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An Approach to the 

Sediment Transport Problem 

From General Physics

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-1

o

p

I?
An Approach to the 

Sediment Transport Problem 

From General Physics

By R. A. BAGNOLD

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-1

From considerations oj energy balance and oj 

mechanical equilibrium, a mathematical expression 

is derived relating the rates of sediment transport 

as bedload and as suspended load to the expenditure 

of power by a statistically steady flow of water

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 - Price 35 cents (paper cover)
CONTENTS

Introduction _______________________________________

General considerations_____ ---.._ .__._______

Essential features of granular flow._______________

Implication: the existence of an upward supporting

stress, ________----__---_-_____-____________

Restrictions of conditions to be considered. ________

Bedload and suspended load and the work rates of their

transport- _ __________________________________

Definitions- _. _ _______________________________

Transport work rates..________________________

Principle of solid friction and bedload work rate____

Suspended-load work rate.--.-________________

Transport work rates and available power; the general

sediment-transport relationship_______________

The general power equation.___________________

Available stream power_______________________

The general transport relationship in outline form___~ 

Bedload transport efficiency et> at high transport stages.- 

Concept of a moving flow boundary.____________

Critical stage beyond which the bedload efficiency

66 should be predictable.______________________

Effect of inadequate flow depth on the values of b~ 

Variation of the dynamic bedload friction coefficient

tan a. _______________________________________

The suspended transport efficiency es of shear turbulence. 

Critical flow stages for suspension_________________

The effective fall velocity V for suspension of

heterogeneous sediments.____-_-_-______-____

The effective mean grain size D for heterogeneous

bedloads.___- _--___--____----_____--____- 

Washload._____ _ ___-____---_____--___________

Page

II

2

2

2

3

3

3

4

4

5

5

5

5 

G

6

6

8

9

10

12

15

17

18

18

Page 

The final transport rate relationship_.--_---_________ 118

Existing flume data_________--___------_-----___--_. 19

Limitations and uncertainties..____--_---______-- 19

Wall drag and flow depth._____________________ 20

Estimation of appropriate fall velocity V. _________ 21

Comparison of the theory with the experimental flume

data.-...-.________._---.-_._._.____ 21

Gilbert 5.0-mm pea gravel-_______-----_-_______- 21

Gilbert 0.787-mm sand__ . __ __________ 21 

Gilbert 0.507-mm sand____-___-____._______ 21

Simons, Richardson, and Albertson 0.45-mm sand_ 26 

Gilbert 0.376-mm sand__________-_-_______ 26

Gilbert 0.307-mm sand________________ 26

Barton and Lin 0.18-mm sand--_---_-_----___--_- 26

Laursen 0.11-mm silt._________________________ 26

General comments.______-_____-_-_-_-_____._--- 26

Experiments by Vanoni, Brooks, and Nomicos__ __ 27 

Relation.between i and co over the lower, transitional 

stages-----_----_____-_--_------------_____-- 28

Transport of fine silts in experimental flumes...____ 28

The available river data, nature and uncertainties. _____ 29

Energy-slope estimation...... .__.__. ... 29

Bedload and suspended load._____--______--___-_ 29

Deficiency of sediment supply___.__.___._________ 29

Values of the stage criterion 6. ___________________ 30

Comparison of predicted with measured river transport 

rates____________________________________________ . 30

Comparison of river transport data with data for windtransported sand--_--__-________________________ 35

Conclusion _ __ ____________________________________ 35

References-_______^________________________________ 37

FIGURE 1. Diagram of the relation of normal force or stress to tangential force or stress between solids in moving contact- . 

2-5. Graphs:

2. Flow relative to moving boundaries..__.__-___._-._-__._._-__.________.____,

3. Values of theoretical bedload efficiency factors, in terms of mean flow velocity, for quartz-density solids 

in an adequate depth of fully turbulent water_____-_-_---___--_______-___-_-_-__----_______-

4. Values of the solid-friction coefficient tan a in terms of the bed-stress criterion 6=Tl(<r p)gD for quartzdensity solids of various sizes in water, derived from figure 5, and the predicted critical values of 

6 beyond which the theory should be applicable------___---_--___-__---___-_-----_---_-____-_-

5. Values of the solid-friction coefficient tan a in terms of the Reynolds-number criterion G for granular 

shear. . _____________.___________________________________________________

6. Diagram of characteristic fluid motion close to the boundary.._________________________________________

7. Schematic diagram showing postulated asymmetry in the normal eddy velocity components of shear turbulence.

8. Graph showing theoretical values of the suspension criterion 8a =V2/gD together with Shields' values of 0 at 

the threshold of bed movement________________________________________________________________

Pago 

14

11

13

13

16

in
IV CONTENTS

FIGUBES 9-13. Comparative plots of theoretical and experimental transport rates (flume experiments): Page

9. Gilbert 5.0-mm gravel and 0.787-mm sand__,__-____-____ _________________ _._ .._ 122

10. Gilbert 0.507-mm sand and Simons, Richardson, and Albertson 0.45-mm sand._____________________ 23

11. Gilbert 0.376-mm and 0.307-mm sand________.________ _.__-.__ _ __ __ 24

12. Barton and Lin 0.18-mm sand and Laursen 0.11-mm sand______________________________________ 25

13. Vanoni and Brooks 0.14-mm sand, Nomicos 0.172-mm sand, and Laursen 0.04-mm silt __ 27

14. Graph of transport-rate discrepancies (predicted/measured), from table 1, grouped according to river

and station______-____-____-- ________________________________________________________ 34

15. Graphs showing examples of anomalous size distributions of assumedly suspended loads, associated 

with anomalous transport rates predicted from them (suggesting that the sampling had included a 

proportion of coarser bedload material)______-____.,____-__----_..----_---------_----_--------- 36

TABLE

TABLE 1. Data from 146 individual river measurements comparing measured with predicted transport rates.

Page 

130

SYMBOLS

Symbol meanings preceded by f refer to quantities involving 

the gravity acceleration g. These quantities have ambiguous 

dimensions and values depending on the ambiguous definition of 

weight. In relations between such quantities, g cancels out and 

therefore has been omitted, leaving the quantities expressed in 

practical units. However, g is necessarily retained in relations 

between such quantities and, for example, velocities. The 

shear stress T, for instance, is sometimes expressed as p gdS and 

sometimes as p dS, according to the context. There is no real 

ambiguity, because all relations are dimensionally correct.

Meanings of dimensionless symbols are preceded by {.

Symbol 

b

c 

D

d 

e b

ec, et 

e.

Where introduced

I 14 

19

G

Meaning 

{Ratio F/w*. 

{Transport concentration by immersed

weight. 

9 {Static bed concentration by volume

= (1 porosity). 

14 {Flow coefficient u/u*. 

7 Grain size.

5 Flow depth.

6 {Bedload transport efficiency 

_ bedload work rate. 

~ stream power

7 {Factors of e_. 

. 6 {Suspension efficiency

suspension work rate 

~~ steam power 

14 {Residual upward momentum flux, an

internal fluid stress. 

IV Gravity acceleration. 

10 {Special Reynolds number for solids

sheared in fluid= »/^ ij V X

4 fTransport rate of solids by immersed 

weight per unit width =- - gj.

Symbol 

I

Uf

u*

Where introduced

14

Meaning

fTransport rate of solids by immersed 

weight per whole width of flow

Transport rate of solids by dry mass

per unit width. 

 { Transport rate of solids by dry mass

per whole width of flow. 

fMass of transported solids over unit

bed area.

4 flmmersed weight is m'g= mg.

IT

4 Stress transmitted from solid to solid

normal to the plane of shear. 

17 {Proportion of unity, by weight, of a

given size grade in a sample of

heterogeneous granular material. 

5 Discharge. 

7 {Conventional Reynolds number for

fluid flow. 

5 {Energy slope. 

4 fStress transmitted from solid to solid

tangential to the plane of shear. 

4 Mean transport velocity of solids. 

7 Transport velocity, relative to ground,

of a notional nongranular moving

flow boundary. 

6 Mean flow velocity

Q 

cross-sectional area

13 Root-mean-square velocity components of turbulent fluctuations.

14 Shear velocity.

13 Root-mean-square value of the downward part of t'.

13 Root-mean-square value of the upward part of "'.
SYMBOLS

Symbol 

V

tan a

Where introduced

15

4

10

9

10

10

4

4

5

5

Meaning

Fall velocity of suspended solids. 

Effective fall velocity of heterogeneous solids =pVj,. 

J Coefficient of solid friction = T/P. 

Absolute viscosity of fluid.

JDimensionless shear stress =

(*-p)gD

tLinear concentration

mean D of solids

mean free distance between solids 

Kinematic viscosity of fluid. 

Density of fluid. 

Density of solids. 

fMean boundary shear stress. 

fStream power per unit length of 

whole stream=pgQS.

Symbd 

co

6 

c

A, a, n, n'

E, F, R, I, t h

Where introduced

15

17

Meaning 

fStream power per unit boundary

area=rtt.

Subscript pertaining to bedload. 

Subscript pertaining to a moving

boundary. 

Subscript pertaining to conditions

within stationary bed. 

Subscript pertaining to the grain size

present in proportion p. 

Subscript pertaining to suspended

load.

Subscript pertaining to critical value. 

Casual symbols confined to isolated

contexts and denoting dimensionless

parameters. 

Other casual symbols confined to

isolated contexts.
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

AN APPROACH TO THE SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS

By R. A. BAGNOLD

INTRODUCTION

During the present century innumerable flume experiments have been done, and a multitude of theories have 

been published in attempts to relate the rate of sediment 

transport by a stream of water to the strength of the 

water flow. Nevertheless, as is clear from the literature, no agreement has yet been reached upon the flow 

quantity discharge, mean velocity, tractive force, or 

rate of energy dissipation to which the sediment 

transport rate should be related.

Consequently there is as yet no agreement on method 

of plotting measured transport rates against the transporting flow; for this reason little serious attempt has 

been made to correlate the results of laboratory experiments, either between themselves or with field data, or 

to discriminate between diverging theories by this 

means.

Indeed, the outlook is obscured by the very number 

of published theories. For example, the engineer 

research student, encouraged by degree thesis requirements to advance a new theory rather than to verify 

existing concepts, is able to find somewhere in the 

literature, and to quote, support for an assumption 

which best suits his argument.

The root cause of this unfortunate proliferation and 

diversity of ideas is evidently the absence of any sound 

and indisputable quantitative basis of reasoning compatible both with the facts and with the laws of nature 

and, therefore, commonly acceptable. Consensus of 

opinion, which varies from decade to decade, is no 

satisfactory substitute.

In most if not all other fields of modern technology 

such a general basis of reasoning has been supplied by 

the parent natural science, in the form of accepted 

relationships between quantities * which are in some 

degree idealized. The technologist has merely to find 

by experiment how to adapt it to real measured 

quantities.

No established branch of physics has, however, 

interested itself in two-phase (fluid-solid) flow, which is

involved in sediment transport. For the fluid-dynamicist has been reluctant to tackle its special problems 

until he has mastered the problems of fluid flow, and of 

fluid turbulence in particular, and can explain them 

quantitatively in terms of established natural laws. 

. Therefore, the hydraulic engineer, concerned with 

specific practical problems rather than with general 

scientific explanations of natural processes, solves his 

problems by empirical reasoning from past experience 

of like conditions. Thus different working formulae are 

found, each an approximation to the truth over a 

different limited range of conditions, within the manmade bounds of professional practice.

Attempts to weld these formulae together into a 

general empirical relationship, applicable under all 

conditions, have failed. This is to be expected because 

(a) reasoning from the particular to the general presents 

obvious difficulties, and (6) the test of success has been 

immediate applicability to specific practical problems 

rather than generality of agreement with the facts.

The following is an attempt made from the opposite 

direction, from the basis of reasoning deducible from the 

principles of physics. The measure of success is to 

be tested, not by immediate practical applicability, but 

by the extent of the range of conditions, practical or 

otherwise, over which it is found reasonably true. By 

"reasonably true" I mean that discrepancies are 

capable of rational explanation such as uncertainties 

of measurement by conventional methods.

The theory makes no pretense of being complete. 

Moreover, some conclusions run counter to conventional 

ideas. Nevertheless, the measure of agreement with 

the generality of fact over a very wide range of conditions 

suggests the likelihood that the general framework is 

sound in principle.

A theory is of doubtful value unless demonstrated 

to fit the facts. It has been necessary, therefore, to 

devote considerable space to the assembly and discussion of as many relevant facts as possible, so that the 

reader may judge the theory for himself. Such a presI I
12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

entation of the facts, in the form of experimental plots 

and of tabulated river data, also serve the useful 

purpose of bringing to light the deficiencies in the 

experimental coverage.

The uncertainties about turbulence effects such as 

those of boundary roughness, form drag, and sediment 

transport on the flow resistance have been avoided 

by the simple expedient of treating both the mean flow 

velocity and the tractive force as independent, or 

given, variables. This seems wholly justifiable since 

the objective is to predict the transport of solids by 

the fluid flow and not to attempt to predict fluid flow 

itself, which lies within the proper province of the 

hydraulic engineer.

A number of theoretical aspects have already been 

discussed in a previous paper (Bagnold, 1956), to which 

frequent reference is made in this paper. However, 

by confining the conditions to those of transport by 

the open gravity flow of a liquid I have been able to 

make the previous theory considerably simpler and 

more directly applicable. Certain of the former 

tentative conclusions have, moreover, been modified.

Since the terms and symbols standardized in hydraulic 

textbooks are inadequate to deal with the transport 

of solids, the well-known shortage of suitable symbols 

has necessitated some revision of their conventional use.

Finally, in presenting any general theory about 

sediment transport some embarrassment arises from 

the great number of existing theories which have appeared during the present century in widely scattered 

publications. So many aspects have been discussed 

from so many viewpoints that it is doubtful whether 

any one person has read them all, let alone digested 

them sufficiently to appreciate all the implications of 

each. It is likely, therefore, that some of the notions 

here embodied have already been suggested by previous 

authors. My apologies are extended to any authors 

to whose work I have unwittingly omitted reference. 

At the same time it should be borne in mind that if the 

objective is a genuine quest for the truth it is immaterial whether the truth is new or old.

GENERAL CONSIDERATIONS

ESSENTIAL FEATURES OF GRANULAR FLOW

The bulk movement, or "flow," of granular solids 

occurs in a wide range of phenomena, from the mainly 

fluid-impelled transport of solids, naturally by wind 

and by water streams and industrially through closed 

pipes, to the spontaneous flow of supersaturated soils, 

and finally to simple granular avalanching or pouring 

in which the impulsion or traction is by the direct 

action of gravity on solids alone, unaided by any fluid 

traction.

The distinctive character of this class of phenomena 

is defined by the following essential features:

1. The motion is a shearing motion wherein successive 

layers of solids are sheared over one another.

2. An impelling or tractive force, applied in the direction of motion, is necessary to maintain the 

motion.

3. The array of solids is immersed in some pervading 

fluid, either a liquid or a gas, and this fluid also 

is under shear.

4. The solids are heavier than the fluid, and are therefore pulled downward toward a lower boundary 

or bed.

5. In steady continuing motion the forces acting on 

every layer of solids must be in statistical 

equilibrium.

IMPLICATION: THE EXISTENCE OF AN UPWARD 

SUPPORTING STRESS

Any general shearing of an array of solid grams 

requires some degree of dilation or dispersion of the 

array, in order to allow sufficient freedom for the 

shearing to take place. This was strikingly demonstrated by Reynolds (1885), who showed that when a 

mass of loose grains at rest is prevented from dilating 

it behaves as a rigid undeformable body.

When the solids are sheared over a gravity bed the 

dispersion must necessarily be upward, against the 

normal component of the gravity force. To maintain 

such shear thus requires the maintained exertion of an 

upward supporting stress of some kind on every granular shear layer.

Whatever may be the degree of upward dispersion, 

equilibrium demands that across any shear plane the 

magnitude of this supporting stress, of whatever kind, 

must be equal to the immersed (excess) weight of the 

solids present above that plane.

The above points are indisputable. The existence of 

an upward supporting stress equalto the immersed 

weight of the superimposed solids and the mechanisms 

by which it is maintained emerge as the nub of the 

problem. This important aspect has often been 

neglected because the traditional approach has been 

largely kinematic.

In a problem such as the present one, which concerns 

the effects of a density difference, any kinematic approach seems to be inherently self-defeating. For by 

eliminating the concepts of mass and force, the effects 

of forces due to density differences are removed from 

consideration. The problem immediately becomes 

indeterminate, because any determinate relationship 

exists only by virtue of these effects. If there is no 

density difference, the transport rate then depends 

only on the availability of transportable material.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 13

The introduction of the density difference indirectly, 

as implicit in the fall velocity of a representative solid, 

is clearly inadequate, for this approach fails to introduce 

the quantity of such solids transported. Since the 

quantity is the core of the problem, the problem 

remains as indeterminate as before.

There can therefore be no alternative to facing the 

following questions of dynamics: By what mechanism 

can the necessary upward supporting stresses be 

transmitted from the bed to the dispersed solids? How 

are these stresses maintained? And what determines 

the magnitude of the stresses?

The first question can be answered at once. There 

are only two possible mechanisms by which mechanical 

forces can be transmitted:

1. By the transfer of momentum from solid to solid by 

continuous or intermittent contact. For example, 

whether I hold a ball in my hand, or repeatedly 

throw it up and catch it, or repeatedly pick it up 

and drop it, or throw it away so it bounces repeatedly over the ground, the weight of the ball is 

supported only by solid contact. 1 When the ball 

is apparently unsupported it is in fact supported 

by the impulse of a previous contact. And over a 

period of time the weight of the ball is precisely 

equal to the mean upward flux of momentum per 

unit of time, irrespectively of what I do with 

the ball.

2. By the transfer of momentum from one mass of 

fluid to another and thence to the otherwise unsupported solid. For example, an overweighted 

(nonbuoyant) balloon can maintain its height 

above the ground if it happens to drift from one 

upward air current to another. If this were to 

continue indefinitely the excess weight of the 

balloon would be found to be precisely equal to the 

mean upward flux of fluid momentum transferred 

to it.2 Continued support by this mechanism 

requires the continued maintenance of upward 

fluid currents whose effects exceed those of corresponding downward currents. The implications of this requirement are discussed later. 

