Flume Experiments on 

the Transport of a 

Coarse Sand

GEOLOGICAL SURVEY PROFESSIONAL PAPER 562-B
Flume Experiments on 

the Transport of a 

Coarse Sand

By GARNETT P. WILLIAMS

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 562-B

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1967
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 30 cents (paper cover)
CONTENTS

Page

Symbols.__________________________________________ iv

Abstract____________________ __________________ Bl

Introduction. ______--_____-___----_____--_____-___. 1

Equipment and application._________________________ 2

Flume______.______________________________ 2

Water circuit and discharge measurement________ 2

Sediment infeed._______________________________ 2

Sediment-transport measurement_._______-__-_-__ 3

Sediment._________________________________________ 3

Procedure.________________________________________ 3

Initiation of runs and attainment of equilibrium conditions. _____________________________________ 3

Measurements__ _ _____________________________ 5

Page

Results..______________________________________ B6

General__ ___________________________________ 6

Sediment-transport rates_______________________ 6

Relations of variables.__________________________ 13

Surface velocity._______________________________ 16

Comparison with data of Gilbert and Murphy______ 18

Bed forms,____________________________________ 20

Interrelations between characteristics________ 22

Characteristics related to fluid flow strength__ 23 

Characteristics related to sediment-transport

rates._ __________________________________ 25

Conclusions._______________________________________ 28

Description of bed configuration, by run.______________ 28

References.________________________________________ 30

ILLUSTRATIONS

FIGURES 1-16. Graphs:

1. Size analysis of sand_________________________________________-__-__--_------------------ B3

2. Power versus transport relation.__________________________-___-_--_----_------------------- 9

3. Shear (yDS) versus transport relation_____.______________-_--_--__--_-_-_----------------- 9

4. Shear (yRS) versus transport relation_____.______-____-______---_------------------------- 10

5. Mean velocity versus transport relation.__.____ ________________-__-_--_------------------- 10

6. Regime bed factor versus transport relation.______________-__-_---_-------_----------------- H

7. Transport relations adjusted for flume wall drag (Einstein method)________-__--_-_-_----------- 14

8. Discharge versus transport relation.___________________-_____------_------------------------ 14

9. Slope versus transport relation________________________________-_-__------_----------------- 15

10. Relations of slope to discharge and mean velocity._______--_-_-----_------------------------- 16

11. Surface velocity versus transport relation__________________________-_-_------------------- 17

12. V/Va relation for various depths-________-__-__--_____-----_------------------------------- 17

13. Relations of surface velocity to slope and discharge-...____________.__-__-___---_------------- 18

14. Comparison of stream power versus transport relations with data of Gilbert and Murphy.________ 19

15. Comparison of shear versus transport relation with data of Gilbert and Murphy_______--_------_ 19

16. Mean velocity versus shear (D/S) relations compared with data of Gilbert and Murphy _______ 20

17. Sketch showing pattern of typical meandering scars.______________---__------_----------------------- 21

18-25. Graphs:

18. Variation in bed-form height with travel velocity___-______-----_--------------------------- 23

19. Variation in bed-form wavelength with travel velocity-____________-__----__------------------ 24

20. Bed-form height versus measures of flow strength-____-_-----_-----_------------------------- 25

21. Bed-form wavelengths versus measures of flow strength,_______.___-__-_---__----------------- 26

22. Bed-form velocity versus measures of flow strength____-__-_--_-----_-----_------------------- 26

23. Relation of bed-form height to sediment-transport rate_____________._--__-_----------------- 27

24. Variation in bed-form velocity with sediment-transport rate___________----_-_----------------- 27

25. Comparison of computed versus measured transport rates..----------------------------------- 29

m
IV CONTENTS

TABLES

Page 

TABLE 1. Experimental results._______________________________________________________________________________ B7

2. Computed quantities.______________________________________________________________________________ 8

3. Hydraulic radius, shear stress, and stream power corrected for wall drag by methods of Einstein and of Johnson

and Brooks ___________________________________________________________________________________ 13

4. Bed-form measurements.___________________________________________________________________________ 22

SYMBOLS

Symbol Dimensions

a Coefficient in velocity-shear equation_______.____._______________________________ I^I2 T~1

b Exponent in velocity-shear equation.

c Bed-form velocity or rate of travel_______________-_________.._-___--______-_-_- LT"1

C Chezy C____-___________________-_______-____-________-__--_-----_-_-_-_---_- Ll*T-1

D Depth of water_____________________________________________________________ L

f Darcy-Weisbach /.

F Froude number.

g Acceleration due to gravity___-___-____--_____--__-_____-_----_----------_--__- LT~2

h Bed-form height_______________-__-_____-___-__-___-___-__----_-__----_--__- L

i Sediment-transport rate.__________________________________-__--_----_-_-__-__- FT~lL~l

I Bed-form length (crest to crest) ________________________________________________ L

n Manning n____--__-____________-___.___________-____________-_--__-____-____ L1/e

Q Water discharge________________________________________________________ L*T~l

Qb Discharge adjusted to eliminate that part of flow affected by sidewalls______________ L*T~1

R Hydraulic radius,_______________ _____________________________________________ L

Rb Hydraulic radius of bed (a theoretical hydraulic radius excluding that part of flow cross

	section which is influenced by sidewalls) ________________________________________ L

8 Slope (ft per ft).

t Temperature (degrees centigrade).

V Mean water velocity_________-___________________________--__-_____--_-__--___ LT"1

Vs Surface velocity of water______________________________________________________ LT"1

W Width of channel___________________________________________________________ L

7 Specific weight of water (=62.4 Ibs per cu ft)____________________________________ Fir3
SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND

By GARNETT P. WILLIAMS

ABSTRACT

A newly constructed 52-foot laboratory flume was used to 

study sediment transport in a series of 37 runs with a coarse 

(1.35 mm) sand at water depths of 0.1, 0.3 and 0.5 foot. The 

data obtained during these runs show that relations between 

variables can be clarified by maintaining a constant flow depth. 

For the range of conditions examined, unique relationships were 

found between any two variables as long as depth was constant. 

The bed forms ranged from an initial plane bed to antidunes, 

but ripples and a plane-bed transition from dunes to antidunes 

did not occur. The data obtained by Gilbert in 1914, using a 

comparable grain size, agree with the results of the present 

study. One method of determining the critical flow strength 

needed to initiate sediment movement indicates that, with the 

flume and sediment used in this study, the critical flow strength 

is determined not only by such factors as sediment characteristics 

and bed slope but also by water depth. (An exception to this 

last statement was found when shear stress expressed as yRS 

was used as the measure of flow strength.) Surface velocity is 

related to all other variables, particularly to rate of sediment 

transport.

INTRODUCTION

A laboratory flume was constructed in 1964 by the 

U.S. Geological Survey in Washington, D.C., to explore various problems in sedimentology, geomorphology, and hydrology; its initial use is in the field of 

sediment transport. Studies of the movement of 

debris by water aid in the interpretation of numerous 

geologic processes, pollution, filling of dams and reservoirs, and other phenomena. Use of the laboratory 

permits the control of many variables which usually 

cannot be controlled in the natural environment.

Although rivers frequently carry coarse sand and 

gravel, most previous laboratory studies have dealt 

with medium and fine sands, with particle diameters 

less than 1.0 mm. Therefore, coarser grain sand was 

used in the present sediment-transport experiments. 

Many of the available data on transport of coarsegrained sand are from the work of G. K. Gilbert (1914) 

and of E. C. Murphy, who performed most of the 

experiments in the Gilbert study.

