubey relation would, for a given velocity, reduce the 

load of coarse sediment.

In the Durand equation, when the head loss becomes 

constant for a given pipe, the Rubey relation of 

LccpV^fw holds. However, there is considerable additional evidence that load and velocity are uniquely 

related. Lacey (1940), Inglis (1949), Vanoni and Brooks 

(1957), Brooks (1958), Kennedy and Brooks (1963), 

Nordin and Beverage (1965), and Colby (1964) all 

have noted the relation between velocity and load. 

The same relation is implied in studies by Vanoni and 

Nomicos (1960) and Kennedy (1961), Colby (1964) 

recommended the use of velocity as the dominant 

measure in expressing bed-load movement and stated 

that it is as accurate as other measures and certainly 

the simplest to use.

Bogardi (1965) stated that at one time European 

practice was to consider the rate of sediment transport 

as proportional to the fourth power of the velocity. He 

referred to Velikanov (1949) as the author of the 

relation

Locmd(V-Vc),

in which L is the rate of sediment movement per unit 

of width, V and Vc are, respectively, the mean velocity 

and critical velocity at which sediment motion b egins, 

m is proportional to Vs, and d is the particle eize. This 

type of relation differs from that proposed by Rubey 

(1938); hence, it appears that the relation between 

velocity and load cannot be described by one simple 

equation. This conclusion is confirmed by data from 

flumes having a wide range of characteristics.

The analysis of flume data as presented herein is 

divided into two major parts: that for unigranular 

material, and that for mixtures of sediment sizes. The 

relation between velocity and discharge of sediment per 

foot in both unigranular material and mixtures had to 

be approximated by using three characteristic ranges 

called the mid-velocity range, the low-velocity range,

0.1

0.001 0.01 0.1 

LOAD, IN POUNDS PER FOOT PER SECOND

1.0

FIGURE 1. Relation between velocity and load, sand B, Gilbert (1914) experiments.
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A5

and the high-velocity range. These are presented in the 

order of their importance in natural streams and canals. 

The relations developed from flume data will be tested 

with field data from a few locations and compared with 

other equations for rates of sediment movement

UNIGBANULAB MATERIAL

Because of the uncertainties in the behavior of sediment mixtures, data from experiments with unigranular 

material were analyzed first. The best known and most 

widely used data of this kind are those from the studies 

of Gilbert (1914). I used some data from studies at the 

Waterways Experiment Station (1935), from Casey 

(1935), and from more recent experiments by Williams 

(1967) as supplements to the Gilbert data.

The relation between velocity and load for Gilbert's 

size B is shown in figure I.1 The plot gives the relation 

VccL 1/4, which indicates a 1:1 relation for F6 and V in

1 Every other one of Gilbert's data on size B are plotted in figure 1, as is done when 

these same data are plotted in other figures. The same procedure is followed in plotting 

data from Gilbert's size C. All the data from Gilbert's sizes A, D, E, F, G, and H are 

plotted unless otherwise noted. Data from runs having width-depth ratios from less 

than 4j^ to 1 have not been plotted unless no other data are available in that size 

range. Load is total load; there is no distinction between bed load and suspended load 

if the bed material is made up of the same sizes as the sediment load.

Rubey's analysis. The same result holds true for 

sizes A, C, and D.

However, for size E (fig. 2), the data for low velocities 

tend to depart from the VccL1 ^ relationship. This departure is even more pronounced for Gilbert's size G 

(%. 3).

Careful observations by Williams (1967) using the 

Geological Survey flume at the Washington Gun 

Factory indicate that with a restricted range in sediment size the break between different velocity-lo^.d 

relations is well defined (fig. 4). Thus the AB part of 

the curve will be called the low-velocity range, and the 

BC part will be called the mid-velocity range.

The complete range of the velocity-load relations for 

a sediment mixture with a mean grain size of 0.19 rrm 

is shown in figure 5; the CD part of the curve is des'^- 

nated as the high-velocity range. These data resulted 

from experiments by Simons and Richardson (1961) 

using the 8-foot recirculating flume at Colorado State 

University.

The relations between the Gilbert (1914) data and 

the Waterways Experiment Station (1935) data are 

shown in figure 6. The mean grain sizes are the same.

0.001 0.01 0.1 

LOAD, IN POUNDS PER FOOT PER SECOND

FIGURE 2. Relation between velocity and load, sand E, Gilbert (1914) experiments.
A6 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10

l.O

0.1

0.001 0.01 0.1 

LOAD, IN POUNDS PER FOOT PER SECOND

1.0

FIGURE 3. Relation between velocity and load, sand G, Gilbert (1914) experiments.

The data from the Gilbert (1914), Williams (1967), 

Waterways Experiment Station (1935), and Simons 

and Richardson (1961) experiments all show the same 

pattern for the relation between velocity and load. 

(Compare figs. 1-6.)

MID-VELOCITY RANGE

If all the Gilbert data in the mid-velocity range are 

plotted in terms of V/w instead of VJw, as developed 

by Rubey, then a correction factor must be introduced 

into equation 2. The result is shown in figure 7 where

V/w is plotted against Z/pw3 ( ^ j. I emphasize that in

this report velocity is a dependent variable of sediment load and discharge; hence, the correction is related 

to discharge. The assumption or conclusion that shear 

velocity, (yDS) 1/2 , is directly related to the sediment 

load, moving either as bed load or suspended load, is 

specifically rejected. Furthermore, no distinction between modes of transport is made.

The complete equation for the mid-velocity range of 

the Gilbert data is (fig. 7),

,.,1/16-,1/16 

1 /4 w I 

,,5/16 (3)

To accommodate data from experiments with grain 

sizes smaller than those used by Gilbert, anothe^ correction factor must be used. This is a coefficient, ki, and

Cot->(-5.55+3.8

The equation of the line shown in figure 8 is

(4)

(5)

A graph of the solution of equation 4 is shown in 

figure 9; the change in the value of ki falls between 

particle sizes of about 0.50 mm and 0.04 mn:. This is 

very close to the range of sizes where the relation 

between the fall velocity of the particle and its mean 

diameter changes from the square root of the diameter 

to the relation found under laminar conditiors insofar 

as the flow around the sediment particle is concerned.
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

10

1.0

..velocity ran&__ 

,_- -X'

o.i

0.001 0.01 0.1 

SUBMERGED LOAD, IN POUNDS PER SECOND

07

FIGUBE 4. Relation between velocity and load. Data from Williams (1967) experiments. Sediment size: 0.25-2.00 mm, 1.40 mm

mean. Width, 1 foot.

It is not unlikely that a change in the Vh <xV relations 

should occur simply because of differences in bed 

forms. However, there is no prior reason why the 

cotangent function should be used, but it seems to fit 

the data best. I must emphasize that all the equations 

presented are empiric, although they are dimensionally 

correct.

The velocity-load relation in the mid-velocity range 

has been extended to much smaller sizes than heretofore supposed possible to the 0.01-mm sand of 

Kalinske and Hsia (1945) and the 0.04-mm sand of 

Laursen (1958) by the introduction of the coefficient 

ki. It results in a much larger load for a given velocity, 

relatively speaking, for a small size than for a large 

size of sediment in motion. These conclusions are 

limited to unimodal mixtures with standard deviations 

of less than 1.6 in the size composition of the sand 

mixture; no data are available on bimodal size distributions.

IOW-VEIOCITY RANGE

The relations between velocity and load in the 

low-velocity range were initially determined from the

Simons and Richardson (1961) data for a number of 

sediment mixtures. However, the form of the equation 

proved wholly applicable to the Gilbert (1914) and 

Williams (1967) data for essentially unigranular 

material. The basic form is also applicable to the 

Waterways Experiment Station (1935) data.

Because there is a velocity at which sediment ceases 

to move, velocity is not solely a function of load or 

vice versa. Consequently, in the analysis of the lowvelocity range, velocity is expressed in terms of an 

effective load, Le , defined as follows:

(6)

Then for unigranular material,

. W ' \pwP

Equation 7 in the form of

V plotted against ^ ( j

(7)
0.0001 0.001 0.01 0.1 

LOAD, IN POUNDS PER FOOT PER SECOND

FIGURE 5. Relation between velocity and load. Data from Simons and Richardson (1961) experiment with 0.19-mm sand in an 8-foot flume

mid-velocity range, has the relation VocL1/4 .
BEHAVIOR OP STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A9

xrV^

EXPLANATION

o Waterways Experiment Station (1935) Sand 4 

x Gilbert (1914) Sand C

0.01

0.00001 0.0001 0.001 0.01 

LOAD, IN POUNDS PER FOOT PER SECOND

0.1 1.0

FIGURE 6. Relation between velocity and load in the Waterways Experiment Station (1935) studies and in the Gilbert (1914)

experiments, both using sand of 0.506-mm diameter.

is shown in figure 10, which is a plot of the Gilbert 

(1914), Williams (1967), and Casey (1935) data.

When concentration of sediment becomes zero, 

equations 6 and 7 reduce to

(8)

Equation 8 will be recognized as the well-known 

sixth-power law relating the weight of the particle 

to the velocity of the stream but with the addition 

of a velocity-depth correction.

Equation 7, for unigranular material and zero sediment concentration, can be expressed as

V=Q.6d°-375D°-125 . (9)

It can be compared with Laursen's (1958) relation of 

F=5.47 GdlKDlK for zero sediment transport, where 

C is SL coefficient; with Kondrat'ev's (1962, p. 10) 

relation, which, when converted from the metric 

system, is V=6.Q5dl/zD1/6 ; with Ho's (1935) relations, 

V=5.1d°-5D°- 13 and V=3.9d° A7D°-22, in which d is in 

inches and D is in feet; and with the Meyer-Peter, 

Favre, and Einstein (1934) relation, V oc rf3/8 D1/8,

if the friction factor is constant. The relations of 

equation 9 are representative of the condition when 

sediment movement ends because they are an extrapolation on a specific type of bed one deformed by 

general sediment motion.

Equation 9 can be written as

d=0.0024 ^0. (10)

Fahnestock (1963) reported that the maximum size 

of material in transport found in the White River 

below Emmons Glacier, Wash., varied with velocity 

to the 2.6 power. Equation 10 fits Fahnestock's observations reasonably well.

The point of intersection of the low-velocity load 

curve (eq 7) and the mid-velocity load curve (eq 5) is

A12/7 ArA1 

P /

(11)

where A is the intercept of the low-velocity load cur^e 

and B is the intercept of the mid-velocity load curve. 

In computing the above value of velocity, the correction 

factor 7.5(pty2/A7c?) 3/2 in equation 6 has been ignored
100

0.001 

FIGURE 7. Dimensionless plot of mean velocity, fall velocity, load, and discharge per foot of width for sands A through G, Gilbert (19

mid-velocity range.
1U

G

z

0 

0

LJ

(/)

CC 

LJ

Q.

ti i o UJ 1 ' u 

LL.

z

t

o 

o

UJ

>

^*"

~*~~^

"

-

-

.

i i i i i i i i

. + , j*

 *> "+ nj x + 

o 0 * *°*

' '^ «. -* ** o o

 

1 1 1 1 1 1 1 1

.

 '

 MA (rfrfif»TC

^^^^jLufcj^S** *

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, , i 4 i i i i

t\on^-^

£0^2-^^^"

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+

EXPLAf

-"-^ 

MAT

Symbol Sand 

x Gilbert A 

o Gilbert B 

 Gilbert C 

+ Gilbert 

» Gilbert 

4 Gilbert 

 Gilbert G 

4 Laursen 

o Laursen 

o Kalinske and

Hsia 

* Williams 

i i i i i i i i 

0.1 

10-1 10 101 . . , 102 103 

L ^ (gq)^4 ( ) /4 1. 4 

pw"> P 1 

FIGURE 8. Relations between velocity and load, unit discharge, and size of sediment for unigraiiular material in the mid- velocity range. 

Laursen (1958), Kalinske and Hsia (1945), and Williams (1967). Plotting of the more than 700 points was so dense that only aber
A12

100

BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10

1.0

0.110-1 10 101 102 103 104

FIGURE 9. Values for the coefficient ki in equations 4 and 5 with variations in the diameter and fall velocity of sediment particles.

as being unimportant. In each computation, the value 

of the correction factor at the point of intersection 

has been less than 5 percent of the value of the sediment 

concentration, C. For all practical purposes the velocity 

at this point is a function only of size of sediment. The 

velocity is at a minimum for a sediment in the sand 

size and increases with both smaller and larger sediment 

sizes.

HIGH-VEIOCITY RANGE

For some time there has been field evidence of a 

limiting high velocity one beyond which a channel 

becomes unstable. Such a high velocity was not readily 

discernible from the analysis of the Gilbert (1914) data. 

The Simons, Richardson, and Albertson (1961) data 

showed some evidence of a limiting high velocity, and 

Stein (1965) confirmed this. Further study however, 

indicated that some of the data from Gilbert's (1914) 

sand sizes A, B, C, and D belonged in a high-velocity 

range (fig. 11). The equation of the line in figure 11 is &r- (12)

Equation 12 can be written in the form

and can be compared with Craven's (1953, p. 73) 

equation for the flow of sediment in pipes of

3=0.808

,2/3

where Qs is the sediment discharge and Q t}^ n, water 

discharge. In the two equations the exponents of the 

ratio of sediment discharge to water discharge (concentration) are almost identical. Both equations indicate that, for coarse material, slope is a function of 

sediment concentration.

Equation 12 seems valid for high concentrations and, 

therefore, is particularly applicable to small flumes 

with low discharges on steep slopes. The relation in 

equation 12 is not the equivalent of the limiting highvelocity range (fig. 4), which appears to be what Stein 

called a "breakaway" velocity and is related to loads
EXPLANA

Gilbert

Gilbert

Gilbert

Gilbert

Gilbert

Gilbert

Gilbert

Gilbert

Williams

Casey

10-6

FIGURE 10. Relations between velocity and effective load and size of sediment in the low-velocity range. Sands A through H, Gilbert (

Casey (1935) experiments; and 1.40-mm sand, Williams (1967) experiments.
A14 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

100

10

1.0

EXPLANATION 

Symbol Sand (Gilbert, 1914)

0.01 0.1 1.0 

LOAD, IN POUNDS PER FOOT PER SECOND

10

FIGURE 11. Relations between load and discharge, slope, and sediment size in the high-velocity range; sands A through D, Gilbert

(1914) experiments.

derived from the stream bed. A field example of a 

limiting velocity is shown in figure 31, where data are 

plotted for the Colorado River at Yuma, Ariz. There, 

maximum velocities under natural conditions were 

between 6 and 7 feet per second for discharges between 

25,000 and 100,000 cfs (cubic feet per second). The 

question of a maximum velocity will be discussed at 

greater length in the following section on mixtures of 

sediment sizes.

MIXTURES OF SEDIMENT SIZES

Although the relations describing the movements of 

unigranular sediments are good, those describing the 

movements of sediment mixtures are indeterminate. 

In fact, this is the first area of indeterminacy encountered in the study of the behavior of alluvial channels. 

In most, but not all, flume studies the size distributions 

of the transported material and of the bed material are 

the same. In most natural streams these two size distributions are not the same, and the differences between 

them may vary widely. Even if the size distributions

were the same in natural streams, the fact would be 

difficult to substantiate because the size distribution of 

material caught in a suspended load sampler is always 

deficient in some of the coarser sizes of the sediment 

load. These pass beneath the sediment sampler and 

constitute much of the unmeasured load of the stream. 

Consequently, in some flumes and in most natural 

streams, the selection of a proper size is very difficult 

on the basis of available information. This difficulty 

increases as sediment concentrations decline and as the 

range of sizes increases.

The effect of size mixtures on the velocity-load 

relation is indicated in figure 12. The mixture of sediment sizes seems to have the effect of extending the 

relation V°cLe1/9 by permitting, for a given mean 

velocity of fluid, an increased dune velocity, an increased dune height, or both. In figure 12 the velocityload relation of the mixture eventually comes back to 

the line BC; there is insufficient data to establish the 

relation for the high-velocity range, the curve CD, but
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A15

100

0.1

0.0001 0.001 0.01

LOAD

FIGURE 12. The velocity-load relation for unigranular material and a unimodal mixture. Solid line ABCD, relation between 

velocity and load for a unigranular material. Dashed line A'B'CD', relation between velocity and load for a unimodal mixture 

with a mean diameter equal to the diameter of the unigranular material.

the slope of the line CD' is presumed to be a function 

of the size composition of the sediment mixture. The 

distance from B' to the line BC appears to be related 

to the range of sizes in the mixture a wide range of 

sizes, a wide departure and particularly to the range 

of particle-fall velocities.

Thus, the behavior of mixtures of sediment sizes in 

alluvial channels is considerably more complex than 

that for unigranular material. Rubey (1938) assumed 

that in any mixture with a wide range of sediment 

sizes, the largest size moved would vary approximately 

as the bed velocity. Because the largest size would have 

an important bearing on the characteristic settling 

velocity of the sediment mixture, there would be some 

degree of relation between the bed velocity and the 

fall velocity. Rubey then concluded that, roughly 

Vj, ocw and Vb3 acL.

Rubey (1938) undoubtedly was influenced by Gilbert 

(1914) for, as shown in figure 13, if all of Gilbert's data 

are taken together, they do show the same rough relation. Thus, Rubey's conclusion is most applicable to 

the mid-velocity range because the Gilbert data are 

typical of it. However, even in the mid-velocity range 

the effects of the range of sizes within a mixture and

of size variation with time are still uncertain, and in the 

other velocity ranges these effects are even more uncertain.

MID-VELOCITY RANGE

The relation between velocity and load for sedimert 

mixtures in the mid-velocity range is shown in figure 14. 

Clearly, equation 5 can be used only with caution and 

with a change in coefficient. The equation of the lire 

shown in figure 14 is

.1/16 (wg) 1/16

(13)

Because of the small influence of the size parameter 

in equation 13, it is doubtful if the fit of the equation 

can be improved by trying to adjust the mean diameter 

of the mixture. The better approach seems to be to 

adopt the Rubey (1938) hypothesis and ignore the sir;e 

parameter in mixtures. The results are almost as good 

as those for unigranular material, and the procedure is 

far simpler.

The resultant equation, which is simply

i/3

(14)
LUU

5 

>

iio>

j

-

T

 .'< * ". *'?

°t° T '

' »!* Ifi'^H^""

 ». *:H * 

I Aq

;°f* ^-^

,*'»^, *

*

i ^^, v * -^ ^^^'^

 ^ ^- ^ MO^

-^~^^ ° ft n,ra^ °^

 ^T^ A ° ACsl "^9!°°'

B o n n go °

A X X XVX^X 1

*^i ^ ^X 4! * X a feP ^i ^

^Xia A ^C^! A " A "" ^

o wOaK2>° ° ° o

p Jfa If A

1.0 -- - - 

0.001 0.01 0.1 1.0 10 100 

LOAD PER UNIT WIDTH 

(FALL VELOCITY) 3

FIGURE 13. Dimensionless plot of velocity, fall velocity, and load for sands A through H, Gilbert (1914) experiments. Relation oc

hence, FocL1 '3 . w L

p

/ o 

(Fa
iU

Q

Z

O

o

UJ 

C/5

QL 

UJ

Q.

1-

uj i n UJ 1 - U

u.

Z

t 

o 

o

_l

UJ

>

0.1 

i i i i i i i i

-

-

-

~

e

o* ____-^ - '

-

-

.

i i i i i i i i

i i i i i i i i

o * ° D °*»

o AA o*. v, 00 ^.!

o I^o * *

 ^^_ <f 0

 ^^^

-o

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 

» / o eou

° O& °£> A ^ * *-~^~~'*~'

0>$ aftt* ^-3t """""

*°t * t^3^"^'^

 ** »ir °

A O^ 4. ^^O X

 . A ^ 

^ J oife y ^

,"*3** *^°o^ x °

O o***

o

1 1 1 1 1 1 1 I

a

o

EXPLA

Symbol 

Simons and 

x 

+ 

* 

 

o 

o 

a 

Broo

» 

« 

Vanoni an

 

Nomicos (

a 

Barton a

« 

i i i i i i i i 

10-1 10 101 £ V4/A7£/V*£4 102 1Q3 

 </lt \ ) \ ) i 

FIGURE 14. Relations between velocity and load, unit discharge, and mean grain size for sediment mixtures in the mid-velocity range. Size
BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

EXPLANATION

Symbol Sand 

Simons and Richardson (1961)

0.19 mm _________ 

o 0.27 mm 

A 0.28 mm 

+ 0.32 mm

* 0.47 mm 

» 0.54 mm

* 0.93 mm

Brooks, Vanoni, and Nomicos (in Brooks, 1958) 

D No. 1

* No. 3

* No. 4

* No, 5

Barton and Lin (1955) 

$ 0.18 mm

0.001 0.01 0.1 

_4_ (LOAD. IN POUNDS PER FOOT PER SECOND) 

P (DENSITY OF WATER)

1.0

FIGURE 15. Relations between velocity and load for sediment mixtures. Flume data. Sizes given are mean diameters.

is shown in figure 15 together with data from recirculating flumes.

In comparison with unigranular material the load of 

a sediment mixture is higher, for a given velocity, in the 

lower part of the mid-velocity range and there is little 

difference in loads in the higher velocities of the midvelocity range.

The mid-velocity range of sediment of any specific 

size encompasses the velocities usually found in nature 

and is particularly important in canal design. We must 

remember, however, that there is an upper and a lower 

limit for each size. Therefore, it is never possible to have 

the same load for the same velocity of, say, a 0.10 mm 

and a 1.0 mm sand over a wide range of velocities. Some 

parts of the velocity and load-size spectrum will coincide 

but not all.