The second question also can be answered at once. 

The support mechanism is maintained by the shearing 

motion, which in turn is maintained by the applied 

tractive force. The solid-transmitted stress arises 

from the shearing of the solids over one another, for 

this stress can be shown to exist in the absence of any

i Although there may be local and transient exchanges of momentum between the 

ball and an immersing fluid whose internal motion, if any, is isotropic, the normal 

components of these cancel out. The fluid plays no overall part in the support, 

which is transmitted solely by solid contact.

' Continued support in this way would be impossible were the random internal 

atmospheric motion to be isotropic. An anisotropic fluid motion is required, such as 

required, such as that of shear turbulence.

208-57& 60 2

fluid at all when a mass of solids is sheared by allowing it to avalanche or pour in an evacuated vessel. The 

fluid-transmitted stress must clearly arise from the 

shearing of the fluid, for it is this which maintains the 

necessary internal turbulent motion.

The answer to the third question, as to the magnitude 

of the supporting stresses and therefore of the immersed 

weight of solids in transit over unit bed area, is given 

later in this paper.

RESTRICTIONS OF CONDITIONS TO BE CONSIDERED

The application of dynamics to any problem requires at the outset a clear definition of the system to 

be considered. For simplicity the conditions are 

restricted to the following:

1. Steady open-channel liquid flow by gravity.

2. Unlimited availability of transportable solids.

3. The concentration of transported solids, by immersed weight, is sufficiently small that the contribution of the tangential gravity pull on the solids 

to the applied tractive stress can be neglected.

4. The system considered is defined as statistically 

steady and as representative not of conditions at a 

single crosssection but of average conditions along 

a length of channel sufficient to include all repetitive irregularities of slope, crosssection, and 

boundary.

Restriction 1 excludes transport by atmospheric 

wind because the motive power of the wind is less 

readily definable than that of a liquid stream of finite 

depth. Restrictions 1 and 2 together exclude the industrial transport of solids through closed pipes, wherein a granular bed is nonexistent and the impulsion is 

by a pressure gradient instead of by gravity. Restriction 3 is not essential; it merely avoids complications. 

Restriction 4 is essential, for it permits energy 

considerations to be precise.

These restrictions leave the reasoning applicable to 

nearly all the conditions likely to prevail both in 

laboratory flumes (whether the flow is turbulent or 

laminar) and in most natural rivers.

BEDLOAD AND SUSPENDED LOAD AND THE WORK 

RATES OF THEIR TRANSPORT

DEFINITIONS

The excess weight of solids in transit over unit bed 

area must be supported by one or the other of the two 

before-mentioned momentum transfer mechanisms. If 

shear turbulence exists, many individual solids are 

likely to be supported partly by one mechanism and 

partly by the other. We can, however, without any 

loss of generality treat a large number of solids statistically, and divide their excess weight into two discrete 

parts.
14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

If the dry mass of the solids is m and their density o-, 

the immersed weight is m'g -mg, where p is the

<7

density of the fluid. We can divide m' into two parts 

m'b and m', defined as follows:

The bedload m'b is that part of the load which is 

supported wholly by a solid-transmitted stress mb g, 

and the suspended load m', is that part which is 

supported by a flu id-transmitted stress m',g.

No experimental means having yet been found by 

which to measure separately the magnitude of either 

m'b or m,',, or even to measure the total load m', these 

quantities have attracted little attention. They are 

nevertheless real and relevant.

Similarly, no experimental means have been found 

by which to measure the mean transport velocities Z76 

and Us of the solids.3

However, the total transport rate is directly measurable 

under laboratory conditions. Denoting the conventional transport rate that is, the dry mass passing in 

unit time over unit width of the bed by the symbol j, 

evidently _ _ _

j=mU=m6 Ub+ms Us (1)

But since we are concerned with actual dynamic 

stresses the transport rate for dry mass is clearly 

inappropriate and should be replaced by the transport 

rate for immersed weight. Denoting this by the 

symbol i,

i= -gmU=il) -}-is=m'b gUi> -}-m'll gUs (2)

TRANSPORT WORK RATES

It can be seen that the dynamic transport rates j b 

and is have the dimensions and quality of work rates, 

being the products of weight force per unit bed area 

times velocity. As they stand, however, they are not 

in fact work rates, for the stress is not in the same 

direction as the velocity of its action.

The dynamic transport rates become actual work 

rates when multiplied by notional conversion factors 

Ab and A,, each defined as the ratio

tractive stress needed to maintain transport of the load 

normal stress due to immersed weight of the load

Both AD and Ag must necessarily have finite values. 

For if either were zero the fundamental principle underlying the so-called second law of thermodynamics 

would be violated; perpetual motion would be possible.

? Velocities of solids in the direction of transport are denoted by U and velocities 

of the fluid by u. The conventional symbol G for the mass transport rate is unsatisfactory, since it may ambiguously refer to the whole transport or to that per 

unit width. Small i and j are here used for transport rates per unit width, leaving 

/and ./for the transport rates over the whole width of the bed.

In fact, no form of continuing material transport can 

possibly take-place under the influence of a continuing 

transverse pull on the material without accompanying 

energy dissipation in resisting the transverse pull. Some 

theories have overlooked this.

PRINCIPLE OF SOLID FRICTION AND BEDLOAD WORK

RATE

Although both solid and fluid friction dissipate 

energy, they differ greatly in character. Solid friction, 

treated in elementary physics, is omitted from hydraulic 

textbooks.

When two solid bodies in contact under a normal 

force P are sheared over one another, a tangential 

shear force T is required to maintain relative motion 

(fig. I A). The coefficient of solid friction is defined as 

the ratio T/P. This ratio is found to be a constant 

for the materials of the contact surfaces. Unlike fluid 

friction, solid friction in general is independent of the 

relative velocity of shear so long as the energy dissipation as heat is insufficient to change the character of 

the contact surfaces. "Stick-slip," as between violin 

bow and string, is exceptional. With the same proviso, 

the ratio T/P is independent of the absolute values 

of T and P.

The coefficient of solid friction is most easily measured by resting one body upon the other and increasing 

the angle a of inclination of the shear plane until 

shearing begins by gravity (fig. IB}. Then since 

T Mg sin a, and P=Mg cos a, the coefficient TjP 

tan a.

The limiting static coefficient for a mass of cohesionless granular solids is measurable in the same way, from 

the limiting angle of repose at which the surface of the 

mass stands without shearing. The angle for most 

sands is approximately 33°, for which tan a=0.63.

The dynamic condition when the mass of grains is 

under continuing shear, with mutual jostling motions 

in all directions, was investigated experimentally 

(Bagnold, 1954), and the application of the results to 

bedload transport was discussed (Bagnold, 1956).

FIGURE 1. Relation of normal force or stress to tangential force or stress between 

solids in moving contact.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 15

For the moment it is sufficient to state that the dynamic friction coefficient or stress ratio TIP across 

shear planes of an array of solid grains is of the same 

order as the static coefficient, not only when the grains 

are closely packed but also when they are considerably 

dispersed. The above work established the existence 

of a direct frictional opposition to the impulsion of a 

bedload, the opposing stress, in the direction of motion, 

being proportional to the excess weight of the load.

The factor Ab immediately becomes the friction 

coefficient tan a, which is a measurable parameter. 

Substituting the immersed load stress m'b g for P, the 

opposing stress in the direction of motion is m'bg tan 

a, and

bedload work rate=m^?76 tan a=il3 tan a (3)

Here the angle a is associated with the average 

angle of encounter between individual grains, and tan 

a is the ratio of the tangential to the normal components of grain momentum resulting from the encounters.

SUSPENDED-LOAD WORK RATE

The suspended-load work rate can be inferred simply 

and indisputably. The suspended solids are falling 

relative to the fluid at their mean fall velocity V. 

But the center of gravity of the suspension as a whole 

does not fall relative to the bed. Hence the fluid must 

be lifting the solids at the velocity V. The rate of 

lifting work done by the shear turbulence of the fluid 

must therefore be

. V 

suspended-load work rate=m^F=i== (4)

Us

The factor Ai=V/Us may be regarded as the counterpart of tan a. The fluid shear turbulence is in effect 

pushing the suspended load up a notional frictionless 

incline V/US .

TRANSPORT WORK RATES AND AVAILABLE POWER; 

THE GENERAL SEDIMENT-TRANSPORT RELATIONSHIP

THE GENERAL POWER EQUATION

When any kind of continuing work is being steadily 

done the principle of energy conservation can be expressed in terms of the time rates of energy input to, 

and output from, a specified system by the equation

rate of doing work=available power unutilized power 

or in an equivalent alternative form

rate of doing work=available power X efficiency (5)

This basic equation has long been familiar to engineers 

in other fields such as mechanical, liyclromechanical, 

and electrical concerned with working machines and 

Avith power transmission. It has attracted but .little 

attention, however, in the field of channel hydraulics, 

and no conventional symbol has been allotted to the 

quantity power.

The power equation appears first to have been applied to sediment transport by Rubey (1933) and later 

by Velikanov (1955). It was again suggested by Knapp 

(1941), and was later introduced by me in a paper 

(Bagnold, 1956) wherein the flowing fluid was regarded 

as a transporting machine. But none of these ideas 

have been followed up by taking the logical step of 

plotting experimental transport rates against stream 

power.

AVAILABLE STREAM POWER

The available power supply, or time rate of energy 

supply, to unit length of a stream is clearly the time 

rate of liberation in kinetic form of the liquid's potential 

energy as it descends the gravity slope S. Denoting 

this power by 12,

where Q is the whole discharge of the stream.

The mean available power supply to the column of 

fluid over unit bed area, to be denoted by w, is therefore

ft pqQS Ji rJrr=5 ^-^jrr = 

flow width flow width

The kinetic energy w liberated in unit time and subsequently dissipated as heat must not be confused with 

the energy stored within the fluid at any given instant. 

The liberated energy is as little related to the stored 

energy as is the inflow and outflow of water through a 

reservoir related to the quantity of water stored in the 

reservoir.

Confusion is sometimes caused by the seeming paradox that at a given tractive or motive force an increase 

of resistance has the effect of reducing the power dissipation even though the extra resistance element itself introduces an added element of power dissipation. 

The reason is immediately clear from the electrical 

analogy. The electric power dissipated at a given 

voltage E is PR=E2/R. Hence if we increase the resistance R to R+R', the power dissipation is decreased 

in the ratio R/(R-\-R') even though the extra series 

resistance R' itself dissipates a new power element PR'.

Uncertainties as to the precise energy state existing 

at any point or cross section have led many to prefer 

momentum to energy considerations. However, in 

statistically steady channel flow no such uncertainties 

arise as to the time rate of energy supply and dissipa-
16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

tion, provided the system considered is defined as representative of the flow along a length of channel sufficient 

to include all repetitive irregularities of slope, cross 

section, and boundary, and the consequent periodic 

energy transformations.

The available power and the transport work rates, 

regarded as so defined, have precise average values 

along a channel although their values at any specific 

cross section may be imprecise.

The available power has the same dimensions and 

quality as the rate of transporting work done. And 

since the dynamic transport rate i of the solids also has 

these dimensions and quality, there can be no reasonable doubt that the transport rate is related primarily 

to the available power, as in all other modes of transport.

THE GENERAL TRANSPORT RELATIONSHIP IN 

OUTLINE FORM

The available power w constitutes the single common 

supply of energy to both the transport mechanisms. 

Applying equations 3 and 5 for bedload transport, we 

have

tan a=< or (7)

where e b is the appropriate efficiency, necessarily less 

than unity.

The work power e e co is dissipated directly into heat, 

in the process of solid friction. It has gone. Hence, 

for turbulent fluid flow, the power available to do the 

work of supporting a suspended load is co(l e h ).

Applying equations 4 and 5 for suspended-load transport, we have therefore

is =r=

Us

=eo)l e or s= (1 (8)

Adding equations 7 and 8 the expression for the 

whole transport rate i is

(9)

The derivation of this general outline relationship 

seems to me straightforward and logical. It involves 

neither assumption nor approximation, and it is applicable both to turbulent and to laminar fluid flow.

In laminar flow the second term disappears, for suspension as here defined cannot occur, all the solid 

material being transported as bedload.

We have now to infer_ the values of the four parameters e&, tan a, et, and U3 .

By an approximation which should be reasonably 

close for most purposes, we can at once reduce the 

number of parameters to three. For since the travel

of the suspended solids is unopposed, they can be 

assumed to travel at the same velocity as the fluid 

surrounding them. The error introduced by substituting thejnean velocity u of the fluid for the mean 

velocity Ug of the suspended solids lies only in the 

differing distributions of fluid and solid discharges with 

distance from the bed. The possible small error introduced into what follows by making this substitution 

should be borne in mind.

BEDLOAD TRANSPORT EFFICIENCY eb AT HIGH 

TRANSPORT STAGES

CONCEPT OF A MOVING FLOW BOUNDARY

The conventional kinematic definition of a boundary 

to shear flow is a surface at which the flow velocity is 

zero relative to the boundary. In channel hydraulics 

the boundary is assumed axiomatically to be fixed relative to the ground. An equally valid, and more general, dynamic definition of a boundary to shear flow is a 

surface, or a zone of finite thickness, at or within which 

the fluid shear stress is reduced to zero by transfer to 

another medium. The boundary medium may or may 

not be fixed relative to the ground.

At the threshold of motion of the solids both e b and 

es are evidently zero. The solids are all stationary on 

the bed. So the flow of the fluid is relative to a boundary which is stationary relative to the ground.

As the available power is increased, however, more 

and more solids move over the bed as bedload, and consequently more and more of the boundary shear stress 

is applied to the stationary bed indirectly in the form 

of the solid-transmitted frictional stress T via the moving bedload solids.

In this transitional range of flow stage, therefore, the 

fluid flow is relative to a boundary which is partly 

moving and partly stationary. Both the internal 

structure of the flow and the velocity distribution are 

likely to be complex.

As the bedload increases, however, a critical stage 

must be reached at which so great a number of bedload 

solids cover the stationary surface as a moving layer 

that in effect this layer occludes the stationary surface 

from the fluid flow above it.

At and beyond this critical stage, therefore, the 

fluid flow is wholly relative to a boundary which is 

itself moving relative to the ground. The fluid shear 

stress T at this moving boundary may be regarded as 

disappearing, the fluid-transmitted shear stress being 

converted progressively through the thickness of the 

layer into the solid-transmitted shear stress T.4

4 Experiment (Bagnold, 1956, p. 242) has indicated that at granular concentrations 

prevailing immediately over a stationary grain bed at the higher flow stages less 

than 1 percent of the shear stress is maintained by the intergranular fluid, the remainder being maintained by the solid-transmitted stress T.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 17

As a first step toward arriving at the efficiency CD 

under these conditions, let the moving granular flow 

boundary be replaced by a continuous carpet. The 

carpet is supposed in contact with the stationary bed, 

and has a varying weight m'g proportional to the applied 

shear stress. The motion of the carpet is opposed bysolid friction at its underface.

The fluid flow above the carpet applies a fluid shear 

stress T to it. Provided the carpet is thin compared 

to the depth of the flow, the resisting shear stress T 

at its underface is equal to T above it. The conditions 

are as sketched in figure 2A.

The problem is now as follows: If the flow is of 

given depth and has a given mean flow velocity u 

relative to the ground, what is the maximum limiting 

rate at which the flow can do work in transporting its 

boundary along the ground?

The mean velocity of the flow relative to its own 

boundary is u Uc> where Uc is the transport velocity 

of the boundary.

For generality, let the flow law be

where n is 2 for fully turbulent flow and 1 for laminar 

flow.

If the thickness of the carpet boundary is negligible 

compared to the flow depth, T=T and the rate of 

transporting work done is

This function has two zero values, when Uc equals 

zero and u, and one maximum value, when

FIGURE 2. Flow relative to moving boundaries. A, Flow relative to a moving 

carpet. B, Flow relative to a moving granular boundary of negligible thickness. 

C, Effect of inadequate Dow depth and appreciable boundary thickness.

Hence the maximum transport efficiency is as follows:

1 : (10) rU

which is one-third for fully turbulent flow and onehalf for laminar flow.

These results are unaffected by the replacement of 

the single carpet layer by a number of superimposed 

carpet layers in solid contact with one another, [provided the total thickness is negligible in comparison to 

the flow depth. Let the superimposed layers each have 

some unspecified thickness t proportional to the number 

of real granular solids each represents. The shear force

on any layer th, moving at a velocity Uh , is T -^h' and

2 i t

the work rate of moving the layer is T *= 

work rate is therefore

r ^. h rU

As a second step consider the essential difference of 

condition between that of a continuous-carpet flowboundary and that of a boundary consisting of dispersed solids. This difference lies in the fact that 

whereas the fluid shear stress T is applied directly to a 

continuous boundary, its application to a dispersed 

granular boundary necessarily involves relative motion 

between the constituent grains and the fluid in their 

immediate neighborhood. Hence the transport velocity 

C76 of the boundary material is less than the velocity ut 

of the boundary it constitutes (fig. 2-B).

The transfer of stress from fluid to solids involves a 

local dissipation of energy. This introduces a further 

efficiency factor eg=UbJuc , so that e h=ec - eg, where, as 

we have seen, ec is one-third for fully turbulent flow.

The limiting value of e s follows from the same line 

of reasoning as before, applied to the local flow in the 

neighborhood of the bedload grains rather than to the 

whole flow. Consider this time a single representative 

bedload grain. Under steady average conditions the 

fluid force F urging it along is in equilibrium with an 

equal mean friction al force applied to it intermittently 

by the bed.