All laboratory investigations require a decision as to 

which of the pertinent variables should be controlled by 

the operator. In flume studies, which attempt only to

simulate natural alluvial conditions, the large number 

of possibly relevant variables can usually be limited to 

sediment characteristics (such as size, size distribution, 

shape, density), sediment-transport rate, water discharge, mean velocity, depth, and bed slope and roughness. Other factors, such as water temperature, may 

have some influence under certain conditions. The 

sediment characteristics are usually predetermined by 

the operator and, thus, must be classified as independent 

variables. In the present study, the sediment-transport rates and the water depths (0.1, 0.3, and 0.5 ft) 

were also controlled by the operator. The dependent 

variables, therefore, were water discharge (or mean 

velocity, at constant depth and width), slope, and bed 

roughness. (True independence or dependence of variables in sediment transportation is questionable, for 

the role of each variable in governing other variables 

or merely reacting to changes in other variables is still 

poorly understood. Thus, describing a variable as 

completely dependent or completely independert is 

both difficult and risky.)

The purposes of this report are to describe the ecuipment and procedures used in a series of sedimenttransport experiments, to present the data of the initial 

group of a continuing series of experiments, and to 

examine some of the observed relationships between 

variables. Certain features of these experiments differ 

from most previous flume studies on sediment transport, and the data collected during this study will 

therefore supplement the published information. For 

example, the use of coarser grained sediment and constant depth in a nonrecirculating flume in this study 

will provide new data. Although a constant depth is 

commonly maintained in flume studies in which the 

water and sediment are continuously recirculated, the 

writer knows of no studies involving nonrecircutating 

flumes in which the water depth was held constant.

This investigation was begun under the supervision 

of Luna B. Leopold, to whom thanks are given for 

valuable assistance and suggestions throughout the

Bl
B2 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

study from design and construction of equipment to 

completion. William W. Emmett assisted in planning 

the laboratory and performed a major role in the design 

and construction of the laboratory equipment. Ralph 

A. Bagnold provided valuable suggestions on various 

aspects of the investigation. Additional assistance in 

construction and in some preliminary runs was given 

by O. Lehn Franke.

EQUIPMENT AND APPLICATION

FLUME

The test section of the flume was 52 feet long. The 

maximum usable width was 4 feet, but for the present 

experiments an inner trough 1.0 foot wide was used. 

This trough had transparent plexiglas walls 15 inches 

high and a wood floor.

At a point 2.0 feet from the downstream end of the 

flume the plexiglas walls were hinged vertically. 

Known collectively as the tailgate, these hinged sections could be converged at the end of the flume to 

dam the water to any desired level. This method was 

used to make the water depths as uniform as possible.

On top of each of the outer (permanent) flume walls 

was a rail traversing the entire flume length. On these 

rails rode a carriage to which was fixed a point gage for 

depth measurements. The height of the gage tip relative to the rails was read directly from an adjacent 

scale, to within 0.001 foot.

The slope of the flume was adjustable from horizontal 

to about 0.02 foot per foot. The test section was 

hinged at the upstream end. Two large chain hoists, 

suspended from an independent permanent steel bracing, bore the weight of the flume at the downstream 

end. The desired slope was obtamed by raising or 

lowering the downstream end of the flume, using these 

chain hoists.

WATER CIRCUIT AND DISCHARGE MEASUREMENT

A large sump located below the floor level contained 

the water supply for the flume. From this sump the 

water was pumped to a constant-head tank located 

high on the wall of the building. The water then 

flowed to the flume by gravity through either or both 

of two pipes (one 6 in. in diameter, the other 4 in.). 

The amount of water was regulated by a valve in each 

pipeline. Excess water was allowed to flow from the 

constant-head tank directly back to the sump.

Also placed within each line, between the constanthead tank and the flume, was a bend meter for measuring the discharge. This was a common pipe elbow 

with holes drilled through the wall at the inner and 

outer vertices of pipe curvature. These holes were 

connected by tubing to a manometer, and the difference 

in pressure between the inside and outside parts of the

bend was a direct indication of the discharge flowing 

through the elbow. 1

From the constant-head tank, water passed through 

pipes to a stilling tank (4 ft X 4 ft X 3.4 ft deep) at the 

head of the flume. From this tank it flowed through 

a horizontal preliminary channel 1 foot vide and 4 

feet long, on the same level as the upstream end of the 

flume. In passing through this preliminary channel 

the water was screened by wire mesh to eliminate largescale disturbances in the flow. The water next went 

through a 4-foot-long zone where the sediment was fed 

into the system, and then entered the 52-f oot-long flume 

test section.

As the water-sediment mixture left the flume at the 

downstream end, it was directed into a collection box. 

This box had compartments in which the sediment was 

trapped. The excess water flowed over a lower wall of 

the box and was channeled back to the sump, thereby 

completing its circuit.

SEDIMENT INFEKD

Sediment was fed into the stream by gradually 

raising a supply of sand above the channel level and 

permitting the flowing water to scour the material off 

the rising floor and into the test section. Tl e technique 

used proved to be satisfactory, except for a minor 

amount of sediment leakage.

The sediment infeed area was a bin 4 feet square and 

3.5 feet deep located immediately upstream from the 

flume test section. Within the bin was a false bottom 

which could be continuously raised or lowered by means 

of two screw jacks run by a motor above the bin. 

Reduction gears between the motor and the screw 

jacks and a speed control on the motor provided a 

wide range of rates at which the false bottom could be 

Uf ted. The dial setting on the motor-speed control was 

a direct indication of the rate at which the false bottom 

was being raised.

Two wooden walls, 4 feet long and 1 foot apart, were 

fixed onto the false bottom. These were placed in 

such a position that a straight channel 60 fret long was 

obtained; the last 52 feet was the flume test section, 

and the first 8 feet was the preliminary charnel and the 

wooden walls of the inf eed section.

Sand was loaded into the 4-square-foot section between the walls of the inf eed bin. After the loading, 

the top of the sand was placed at the same level as the 

top of the adjacent sand surface in the flume test section. As the area within the wooden walls and the 

rate at which this area (that is, the false bottom) was 

being raised were known, it was easy to compute the

i Calibration was done at the Hydraulics Laboratory of the Georgia Institute of 

Technology, and the author gratefully acknowledges that departnent's kindness in 

permitting the use of their facilities.
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B3

rate at which sediment (immersed weight per unit time) 

was bein^ exposed to the flow. The dial setting on the 

motor-speed control was therefore directly related to 

rate of sediment infeed.

As the false bottom slowly rose, some sand leaked out 

between the upstream and downstream ends of the 

false bottom and the walls of the bin. During the 

course of the investigation this condition proved impossible to correct and served to detract somewhat 

from an otherwise satisfactory infeed technique. It is 

believed, nevertheless, that the sand was being introduced into the flume at an approximately constant rate 

so that the disadvantage was an inability to state 

exactly what the rate of infeed was at a given dial 

setting.

SEDIMENT-TRANSPORT MEASUREMENT

The water-sediment mixture left the flume and was 

directed into a partitioned collection box just below the 

downstream end of the flume. The sediment settled 

to the bottom of this box, and the excess water flowed 

over the top of one wall and back into the main circuit. For some of the runs, the water-overflow region 

of the collecting box was covered with a fine wire screen 

to insure that no sand escaped from the box. (Little 

difficulty was encountered, however, because the sand 

used in the tests was fairly coarse.)

The measured sediment-transport rate represented 

the total quantity of sand transported, and no distinction could be made between bedload and suspended 

load. The collection box rested on a large floor scale, 

and the amount of sand and water in the box was 

weighed directly at the beginning and end of each run. 

The gain in sediment (immersed weight) and the time 

interval between weighings provided the rate of transport. This rate, accordingly, is expressed in pounds 

(immersed weight) per second per unit channel width. 

During the actual weighing process, the water-sediment 

mixture leaving the flume was diverted into a by-pass 

trough so that the weighing could be done undisturbed.

SEDIMENT

The sediment used in the experiments was nearly 

100 percent quartz sand of a rather narrow size range 

(fig. 1). This range was about 0.60-2.50 mm, in terms 

of sieve aperture, and the median size was 1.35 mm. 