A word of caution must be introduced about the use 

of the velocity-load relations, particularly equations 13 

and 14. Load is a function of some power of velocity. 

Consequently, if the velocity distributions throughout 

the water prism vary greatly, the coefficients of the 

equations must vary to some unknown extent. The

situation is somewhat analogous to the determination 

of the energy coeffi(cient. Chow-Ven (1959, p. 28) indicated that the value of the energy coefficient for natural 

streams (and presumably this would be true of canals) 

exceeds that for flumes. The coefficients of equations

13 and 14 should have an inverse relation to the energy 

coefficient; consequently, the coefficient of equation 13, 

and particularly equation 14, should be reduced for 

large streams and canals. Data indicate that equation

14 for canals and many rivers should be

i/3

(14a)

Because at this time there is no way of evaluating 

the effect of change in velocity distribution in the 

water prism, the coefficient of equation 14 should be 

determined on the basis of local evidence.

LOW-VELOCITY RANGE

The velocity-load relations in the low-velocity range 

for the Simons and Kichardson (1961) data, using the 

effective load as defined in equation 6, are
BEHAVIOR OP STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A19

w

>l/2

(15)

This equation is shown in figure 16, where the Simons 

and Richardson (1961) data are plotted as

T7 1 ** J ,/i.\/AW\9/2 " plotted against f 3 j ( j 

In the low-velocity range the effect of a reduction in 

the fall velocity is a reduction in the sediment load if 

the velocity of the fluid remains constant (eqs 7 and 15). 

Thus, part of the variation in the coefficient in these 

equations and the spread of the data plotted in figures 

16 and 17 can be eliminated by the use of fall velocity 

weighted in accordance with the variation in sizes of 

the mixture. However, as indicated by data for the 

0.45-mm runs of Simons and Richardson (1961) in 

figure 16, the size composition of the material actually 

being moved may change considerably from time to 

time, but when the size composition of the material in 

the bed and that of the material in motion are the 

same the relations are as good as those for unigranular 

material. Variation between the sizes of the bed material 

and the transported material was also observed in the 

Waterways Experiment Station (1935) studies. The relations velocity and effective load and mean grain size 

are shown in figure 17.

Two problems exist in dealing with sediment mixtures: The first is to establish the relation between the 

mean diameter of the mixture and the variations in 

sediment size. The second is to predict the variation 

between the sizes of material in motion and the sizes 

of the material in the bed that is, the sorting problem.

In view of the uncertainties about the correct value 

of the characteristic fall velocity of a group of particles 

and other unknowns about sediment mixtures, such as 

sorting and size separation, it is considered adequate, 

for the present, to express variations in the velocityload relations in the low- and mid-velocity ranges by 

two sets of equations (eqs 5 and 14 for unigranular 

materials and eqs 7 and 15 for sediment mixtures). 

Fall velocities used in this report are those for quartz 

spheres of a diameter d.

As will be shown later, low sediment loads are not 

important in determining velocity. Thus, errors are 

likely to be concentrated in the estimates of the rate 

of sediment movement rather than the estimates of 

velocity. Fortunately, such errors, while large percentagewise, are small in terms of the actual quantity of 

sediment moved by a stream at various discharges. 

There are large changes in the amount of sediment in 

movement for small variations in velocity, and the 

stream adjusts readily to variations in load with small 

changes in the velocity. Low sediment loads are not

329-510

important in canal design. Under such conditions, 

sediment size becomes a dominating characteristic. 

Except for flumes having small discharges, there is 

not much interest in either the determination or the 

effect of very low rates of sediment movement.

HIGH-VELOCITY RANGE

As noted in the section on unigranular material, 

little information is available about the high-veloc; ty 

range. An analysis of the Simons and Richardson (1961) 

data, confirmed in part by the use of data from Stein 

(1965) with respect to his "breakaway" velocity, indicates that the limit of the velocity parameter is

which reduces to

F=250 ,1/3

(16)

(17)

It is hypothesized that in a channel with a bed of 

incoherent unigranular material which, with increasing 

discharge, is to be a source of sediment there if a 

velocity above which channel stability will not exist 

that is, a channel will simply erode and enlarge its 

sections, keeping velocity essentially constant. Equations 16 and 17 are based on data from mixtures. In a 

mixture, however, each size should be considered 

separately, and the actual limiting velocity should be 

based on the largest sizes present in appreciable 

amounts. The slope of the high-velocity load curve 

CD in figure 5 is presumed to be dependent on the 

range of sizes of sediment present in the sediment 

mixture.

According to equation 17, maximum velocity vdll 

be constant at about 4 feet per second for all sediment 

sizes of less than about 0.10 mm and will be proportional 

to d112 for sizes greater than about 5 mm. According to 

my own estimate, based on the relations in equation 17, 

the size of sand that stabilized the predam flow of the 

Colorado River at Yuma, Ariz., was about 0.8 mm. The 

mean size was smaller than this, but, presumably, the 

largest size appearing in any quantity in the mixture 

was about 0.8 mm. The behavior of the sediment load 

of the river is shown in figure 18.

Because there was little variation of width with discharge at the Yuma gaging station, the sediment concentration decreased with increasing discharge and the 

load remained constant. Thus, the load was constant 

while the depth-slope product increased considerably. 

Hill (1925) excluded flow from the Gila River in his 

data on the Colorado River, and at that time the Gila 

River was largely uncontrolled. A suggested reason for
10-7 10-6 10-* 10-4 

1U

a

z 

o

o

UJ

C/3

CC 

UJ

a.

iLJ , n

u 1.0

Uz

> "

o

o

_J

UJ

n i

-

.

-

-

x

X X + A """o"" 0

X* o o 

o

-

-

-

-

1 1 1 1 1 1 1 1

A D X ^

_ ,: °* ><r j

S " 0

o

1 1 1 1 1 1 1 1

+ 

'+ & 6 "° o

1 1 1 1 1 I 1 1

A A A^ S~~

 .D -.V

EXPLANATION

Symbol Sand (mm)

Simons and Richardson (19

x 0.19

+ 0.27

A 0.28

* 0.32

o 0.45

 0.47

° 0.54

A 0.93

> i i i i i i i

Equa

51)

FIGURE 16. Relations between velocity and effective load and mean grain size for sediment mixtures in the low-velocity range. Data fro

(1961). Sizes given are mean diameters.
1U

Q

Z

0

o

LU

C/5

o:

LU

Q.

1-

Lji-O 

UZ

1-

o 

o

_J 

uu

0.1 

^

_

-

-

o °_ "i$*7.+

 + 9 -!_ *

"+ ? + ++ 71° +

o

o "

-

"

1 1 1 1 1 1 1 1

 Iff!

___ _ ,. ^"A" *"* q£**^*^ ** 0°°

**oo °+ |#- *o ° ° °

j O * O

1 t 1 1 1 1 1 1

ox £o °

 1* & i^? x *Sc x x x ^

3 ^* « *KX ** O

o»- x c^f * "o

0 *

1 1 1 1 1 1 1 1

A A

_ __ . * \ *

__ - - "" *

EXPLANATION

Symbol Sand Diameter (mm)

Waterways Experiment Station (1935

« 1 0.586

o 4 0.506

 6 0.347

+ 7 0.310

o 8 0.205

9 4.077

i i i i i i i i

g

- 

10-8 10-7 10-6 10-5 10-4 

 ( ) 2 

pw3 V p '

FIGURE 17. Relations between velocity and effective load and mean grain size for sediment mixtures in the low-velocity range. Data fro

ment Station (1935). Sizes given are mean diameters.
A22 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

Q 400

Z 

O

(J ,-UJ

^co 300 

(75 cc

LU UJ I-

^<-> §1(J

200

100

± 0

Petition of percentage 

of silt to depth

10 20 30 

MEAN DEPTH, IN FEET

40

0 20 40 60 80 100 120 140 160 

DISCHARGE, IN THOUSANDS OF CUBIC FEET PER SECOND

B

FIGURE 18. Behavior of the sediment load of the Colorado 

River at Yuma, Ariz. A, Relation of percent of sediment and 

mean velocity to mean depth (from Hill, 1926, p. 963). These 

curves are based on about 3,000 measurements at the Yuma 

gaging station, taken when there was no flow in the Gila 

River. B, Relation between sediment discharge and water discharge (from Hill, 1925, p. 420). The curve is based on about 

3,000 samples taken during the period 1908-16 at the gaging 

station, Yuma, Ariz., at times when there was no flow in the 

Gila River. One cubic foot of deposited silt assumed as 

weighing 83 pounds.

the exclusion is that the Gila River had a slope about 

three times that of the Colorado River, and the Gila 

moved considerably larger sizes of sediment. Consequently, floods from the Gila River at the Yuma gaging

station had higher velocities, for the same d '^charge, 

than did the Colorado River.

The coefficients of equations 16 and 17 are based on 

the results of a small number of experiments, Consequently, the coefficients are subject to error and, perhaps, to a difference in interpretation, but the equations 

seem to describe field conditions and should b°, worthy 

of further study.

Testing the Simons and Richardson (1961) data for 

the choked-flow conditions yields

(18)

which should be compared with equation 12 from the 

Gilbert data (p. A12).

The scatter of the Simons and Richardson data shown 

in figure 19 is much greater that that of the Gilbert data, 

which indicates, perhaps, that many of the Simons and 

Richardson runs were not in a choked-flow state. 

What is clearly indicated is that mixtures result in 

much higher loads than do unigranular materials for a 

given discharge, slope, and median grain size of sediment; 

this result was not unexpected.

EXTRAPOLATION FROM FLUME STUDIES

Sediment movement in alluvial channels has been 

related to three parameters a shear stress, yDS; a 

mean velocity; and a combination of the two, a stream 

power, y%S. Field evidence led Leopold and Maddock 

(1953a, b) to conclude that load was not related to 

either yDS or yyS, which was simply a confirmation 

through observations at other locations of what Hill 

(1925, 1926) had observed almost 30 years earlier. 

Flume evidence of the nonuniqueness of the yDS 

parameter was first reported by Brooks (1958) and was 

confirmed by Stein (1965) in detail. Thus, the relation 

between velocity and load is the only one that, to any 

degree, can be said to be unique. The com^exity of 

this relation is obvious from the preceding discussion. 

However, the fact that load is not uniquely related to 

slope means that for a given discharge on a given slope 

different loads will be moved if it is possible to change 

velocities accordingly. The effects of different loads are 

very difficult to determine in small flumes having 

shallow depths. The introduction of sediment into a 

small flume already moving sediment blocks the whole 

sediment load and scour occurs downstream (R. E. 

Rathbun and H. P. Guy, unpub. data, 1967). Thus, 

the behavior of flumes and natural streams can be 

very different.

Particularly, the different ways of measuring velocity 

and load in flumes and natural streams mus* be considered. Velocity in a flume is usually calculated by 

dividing the discharge by the product of the width 

and the average depth over a considerable length of
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A23

100

10

EXPLANATION 

Symbol Sand (mm)

o 0.19 >

x 0.27 {

o 0.28 l"

0.47 J

 0.32 \ 

+ 0.54 j'

8-foot flume

2-foot flume

1.0

0.1 1.0 10 

LOAD, IN POUNDS PER FOOT PER SECOND

100

FIGURE 19. Relations between load and unit discharge, slope, and sediment size for sand mixtures in the high-velocity range.

Simons and Richardson (1961) data.

the flume. Load is determined by frequent samplings 

during a long period of time. The mean velocity of a 

natural channel is usually computed by dividing the 

discharge, determined by summing the products of 

depth and velocity (determined by current meter) of 

a group of discharges of unit width, by the product of 

the width and average depth. The sediment load is 

determined by measuring the concentration of a small 

group of water samples taken at one cross section. 

Considering the potential for variation, the results of 

sampling are remarkably consistent. In terms of actual 

variation, however, a half log cycle variation is the 

most that can be expected under normal conditions. 

Consequently, considerable variation is to be expected 

in comparing flume and field observations.

At the present time, it appears that the equations 

developed from flume studies may be extrapolated to 

larger streams if the following conditions are met: 

(1) The sediment in motion is unigranular in size or 

the sizes of the material in the bed and the sizes of 

the sediment in motion are the same; (2) the vertical 

velocity distributions are symmetrical about the center

line of the stream cross section; and (3) the sediment 

is moving in the same manner across all the bed.

Where vertical velocity distributions are not symmetrical about the center of the stream but are skewed, 

the stream will carry a relatively larger load for a 

given mean velocity in the mid-velocity range and a 

smaller load for a given mean velocity in the lowvelocity range. Thus, for most natural streams that 

have nonsymmetrical velocities, the exponents of 

equations 13 and 14 must be decreased and increased, 

respectively, by about 10 percent.

One must realize, however, that the average rate of 

sediment movement is computed from flume data. 

Actually, the different instantaneous measuremerts 

that make up the average vary considerably. Consequently, sediment measurements must be taken over 

fairly long periods of time to provide an adequate 

estimate of the rate of sediment movement in large 

natural streams. Until there is some certainty about 

the reliability of present estimates, the proper evaluation of the coefficients of equations 13 and 14 will 

be in doubt.
A24 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

The relation between discharge of water and sediment in natural streams is usually a poorly defined 

sediment rating curve. Nevertheless, if the relation 

between velocity and discharge or depth and discharge 

is good the relation between water discharge and 

sediment discharge will be reasonably good also.

In the next section of this report a means will be 

given for estimating velocity, given discharge per unit 

of width, slope, and size of bed material. This is for an 

average condition because no great variation in slope 

is possible on a given reach of a stream and changes 

in sediment load are accommodated by changes in the 

friction factor. These change the velocity-unit discharge relation.

The relations between discharge per foot of width 

and depth for Mountain Creek, N.C., and West Goose 

Creek, Miss., are shown in figures 28 and 29 respectively. Also shown in each of the graphs is a relation

computed using the Einstein-Barbarossa (1952) method 

and another relation computed from equation 35. Using 

the equations for the two locations, relations between 

sediment discharge and water discharge were computed 

by use of equations 13 and 15. The results ar^ shown 

in figure 20, together with computations from various 

authors as noted. These graphs as well as the graphs 

showing the results of the Einstein-Barbarossa (1952) 

method are found in Vanoni, Brooks, and Kennedy 

(1961), and their use is greatly appreciated.

Another field example is the water and sediment 

discharge of the Colorado River at Yuma, Arz., prior 

to the construction of major dams. Figure 34 shows the 

velocity-discharge relation as V 0.046 Q1/2 . Assuming 

a constant width of 500 feet, this reduces to V= 1.04 g1/2 . 

Substituting into equation 13 with a mean grain size 

of 0.10 mm gives Z=0.015 g1 -78 in pounds per foot 

per second. This is reduced to tons per day for the

0.001

0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 

WATER DISCHARGE, IN CUBIC FEET PER SECOND PER FOOT

0.1 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 

WATER DISCHARGE, IN CUBIC FEET PER SECOND PER FOOT

FIGURE 20. Relation between sediment discharge and water discharge. From Vanoni, Brooks, and Kennedy (1961). Left, Mountain 

Creek near Greenville, N.C., showing observed values (black dots) and values computed by equations 11 and 12 and by methods 

of authors as noted (solid lines). Slope, 0.00157 foot per foot; geometric mean, 0.86 mm; <rg, 1.8; temperature, 78°F. Right, West 

Goose Creek near Oxford, Miss., showing observed values (black dots) and values computed by equations 13 and 15 and by 

methods of authors as noted (solid lines). Slope, 0.00305 foot per foot; geometric mean, 0.287 mm; ag, 1.5; temperature, 68°F. 

From Vanoni, Brooks, and Kennedy (1961, p. 7.8 and 7.9).
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS /25

whole width of the stream, and the result is shown in 

figure 21 together with data on the suspended load of 

the Colorado River at Yuma in 1933 and 1934 as given 

by Corfitzen (1940). Samples at Yuma and samples 

from flow over the crest of Laguna Dam upstream 

from Yuma showed essentially the same sediment content; hence, it is presumed that the suspended load 

measurements closely approximate the total load.

10,000,000

1,000,000

100,000

10,000

1000 I I

1000 10,000 100,000 

DISCHARGE, CUBIC FEET PER SECOND

FIGURE 21. Relation between discharge of coarser grades 

(Nos. 1, 2, 3, 4, and 5; 0.25-0.01 mm) of sediment and water 

discharge of the Colorado River at Yuma, Ariz., showing 

observed values (black dots) and values computed by equation 

13 (solid line). From Corfitzen (1940).

The Colorado River at Yuma is an example of a 

stream in which the relation between velocity and discharge is controlled by the load. As will be noted later, 

a constant slope stream is expected to have a V<xg° A 

relation if load is a dependent variable. The exponent 

of "%" indicates that sediment load is increasing 

unusually rapidly for an increasing discharge. That this 

increase in load falls off with increase in discharge 

above about 30,000 cfs has been discussed previously.

TWO-DIMENSIONAL FLOW

The hydraulics of channels with rigid boundaries and 

high width-depth ratios has been established reasonably 

well in terms of the relation between the friction factor 

and the ratio of the water depth to the height of roughness elements. The effects of the spacing and the form of 

the roughness elements, particularly when they are large 

in terms of depth, are still not completely known. Nevertheless, given any two of depth, velocity, and slope, the 

other may be computed with reasonable accuracy on 

the assumption of a channel roughness.

The problem of hydraulics of channels with movr.ble 

beds is that even if the relative roughness of a specific 

bed form is known, there is no assurance that this 

bed form would exist given any two of depth, velocity, 

and slope. Thus, although it has been shown that tl °-re 

is a reasonably good and consistent relation among 

velocity, depth, and load, there is no fully consistent 

relation among these values and slope. As long as flume 

experiments are performed in precisely the same manner, relations tend to be consistent, however, the 

results of different kinds of experiments are considerably 

different.

Some of the procedures and results of different 

experiments will be considered. Langbein's (1964, 1C15) 

procedure for identifying the behavior of dependent 

variables associated with changes in independent variables will be discussed. Then the principal constraints 

upon this relation will be evaluated; the most important 

of these is the tendency of the friction factor and the 

shear stress to be minimized, either jointly, when constraint is a minimum, or individually, when the response 

is limited by flume operation. Each type of constraint 

evaluated by Langbein's procedure will then be tested 

by field data. The discussion of two-dimensional flow 

will be closed by a summary of slope as an element in 

stream behavior.

SLOPE

Consider the differences in the manner in which the 

different flume studies have been conducted. Gilbert 

(1914) introduced water and sediment into a flume and 

let the bed and water surface seek an equilibrium slope. 

An unknown degree of regulation by a tail gate, affected 

the final result. In the Waterways Experiment Station 

(1935) experiments a channel was constructed en a 

given slope. Water was introduced in increasing 

amounts into the channel until erosion began. When 

erosion was noticeable, just enough sediment was 

introduced into the channel to keep the original bed 

and water-surface slopes; depth was free to adjust. 

Once equilibrium was established, discharge was 

increased, and, again, just enough sand was introduced 

to maintain equilibrium. The procedure was carried
A26 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

on for a number of discharges. The Casey (1935) 

experiments were performed in essentially the same 

manner, except that measurements were made with 

both increasing and decreasing discharges without 

modifying the bed configuration. In the Williams (1967) 

experiments, values of depth and load were selected. 

Estimates of a probable discharge and an equilibrium 

slope were made. The flume was set at the estimated 

slope, and sediment was spread evenly over the floor 

of the flume. The sediment infeed and the discharge 

were started. If the flow depth was not within the range 

selected, discharge was altered to raise or lower the 

depth. Once the desired depth and a uniform flow had 

been established, the slope was allowed to adjust until 

it became stabilized. For the majority of runs, no 

changes in slope occurred, which indicates that the 

equilibrium slope was the slope of the flume itself.

Three different sets of relations among velocity, 

discharge, and slope are found in the four experiments 

just described. The first, from the Gilbert experiments, 

is shown in figure 22 and has the equation

.1/8

(19)

The second, represented by Waterways Experiment 

Station sands 1 and 9 and by Casey's sand h, is a condition where grain roughness appears to control the relations (fig. 23). Williams data represent a condition of 

grain roughness control at small discharges or the effect 

of flow at a controlled depth (fig. 24). The third set of 

relations is shown in figure 23 by data from Waterways 

Experiment Station sands 4, 6, 7, and 8. The equation 

for these groups of data is

(20)

and it represents the situation when the channel roughness is controlled by the deformed bed.

In the Gilbert data and particularly in the Williams 

data shown in figures 22 and 24, there is no variation 

in the relations between velocity and discharge and 

slope for the high- and low-velocity ranges. Although

EXPLANATION 

Symbol Sand

0.01

FIGURE 22. Relations between velocity and unit discharge, slope, and size of sediment for unigranular material in the mid-velocity 

range. Data from Gilbert (1914). Discharge and load are independent variables.
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A27

0.1 L_

0.0001

FIGURE 23. Relations between velocity and unit discharge, slope, and mean grain size of sediment mixture in the low-velocity range. 

Data from Waterways Experiment Station (1935) and Casey (1935). Slope is an independent variable.

the sediment load is explicitly stated in these experiments, it has no direct association with the velocitydischarge-slope relation and, consequently, with the 

velocity-depth-slope relation.