The force F varies as (uc Ui,)n ' where n' varies 

between 1 and 2 according to the local Reynolds number R==(uc Ut}D/v, being 1 in the ultimate Stokes 

law region and 2 for large grains. The work rate 

UD(UC £/&)"' has a limiting value when Ubluc ~eK=

The exponent n' for a given grain size D and a given 

R, that is, for a given slip velocity uc £7&, can be obtained from the slope of the experimental log curve of
18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

drag coefficient versus R. If the slope at any point on 

the curve is a, then n' =2 a. Since

uc= (uc n

fl

, l- and ut=- u 

o

for fully turbulent flow, values of eg and eb can be 

found corresponding to a range of arbitrary values of 

the mean flow velocity u. These efficiency values are 

shown in figure 3 for various grain sizes D (quartzdensity grains in water). The bedload efficiency eb 

should range from 0.11 for large grains and large flow 

velocities to 0.15 for very fine grains and low flow 

velocities.

The whole rate of energy expenditure in transporting 

the bedload consists of the work rate e 6 co+0c -6gw plus 

the ineffective power dissipation ec co(l eg ) involved in 

the local transfer of stress from fluid to solids. It 

might at first sight be thought that in the transport of 

large grains since this latter dissipation takes place 

through the creation of wake eddies some of it may 

be available to maintain the turbulent suspension of 

smaller grains. But, as later becomes apparent, this 

kind of turbulent eddy is unlikely to possess the essential quality of boundary shear turbulence necessary 

to maintain a suspension. Therefore, the overall power 

loss attributable to the bedload transport is eew=|co, 

so that only f o> remains available to maintain a suspended load.

CRITICAL STAGE BEYOND WHICH THE BEDLOAD 

EFFICIENCY e b SHOULD BE PREDICTABLE

The foregoing argument is restricted to conditions 

in which the moving bedload solids are sufficiently 

numerous to interpose an effective flow boundary between the free fluid flow above and the stationary 

bed below. By the foregoing dynamic definition of a 

flow boundary the above condition is fulfilled when 

virtually the whole applied fluid stress is transferred 

to the moving bedload solids.

The critical bedload stage appears therefore to be 

definable by a critical value of the bed stress T.

A rough estimate of this critical value can be obtained 

as follows. The topmost stationary grain layer of the 

bed effectively occludes the fluid flow above it from 

the stationary layer immediately beneath it. Hence

0.5

0.4

0.03

0.3

0.2

0.18

0.16

0.14

0.12

0.1

1.5 2 3456 8 10

MEAN FLOW VELOCITY OF FLUID (u), IN FEET PER SECOND

FIGURE 3. Values of theoretical bedload cfllciency factors, In terms of mean flow velocity, for quartz-density solids In an adequate depth of fully turbulent water.
APPBOACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 19

this occlusion can be expected to persist if the whole 

topmost layer is set in motion as bedload over the layer 

beneath, which has now become the topmost stationary 

layer.

The immersed weight of the original topmost layer, 

which is now in motion, is (a p)gD -C0 where D is 

the mean grain size, and C0 is the static volume concentration, that is, (1 porosity), whose average value 

may be put at 0.6 to 0.7.

The bed shear stress r required to maintain the 

motion of this load is

T=(<r p}gDC0 tan a

Introducing 6 as the dimensionless bed shear stress 

TJ(cr p)gD, the critical stage should occur when, 

approximately,

dx=CQ tan a (11)

From experiment, tan a appears to vary according to 

the conditions of shear from 0.375 to 0.75 owing to the 

variation of fluid-viscosity effects with variation of 

grain size and mass. The estimated range of dx from 

about 0.5 for grains smaller than 0.3 mm to about 0.25 

for grains larger than 2.0 mm is sketched in figure 4. 

This range is derived from a more basic relationship, 

figure 5, introduced under the later heading "Variation 

of the dynamic bedload friction coefficient tan a."

It should be realized that any direct manifestation of 

this critical stage concerning the bedload by a change 

in the trend of experimental plots of total transport rate 

against the power may, for fine grains, be masked by a 

more pronounced change resulting from the attainment: 

of a corresponding critical stage at which the suspended 

load becomes fully developed.

However, as suggested previously (Bagnold, 1956, 

p. 256), the critical bedload stage Bx is likely to be: 

broadly associated with the value of 0 at which bed 

features disappear or at any rate cease to create appreciable form drag. The experimental evidence (figs. 

9-13) shows the correspondence to be moderate. :

This evidence is somewhat indefinite. The change 

from large-scale dunes to fiat beds occurs gradually over 

a considerable range of stage, and different observers 

tend to discriminate differently. The experimental 

scatter adds to the uncertainty. As can be seen, some 

reported dunes widely overlap with reported flat beds.

More precise experimental work is needed before a 

satisfactory criterion is found by which to predict for a 

given sediment the critical stage at Avhich the change of

end occurs in the transport-rate plots.

Wholly viscous conditions

<0.25

0.3 

0 1 0.2 0.3 0.4 0.6 0.8 1.0 2

DIMENSIONLESS BED SHEAR STRESS (0= T =\ \ (ff-p)g D'

FIGURE 4. Values of the solid-friction coefficient tan a in terms of the bed-stress 

criterion 0=ri(a p}gD for quartz-density solids of various st7.es in water, derived 

from figure 5, and the predicted critical values of 9 beyond which the theory 

should be applicable.

EFFECT OF INADEQUATE FLOW DEPTH ON THE 

VALUE OF e 6

In a previous subsection of this paper the evaluation 

of the bedload efficiency factor ec as being l/(?i+l) = l/3 

for fully turbulent flow depended on the assumption 

that the thickness of the conceptual moving carpet 

could be neglected in comparison to the flow depth.

As the flow depth is progressively reduced, a point 

must clearly be reached at which this assumption becomes invalid. Consider the extreme case when the 

water surface coincides with the height reached by the 

saltating solids (fig. 20}. The fluid shear stress is now 

transferred to the solids throughout the flow. The 

moving-carpet concept entirely breaks down, and e c approaches unity. The bedload efficiency e h is nearly 

three tunes greater, and approaches eg in value.

At a given power, therefore, the transport rate of 

bedload would be three times greater than the rate to 

be expected at greater flow depths.

This immediately raises the question, what is the 

effective thickness of the bedload zone at high stages 

when the number of moving solids is equivalent to the 

number in several grain layers of the bed? This question is of some importance; for if flume experiments are 

to be representative of river flow the experimental flow
110 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

depths must exceed some multiple, say 10, of this thickness. Otherwise the experimental transport rates 

would be expected to be too large by an appreciable 

factor.

It appears that no reliable factual knowledge exists 

on this point. The conventional assumption that the 

bedload zone is but 2 to 3 grain diameters thick is 

based solely on visual observation of individual saltation paths at such very low stages close to the threshold 

of motion that the individual paths can be picked out.

At the higher stages this assumption is untenable on 

spatial considerations alone. The probability that the 

saltations will be interrupted by collisions between 

rising and falling grains becomes rapidly greater as the 

number of saltating grains increases.

Moreover, whatever may be the actual nature of the 

forces initiating a saltation, these forces, and with 

them the height attained, would be expected to increase approximately as the square of the flow velocity 

u. And since the number of grains moving over unit 

bed area increases as the shear stress T, the effective 

thickness of the bedload zone would be expected to 

increase as a function of the power omega. Hence, if 

the flow depth were inadequate, 06 would increase with 

increasing power instead of remaining virtually constant 

over the higher range of stages.

This consideration is directly relevant to a later discussion of the marked discrepancy between Gilbert's 

experimental transport-rate data, obtained at small 

flow depths of 0.1 to 0.3 feet and other comparable 

data obtained at flow depths several times greater. 

It is also directly relevant to the vexing problem of the 

proper allowance to be made for the effects of wall 

drag in laboratory flumes.

It would seem impossible to gage the height to 

which the bedload saltation rises at high stages of 

turbulent water flow, because of the presence of a 

concurrent suspended load. However, the saltation 

mechanism cannot depend on turbulence (unless we 

postulate two different mechanisms having identical 

effects), for saltation persists unchanged in appearance 

when the flow is made purely laminar. Much could 

therefore be learned about the effective thickness of 

the bedload zone at high stages by increasing the 

viscosity of the fluid till both turbulence and suspension disappear. A light machine oil of viscosity about 

2 poise would seem suitable.

When a correction is made for the effect of increased 

viscous drag on the rising grain, I would not be surprised if the estimated thickness of the bedload zone 

in water at stages corresponding to 6 values between 

say 0.5 and 2 was found to be as large as 20 to 50 grain 

diameters.

The basic experiments discussed previously (Bagnold, 1954, 1956) showed that the dynamic friction 

coefficient 7yP='tan a varies over a factor of 2 according to the relative effects of grain inertia and fluid 

viscosity on the internal grain motions.

The grain stress ratio T/P may be regarded as the 

ratio of the tangential to the normal components of 

grain momentum associated with the impacts between 

grains as they are sheared over one another within 

the fluid.

The normal momentum component created at an 

impact between solids must necessarily be outward, 

tending to disperse one from another, as every billiard 

player knows. If the grains are assumed to be elastic, 

this* normal component consists of two equal elements 

associated respectively with approach and rebound. 

In a viscous fluid all the rebound element may be transferred to the fluid during the time of passage of a grain 

to the next impact. But in a nonviscous fluid no such 

transfer takes place. Hence the effect of a change 

from viscous to inertial conditions is to double the normal momentum component. The tangential component on the other hand is not affected, since the mean 

forward motion of the grains remains the same, being 

determined by the imposed rate of shear. Hence the 

change from viscous to inertial conditions has the 

effect of doubling the dispersive grain stress P and 

therefore of halving the friction coefficient TjP.

Any change from viscous to inertial conditions of 

motion of either solids or fluids is definable by a criterion 

of the form of a Reynolds number. For solids the 

number was found to take the form

aT_ 

X

(12)

where T? is the dynamic viscosity, a is the density of the 

solids, T is the solid-transmitted shear stress, and X 

is the linear spatial concentration denned by the ratio

X= mean diameter D of solids 

mean free distance between solids

It can be seen that in its equivalent form

 T^

flfcr \ a\

G is very closely analogous to the fluid Reynolds 

number

D
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 111

0.8

0.7

0.6

0.5

0.4

0.3

20 30 40 60 80 100 200 300 400 600

- <rD2 r

~ : I4ij2

800 1000 2000 3000 4000 6000 10,000

FIGURE 5. Values of the solid-friction coefficient'tan a in terms of the Reynolds-number criterion 0 for granular shear.

Experimentally, the change of conditions of motion 

was found to occur within a range of G between 10 and 

90. The corresponding change in the numerical 

values of T/P=tan a are shown graphically in figure 5. 

From this the values of tan a at and above the critical 

stage for bedload movement over a gravity bed can be 

calculated on the assumptions that (a) over the higher 

stages T T and (6) the concentration Xo at the bed 

surface remains constant at 14, which is the limiting 

concentration at which a sheared array of solid grams 

was found to cease to behave as a Newtonian fluid, 

and to begin to behave as a rheological "paste."

On these assumptions

<rZ?3 (t7-p)0 (13)

where 0=- and T is in engineers weight-force

units and does not involve g.

The resulting values of tan a in terms of 6 are shown 

by the family of identical curves in figure 4.

It should be noted that equation 13 is inapplicable 

to the lower transitional stages below the critical stage, 

because in equation 12 T is not equal to T and X is not 

constant. Both must dwindle to zero at the threshold of bedload motion. Grounds were given previously 

(Bagnold, 1956, p. 260) for assuming that over the

208-B78 «6 3

transitional stage G may remain appreciably constant 

and thus make tan a constant as shown in figure 4.

Some support for these quantitative inferences is 

provided by the following:

1. Visible saltation of quartz grains in water as a 

manifestation of the effects of grain inertia has 

been reported (Durand, 1952) to fade out progressively as the grain size is reduced from 2 to 0.2 

millimeter. As can be seen from figure 4 this is 

precisely the range of the change from wholly 

inertial to wholly viscous conditions.

2. According to the reasoning given previously (Bagnold, 1956, p. 254), instability at the bed-surface 

interface resulting in bed features, such as ripples, 

should occur only when the dynamic value of tan a 

within the moving bedload exceeds the static value 

tan a that is, approximately 0.63 within the 

stationary bed. As can be seen from the broken 

curves in figure 4 this result should restrict the 

occurrence of spontaneous bed features, in water, 

to beds of mineral-density grains smaller than a 

size between 0.7 and 0.8 mm, in good agreement 

with observation.5

' Long flat scalelike features do occur on beds of much larger grains. But these 

features appear to be transient and to result from some disturbance which has caused 

a temporary local deposition for example, a temporary excess of sediment fed to a 

noncirculating flume.
112

3. As reasoned on page 256 of the above paper, spontaneous features on beds of grains below the 

critical size should flatten out when the residual 

fluid-transmitted shear stress T exerted directly on 

the stationary bed disappears that is, when 

T = T at the critical stage. The reader can judge 

for himself later how far this prediction is borne 

out experimentally.

4. Perhaps the strongest evidence is offered by the 

behavior of windblown sand. Here, owing to the 

relatively small value of the viscosity 77, 1.3X10"4 

poise instead of approximately 10~2 , all natural 

dune sands behave inertially (G2 > 8000, fig. 5). 

We can, however, reduce the value of G2 by 

reducing the grain size D in equation 13. The 

theory would predict that when G2 is reduced by 

giving appropriate experimental values to D and 

0 to the critical value of approximately 530, for 

which tan a = tan a0 = the static value 0.63, conditions at the dry bed surface should become 

similar to those under a water stream. The 

typical ballistic pattern of wind ripples should then 

change to the very different water-stream pattern. 

Experiments in a wind tunnel confirmed this 

(Bagnold, 1956, p. 262 and pi. 2). The change 

took place abruptly, and at the predicted values 

of D and 6.

The original experiments, on the results of which 

figures 4 and 5 and equation 12 are based, were done on 

the shearing of artificial grains of zero excess density in 

water and other fluids, contained within the annular 

space between rotating drums (this being the only 

practicable way of measuring the intergranular stresses). 

The above applications to transport over a gravity bed 

over so great a range of density differences therefore 

constitute a severe test of the general correctness of the 

various inferences.

The original experiments unfortunately have not as 

yet been repeated by others using improved apparatus. 

So the curves of figures 4 and 5 may need some modification in detail. In default of any other information, 

however, the values taken for the friction coefficient of 

bedload transport in the following discussion are those 

given by figure 4.

THE SUSPENDED TRANSPORT EFFICIENCY e, OF SHEAR 

TURBULENCE

Mass-transfer theories on the suspension of solids by 

fluid turbulence are based on the analogy of momentum 

transfer. Since no density differences are involved in 

the latter, the reasoning can justifiably be kinematic. 

To extend the same reasoning to solids having an 

appreciable excess weight it is necessary to define the 

solids by their kinematic property of fall velocity.

This is in effect to treat the solids as little fishes of zero 

excess weight swimming perpetually downward relative to the fluid at the given fall velocity. With the 

aid of certain assumptions about the turbulence, this 

kinematic approach succeeds in predicting the decrease 

in concentration of the fishes with increase of distance 

from the boundary.

This approach fails however, inevitably, to predict 

any limit to the total weight of fishes which a given 

turbulent flow can carry in suspension. For it ignores 

the reality that to maintain a suspension of heavier 

solids the turbulence must exert an upward stress equal 

to the excess weight of all the solids present over unit 

bed area. Clearly, no such upward stress is necessary 

to support the fishes.

That a limit to the suspended load must exist is 

indisputable. Otherwise rivers as we know them could 

not exist, for no deposition of suspended material could 

occur and no suspendable bed material could remain 

unsuspended. From the dynamic viewpoint it is 

evident that the suspension of heavier solids requires 

some limited proportion of the internal energy of the 

turbulence to be organized in such a way that it is 

capable of doing work in supporting the excess weight 

of the suspension. This means that es must have a 

certain limiting value.

In an attempt to predict this limit, conventional 

theory assumes that the necessary datum concentration 

close to the gravity boundary is given by that of the 

solids transported along the boundary as bedload, that 

is, by another mechanism. This assumption implies 

that a maintained suspension requires the presence of 

a bedload and cannot continue without it. If so, the 

hydraulic transport of solids in suspension through 

closed ducts would become inexplicable. For here the 

development of a bedload is deliberately prevented by 

working within a limit of concentration beyond which 

a bedload begins to develop.

A further weakness of the above assumption is that 

it makes serious difficulties when the suspended 

material is small in size. Such material has to be 

arbitrarily excluded from consideration, as "washload."

The following alternative dynamic concept avoids 

these difficulties and at the same time predicts a 

quantitative limit to es of approximately the required 

value.

Isotropic turbulence cannot by definition be capable 

of exerting any upward directed stress which could 

support a suspended load against gravity. For any 

suspended solid must experience over a period of time 

a downward flux of eddy momentum equal on the 

average to the upward flux. A swarm of solids would 

be dispersed equally in all directions by diffusion along 

uniform concentration gradients, but the center of grav-
113

ity of the swarm would continue to fall toward a distant 

gravity boundary.

The center of gravity of a swarm of solids suspended 

by shear turbulence, on the other hand, does not fall 

toward the gravity shear boundary. The excess weight 

of the solids remains in vertical equilibrium. It follows 

therefore that the anisotropy of shear turbulence must 

involve as a second-order effect a small internal dynamic 

stress directed perpendicularly away from the shear 

boundary. In other words, the flux of turbulent fluid 

momentum away from the boundary must exceed that 

toward it.

One might infer intuitively, on the analogy of solid 

physics, that the magnitude of this stress would be of 

the same order as that of the shear stress causing it. 

Although a stress of this order would account for the 

observed phenomenon of suspension, it would be too 

small to have been distinguished instrumentally from 

the far greater superimposed fluid pressures, static and 

dynamic.

Instruments used to measure the three components 

u', v', and w' of the internal velocity fluctuations are 

incapable of distinguishing to from fro. They give only 

the overall root-mean-square values. And for mathematical simplicity it has been assumed that the fluctuations in shear turbulence are as symmetrical in the 

to-and-fro sense as they must be in isotropic turbulence. 

That is, the root-mean-square values of the separate 

positive and negative parts of a complete velocity fluctuation have been assumed symmetrical.

This symmetry seems unlikely to be true of motion in

Up 

A

 o

UJ

'V

Down

Fluid masses partaking 

isotropic turbulence

boundary shear turbulence in the direction normal to 

the boundary. Photographs such as that in figure 

3.6 of Prandtl (1952), though admittedly of artificially 

induced turbulence, show the normal motion to be 

highly asymmetrical.