A visual-accumulation-tube size analysis was also performed, in which the sizes are expressed in terms of fall 

diameter (that diameter of a quartz sphere having the 

same fall velocity as a sand grain). The range measured by this method was 0.46-1.25 mm, and the median 

fall diameter was 0.76 mm. The specific gravity and 

porosity of the material were 2.66 and 0.445, 

respectively.

100

80

u- 40

20 -

Visual -accumulationtube analysis

Sieve analysis

0.5 1.0 2.0 3.0 

PARTICLE DIAMETER, IN MILLIMETERS

FIGURE 1. Size analysis of sand.

The fall velocity was determined in a water-filled 

plastic tube that was 1.35 meters long and had an inner 

diameter of 29.6 cm. Particles were timed individually 

over a distance of 100 cm, after allowing 20 cm for the 

period of acceleration. The average rate of fall for 75 

randomly selected grains was 14.5 cm per sec (centimeters per second) at a water temperature of 31.5° C. 

A correction for temperature (U.S. Inter-Agency Comm. 

Water Resources, 1957) would adjust this rate tc 14.2 

cm per sec at 24° C.

PROCEDURE

INITIATION OP RUNS AND ATTAINMENT OF 

EQUILIBRIUM CONDITIONS

To begin a run, the operator first decided upon a 

sediment transport rate and depth (the independent 

variables). An estimate of the probable discharge and 

equilibrium slope was then made, and the flume was 

set at this estimated slope to shorten the time needed 

to reach equilibrium conditions. Sediment was spread 

evenly over the floor of the flume to a depth of about 

inches.
B4 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

The sediment infeed and discharge were then turned 

on. By means of several vertical scales at the flume 

wall a rough estimate of the depth and downstream 

change (if any) in depth was obtained. If the flow 

depth was obviously not within an appropriate range 

(described below), the discharge was altered to change 

the depth. If, on the other hand, the depth appeared 

to be within the proper range, no change in discharge 

was made. The tailgate was adjusted when necessary 

to raise or lower the water level at the downstream end, 

until the depth appeared to be uniform throughout the 

flume.

The next step hi the procedure was a rather detailed 

measurement of depth, slope, and degree of uniformity 

of depth throughout the flume. This was done by 

making longitudinal profiles of both the water surface 

and the bed surface along the center line of the channel. 

Depth readings were taken with the point gage on each 

of these surfaces. The horizontal interval at which 

readings were taken was generally 4 feet, though sometimes 2-foot intervals were used. The readings of 

depth and distance downstream were plotted on arithmetic scales, and straight lines of best fit were estimated 

visually for the bed and water surfaces. If these 

straight-line profiles diverged or converged hi a downstream direction, the tailgate opening was changed to 

make depth uniform. After any change in tailgate 

setting, additional profile measurements were made, 

until the lines on the graph indicated a uniform flow.

The perpendicular distance between the two parallel 

profile lines was taken as the depth of flow. The range 

of depth tolerance was about ±7 percent of the desired 

depth because experience has shown that errors involved 

in depth measurements at the higher manageable discharges did not justify attempts at greater accuracy. 

If necessary, the discharge was adjusted to bring the 

depth into the range of tolerance; however, it was 

frequently unnecessary to change the original setting.

Once the desired depth and a uniform flow were 

established, the slope was allowed to adjust until it 

became stabilized. Changes in slope were determined 

by making profile measurements at intervals ranging 

from 20 minutes to several hours. In many runs no 

change in slope occurred, indicating that the equilibrium 

slope was the same as that of the flume itself. In 

most of the other runs a slight increase in slope, compared to that of the flume, took place.

If the slope changes brought about a decrease or 

increase in water depth so that depth exceeded the 

tolerance range, the discharge was adjusted accordingly. 

Any such alteration in discharge necessitated additional 

profile measurements, possible tailgate changes, and 

periods of waiting for any further slope changes, until 

equilibrium was established.

On a few occasions the slope ceased changing and 

equilibrium appeared to have been reached; the subsequent profiles, however, indicated that the bed was 

aggrading or degrading uniformly throughout the 

flume. Repeat profiles were therefore made in nearly 

all runs to insure not only that the slope had ceased 

changing but that the bed and water sureaces were 

occupying the same spatial elevation with time. The 

bed elevation (relative to the flume rails) at a given 

downstream location had to be within ±3 percent of 

the elevation obtained from the previous profile. The 

criteria for equilibrium, in other words, were (a) constant slope with time and (b) no net gain or loss of 

sediment in flume, with time. Both of these factors 

were deduced from the profiles.

If the profiles indicated that the flume was gaining 

sediment even though the bed slope was not changing, 

this was interpreted as a need for greater flow strength 

to accommodate a given sediment infeed. The usual 

procedure was to open the tailgate. This lowered the 

water depth downstream and caused scour, eo that the 

overall bed slope was increased. Conversely, if the 

flume was losing sediment in spite of a constant slope, 

the tailgate was closed. Usually when this problem 

occurred, the sediment infeed became depleted before 

equilibrium conditions could be established, and a new 

run had to be made. The usefulness of the attempted 

procedure for arresting bed-elevation changes at constant slope was therefore not conclusively determined.

Most instances of bed-elevation change at constant 

bed slope occurred in runs at the 0.1-foot depth; in 

fact, it was difficult to obtain equilibrium conditions 

at this depth, especially at lesser sediment transport 

rates. (Rate of sediment supply of course h as a direct 

influence on time required for slope readjustments.) 

At the 0.1-foot depth, the water-surface profile was 

particularly sensitive to minor changes of the tailgate 

and discharge. In a few runs, many days were needed 

to attain equilibrium at a depth of 0.1 foot. Discharge could generally be adjusted rather e^rly in the 

experiment, whereas the correct tailgate setting often 

required considerable time and trial-and-error methods. 

After some points on the sediment-transport graph 

had been obtained, the process was somewhat easier, 

because approximate tailgate settings could be extrapolated before the water was turned on. It was 

necessary, therefore, to establish a consistent experimental procedure. The writer suspects that the correct tailgate setting can be influenced by the depth to 

which the sediment is laid down on the flume bottom 

prior to a run.

Generally, the time needed to reach equilibrium in 

the experiments varied from about one half hour to 

many days. (This excludes nights, at which time
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B5

the pump was always stopped.) Often a run could be 

completed in a single day or less. The following 

factors greatly facilitated the attainment of equilibrium:

1. A high rate of sediment transport (a high value of 

the fluid flow strength).

2. Depths of 0.3 and 0.5 rather than 0.1 foot.

3. An accurate estimate of the flume slope setting.

Secondary factors which could expedite attainment of 

equilibrium were an appropriate initial choice of water 

discharge to provide the desired depth, and good judgment in determining the tailgate opening. By far the 

greatest hindrance to a prompt attainment of equilibrium was a shallow depth (0.1 ft) or some feature 

associated with this depth.

MEASUREMENTS

As soon as equilibrium conditions had been realized, 

the run was started. This involved a measurement 

of the following:

1. Discharge, as read from the manometer. This was 

recorded at the start and finish of each run.

2. Water temperature, also measured at the start and 

finish of each run.

3. Slope:

(a) The slope of the flume (or more precisely, that 

of the flume rails upon which the point gage 

rode) was determined by setting up a surveyor's 

level independent of the flume structure and 

measuring the vertical drop in a known distance downstream. The average of two measurements was taken as the final value.

(b) The slope of the bed (and water surface) relative 

to the flume rails was taken from the plotted 

profiles. Horizontal profiles on the graph indicated that the bed and water surfaces were 

parallel to the flume rails. The extent to 

which the lines departed from the horizontal 

on the graph reflected the extent to which their 

inclination departed from that of the flume 

rails. Slope relative to flume rails and slope 

of flume rails themselves were combined to 

give the actual bed (and water surface) slope.