Figure 23 also shows two envelope curves which 

represent the maximum and minimum velocities to be 

expected with a fully deformed bed. These curves are 

for a size of sediment of about 0.3 mm. Their position 

will vary with the size of sediment. There is a considerable degree of uncertainty about the actual 

nature of the envelope curves, and future experimental 

data may yield different results.

Recirculating flumes, in general use since World 

War II, are supposed to be more representative of the 

behavior of natural channels than the older experiments of Gilbert and the Waterways Experiment 

Station. Actually, they are not because too much 

depends on how the flume is operated and on flume 

characteristics.

Laursen (1958), Brooks (1958), Vanoni and Brooks 

(1957), Vanoni and Nomicos (1960), and Stein (1965)

329-^510 O 69 5

all operated flumes at a constant depth under varying 

discharges with slope free to vary. They found slop*. 

to be constant over a wide range of discharges.

Simons and Richardson (1961) operated recirculating flumes under a range of slopes, discharges, and 

sizes of sediment. The procedure was described by 

Guy, Simons, and Richardson (1966, p. 14) as follow:?:

The procedure followed for some runs involved recirculating a 

given discharge of water-sediment mixture in the flume at a 

preselected slope until equilibrium conditions were established. Equilibrium here should be denned as a condition of 

statistically uniform velocity, concentration and slope with 

respect to time and space in the flume where the variables are 

observed. Even though discharge can be held relatively constant, 

depth, slope, and sediment discharge condition (several hours 

or days after the start of the run) could be considerably differert 

from, those in the initial condition. For some runs depth wf,s 

kept constant either by adjusting the tailgate slope [sic] or discharge. For some other runs slope was kept constant by adjusting the tailgate or the discharge, or both.

The independent and dependent variables are most 

uncertain, and consequently, the relations among the
A28 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10

i.o

I 1 I I I I I I

o *

o o 

X

\ I I I T I I I I I I I I I I

EXPLANATION

Symbol Depth (ft)

o 0.3

x 0.1

+ 0.5

0.1

0.001 0.01 0.1 1.0

FIGURE 24. Relations between velocity and unit discharge, slope, and mean grain size of sediment mixture in the low-velocity range. 

Flume data from Williams (1967). Depth and load are independent variables.

variables are uncertain. These relations will be discussed later.

Guy, Rathbun, and Richardson (1966), experimenting in a 20-cm-wide flume with a 0.3-mm median 

fall diameter using both a recirculating and a sand-feed 

system, concluded that there was no difference between 

the two systems. The relation of equation 19 prevailed.

Clearly there are wide variations in flume behavior 

which are related to slope; the problem is to find an 

explanation for such behavior. A likely hypothesis 

appears to be that for any specific sediment mixture 

exposed to increasing unit discharges, there is a relation 

between velocity and the product of depth and slope. 

The actual relation is dependent on the width-depth 

ratio of the channel, the median-diameter particle 

size of the sediment mixture, and the maximum size 

of particles found in appreciable amounts in the sediment in motion. Thus, at low discharges, the flume 

and the grain size control roughness. Once the product 

of unit discharge and slope exceeds a specific value 

and the width-depth ratio exceeds a value of about

four, the relation between the velocity and the product 

of depth and slope for a given sediment mixture is 

indeterminate. The boundaries of this indeterminacy 

are limited by the range in roughness of the deformed 

and plane bed. How can this statement of indeterminacy 

be reconciled with the statements of equatiors 19 and 

20? The answer is that in nature certain constraints 

are in operation that make some relations more probable 

than others. Langbein (1964, 1965) concluded that 

these constraints are the results of minimization or 

maximization processes and that the results will differ 

depending on the selection of independent and dependent variables. Combining the idea of minimization or maximization of the functions of certain 

dependent variables with the results of the previous 

section on the relations between velocity r,nd load, 

the behavior of flumes can be predicted as described 

in the following section.

INDEPENDENT VERSUS DEPENDENT VARIABLES

If sediment sizes remain constant, the pertinent
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A29

variables in two-dimensional flow are: discharge, 

velocity, depth., slope, and load. Discharge nearly always is independent that is, the discharge is selected 

or determined independent of the channel. This is 

always true in natural streams, but there is no prior 

reason why discharge should be independent in flume 

studies. Potentially, there are four degrees of freedom 

among the remaining variables. However, these are 

constrained by interrelationships; for example, velocity 

X depth= discharge. Also, velocity and load have 

specific relations, as previously discussed. Considering 

these two constraints, only two degrees of freedom 

remain. If the degrees of freedom are to be reduced to 

one, only one of velocity, depth, slope, and load also 

may be an independent variable. Because of the nature 

of the constraints, this statement can be narrowed 

down to the following. If discharge is an independent 

variable, one of velocity, depth, or load may be chosen 

as the other independent variable. The two remaining 

variables become dependent variables, and the solution 

is unique. The problems of channels on movable beds 

are due to the fact that when slope is a variable, either 

dependent or independent, the solutions are not unique 

but are subject to other constraints.

Table 1 shows the behavior of flumes under twodimensional flow with different independent and dependent variables but with unchanging sediment 

sizes. In each example, two independent variables are 

chosen, and the table indicates whether the solution 

is unique or, if not, what is required for a unique 

solution. Meaningful relations are those that occur

TABLE 1. Behavior of flumes of unit width and same sediment sizes 

with different independent and dependent variables

No.

U

22

3

»4

5

6

7

*8

9

Variables

Inde- Depend- pendent variables 

pendent ent

q,L

q,D

V,D

q,8

D,S

V,S

L,S

D,L

V,L

V,D, S V and D unique, S usable only if

some /(S) is minimized or maximized.

V, L, S V and L unique. D8 and F each

becomes constant for certain

discharges.

q, L, 8 q and L unique. D8 becomes constant for certain discharges.

V, D, L Only if a function of V, D, or L is

minimized or maximized.

q, V, L Only if a function of V, D, or L is

minimized or maximized.

D, q, L Load essentially unique. D or q

only if a function of D or q is

minimized or maximized.

q,V,D V essentially unique. D or q only

if a function of D or q is minimized or maximized.

V, q, S V and q essentially unique. F and

D8 each become constant for

certain discharges. Same as 2.

D, q, S None possible.

Required for 

unique solution

T, Sor F.

Nothing.

Do.

Any one dependent

variable.

Any one dependent

variable.

D or?.

D or q.

Nothing.

D and S, q and S.

1 Gilbert (1914) experiments. /(S) minimized, relations consistent.

2 Brooks (1958), Kennedy (1961), Laursen (1958), and Stein (1965).

s Waterways Experiment Station (1935), sediment sizes 3-8. /(S) minimized. For 

sizes 1-2 and 9, grain size controlled results. Some data from Simons and Richardson 

(1961) confirmed this behavior. Usable relations doubtful.

* Williams flume (1967).

NOTE. Unique means fully predictable by two independent variables.

without a forced constraint caused by flume characteristics or method of flume operation.

Langbein (1964) postulated that the observed character of flow is such that changes in the independent 

variables are reflected by the minimization, not of 

functions of slope themselves, but of the variance of 

these functions. He also proposed an effective way to 

evaluate the variance of any function of slope.

The problem is to express the width, depth, and 

velocity of a stream in terms of the applicable independent variables, one of which is discharge. Because 

of the interrelations of the variables and their product, 

it appears that an exponential relation between each 

dependent variable and the independent variable, as 

used by Leopold and Maddock (1953a), is a simple 

form that proves satisfactory. If discharge is the independent variable, then width, depth, and velocity may 

be expressed as functions of discharge. Thus,

velocity cc Qm, 

depth oc Qf, and 

width oc Qh .

Since

discharge= widthX depthX velocity, 

from equation 21,

and

(21)

(21a)

Equation 21 a can be stated as

log discharge=log width+log depth+log velocity. 

Although these equations are all true in a hydraulic 

sense, they also have a purely statistical significance.

For example,

log width = log a+b (log discharge),

log depth=log c+/ (log discharge), (22)

log velocity = log k-\-m (log discharge).

Because a, c, and k are found by experiment to be 

essentially constant and have little variance, the 

standard deviations can be written as

o-log Wccbalog Q, o-log Focmo-log Q, etc.,

and finally

<rlog

~<rlog Q 

<rlogZ>

<rlog Q

, o-log V, 

"o-log Q

(23)

in which the population is all observed values of Q 

and their respective values of V, D, and W. Langbein 

(1964) shows that this reduces to
A30 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

l=6+/+m (21b)

in a purely statistical sense.

The exponents of the independent variables describing the elements of discharge therefore, are indices 

of the standard deviations of the dependent variables. 

Further, the variance of the dependent variable can be 

expressed by the square of the indices or the proper 

combinations thereof. For example, the variance of 

velocity can be expressed by m2 , and, because m= 1 6 / 

in equations 21a and 21b, the variance may also be 

expressed as (1 6 f)2 . The variance of a Froude 

number can be expressed as (2m /)2 or (2 26 3/) 2 .

The measures of the relative changes in velocity, 

width, and depth, as represented by equation 2 la, 

are most nearly equal when the sum of the squares 

of the discharge indices that is, the sum of the variances is a minimum. Under these conditions, m2 +/2 

+ 62 would be a minimum. We can easily show that 

if all elements are free to vary and have the same 

probability, m=/=6=%. The continuously reforming 

streams on the valley train beneath Emmons Glacier 

on Mount Rainier, Wash., are representative (Fahnestock, 1963).

In flumes, width is a constant, and in most streams 

width is great enough to consider the flow as two-dimensional. Under these conditions, the unit discharge, 

q, is used instead of the total discharge Q. Thus,

and V and D are expressed as functions of q. Hence, V 

ocgm and D oc Qf. When q is a variable, m+/=l, and 

when q is a constant, m= /. Expressing the variance 

of V as m2 and the variance of D as f*, the variance of 

the Froude number, if unit discharge is the independent 

variable, is (/ 2m) 2 or (1 m 2m) 2 . If the variance of 

the Froude number is to be minimized, the first derivative of (1 3m) 2 must be set equal to zero. Then

18m 6=0 

TO =1/3.

Thus, the variance of the Froude number is minimized 

when V=ql/3 . Since D=q2/3 when V=ql/3, it follows, 

Foc<f /3/<f/3 or F= constant. When the Froude number is 

constant, its variance is zero.

Discharge is but one o± the independent variables, 

and, as already stated, there must be a second. We now 

assume that any dependent variable can be expressed 

as any independent variable raised to some exponent 

in precisely the same way as we assumed with discharge 

and that the evaluation of the relations between these

exponents and the variance of the dependent variables 

will be the same as was developed for discharge.

Definite experimental evidence exists that the 

variance of slope itself is, never minimized in flumes. 

The two important functions of slope are the shear 

stress, yDS, and the friction factor gDS/V2. Analyses 

of the Gilbert (1914) and Waterways Experiment 

Station (1935) data indicate that neither the shear 

stress nor the friction factor alone were minimized in 

the performance of the experiments (eqs 19 and 20). 

This is in accord with the theory presented herein. The 

effects of any change in the independent variable 

affecting both the shear stress and the friction factor 

should not be concentrated in either one but should be 

split between the two. Only if one of the two terms is 

held constant should all the change be concentrated in 

the other. Furthermore, according to the theory presented, the variability of each tends to be a minimum. 

Because both terms cannot be a minimum at the same 

time for the same discharge, the most probpble condition occurs when the sum of the two variances is at a

minimum.

CONSTRAINTS

In accordance with Langbein's (1964) concepts, we 

will analyze the behavior of alluvial channels under 

different combinations of independent and dependent 

variables that are under different kinds of constraints. 

These constraints are (1) the joint minimization of the 

shear stress and the friction factor, (2) the joint minimization of the velocity and the friction factor, and (3) 

the joint minimization of the concentration and the 

friction factor. The constraints are exclusively statistical relations that do not give directions for deciding 

which dependent variables are relevant. Such information, which is to be included in the variance statement, 

can only come from the mechanics of the processes 

involved or as inferred from the data. Following each 

analysis, the applicable flume and field (river and 

canal) observations will be compared with the theoretical forecast.

DISCHARGE AND SEDIMENT LOAD AS INDEPENDENT VARIABLES

Given

q= independent variable, and 

i=load as constant, 

define

Vacq ,

slopeacq*,

loadocg", (because load is constant).

As shown in the first part of this report, load, the 

sediment transport per unit of width, is a function of
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A31

m+f=l.

velocity and discharge that can be simplified to the 

relation

ZocFY- (24)

This relationship always exists in the same sense as 

q=DV, whatever the dependent or independent variables. Because load has a unique relation to velocity and 

discharge and a and,/ are given, then

or

Further,

or

or

~~a J ~ « (25)

Now the variance of the shear and friction factor in 

combination is minimized. First, we note that

variance of shear stress = (f-\-z) 2 , and 

variance of friction factor= (/+2 2m) 2 .

The variances are added and minimized:

a minimum.

Squaring, as indicated, and setting the derivative 

equal to zero,

,=-i_2z,

or

1+2J 

) «. (26)

Now, consider q as constant and L as the independent 

variable. By definition

V<xL 

DocL*,

/P

where

or

and

q=VD,

X 

a (27)

Minimizing the variance of shear and friction factor 

in combination as before,

(z+/3) 2 +(a;+j8 2m) 2->a minimum,

/ 1\2 / 1 2V (|8 ) +(|8 ) >a minimum,

\ a/ \ a a/

or

Thus,

slope ocZ«. (28)

If there is no correlation between the independent 

variables, discharge and load, the two relations above 

may be combined as

Soc

But since

<2 «

V2

 >

2

or

(29)

The exponents disappear in the analysis, although 

they are based on an explicit statement of sediment 

transport in the form LocVaq} . The relation, then, is 

independent of the nature of the transport equation, 

as long as load is a unique function of velocity and 

discharge.

The Gilbert equations have the same independent 

variables, discharge and load, and the same dependent 

variables, velocity, depth, and slope, as were set forth 

in the hypothetical example. Equation 19, developed 

from an empirical analysis of these data was

which confirms equation 29. (Note in figure 22 that 

eq. 19 relates to an average or a most probable condition.) Because this is a statistical evaluation, these 

relations are not expected to be precise. This will be 

true for all relations developed by this type of analyses; 

however, the coefficient of equation 19 is related to 

stream behavior. A high value of the coefficient is 

associated with a stream that is carrying a high load 

for a given slope or a stream that has a tendency toward aggradation. A low value of the coefficient is 

associated with low loads and tendencies toward degradation. Thus, the range in coefficient represents the
A32 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

range of change of roughness due to change in bed 

form. To aid in the solution of equations 19 and 20, 

values of the parameters wl/i/d3/s and (w/d)l/2 forgiven 

values of the particle diameters are shown in figure 

25^4 and B.

Because natural streams exhibit the reaction of load 

and discharge on a long-term basis (Rubey, 1952), 

equations 19 and 29 would be expected to describe 

them.

10

2 

0

I i i i i i i i

i i i i 1 1 1 1

1 1 1 1 1 1 1 1

 - -.

Most common sand sizes

i i i i i i i i

-

-

01 0.1 1.0

PARTICLE DIAMETER (d), IN MILLIMETERS

100

10

Most common sand sizes I I

0.01 O.i 1.0 

PARTICLE DIAMETER (d), IN MILLIMETERS

B

FIGURE 25. Values of sediment size parameters from equations 

19 and 20. A, Values of w1 /4 /^ 78 - B, Values of (w/d) l '°. Values 

for A and B computed for quartz spheres in water at 25° C.

Blench (1966) gave three equations basic to "regime" 

channels. These are V2/D=Fb , V2/gDS=3.&3 

and V3/b=Fs. Combining the last two

V2

gDS

/T/4\l/4

=3.63 (^)

or

and

F2=3.63 il/4

(30)

This is the same basic relation as equations 19 and 

29. As will be shown later, the term 3.63 (Vb/v) 1 '* is not 

a unique measure of the friction factor in rvers. In 

the form of equation 30, it is an expression of specific 

channel behavior through the use of the coefficient Fs. 

Equation 30 differs from equation 19 in that it contains no size parameter. Blench's size parameter is 

contained in his first equation V2/D=Fb . In the author's 

system, V2/D=C, or concentration of sediment in the 

mid-velocity range and size is unimportant.

The Lacey equations are also variants of equation 

19. Let the equation be written

.1/2

gD

Lacey used a silt factor /, which is defined as either 

f<x.V2/D orj2<x.VS. In each equation, J2 ccd. In the sand 

range, the relation between particle diameter and 

w/d3/2 varies from about w/d3/2<xd~l/2 to w/d3/2 ocd~l . 

Consequently, substituting / for w/d3 '2 the range would 

be from j~l to /~2 . Substituting the two silt factors 

when w/d3/2 ccf~2

or

The other relation is

VS

V2

D

Both of these expressions are equivalent to Lacey's

When

w

^°<

T/2

-WCC54

(vs <2

or
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A33

the other relation is

~D

or

The latter is equivalent to Lacey's F=60 

It would appear however that Lacey's general relationship should be VocDS/dl/2 and that B is essentially a 

constant.

From equation 30, Blench's side factor Fg is related 

to B, that is, B^Fs~ll\ Since FS=V*IW and V3&:QC/W, 

FxocQC/W2 and B&:(W2/QC) 1/8. Thus, according to 

Blench there will be very little variation in the coefficient B.

In a discussion of departure from "regime," Inglis 

(1949, p. 99) uses two values of /. Lacey's/=(F2/D) 1/2 

becomes / , and /oc(7?1/2$) 2/3 becomes / . Note that 

fvr/(frs)3/2= Bi:, which appears to confirm Inglis' statement that (/w//rs) 1/2 was a measure of the deviation from 

regime. From a practical standpoint, however, it is 

obvious that the / in the Lacey equations is a most 

elusive parameter.

The relations among the coefficients of these equations will be discussed more fully in the sections on 

regime channels and stable channels.

Recently, Engelund (1966) proposed a series of equations that Maddock (1966) showed to reduce to

(31)

Engelund apparently developed the logarithm term 

from analysis of flume experiments. Consequently, it 

is doubtful if the term will fit all flume and field data. 

Nevertheless, the basic form is the same as that of 

equation 19.

Einstein and Barbarossa (1952) proposed a similar 

but more complicated relation, in which the friction 

factor due to the bed forms is related to the shear 

stress on the grains of sediment and to the size of grain. 

For values of 1.68 D35/R'S in excess of about 2 the 

relation becomes

V gR'S " 1/2

which, for all practical purposes, seems to be another 

variant of equation 19. The feed system flume data of 

Guy, Rathbun, and Richardson (1966) also have this 

same basic relation.

In conclusion, all the standard approaches to the 

roughness of alluvial channels have the specific limitation that the depth, velocity, and slope are all supposed

to vary with changing discharge. This may occur because sediment load is an independent variable, as it 

was in the development of equation 29, or because small 

discharges, narrow flumes, or coarse material limit the 

possible combinations in a recirculating flume. Equation 

29 and the Blench (1964) and Engelund (1966) equations are independent of the form of the relation b°-- 

tween velocity and sediment load; the Lacey (1946) 

and, perhaps, the Einstein-Barb arossa equations rre 

not. No precise relation exists in equation 19 or any 

other equation relating to slope; the relation may be 

the most probable or may reflect channel behavior, and 

consequently these equations are all "regime"equations.

DISCHARGE AND SLOPE AS INDEPENDENT VARIABLES

Variance of shear stress and friction factor

First, consider discharge as an independent variable 

and slope as constant:

Vacg , 

Dccf,

slope ocg3 where z=Q (slope constant),

Minimizing the sum of the variances of shear stress and 

the friction factor

(f+z) 2+(f+z-2m) 2 

(l-m) 2+(l-m-2m) 2

>a mnimum, 

->a minimum.

We perform the necessary operations, 

m=0.40,

(32)

Secondly, consider discharge as constant and slope 

as an independent variable. 

Define

t+u=Q.

Minimizing the sum of the variance of shear stress and 

(u + 1 ) 2 + (u + 1 2f) 2 >a minimum,

and, as before,

=4

(33)
A34 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10

^ 1.0

Approximate limiting velocity

for'size shown 1-0 mm

Maximum velocity with a 

fully deformed bed

Minimum velocity with a 

fully deformed bed

EXPLANATION

Symbol Sand (mm)

X 0.19 

o 0.27

0.28

+ 0.32 

o 0.45 

. 0.47 

e 0.54

0.93

0.1

0.001 0.1

c f w \ qS(T)

0.1 1.0

FIGURE 26. Relations between velocity and unit discharge, slope, and mean grain size of sediment, Simons and Richardson (1961)

data from recirculating flumes.

Again, if there is no correlation between the two 

independent variables, equations 32 and 33 may be 

combined in the form

(34)

Obviously, there are many situations where slope is 

not going to change with discharge, and both flume 

and field data are available to confirm this relation. 

As was previously noted, equation 20 was developed 

from the Waterways Experiment Station (1935) data 

with slope as an independent variable.

The Simons and Richardson (1961) experiments 

provide data for a much greater range in the value of 

slopes than those of depth and unit discharge. (The 

highest value for slope is about 100 times greater than 

the lowest value; the highest values for depth and unit 

discharge are about four times greater than the lowest 

values.) That these data are essentially the result of a 

slope-independent variable operation is shown in figure 

26, where the equation of an average line is

Equations 20 and 35 are identical except for the coefficients, 4.4 and 5.5.

Data from Hubbell and others (1956, 1960), Simons 

(1957), Nordin and Beverage (1965), and the Harza 

Engineering Co., International (1963), are shown in 

figure 27. Again the relation of the data to equation 35 

is evident.