The basic features disclosed by this photograph are as 

sketched here in figure 6. The turbulence appears 

to be initiated and controlled by a process akin to the 

generation of surface waves by a strong wind. An 

upwelling on the part of a minor mass of less turbulent 

boundary fluid intrudes into an upper, faster moving 

layer, where. its crest is progressively torn off, like 

spray, and mingles with the upper layer. Corresponding motion hi the reverse sense are absent or 

inappreciable.

Since there cannot be a net normal transport of 

fluid, the return flow must be effected by a general 

sinking toward the boundary on the part of a major 

mass of surrounding fluid. Hence the velocity «/up of 

the upwelling exceeds the downward velocity v'^ of the 

return flow, as sketched in figure 7.

'///////////////////////////////////////////A//

FIGURE 6. Characteristic fluid motion close to the boundary, as inferred from 

photograph (Prandtl 1952, flg. 3.C) of the onset of boundary turbulence.

Up 

A

ap

v 

Down

Fluid masses partaking 

shear turbulence

FIOUBE 7. Schematic diagram showing postulated asymmetry in the normal eddy velocity components of shear turbulence. In both isotropic and shear turbulence, continuity requires that the total discharge, as given by the positive and negative areas, must be zero. In isotropic turbulence the symmetry makes 

the upward and downward elements of momentum flux equal, but the asymmetry in shear turbulence gives a residual upward momentum flux.
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Consider a representative unit volume of fluid 

within the boundary region where the normal fluctuations are decreasing toward the boundary. Inside 

this unit volume let a minor mass p(% a) be moving 

upward at root-mean-square velocity v'up, and the 

remaining major mass p(K+&) be moving downward 

at a velocity v'An, the asymmetry a being positive.

The total normal momentum must be zero, so

but the normal momentum fluxes, per unit area of a 

shear plane, are unequal. There results a residual 

unidirectional momentum flux /, upward into the body 

of the flow, of magnitude

2a ,..., (14)

balanced below by an equivalent excess of mean static 

pressure at the boundary.

If the velocity v'w is determined by the shear stress 

r, then for a given T the flux / has a limiting maximum value when the asymmetry a=~ (V2 1) =0.207. &

Shear turbulence results, broadly, from a general 

dynamic flow instability. And again broadly, within 

the region of its generation, the intensity of any dynamic instability disturbance proceeds spontaneously 

to the limit set by the approach of the energy losses involved to the energy supply. It is to be expected therefore that this particular energy loss, associated with /, 

would be maintained at its maximum value. So there 

is a reasonable probability that the asymmetry a has 

the critical value 0.207 within the boundary region. 

It must, however, decrease toward zero through the 

body of the flow as the turbulence approaches isotropy 

with increasing distance from the boundary. ;

The relation in equation 14 can be expressed in terms 

of the overall mean measured velocity v' by addition

7'2_ /2 l-2o ,2 l+2a ! ,,

'up n T^dn o V»

whence

l+2a (15)

/=2apy/2=0.414py' 2 on the above assumption

The measured quantity v' has been found to increase 

from zero at the boundary to a sharp maximum, and 

then to decrease progressively with distance from the 

boundary (Laufer, 1954; Townsend, 1955). The ratio 

flmaxMie appears to be in the neighborhood of unity.

Thus the normal momentum flux entering the body 

of the flow above the plane of v'm&x is

This value would be well below the limit of sensitivity 

of Laufer's manometer. So the excess reaction pressure at the boundary could not have been detected. 

Significantly however an energy loss toward the boundary by pressure transport was inferred from the energy 

balance drawn up. (See also Townsend, 1955, p. 217 

and fig. 9.12.)

The propagation velocity of the flux / being V//p, 

the supply of lifting power to the body of the flow is

:'s

'max (18)

It remains to express this relation hi terms of the 

whole power supply W=TU, from the rather inadequate 

experimental data.

According to Laufer's work the ratio "v'ma.i]u* varies 

with the Reynolds number of the flow, and is approximately 1.0 at R=3X104 and approximately 1.1 at 

R=3X105 . This ratio being denoted by b and the 

flow coefficient u/u* by c, the suspension efficiency e» 

should be given by

_/V7/p_n r~ ru ~U< (17)

At flow velocities u of the same order, b would increase with increasing flow depth. And since c increases 

likewise, it is possible that 63 /c may remain constant 

over a wide range of conditions.

Experimental flume conditions being taken as a 

standard, since they cover approximately the same 

experimental range of R, b may be put at say 1.03, and 

c appears to range only between 16 and 20 for highstage flows and for the sand range of grain sizes. 

Putting c=18,

0.266X1.1

18 :=0.016 (17a)

Laufer's measurements refer to flows past smooth 

boundaries, and the effect of boundary roughness on the 

ratio b appears to be uncertain, as is also the effect of 

the presence of transported solids. However, for test 

purposes I decided to ignore these uncertainties and to 

assume that the suspension efficiency es has the universally constant value 0.015 for fully developed 

suspension by turbulent shear flow. This assumption 

gives the numerical coefficient in the second term of 

equation 9 the round figure of f X0.015=0.01.

It may of course be fortuitous that this value makes 

the general transport relation in equation 9 accord 

surprisingly well with the comprehensive range of 

transport data to be presented later. From the river 

evidence the figure might be 25 percent larger but no 

more.

The postulated upward lifting power supplied to the 

flow body from below would also account for the
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 115

hitherto unexplained persistence of secondary circulations in long straight channels, by creating large-scale 

instabilities of the same nature as density instabilities 

due to thermal gradients.

Incidentally, the theory may shed some light on the 

truth of the legend that the surface of a stream is 

slightly raised in the center. If the stream cross 

section is dish shaped or trapezoidal, so that the local 

T is negligible at the sides, and if the stream is unladen 

with suspended sediment, the maximum predicted 

excess height in midstream would be J/pg^QAdS. In 

natural rivers this maximum would rarely amount to 

10~3 feet. It would be still less if the excess head were 

relieved by secondary circulations.

Since writing this section I have seen Prof. S. Irmay's 

highly relevant paper (Irmay, 1960). In the latter, my 

postulated normal stress / is predicted mathematically 

in terms of acceleration, by an entirely different approach 

along the lines of Reynolds' (1895) treatment of the 

Navier Stokes equations.

Irrnay has shown that the Reynolds approach leads 

to a mean acceleration A, the normal component of 

which, in the present notation, is

'

(Irmay's equation 11 3) v

Hence if 7 is the flow depth from the boundary up 

to the no-shear plane, the upward momentum flux / 

is given by

C Y

J=p Aydy=pv' 2 , since v'=Q at the boundary Jo

This flux is shown to be generated in the region of 

maximum v' close to the boundary (see Irmay's fig.4) . 

My own approach appears to explain the mechanism 

of its generation there.

As can be seen from Irmay's figure 2, based on 

Laufer's data for R=3X104,

x. 0.375r

which compares well with my value of approximately 

0.41r.

Thus my conclusions receive strong independent 

support. In the light of these new ideas it would 

seem that the basis of the conventional theory of suspension may need serious reassessment. For the existence, inherent in shear turbulence, of an upward fluid 

stress appears incompatible with theories purporting 

to explain suspension in the absence of such a stress.

Two immediate implications of Irmay's more detailed analysis may be noted here.

1. The normal concentration profile of suspended solids

is related to the rate of decrease with distance

from the boundary of the normal momentum flux

208-578 66 4

jv=pv'v2 as modified by the transfer of upward momentum to the solids.

If Cy is the weight of suspended solids per unit 

volume at distance y from the boundary, the 

work rate needed to maintain the suspension of 

the solids in a layer dy is CvVdy. The fluid lifting 

power is d(js/s)^. Whence

dy V dy

The required modification of the v' profile may be 

deducible from the measurable modification of the 

Karman constant by a suspension of solids.

2. The no-shear region in Laufer's experiments was 

that of the central axis of a closed duct. So the 

normal fluctuations remained finite there. At the 

free surface of an open flow the normal fluctuations 

are inhibited. The dynamic Reynolds stress pv' 2 

must therefore fall sharply to zero as it does at the 

shear boundary below, being converted into an 

excess of static head. Thus the whole integral

C- =p L

Jo

Aydy

needs to be taken to an upper limit just below the 

free surface.

In the thin intervening zone there must exist a 

large negative acceleration, downward away from 

the surface, possibly of the same order as the large 

positive acceleration, found to exceed 2g, at the 

shear boundary.

This seems likely to account for the hitherto 

unexplained phenomenon observable in flume 

experiments on sediment transport, namely the 

inability of transported solids ever to touch the 

actual surface film, together with the existence of 

a thin layer of relatively sediment-free fluid 

immediately beneath the surface.

CRITICAL PLOW STAGES FOE SUSPENSION

It is reasonable to suppose that no solid can remain 

suspended unless at least some of the turbulent eddies 

have upward velocity components v'uv exceeding the 

fall velocity V of the solid. The turbulence has a 

spectrum of such velocity components, of which v'nv is 

the mean. Some eddies have greater upward velocities 

and some less. Hence the stages marking the threshold and full development of suspension should be 

definable by critical values of the ratio v'up/V, the 

threshold occurring 'at some value less than unity and 

full development at some higher value.

From equation 15, v'uv=1.56v' when a=0.207. And 

v' varies as the shear velocity VT/P. The Laufer (1954)
116 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

data being used as before, W appears to have an average 

value of about 0.8w* over the flow depth. Whence as 

a tentative approximation we can write v' = 1.25 Vr/p. 

So unit value of the ratio v'^/V can be defined by 

1.252T=p-72 . Substituting T=6s(ff p)gD,

<r p

V2

=0.4 -~ for quartz-density grain in water (18a)

Since the fall velocity V becomes proportional to ^/D 

for all large grains exceeding about 2 mm, 9S should 

remain constant for all large material irrespective of 

size. V being 20 centimeters per second for D=2 mm, 

0S ~O.S for all such material.

On the other hand, V*/gD decreases rapidly with 

decrease of grain size, as is shown in figure 8.

Here the 6 values for the threshold of bed movement, according to Shields (1936), are also shown. It 

can be seen that suspension is likely to be appreciable 

at the threshold of bed movement when the grain size 

is 0.25 mm or less, and to become fully developed at 

that early stage when the grain size is 0.1 mm or less. 

By contrast, grains larger than 1 mm are unlikely to 

be suspended at all until the bed stress is many times 

greater.

The above applies to grains of uniform size. The 

suspension of natural materials of heterogeneous size 

involves the spectra of both eddy velocity and fall 

velocity. Over the transition range of stages, suspension must begin at an earlier stage than that for uniform material of the same mean size, the smaller 

materials being suspended first.

o.oi 0.02 0.03 0.04 0.06 0.08 0.1 0.2 0.3 0.4 0.6 

GRAIN SIZE, IN MILLIMETERS

0.8 1.0 8 10

FIGURE 8. Theoretical values of the suspension criterion 9t= Vt/gD together with Shields' (1936) values of 0 at the threshold of bed movement.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 117

An interesting and informative test of the generality 

of equation 18a is afforded by applying it to the atmospheric transport of solids. From the derivation given, 

equation 18a can be written as

F2=1.5r/p

where p is now 1.2X10"3 grams per cubic centimeter.

If T is taken as 10 dynes per square centimeter for 

a reasonably strong ground wind over an open sand 

surface, then F=1.14 meters per second. And this is 

the fall velocity in air of 0.14-mm quartz grains. Hence 

suspension should not be appreciable in such a wind, 

for grains larger than say 0.2 to 0.25 mm in size.

We should expect, therefore, that the wind would 

carry away grains smaller than this, scatter them over 

wide areas, and leave behind local dune accumulations 

having a dominant size never less than 0.2 mm (except 

in specially sheltered spots).

This is precisely what is found. The finest dune 

sand found on open country has a dominant size seldom 

less than 0.2 mm usually 0.25 mm.

THE EFFECTIVE FALL VELOCITY V FOE SUSPENSION 

OF HETEROGENEOUS SEDIMENTS

The fall velocity V enters as a factor of the work rate 

m'sgV (equation 4) required to maintain the suspension 

of an excess weight m'sg of solids. It follows that if 

the suspended solids are heterogeneous in size, the work 

rate is given by the arithmetic mean of the fall velocities

(19)

where p is the weight of any individual constituent 

grade whose fall velocity is Vp .

The effective mean fall velocity V of a heterogeneous 

suspension (equal to 2(pVP) when 2p is made unity 

and p becomes a numerical proportion) may be so 

much larger than the fall velocity of a solid of size 

DBO percent °n the conventional cumulative diagram that 

it cannot always be assumed equal to it even as a first 

approximation. It must be remembered that the 

cumulative diagram is merely an arbitrary device for 

representing the size distribution and takes no account 

of any physical effect of grain size as such. It is 

simply Up plotted against DP .

V=2(pVp) is comparable to the arithmetic mean 

grain size

But since Vp is a varying function of Dv , V can only 

be equal to YD* over the narrow range of sizes within 

which Vp is approximately proportional to DP that is, 

around .D0 =0.6 mm for quartz-density grains in water.

Some idea of the magnitude of the discrepancy 

between V=2pVp and VD50 percent can be got from the 

following examples, taken from U.S. Geological Survey 

analyses of suspended river sediments.

Rio Grande at Otowi Bridge, N. Mex. * (5-12-48)

Size limit 

(mm)

4.. .......

2.........

1. . .......

.5.......

.25......

.025.....

.0625.... 

.0156.... 

.0039....

Total...

Percent 

finer

100

98.8

94.0

87

78

58

42 

20 

13

P

} 0.012

} .048

}.«

} .09

} .20

}. 

} .22 

}.07 

2.12

1.00

DP

(inm)

3.0

1.5

.71

.35

.176

.088 

.0312 

.0078

pDv (mm)

0.036

.072

.050

.032

.035

.014 

.0069 

.00054 '

Vf (cm 

per sec)

24.0

22.0

10.0

5.0

1.9

.6 

.08 

.005

j>Vv (cm per sec)

0.29

1.06

.70

.45

.38

.096 

.0176 

.00035

sec

1 D 50 percent =0.088 mm, and VD co percent would be 0.6 cm per sec. Whereas, 

V=3 cm per sec is five times bigger. Dv=D0 =0.245 mm, corresponding to approximately D so percent. The three biggest grades together constitute only 6 percent of' 

the suspended material, but contribute 66 percent to the value of V.

2 Fines neglected.

Elkhorn River at Waterloo, Nebr. : (S-26-62)

Size limit 

(mm)

0.5.......

.25

.125.--..

.0625....

.0312....

.0156....

, .0073....

.0039....

, Total

Percent 

finer

100

96

77

62

49

28

24

18

P

J0.04

}. 

.15

.13

.21

[ .04

<

\ .06 1

2.18

1.00

Dv

(mm)

0.35

.176

.088

.044

.022

.011

.0055

pDp (mm)

0.014

.0335

.0132

.0057

.0046

.00044

.00033

Da =0.0718 mm

VP (cm 

per sec)

5

1.9

.6

.16

.04

.01

.0025

pVj, (cm per sec)

0.2

.36

.09

.0208

.0084

.0004

. 00015

V- 0.68 cm per

sec

1 D 50 Poroent=0.033 mm, and VD 50 percent would be 0.09 cm per sec. Vis 7.6 

times bigger. Dp=0.092 mm, which Is 1.3 times Da . Dv corresponds to approximately D 70 percent.

2 Fines neglected.

Rio Puerco near Bernardo, N. Mex. 1 (8-10-59)

. Size limit 

(mm)

'1.0 -

.5.-- .

.25. . 

.125--. 

.0625. 

, .0312.... 

.0156.... 

.0078-.. 

.0039....

Total...

Percent 

finer

100 

99.8 

99.2 

95.1 

86. 8 

71.5 

61.4 

51.4 

50.0

P

} 0.002 

} .006 

} .041 

} .083 

} .153 

} .101

} .014 

2.50

1 00

(mm)

0.71 

.35 

.176 

.088 

.044 

.022 

.011 

.0055

pDp (mm)

0. 00142 

. 00210 

. 00720 

.00555 

.00670 

.00220 

.0011 

.00007

Da =0.026 mm

Fp (cm 

per sec)

10 

5 

1.9 

.6 

.16 

.04 

.01 

.0025

pVp (cm per sec)

0.02 

.03 

.078 

.049 

.0244 

.0040 

.001 

.000035

F=0. 206 cm per 

sec

1 I>60peroent=0.0039mm, and VDSO percont would be 0.0013 cm per sec. Fis 160 times 

bigger. Dp=0.016 mm. which is smaller than D0 . Dv corresponds to approximately 

Z>75 percent. The four biggest grades together constitute only 5 percent of the suspended material, but contribute 60 percent to the value of F. 

2 Fines neglected.
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

The comparatively large discrepancies between V 

and VD5o percent *& the above examples refer to wide size 

distributions, and to those which extend well into the 

Stokes law range of sizes. Naturally, the discrepancy 

diminishes with decrease of distribution width (see 

following example).

Rio Grande at Cochiti, N. Mex. 1 (6-17-58)

Size limit 

(mm)

2.0

1.0

.5

.25

.125

.0625

Total

Percent 

finer

100

99

74.1

28.6

4.8

P

.01

.249

.455

.238

.012

J.036

1.00

Dv 

(mm)

1.5

.71

.35

.176

.088

pD (mm)

.015

.177

.158

.042

.0011

V, (cm 

per sec)

22

10

5

1.9

.6

pVf (cm per sec)

22

2.49

2.27

.35

.01

sec

1 Dx percent=0.34 mm, and VDM percent would be 4.7 cm per sec. So V is only 1.3 

times larger. Dp=0.37 mm corresponds to approximately Djo percent.

2 Fines neglected.

The effective mean fall velocity V thus depends both 

on the distribution of p versus D and on the position of 

this relative to the curve of fall velocity versus 

D. It therefore bears no general relationship to

VD5Q percent*

It should perhaps be emphasized here that the effective mean fall velocity V=^,pVp is involved in the 

transport of suspended load only. It is not involved 

in the transport of bedload. Moreover, the summation 

of pVp refers to the actual size distribution of the suspended material. V may be considerably smaller than 

the value computed from the overall size distribution 

of the whole material.