4. Depth, from profiles described in 3b.

5. Sediment-transport rate. An effort was made to 

collect sediment in the collection box for as long a 

period as possible. To ensure a representative 

sediment sample, this period was always long 

enough to allow at least several (and usually 

many) bed forms to move out of the flume and into 

the collection box. This procedure helped obviate 

possible error due to variations in the transport 

rate between the trough and the crest of bed forms,

error which was suspected to be the cause of point 

scatter in sediment-transport diagrams.

6. Bed-form characteristics:

(a) General type of bed form and other noteworthy features.

(b) Wavelengths. A rapid traverse was made of 

the flume length, and in this traverse the 

location of every well-defined bed-form 

crest was recorded. From these data an 

average wavelength could be obtained, as 

well as an estimate of the degree of uniformity of wavelengths along the flum^.

(c) Trough-crest height, perpendicular to bed 

slope (usually estimated from a scale on the 

flume wall).

(d) Rate of movement downstream. A single 

representative bed feature was selected and 

timed while it passed over a known distance. 

Occasionally, several bed forms were timed 

individually and an average rate of r^ovement recorded.

7. Surface velocity. Two or three small pieces of 

wood were individually timed for rate of movement as they floated along the water surface. 

(Most pieces were shaved from a stick wHh a 

pocket knife and were 1-2 in. long but relatively 

narrow and flat. In some of the runs at the 

greater depths, small blocks of woods measuring at 

most 3 in. X 1 in. X 1 in. were used.) The 

average velocity of the group was taken as the 

final value. This surface velocity was multiplied 

by 0.80 to provide a rough estimate of the mean 

flow velocity, as is explained in the results section. 

(The actual value of mean velocity was obtained 

by dividing discharge by cross-sectional area, 

based on depth as given in profiles.)

8. Sediment samples. Material in the collectior box 

and material left on the bed were sampled after 

the completion of the run. The samples were 

dried and then sieved on a Ro-Tap. (The average of 36 such analyses provided the sediment-size 

characteristics described.)

Photographs were taken, usually during and after 

the run. Those taken during the run were made from 

a side view, whereas those taken after the water had 

drained out of the flume depicted an upstream view. 

A pitot static tube was used for velocity measurements in some early runs, but this practice was soon 

discontinued. Because the bed and water surfaces 

were often irregular, the vertical height at which the 

readings were being taken was uncertain. Secondly, 

any reading was greatly affected by the proximity of a 

bed form. Finally, the presence of the tube near the 

bed caused scour in the sediment below the tube, and

248-117 O 67-
B6 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

the effect of this scoured region on the velocity value 

could not be determined.

The measured values for several of the variables 

were checked by various methods. The mean velocity 

as obtained from the discharge, and depth was compared with the mean velocity as estimated from float 

measurements. This same estimate of mean velocity 

could be used with the discharge to provide a check on 

the depth as obtained from the profiles. Finally, the 

attainment of equilibrium and the slope value were 

verified by repeated profile measurements. Repeat 

readings were taken on many of the variables as a 

standard procedure.

Runs in which mean flow velocities were greater than 

about 3.5 ft per sec could not be made with the present 

equipment. Faster velocities could not be investigated 

because the supply of sediment in the submerged infeed 

bin was depleted before all of the necessary experimental 

measurements could be completed. Also, at the 0.5- 

foot depth the maximum water capacity of the flume 

system was nearly reached.

No convenient method was available for maintaining 

a constant water temperature. In most runs the water 

temperature ranged from 11° to 27°C. Temperature 

during any run varied only a degree or two, if at all. 

A significant temperature effect on rate of sediment 

transport is doubtful for the grain sizes used in this 

study. According to Colby and Scott (1965), the 

percentage change in bed-material discharge that, 

accompanies a particular temperature change is generally small for particles 1 mm in diameter or larger.

Reproducibility at depths of 0.3 and 0.5 foot was 

excellent. Attempts were not made to reproduce any 

of the runs at the 0.1-foot depth.

Run 23 was conducted with the flume horizontal 

rather than at a slope. This procedural deviation was 

made because colleagues suggested that the setting of 

the flume at some particular slope (prior to the run) 

was prejudicing the final slope value. That is, they 

felt that a range of equilibrium slopes was possible; 

the measured slope value might be a minimum possible 

slope, a maximum possible slope, or some intermediate 

slope, depending on how the operator performed the 

experiments. For a given depth and velocity, the 

various possible values of slope could presumably be 

accommodated by changes in the bed roughness. Run 

23 satisfied the stated prerequisites for equilibrium and 

is therefore included in the results, but no insight was 

obtained into a possible range of slope values. The 

physical conditions within the flume during run 23 

unavoidably differed in one respect from the conditions 

of all other runs. The entrance region in the flume 

was positioned vertically in such a way that after the 

sand wedge had formed in the flume, the water passing

through the infeed section had to flow very slightly 

uphill before coursing over the sand in the flume. This 

difference in conditions was sufficient to causi^ diagrams 

for run 23 to be distinguishable from diagrams for other 

runs made at a similar depth in many of the illustrations 

presented later in this report. The horizontal flume 

showed the following disadvantages:

1. The total time needed to conduct a run increased 

about tenfold. Some of the extra time was required for slope adjustments by the water-sediment mixture, and some was required for temporarily stopping the run to refill the sediir<ait-infeed 

bin.

2. The maximum discharge capacity of the flume was 

drastically reduced by the large amount of sand 

that accumulated in the flume.

3. About 1^-2 tunes as much sediment was needed to 

perform a run, because so much sediment was 

stored in the wedge within the flume.

RESULTS 

GENERAL

The measured and computed values for all runs, 

excluding bed-form data, are listed in table^ 1 and 2. 

In these tables the runs made at each of the three 

depths have been arranged in order of increasing sediment-transport rate. The bed configurations ranged 

from plane bed to antidunes. A plane-bed stage between dunes and antidunes did not occur. (Bed forms 

are discussed later.) Sediment movement T^as always 

hi the form of bedload, according to visual observation, 

and little if any material was transported as suspended 

load. The size distribution of material caught in the 

collection box did not differ significantly from that of 

material remaining on the flume bed.

SEDIMENT-TRANSPORT BATES

Various measures of the strength of flovdng water 

can be related to the rate at which sedimert is transported. Examples of such measures are the following: 

1. Stream power. The power for a whole channel can 

be defined as the product of the weight of fluid 

(in pounds) flowing per second and the energy per 

unit weight of fluid (in foot pounds p-sr pound). 

This product can be written as jQ X energy, where 

7=specific (unit) weight of water ar<i Q=discharge of water. The dimensions are FLT~l . When the flow is uniform the energy lo^s per foot 

of channel length is given by the slope S, so that 

the power per unit length of channel becomes yQS 

(dimensions FT7"1). The stream power per unit 

bed area is then yQS/W, where W= channel width. 

The dimensions are FL~l T~ l, pounds per second
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B7

TABLE 1. Experimental results

Run

l____----_-___-

2____-__.____-_

3____. _________

4____ - .

5_- _ 

6

7__. ___________

8_________.____

9__ ---------

10--- 

11______ __ __

12_ _ __--.-_----

13_. -----------

14___ -------

15___ -------

16___ .- . 

17___ 

18. _.--_----_--

19--- - 

20--- -

21____--__-__--

22__. ----------

23___ -------

24___ --------

25--- 

26--- 

27___ 

28. _-__-_---_--

29__ - 

30-- 

31-- -

32. ______------

33___ 

34_____-_.--__.