The difference between the coefficients of equations 

20 and 35 is attributed to the fact that the procedures 

of the Waterways Experiment Station (1935) studies 

had the effect of minimizing the sediment load and, 

consequently, the velocity. The range in observed 

values about equation 35 represents the change in the 

coefficient due to variation of bed forms largely associated with sediment load. Differences in the sizes of 

transported and bed materials may account for a small 

part of the difference.
10

BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

10

A35

LJ 1.0

LLl

x xxx

Minimum line extended

Maximum velocity vyith a 

fully deformed bed

x o o

 9 » X

^r * * **/ 

Minimum velocity with a 

fully deformed bed

0.1

0.001

EXPLANATION

Marala-Ravi Link canal 

Rangpur canal [ (Harza 

^Engineering 

Lower Chenab canal QO 1953)

Abbasia canal J 

Canal (Simons, 1957) 

Rio Grande

(Nordin and Beverage) 

Middle Loup and Dismal

Rivers (Hubbell)' 

Colorado River at Yuma, Ariz.

(Leopold and Maddock, 1953a)

0,01 0.1 1.0

FIGUEE 27. Relations between velocity and unit discharge, slope, and mean grain size of sediments for certain canals and natural

streams.

The envelope curves shown in figures 26 and 27 are 

approximations of the limit of change in roughness for 

any value of qS for a mean grain size of about 0.3 mm. 

The position of the upper limit at least, will vary with 

changing grain size. Figure 26 also shows the approximate maximum velocities for given sizes of sediment. 

These, together with the envelope curves, circumscribe 

the behavior of any alluvial channel.

Figures 28 and 29 are taken from Vanoni, Brooks, 

and Kennedy (1961) and show the relation between 

equation 35 and observations for two natural streams 

at constant slope. In contrast, figure 30 (Culbertson 

and Dawdy, 1964) shows observations for a stream 

having a variation in slope and in the sediment load. 

The figure shows the potential variation, due to changes 

in slope, in the relation between the hydraulic radius 

and velocity. The other variations as shown by the 

plotted points may be due to changes in load and size 

of the sediment in transit.

Variance of velocity and friction factor

Rubey (1938) noted that for the movement of smaller 

sizes of sediment shear stress seemed to be the governing 

factor, but in the larger sizes, velocity seemed to be 

most important. If this is true, variance in velocity 

possibly should be minimized. Furthermore, it would 

seem natural that a stream would resist variations in 

velocity to minimize change in momentum. Decisions 

as to whether variance in load or velocity are being 

minimized may be difficult. Consequently, the effect 

of minimization of velocity together with the friction 

factor must be determined.

If g and 8 are independent variables, and if q is 

varied while S is constant, and

Vacg ,

q=VD and m -\- f= 1 , 

$=constant=#°.
A36 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

0.1

0.1 0.4 0.6 0.8 1.0 2.0

DEPTH (D), IN FEET

FIGURE 28. Relation between unit discharge and depth for 

Mountain Creek near Greenville, N.C., showing observed 

values (solid circles) and values computed by equation 35 

and by the Einstein-Barbarossa method. From Vanoni, 

Brooks, and Kennedy (1961).

To minimize the variance of V and the friction factor 

in combination

or m2+(l 3m) 2 ^minimum, 

2m+18m 6=0, 

m=0.3,

(36a)

If S is varied where g is constant and 

VocS1,

then

q=VD= constant and t+u=Q, u= t.

To minimize the variance of V and the friction factor 

in combination

or

2^+18^-6=0, 

£=0.3, FocS0 -3 . (36b)

10.0

0.1

0.06 0.08 0.1 0.2 0.4 0.6 

DEPTH (D), IN FEET

0.8 1.0

FIGURE 29. Relation between unit discharge and depth 

for West Goose Creek near Oxford, Miss., showing 

observed values (solid circles) and values computed by 

equation 35 and by the Einstein-Barbarossa method. 

From Vanoni, Brooks, and Kennedy (1961).

Under conditions of independence of g and S, equations 

35 and 36 may be combined, and

(37)

The relation between velocity and %S for stations on 

the Colorado River below Glen Canyon Dam is shown 

in figure 31. The high flows, usually associated with 

periods of scour, are for the Colorado River at Grand 

Canyon, Ariz., from 1961 to 1965. Other data are from 

locations below Davis Dam reported by the Bureau 

of Reclamation. In this section of the river, the stream 

derives its load from its bed and banks, as incoming 

natural load has been cut off by dams. Considerable 

diurnal variation occurs in the releases from Davis 

Dam, and the flow during flood periods was unsteady.

The relation shown in figure 31 is

V=UA(g_S) 0.3 (38)

as might be expected from the analysis leading to 

equation 37.

Figure 32 is taken from Vanoni, Brooks, and Kennedy 

(1961) and shows the adaptation of equation 38 to the
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A37

6.0 

5.0

4.0 

3.0

2.0

1.0

0248 

VELOCITY, IN FEET PER SECOND

FIGURE 30. Relation of velocity to hydraulic radius for San 

Francisco riverside drain near Bernardo, N. Mex. Different 

values of slope computed from equation 35. From Culbertson 

and Bawdy (1964).

Colorado River at Taylors Ferry, Ariz. Figure 33 shows 

the behavior of the Colorado River at Yuma, Ariz., 

before and after the construction of Hoover Dam. The 

original slope was an aggrading, self-formed slope, so 

a high coefficient was used for equation 19. After dam 

closure and with only a small amount of sediment from 

the desilting works of the All-Americah Canal, the 

channel was still self-formed but degrading, hence, a 

smaller coefficient was used for equation 19. The 

higher releases in the post-dam period, however, were 

for sluicing purposes; hence, discharge was irregular and 

sediment load was secured locally. Consequently, 

equation 38 applies as shown.

Variance of concentration and friction factor

Observation indicates that the concentration of sediment tends to be conservative, not the load, which is a 

rate of sediment movement per width of channel. Thus, 

changes in width, which result in changes in load with 

concentration and stream discharge constant, are readily 

accepted by a stream, but changes in concentration 

appear to be resisted. The heads of flumes and even 

canal systems, particularly with low discharges and 

with coarse sediments, seem to illustrate this condition.

If slope and discharge are independent variables and 

if L oc V°tf, then C oc Vaqj~l . If the variance of the

concentration and friction factor are to be minimized 

and V oc qm, the former procedures yield, when slope 

is constant,

minimum,

 > minimum. 

Setting the first derivative equal to zero

2a2m+2at(j l) + 18m 6=0, 

or

m=-

or

V=q ' ^

If S is a variable and q is constant, V oc S* and 

C oc Sat. Further, D oc Su and, because q is constant, 

u= t. Again, to minimize jointly the variance of the 

concentration and the friction factor

(at) 2 + (1 3<) 2 » minimum 

and, setting the derivative equal to zero, 

2o?t- 6+18<=0,

t=

or

140)

With the restriction of complete independence of g 

and S, equations 32 and 33 may be combined in the 

form

3-aQ-l)

V=q «2

The exponents of this equation will vary, depending 

upon which of the velocity-load equations are chosen. 

If equations 10 or 11 are chosen, a=4 and j= % > 

whence

(42)

If equations 6 and 12 are used, a=9 and.?'=0, and

(43)

Because the variance in concentration is minimized, 

these equations show little variation in velocity with 

discharge in the low-velocity range or with slope in 

either the low- or mid-velocity ranges. However, this 

condition seems more likely to be found in the lowvelocity range than in the high-velocity range. In the 

studies of Guy, Simons, and Richardson (1966, p. I 33) 

using 0.45-mm sand, flow was increased from 4.24 cfs 

in run 19 to 12.12 cfs in run 21. Slopes in the two runs 

were 0.00112 and 0.00114, respectively. The report 

states: "This increase in flow caused the depth of flow
A38

10

BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

0.1

0.0001

EXPLANATION

Grand Canyon

Taylors Ferry

RS 30A

RS 33

RS below Needles Bridge

RS 43

RS below Palo Verde weir

(l/<2.5 ft per sec only)

0,001 0.01 0.1

FIGUEB 31. Relations between velocity and the product of unit discharge and slope for locations on the Colorado River at Grand 

Canyon, Ariz., and below Davis Dam. Discharge measurements at Grand Canyon by U.S. Geological Survey; all other data 

from U.S. Bureau of Reclamation (1958), RS, River station.

to more than double and caused the velocity to increase 

from 1.30 to 1.58 fps." This appears to be a confusion 

of cause and effect. The flume by itself could not 

increase the depth. Nearly 5,000 gallons of water had 

to be added to the recirculating system. Undoubtedly, 

a considerable manipulation of the flume's tailgate was 

required. The point is that the slope of run 19 could 

not have been maintained with the discharge of run 21 

without holding the scour at the lower end of the flume 

to a minimum and maintaining sediment concentration. 

Tailgate manipulation had this effect. Actually, the 

sediment concentration did increase from 208 ppm to 

380 ppm, although the mean particle size declined, 

Thus, the velocity rose to 1.58 fps (feet per second) 

rather than to 1.45, which would have resulted if the 

variation in the sediment concentration had been 

minimized.

The Simons and Richardson experiments using a 

0.93 mm sediment also are reported by Guy, Simons, 

and Richardson (1966). The results of these experiments are shown in figure 34. Relations between 

velocity and gS for qS values of less than .001 are 

mostly dependent on grain size or a limited variation 

in concentration with low discharges. Once the bed 

is deformed, however, other considerations take hold. 

Equation 43 in the form

F=2.33 (44)

was applied to runs with values of qS between 0.001 

and about 0.004, which appears to be the end of the 

low-velocity range. The results are as follows:

Velocity

Run

15.

16. 

35. 

17. 

10, 

36.

Observed

1.93

2. 03

1.80

2. 10

1.88

2.04

Computed

2.00

2. 06

1.87

2. 08

1.86

1.96

Difference

+ 0. 07 

+ . 03 

+ . 07

-. 02

-. 02

-.08

Percent

3.61

1.47

3. 87

.95

1. 06

3. 92
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

100

Calculated by

Einstein-Barbarossa

method

4 6 8 10 

DEPTH IN FEET

FIGURE 32. Relation between unit discharge and depth for 

Colorado River at Taylors Ferry, Ariz., showing observed 

values (solid circles) and values computed from observations at Yuma, Ariz., using equation 38 and by the EinsteinBarbarossa method. From Vanoni, Brooks, and Kennedy 

(1961). Slope is 0.000217 foot per foot.

Equation 43 would rarely apply in a natural stream 

because velocities are usually above the low-velocity 

range.

BEHAVIOR OP FLUMES OPERATED AT CONSTANT DEPTH

The cause of much of the confusion in the study of 

alluvial channels can be seen in the behavior of flumes 

operated at constant depth. If depth is held constant, 

that is, not permitted to vary, and discharge is an 

independent variable, held constant for any one run, 

velocity is uniquely determined, and there is no variance 

in either velocity or depth. The only variance is in 

slope. Thus, there may be either minimum variance 

in the shear stress yDS or minimum variance in the 

friction factor yDS/pV2 , but not both. If the variance 

of the shear stress is to be minimized, Soal/D. If the 

variance of the friction factor is to be minimized, 

¥«(#£) 1/2 . Therefore, if D is constant, S must either 

be a constant and not related to velocity or V<x.Sl/z. 

If discharge is held constant through a series of runs 

at different depths in which depth is constant over 

the period of the run, again velocity and depth are

1000

20

- 10

z 

o

500 1000 10,000 100,000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 33. Relation of width, depth, velocity, water-surface 

elevation, and stream-bed elevation to discharge, Colorado 

River at Yuma, Ariz. Samples taken during periods before 

and after construction of Hoover Dam are shown. Circles 

indicate samples taken during 1927-34; X's indicate samples 

taken during 1945-47. From Leopold and Maddock (1953a)

uniquely related, and slope is the only variable. Agair, 

the variance of the shear stress or of the friction factor 

may be minimized, but not both. If the shear stress
A40

10

BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

 1.0

0.1

0.0001

^Maximum velocity with a 

fully deformed bed

^Minimum velocity with a 

fully deformed bed

0.001 0.01 0.1

FIGURE 34. Relations between velocity and unit discharge and slope. Simons and Richardson's 0.93-mm sand in 8-foot 

recirculating flume. Circles represent low-velocity range VocLel'g ; X's represent mid-velocity range

is minimized, Socl/DocV. If the variance of the friction 

factor is minimized, VocS173 .

When the above possible relations are combined with 

others previously noted, at least five different relations 

between velocity and slope can be expected if depth is 

a constant. The first three are where the value of 

depth is arbitrarily selected from an array of data: (1) 

depth is the result of a changing discharge, in which 

depth, velocity, and slope are free to vary Voc(qjS} 1/2 ; 

(2) depth is the result of changing discharge and slope, 

in which only depth and velocity are free to vary 

Voc(qS~) QA ; and (3) depth is the result of changing 

discharge and slope with the constraint of joint minimization of the variance of the sediment concentration 

and the friction factor Foeq°^S°-12 or Foc2°-133S0 - 033 , 

depending upon the applicable velocity-load relation. 

For a selected depth, these reduce to VocS, FocS2/3 , 

FocS0 - 177 or FocS0 - 038. For runs where, by flume operation, depth is held constant, then VocS0 that is, no 

relation or FccS1/2 .

Examples of selection of a constant depth from data 

from the Gilbert (1914) experiments are shown in

figure 35. In these experiments discharge and load were 

the independent variables and velocity, depth, and 

slope were free to vary. Note that the predicted relation, 

VocS, is supported by data from Gilbert's runs with 

sand A, 0.305 mm, and sand B, 0.375 mm. This result 

should be compared with the results from recirculating 

flume data to be presented later.

As noted earlier, the Williams (1967) data result from 

operation of a flume at constant depth; load and depth 

were independent variables. Because of the near-unique 

relations among velocity, discharge, and load, the 

result is the same as if depth and discharge were 

independent variables. As shown in figure 24, Vac(qS) 1/3 

or VocS1 '2 .

Recirculating flumes are particularly applicable to 

the study of channels operated at constant depth. If 

the amount of water in the recirculating system, which 

includes any storage tank, is kept constant, the mean 

depth of the water in the flume must remain the same. 

Thus, through simplicity of operation, many observations of recirculating flumes have been made, perhaps 

unintentionally, from runs made at constant depth.
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A41

10

1.0

0.001

Depth 0.115±ft Depth 0.195±ft

SLOPE 0.01 0.001

GILBERT SAND A (0.305 mm)

SLOPE 0.01

10

1.0

0.001

Depth 0.1 10±ft

/*

X/ *

** y

Depth r\220±ft

SLOPE 0.01 SLOPE 0.01

GILBERT SAND B (0.375 mm)

FIGURE 35. Relations between velocity and slope when depth is approximately constant, Gilbert (1914) sands A and B for depth?

as shown; X's observed values.

These are, of course, in addition to those operated 

specifically to conserve the amount of water in the 

system. Data from a number of small flumes operated 

in this way are shown in figure 36. Data from larger 

flumes are shown in figure 37. As will be noted later, 

not all the data from Simons and Richardson (1961) are 

for runs with depth held constant.

Examples of sets of runs performed at a constant 

depth were given by Vanoni and Hwang (1967) for 

depths of about 0.24 foot and about 0.59 foot (fig. 

38.4) and by Vanoni and Brooks (1957) for depths of 

about 0.24 foot and 0.54 foot (fig. 385). As shown 

in figure 38A and B, sand sizes used by Vanoni

and Hwang were 0.23 mm and 0.206 mm, and Vanoni 

and Brooks used a sand with a geometric mean grain 

diameter of 0.137 mm. Figure 38A also shows data 

from Simons and Richardson (1961) for runs with 

depths of about 1 foot and about 0.57 foot with sard 

sizes of 0.19-mm, 0.27-mm, and 0.28-mm mean grain 

diameter. Although Guy, Simons, and Richardscn 

(1966) did not report any special effort to keep depth 

constant throughout a run, the data shown in figure 

38A (which includes all data from runs using a 1-ft 

depth and a part of those using a 0.57-ft depth) indicate 

that this was probably the situation.

In figure 38^4 and B the predicted relation VccS1/2
A42 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10

a LO

Maximum velocity with a 

fully deformed bed

Minimum veloci''1 ' with a 

fully deformed bed

EXPLANATION

Laursen (1958) 0.1 rrm 

Barton and Lin (1955) 

Vanoni and Brooks(1957) 

Brooks (1958) 

NomicosOn Brooks, 1958)

0.1

0.0001 0.001 0.01 0.1

FIGURE 36. Relations between velocity and unit discharge, slope, and mean grain size of sediment. Small recirculating flumes.

is clearly defined in the lower slopes. The effect of 

the constraint of constant discharge is shown in figure 

38(7. In accordance with the prediction the friction 

factor is constant, Voc/S173 . This relation is particularly 

applicable to the 0.28 mm grain size of Simons and 

Richardson (1961).

When the values for V and $ for different discharges 

(fig. 38(7) are combined, the result is very close to 

Foc(g$) 1/3, or the friction factor is approximately 

constant for a range of discharges. If the values of 

V and S for different depths (fig. 38.4) are combined, 

the result is VocDS1/2 .

It should be emphasized that although a good correlation exists for the two equations the result is meaningless. If discharge is held constant and depth is permitted to vary or if depth is held constant and discharge 

is permitted to vary, there is no fixed relation between 

velocity and depth and slope unless some specific 

condition is implied. In one situation, depth is a 

constant and VocQ; in the other discharge is constant 

and Vccl/D. Thus, no direct relation actually exists 

between velocity and slope at all.

Although the constant friction factor relation is well 

defined in figure 38^4, B, and G, the constant shear stress 

relation is not as clear. The Vanoni and Hwang (1967) 

data for a depth of 0.24-foot (fig. 38.4), the Vanoni and 

Brooks (1957) data for a depth of 0.24-foot (fig. 38J3), 

and the Stein (1965) data for different depthe (fig. 39^4, 

B, (7) all show the shift from a constant friction factor 

to a constant shear stress sharply and clearly. The shift 

is only partially defined in the Simons and Pichardson 

(1961) data for the 0.27-mm and 0.28-mm grain sizes 

in the 8-foot flume and for the 0.33-mm sands in the 

2-foot flume. The shift is not obvious in the Simons and 

Richardson data for the 0.45-mm sand (fig. 3?A, B, (7).

Part of the problem of obtaining definitive relations 

is that in a recirculating flume the shift to constant 

shear stress can take place only after the bed forms have 

reached some critical size, shape, or other characteristic. 

This is an unstable situation, because, in the data 

shown in figure 38 A, B, and C, the friction factor has remained essentially constant at a constant depth of 

flow despite the fact that the height of the bed forms 

has increased with increasing slope. In the data for
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

Maximum velocity 

fully deformed

Minimum velocity 

fully deformed

EXPLANATION

Symbol Depth (ft)

Stein (1965)

o 1.0

 0.6

x 0.8

n 0.4

a Others

Simons and Richardson (1961) 

e Varying

0.0001

FIGURE 37. Relations between velocity and the product of unit discharge and slope for 0.4 and 0.45 mm sand. Data from Steir 

(1965) and from Simons and Richardson (1961). Slopes for constant DS and friction factor are also shown.

1-foot depths (fig. 38^4), there is a maximum range in 

height of bed forms of more than 20 times (0.03ft-0.65 

ft for the 0.19-mm sand) between slopes of 0.00023 and 

0.00165. Increased height of bed form with increased 

slope, although a change of lesser magnitude (of 20 

times), is observed in data shown in figures 38^4, B, 

and C.

Possibly the flume may make some internal adjustment. The point of instability occurs when sand bars 

and sand waves appear in the flume. The average 

depth remains the same, but sediment moves at different 

rates at different depths at different points in the flume. 

This could result in flow in which the surface slope 

could markedly vary and with unknown results and 

could change the situation to one of depth, velocity, 

and slope all varying. In this situation V<x.S. These 

changes at the point of instability may account for 

the behavior of the 0.27-mm, 0.28-mm, and 0.33-mm 

sands (fig. 38Z?) as well as that for the Vanoni and 

Hwang 0.206-mm sand at a depth of 0.59 foot. The 

behavior may also be caused by variation in the method 

of operating the flume, although this would seem doubtful under the special conditions of the Vanoni and

Hwang experiments. The method of operation does, 

however, seem to be the cause of the variation in the 

results of the Simons and Richardson experiments 

with 0.45-mm sand in the 8-foot flume.

The difference in the behavior of the flume operated 

by Simons and Richardson and that operated by Stein 

is shown in figure 39.4, B, and C. The Stein data ar-, 

consistent that is, low and high flows are constrained 

by a constant friction factor, and intermediate flow? 

are constrained by constant shear stress. The Simon^ 

and Richardson data are not consistent; at a depth of 

about 0.4 foot (fig. 39^4), the flume seems to have been 

operated for a few runs in which depth was held constant but for other runs in which slope and discharge 

were independent variables and V<xS2/3 - The same situation is observed for 0.8-foot depth (fig. 395), but the data 

for depths of about 0.6 foot (fig. 39(7) appear to have 

such variation in method of flume operation that one 

run has no specific relation to another. The operation 

of the flume as described by Guy, Simons, and 

Richardson (1966) would permit almost anything to 

happen.
A44 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

1.0

EXPLANATION

Symbol Depth (ft) Sand (mm)'

Simons and Richardson, width 8 feet

 1.03±0.03 0.19

o 0.56±0.02 0.19

A 1.02 0.27

v 1.03±0.03 0.28

a 0.59 and 0.57 0.28

Vanoni and Hwang, width lOVi inches

x 0.240+0.008 0.23 

Vanoni and Hwang, width 435/16 inches 

O 0.590±0.013 0.206 

c 1.216 0.206

0.1

0.0001

B

EXPLANATION

Symbol Depth (ft) Sand (mm)

Vanoni and Brooks, width 33l/z inches

e 0.240±0.003 0.137

o 0.538±0.011 0.137

SLOPE

0.001 0.001 

SLOPE

10

8.91 cfs.