THE EFFECTIVE MEAN GRAIN SIZE D FOB HETEROGENEOUS BEDLOADS

The bedload friction coefficient tan a and the dimensionless bed-stress parameter 6=T/(<r p)gD are both 

functions of D. The questions therefore arise, What 

kind of mean value of D should be taken as representative of the bedload, and on what size distribution 

should this mean be computed?

Since the physical effects of the magnitude of D are 

relevant, the arithmetic mean ^,pDp seems more likely 

to be representative than the arbitrary D5Q percent which 

takes no account of the effects of magnitude. Ideally, 

the size distribution should be that of a sample of the 

bedload itself; but no such sample can be isolated 

experimentally. It seems reasonable to assume the 

closest practicable approximation to be for a river a 

sample of the bed surface material, and for a laboratory 

flame a sample of the whole experimental material.

WASHLOAD

An examination of the final columns of the above 

tabular examples shows that in the first three examples 

the contribution to the total suspension work rate 

made by the whole finer 50 percent of the suspended 

load is respectively 3.8, 1.3, and (by estimation) less 

than 0.4 percent.

These values appear to give the so-called washload 

a rational physical meaning. It is merely that part of 

the suspended load whose contribution to the total 

suspension work rate is relatively negligible. If relative 

negligibility is placed at 1 percent, the distinguishing 

grain sizes hi the above three examples would be 

respectively 0.08 mm, 0.02 mm, and 0.07 mm approximately.

Further, the empirical definition of washload as fine 

suspended material which is not present on a riverbed 

in appreciable quantity is also consistent with the 

foregoing ideas. As can be seen from figure 8 material of this order of size is so readily maintained in 

suspension that it would be unlikely to become deposited even at low river stage anywhere other than in 

stagnant backwater.

It might at first sight be concluded that since the 

finer grades of a suspended load make negligible contribution to the suspension work rate, the transport 

rate of the suspended load must be indeterminate 

But this conclusion would be wrong. Suppose we 

were to double the transport concentration C and 

thereby the transport rate by adding very fine material, which has so small a fall velocity that it contributes 

nothing to the suspension work rate. The effect of this 

addition would be to halve all the frequencies p of the 

larger contributing grades. Hence ^,pVp is halved. 

So, by equation 8 the predicted transport rate would 

be doubled, as required by the original supposition.

THE FINAL TRANSPORT RATE RELATIONSHIP

We can now amplify the outline relationship in 

equation 9 to give it a sufficiently practical form to 

: permit a trial comparison with the measured data from 

both flume experiments and natural rivers. 

' Adequate flow depth and fully turbulent flow being 

assumed, equation 9 reduces to

£=co

\tan a

+0.01 p, (20)

a p

: where i - j=Q.62 times the conventional transport ff

irate by dry weight for quartz-density grains in water; 

co=stream power pdSu in units consistent with i; e^ is 

given in figure 3; tan a is given in figure 4; V== effective 

fall velocity SpFP for the suspended material; and
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 119

the coefficient 0.01 against the second suspension term 

is the theoretical suspension efficiency 0.015 reduced 

by the factor 2/3 on account of the stream power 

already dissipated in bedload transport.

For mean bed grain sizes Da less than 0.5 mm, e 6/tan 

a can as a first approximation be taken as 0.17 for 0 less 

than unity.

Equation 20 can of course be written as

(r^+0-01 ~ 

\tan« V

(20a)

in terms of the whole width of the flow.

If C' =0.62(7 is used for the transport concentration 

by immersed weight, equation 20 may be put in the 

alternative form

' =S

or

(206) 

'

u

The ratio

' /' _susPen(fed transport rate_- ... u tan a , . 

* bedload transport rate ' T7"

u This equation is approximately equal to 0.06 ^ f°r

grains smaller than 0.5 mm.

This ratio should increase with decreasing grain size. 

For a given overall size distribution it should also 

increase with increasing stage, but this increase may 

be masked by a progressive increase in V as larger 

grams are brought into suspension.

Inadequate flow depth, however, should cause the 

bedload rate i& to increase disproportionately with 

increasing stage, by a factor of 3 or less.

on. i j 4.- suspended load ,,,-,, .. c The load ratio ^ TJ -5 > that is, the ratio of bedload

the quantities of suspended and unsuspended solids 

present over unit bed area, appears to be considerably 

smaller. The suspended load is isfu and the bedload 

is i6/C7B, where C76/iZ=e 6 , which is of the order of 0.13 

(fig. 3). The load ratio is therefore

(22)

Clearly, if the bedload travels slower than the suspended load the quantity of bedload present must 

be correspondingly greater. This emphasizes the need 

to distinguish between transport concentrations and 

spatial concentrations. The present looseness of definition tends to a confusion of thought.

Equations 20, 20a, and 206 can be written in the 

alternative dimensionless forms

(20c) = .:_=_:_ __ ° I Q 01

co u u S u tan au

The equivalent terms on the left are now the proportion 

of stream power expended in sediment transport if the 

whole of the load were suspended. This approximates 

to reality when the bedload term on the right is small

compared with 0.01, that is, when V/u is small.

i "V If we neglect the bedload work, the proportion - 

CO U

of stream power expended in sediment transport is 

constant, as was suggested by Kubey (1933, p. 503). 

Kubey's values for this proportion, inferred empirically from river data, were rather larger, around 0.025 

instead of 0.01. This is understandable because the 

bedload work element was not taken into consideration 

and because the effective fall velocity V was estimated 

in a different way.

EXISTING FLUME DATA

LIMITATIONS AND UNCERTAINTIES

Very few of the many laboratory transport-rate 

measurements extend to the higher stages with which 

we are here concerned and which seem to be prevalent 

hi many rivers. Although now half a century old, 

Gilbert's (1914) work still provides the majority of 

the available data.

This data deficiency may be attributed in part to 

a lack of appreciation of the real conditions the experiments were aimed at reproducing; but it seems to be 

due mainly to two outstanding experimental weaknesses: (a) underestimation of the pumping capacities 

required, and consequent inability to maintain adequate 

flow depths at the higher stages; and (6), resulting 

from a, increasingly serious disturbances of the flow 

associated with unnaturally high Froude numbers at 

inadequate flow depths.

Several uncertainties arise in the interpretation of 

the few relevant sets of data available:

1. A large scatter exists within most sets of data. No 

serious investigation has been made into the cause. 

So it has remained uncertain how far this scatter 

may be systematic. A correlation with variations 

of flow depth has long been known qualitatively 

to exist, but unfortunately 

2. We have no reliable way of estimating the proper 

reduction factor to be applied in the evaluation of 

the tractive stress r and of the available power co 

to make allowance for ineffective wall drag in 

rectangular flumes.
120 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

3. Although in some sets of data the Ao percent grain 

size of the total transported solids is given, 

together with that of the whole stock of experimental material, the size distributions of the 

transported solids are omitted in all sets of data. 

Further, even were the transported distributions 

to be given, it would be impossible in such shallow 

flow depths to make any reliable estimate of the 

size distributions of the suspended loads as distinct 

from the bedloads. Hence the effective fall 

velocity V of the suspended loads cannot be 

determined with any accuracy.

In this respect the data obtained from natural 

rivers is likely to be more relevant. For over the 

greater prevailing flow depths the material 

collected by a sampler and its size distribution 

have reasonable likelihood of representing the 

suspended load alone.

Similarly, no information is available as to the 

mean size of the bed-surface grains as an indication 

of the mean bedload size in flume experiments. 

Hence any estimate of the effective value of the

7*

bedload criterion 6=-, r r> is open to some (v p)gD *

doubt. Since the data show that in general the 

- mean transported size is smaller than that of the 

bed stock, it follows that the bed-surface size 

must be correspondingly larger. That is, the 

effective value of 6 is likely to be smaller than that 

given by equating D to the measured mean stock 

size.

WALL DRAG AND FLOW DEPTH

I spent much time working through Gilbert's and 

other data in an attempt to discover what deduction 

factor applied to the tractive stress r and the power w, 

to allow for wall drag, would minimize the scatter in 

the plots of transport rate against w. The factor which 

minimized the overall scatter throughout the range of

. , , , , bed width m, .

stages was found to be T-^J .,., , , . , -=- This b bed width+1.6 depth

factor made a very marked improvement in the general 

alinement of the plotted points. 

However, two significant facts emerged:

1. This reduction factor minimized the scatter in 

Gilbert's data not only over the range of sand 

sizes but also for the 5-mm pea gravel. Such a 

large allowance for wall drag for so rough a bed 

relative to the smooth flume walls appears to be 

unacceptable. Similarly for the finer grades, so 

large an allowance is also unacceptable over the 

lower, transitional stages in view of the much 

larger form drag exerted by the bed features.

2. In spite of this large adjustment of the omega values, 

the plotted points for the smallest depths (less 

than 0.1 ft) remained persistently discrepant. 

The nature of the discrepancy is clear from figure 

$B, to which the points corresponding to these 

flow depths have been added as an example. 

It can be seen that the discrepancy would have 

increased with increasing stage had .the smallest 

flow depths been maintained.

The explanation is immediately apparent from 

what has been said under the heading "Effect of 

inadequate flow depth on the values of e 6 ." Just 

such a discrepancy is predicted by the theory; 

and indeed the theoretical prediction of the bedload efficiency et> would have been invalidated by 

its absence.

This explanation removes any need to make an 

objectionably large allowance for wall drag. It now 

becomes reasonable to assume the relative wall drag 

to be considerably smaller for most experimental width 

to depth ratios.

On the other hand, the impression gained from the 

analysis of Gilbert's data, which includes some very 

small width to depth ratios, is that relative wall drag 

begins to increase rapidly when the width to depth 

ratio is made smaller than, say, 3.

These considerations have led me (a) to discard all 

Gilbert's data relating to the smaller flow depths on 

grounds of inadequate depth and to discard also all 

his data concerning width to depth ratios less than 

approximately 3; and (b) to plot the remaining measured 

transport rates against the power co without making 

any allowance for wall drag. From the other, more 

recent experiments, the data, which all lie within 

these limits, have been plotted in their entirety.

Adequate flow depth being assumed measurable in 

terms of grain diameters, it would seem likely that 

Gilbert's entire data relating to the 5-mm pea gravel 

should be discarded on the ground of inadequate flow 

depth (around 14 diam only). I have however included 

these results for the sake of comparison.

Appreciable errors may well be introduced both by 

ignoring the effect of wall drag altogether and by assuming flow depths of 0.2 feet to be sufficiently large 

in relation to the effective thickness of the bedload 

zone, over the sand range of grain sizes, to validate 

the theory.

It seems to me strange that although research in 

rectangular flumes has been done for the past 50 years, 

no serious steps have been taken to clear up these 

uncertainties.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 121

ESTIMATION OF APPROPRIATE FALL VELOCITY V

As previously pointed out, the experimental data 

show that in general the D^ percent size of the total 

transported material, bedload plus suspended load, is 

considerably smaller than that of the whole bed stock. 

Hence the effective fall velocity V of the suspended 

load should also be smaller than that computed from 

the bed-stock data.

As a systematic approximation I have therefore assumed arbitrarily, in the absence of any factual information, that V has one-half the value given by 2pVp 

for the size distribution of the bed stock.

COMPARISON OF THE THEORY WITH THE EXPERIMENTAL FLUME DATA

In the following figures 9 to 13, the measured transport rates i are plotted against the power u=pdSu. 

The plots are arranged in descending order of grain 

size.

Both the quantities i and « have the dimensions of 

energy rates per unit of boundary area. The units 

used for both are pounds-feet per second-foot2, equivalent to pounds per second-foot.

Superimposed are comparative plots of the theoretical values of i predicted according to equation 20. 

This equation, it should be remembered, was derived 

without recourse to information obtained from any 

flume experiment.

In plotting the factual data I have attempted to 

distinguish between reportedly "flat" or "smooth" 

beds and "dune" or "sand-wave" beds, and between 

these and "antidunes" occurring at the unnaturally 

high Froude numbers obtained in small-scale experiments. The difficulty here is that one condition 

merges gradually into the next, which makes the 

reported distinctions a matter of personal judgment. 

Reported "transition" conditions have been referred to 

the next lower condition. Thus "transition" from 

dune to flat is plotted as dune, and that from flat to 

antidune as flat.

The antidune condition would not have intervened 

had the experimental flow depths been larger, and it 

may be assumed, I think, that the beds would have 

remained flat. I have distinguished the antidune 

condition merely as an indication that the violent 

disturbances of the flow are likely to have rendered 

the measurements less reliable.

To complete the information, the corresponding 

values of u and 6 are plotted in the following illustrations. In the absence of any information as to the 

actual mean sizes of the grains in transit close to the 

bed surface, the bedload grain size D in the denominator of 5 has been given the value corresponding to

the overall mean size of the material stock. The 

actual size D in 0' may well be larger by a factor 

approaching 2.

GILBERT (1914) 5.0-mm PEA GRAVEL 

Figure 9,4

In terms of 6 the highest stage reached is only 0.21, 

whereas according to figure 4 the critical stage should 

not have been reached until 5=0.26. Hence this 

whole plot would seem to lie within the lower, transitional stages.

According to figure 8 it seems very likely that 

suspension would be negligible, the entire transport taking place as bedload. In consequence, the predicted 

values of i have been calculated using the first term only 

of equation 20.

The flow depth being around 14 grain diameters only, 

it would certainly have been inadequate at the higher 

stages. One would therefore expect the measured 

transport rates to have been too large by a progressively 

increasing factor, consistent with the plot.

GILBERT (1914) 0.787-mm SAND 

Figure 9£

Depth range 75 to 110 diameters Approaches adequacy? Max 5=0.8. Suspension possibly approaching 

ing full development (fig. 8). F=0.18 ft per sec. 

Average flow velocity u=3.5 ft per sec over the higher 

stages. Ratio of suspended to bedload transport 

rates (ig/ij>) approx 0.86 according to equation 21. That 

is, bedload transport rate still exceeds that of suspenT j 4.- suspended load i» _ , _ sion. Load ratio *, j eb =0.15. bedload \

Critical stage: predicted from figure 4, 5=0.4. From 

plot, 6 between 0.3 and 0.5, as indicated by onset of 

flat beds, and between 0.4 and 0.5, as indicated by 

change of trend in data plot.

GILBERT (1914) 0.507-mm SAND

Figure 1QA

Depth range 120 to 200 diameters. Probably adequate. Max 5=1.2. Suspension probably fully develop 0d (fig. 8). (F=0.13 ft per sec. Ratio is/ib=approx 

1.6 over the higher stages. Suspension beginning to 

dominate. Load ratio 0.29.

Critical stage: predicted from figure 4, 5=0.5. From 

plot, flat beds, 5 between 0.4 and 0.7; change of trend, 5 

between 0.5 and 0.6.
122

10

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS 

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POWER (w), IN POUNDS PER SECOND FOOT

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FIQDEE 9. Comparative plots of theoretical and experimental transport rates (flume experiments). A, Gilbert 5.0-mm gravel. B, Gilbert 0.787-mm sand.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 123

1.0 10' 10" 2 34 6 8 10" 234 68 2 34 68 10""' 2 34 6 8 1.0

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 3-

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Width 8 ft xx

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Predicted by equation 20

 U, IN FEET PER SECOND

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POWER (u), IN POUNDS PER SECOND FOOT

Fio UKE 10. Comparative plots of theoretical and experimental transport rates (flume experiments). A,G ilbert 0.507-mni sand. B, Simons, Richardson, and Albertson

0.45-mm sand.
124 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

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POWER (co), IN POUNDS PER SECOND FOOT

23468 10' 1 2 34 6 8 1.0

FIGURE 11. Comparative plots of theoretical and experimental transport rates (flume experiments). A, Gilbert 0.376-mm sand. B, Gilbert 0.307-mm sand.
TRANSPORT RATE (i ), IN POUNDS PER SECOND FOOT (IMMERSED WEIGHT)

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126 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

SIMONS, BICHABDSON, AND ALBEBTSON (1961) 0.45- 

mm SAND

Figure 10B

Depth range 140 to 700 diameters. Adequate? No 

evident correlation with flow depth, in spite of the wide 

depth range. However, there is an unfortunately large 

uncorrelatable scatter. This is probably due to too 

infrequent measurements of the transport rate during 

the rims. The scatter is noticeably greater over the 

lower, transitional stages, when large long-period 

fluctuations in the discharge of solids from the flume are 

to be expected as alternate dunes and troughs pass the 

end of the flume. Significantly, the transport-rate 

measurements were made by periodic sampling at the 

immediate exit from the flume. Max 0 1.2. F=0.1 

ft per sec. Average ratio isjih=2.2 over the higher 

stages. Load ratio 0.4.

Critical stage: predicted from figure 4, 0=0.5. From 

plot, flat beds, 0=approx 0.5; change of trend, 0=approx 

0.5.

QILBEBT (1914) 0.376-mm SAND 

Figure 11 A

Depth range 160 to 240 diameters. Max 0=1.5. 

y=0.086 ft per sec. Average ratio is/ih=2.3. Load 

ratio 0.42.

Critical stage: predicted 0=0.5. From plot, flat beds, 

0= approx 0.6; change of trend, 0 between 0.5 and 0.6.

QILBEBT (1914) 0.307-mm SAND

Figure 11B

Depth range 200 to 300 diameters. Max 0 = 1.1. 

V=0.066 ft per sec. Average ratio /i/i& =2.6. Load 

ratio 0.47.

Critical stage: predicted 0=0.5. From plot, flat 

beds, 0 between 0.5 and 0.6; change of trend, 0 between 

0.6 and 0.7.

.BABTON AND LIN (1955) 0.18-mm SAND

Figure 12.4

Depth range 500 to 2700 diameters. Max 0 = 1,4. 

V=0.035 ft per sec. Average ratio is/i6 =4.8. Load 

ratio 0.86.

Critical stage: predicted 0 = 0.5. From plot, flat 

beds, 0 between 0.5 and 0.6; change of trend, 0 between 

0.5 and 0.7. Inexplicable scatter.