35_____ _ ___ 

36. ____________

37_. -----------

(cfs)

0. 127 

. 142 

. 158 

. 154 

. 166 

. 191 

. 220 

. 233 

. 292

. 405 

. 440 

. 440 

.438 

.452 

. 470 

. 498 

. 503 

. 512 

. 530

. 551 

.570 

. 650 

. 790 

.741 

.794 

.860 

1. 050 

1. 120

.845 

.795 

. 866 

.905 

.960 

.958 

1. 050 

1. 100 

1. 320

s

(ft per ft)

0. 00411 

. 00495 

. 00590 

. 00751 

. 01082 

. 01280 

. 01505 

. 01992 

. 02218

. 00110 

. 00121 

. 00136 

. 00162 

. 00182 

. 00200 

. 00210 

. 00211 

. 00236 

. 00272

. 00318 

. 00397 

. 00509 

. 00471 

. 00557 

. 00643 

. 00721 

. 00824 

. 01088

. 00137 

. 00133 

. 00144 

. 00172 

. 00184 

. 00216 

. 00251 

. 00314 

. 00416

D

(ft)

0. 100 

. 100 

. 102 

. 101 

. 095 

. 101 

. 099 

. 094 

.097

. 297 

. 314 

. 313 

. 294 

. 294 

. 300 

. 303 

. 304 

. 295 

. 294

. 290 

.288 

.287 

. 298 

. 297 

. 306 

. 307 

. 324 

. 321

. 509 

.485 

. 505 

. 517 

. 512 

.502 

.517 

. 504 

. 510

V 

(fps)

1. 27 

1. 41 

1. 55 

1. 53 

1.75 

1. 89 

2. 22 

2. 48 

3. 01

1. 36 

1. 40 

1.41 

1.49 

1. 54 

1.57 

1.64 

1.65 

1.74 

1. 80

1.90 

1.98 

2. 26 

2.65 

2.49 

2. 59 

2.80 

3.24 

3. 49

1. 66 

1.64 

1.72 

1.75 

1.87 

1.91 

2. 03 

2. 18 

2. 59

Sediment transport

Measurement 

period 

(sec)

17, 940 

10, 140 

5,400 

2,460 

1,440 

1,684 

2, 183 

960 

473

24, 180 

26, 040 

23, 820 

23, 400 

12, 240 

14, 040 

6, 360 

2,400 

5, 040 

3,620

2, 340 

2, 160 

1,980 

2,940 

2, 130 

1, 353 

960 

720 

510

14, 040 

9, 180 

10, 020 

8,460 

18, 420 

3,840 

6,300 

2,640 

1,800

i 

Qbs per sec 

per ft)

0. 00145 

. 00291 

. 00593 

. 00935 

. 01980 

. 03444 

. 05772 

. 11200 

. 16810

. 00025 

. 00031 

. 00054 

. 00107 

. 00171 

. 00256 

. 00361 

. 00375 

. 00585 

. 00925

. 01090 

. 01597 

. 03156 

. 03262 

. 03826 

. 04805 

. 06875 

. 09722 

. 16280

. 00203 

. 00207 

. 00288 

. 00479 

. 00610 

. 00716 

. 01128 

. 01970 

. 03333

R

(ft)

0 083 

083 

085 

084 

080 

084 

083 

079 

081

186 

193 

193 

185 

185 

188 

189 

189 

186 

185

183 

183 

182 

187 

186 

190 

190 

197 

194

252 

246 

251 

254 

253 

250 

254 

250 

252

t (°C)

18. 5 

19.2 

21. 0 

12. 5 

11. 9 

17.6 

15. 5 

23. 5 

24. 7

21. 3 

22.7 

19. 8 

18.6 

16. 5 

18.4 

17. 2 

17. 8 

17. 1 

17.8

16. 1 

16. 2 

15.9 

30.8 

12. 3 

15. 5 

21.8 

23. 5 

25. 0

21. 0 

26. 0 

22.9 

23.6 

23. 2 

17.9 

18.7 

20. 8 

20. 8

V.

(ftw>

1.68 

1.89 

2.08 

2.22 

2.41 

2.67 

2. 86 

3.39 

3.85

1. 56 

1. 67 

1. 76 

1.85 

1.87 

2. 06 

2. 16 

2. 10 

2. 14 

2.30

2. 31 

2. 50 

2.92 

2. 86 

2.92 

3.05 

3. 33 

3.67 

3. 74

1.91 

1.92 

1.97 

2. 02 

2. 25 

2. 20 

2.38 

2. 54 

3. 14

per foot; these same units are used to measure the 

sediment-transport rate.

2. Shear stress based on depth. The shear, or tractive 

force, exerted on a unit bed area is commonly 

defined as the weight of the water above the unit 

bed area times the slope of the channel. The 

volume of water (area X depth) above a unit bed 

area will be numerically equal to the depth of the 

water D, and the weight will therefore be given by 

yD (dimensions FL~2). The shear per unit bed 

area is therefore yDS, in pounds per square foot.

3. Shear stress based on hydraulic radius. For very 

wide channels the depth is equal to the hydraulic 

radius R, so that the shear stress over a unit 

bed area would be yRS. The shear stress computed in this manner is assumed to represent an 

average shear per unit area of channel boundary. 

However, for narrow sediment-bearing channels 

with smooth sidewalk the shear probably is not

equally distributed over the walls and floor of the 

channel; the computed shear stress is then of 

questionable value.

4. Mean water velocity V, defined as Q/WD.

5. Regime theory bed factor, V2 /D. (Except for a 

gravitational constant g, this is the square of the 

Froude number F, where F= V/^/gD.)

No single measure of flow strength has yet gained 

widespread acceptance. A meaningful comparison 

might be obtained by relating the sediment-transport 

rate (i) to each of the measures listed above. The 

relations are shown in figures 2 6.

Several features are immediately noticeable in these 

diagrams. First, if depth is held constant, in any plot 

the data for any chosen depth fall on a single curve and 

show very little point scatter. This fact shoulc1 be 

emphasized because it suggests that permitting the 

depth to vary may well cause much of the scatter 

found in sediment-transport diagrams. (Although it
B8 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS 

TABLE 2. Computed quantities

Run

!_____. ..

2__. _______

3-_-------

4___-_--___

5_---_---__

6.-- -----

7_ .

8 __

9__ .

10---------

11__. ______

12_-_---___

13--------.

14__-_---__

15 ----- _

16 -------

17 

18.--------

1Q

20._-------

21__-_--_-_

22__---_-_-

23---------

24__-____._

25 

26.--------

27 _

28 

29_. _______

30--------.

31__------_

32_----_-_-

33---------

34_________

35__-_----_

36 ---.

37__-___-__

QS (X 103)

0. 522

. 703

. 932

1. 16

1.80

2.44

3. 31

4.64

6. 48

.445

.532

. 598

. 710

.823

.940

1. 05

1. 06

1.21

1. 44

1.75

2. 26

3. 31

3. 72

4. 13

5. 10

6. 20

8.65

12. 19

1. 16

1. 06

1.25

1.56

1.77

2. 07

2.64

3. 45

5. 49

RS (X 103)

0. 342

.412

. 500

.631

. 863

1.075

1. 243

1. 576

1. 800

. 205

. 233

. 262

. 300

.337

.375

. 396

. 398

. 438

. 504

. 584

. 725

.928

. 880

1. 038

1. 220

1.371

1. 620

2. 100

. 346

. 327

. 362

A.V7

4-fi^

KA1

.638

. 788

1. 050

DS (X 10")

0. 411

. 495

. 602

. 758

1. 03

1. 29

1. 49

1. 87

2. 15

. 327

. 380

. 426

. 476

. 535

. 600

.636

. 641

.696

. 800

. 922

1. 14

1. 46

1. 40

1. 65

1 07

2. 21

2. 67

3.49

. 697

. 645

. 727

.889

. 942

1. 084

1. 30

1. 58

2. 12

V2/D

16. 1

1Q Q

23. 6

23. 2

32. 2

35. 4

4.Q 8

65. 4

no A

6.2

6.2

6.4

7.6

8. 1

8 2

S Q

Q 0

in °>

11. 0

12. 4

13. 6

17. 8

23. 6

20.9

91 Q

25. 5

32 4

37. 9

x, 4.