X 7.76cfs

EXPLANATION

Symbol Sand (mm)

Simons and Richardson, width 8 feet

x 0.28

o 0.27

 0.19

Flume discharge, in cubic feet per 

second as shown

0.1

0.0001

.0.59

EXPLANATION

Symbol Sand (mm) 

Simons and Richardson, width 8 feet

 0.28

 x 0.27 

Simons and Richardson, width 2 feet

o 0.33 

Depth, in feet as shown

SLOPE 0.001 0.001 SLOPE

FIGURE 38. Relations between velocity and slope for various depths of flow and different sizes of sand. Data from 

Simons and Richardson (1961), Vanoni and Hwang (1967), and Vanoni and Brooks (1957). Dashed lines used 

so as not to obscure plotted points.
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A45

1U

SECOND

(r

LJ 

Q.

Ld 

U.

Z

t 1.0 

o 

o

Ld

Z

LJ

n 9

-

0.'

0.46/

/S^

X 0.35 X 0' 35

-

-

A +<& 0.43 x v!xx^

0.40 x °

0

o

\| JX^o 42^^ '^ ^J^^^ '^^"^

EXPLANATION

Symbol Depth 

Stein (1965) 

o 0.4 ft

Simons and Richardson (1961)

\

Depth, in feet, as shown 

i i i i i i i i

* O -

o

-

-

-

-

-

10-4 10-3 10-2

SLOPE

10

LJ 1.0

0.1

B

*0.81

EXPLANATION

Symbol Depth 

Stein (1965) 

o 0.8 ft 

Simons and Richardson

(1961) 

x 

Depth, in feet, as shown

EXPLANATION

Symbol Depth

Stein (1965)

 0.6ft

Simons and Richardson (1961)

A 

Depth, in feet, as shown

10-3 10-3 10-2

SLOPE

FIGURE 39. Relation between velocity and slope for certain depths. Data from Stein (1965) for 0.40-mm sediment 

and from Simons and Richardson (1961) for 0.45-mm sediment.
A46 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

In the Stein (1965) data, with constant slope, there 

is a variation of about two times in the velocity, and 

the actual velocity increases somewhat with increasing 

depth. Therefore, if discharge is held constant and 

depth varied, the range in which the depth-slope 

product will be constant is also about 2. This is shown 

in figure 40, which shows the relation between depth 

and slope for a discharge of 7 cfs.

Because of the limitation on the range of roughness 

by changing bed forms, the variation in shear stress 

with wide ranges in discharge cannot be ignored. Under 

these conditions, the value of yDS cannot be constant. If 

values of constant slope are selected from figures 39A, 

B, and C, clearly yDS is not constant. It increases with 

increasing discharge, but not as much as it would if the 

shear stress and the friction factor were minimized 

jointly.

Stein also made a set of runs in which he appears to 

have attempted to maintain a constant velocity. All 

equations developed herein imply the form S=K^~l 

under such circumstances, but the relation OSocg"1 ) 

from flume data is indeterminate because of the variation in the coefficient. The runs with constant velocity 

were consistent with those with constant depth, and

i.o

\

0=7 cfs(Stem, data 1965)

0.1

0.001 0.01

SLOPE

FIGURE 40. Relation between depth and slope at constant 

discharge.

the relation can be determined from figure 39A, B, 

and (7.

Further information on the behavior of channels at 

constant depth is shown in figure 41. Simons and 

Richardson (in Guy, Simons, and Richardson, 1966) 

studied a 0.33-mm uniform sediment and a mixture 

having a median diameter of 0.30 mm under a flow of 

about 0.5-foot depth in a 2-foot flume. As shown in 

figure 41, there is a considerable difference in the behavior. The uniform material acts in the same way as 

a coarser sand in the 8-foot flume (fig. 38Z?). However, 

the sediment mixture shows the expected relation 

that would result if discharge and slope were independent variables and velocity, depth, and load were dependent variables. At least part of the variation in 

behavior is due to the differences in the way the flume 

was operated. Note that for values of qS of 0.0042 and 

0.0043, the uniform sand gave velocities of 2.57 and 

3.43 fps. This would be expected if the flume were 

actually operated with depth as an independent 

variable. Because the record of the flume operations 

during the different runs is not clear, the actual difference between stream behavior with uniform material 

and mixtures will be unclear.

Guy, Rathbun, and Richardson (1966) made studies 

on a d50=0.30-mm sand in a flume 20 cm vide. The 

results are shown in figure 41 along with date, from the 

2-foot flume. The studies were made to determine if 

there was any difference in the behavior of recirculating 

flumes and behavior of flumes in which the sediment 

load was introduced by a sand feed system. Their 

conclusion was that either system gave the same results 

because their data so indicated.

Unfortunately the Guy, Rathbun, and Richardson 

recirculating flume was operated differently from other 

recirculating flumes. In other flumes the sediment load 

was determined by sampling the water-sediment 

mixture, measuring the concentration of the sediment, 

and computing the sediment discharge as the product 

of the sediment concentration and water discharge. 

Guy, Rathbun, and Richardson trapped all of the sediment leaving the end of the flume during a measuring 

period. During this time sand was fed into the flume 

at an estimated rate. During a critical period of operation the so-called recirculating flume was operating 

under a feed system for about one-half the time.

In fact, the recirculating flume probably never behaved like anything other than one with a sand feed 

system as the data indicate simply because the feed 

system was dominant. Nevertheless there is one other 

possibility that bears investigation.

Given the relation
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A47

10

EXPLANATION

x Guy, Rathbun, and Richardson (1966) 

Mixture <//50 = 0.30 mm

oGuy, Simons, and Richardson (1966) 

Uniform d =0.33 mm

 Guy, Simons, and Richardson (1966) 

Mixture <//50=0.33 mm

. 

0.0001 0.001 0.01 0.1

DISCHARGE X SLOPE 

WIDTH

FIGURE 41. Relation between mean velocity and qS for uniform and mixed sediments of about 0.3 mm mean diameter in flume 

experiments. Depth is constant. Dashed line used so as not to obscure plotted points.

where Bf is a bed form. It has been noted that given D 

and C as invariant independent variables, either the 

variance of the shear stress or the friction factor tends 

to be minimized, because under these conditions there 

is no variance of D or V. But, if either by chance or by 

the method of flume operation, some perturbation should 

be introduced into C so it is not invariant, then the 

variances of the shear stress and the friction factor in 

combination would be minimized. The situation would 

be as follows. Depth and concentration are independent

variables. In addition L<xVaqJ. Then CocVa+J~lDJ~ l ,

-j-i

and when C is a constant V<x.D a+J-l } and when D is

a constant, V<xC°+J-i. Let SccD'C*. Then when D is

constant, S<x.Ch, and when C is constant, S<x.De . In 

accordance with the past procedures when D is constant variance of the shear yDS is expressed as h2 and

/ 2 V the friction factor, yDS/pV2, as ( h -^ - ) Then

\ a-\-J I/

the sums of the variances are minimized

/ 2 \2

h?-\-( h . =: - } ^minimum. 

V a+J I/

The first derivative of the function is set equal to zero

2h+2h 

h=

a+J-l 

I

=0,

or

When C is constant the variances of the shear streps 

and the friction factor can be expressed, respectively

as (1+e) 2 and \l+e+2a+J-1 ). Then the sums of 

the variances are minimized

^minimum.

The first derivative of the function is set equal to zero

j-i

2e+2+2e+2(l+2«+-7-1)=0,

i J~l e= l 

and

But <7ocFtt+/-1 ZX-1 ; 

hence,

J~l

a + J-l J-l l_

hence, 

and

VxDS.

This is the relation found in the Guy, Rathbun, and 

Richardson experiments. The perturbation in load 

caused by a change from a recirculation system to a 

feed system or vice versa may have introduced a 

variance into C. This relation is offered to explain the 

V<x.DS relation found when sand waves appear in 

recirculating flumes operated at constant depth such 

as in the Vanoni and Hwang experiments. Under these
A48 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

conditions the sediment discharge varies considerably 

with time.

The so-called transition between a "lower regime" 

and "upper regime" seems to be directly associated 

with a recirculating flume that, during each run, is 

operated at a constant depth. Because there is a natural 

tendency to operate any recirculating flume in this 

manner, all recirculating flumes tend to have this 

transition.

Because the sediment discharge for a unit width of 

channel may be approximated by the product of the 

average height of the bed form and its velocity and 

because the load of a stream is associated with its mean 

velocity, the height of the dunes and the dune velocity 

also must be associated with the mean velocity. The 

spacing of the bed forms, which is essentially unrelated 

to the load, has a marked effect on the friction factor. 

Therefore, if the friction factor remains constant in a 

recirculating flume operated at constant depth, there 

appears to be one specific but undescribable relation 

by which dune height, dune velocity, and dune spacing 

are linked with velocity and slope. One of the elements 

of this relation seems to be an increase in dune height 

with increasing discharge. Because this means an increase in the dune height-depth ratio, which normally 

would result in an increase in the friction factor, dune 

spacing and dune velocity must be modified to make 

up for the increased dune height.

When the bed forms have reached the maximum 

height possible for a given depth of flow, further increases in discharge, with a consequent increase in 

velocity and load, require that the increased load be 

taken care of by a rapid increase in. the velocity of the 

bed form relative to the mean velocity of the stream. In 

this phase of the behavior of a recirculating flume 

operated at constant depth, the constraint of constant 

friction factor ceases and the flume operates under the 

constraint of constant shear stress. The bed forms move 

faster and the spacing between them increases. Finally 

there is a reduction in the height of the bed forms so 

they appear to "wash out." The bed then becomes 

smooth. With further increase in discharge this phase 

is succeeded by another wave form antidunes and 

the value of the shear stress no longer approaches a 

constant.

In conclusion, at least two items determine the 

friction factor for a given depth of flow: the slope and 

the sediment concentration. Were the data available, 

the friction factor of a stream probably could be related 

to the slope, dune height, dune spacing, and dune 

velocity. The complexity of such a procedure to determine the friction factor of a stream leaves nothing 

to recommend it. The idea that given a characteristic 

sediment for a single depth and single slope there is

but one velocity and one load must be flatly rejected. 

Neither are there two velocities and two loads. The 

interrelations among velocity, depth, load, r.nd slope 

are mutually determined under a specific se* of constraints. There are no unique relations, even among 

velocity, unit discharge, and load. The average or 

most probable condition is the best that can ba- defined.

SUMMARY OF SLOPE AS AN ELEMENT IN STREAM BEHAVIOR

The statement that

V=$(D, S, d, c, p, g, w, ps, M) (45)

is frequently made, but it is incorrect if it implies a 

continuous unique relationship. Equally incorrect is the 

statement that given an identical cross section and 

identical sediment characteristics, a steady two-phase 

flow phenomenon is unequivocally defined if the values 

of v, p, D, ps, d, S, and g are all stated. Neither is it 

possible to substitute bed form for one of the variables 

in equation 45.

A more precise statement of the variables involved is

$[Q, Qs, S, D, V, W, C, d, w, <r, g, v, p, Ps, Bf,

Sc , Sp , Sf]=Q, (45a)

where Qs is the total sediment load of the stream, a is 

a measure of the size composition of the sediment, 

ps is the density of the sediment, Bf is a bei form or 

bed configuration, Sc is a shape factor of the chs.nnel cross 

section, SP is a shape factor for the sediment particles, 

and Sf is a sorting factor which describes that portion 

of the sediment concentration actually effective in 

determining the characteristics of the channel.

Given a specific fluid, a specific sediment mixture, 

and a specific temperature, then d, w, a, v, p, ps, Sp , 

and Sf are all constant as is the acceleration of gravity. 

If two-dimensional flow or flow in a fixed width flume 

is to be studied, W is constant. This leaves

l.,S,D,V,C,B,,SA=Q. (45b)

There are fundamental relations among these 

variables.

QIW=VD, QSIW=L, QS/Q=C, q= 

which are generally recognized. In addition tl ^ relation

is equally deterministic. If two-dimensional flow prevails or if velocity distributions are essentially symmetrical about the centerline of the channel, Pc may be 

ignored. Consequently equation 45b reduces to .three 

basic relations
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS

, L, S, B,] = Q 

, C, S, 5,] = 0 

*[q,D,S,Bf] = Q.

(45c) 

(45d) 

(45e)

For all practical purposes, equations 45d and 45e 

are alike. In each of these equations g and L are defined 

and consequently are linked. One cannot be changed 

without changing the other. In equation 45c, q and L 

are independent of each other. Consequently, equation 

45c represents a less constrained situation than do 

the 45d and e.

If the first two variables in each of the equations are 

independent, then V, D, G, g, and L or Q and Qs are all 

determined; experiment will yield consistent relations 

between V, D, G, and S because natural constraints 

prevail. The relations of equations 45d and 45e are the 

same but differ from those of equation 45c. Selection 

of different independent variables and different constraints yield different relations as has been previously 

discussed.

Nothing in these relations indicates the nature of the 

bed forms. All that is secured is a number representing 

a kind of flow behavior. It is known, of course, that this 

number can vary only within relatively narrow limits. 

Nevertheless this variation is the key to the behavior 

of streams with movable beds.

If, as it frequently is in natural streams, sediment 

discharge is an independent variable, the bed form and 

the sediment load must be congruent. This automatically results if discharge is the other independent 

variable. If sediment load is a dependent variable in a 

flume, bed form may be substituted only if there is, in 

addition, a precise statement on how the flume is to be 

operated.

If unit discharge and load are independent variables, 

the response to change will be dependent *ipon whether 

slope is free to vary or whether it is fixed. If it is free to 

vary, V and D will be dependent on load, and S will be a 

dependent variable. The constraint is the minimization 

of the shear stress and the friction factor in combination. 

If slope is not free to vary, V and D will be dependent 

on load to the extent that the bed forms can be changed 

to provide the necessary friction factor. If this is not 

possible channel instability results.

If unit discharge and slope are independent variables 

and load is a dependent variable, the response to change 

will be dependent on the size and construction of the 

flume and the method of operation. If the variation in 

concentration is resisted by the flume, change in slope 

has a small influence, and change in unit discharge will 

have a large influence. If the slope is held constant 

and the depth is permitted to vary and there is no 

resistance to change in concentration, then the constraints will be the minimization, jointly, of the variances of the shear stress and the friction factor.

If discharge and depth are independent variables 

and qS is in excess of a specific amount related to grain 

size, within the range of change in roughness due to 

bed forms, slope will be constant.

Much of what has been considered to be functional 

relations among the hydraulic variables of flow in 

alluvial channels is simply the operation of different 

constraints. Too often the constraints have been flume 

size, discharge, and method of operation. The result is 

that any supposedly functional relation involving slope 

is suspect, and this includes both velocity and sediment 

load, unless the applicable constraints are known. 

Much about the variation of flow in alluvial channels 

is unstudied and unknown. In fact, it is probably a good 

guess that a great deal of variation is wholly unobserved. Consequently, if the character of an alluvial 

channel is dependent on the minimization of various 

unknown parameters, the controls over particular or 

local conditions have not even been touched by tl is 

report.

A consistent relation between discharge and load 

develops in many natural streams. Under these conditions, it is possible to find consistent relations among 

discharge, slope, and depth (figs. 22, 23, and 41). But 

if the relation between discharge and load is not consistent, neither are the relations among discharge, slop?., 

and depth. The results of inconsistency are generally 

found in the discontinuous rating curves of Dawdy 

(1961) or of Culbertson and Dawdy (1964) and in the 

shifting controls at gaging stations on alluvial streams. 

The variation in the velocity-depth relations with 

changing sediment loads was demonstrated by Leopold 

and Maddock (1953a, b).

The great difference between a river and a flume is 

that in a river all natural actions at a single point on 

its course tend to minimize the variance of slope. In a 

flume, the variation in slope is minimized only uncSr 

certain conditions. Consequently, the effects of varying 

discharges and loads with slope constant are readily 

observed in natural streams or even canals, but rot 

in flumes.

It would appear that, at least with our present 

knowledge of changing roughness caused by changing 

bed forms, the only parameters defining velocity with 

any degree of accuracy are load and unit discharge. 

Slope becomes redundant. At the present time, we 

consider slope allied to width and to channel behavior. 

If load and discharge are independent variables (and 

they usually are) either equation 18 or equation 42 

may be used to determine slope if the coefficient is 

subject to change. For average conditions use the 

coefficients shown in equations 18 and 42. If a tendency
A50 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

toward aggradation is desired, increase the coefficient; 

decrease the coefficient for degradation. Changes in 

the coefficient, however, must conform to the observed 

ranges for the particular sediment mixture. It is doubtful 

if the average coefficients are satisfactory for flows 

with high Froude numbers and where much energy is 

dissipated by wave action. Neither are they satisfactory for three-dimensional flow, which also increases 

the rate of expenditure of energy. Under both these 

conditions, coefficients should be decreased. However, 

little information is available on such conditions.

THREE-DIMENSIONAL FLOW

The presentations made thus far have been in terms 

of unit discharges of water and sediment that is, in 

rates per foot of width. The variables, therefore, have 

been velocity, depth, slope, and load. Channels, particularly those that are self-formed, have a relation 

between discharge and width or between slope and 

width, as well as between discharge and the other 

variables that have been noted.

The introduction of the width term brings up a 

number of problems that have not been discussed heretofore, particularly the stable channel problem, which 

is generally associated with the maintenance of channel 

banks. Channel width is also associated with meandering. Meandering, as such, will not be discussed, as the 

subject is restricted to straight channels. Nevertheless, 

the three-dimensional bed forms that lead to meandering 

also are related to the stable channel problem and must 

be considered.

Width may be an independent variable, but obviously 

if a selected width is one that would be developed by a 

self-formed channel having the same discharge, slope, 

and load, a high degree of channel stability may result. 

Thus, it is important to determine the role of width as a 

dependent variable. However, given a discharge, width, 

slope, and load, the characteristics of a channel are 

fixed no freedom is left consequently, the effect of 

the relaxation of one variable on all the other variables 

must be determined.

In discussing three-dimensional flow, cross-channel or 

helicoidal flow will not be considered as such. The 

presence of such flows will be accepted as being related 

to bed forms and the way in which a stream spends its 

energy.

In this part of the report, width will be considered 

first as an independent variable and then as a dependent 

variable. The first consideration is rather simple, but 

the second is complex in that the effects of sediment 

concentration and sediment size are important. A basis 

for the development of the Lacey (1946) equations will 

be given for streams transporting sediment. Another 

"regime" equation will be developed for streams with

beds of infrequently moved coarse material. Finally, 

the stable channel problem will be discussed with 

particular attention given to field observatiors.

WIDTH AS AN INDEPENDENT VARIABLE

In any stream a close relation exists between width 

and slope. This relation, quite ubiquitous, can be established by the procedure of minimizing the variance of 

the shear stress and the friction factor under the 

constraint of uniform sediment concentration.

The relation among velocity, discharge, and load in 

the mid-velocity range has been expressed in two 

equations: one for a sediment of uniform size and 

another for sediment mixtures. For a constant sediment 

concentration, the relations between velocity and depth 

become VocD5 /11 for the uniform sediment size and 

V<x.D1/2 for the sediment mixture.

If width is an independent variable, velocity, depth, 

and slope can be written in terms of width so that 

V oc Wm, DocWf,SocWz ; then, if the stream discharge 

is constant, VWD=Q= constant, andm+/+J=0. For 

sediment of uniform size m= 5/1 I/. Then, /= 11/16, 

m= 5/16, the variance of the shear stress equals

( z -^ ) , and the variance of the friction factor equals

ll.lO 

IG+ 

Minimizing the combined variances

Setting the derivative equal to zero

2=3/8,

and

(47)

whence,

If the operation is repeated for a sediment mixture, 

w = l/2/,/=-2/3, and m=-l/3,

SacW1 '3 . (48)

The same results can be achieved by equating 

equation 19 with equations 5 and 14. For unigranular 

material the result is

(49)

and for sediment mixtures
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A51

or

in which A and Al are the coefficients of equations 5 

and 14, respectively, and B is the coefficient of equation 

19. C is the concentration of sediment in parts per 

million. There is little variation in the values of A and 

Ai as long as the flow is two dimensional. A change is 

to be expected, however, under three-dimensional 

flow that is, when the thread of the main current 

swings back and forth across the stream. The coefficient 

B represents an index of bed deformation and, therefore, 

its variation has a maximum of about three times at 

high discharges and wide channels but has a very small 

variation at small discharges and narrow channels.

Equations 49 and 50 should be compared with 

equation 45. A function of V and D can be substituted 

for C, and Q may be divided by W, yielding VD. Then

=$ , V, D, Ap, P, g, d, (51)

The difference between equation 51 and equation 45 

is the inclusion of Ai/B and the elimination of the viscosity and size variation, a. The resolution of the effect 

of sediment mixtures requires two equations. The 

influence of the viscosity and the shape of the sediment 

particle appears in the fall velocity, w. As neither A, 

AI, or B vary greatly in very small flumes with small 

discharges, the premises of equation 45 may be satisfactory under such conditions. But in natural streams 

and in larger flumes, these coefficients are indispensable 

and must be included, as shown in equation 51.