LAUBSEN (1957) 0.11-mm SILT 

Figure 125

Max 0 = 1.6. V- 0.0175 ft per sec. Average ratio 

i,/i& = 6.1. Load ratio 1.1.

Critical stage: predicted 0 = 0.5. From plot, flat 

beds, value of 0 uncertain; beginning of trend coincidence, 0 between 1.1 and 1.4.

GENEBAL COMMENTS

The mutual consistence between the experimental 

results is remarkable in view of the differences in the 

experimental conditions different experimenters and 

methods of measurement, different apparatus from 

nonrecycling to recycling flumes, and different grainsize distributions.

There is also a surprising general agreement, all 

differences being within the limits of experimental 

error, between these results and those predicted by the 

present theory, over the 50-fold range of grain sizes 

covering nearly the whole range of transport modes 

from transport as bedload alone to transport in which 

suspension greatly predominates.

The scatter tends to obscure evidence of the critical 

stage at which the theory becomes operative. This 

stage appears to be predictable to a fair approximation 

by a critical value of the bedload criterion 0 between 

0.5 and 0.6 for Gilbert's data. But these data refer 

to narrow grain-size distribution. Other plots suggest 

that the critical stage occurs at considerably larger 

values of 0. However, this discrepancy may well be 

apparent rather than real. For with a wider size distribution the finer grades tend to be removed into 

suspension, leaving the bedload consisting of grades 

appreciably coarser than the mean stock size. Thus, 

were 0 to be based, as it theoretically should be, on 

the mean bedload size, its value would be appreciably 

smaller. It would however be unwise to speculate 

further about this until experiment is improved toward 

repeatability by the avoidance of the scatter, and until 

methods are devised for measuring the size distribution 

of the grains in transit close over the bed.

Obviously a comprehensive investigation is needed 

to determine the causes of this unfortunate scatter, 

including the relative effects on the transport rate of 

(a) wall drag at various flow stages and for various 

grain sizes, and (&) the real effective height to which 

the bedload rises at high-flow stages.

Further, for more accurate application of the theory, 

it is evident that more attention should be paid to the 

size distributions of the suspended load, so that the effective value of V can be better estimated.

With regard to the scatter, the plots show that it 

is far narrower in the predicted transport-rate values 

than in the experimentally measured values. Since 

the predicted values are based on the same experimental values of the flow quantities co and u and on a 

systematically estimated value of V, it follows that 

either there were large variations in the real effective 

value of V from run to run, which seems unlikely, or 

alternatively the experimental scatter originated in the 

measurement of the transport rates.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 127

It is significant that the experimental scatter is worse, 

on the whole, for modern experiments done in recycling 

flumes, in which this measurement has to be by sampling, than for Gilbert's experiments done in a nonrecycling flume, in which the transport-rate measurement 

was direct, by integrated weight.

However accurately the sampling method may measure the instantaneous discharge of solids, it cannot 

make accurate measurement of the mean value of a 

spontaneously fluctuating discharge unless samples are

taken systematically over a period covering several 

fluctuations.

EXPERIMENTS BY VANONI, BROOKS, AND NOMICOS

That the scatter which mars most of the experimental results can be greatly reduced by proper experimental design is shown in figures 13 A and B by the 

results of experiments carried out at California Institute 

of Technology (Brooks, 1957).

8 x

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II II III 11 11 11

~ Vanoni and Brooks 0.14 mm Nomicos 0.17 mm

o Depth 0.24-0.3 ft B Depth 0.24 ft 

D Depth 0.53 ft Q Width 0.875 ft o» x 

Width 2.7 ft °OX

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EXPLANATION

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x

~~ Predicted by equation 20

A B

1 I II 1 1 -1 . 1 1 II II

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Depth 0.38-0.67 ft 

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POWER (w), IN POUNDS PER SECOND FOOT

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FIGURE 13. Comparative plots of theoretical and experimental transport rates (flume experiments). A, Vanoni and Brooks 0.14-mm sand, B, Nomlcos 0.172-mm sand,

C, Laursen 0.04-mm silt.
128 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

These experiments differ from all others in that 

efforts were made to maintain constant flow depth, 

that is, to vary the tractive conditions alone, without 

varying the boundary conditions simultaneously.

Figure 13-4 shows the measured transport rates of 

the same sand, D50 perc ent=0.14 mm, at two different 

flow depths of approximately 0.24 and 0.54 feet in the 

same recycling and tilting flume 2.8 feet wide.

Though each of the two superimposed plots is highly 

consistent within itself, there is a systematic discrepancy 

between them. This discrepancy emphasizes the need 

to find out by well-designed and critical experiment the 

answers to the following questions: Is the discrepancy 

due to differences in relative wall drag; or is it due to 

excessive transport of bedload at the smaller depth, 

this depth supposedly being inadequate in the sense 

already discussed; or is it inherent in the arbitrary 

method of estimating the mean flow depth, as a factor 

of w, over a rippled and highly irregular flow boundary; 

or is it perhaps a combination of all three?

In figure 135 are plotted the transport rates of 

another sand of considerably wider size distribution, 

Da=0.172 mm, measured by Nomicos in a narrower 

flume but 0.875 feet wide. The flow depth was kept 

at 0.24 feet.

The results are again consistent, and this consistency 

discloses a clearer picture than before of the abrupt 

change of trend which occurs at the critical stage.

Superimposed, are the theoretical transport rates 

calculated from equation 20 on the same arbitrary 

assumption as before that the effective suspension fall 

velocity V=y2 'ZpVp where the summation is taken over 

the given size distribution for the whole stock of 

material.

RELATION BETWEEN i AND a OVER THE LOWER, 

TRANSITIONAL STAGES

It is not the purpose of this paper to consider conconditions over the lower, transitional stages between 

the threshold of motion and the critical stage. The 

very marked linearity of the logarithmic plots in this 

region is nevertheless noteworthy. It appears in all 

the relevant plots in which transport by suspension 

predominates: figures 105, 12A, 125, ISA, and 135. 

It is particularly striking in the last two figures.

The transport rate i increases as cow , where the mean 

value of n approximates 3. I offer no explanation.

TRANSPORT OF FINE SILTS IN EXPERIMENTAL 

FLUMES

Laursen repeated his own experiments on the 0.11-mm 

sand, using a still finer material having a mean size of 

0.038 mm. The measured transport rates are plotted

in figure 13(7 together with the theoretical rates given 

by equation 20 on the same standard assumption that 

V=yzIlpVP over the size distribution of the stock 

material. On this basis y=0.0047 ft per sec, corresponding to a uniform Z>V=0.04 mm.

As can be seen, the measured rates are too large by 

an order of 10, and this might at first be taken to 

indicate the breakdown of the theory. A study of the 

report, however, suggests that such an anomaly is 

rather to be expected in view of the particular 

experimental conditions.

The solids enter a recycling flume more or less 

uniformly dispersed throughout the flow depth. The 

coarser grades fall to the neighborhood of the bed, 

where under normal experimental conditions they are 

transported along the bed as bedload. At the exit 

they are remixed with the circulating load.

However, at the high transport concentrations prevailing for fine materials having a small fall velocity 

V concentrations approaching 10 percent in Laursen's 

experiments the bed boundary is invisible. So the 

flow has to be stopped to allow the suspension to settle 

out before depth measurements can be made.

This deposition had the reported effect of blanketing 

the bed to the extent that the ripple features were 

partly obliterated. As a result, it seems reasonable 

to suppose, the subsequent transport of the coarser bed 

grains being prevented, the circulating load would 

become progressively finer as the coarser constituents 

were progressively trapped. (It is significant that 

deposition over the first 70 ft of the flume was reported 

to be continuing while transport measurement^ were 

made over the final 20 ft.)

No analysis, unfortunately, appears to have been 

made of the size constitution of the circulating load. 

It may well have consisted mainly of the 15 percent 

of the bed stock which was less than 0.02 mm in size, 

for which the effective fall velocity V would be of the 

order of 0.0007 ft per sec.

The flow velocities being around 2 ft per sec, this 

value of V inserted into equation 20 would give predicted transport rates of the same order as those 

measured.

The theory does indeed become inapplicable when 

the effective grain size is reduced below, say, 0.015 mm, 

as is evident from considerations of spatial concentration. Two-phase flow at very high concentrations is 

beyond the scope of the present paper. It has already 

been discussed in previous papers (Bagnold, 1954, 

1955, 1956).
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 129

THE AVAILABLE RIVER DATA, NATURE, 

AND UNCERTAINTIES

The considered river data consist of 146 sets of records 

taken on various dates at various stations on certain 

rivers in western conterminous United States by the 

U.S. Geological Survey. These sets were selected some 

years ago by L. B. Leopold, before the present theory 

had been conceived, as being most likely to be reasonably reliable.

Each set comprises the flow data, the transport 

rate measured by sampling the flow, and integrated 

size analyses both of the sampled solids transported 

and of samples taken simultaneously from the bed 

surface.

The factual information is in this respect more 

complete and more readily comparable with the theory 

than is the experimental .evidence. Further, both the 

width to depth ratios and the flow depths are much 

larger; so the uncertainties arising from the influence 

of side boundaries, from the likelihood of inadequate 

flow depth, and from Froude number disturbances are 

absent or greatly reduced.

There are, on the other hand, a number of other 

uncertainties.

ENERGY-SLOPE ESTIMATION

In many sets of data all reference to slopes was 

omitted from the original records. So the necessary 

information had to be obtained subsequently from 

map contours. Provided no discontinuities of profile 

at local rapids have been overlooked, and provided the 

river flow is not artificially constricted locally for 

example, by bridge works this map slope may be 

assumed to coincide with the true-energy slope at 

bankfull or flood stage.

In other data sets the recorded slope is the directly 

measured slope of the local water surface. Comparison 

with the map slope shows that in general the slope of 

the local water surface may be smaller at low river 

stages by a factor of 2 or more, owing to local irregularities of bed profile.

Only at one station have the true energy slopes been 

computed from measurements of both water surface 

and bed profiles. Naturally such costly and timeconsuming measurements are unlikely to be made 

unless a need for them is clearly established.

BEDLOAD AND SUSPENDED LOAD

Unlike flume experiments in which it is possible to 

measure the total transport rates directly, the transport 

rates obtained for rivers are derived from samples 

taken from the body of the flow, integrated over the 

cross section. So there is an indeterminate deficiency in

respect to the unsampled transport passing close over 

the bed.

This unmeasured transport is often referred to 

loosely as that of "bedload." Since, however, the 

magnitude of it depends entirely on the inadequacy of 

the humanly devised method of measurement, it is 

unrelated to the magnitude of the real bedload transport as defined by its dynamic mechanism.

As already pointed out, the maximum height above 

the bed to which the real bedload may rise, at high 

flow stages, is not known. So how much of it may 

have been included in the sampling and how much 

excluded cannot be determined. It is known for 

instance that by placing large-scale obstacles on a 

riverbed the local turbulence is increased to the extent 

that a large proportion of the normal bedload is thrown 

up into temporary suspension and can thus be included 

in the sampling. Since at high river stages the bed is 

frequently invisible, there may be undetected obstacles upstream doing the same thing.

In face of this uncertainty I have had to assume, 

as an arbitrary systematic assumption, that the real 

and fictitious bedloads are the same. Consequently, 

the recorded transport rates are assumed to refer to the 

suspended load alone. Consistently, the comparable 

predicted rates have been computed from equation 20 

using only the second, suspension term 0.01 u/V.

Since the first, bedload term e 6 /tan a is nearly 

constant at around 0.18, the effect of ignoring the 

bedload becomes serious only when the suspension 

term is of the same small order. The error may then 

mount to a factor of, say, 1.5 either way. The 

suspension term in many data sets is, however, much 

larger.

The above assumption also introduced possibly more 

serious error. The effective fall velocity V = 2pVP is 

computed directly from the recorded size distribution 

on the assumption that this refers to the suspended, 

load. But the undetected inclusion of even a small 

proportion, say 5 percent only, of the coarser bedload 

may so alter the pattern of the size distribtuion as to 

have a profound effect on the computed value of T>. 

The value of V would be too large, and the predicted 

transport rate may in consequence be several times too 

small. As can be seen from the sample computations 

of V given earlier, the summation is very sensitive to 

small changes in the proportions of the few largest 

grades.

DEFICIENCY OF SEDIMENT SUPPLY

It may well happen, on the other hand for example, 

after a flood stage has removed much of the transportable material from the riverbed that the river transports less sediment than it could if more transportable
130 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

sediment were available. The predicted rates might 

then be considerably larger than the actual measured 

rates.

VALUES OF THE STAGE CRITERION 6

The size analysis of the bed surface being given, 6 

should be capable of direct determination from it by 

putting D=Da = IlpDp . However, some uncertainty 

arises from the fact that, again unlike laboratory conditions, the largest bed grains according to the analyses 

are often greater by a factor of 30 or more than the 

largest transported grains. So it is open to some doubt 

whether or not the whole bed surface can be assumed 

to be mobile.

Again, though the value of 6 is an approximate guide 

in default of a better one, it is not, as pointed out 

earlier, a precise criterion for either the disappearance 

of dune features on the bed or the change of trend in 

the transport rate versus power curves. As is apparent 

from figures 9 to 11, dune features often persist at the 

higher flow stages where the theory agrees with the 

data. Accordingly agreement with the river data is 

found even when dunes are known to be present.

In view of these various uncertainties, many discrepancies, in both directions, must be expected when

comparing the predicted with the measured transport 

rate.

COMPARISON OF PREDICTED WITH MEASURED RIVER 

TRANSPORT RATES

In contrast to laboratory conditions, hi which the 

same stock of sediment material taken from some 

natural deposit of limited size distribution is transported at progressively increasing flow stages, the material transported by a river, together with that of the 

local bed surface, changes from season to season, from 

day to day, and often from hour to hour, according to 

the ever-changing tributary influxes upstream.

A group of records taken at the same gaging station 

on different dates would therefore show little if any 

correlation. So it would be unprofitable to plot the 

transport rates. Instead, each record should be treated 

as though it were the result of a separate and independent experiment done on the transport of a different 

sediment stock.

Accordingly I have tabulated the river records (table 

1) so as to compare the transport rates measured on 

each separate occasion with the rate predicted for that 

occasion from the flow and sediment conditions then 

prevailing.

TABLE 1. Data from 146 individual river measurements comparing measured with predicted transport rates

[Slope: M, map slope; L, measured slope of local water surface; ES, estimated true-energy slope. Basic data compiled by L. B. Leopold from various sources, mostly measure - 

merits by U.S. Qeol. Survey. The compilation included arbitrarily chosen individual measurements to illustrate conditions over a range of discharges. Measuremen t 

stations in the tabulation were limited to those for which complete discharge and suspended-load data were available. Data are available in the flies of the U.S. Qeol. 

Survey, Washington, D.C.]

Date

USQS 

serial 

No.

Discharge 

(cfs)

Depth 

(feet) 

d

Slope 

S

Mean 

velocity 

(ft per sec) 

u

Bed

Grain size (mm)

Maximum 

D

Mean

Da

Bed 

shear 

stress 

0

Suspended load

Fall 

velocity 

V=2pVp 

(cm per sec)

Concentration 

C

Transport rate

Measured 

C' i.

S <a

Predicted

i. 0.01 U

0) V

Discrepancy 

predicted 

measured

BRAZOS RIVER 

Richmond, Tez.

5- 7-57 

4- 3-57 

5-14-57 

5- 9-58 

4-24-57 

10-18-57 10-24-57

1

2 

3

4 

5 

6 

,7

107, 000 

3,460 

59, 000 

31, 000 

31,760 

83,600 

51,300

28.2 

9.5 

26.5 

18.8 

14.9 

27.5 

23.3

1. 59X10-* (M) 

1. 59X10-* (M) 

1.59X1<H(M) 

1. 59XHH (M) 

1. 59X10-* (M) 

1. 59X10-* (M) 

1. 59X10-* (M)

6.83 

2.83 

4.7 

3.96 

5.47 

6.3 

5.0

>16 

16 

8 

16 

4

16

0.4 

.24 

.9 

1.9 

.36

1.1 

.91 

.2 

1.4

.7

0.72 

.083 

.159 

.24 

.122 

.215 

.175

2. 24X10-8 

3. 88X10-8 

3.0 XlO-s 

3. 19X10-' 

6.76X10-3 

7.07X10-' 

4. 24X10-3

8.75 

15.2 

11.7 

12.5 

26.4 

24.8 

16.5

2.9 

10.0 

8.9 

5.0 

13,5 

9.0 

8.9

0.33 

.66 

.76 

.42 

.51 

.36 

.54

COLORADO RIVER (OP THE WEST) 1 

Taylors Ferry, Ariz.

6- 2-56

9- 6-56

9-20-56

10- 6-56

9-15-55

12-15-55

12-29-55

3- 5-56

3-21-56

4- 3-56

5-17-56

5-31-56

8-21-56

248

218

219

220

208

209

210

211

212

213

214

215

216

217

7,900

9,880

6,625

6,400

10,845

4,200

4,760

7,762

7,767

10,733

9,533

7,409

8,480

10,480

8.57

9.41

7.45

7.38

9.9

4.92

5.4

7.02

7.7

9.81

9.25

7.67

7.61

9.7

1. 73X10-* (L

1. 47X10-* (L

1. 73X10-* (L

1. 78X10-* (L

2. 16X10-* (L

3.3 X10-*(L

2. 6 X10-* (L

1.9 X10-*(L

2. 16X10-* (L

2. 07X10-* (L

2. 24X10-* (L

2. 27X10-* (L

1. 87X10-* (L

2. 33X10-* (L

2.63

2.99

2.55

2.49

3.08

2.49

2.53

3.17

2.89

3.12

2.93

2.76

3.17

3.08

2

8

4

2

2

1

1

2

2

2

1

4

16

2

0.4

.64

.8

.45

.35

.31

.17

.2

.19

.37

.19

.36

1.2

.5

0.7

.4

.3

.55

1.15

1.0

1.5

1.25

1.65

1.03

2.04

.9

.215

.85

1.75

1.75

2.83

1.17

1.73

2.0

1.96

1.78

1.93

1.44

2.1

2.26

1.77

1.66

7. 6 X10-5

1. 18X10-*

2. 07X10-*

2. 69X10-*

1. 59X10-* 9.4 XW-s

1.4 X10-*

1. 16X10-*

1. 51X10-*

2. 44X10-*

9. 39X10-8

1. 51X10-*

7. 7X10-8

5.0 X10-5

0.27

.5

.72

.935

.45

1.75

.33

.38

.43

.73

.26

.38

.255

.13

0.45

.515

.44

.675

.53

.37

.39

.53

.45

.64

.42

.37

.88

.57

1.67

1.03

.61

.68

1.16

.21

1.17

1.4

1.05

.88

1.6

.96 24.1

4.4

See footnotes at end of table.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS I 31

TABLE 1. Data from 146 individual river measurements comparing measured with predicted transport r.ates Continued

Date

USGS 

serial 

No.