5

c

x> Q

x> Q

6. 8

7 °.

o n

9. 4

13. 2

W/D

10. 00

10. 00

9.80

9.90

10. 52

9. 90

10. 10

10.63

10. 30

3. 36

3. 18

3. 19

3.40

3. 40

3. 33

3. 30

3. 28

3. 38 °. 4.0

3. 44

3.47

3.48

3. 36

3. 36

3. 26

3. 25

3. 08

3. 11

1 Qfi

2. 06

1.98

1 Q4.

1.95

1 QQ

1.94

1.98

1.96

v/v.

0. 76

.75

.75

.69

.73

.71

.78

.73

.78

.87

.84

. 80

. 81

.82

.76

. 76

7Q

.81

.78

. 82

.79

.77

.93

.85

.85

. 84

.88

.93

.87

.85

.87

.87

. 83

.87

. 85

.86

. 82

Chezy C

68. 7

69. 5

69. 2

61. 0

59. 5

57.6

63. 1

62. 5

71.0

95. 1

91. 5

87. 0

86. 1

83. 7

80. 9

84. 5

82.9

83. 3

80. 4

78. 5

73. 6

74. 1

89. 2

77.3

74.2

75.7

80. 6

76.2

89. 3

90. 6

90. 5

83. 7

86. 6

00 0

80. 2

77.9

79. 9

DarcyWeisbach /

0. 0546

. 0533

.0537

. 0692

. 0727

. 0775

. 0647

. 0660

. 0511

. 0284

. 0307

. 0339

. 0347

. 0367

. 0393

. 0360

. 0374

.0371

. 0398

. 0417

. 0475

. 0469

. 0323

. 0430

. 0467

. 0449

. 0396

. 0443

.0323

. 0313

. 0314

. 0367

. 0343

. 0379

. 0400

. 0424

. 0403

Manning n

0. 0142

. 0141

. 0142

. 0161

. 016S

. 017C

. 015£

. 015E

.0137

. 0117

. 012?

.012£

. 013C

. 013?

. 013?

. 013?

. 0135

. 0134

. 013P

. 0142

. 0152

. 0151

. 0125

. 0145

. 0151

. 014?

. 0140

. 014?

. 0132

. 012?

. 0130

. 0141

. 0136

.0143

. 0147

. 0151

. 0147

Froude No. 

F

0.71

70

86

.85

1. 00

1. 05

1. 25

1.43

1. 70

.44

.44

.44

. 48

.50

.51

.53

.53

. 56

. 59

.62

.65

.74

.86

. 81

.83

.89

1. 00

1. 09

.41

.42

.43

.43

. 46

.47

.50

.54

.64

may be possible by interpolation to sketch lines of 

constant depth for data from runs in which depth was 

allowed to vary, such a procedure would certainly be 

much less precise.) Thus, although the general depth 

effect could be perceived regardless of how the experiments were run (depth constant from run to run or 

depth varying), the maintenance of a constant depth 

permitted more accurate determination of the relations. 

A second characteristic of the graphs is that, with 

the exception of the yRS plot, a unique relationship 

does not exist between the measures of flow strength 

and sediment-transport rate. A given rate of sediment 

movement, as the independent or imposed variable, 

can be accommodated by various values of yQS, V, 

yDS, and V2/D. Conversely, according to figures 2, 

3, 5, and 6, a given flow strength can transport sediment 

at various rates. The determining factor on these 

diagrams is the water depth. Surprisingly, if shear 

expressed as yES is used as the measure of flow strength, 

the effect of depth is practically eliminated, and all the 

data can be described by a single curve (fig. 4). On 

the other four diagrams, however, the depth effect is

evident, though to differing extents. T1 Q. velocitytransport relation suggests that the effect of depth 

might be eliminated at slightly greater depths on this 

particular plot. There may be a tendency on the V2/D 

graph for the lines of greater depths to converge, but 

this plot has magnified the depth effect mo^e than any 

of the others. Such an influence may or may not be 

desirable. The diagrams of stream power and shear 

(yDS) do not seem to show any tendency to eliminate 

the influence of depth for the range of conditions 

examined. Power and shear (yDS) both involve a 

depth-slope product, so that in a laboratory flume the 

main distinction between the two is the inclusion of 

mean velocity in the computation of stream power. 

Significantly, the same depth-slope product (or velocitydepth-slope product) can give different sedimenttransport rates. As depth increases, with a corresponding reduction of slope so that DS remains constant, 

the sediment-transport rate decreases. r^he general 

depth effect on sediment-transport rates was discussed 

by Colby (1964b, p. A25-A29).
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND

 I I I I II I I I I I i I I I I I I I 

B9

i i r

EXPLANATION

Symbol Depth, 

in feet

A 0.1

o 0.3

a 0.5

0.1

0.01 J__L.

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 2. Power versus transport relation.

TT

o.i

EXPLANATION

Symbol Depth, 

in feet

0.01

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 3. Shear (yDS) versus transport relation.
BIO SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

0.1

0.01

EXPLANATION

Symbol

A 

o 

D

Depth, 

in feet

0.1

0.3

05

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 4. Shear (yRS) versus transport relation.

0.0001 0.001 0.01 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER

FIGURE 5. Mean velocity versus transport relation.

o.i

FOOT WIDTH

The fact that a given flow strength, as defined by 

yQS, V, jDS, and V*/D, can be related to various 

sediment-transport rates may be a reason for some of the 

difficulties encountered in seeking reliable sedimenttransport formulas.

What determines the particular DS or QS combination that will be derived for a given sediment load? 

When sediment suddenly enters a reach of stream at a 

faster rate, does the slope remain constant and the 

depth increase to provide the necessary flow strength? 

A current hypothesis (Langbein, 1964) asserts that each 

dependent variable will change in value by equal and 

least possible increments, insofar as permitted by 

limitations within the system.

A single straight line cannot be fitted to the plot 

of data for the present runs at a given depth, regardless 

of which measure of flow strength is involved. All 

the relations exhibit a change in the trend of the data

plot near 0.004 <i< 0.01 Ib per sec per ft. A straight 

line fits the points below this range, and a steeper 

straight line fits the data above this ran^e. If the 

zone where the trend changes is regarded as a boundary 

between a low transport range and a highe^ transport 

range, the relations shown in figures 2 6 a,re approximately as follows:

Low transport range Higher transport range

yOCiO.09 VOCtO-25

All the relations show the change in trerd, so there 

is little probability that this particular feature is due 

to any selected measure of flow strength. Rather, it 

would seem that within the indicated zone some 

physical change actually occurs in the mechanics of
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND Bll

100 EXPLANATION

Symbol Depth, 

in feet

A 0.1 

o 0.3 

D 0.5

10

_L I I I I I I

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 6. Regime bed factor versus transport relation.

the water-sediment relationship. The present data 

do not explain this phenomenon.

The steepness of the fitted lines indicates that, for the 

conditions examined in this study, the measure of 

flow strength which seems to be most sensitive to 

changes in sediment-transport rate is the mean velocity. 

A small change in V is associated with a large change in 

i. The least sensitive measure appears to be stream 

power. A 10-percent error in stream power, in other 

words, would provide a more accurate estimate of a 

given sediment-transport rate than would a 10-percent 

error in mean velocity. This is true whether or not the 

water depth is known. An evaluation of power, 

however, involves the measurement of an additional 

variable (slope); thus, there is a greater likelihood of 

error in the power value. Colby (1964a; 1964b) 

compared several of the currently popular measures 

of flow strength in connection with sediment transport.