If both sides of equation 50 are multiplied by Q, 

the result is

where Qs is the total sediment discharge of the stream. 

This should be compared with Rubey's (1952, p. 132) 

equation

/i,,2/3 Ji/a

(53)

in which K^ is a constant and XA is an optimum form 

ratio D/W. Rubey was trying to express what would 

now be considered a "regime" relation, but the correspondence between equations 52 and 53 is evident. 

The relation proposed by Lane (1955), QS oc Q4, 

must be considered as nothing more than a general 

statement of behavior that has been recognized for 

some time. That is, streams moving coarse material 

or high sediment loads flow on the steepest slopes.

The appearance of the term Ayd/pw2 in equations 

49 and 50 complicates the effect of size of materiel; 

for coarse material this term is a constant. The terms 

A7(#/ptt>2 (^4/3) or (Ayd/fM)2) (d2) have a small variation 

in the sand size, but each becomes larger for material 

both larger and smaller than this size. There is a lower 

limit to the size to which these relations are appUcable, because very small particles become coherert.

Two examples of field behavior will be given. Tv?o 

sections of the rectified channel of the Rio Grande 

near Vinton, Tex., are a short distance apart. Both 

sections are tangents of more than 1,000 feet long, but 

one is 100 feet wide and the other 200 feet wide, and each 

carries the same discharge of water and sediment. 

Both are very stable and have no maintenance problems. The relation between width and slope is shown in 

figure 42 and confirms equations 47 and 48.

A second example is taken from Rubey (1952, p. 

126). Rubey stated that the Illinois River in flowing 

around islands in its channel increases its total width at 

bank full stage by about 16 percent. The material on

400 r

300

LJ 

LL.

- 200

I

Q

100

5 678 

SLOPE, IN FEET X 104

10

FIGURE 42. Relation between width and slope, the R: o 

Grande near Vinton, Tex. Solid line, equation 47; dashed 

line equation 48. (T. Maddock, Jr., unpub. data.)
A52 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

the bed is a fine uniform sand. As discharge and sediment concentration are the same, equation 47 is applied, 

VorW-w, Z>ocTF-11/16, and SacW3/*. Then the changes 

expected would be a decrease of 4.7 percent in V, a 

decrease of 11 percent in D, and an increase of 6.1 percent in slope. Rubey stated that the depth decreased 

about 10 percent, that the mean velocity decreased 

about 4 percent, and that the slope was difficult to 

determine but increased between 5 and 10 percent.

The relations used in the Rio Grande and Illinois 

River examples when combined yield gDS/V2<x.W5/H, 

when concentration and discharge are constant. Blench 

(1966) stated that V2 /gDS=3.63 (VW/v) 1 '*. Insofar as 

width is concerned, this is precisely opposite the relation found in the two examples. We conclude that the 

Blench equation is of very limited application and 

should be used only in a specialized situation.

Slope, width, and the coefficients A and B from either 

equation 49 or 50 can remain unchanged if a specific 

relation exists between G and Q. For equation 49 this is 

<?oc<23/* and for equation 50, CccQ1 '2 . In terms of sediment discharge this is Qs<x.Q7/* for equation 49 and 

QsacQ?/2 for equation 50. Such relations are commonly 

observed in straight-walled channels moving heavy 

sediment loads.

Two other relations may be developed using the lowvelocity range equations for large and small sized sediments in combination with equation 19. The first is for 

when the bed material is coarse, Ayd/pM2 is constant, 

and the sediment moves at just about bank full stage. 

The equation is

.1/4

(54)

or

At is the coefficient of equation 7 when concentration is 

zero. When the size of material is small and the sediment concentration is greater than about7.5 (pw2f&yd')s/2 , 

the combination of the two equations is essentially

-WY Y C

/9

or

Az is the coefficient of equation 7 under the specified 

sediment concentration.

If equations 50, 54, and 55 are compared, it will be 

seen that for a given unit discharge and a given value 

of A/B, slope is greatly affected by size at the beginning 

of motion, but this effect decreases with increasing

sediment concentration. In the mid-velocity range 

slope is more responsive to sediment concentration than 

to sediment size.

The size parameter, once sediment begins to move, is 

^1/4^3/8 Ag snown jn figure 25, this parameter has a 

maximum value at about 0.23 mm and increases with 

both increasing and decreasing sediment size. For the 

same sediment concentration and the same slope, a 

channel moving 0.25 mm sediment would have a width, 

according to equation 55, about 45 percent greater than 

a channel moving 0.05 mm sediment.

Inglis (1949) reported, in accord with such a relation, that in the Sind of Pakistan canrls move 

sediment much smaller in size and in concentration 

and, as a consequence, the canals are as much as 50 

percent narrower and are also flatter than those in 

the Punjab. As would be expected from equation 50, 

canals in the Imperial Valley of California moving 

heavy sediment loads are narrower than the canals in 

Pakistan and, as also expected, flow at higher velocities .

WIDTH AS A DEPENDENT VARIABLE

If, in a natural stream or canal, the water and sediment discharge were constant, there would be no 

alluvial channel problem. From equation 50, given a 

discharge and an unchanging size composition of

sediment, 2 2 /3 (56)

The constraints are (1) a limited range in the coefficient B, (2) an even more limited range in the 

coefficient AI, (3) a maximum velocity limited by the 

size of the sediment, (4) a minimum velocity representing the lower limit of the mid-velocity rr.nge, and 

(5) an undefined constraint that when a stream becomes so wide and steep it wil] either break up into a 

series of channels or will meander. The last two conditions result when sediment is no longer moved over 

the bed with some degree of uniformity. Therefore, 

the relation between S and W is limited, and, more 

importantly, AI and B are not completely independent 

of some combinations of W and S. Width is not a completely independent variable but has to b°.ar some 

relation to discharge.

The possible combinations of equation 54 yield an 

equal number of differing characteristic flow patterns. 

Those patterns that are stable tend to persist; those 

that are unstable tend to disappear. Therefore, the 

geometry of a stream system represents a sort of 

"survival of the fittest," which appears to be a constraint on equation 56.

The effect of changing width is to change the way a 

stream consumes its energy. For example, from equa-
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A53

tion 50, if discharge and sediment concentration are 

constant

QSacW1 *, 

and

Thus, there is a considerable difference in the relative 

rates of expenditure of energy per unit of length and 

per unit of bed area with a simple change in width. 

This redistribution in the method of energy dissipation 

is brought about by changes in bed forms.

The hypothesis is made that those persistent patterns 

that are observed in natural streams, which may be 

called a "regime, "are the result of similar behavior with 

respect to rate of energy dissipation. Therefore, a 

"regime" is defined as a set of relations among width, 

depth, velocity, and slope that exists when the variance 

of different rates of internal energy expenditure jointly 

are minimized under a constant or specified sediment 

concentration.

If width, depth, velocity, and slope are expressed in 

terms of the total discharge instead of the unit discharge, WacQ", VacQ , DacQf, and SacQ*. If sediment 

concentration is constant, from equation 14, m=%f. 

As before, jointly minimizing the variance of the 

shear stress and the friction factor,

 »mnmum, 

/2+2/2+22+22 ^minimum.

Differentiating with respect to z and setting equal to 

zero,

2/+ 22+22=0,

Langbein (1965) listed five aspects of rates of energy 

expenditure or stream power. Three of these can be 

expressed in terms of width, discharge, and slope. 

They are (1) power per unit length per unit discharge 

or S, (2) power per unit of area of channel bed or 

qS, and (3) power per unit length of channel or QS. 

The variances of the three different stream powers 

are, respectively, z2 , (m+/+s) 2 , and (1+2) 2 . These variances are now written in terms of/ and are jointly minimized. Thus,

Expanding and setting the derivative equal to zero

From the other relations m=%, b=Y2 , and 2= 

Thus, there is a set of regime equations, in which

SacQr

l/2

1'6

These are the well-known Lacey relations.

As it was stated that a "regime" equation would 

exist for a specified as well as a constant sedimert 

concentration, a set of "regime" equations may bo 

written if concentration is some function of stream 

discharge, such as CacQ*. Then, Voc(QO/W} 1/3 from 

equation 14 and CacQ*; then,

i+x 1/3

Substituting these values into equation 19, the result

s

Now the combined variances of the three kinds of 

stream power are minimized, writing S, V, and D in 

terms of the variables Q and the unknown W. The?i,

WocQ1/2 .

Thus, what can be called the generalized Lacey equations can be written when concentration varies with 

discharge to some power. 

These are

WocQ1 '2,

(6P)

Note that the WocQ1/2 relation is independent of the 

change in load with discharge. The only requirements 

are that V2 ocDC and VocDS.

The Lacey (1946) equations were developed from 

equation 14 for a mixture of sediment sizes. A comparable set of equations may be secured for a unifora 

grained material by using equation 5. These are

WocQ°A5,

FocQ0 - 17 ,

DocQ0 - 38 ,

(59)

and a set of generalized equations also could be developed. However, the Lacey equations have been in 

existence for a long time, and a considerable amount 

of information regarding their use is available. Conse-
A54 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

quently, it does not seem advisable to propose another 

set of equations that give essentially the same results 

unless the pertinent sediment sizes are outside the 

limit of applicability of equation 14. Again, it is desirable to review the limitations of the Lacey equations 

in accordance with the manner of their derivation.

Slope is a dependent variable. For every change in 

discharge, slope is expected to change. This is important 

to geomorphologists and others who are concerned 

about misfit or relic streams flowing in valleys obviously 

formed by a larger stream in some previous geologic 

time. If slope is constant and if stream discharge is to 

be estimated from some value of width, equation 49 

must be used. Thus, Q is proportional to W and not 

Q proportional to W2 as stated in the Lacey relations or 

those noted for natural streams by Leopold and Maddock (1953a).

The velocity-load relation used in the derivation 

applies to sediment mixtures in the mid-velocity range 

only. Although a "regime" equation could be derived 

for the low-velocity range, this is usually not of importance except for zero sediment transport. Nevertheless, the use of equation 47 or others of the same form 

may yield velocities too high for very large canals or 

velocities too low for small canals or small concentrations of sediment. Whether or not velocity is too high 

will be dependent upon the magnitude of the largest 

size found in appreciable quantities in the sediment 

mixture.

As long as the sediment is in the sand range and the 

flow is in the mid-velocity range, the influence of size 

of sediment is minimal. When sizes in the range of 

Aryd/pw2 are constant or sizes less than about 0.10 mm 

are involved, size cannot be eliminated from consideration, and equation 5 should be chosen in place of the 

Lacey equations.

If equations 57 or 58 are written in the form W=aQl/2 , 

D=cQ1/3 ,V=kQ1/6, and S=tQ~ l/G, then certain relations 

may be established. Because V2/D=A13C, then k2/c 

=A13C, oTk=Al3/2c l/2Cl/2 .Vutack=l, soc-l/a2/3 (71/3^1 . 

There is also a built-in relation between t and a. From 

equation 50,

*Q- 1/6H ^

2/3

io6J LP 2 J

9 d

1/2 raQi1/2/3

which for coarse sizes becomes

These relations will be discussed in more detail in the 

section on stable channels.

The Lacey equations apply only to streams in which 

the bed material is in general motion. The hydraulic

roughness is controlled by the bed forms. The sediment 

load is an independent variable.

SEDIMENT SIZE AS DEPENDENT OE INDEPENDENT VAH TABLE

In contrast to the Lacey regime channels, other 

streams flow over beds of material so coarse that there 

is general movement only at high discharges, usually 

about bank full stage. Thus, the size of the bed material 

controls the stream characteristics and another kind of 

"regime" exists.

Under such conditions, equation 54 applies, and

\7/9**

if docQ",

S=Q7/6n ~ 7/gW7/g .

Then, if FocQw, D<xQf, TFocQ6, and SocQ*, the following 

relation exists:

m+/+6=l, 2=7/6 - 

From equation 7,

n n

Now, as was done previously, the variances of three 

stream powers z2 , (m+f+z*) 2 , and (l+z) 2 are written in 

terms of b and jointly minimized to find the values of 

b, m, /, and z. The result is

^170^0-382-1.008^

yocQo.oeg+o W

£)K QO .54 9+0.563 « 

).48+0.383n (60)

If size decreases downstream so that docOr1/10 , the 

results are TFccQ0 -483 , FocQ0 - 024, AxQ0 -493 , and S<xQ-°-5ls, 

relations that approximate those frequently observed 

in the headwaters of some clear flowing streams. 

However, equation 60 is used if the size of the material 

in the stream bed is not a function of stream discharge, 

as shown by Hack (1957). If size is constant,

2 , FocQ0 - 069 , DocQ° Mg, and S<xQr° A8 .

(61)

One of the criteria for the development of the 

"regime" relations of equation 60 was that the variance 

of the power, QS, tends to be minimized that is, QS 

approaches a constant and Qo^S'1 . Then if W/Q°-382 is a 

constant and QS approaches a constant, the relation 

SOM8 ocd1 -5 is approached, providing, of course, that B/A 

is a constant. This relation may be expressed as Socd2 -26 .

If equation 54 is rearranged,

118/7 ^- QSg/7d- 3 '2 .
BEHAVIOR OP STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A55

Wolman (1955), in his study of Brandywine Creek, 

Pa., provided information that permits a test of the 

validity of such an analysis. Figure 43 shows the 

relation between width and discharge, slope and 

discharge, and slope and size of bed material. The 

agreement between the "regime" relation and that 

proposed by Wolman is clear. It must be remembered, 

however, that these relations are not unique, but they 

are statistically the most probable.

If the coefficients of the three relations are observed 

in the field, the whole system of "regime" equations 

may be developed with the use of equation 54. In the 

Brandywine Creek basin these relations are

S=Q.QQd2 -25, 

F=10.3#-°-079, 

#=0.021 #°-696.

(62)

The relations of velocity and depth to discharge 

from this set of equations are compared with Wolman's 

in figure 44. Velocity and discharge have the greatest 

variance because they represent conditions at a specific 

cross section, and the other relations are more applicable to reaches of a stream.

200

100

,0.005

0.005

0.001

0.001

0.0005

0.0005 -

0.0001

0.5 1.0 5 

BED MATERIAL, MEDIAN DIAMETER 

IN INCHES

1000 10,000 

DISCHARGE, IN CUBIC FEET PER SECOND (BANKFULL STAGE)

FIGURE 43. Variations of width and slope of the water surface with discharge and relation of slope of water surface to size of bed 

material, Brandywine Creek, Pa. Dashed lines, Wolman's (1955) relation; solid lines, "regime" relation.
A56 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10

Li- 5

z

1000

DISCHARGE, IN CUBIC FEET PER SECOND 

(BANKFULL STAGE)

10,000

FIGURE 44. Variations of mean depth and mean velocity with 

discharge in the downstream direction at the bank full stage, 

Brandywine Creek, Pa. Dashed lines, Wolman's (1955) 

relation; solid lines, "regime" relation.

Ill this basin, discharge is a function of climate and 

drainage area and is, thus, truly independent. Slope to 

some extent is the result of stream adjustment; however, 

because slope is constrained by relief it too is, at least 

in part, an independent variable. Size of material is 

related to weathering and rock type and, hence, is 

related to climate and to geology. Nevertheless, it too 

is constrained by relief. Furthermore, the equations 

were developed with the premise that a specific relation 

between S and d existed such that QS was constant. 

This is not a usual situation.

The set of equations is, therefore, of limited application. Note that a stream having 1 cfs flow would have 

a slope of 2.6, be 4.6 feet wide and 0.021 foot deep, have 

a velocity of 10 fps, and move material with an average 

gain size of 4.5 feet, obviously an impossibility.

SUMMARY OF WIDTH AS A DEPENDENT VARIABLE

A discussion of widths of alluvial streams inevitably 

becomes a discussion of "regime" channels. The word 

"regime" has different connotations to different workers 

in the fields of channel behavior and sediment transport. 

In this paper, "regime" is defined as a set of relations 

among width, depth, velocity, and slope that exists 

when the variance of different rates of internal energy 

expenditure jointly are minimized under a constant or 

specified sediment concentration. "Regime" channels 

expend their energy in similar ways.

Because sediment load is an inherent element in 

channel behavior and because no single function can be 

used effectively to express the relation of the sediment 

load to other variables, no one set of "regime" equations 

is to be expected. Furthermore, even if a set of equations 

can be developed they are only the most probable 

relations and do not express a unique relation..

"Regime" channels must be free to make adjustments. 

The dependent variables width, depth, velocity, and 

slope must respond to changes in the independent 

variables discharge, sediment concentration, and size 

of sediment in motion. Thus, all "regime" channels 

must have some movement, even if it is very small, of 

nearly all the sizes of sediment making up the- material 

on the bed of the stream.

All dependent variables do not respond to changes in 

the independent variables with the same rapidity; 

hence, a "regime" channel usually reflects a long-term 

average condition. Furthermore, some variables, particularly slope but also sediment concentration and size 

of bed material, may not be wholly dependent or 

independent. Thus, in a way, "regime" conditions may 

actually be somewhat ephemeral. At best, they represent 

a mean condition about which all observed values of 

the hydraulic parameters tend to deviate. Tlur. there is 

a distinction in this paper between "regime" channels 

and "stable" channels. This will be discussed in the 

next section because stability must be considered under 

conditions where constraints are such that dependency 

or nondependency of variables changes, either in place 

or in time.

STABLE CHANNELS

A stable channel, strictly speaking, would be one in 

which the bed and banks do not change. Such a channel 

requires constant discharges of water and sediment. 

Consequently, it would have to be an artificirl channel 

because such uniform conditions do not exist in nature. 

Most channels found in nature, and this includes canals 

that are called stable, are those whose banks are essentially fixed or change very slowly but whose beds may 

rise and fall with variable discharges and loads, about 

some unchanging equilibrium position. Such a channel 

is called a "regime" channel by Blench (1952). The set 

of relations that exists among a group of such channels, 

if any, is called a "regime" in this paper. Thus, stable 

channels may, in the broadest sense, be "regime" 

channels, but all "regime" channels may not be stable 

because their banks may not be fixed.

There are two basic types of stable channels. The 

first is one that transports no sediment; the bed material 

does not move during most discharges. The second is 

one in which both water and sediment are in motion, 

but the bed forms are such that no persistent thread of
BEHAVIOR OF STRAIGHT OPEN CHAN3STELS WITH MOVABLE BEDS A57

flow is directed against the banks. If sediment load and 

water discharge vary, channel scour and fill must be 

expected. How this can be accommodated with bank 

stability is the essence of the stable channel problem.

GRAVE! CHANNELS

An examination of the problems of stable channels 

may begin with a study of the conditions of no sediment 

loads. Most channels moving no sediment are armored 

that is, the material on the bed is an accumulation of 

coarse material left by stream sorting. If this is the 

situation, the slope is usually fixed. However, if the 

stream has a very small sediment load, the material on 

the bed may actually be in transit in some flows. Consequently, the slope may have been self-formed. Thus, 

two equations for slope are available, equations 19 and 

35. These two equations can be combined with equation 

7 to yield, respectively, for zero concentration

7L>£=0.066(7#) I/8 (A7d) 7/8

/4

These equations may also be expressed as

and

(63)

(64)

(63a)

(64a)

The much greater coefficient for equation 63 than 

for equation 64 would indicate that streams forming 

their own slopes are hydraulically rougher than those

(7£>S ),IN POUNDS FORCE TRACTIVE 

FOOT SQUARE PER _ 

P D 

^ co 

~

Bra 

100 

40C

4-ft depth-x; 

-^^1-ft depth

ndywine Creek,

0 cfs bankfull 

X'

-^"l

^1_ :

X' ^

:

-

0.01 0.1 1.0 10 100 

MEAN DIAMETER OF BED MATERIAL, IN MILLIMETERS

FIGURE 45. Relation between tractive force and sediment size 

with slope a dependent variable. Lines are plots of equation 63 

at depths indicated. Data for Brandy wine Creek, Pa., from 

Wolman (1955).

where slope is an independent variable. Graphs of 

equation 63 for depths of 1 and 4 feet are shown in figure 

45. Also plotted are values from Wolman's (1955) data 

for Brandywine Creek, Pa. This stream, presumably 

self-formed, has a much lower value of yDS at bankfull 

stage than indicated in the figure. However, the same 

graphs of equation 63 are shown in figure 46 taken 

from Lane (1955, p. 1253). The graphs of equation 63 

fall within the broad limits of the recommendations 

of different authorities.

Graphs for 1 and 4 feet depths for equation 64 ar? 

shown in figure 47. Also plotted in the figure are data 

from the Bureau of Reclamation studies by Lane and 

Carlson (1953) on canals in the San Luis Valley, Colo. 

Other data are from Simons (1957). Hata for canah 

moving low sediment loads of finer material should bs 

compared with the graph showing relations between 

shear stress and mean grain size of sediment for a 

sediment concentration of 50 ppm (parts per million).

Data presented herein show a wide variation in 

values of 7!)$ for a given size of material. There are a 

number of reasons for this variation. Both a characteristic size of sediment and its fall velocity must b^ 

determined; the average or median grain size may lo«. 

found in different ways. Wolman (1955) tried to measure 

the distribution of grain sizes actually exposed on the 

bed. The Lane and Carlson (1954) sizes are those for 

bed material, but how samples were taken is net 

stated. Lane (1955) sampled the composition of the 

materials through which the canals were constructed 

and designated the characteristic size as one of which 

25 percent by weight of the materials was larger 

The influence of the shape factor and the imbrication 

of the coarse material on the bed of the stream makes 

the settling velocity most difficult to determine. 