Discharge 

(cfs) 

Q

Depth 

(feet) 

d

Slope 

S

Mean 

velocity 

(ft per sec) 

a

Bed

Grain size (mm)

Maximum

D

Mean

Da

Bed

shear 

stress

0

Suspended load

Fall velocity 

V=ZpVp

(cm per sec)

Concentration

C

Transport rate

Measured 

C"= i.

Predicted 

f. 0.0117

co" V

Discrepancy 

predicted

measured

COLORADO RIVER (OF THE WEST) Continued 

Below Needles, Ariz.

10-17-56

7-31-56

8-10-55

9-30-55

12- 8-55

1- 5-56

3- 7-56

4-13-56

239

240

250

251

252

253

254

255

8,600

10, 365

12,316

12, 670

3,855

7,024

7,780

10. 357

8.61

10.77

11.69

11.86

5.24

7.73

10.0

10.88

1. 77X10-* (L)

1. 53X10-* (L

1. 96X10-* (L

2. 2 X10-* (L

2. 77X10-4 (L

2. 77X10-*

2. 13X10-*

1. 93X10-*

L)

L)

L

3.02

2.86

3.07

3.17

2.44

2.81

2.3

2.75

2

16

0.5

1

1.5

2

3

8

0.34

.58

.3

.3

1.0

.25

.3

.35

0.94

.53

1.46

1.63

.27

1.6

1.3

1.15

0.893

1.89

2.94

3.02

3.34

2.48

1.8

3.71

3. 79X10-*

1. 13X10-4

1.1 X10-*

2. 44X10-*

2.6 X10-*

1. 92X10-*

1.52X10-*

4.7 X10-*

1.32

.455

.35

.69

.58

.43

.44

1.55

0.67

.455

.31

.5

.22

.34

.38

.31

0.51

1.0

.89

.72,

.38

.79

.86

.2

Lees Ferry, Ariz.

10-16-55

6-25-58

4-14-56

3, 560

33, 100

10, 000

 4.3

18.1

16.6

7.73

2.23X10-3(M)

2. 23 X 10-3 (M)

2. 23X10-3(M)

2. 23X10-8(M)

2.47

5.21

3.55

1

1

6

0.20

.32

.21

.33

9

23

30

11

0.265

1.37

.48

.52

1.01x10-3

6. 25X10-3

1.8 X10-8

3.0 X10-3

0.27

1.74

.5

.835

2.8

1.97

3.24

2.04

10

1.13

6.48

2.45

Grand Canyon, Ariz.

4-12-56

5-31-56

6-17-55

C

K

F

11,300

54,300

27, 110

9.73

20.1

14.6

1.89X10-S(M)

1.89X10-3(M)

1. 89X10-«(M)

4.08

9.02

6.29

0.9

4.0

.5

0.34

.45

.2

10

16

27

0.25

' 1.19 .7

4.57X10-8

9.0 X10-3

8. 88X10-3

1.5

2.95

2.9

4.9

2.27

2.7

3.25

.77

1.06

Palo Verde Weir, Ariz. (1.4 miles below)

5-31-56 

5- 2-56 

5-17-56

8-21-55

9- 6-56 

9-26-56 

10- 5-56 9-16-55 

12-14-55 

12-28-55 3- 6-56 

3-21-56

4-2-56

4-19-56

203 

201 

202

204

205 

206 

207 

241 

242 

243 

244 

245

246

247

12,180 

12, 170 

10, 660

11,860

0,880 

10, 200 

8,440 

9,461 

4,412 

4,174 

6,778 

9,633

12,819

10,900

9.2 

8.9 

8.67

9.74

8.95 

8.8 

8.25 

9.93 

6.87 

6.9 

7.74 

9.25

10.87

9.00

1.6 X10-*(L 

1.13X10-*(L 

1. 47X10-*(L

6.0 XHH(L

1.53X1(H(L 

6.6 X10-KL 

1.27XHH(L 

5.3 X10-«(L 

1.06XNHCL 

1. 13X10-*(L 

1.4 X10-*(L 

1.03XHH(L

1.27X10-*(L

1.2 X10-HL

)

)

) 

)

)

2.87 

2.98 

2.78

2.64

2.41 

2.58 

2.31 

2.93 

2.04 

1.93 

2.66 

3.08

3.27

2.75

16 

16 

>16

>16

16 

3 

4 

8 

8 

8 

>16 

>16

>16

>16

1.2 

1.3

1.6 

0.28 

.22 

2.2 

.63 

.86 

1.5

0.23 

.14

.17 

.57 

.9 

.05? 

.2 

.17 

.14

2.86 

2.11 

2.98

3.6

1.55 

1.43 

1.03 

1.6 

2.0 

2.75 

2.1 

6.47

2.85

1.76

9. 27X10-8 

1. 09X10-* 

2. 14X10-*

2.0GX10-*

1. 42X10-* 

1. 46X10-* 

2. 69X10-* 

7.1 X10-{ 

5.2 X1(H 

1. 12X10-» 

1. 29XHH 

1. 68X10-*

1. 69X10-*

6.3 X10-«

0.35 

.64 

.9

.97

.65 

1.38 

1.3 

.83 

.3 

.062 

.58 

1.0

.84

.32

0.30 

.43 

.29

.70

.48 

.54 

.69 

.55 

.3 

.2 

.38 

.14

.35

.47

0.85 

.66 

.32

.73

.74 

.39 

.53 

.66 

1 

3.2 

.65 

.14

.42

1.46

COLORADO RIVER (OF TEXAS) 

Columbus, Tex.

4-2-57

5-9-57

9-26-57

4-9-59

9-25-57

5- 6-57

9-22-58

10-17-57

9-25-57

5- 4-58

4-27-57

8

9

10

11

12

13

14

15

16

17

18

2,250

13, 000

16, 100

8,500

4, 600

31,800

24,600

36, 700

18, 500

2,460

31,550

5.06

11.34

21.7

6.8

4.5

16.18

11.2

18.2

10.1

13.0

12.86

2.3 X10-*(M

2.3 X10-«(M

2.3 X10-*

2.3 X10-*

[M

[M

2.3 X10-*(M

2.3 X10-* M

2.3 X10-*(M

2.3 X10-*(M

2.3 X10-*

2.3 X10-*

2.3 X10-*

M

M

M)

2.04

2.64

5.34

2.93

2.48

3.93

4.57

3.95

4.12

3.37

5.0

2

16

8

1

2

12

1

16

.5

8

1

0.36

1.17

.5

.3

.35

.55

.34

1.5

.24

.36

.3

0.62

.4

1.9

.98

.55

1.25

1.4

.52

1.8

1.6

1.85

0.030

.23

.92

.89

.08

.59

.2

.15

.83

.41

1.0

1. 57X10-8

4. 48X10-*

1. 94X10-3

1. 68X10-3

1.97X10-3

6. 87X10-*

2. 62X10-3

2. 57X10-3

5.25XHH

2.34X10-3

4,38X10-3

4.25

1.2

5.2

4.65

5.3

1.85

7.0

6.9

14.2

6.2

12.8

20

3.3

1.7

1.0

9.3

2.0

6.8

5.9

1.5

2.4

1.5

4.7

2.7

.33

.22

1.75

1.08

.98 °.86

.104

.39

.12

ELEHORN RIVER 

Near Waterloo, Nebr.

3-26-52 

4- 1-52

4- 2-52 

4- 3-52 

4-18-52 

5- 1-52 

5-15-52 

5-21-52 

6^30-52 

6-26-52 

7- 2-59 

7- 2-59 

7- 2-59 

7- 2-59 

7- 2-59 

7- 2-59

257 

258 

259 

260 

261 

262 

263 

264 

265 

266 

267 

268 

269 

270 

271 

272

2,830 

8,850 

7,860 

6,520 

2,860 

2,150 

1,500 

1,500 

1,820 

6,960 

2,722 

2,272 

2,387 

2,387 

2,240 

2,240

3.07 

6.08 

5.73 

5.46 

2.9 

3.02 

2.84 

2.84 

3.44 

4.38 

3.92 

4.0 

3.74 

3.8 

3.7 

3.78

4.03X10-*(ES) 

3.7 X10-*(ES) 

4.33X10-*(ES) 

4. 75 X 10-* (ES 

3.72X10-*(ES 

4.07X1<H(ES 

4. 38 X 10-< (ES 

4. 24 X 10-* (ES) 

3. 64 X 10-* (ES) 

4. 67 X 10-* (ES) 

2.68X10-*(ES) 

2.68X10-*(ES) 

2. 68 X 10-* (ES) 

2.68X10-*(ES) 

2. 52 X 10-* (ES) 

2. 52 X 10-* (ES)

3.79 

5.51 

5.23 

4.55 

3.68 

2.78 

2.08 

2.14 

2.05 

6.00 

3.02 

2.43 

2.77 

2.26 

2.64 

2.13

0.5

2 

1

8

1

0.075

.29 

.21

1.2

.21

1.33 

.93

.15

.83

0.68 

.48 

.48 

.46 

.59 

.65 

.62 

.43 

.08 

.18 

.12 

.087 

.086 

.086 

.0855 

.079

3.43X10-3 

6 X10-3 

6. 01 X 10-8 

4. 14 X 10-8 

1.52X10-3 

1 X10-8 

6 X10-* 

6, 98 X 10-* 

3. 41 X 10-3 

2. 1 X 10-2 

3. 7 X 10-3 

3. 42 X 10-3 

3. 3 X 10-3 

3. 28 X 10-8 

3. 41 X 10-3 

3. 08 X 10-3

5.25 

10.0 

7.2 

5.4 

2.5 

.95 

.85 

1.02 

6.8 

28.0 

8.6 

7.9 

7.2 

4.75 

8.4 '7.6

1.7 

3.4 

3.3 

3.0 

1.86 

1.27 

1.0 

1.53 

7.5 

10.0 

7.6 

8.4 

9.6 

7.9 

9.4 

8.1

0.32 

.34 

.45 

.55 

.75 

1.34 

1.18 

1.5 

1.3 

.36 

.88 

1.06 

1.33 

1.66 

1.12 

1.07

See footnotes at end of table.
I 32 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

TABLE 1. Data from 146 individual river measurements comparing measured with predicted transport rales Continued

Date

USOS 

1 serial 

No.

Discharge 

«ft)

Depth 

(feet) 

d

Slope 

S

Mean 

velocity 

(ft per sec)

a

Bed

Grain size (mm)

Maximum 

D

Mean 

Da

Bed 

shear 

stress

e

Suspended load

Fall 

velocity

V-SpVp 

(cm per sec)

Concentration 

C

Transport rate

Measured 

C' I.

S CD

Predicted 

I, 0.01 U

w V

Discrepancy 

predicted 

measured

GALISTEO GREEK 

Domingo, N. Mex.

F

E

448 

381

0.71

.77

4. 44 X 10-3 (M) 

4. 44 X 10-3 (M)

4.23 

4.74

16 

8

1.2

.62

0.49 

1.05

0.2 

.11

1.1 XlO-i 

7. 0 X 10-2

15.4 

9.8

6.3 

12.9

0.41 

1.31

RIO GRANDE

Bcrnalillo, N. Mex.

4-25-52 

4-25-52 

5-12-52 

5-12-52 

6-17-52 

6-17-52 

6-20-52 

6-20-52 

6-26-52 

6-26-52 

7-24-52 

7-24-52 

6-18-58 

6-13-58 

6-10-58 

6-25-58 

5-8-58 

6-4-58 

5-13-58 

5-27-58

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

84

2,820 

2,820 

6,440 

6,440 

6,120 

6,120 

4,775 

4,775 

2,800 

2,800 

2,030 

2,030 

4,000 

4,340 

5,800 

6,040 

6, 860 

8,160 

8,320 

10,100

2.94 

1.18 

3.55 

2.09 

3.72 

2.18 

3.44 

2.51 

2.75 

2.42 

2.69 

1.58 

2.76 

2.67 

3.43 

3.40 

3.68 

4.34 

4.46 

4.8

9.0 X10-*(L 

1. 03 X 10-3 (L 

9.6 X10-«(L 

8.8 X10-*(L 

8.6 X10-<(L 

7.5 X10-*(L 

8.4 X10-*(L 

8.6 X10-*(L 

7.9 X10-*(L 

1 X 10-3 (L 

9.6 X10-*(L 

9.3 X1(H(L 

1.14X10-3(L 

1. 15 X 10-s (L 

1.2 X10-3(L 

1.2 X10-3(L 

1.27X10-3(1 

1. 15 X 10-3 (L 

1. 27 X 10-3 (L 

1. 2 X 10-» (L

) 

) 

) 

) 

) 

)

4.14 

3.72 

6.52 

5.4 

5.95 

5.73 

5.04 

5.43 

3.74 

3.11 

2.77 

2.2 

5.06 

6.10 

6.27 

6.5 

6.91 

6.92 

6.88 

7.71

16 

12 

12 

12 

12 

8 

16 

12 

, 8 

16 

8 

12

12 

16 

12 

16 

1 

12 

12

0.32 

.4 

.6 

.6 

.38 

.45 

.62 

.48 

.40 

.73 

.36 

.53

.54 

0.62 

0.44 

.86 

.32 

.48 

.41

1.4

.55 

1.1 

.58 

1.6 

.68 

.88 

.84 

1.0 

.65 

1.35 

.52

1.07 

1.25 

1.6 

1.02 

2.8 

2.2 

2.6

0.72 

.84 

1.38 

.94 

2.21 

1.45 

1.7 

1.77 

1.86 

1.35 

.58 

.5 

1.67 

1.53 

2.87 

2.77 

1.16 

1.64 

1.21 

1.39

3. 32 X 10-3 

2. 82 X 10-3 

3. 71 X 10-3 

2. 84 X 10-3 2.36X10-3 

1. 75 X lO-8 

1. 66 X 10-3 

1. 57 X 10-3 

9. 42 X 10-« 9.46X10-< 

2. 74 X 10-3 1.94X10-3 

5. 98 X 10-8 

2. 08 X 10-3 

3. 48 X 10-3 

5. 41 X 10-3 4.42X10-3 

2. 58 X 10-3 

4. 74 X 10-3 

3. 04 X 10-3

2.3 

1.7 

2.38 

2.0 

1.7 

 1.45 

1.23 

1.13 

.74 

.58 

1.77 

1.29 

3.26 

1.12 

1.8 

2.8 

2.16 

1.4 

2.3 

1.57

1.73 

1.32 

1.42 

1.72 

.81 

1.18 

.89 

.92 

.6 

.69 

1.43 

1.32 

.91 

1.2 

0.65 

.71 

1.78 

1.26 

1.73 

1.66

0.76 

.77 

.6 

.86 

.47 

.81 

.72 

.82 

.81 

1.2 

.81 

1.02 

.28 

1.07 

0.37 

.25 

.83 

.9 

.75 

1.06

San Antonio, N. Mex.

6-26-58

6-19-58

6-9-58

6-11-58

5- 8-58

6- 5-58

5-29-58

6-19-52

5-13-58

5-22-58

85

86

87

88

89

90

91

92

102

103

378

3,080

4,090

4,260

6,180

6,570

8,500

4,850

6,940

7,740

1.21

2.59

3.28

3.26

3.87

4.23

4.98

3.17

4.29

4.64

34.5 X10-<(L)

33.5 X10-*(L)

> 5.0 X10-*(L)

'6.0 X10~<(L)

35.5 XNH(L)

35.5 X10~*(L)

35.6 X10-«(L)

7. 9 X 10-* (M)

« 6. 5 X 10-* (L)

«5.5 X10-*(L)

1.58

5.24

5.44

5.66

6.79

6.61

7.26

5.89

6.81

7.04

8

.5

8

1

1

1

4

8

1

1

0.30

.173

.30

.24

.69

.22

.25

.27

.14

.21

0.34

1.0

1.02

1.5

.58

2.0

2.1

1.74

3.72

2.3

.97

.6

.72

.71

.61

2.15

.8

.83

.91

.54

9. 04 X 10-*

5.38X10-3

4. 62 X 10-3

4. 56 X lO-3

1. 26 X 10-2

6. 07 X 10-3

7. 61 X 10-3

3. 8 X 10-3

1.15X 10-2

9. 28 X 10-3

1.24

9.4

5.7

4.7

14.1

6.85

8.6

3.0

11.0

10.4

0.48

2.6

2.26

2.4

3.3

.92

2.72

2.14

2.24

3.9

0. 39 (* . 68)

.28 0-63)

.4 (<.63)

.51 0.67)

. 24 0 . 34)

. 13 0 . 18)

.320-46)

.71

.2 ^.24)

.370-53)

St. Marcia] flood way, N. Mex.