Critical, or threshold, flow strength is the minimum 

flow strength that will transport sediment. Two 

methods are known to the writer for determining this 

value. One method is by visual means and is very 

subjective; owing to irregularity of particle movements, 

this approach is generally considered difficult. The 

second method is to extrapolate a sediment-transport 

curve down to zero or near-zero values of sediment 

transport and to accept the indicated value of flow 

strength. For the present purpose, suppose that the 

accepted lowest rate of continuous sediment transport 

is 0.0001 Ibs per sec per ft. (Visual observations 

during the actual runs suggest that this is a reasonable

figure.) Extrapolating the stream-power lines (fig. 2) 

down to this transport rate shows the critical pcwer to 

be about 0.0122, 0.0215, and 0.0245 Ib per sec per ft at 

depths of 0.1, 0.3, and 0.5 foot, respectively. The 

same procedure can be applied to the plots of shear 

expressed as yDS, mean velocity, and the bed factor (figs. 

3, 5, 6). The inescapable conclusion is that, at le<\st for 

the grain size used in this flume, unique values for such 

factors as critical power and critical shear (yDS) do 

not exist. Instead, a different value of critical flow 

strength is obtained for each depth examined. However, if yRS is used as the measure of flow strength 

(fig. 4), only one value of threshold shear exists, 

regardless of depth.

Because the same flow strength (except th<?. yRS 

data) moves different quantities of sediment at different 

depths, it is likely that any expression of the critical 

flow strength (except yRS) will have to refe^ to a 

specific water depth, at least for channels and sediment 

similar to those studied here. This depth influence 

has been considered only in some proposed definitions 

of critical flow strength. (See Sundborg, 1956, p- 177.) 

For the present data (unless yRS is used as the measure 

of flow strength), any definition of threshold flow 

strength that is based only on sediment characteristics, 

and perhaps on bed slope, and which is not adjusted to 

resolve the depth effect, would appear to be inadequate. 

Such expressions may be in error in proportion to a 

certain range of water depths. According to this 

investigation, such a depth range generally extends 

from 0.1 to 0.5 foot, and the range is probably somewhat
B12 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

greater than this. Note that the problem is not necessarily solved by including depth in the computation of 

the flow strength ( yDS). The present data imply that, 

for comparable sedimentary material, the only possibility of avoiding the determination of a different 

critical flow strength for each depth in a channel 1 foot 

wide lies in the use of yRS as the measure of flow 

strength, or in the hope that depth suddenly ceases to 

play a major role in sediment transport at extremely 

low transport rates. The latter possibility seems remote, according to the curves in figures 2, 3, 5, and 6. 

Why are the data in most of the plots segregated 

on the basis of depth? This question would seem to 

warrant some attention in future laboratory research. 

One possible influence, suggested by Einstein and by 

Bagnold in written communications (1965) is the drag 

exerted by the smooth sidewalls. A greater depth 

means that a greater side wall area is in contact with 

the moving fluid, so that the wall drag increases as 

depth increases. Thus for given values of stream 

power, mean velocity, and such factors, the additional 

wall drag associated with a greater depth would result 

in a greater overall retardation of the movement of 

water and of sediment. In other words, for a given 

flow strength, a greater depth would mean a lesser 

rate of sediment transport. If wall drag is the sole 

cause of the segregation of points, then presumably if 

the same 37 runs were conducted in a much wider 

channel, all the plotted points would fall on a single 

line regardless of flow depth. (On the other hand, the 

question then arises as to how such data would plot 

if yRS is used as the measure of flow strength.) Another possible cause of the segregation, suggested by 

Bagnold, is the height to which the saltating grains 

rise into the body of the flow; that is, the ratio of 

saltation height to flow depth. The transport rate 

might be expected to be greater at shallow flow depths 

because the saltation zone occupies a larger proportion 

of the whole flow depth. As Bagnold (written 

commun., 1965) pointed out, if this is true, the variation with depth is a small-scale phenomenon only, and 

would disappear progressively as depth is increased 

(provided of course that any wall-drag effect in narrow 

channels is also taken into account). Another proposal (Colby, 1964b) focuses attention on the vertical 

distribution of velocity. In comparing a shallow 

depth and a deeper one, the same value of mean velocity for both depths would require that the velocity 

increase much faster with height above the bed for 

the shallower depth. Consequently, the giv,en mean 

velocity would occur much closer to the sand bed in 

the shallower depth. In other words, velocities at a 

given distance from the bed (in the lower part of the 

full depth, at least) would be greater in runs at the

shallower depth, and, hence, more sediment would be 

moved.

The sidewalls of the flume undeniably exert some 

drag on the flow, and curiosity naturally arises as to 

whether the same values of flow strength would be 

required to move sediment at the present transport 

rates in a channel of "infinite" width that is, a channel 

wide enough that any sidewall influence is negligible. 

Two theoretical proposals for adjusting datr, obtained 

in narrow channels to eliminate the wall-drag: effect are 

the method of Einstein (1942) and the method of 

Johnson as modified by Brooks (1954) herein called 

the Einstein and the Johnson-Brooks methods. Briefly, 

these methods divide the flow cross section into that 

part influenced by sidewalls and that part in the center 

of the channel, unaffected by sidewalls. A hydraulic 

radius Rb is calculated for the latter part. A bed shear 

stress yRbS can then be computed which presumably 

represents the shear acting on the bed and omits that 

part of the shear acting on the sidewalls. Furthermore, 

because the hydraulic radius R for an infinitely wide 

channel is equal to Rj> (wall drag negligible) and is also 

equal to the mean depth, the computed R* r> equal to 

a depth Db which would be obtained if that p<\rt of flow 

affected by sidewalls were removed. If mean velocity 

and bed width are considered to be unchanged, a new 

discharge Qb can therefore be computed, where Qt,= 

VDbW or Qb=VR*W. This would be the d'scharge if 

the velocity and bed width were to remair but flow 

affected by sidewalls were excluded. Such a discharge 

can presumably be used in sediment-transport relations 

to represent the discharge free of wall drag. Thus, 

stream power unaffected by sidewalls would b^ yQi>S/W. 

The main distinction between the two methods is that 

Einstein's is based on the Manning formulp,, whereas 

the Johnson-Brooks approach uses the Darcy-Weisbach 

relation. Both methods have been applied to the 

present data, and the computed values are listed in 

table 3.

It can be noted from table 3 that at Z>=0.1 foot the 

two methods produce almost the same val'ies of Rb . 

At the other two depths the Johnson-Brooks procedure 

yields values which are about 3-6 percent greater than 

the values obtained by the Einstein method, and differences of this magnitude are considered rather slight. 

For both methods the calculations are not difficult and 

require about the same amount of time. Th°* adjusted 

values of flow strength obtained using the Einstein 

method are plotted against sediment-transport rate in 

figure 7. (The constant 7 has not been included in 

the computations of flow strength.) In regard to the 

power-transport relations, the lines for the various 

depths are slightly closer together compared with the 

yQS/i diagram of figure 2. There still exists, however,
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B13 

TABLE 3. Hydraulic radius, shear stress, and stream power corrected for wall drag by methods of Einstein and of Johnson and B-ooks

Run

!_._____. ____________________

2. ______---_--_-___-_.___-__

3___-_______________________.

4_ _

5

6_

7__..____-___________________

8..- ._ _ -

9 ___ _______________________

10_. _____-.-_-_---_-.______--

ll__.__ _ _--__-.___-___-.__

12.______._ ____ _ ____________

13____. __-.-_----_._-_-______

14__ -- __ 

15______-. ___________________

16____-__-_-. ________________

17.. _ ______

18__ -_- _ 

19._____--_-.-___. ___________

20__ ____-______--___.____.___

21__. ____--_-___--_.__-__-___

22_______-____.__. ___________

23... _ - - 

24 _ __-_-__-.___..____.___-.