Finally, bed material may not move at bankfull stages, 

although some evidence indicates that it does.

The differences caused by measurement techniques 

are not enough to account for differences in the coefFcients of equations 63 and 64. Variation in the value of 

the coefficients of equations 19, 35, and 7 must be the 

reason. Because it appears that the coefficient of 

equation 7 does not vary greatly, most of the variation 

must be found in the coefficients of equations 19 and 35.

As was previously stated, the difference in the coefficients of equations 63 and 64 represents a change in 

the friction factor rather than a measure of shear at 

the beginning of motion because the movement of the 

sediment is related to the velocity. Note that equations 

63a and 64a tend to approach each other as the depth 

becomes very great. This leads to the conclusion thr.t 

streams with very few, if any, definable bed forms w;ll 

have low coefficients for equations 63 and 64. Streams 

having observable bed forms will have higher friction
A58 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

10,000

8000

6000

4000

2000

1000

800

600

400

200

100

80

60

40

Tractive force (Ib per ft2) = y2 diameter (in.) or

Tractive force (kg per m2)= diameter (cm)

Eq 63, 4-ft depth 

Canals, with 2.5 percent colloids in water, Russia

Eq 63, 1-ft depth

30

Recommended for canals with high content^ 

of fine sediments in water

Recommended for canals with low content, 

of fine sediment in water

Recommended by S. Fortier and F. C. Scobey for canal; 

in fine sand with water containing colloids

Recommended for canals with clear water

Fcecommended by A. Schoklitsch 

for canals in sand

II

^

Canals, with 0.1 percent 

colloids in water, Russia

Recommended for canals in coarse 

noncohesive. Material size 25 

percent larger

Values of the critical tractive 

force determined by L. Straub

^Canals with clear water, Russia

Recommended by S. Fortier and F. C. Scobey 

for canals in fine sand with clear water

0.1

1

0.8 

0.6

0.4

0.2

0 1

008

0.06

004

O

0.02 t 

a: 

O

0.01

0.008

0.007

0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 

MEAN DIAMETER, IN MILLIMETERS

20 40 60 80 100

FIGURE 46. Recommended limiting tractive forces (or shear stresses) for canals. Modified from Lane (1955, p. 1252).

factors and, of course, larger coefficients. Because bed 

forms require resistant banks, channels whose bank 

materials are low in clay, yet are made up of sizes considerably smaller than those making up the bed, will 

have low values of the coefficients of the two equations.

In view of the paucity of data upon which any conclusions could be based, I suggest that the Wolman data 

for Brandywine Creek be used for minimum values for 

the coefficient of equation 64a. Brandywine Creek 

appears to represent a stream with weak banks, even 

though the banks seem to be protected by trees. A value 

of about 0.021 is suggested. Lane's (1955) data from the 

San Luis Valley might be used to provide for the situation of coarse material in both bed and banks. Equation 

63 and its coefficient would be used only on relatively 

deep and narrow streams, most of which would be 

expected to meander. If immobility of banks is desirable, 

equation 64 would seem to represent a good average 

condition.

This is the two-dimensional approach. Given a size 

of material, velocity is computed. Then for every value

of depth there is a corresponding value of slope. If the 

selection of a slope is constrained by topographic or 

other conditions, depth is computed, fixing the unit 

discharge. The required total discharge then establishes 

the width. The relations between width and slope, for a 

given stream discharge and size of bed mater HI, will be 

dependent on the selection of the coefficient.

Equation 54 can be rewritten in terms of width, 

stream discharge, slope, and composition of bed material as

*-(£} P1/2T

(65)

If a "regime" condition exists, then WocQ0 -382 and

W =(B\.18/7 0.618 9/7 ,,1/2

Q *

PT ,9/28

Testing the Lane and Carlson (1954) date, from the 

San Luis Valley, we find that there is a "regime" 

relation, W=3.8Q°-zs2, and that it is not a function of
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A5S

1.0

Brandywine Creek, Pa., 

4000 cfs bankfull 

1000 cfs bankfull

Equation 64^ 

4-ft depth J/

X

sample for size analysis

£0.01 

O

No bed forms50 ppm concentration 

"---..l-ft depth.

EXPLANATION 

 Simons (1957) data

X Lane and Carlson (1953) data 

O Wolman (1955) data

0.001

0.01 0.1 1.0 10 

MEAN DIAMETER OF BED MATERIAL, IN MILLIMETERS

100 200

FIGURE 47. Relation between tractive force and sediment size with slope an independent variable. Many samples taken to determine 

size for all values except those indicated. Dashed line, plot of equations 7 and 35 using a concentration of 50 ppm.

size of material (fig. 48). The right-hand side of equation 

65 must also equal a constant, 3.8, and QOA8Socd7/6 

(A2/B) 2 . As shown in figure 49, there must be a variation in the coefficients A2 and B because there appear 

to be two relations, d=l.8QOAlS6 '7 and d=3.lQ°-*lS6 '7 . 

These two equations are matched by two others, V= 

7.45 (Q/Wy'9dl/3 and F=6.58 (Q/WY'W*. Thus, there 

are two sets of "regime" equations in the San Luis 

Valley canals data:

W=3.8Q°- 3S2,

F=7.45 (

1/9

or =6

and

(66)

F=6.58

1/g

or =

(67)

These differences probably were caused by different 

design criteria. Note however, that the canals at the 

upper ends of the system, and therefore the largest, are 

relatively steep. This is a desirable situation and should 

be a general practice.

Except for the width-discharge relation, equations 62 

and 66 are each special cases of "regime" systems. The 

coefficients of the equations are not constant, but thil 

result is to be expected. Because the mean velocity of 

the stream is related to a size of sediment grain, the 

shape of the channel may have an influence in addition 

to the various factors heretofore discussed. Finally the 

relation between velocity and grain size and unit 

discharge was determined in relatively small flumes witl 

fairly small grain sizes. Consequently, the extrapolation 

of the velocity equation to large cobble beds demands 

a minimum of depth of flow to give good results.

To some extent the agreement between the Lane and 

Carlson data and the Wolman data is fortuitous because 

different methods were used for determining the size 

of bed material. However, if stable channels can be
AGO BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

observed in the field any consistent method of determining sizes may be used it even may be size of bed 

material or, as Lane and Carlson (1954) suggested, the 

selection of some coarse fraction of the native material 

through which a canal is built. The velocity equation 

may be developed from local data to fit any specific 

condition. Nevertheless, for the present and in the 

absence of other data it is suggested that for the 

velocity-discharge-sediment size relation and for the 

slope-discharge-sediment size relation equation 66 is 

satisfactory for canals designed for flows of more than 

500 cfs and equation 67 for canals designed for discharges of less than 500 cfs.

The value of the coefficient of the discharge-width 

relation appears to be uncertain. For artificial channels 

a value of 3.8 seems to be satisfactory because it 

applied to the San Luis Valley canals. A value of 

from 4.6 to 5 seems best for natural channels subject to 

over-bank flow. There appears to be considerable 

margin for error width does not have to be determined 

closely, and channels appear to readily adjust to local 

conditions.

The criteria for gravel channels are somewhat

100

10

10 100 1000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 48. Relation between width and discharge, San Luis 

Valley, Colo., canals: X's. data from Lane and Carlson 

(1954).

0.01 0 °-48S

0.10

FIGURE 49. Relations between sediment size and discharge 

and slope, San Luis Valley, Colo., canals; X's, data from 

Lane and Carlson (1954).

specialized. Such channels are relatively straight and 

uniform, with practically no bed forms. These specialized 

criteria should not be used for channels obviously of 

a pool and riffle configuration or where there is obviously a considerable amount of sediment in transport 

at the bank full stage. The assumption made in the 

beginning of the analysis was that although movement 

at bank full stage is required it is small in amount. 

For discharges below bank full stage, the^e channels 

operate as rigid-boundary channels under conventional 

hydraulics.

Although channels whose characteristics are controlled by grain size are frequently encountered in 

humid regions and are of course associated with low 

sediment loads, they are not often found in arid 

regions. In the arid regions high sediment loads seem 

to determine the aspect of the stream system.

SAND CHANNELS

The stabilization of channels moving sediment 

presents problems differing from those of channels 

whose beds are made of a coarse materal that is 

immobile during most discharges. Channels in which 

the discharges of both water and sediment vary must 

aggrade or degrade at some time. In a stal T e channel 

this must not be accompanied by any patterns of 

meandering flow, which would lead to bank erosion.

If discharge is kept constant in an artificial channel, 

such as a canal, there will be a tendency for the bed to 

rise during periods of high sediment inflow and to fall 

in periods of low sediment movement. If the change 

between high and low sediment load takes place 

quickly, the bed may not degrade rapidly, and the 

discharge with a low sediment load will have a higher 

gage height than did the same water discharge during 

the period of heavy sediment movement. Channel 

stability, therefore, requires that removal equal deposition. Were it not for the fact that removal proceeds 

at a slower rate than deposition it would H assumed 

that a balance would be maintained ratter simply. 

In practice the balance must be weighted toward 

degradation unless periodic canal cleaning is to be 

adopted as a continuing practice.

Examples of this type of flow are showr in photographs of the Colorado River downstream from Davis 

Dam near the Nevada-California boundary and of the 

Rio Grande near Vinton, Tex. (figs. 50 and 51). Both 

channels are lightly riprapped with rock. Without it 

bank erosion would take place, and the stream would 

eventually meander.

The bars that create meandering of flow are formed 

at high discharges with low sediment loads. The bars 

are systematic, alternating from one side of the channel 

to the other, and have a wave length of "ZirW. When
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A61

FIGURE 50. Aerial photograph of the Colorado River near the Nevada-California boundary showing systematic bed forms (light

areas) under water.

FIGURE 51. Strip aerial photograph of the Rio Grande near Vinton, Tex., at low flow showing river reworking systematic bed forms

(light areas) that are relics from higher discharges. Flow from right to left.

the discharge declines quickly, the bars are left as 

relics of the previous discharge and appear above the 

water surface. Low flows simply erode the bars and 

redistribute the sediment locally. There is little general 

sediment movement in a downstream direction at low 

flow.

"REGIME" CHANNELS

The designer of a canal hopes to occupy that part of 

the spectrum of canal regimes that permits the removal 

of material deposited under high sediment loads yet 

does not result in a canal width and slope associated 

with bank erosion during periods of flow with low 

sediment concentrations.

For areas of similar climatic, topographic, and 

geologic conditions equations relating to networks of 

channels may be developed. These are "regime" equations, such as the Lacey (1946) equations, with the 

inherent assumption that width, depth, velocity, and 

slope of channels all will change with changing discharge or changing sediment concentration. Consequently, this is a basic system that provides a mean 

condition. Obviously, should local conditions depart 

from this mean there must be some local variation from 

the basic pattern.

If the "regime" system approach is to be used, the 

relations among the various coefficients must be fully 

understood.

If the two equations V=A(L/dl/s and V=B(qS) 1/2 

(jg/Ajd)l/*(pid2/^rYd) l/s are combined to eliminate V, 

the result, as has been noted before, is

>3/4

If, from the "regime" relations, W=aQ1 '2 and S=tQ-1/G, 

then

(68)

in which C is a system concentration and d and w are 

related to sediment sizes in the system.

If all the coefficients in equation 68 were independent 

and uncorrelated, enormous variation in the coefficient 

a would be expected. Actually, little variation is observed so that a high degree of correlation must exist. 

However, the objective of a regime approach, from the 

standpoint of analysis made herein, is to reduce the 

variation in the values of the coefficients A and B. 

Because the coefficient A does not vary greatly under 

normal conditions, the important fact is that B be little 

changed because it represents a specific kind of energy 

consumption in two-dimensional flow. The end result 

of the "regime" analysis would be expected to be

aoct3 w3/z

(69)
A62 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

The Lacey relations appear to be a variant of equation 69. The Lacey slope equation, when D is substituted for E, is y=16#2/3S1/3. This may be expressed 

as F5/3=16 q Sl/3 or Fccg2/5S1/5. Lacey also states that 

Foe (fD) 1/2 orF^oc (Jq) 1/2 and Foe (/g) 1/3. Then/^ocg1 /15^1 /5 

and/5 oc(#/TCOS3 . If WocaQl/2J5KQl/2S3la. BuUoc()1/6S; 

hence, .foctf/a or, if o is constant, tac[A/B]2C2/3dmfu?/2. 

Therefore

But/must be proportional to C; hence, <?3/5ocC<4/jB) 6/5 

(d27/20/w9/1°) or, in the Lacey equation, Boc(A/C1/2)(d9/s/ 

w374). Because the term dg/B/u?/4: is essentially constant 

for sand sizes in the Punjab, as given by Inglis (1949), 

and because A must be a constant, jBocl/C172. This 

value should be compared with the one developed in 

the section on slope. Both values indicate, however, 

that Lacey channels may be unduly flat if designed 

for small sizes of bed material. The existence of a 

correlation between size of bed material and sediment 

concentration must be assumed. This may be true for 

canal systems and the higher velocities associated with 

greater concentrations may result in coarser material 

being found on the bed. However, within a single canal 

system, which tends to have a uniform concentration 

and a uniform value of V2/D, there is a small increase 

in size of bed material associated with the increased 

velocity of higher discharges.

Although Lacey (1940) considered the coefficient of 

the width-discharge relation to be essentially constant, 

Blench (1952) expressed width in terms of a side factor, 

Fs, in which FS=V3/W. Now if W=a@/2, it follows 

that W<x.a(QO) 1/2 because the sediment must move 

with the water in a "regime" system. Then W1/2cca(QC/ 

W) 112 . But Q/W=q, and from the velocity-load relation 

for mixtures, (qCy /2<xV3/a. Hence, Wl/2ocaV3/2, ory3/T^oc 

l/o2, or Fsccl/a2 . Thus, Blench's side factor, Fn is inversely related to the square of the coefficient of the 

width-discharge relation. The three fold range in variation in the value of Fs noted by Blench would provide 

a variation of about 70 percent in the value of the 

coefficient a. Blench stated that V2lgDS=3.63 (VWfp) 1 '* 

or, substituting V3/FS for W, F2/#Z>S=3.63 (VV*/Fp)lf*, 

and F=3.63 gDSI(FJ) 1 ". But from equation 19, 

F=£2Z>S(75f/A7d) 1/W/A7d) 1/4. Hence, B of equation 

68 is related to Fs by

w1 (69b)

For a constant sediment size, coefficient B is proportional to F~l/8. Thus, the premise stated herein that a 

"regime" relation presupposes a small variation in the

Blench's equation is

coefficient of equation 19 seems to be supported by 

Blench.

Blench stated that F2/Z>=F6 and assigned it a variation with size of material. Setting the Blench relations 

in terms of 5, Foc(2FB) 1/3 and Foc(2S) 1/2/Fg1/8, and if V 

is eliminated, 21/6S1/2ocFB1/3Fs1/8 or QS3IW*F,?F 3/*. 

Then if W=aQ1/2, Q1/2S3/aacFv2F3/*. Since Q1/2S3<xt3 , 

the result is f<xaFb2Fs3/4 . If a is constant, Fs must be 

constant and ^ocFB2. This would yield the same relation 

as equation 69 and FbccC. If a is not constant, 

,1 '*, whence

(69c)

(69d)

The similarity of equations 69, 69c, and 69d is 

apparent. However, the Blench equations result in a 

decrease in slope with increasing width and increasing 

concentration, neither of which are confirmed by 

laboratory or field observations.

The effect of the size parameter on the two coefficients 

a and t is interesting. The parameter has a maximum 

value at a size of about 0.20 mm, but between 0.06 mm 

and 0.40 mm the size variation is only about 20 percent. 

However, the magnitude of the parameter decreases 

rapidly with both increasing and decreasing size of 

sediment. Consequently, maximum channel widths 

should be expected with grain sizes of 0.2-0.3 mm. Also 

to be noted from equation 69 is that channels decline 

in width with increased sediment concentrations and 

with decline in relief (as represented by the coefficient 

0 if size of sediment does not vary greatly.

These generalities concur with observations made in 

the field. Inglis (1949) noted that in the Sind of Pakistan, where most sediment is in the silt size, canal 

widths are less than in the Punjab of Pakistan where 

the sediment is in the sand size. Values of the coefficient 

a are about 2.6 in the Punjab in contrast to about 2.1 

in the Sind.

Simons (1957) shows a variation of about two times 

in the value of coefficient a, from about 3 to about 

1.5. The highest values are for channels with sand 

sized material, low concentrations, and a fairly great 

relief in the canal system. The lowest values, about 

1.5, are from the Imperial Valley with low relief, high 

sediment concentrations, and sediment in the silt size. 

Canals with beds of very coarse material also have low 

values of the coefficient a.

Rarely is the effect of the change in one variable 

concentrated in a single remaining variable. For example, a change in the sediment concentration in the
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A63

system results in an appropriate change in both the 

width, and slope coefficients. A change in size affects 

the other variables in the same way.

According to the argument presented herein, both 

the Blench and the Lacey relations require an inherent 

correlation between sediment concentration and size of 

material. Blench does provide a correction for sediment concentration but its result is questionable. 

Note that Blench does not relate Fs with size; his size 

parameter is carried in a bed factor, Ft, which is equal 

to V2/D. Data presented herein show that for the 

normal sand-size ranges VZ/D is nearly independent of 

size in sediment mixtures but is related to sediment 

concentration. Furthermore, data do not support 

Blench's assumption that the friction factor varies 

inversely with width because of a hydraulic principal. 

If there is anything certain in fluvial hydraulics it is 

that any statement of uniqueness with respect to the 

friction factor is impossible to make.

The Blench and Lacey equations are restrictive in so 

far as variation in width for a given discharge is concerned. The equations presented herein are considerably less restrictive. However, the role of sediment 

concentration requires further consideration. If equation 68, for example, is to be used with high sediment 

concentrations, a smaller than average value of B 

should be selected. This is because if high sediment 

concentrations are expected, with consequent deposition, slope must be high enough to maintain channel 

capacity during periods of low sediment discharge 

when the channel is scouring.

Equation 68 presumes a velocity-load relation of the 

mid-velocity range. However, equation 19 can be 

reduced to a relation between velocity and slope if 

depth, any combination of velocity and depth, or the 

values of a and Q in the relation W=aQl/2 are known. 

In each, a specific concentration is implied in any 

relation between slope and velocity.

Using a value of 8.2 for w1/4/d3/8 in equation 19, the 

following relations are found

0.009 V 

'' B2 D

When V2/D=k, S=0.009fc 

B2V '

When W=aV», S=° *.

Apparently neither the limiting velocity nor the 

limiting tractive approach to canal design is satisfactory for a range of discharges; however, if V2/D is 

constant, VS is constant for a constant value of B. 

It should be observed that this is in agreement with

two of Blench's "dynamical statements," (Blend, 

1966, p. 89). These are (1) channels of the same water 

sediment complex tend to acquire the same Froude 

number in terms of a suitable depth, and (2) channel^1 

with the same water sediment complex and the same 

measure of erosive attack on the sides tend to adjust to 

the same dissipation of energy per unit mass per unit 

time. These statements should be restricted to flows 

in the mid-velocity range.

A set of regime channels was defined as one in which 

the relation among width, depth, velocity, and slope 

are such that the different rates of internal energy 

expenditure tend to remain constant under a constant 

sediment concentration. Three of Langbein's power 

statements were used to develop the regime relations. 

The term VS is also one of his power statements. His 

fifth, power per unit time of stream travel QVS, is not 

applicable.

The coefficient B is considered to be an indirect 

measure of the erosive attack on the sides. Apparently 

Blench's term ppV^/W does not yield any effective 

information. As has been shown previously, if V2/DccC 

and W=aQl/2, V^/W is simply a statement that tl e 

sediment moves with the water.

An average or regime value of B is about 4; variations 

of plus or minus 10 percent are frequently encountered. 

This average value is in comparison with a value of 4.7 

from the Gilbert data. Some obviously unstable 

natural streams have values as low as 2; however, a 

range in the value of B from about 3.3 to 7.7 has been 

observed in canals. Because all these channels of 

canals appear reasonably stable, obviously some 

means must be found to limit the range in variation 

of the stability parameter.

Any stream flowing in alluvium has a pattern of 

bed forms and velocity distributions in the water prism. 

In most natural streams this pattern is constantly 

changing because of the variation in discharges of 

water and sediment from tributary streams. Widths 

are rarely constant and channels seldom are straight. 

Although the equation of continuity may be valid for 

water discharge, it is not valid for sediment discharge 

except as an average over a long period of time. Thus, 

in water discharge it may be possible to ignore the 

storage factor. This cannot be done with sediment discharge on beds with highly developed bed forms. 

Therefore, alluvial channels do not have constant 

slopes; in what appears to be a uniform reach of stream, 

slopes over long distances may vary considerably from 

the average slope.

Heads of canals are locations of varying combinations of water and sediment discharge. The parent 

stream has a wide range of combinations of water at:d 

sediment discharge; consequently, the sediment con-
A64 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

centration at the intake will vary considerably. Even 

if the discharge of water and sediment in the parent 

stream were constant, sediment movements into the 

canal would vary as the moving bed forms altered flow 

patterns at the canal heading. Consequently, there is 

a considerable variation in slope of the upper ends of 

canals as well as changes in the topography of the bed.