6-27-58

6-20-58

6-10-58

6-12-58

5- 8-58

5-12-58

6-6-58

5-22-58

5-28-58

93

94

95

96

97

98

99

100

101

1,990

3,810

4,230

4,570

6,170

6,420

6,870

7,710

8,680

2.73

3.62

4.07

4.06

4.97

5.03

5.39

6.23

6.75

«4.5 X10-*(L)

«3.5 X10-*(L)

8 6. 0 X 10-* (L)

» 5. 0 X 10-* (L)

»5.0 X10-*(L)

» 3. 0 X 10-* (L)

' 2. 0 X 10-* (L)

«4.0 X10-*(L)

«4.0 X10-*(L)

3.99

5.6

5.59

5.95

6.43

6.62

6.61

6.48

6.53

.5

.5

.5

1

1

1

.5

.5

.5

0.115

.157

.156

.25

.14

.13

.15

.16

.17

2.0

1.51

2.9

1.53

3.3

2.1

1.35

2.9

3.0

.344

- .36

.55

.50

.36

.3

.51

.33

.38

3. 73 X 10-3

4. 21 X 10-3

4.37X10-3

4. 08 X 10-8

9. 6 X 10-s

7. 83 X 10-8

5. 95 X 10-3

6. 08 X 10-3

5. 45 X 10-3

5.1

7.5

4.5

5.1

11.9

16.1

18.5

9.4

8.5

3.5

4.7

3.1

3.6

5.4

6.6

4.0

5.9

5.2

0.68(<0.95)

. 62 (* 1. 1)

.680.71)

.7 0.88)

.450.57)

.41 0-86)

. 22 0 . 68)

.620.98)

. 61 0 96)

Otowi, N. Mex.

5-12-58

5- 6-58

6- 3-58

6-12-58

6-17-58

6-24-58

5-26-58

276

277

278

279

280

281

282

9,340

7,320

8,590

5,000

2,240

1,130

10, 100

10.23

8.11

8.96

7.19

4.69

1.44

10.21

2. 46 X 10-8 (L)

2.3 X10-3(L)

2.4 X10-3(L)

2. 31 X 10-3 (L)

1. 63 X 10-3 (£,)

1. 31 X 10-3 (L)

2. 35 X 10-3 (L)

7.02

6.84

7.1

5.56

4.23

3.35

7.08

8

>16

>16

>16

16

»16

>16

0.54

2.13

3.51

2.3

1.1

?

1.54

8.7

1.65

1.15

1.35

1.3

?

1.0

3.0

2.37

1.8

2.7

7.2

2.79

1.62

5.02X10-3

7.44X10-8

1.72 X 10-3

1. 11 X 10-3

1.86X10-3

3. 16 X 10-*

2. 99 X 10-3

1.26

2.0

.44

.3

.73

.15

.79

0.7

.87

1.15

.62

.17

.36

1.3

0.5£

.43

2.6£

2.0

.24

2.4

1.6

Cochita, N. Mex.

6-24-58

6-17-58

6-17-58

6- 3-58

5-12-58

5-20-58

6-26-58

283

284

285

286

287

288

289

1,000

2,040

5,060

8,680

8,900

8,920

9,810

1.71

2.24

4.03

4.85

4.09

4.34

4.39

1. 18X10-3 (L)

1. 18X10-3 (L)

1. 13X10-3 (L)

1. 27X10-3 (L)

1.2 X10-8 (L)

1.2 XIO-3 (L)

1.27X10-3 (L)

2.23

3.2

4.22

6.07

6.64

6.51

6.68

8

8

>16

>16

>16

>16

»16

2.1

1.0

2.3

3.5

2.23

5.0

0.18

.5

.37

.33

.42

.2

1.18

5.4

8.1

3.5

1.34

1.33

2.0

2. 47X10-*

5. 24X10-3

8. 84X10-3

3.11X10-3

4.55X10-3

3.11X10-3

3. 11X10-3

0.13

2.75

4.9

1.5

2.35

1.6

1.5

0.56

.18

.16

.51

1.48

1.47

1.0

4.3

.065

.032

.34

.63

.92

.66

See footnotes at end of table.
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 133

TABLE 1. Data from 146 individual river measurements comparing measured with predicted transport rates Continued

Date

USDS 

serial 

No.

Discharge

(cfs)

Q

Depth

(feet)

d

Slope 

S

Mean

velocity

(ft per sec)

a

Bed

Grain size (mm)

Maximum 

D

Mean 

D,

Bed

shear

stress

6

Suspended load

Fall 

velocity

(cm per sec)

Concentration 

C

Transport rate

Measured Predicted 

i. O.Oltf u= V

Discrepancy 

predicted 

measured

RIO GRANDE Continued 

San Filipe, N. Mex.

6-24-58

6-17-58

6-17-58

6-9-58

5- 7-58

5-12-58

6-3-58

5-21-58

5-26-58

290

291

292

293

294

295

296

297

298

1,020

2,200

6,010

5, 120

7,520

8,200

8,590

9,140

9,720

2.17

2.73

4.47

4.49

5.67

5.56

6.0

5.86

6.17

1. 0 X10-» (L)

9. 1 X10-< (L)

1. 51X10-' (L)

1. 51X10-' (L)

1.52X10-' (L)

1. 76X10-' (L)

1. 68X10-' (L)

1.8 X10-» (L)

1. 93X10-' (L)

2.6

4.44

5.99

6.06

6.7

7.19

7.16

7.43

7.53

»16

16

»16

16

(6)

»16

16

»16

»16

1.27

1.0

(«)

2.1

0.36

1.3 («)

0.9

1.33

2.66

5.3

6.5

1.4

2.9

2.3

1.56

1.53

5.77 XHM

1.1 X10-»

2.74 XHH

6.3 X10-'

4.62 X10-'

4.41 X10-»

2.430X10-'

2.79 XIO-'

2.58 X10-'

0.36

.75

1.12

2.6

2.6

1.55

.9

.96

1.16

0.59

.5

.29

.28

1.44

.75

.94

1.4

1.46

1.6

.66

.26

.11

.58

.48

1.05

1.4

1.3

RIO PUERCO

Bernado, N. Mex.

8-10-59

8-26-59

11- 3-59

1

2

3

1,600

2, 010

31.8

2.62

2.64

.97

1. 05X10-'(M)

1.05X10-'(M)

1. 05X10-'(M)

6.58

7.31

1.37

1

2

2

0.29

.33

.24

1.8

1.6

.8

0.20

.22

.0074

1. 41X10-'

1. 65X10-'

4.8 XlO-a

83

97

28.4

10

10

56

0.12

.10

2

SAN JUAN RIVER

Shiprock, N. Mex.

6-31-51 275 6,120 5.49 4.1 XHH (L) 6.3 («) («) («) 1.9 2. 07X10-' 3.12 1 0.32

SCHUYLKILL. RIVER 

Philadelphia, Pa.

9-3-59 256 13, 000 8.08 5.3 XHH (L) 4.9 »16 0.27 4.88X10-« 0.57 6.6 9.6

1 Data from Colorado Biver (of the West) unpublished, from U.S. Bur. of Reclamation, on file in U.S. Qeol. Survey, Washington, D.C.; locations shown on map published 

by the U.S. Inter-Agency Committee on Water Resources (1961).

2 Below critical stage.

' Map slope is 7.9X10-*. Local pool?

« Discrepancy using map slope.

« Map slope is 6.3X10-*. Local pool.

a No bed.

The comparison is given in terms of the dimensionless 

ratio C'IS=i8lu>, where C' is the transport concentration 

by immersed weight and is assumed to refer to suspended load only.

The effective fall velocity V 2pVp is given in 

centimeters per second for easy reference to standard 

values such as those plotted in Report 12 of the U.S. 

Inter-Agency Committee on Water Resources (1958). 

The predicted ratios i3/u=Q.Qlu/V have, however, 

been calculated in consistent units of u and V.

The prevailing values of the stage criterion 6 have 

been added, together with the maximum and mean 

grain sizes of the bed.

In summary of the comparative transport rates 

(table 1), the predicted rates are within a factor of 

1.5, either way, of the measured rates in 49 percent of 

the data sets and within a factor of 2 in 65 percent of 

the data sets. Moreover the overall geometric mean 

of all the 146 discrepancy ratios is as near parity as 

0.73. When all the uncertainties are considered this 

measure of agreement is very promising.

The discrepancies given hi table 1 are analysed in 

figure 14 according to river and gaging station. There

is, as figure 14 shows, a marked correlation b.etween 

the discrepancies and their relevant stations. Although for most of the data the points are significantly 

concentrated close to the parity line, the results from 

some stations, for example, Colorado River of the 

West at Lees Ferry, Ariz., and Colorado River of 

Texas at Columbus, Tex., are very erratic.

It should be remembered that energy effects calculated from data obtained from a single cross section 

of a river are likely to be unrepresentative. In addition 

most of the gaging stations involved were permanently 

sited for the original and simpler purpose of gaging 

water discharge only; for that purpose the effects of 

local energy interchanges are immaterial. The Lees 

Ferry station, for instance, is sited just downstream 

of an abrupt widening of the river channel.

The discrepancies shown above the parity line in 

figure 14 indicate too large a predicted value; these 

discrepancies, however large, can readily be explained 

by concurrent infilling of a local scour, by a general 

dearth of suspendable material, or by the flow stage being subcritical. In many plots where the predicted rates 

are several times too large, the stage as indicated by 9
134 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

COLORADO 

(OF THE WEST)

COLORADO 

(OF TEXAS) RIO GRANDE

10 -

9 -

O

UJ c

<r 5

D

2 .

s 4 v.

a

UJ

Mo

'- ' ' I 0 \

°£

3 JO 

(0 P

£ a

3 "5 

CD 0)

a:

o

. U

FIGDRE 14. Transport-rate discrepancies (predicted/measured), from table 1, grouped according to river and station.

either is below the critical value or may well have been 

had the actual bed-surface conditions been better 

known. Big discrepancies in this direction do not 

occur when 8 is large.

The many large discrepancies in the other direction 

indicate too small a predicted value and are of more 

interest. A search for their probable cause led me to 

inspect the recorded size distributions of the sampled 

material transported, for some common abnormality.

While the sampler is lowered through the river to 

final contact with the bed and is then withdrawn, it 

seems inevitable that some of the coarser bedload grains 

become included in the integrated sample. The proportion in most samples may be small, but hi the presence of large-scale bed features such as boulders, gravel 

bars, and old bridge debris it may be very appreciable.

The inclusion of bedload transported by another 

mechanism, and therefore likely to have a different
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 135

size distribution, should, it seem reasonable to suppose 

be manifested in the combined size distribution.

The conventional integrated percentage-less-than 

method of presentation of size distributions discloses 

but little useful detail. Indeed it is designed to smooth 

out irregularities. When, however, the distribution is 

converted back into its original grade proportions p, 

and these are plotted as log p against log D, distinguishing features can readily be spotted.

Exceptionally low predicted transport rates appear 

to be associated with size distributions having an actual 

or inferable second hump toward the larger end of the 

size scale. Conspicuous examples are shown in figures 

15^1 and B. When these curves are adjusted by the 

removal of the second hump and the appropriate increase of the remaining # values, the summation ^pVv 

is reduced to the extent that the predicted value of 

O.Olw/F is increased to the right order of magnitude.

In some plots the second hump is merged into the 

main one to the extent that its presence is difficult 

to distinguish. Figure 15(7 gives an example. Here 

again, a reasonable adjustment based on the rather 

slight indications is found to remove the discrepancy 

in the transport rates.

It should be a simple matter to test this hypothesis 

by duplicating the sampling. If a sample taken over 

the full depth, to contact with the bed, were to show 

the second hump as in figures 15^1 and B, owing to the 

inclusion of bedload material, another sample taken on 

the same occasion down to say only two-thirds of the 

depth should show the second hump to be absent or 

considerably reduced.

In this connection, although the majority of the 

discrepancies in table 1 are comparatively small, those 

in the direction of too low a predicted value are markedly preponderant. In view of the close agreement of 

the laboratory data, it would seem advisable, before 

attempting to apply some empirical correction to the 

coefficients of equation 20, to investigate the extent to 

which responsibility for this general tendency can be 

attributed to the inevitable inclusion of coarser bedload 

material in the samples of supposedly suspended loads.

Considerably clearer inferences could be drawn from 

distribution graphs such as those of figure 15 had the 

size analyses been made in grade intervals of -ft, 

instead of the wider intervals of 2 and sometimes 4.

Some further points are noteworthy in connection 

with table 1. Although the 8 values shown cannot be 

regarded as precise, they do appear to indicate that in 

the types of river sampled, at any rate, conditions 

usually are those of the high transport stages to which 

the present theory is applicable.

Since the present theory takes both modes of sediment transport into consideration, it should be applicable

to rivers which transport relatively large materials pebbles, gravels, and boulders mainly as bedload. None of these types of river, however, are 

included in the table, for the reason that no data are 

available owing to lack of means of measurement. 

Hence the bedload part of the theory remains untestable 

except by Gilbert's laboratory experiments done at 

clearly inadequate flow depths and over a too limited 

range of transport stages. :

Lastly, the need is evident for accurate determinations of the true-energy slopes of rivers.

COMPARISON OF RIVER TRANSPORT DATA WITH DATA 

FOR WIND-TRANSPORTED SAND

From the viewpoint of general physics, a broad 

theory of the present kind would be expected to be 

consistent with the facts over a still wider field. For 

it is contrary to experience that Nature restricts the 

operation of her basic principles to particular phenomena, such as the transport of solids by a particular 

fluid. If the theory is soundly based, therefore, it 

should be consistent with evidence on the transport 

of windblown sand.

This evidence indicates in the first place that the 

transport of sand as opposed to fine dust over the 

ground is by the bedload mechanism alone, suspension 

by air turbulence being negligible over the range of 

wind speeds commonly experienced. Hence the first 

term of equation 20 should alone be operative.

Owing to the very low dynamic viscosity of ah*, the 

conditions of the bedload motion are wholly inertial; 

so tan a should be constant at its lower value.

The theory would therefore predict that the transport 

rate of a given windblown sand should, to a close 

approximation, be proportional to the available power.

The quantitative data from wind-tunnel experiments 

is definite on this point (Bagnold, 1941; Zinggj 19,50). 

The transport rate is indeed proportional to the power. 

So to this extent the theory is entirely consistent. 

CONCLUSION 

The foregoing theory constitutes an attempt to 

explain the natural process of sediment transport along 

open channels quantitatively, by reasoning from the 

general principles of physics and from the results of 

certain critical experiments. General relationships 

have been derived independently of any quantitative 

data drawn from experiments on channel flow. To 

this extent the theory is rational rather than empirical.

Consideration has been confined to transport conditions at the higher stages of flow. For here the process 

appears to be much simpler than over the lower, transitional stages. An understanding of this simpler process is directly relevant to much of riverflow, and should 

be relevant indirectly to the more difficult problems 

presented by the lower, transitional stages.
136 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0.1

0.01

56 percent unanalysed, but negligible 

contribution to SpVp

0.1

0.01

O 0.00045

0.3

0.1

0.05

0.0009 I 0.0019 0.0039

GRAIN SIZE, IN MILLIMETERS 

0.0078 0.0156 0.0312 0.0625 0.125 0.25 0.5

60 percent unanalysed

0.1

0.01

0.0625 0.125 0.25 0.5 1.0 2

FIGURE 15. Examples of anomalous size distributions of assumedly suspended river loads, associated with anomalous transport rates predicted from them (suggesting

that the sampling had included a proportion of coarser bedload material).

Within the flow region considered, the derived 

relationships are, I believe, fairly and evenly consistent 

with the available facts over a range of conditions far 

wider than for any previous theory.

Wide consistency of this kind tests the general form 

of a theoretical relationship in a way that consistency 

over a narrow range of conditions connot do. If the

general form appears sound, the relationship is worth 

additional study by others who may modify the parameters in the light of further factual knowledge to bring 

the relationship into closer approximation everywhere 

without detriment to its generality.

No theoretical results can, however, be properly tested 

unless the facts against which they are tested are
APPROACH TO SEDIMENT TRANSPORT PROBLEM FROM GENERAL PHYSICS 137

themselves both adequate and certain. If the facts, 

are in doubt, then a fair approximation is all that can 

be expected. Many serious factual inadequacies and 

uncertainties in the existing knowledge about sediment 

transport have already been pointed out. The majority 

could undoubtedly be removed by experimental researches of a critical kind, if these were imaginatively 

and scientifically designed and carried out for this 

specific purpose only, regardless of either convention 

or immediate practical utility. Given adequate facilities, each of the researches I have in mind should be 

capable of completion within a short period of say 2 

years. Continued tolerance of longstanding factual 

uncertainties cannot but have an adverse effect on the 

status of research in this field.

The most serious factual inadequacy in the field of 

sediment transport is, I suggest, our lack of data on 

the unsuspended transport of bedload by a turbulent 

fluid. The reason is of course our inability to separate 

this transport experimentally from a concurrent transport in turbulent suspension. Consequently we cannot 

check theoretically predicted transport rates by either 

mechanism separately. So any theory must extend to 

cover the prediction of both transport rates before any 

verification is possible.

This difficulty remains insuperable so long as the 

belief prevails that bedload, although unsuspended, is 

yet somehow activated and transported by the agency 

of turbulence. The present theory however denies this 

belief, on the direct visual evidence that the saltating 

motion characteristic of bedload transport persists also 

under laminar flow in the entire absence of turbulence. 

The bedload transport relationship it derives is 

applicable both to turbulent and laminar conditions.

This immediately opens the possibility that the 

transport of bedload can be studied quantitatively 

under conditions of laminar flow in a way that is 

impossible under practical turbulent conditions except 

within the narrow range of large grains at very low 

flow stages.

Unfortunately no quantitative experiments on sediment transport by laminar flow have ever been done; 

for on the above belief, such experiments would be 

irrelevent and unpractical. For the same reason even 

simple qualitative experiments have been so rare that 

few if any present-day workers have had opportunity 

to observe the reality of transport by laminar flow 

and the closeness of its similarity to transport by 

turbulent flow.

It is hard to avoid the conclusion that progress in 

the field of sediment transport has been retarded by 

a refusal to appreciate the significance of experimental 

results which may appear superfically to be unpractical.

REFERENCES

Bagnold, R. A., 1941, Physics of blown sand: New York, William 

Morrow.

 1954, Experiments on the gravity-free dispersion of large 

spheres in a Newtonian fluid under shear: Royal Soc. [London] Proc. A 225, 49.

 1955, Some flume experiments on large grains but little 

denser than the transporting fluid, and their implications: 

Inst. Civil Engineers Proc., Pt. 3.

 1956, Flow of cohesionless grains in fluids: Royal Soc. 

[London] Philos. Trans., v. 249, p. 235-297.

Barton, J. R., and Lin, P. N., 1955, A study of sediment transport in alluvial channels: Fort Collins, Colorado State Univ., 

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