Given an alluvial channel of considerable length and 

reasonably uniform width and some limit on the variation of discharges of water and sediment, however, the 

channel will approach a steady flow. In a way, the 

water and sediment will "get in step." This is easy to 

observe in flow ot canals, but it happens, to some 

extent, in natural streams. As the size of a drainage 

basin increases, the distance between major tributaries 

declines and variations in the discharge of water and 

sediment tend to decrease. The net result is a change 

not only in the values of coefficient B but also in the 

coefficient A of the velocity-load relation. But it 

should be clear that any value of B selected for a 

"regime" equation must represent an average of slopes 

changing over a long reach of a canal.

Oddly enough, both near-maximum and near-minimum values of B for a reasonably stable canal have 

been found on the Marala-Ravi Link Canal in Pakistan. 

This canal was designed for a capacity of 22,000 cfs 

with a velocity of 4.28 feet per second, a depth of 14.6 

feet, and a slope of 1 foot per 10,000 feet. This yields 

a B of 5.15 and a V*/D of about 1.25.

When put into operation the canal filled over much 

of its length and apparently widened slightly in its 

uppermost reach. The maximum capacity has been

reduced to about 15,000 cfs; the fill at the intake is 

about 5 feet. Twenty thousand feet from the intake 

the fill is about 4% feet, and at 70,000 feet it is over 3 

feet. One hundred sixty thousand feet from the heading, 

the canal is about on its original grade. These values of 

filling are by no means static. The average bed elevation at any one section in the canal may change as 

much as 2 feet in a short time.

Data from the West Pakistan Water and Power 

Development Authority Canal and Headworks Observation Program (Harza Engineering Co., 1963) relating to actual canal behavior for three reaches of the 

Marala-Ravi Link Canal are given in tables 2 and 3.

Table 2 shows values of V2/D and the coefficient B 

from equation 51b. The question is, does the increase 

in the value of V2/D between river sections 2C ,000 and 

160,000 indicate an increase in the sediment concentration? Although sediment samples and observations of 

scour and fill indicate some increase or decrease in 

concentration, there is no reason to assume that this 

variation is enough to account for the changes in the 

value of V2/D. Thus, the variation in V2/D imrt be assigned to a variation in the coefficient A\ in the equation V=A1 (L/p)lf3. If this is true, V2/DocA13C. Substituting this for k in equation 51b, SccAi3C/B2V and 

VSccAi3C/B2 . However, there is a correlation between 

AfC and B2 in the form Bac(Al*C) 1 '*. Consequently, 

concentration is the same in the three sections of the 

Marala-Ravi Link Canal, and there is a tendency for 

VS to be constant.

In the section on sediment transport it was stated 

that the value of the coefficient AI is uncertain and, at

TABLE 2. Data for the Marala-Ravi Link Canal relating to behavior 

[Data from Harza Engineering Co. (1963)]

Behavior at indicated distance from headworks (ft)

Date

6-18-62

7- 5-62

8-22-62

9- 6-62

3-29-63

4- 5-63

5- 8-63

5-21-63

6- 6-63

6-23-63

7- 6-63

7-25-63

8- 6-63

8-26-63

9- 9-63

5-15-64

6-11-64

7- 7-64

9-10-64

20,000-26,000

Q (Cfs)

15, 500

14, 500

13, 300

12, 800

7,420

8,250

13, 300

11, 900

15, 000

14, 300

15, 100

13, 300

13, 800

11, 900

11,400

12, 900

11, 900

12, 800

12, 500

66,000-70,000

VI D

2.31

1.4

1. 14

1.17

1.23

1.24

1.39

.90

.966

1.02

1.20

1.01

1. 19

.96

1.23

1. 13

1.02

1.19

1.15

B

4.0

3.5

3.74

4.39

4.71

4. 19

3.94

3.77

3.35

3.57

3.85

4.18

4. 12

4.0

4. 14

3.68

3.62

4.0

3.92

Date

4- 6-63

4-29-63

5- 8-63

5-21-63

6- 7-63

6-24-63

7- 7-63

7-26-63

8-10-63

8-26-63

9-10-63

5-16-64

6-12-64

7- 6-64

9-11-64

Q(cfs)

6,090

13, 350

13, 000

12, 300

15, 400

16, 300

15, 100

13, 200

13, 300

12, 100

11, 000

11, 900

12, 600

14, 100

12, 300

V*ID

3.19

3.99

3.81

2.57

1.83

1.65

1.59

1.82

3.38

1.37

.87

.98

1. 15

1.34

1.43

B

6.3

6.85

7.14

5.97

5.87

5. 16

4.83

5.6

6.6

3.76

4.41

4.58

4.22

4.78

6.0

Date

6-19-62

7- 6-62

8-23-62

9- 7-62

3-29-63

4- 7-63

4-28-63

5- 9-63

5-22-63

6- 8-63

6-25-63

7- 8-63

7-26-63

8-11-63

8-27-63

9-10-63

5-16-64

6-13-64

9-12-64

154,000-160,000

Q(cfs)

15, 000

14, 100

13, 400

12, 400

7,610

6,020

11, 430

13, 500

11, 800

14, 600

15, 000

14, 600

12, 400

12, 700

12, 500

11, 000

12, 300

13, 400

10, 600

V»/D

3.52

3.44

3.88

3.35

2.83

2.37

3.26

2.97

2.14

2.29

3.0?

3.28

3. 1

3.22

2.47

1.38

2.57

2.73

2.33

B

7.15

7.5

7.72

7. 1

7.76

7.45

7.32

6.90

5.42

5.51

7. 15

7. 15

7.40

7.20

6.66

4.91

6.6

6.8

8.0
BEHAVIOK OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A65

TABLE 3. Data for the Marala-Ravi Link Canal 

[Data from Harza Engineering Co. (1963)]

Date

6-18-62

7- 5-62

8-22-62

9- 6-62

3-29-63

4- 5-63

5- 8-63

5-21-63

6- 6-63

6-23-63

7- 6-63

7-25-63

8- 6-63

8-26-63

9- 9-63

5-15-64

6-11-64

7- 7-64

9-10-64

20,000-26,000

Discharge (cfs)

15, 500

14, 500

13, 300

12, 800

7,420

8,250

13, 300

11, 900

15, 000

14, 300

15, 100

13, 300

13, 800

11, 900

11, 400

12, 900

11, 900

12, 800

12, 500

Slope

3,460 __

3,410 _.

4, 620 _ _

6,200 __

5,280

4,680

4, 510

5,300

4,330

4,620

4,940

5,260

4,980

5, 140

5, 110

4, 540

4, 510

4,570

4,474

Behavior at ir

Date

4- 6-63

4-29-63

5- 8-63

5-21-63

6- 7-63

6-24-63

7- 7-63

7-26-63

8-10-63

8-26-63

9-10-63

5-16-64

6-12-64

7- 6-64

9-11-64

idicated distance from

66,000-70,000

Discharge (cfs)

6,090

13, 300

13, 000

12, 300

15, 400

16, 300

15, 100

13, 200

13, 300

12, 100

11, 000

11, 900

12, 600

14, 100

12, 300

headworks (ft)

Slope

5, 160

6,800

7,570

6,530

8,700

7,350

6, 620

7,680

7,090

3,940

7,300

7,310

5,880

7,020 __.

9,091

Date

6-19-62

7- 6-62

8-23-62

9- 7-62

3-29-63

4- 7-63

4-28-63

5- 9-63

5-22-63

6- 8-63

6-25-63

7- 8-63

7-26-63

8-11-63

8-27-63

9-10-63

5-16-64

6-13-64

9-12-64

154,000-160,000

Discharge (cfs)

15, 000

14, 100

13, 400

12, 400

7,610

6,020

11, 430

13, 500

11, 800

14, 600

15, 000

14, 600

12, 400

12, 700

12, 500

11, 000

12, 300

13, 400

10, 600

Slope

8, 4PO

9,200

8,800

8, 040

9, 2«0

8, 820

8, 510

8,440

6, 2PO

6, 6f 0

9, If 0

8,700

9, 320

8, 5PO

8, 550

6, 6fO

8, 4?0

8,540

12, srs

least, ranges from 5 to 6. This gives a range in the 

value of Ai3 of about 1.7 times. A change of 36 percent 

in the value of Av will give a change of A? values of 

250 percent, which would seem to be a reasonably 

expected degree of variation. If JBoc(^li3 (7) 1/2 , a change 

of 36 percent in the value of AI would give an 85 

percent change in the value of B. This is to be compared with an observed change of about 300 percent 

in the value of the coefficient B. Thus, most of the 

change in the coefficient B seems to be related to 

changes in the sediment concentration.

Because the coefficient A appears to be associated 

with velocity distributions within the water prism, and 

particularly three-dimensional flow, a restriction in its 

variation would seem essential in the design of stable 

channels. It should be noted, however, that the relation 

between the coefficients is still unclear. For example, 

it has been noted that the Lacey relationship VocB2/3SlfZ 

requires that JBocl// or JSocl/^VO) 1 /2. Thus, from 

equation 51b

A 3 (7-\ or> replacing A?C with V2/D,

jf, and or VocD2/3Sl/3 .

If, in the Lacey relationship A! is constant, an increase 

in concentration increases the slope. This is in contrast 

to the situation when concentration is constant and 

AI is variable where an increase in Al decreases the 

slope.

In natural channels the coefficient B tends to declir e 

in a downstream direction; however, the coefficient A 

in the velocity-load relation also tends to decline. 

Consequently, in a natural stream system VS is not 

constant. The result of all the possible variations 

in the coefficients of the velocity-slope and the velocityload relations is a wide spectrum of VS values, all 

related to changing bed forms and ail having different 

degrees of stability. The purpose of "regime" analysis 

is to regulate this spread of values.

The problem of the Marala-Ravi Link Canal seeirs 

to be that it was built on too flat a slope. A slope of 

1 foot per 8,000 feet would have been much safer 

for a canal of 22,000-cfs capacity that at times had to 

scour its bed. This is consonant with a B value of about 

4.5. The second factor is that the canal carries only 

7,000-8,000 cfs in the early spring months. The cans,! 

will not transport a normal sediment concentration at 

this low discharge on such a low slope. Further, tte 

discharge of the Chenab River is low during the early 

spring months, and a relatively large portion of the 

discharge is diverted; thus, the possibility of excessive 

diversion of sediment is increased. Finally, the fact 

that the canal is usually dry during the winter msy 

have an adverse effect on bed mobility.

Much has been said about the "dominant" or channelforming discharge in discussions of "regime" channeK 

Insofar as water is concerned, this discharge may l^ Q. 

a satisfactory description, but it is important to note 

that the slope and other characteristics of a channel 

are dependent not only on the water discharge but 

also on the sediment concentration.
A66 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

NONBEGIME CHANNELS

In a sense, a "regime" channel becomes a nonregime channel the moment it is built. The reason is 

that it cannot change its slope with changing discharge 

and loads. Although natural streams do change their 

slopes with varying discharges, the amount of change 

is limited. Hence, natural streams, from the short-term 

view, are also iionregime channels.

If slope is an independent variable, the relation between velocity and discharge and slope is

T4 . J '

if this is combined with V=*

1/3

w \ 

B, <71/3 \AydJ

1/5

(70)

When S is a constant and concentration a variable, 

the adjustment in the stream is made by a variation in 

the coefficients AI and B1} nearly all of which is in BI. 

A range of a little over two times in the value of B 

will provide for a tenfold range in the sediment concentration. This range is observed on many streams. 

The effect of width is so small in equation 70 that it

0.0001 0.001 0.01

SLOPE 

A

100 1000

CONCENTRATION, IN PARTS PER MILLION 

B

10,000

FIGURE 52. Relations in equation 70. A, Relation between the coefficient BI and slope for a constant concentration 

and particle diameter and different unit discharges. B, Relation between coefficient Bl and concentration for a 

constant width, slope, and particle diameter and varying discharge.
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS A67

may be neglected insofar as the relations among concentration, slope, and size of material are concerned 

because of the overriding importance ot the coefficient 

BI. For example, with discharge and concentration 

constant a change in width of 10 times would result 

in a change in the coefficient BI of about 17 percent. 

The relations in equation 70 are shown in figure 52A 

and B. Both figures show the relative unimportance of 

the width (or the unit discharge) as a parameter. Both 

figures show that if slope is constant, changes in sediment concentration are taken care of by adjustment in 

the coefficient BI. Relations between the coefficient BI, 

sediment load, and slope are shown in figure 52A

A considerable error may be made in the estimate of 

the width of a channel, but the relation between slope 

and sediment concentration must be fairly precise if 

the variation in the coefficient BI is to be held within 

close limits. For practical purposes the Lacey equation, 

W=2.QQ1/2, may be used as an initial estimate of width 

of a channel. Such a channel width is then tested to 

determine the effect of the different combinations of 

discharge and sediment concentration that may be expected. Usually, there is little choice in slope. (If there 

is a choice of widths and slopes, a "regime" procedure 

should be used.) Thus, the object of canal design is to 

pick the kind of section that will give the least trouble 

or the lowest cost of operation.

The average value of the coefficient BI is about 5.5. 

The maximum and minimum for stable channels are 

about 9 and 3.4, although some unusual channels have 

values greater and less than these. Nearly always, the 

high values are associated with aggradation and the 

low values with degradation or bank erosion. Streams 

moving rapidly varying sediment loads should have 

lower than average values of the coefficient BI. The 

upper ends of canals in particular have this characteristic. As was noted in the discussion of "regime" 

channels, the amount of sediment entering the canal 

not only changes with variation in the overall average 

rate of sediment movement in the main stream, but it 

also changes, at any main-stream discharge, with the 

passage of bars in front of the diversion works. Even 

with good design of the canal heading, the upper 1 mile 

of a canal moving appreciable amounts of sediment 

ought to have a coefficient of about 4.7, and the next 

2-4 miles, a value of about 5. Any stream that has 

rapidly varying discharges of water and sediment 

throughout its length should have a value of about 5 

or somewhat less for the coefficient. Otherwise, scour 

between periods of deposition is not likely to take place.

CONCLUSION

The 10 variables associated with flow of water and 

sediment in alluvial channels are discharge of water,

discharge of sediment, size composition of the moving 

sediment, size composition of the material making up 

the bed, the fall velocity of a characteristic sediment 

particle, width, depth, velocity, slope, and a characteristic of the pattern of streamflow. Masses and weights 

are assumed constant. These 10 variables can be reduced 

to nine by the equation of continuity discharge 

equals the product of width, depth, and velocity.

Nine equations are required for the complete solution 

of the problems of flow in alluvial channels. The 

equations do not exist and probably never will. Therefore it is not possible, and probably never will be 

possible, to provide a precise answer to all flow problems. The question is, what is needed to provide 

approximate answers to the problems? If there is no 

movement of bed material, conventional hydraulics 

will apply. Such a statement does not provide a high 

degree of certainty, but it at least provides consistencj". 

If an appreciable amount of sediment is being transported by the stream and is in motion over the bee?, 

conventional hydraulics are inapplicable because many 

of the variables are mutually determined and are net 

unique.

The best and most consistent of all relations in 

channels with movable beds is a complex one between 

velocity and unit discharge and load for sediment cf 

material of uniform size. The relation is not simple and 

does not lend itself to a simple mathematical expressior. 

There is less certainty about the relations for a mixture 

of sediment sizes; but, fortunately, for the range of 

velocities commonly observed in nature, the effect of 

size composition of mixtures is small for a considerable 

range of sizes. Therefore, although it is impossible to 

get precision in the estimate of the rate of sediment 

movement, the magnitude of the mean velocity can b«- 

determined with assurance if the unit discharges of 

water and sediment are known.

The common characteristic of natural channels in 

alluvium is the lack of stability in the relations among 

width, depth, velocity, and slope. Most of this instability 

is caused by variation in the water and sediment discharge and by changes in the size composition of the 

sediment mixture. The variability of natural channel^ 

cannot be studied in flumes because of constraints 

placed upon the possible combinations of width, depth, 

velocity, slope, and load that are inherent in flume 

operations.

In natural channels and in canals, for a given discharge and with slope essentially a constant, within 

the range of adjustment of hydraulic roughness through 

changes in bed forms, filling will occur with high velocities and scour with low velocities. This statement is 

fully justified by field observations and is in accord
A68 BEHAVIOR OF NATURAL ALLUVIAL CHANNELS

with the presentation made in this report. An increase 

in concentration of sediment requires a higher velocity 

to move the increased load. At constant discharge and 

width this means a lower depth. If the slope is unchanged, the friction factor must decrease, and this 

decrease is associated with sediment deposition on the 

bed. A decrease in sediment concentration yields the 

converse of the previous situation. Velocity must 

decrease; hence, depth must increase. This is accompanied by scour which is associated with a greater 

friction factor.

If the friction factor is already at the lowest possible 

value, an increase in sediment concentration can only 

be accommodated by a continuous deposition of material on the bed and an increase in the slope of the stream. 

It is practically impossible to increase velocity and depth 

quickly by decreasing width by natural processes. If 

the friction factor is already great, a decrease in concentration may result in a decrease in slope, but more 

likely the result will be an increase in the width of the 

stream, braiding channels, or meandering. Streams 

with well-developed bed forms usually do not degrade 

their beds rapidly; hence, a sudden decrease in sediment 

load usually results in a rise in the stage-discharge 

relation. This condition is frequently observed at the 

upper ends of canals in the Indus River basin.

Although the premise of a velocity-sediment-load 

relation will be congruent with these behavior patterns, 

any relation between a parameter containing a depthslope product and sediment load will not be satisfactory 

under the conditions of a constant slope and of sediment 

and water discharge as independent variables. Thus, 

no relation between yDS, jRS, or jqS and either velocity 

or load can be unique; hence, it cannot be wholly 

acceptable.

The reason for the adoption of such parameters in 

the past is that their apparent rationality is supported 

by any flume experiment operated in a consistent 

manner. For example, there is a consistent relation between load and qS in the Gilbert data. Like the relation 

between load and velocity based on the same data, it 

cannot be expressed in any simple mathematical function. But such a relation will not fit one developed from 

data from Brooks (1958), Vanoni and Brooks (1957), 

or Vanoni and Hwang (1967), which are consistent 

among themselves.

With the conclusion that an effective relation can 

be developed between velocity or depth and unit discharges of water and sediment, width, slope, and a 

characteristic river behavior remain as unknowns. As 

is to be expected from the preceding discussion, the 

relation between velocity and unit discharge and slope 

is complex. It depends upon the constraint placed upon 

depth and slope. Expressing this relation in the form

V=BqxSv, the range in the coefficient B is dependent 

on the variation in the roughness due to bed forms. The 

use of a bed form parameter is not a solution of the 

equation for two reasons: first, at the present time it 

is impossible to evaluate bed forms in this way, and 

second, a bed form has to be associated with an amount 

of sediment in motion. Therefore, given B, q, and S, 

velocity cannot be determined unless the sediment 

load is known because x and y are unknown. If sediment load is an independent variable, the velocity 

cannot be forecast from known values B, q, and S.

Nevertheless, given the proper combination of independent variables, it is possible to provide a solution 

to the equation V=BqxSv . The most probable condition 

is represented by one value of B. If B is less than this 

amount, the tendency is toward degradation. If B is 

more, the tendency is toward aggradation. The only 

reason this is possible is because of natural constraints 

operating the most usual being the tendency of the 

variances of the shear stress and the friction factor, 

jointly, to be minimized.

The two velocity relations apply to two-dimensional 

flow; hence, the result is a suite of values in wtich width 

and slope are two variables. The pattern of flow in a 

channel appears to vary with each combination of 

sediment discharge, width, and slope. This pattern 

again is reflected by change in bed forms. Some of these 

patterns lead to a considerable amount of cross-channel 

flow. Because they are associated with ban!" erosion, 

they must be avoided or controlled; this is tie crux of 

the problem of channel stabilization.

Given a stated relation between sediment concentration and discharge, it appears that a similarity of 

energy dissipation processes is developed when width 

is proportional to the product of a coefficient and the 

square root of the stream discharge, provided width, 

depth, velocity, and slope are all free to vary. If slope 

is fixed, it is not possible to secure this similr.rity.

If, over a considerable geographic region, concentrations and sizes of sediment in natural streams are of 

about the same magnitude and channels of varying 

discharge capacity have a type of bed form which 

does not lead to bank erosion, a set of regional equations 

can be developed by a study of existing stable channels. 

One such set is the Lacey equations.

Channels whose beds are made up of coarse material 

that moves only at the higher discharges (hence, also 

are self-formed) are also subject to analysis. But a set 

of these channels will not have the same characteristics 

as will a set of channels with sediment transport at 

the velocities most encountered.

It must be remembered that the presumption of the 

well known "regime" channel relations is that slope 

varies with discharge. If slope is fixed, changes in
BEHAVIOR OF STRAIGHT OPEN CHANNELS WITH MOVABLE BEDS AGO

discharge and load are accommodated by changes in 

width or changes in the friction factor. The latter is 

the most common.

Clearly, there is a great range of possible adjustments 

a stream may make yet remain in equilibrium as far 

as slope is concerned. These are all associated with 

changes in the size distribution of the sediment discharge or changes in the bed form. The easily observed 

behavior factors of an alluvial channel scour, fill, 

bank erosion, overbank flow or anastomosis are all 

related to these two highly complex phenomena. In 

view of this complexity, it will never be possible for an 

alluvial channel to remain stable and still pass a wide 

range of discharges of sediment and water. It appears, 

however, that given a good appraisal of local behavior, 

much can be done to mitigate the ill effects of any 

specific set of conditions.

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U.S. GOVERNMENT PRINTING OFFICE: 1969 O-329-510