25-______-_---_---______.--_-

26.. _ --- 

27_______-__------__-__-____-

28._._____-_--__. ___________

29___-___-___-__--__-_____._-

30.. _-

31 _ __-_-_---__--___-___.___

32__.____- ___________________

33 - __ 

34.. --- _ --___ 

35. _ -__ -

36._ - - 

37 -__ _-_- 

*i

(ft)

0. 091

. 091

. 092

.093

. 088

DQ4

. 091

.087

. 088

. 215

. 230

.235

. 224

. 227

. 229

. 234

. 235

.228

. 231

.230

. 235

.233

. 223

. 237

. 247

. 246

. 251

. 255

348

. 331

. 342

.368

. 357

. 363

.376

. 375

. 373

Einstein

RbS 

(X103)

0. 374

. 450

. 543

. 698

. 952

1. 203

1. 370

1.733

1. 952

. 237

. 278

. 320

.363

.413

. 458

4Q1

.496

.538

. 628

.731

.933

1. 186

1. 050

1. 320

1. 588

1. 774

2. 068

2.774

477

440

492

633

657

784

944

1 178

1 552

QiS=VRiS 

(X10»)

0. 475

.635

. 842

1. 068

1.666

2. 274

3. 041

4.298

5.876

. 322

. 389

. 451

. 541

. 636

. 719

.805

.818

.936

1. 130

1. 389

1.847

2.680

2. 783

3.287

4 113

4967

6. 700

9.681

. 792

. 722

.846

1. 108

1. 229

1. 497

1.916

2. 568

4. 020

.Rb

(ft)

0 092

092

094

095

090

095

093

089

090

221

238

244

232

235

243

241

243

238

241

239

244

243

237

248

259

258

268

269

363

346

359

385

377

381

398

394

397

Johnson-Brooks

RiS

(X10«)

0.378

. 455

. 555

.713

.974

1. 216

1. 400

1.773

1.996

. 243

.288

.332

.376

. 428

. 486

. 506

. 513

. 562

.656

. 760

.969

1. 237

1. 116

1. 381

1.665

1.860

2. 208

2.927

. 497

. 460

. 517

.662

.694

.823

.999

1.237

1.652

QkS=VK kS 

(Xl-w)

0. 481

.642

. 860

1.091

1.705

2. 298

3. 108

4 397

6. 008

.330

. 403

. 468

. 560

.659

.763

. 830

. 846

Q7C

1. 181

1. 444

1. 919

2. 796

2.957

3. 439

4 312

5. 208

7. 154

10. 215

.825

. 754

.889

1. 159

1. 298

1. 572

2. 028

2.697

4 279

a definite segregation of fitted lines on the basis of depth. 

For the shear (yRS) versus transport relation it will 

be recalled that the unadjusted data could be described 

by a single line (fig. 4). The adjusted shear values 

show no prominent depth segregation at low transport 

rates, where the slope of the fitted line is small. As i 

increases, a slight depth influence becomes noticeable; 

and in this higher transport range, it appears that a 

separate line should be assigned to each water depth. 

A depth segregation would be slightly more pronounced 

if the Johnson-Brooks figures were used.

In general, the application of the two methods for 

eliminating the wall-drag influence tends to support 

the observation that, for a given value of flow strength, 

the transport rate increases as depth decreases, for the 

present sediment and flow conditions. A shear (jRS) 

versus transport diagram may show a depth segregation 

for wide channels, according to figure 7.

RELATIONS OF VARIABLES

The relation of mean velocity to the independent 

variables transport rate and depth is shown in figure 5. 

Prom the given sediment-transport rate and the depth, 

velocity was uniquely determined.

Discharge was also uniquely determined by sed:menttransport rate and depth (fig. 8). The slope of the 

fitted line for any depth was not constant for the whole 

range of conditions studied. At low sediment-transport 

rates Q varied as ^0 10. But the slopes seemed to 

undergo a change near ««0.01 Ib per sec per ft, becoming slightly steeper (Q<x.i°-25) for transport rates 

somewhat greater than this. The break in the relation 

appeared to occur at lesser values of i as depth was 

increased.

Slope was also uniquely related to transport rate 

and depth, as can be seen in figure 9. The plot again

248-117 O 67-
B14 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

0.005 -

0.001 -

0.0005 -

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 7. Transport relations adjusted for flume wall drag (Einstein method).

- o.ooi

0.0005

a 1.0

I I I i i r TTI

EXPLANATION

Symbol Depth, 

in feet

0.1 

0.3

0.5

I I I

0.1

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 8. Discharge versus transport relation.
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B15

0.01

0.001

I I I I I I

EXPLANATION

T I I I I I

Symbol Depth,

in feet

0.1

0.3 

0.5

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 9. Slope versus transport relation.

seems to suggest the presence of two regimes. At low 

sediment-transport rates, slope was approximately 

proportional to i°-26 . At transport rates greater than 

about 0.06 Ibs per sec per ft the relation changed to 

approximately Soci°A*.

The reason for the break in the Qfi and S/i relations 

is unknown. The same break was noticed in the 

diagrams relating flow strength to transport rate 

(figs. 2-6), all of which involve either or both Q (or 

components of Q) and S.

The resistance to flow was computed using the three 

common measures currently in use: Chezy C(=Vj 

VB§), Manning n(=1.486 E°-67 S°- 50/F), and DarcyWeisbach/^StfSS/F2). The calculated values (table 

2) indicate that the resistance to flow did not change 

much within the range of conditions investigated; 

a rather slight increase in resistance occurred with 

increasing sediment-transport rates.

Unique relations were also found between slope and 

water discharge, and therefore between slope and 

velocity, for a given depth (fig. 10). (For a constant 

depth and flume width, velocity varies directly with 

discharge.) Slope increased about as the 2.5 power 

of water discharge or velocity. Other investigators, 

however, have reported different findings. Brooks 

(1958) and Vanoni and Brooks (1957) found that when 

bed forms (dunes) began to be swept away during the 

transition to plane-bed stage, the S/V trend for a given 

depth was reversed so that an increase in velocity was 

accompanied by a reduction in slope. This reversal 

was possible because of the decrease in flow resistance 

resulting from the gradual disappearance of the bed

forms. After the plane-bed stage was attained in the 

studies just cited, slope stopped decreasing and began 

increasing again with further increase in velocity. 

The resulting S/V curves thus showed that more than 

,one velocity was possible for a given slope and depth 

within a certain range of bed configurations. The 

absence of any such break in the present curves leads 

to the question of what differences existed in th°. two 

groups of experiments.

The chief differences seem to be that Brooks and 

Vanoni used relatively fine grain sizes (about 0.08-0.15 

mm median diameter), recirculating flumes, fMines 

having dimensions different from the present flume, 

and dependent and independent variables differing from 

those of the present study. Flume dimensions seem 

to be the least significant factor because two fumes 

were used in the Brooks and Vanoni work (flume A was 

10.5 in. wide X40 ft long and flume B was 33.5 in. 

wide X60 ft long) and the present apparatus is intermediate in size between these two flumes. Little information is available to indicate that differences 

should be expected in data obtained in recirculating 

versus nonrecirculating flumes. In both groups of experiments depth was an independent variable. Sediment-transport rate, however, cannot be an independent 

variable in a recirculating flume; discharge (and consequently, velocity) was independent in the Brooks and 

the Vanoni-Brooks studies. The same relations between variables might be expected to exist regardless 

of which factors were dependent and which were independent during the runs. The ranges of flow conditions 

were comparable for the two groups of experiments.
B16 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

0.01

2 0.005

0.001

EXPLANATION

Symbol Depth,

in feet

0.1

0.3 

0.5

0.1 0.5 1.0 

DISCHARGE, IN CUBIC FEET PER SECOND

2.0

MEAN VELOCITY, IN 

FEET PER SECOND

FIGURE 10. Relations of slope to discharge and mean velocity.

The most likely cause of the differing results in the 

slope-velocity trends would seem to be the grain size 

and the associated differences hi bed behavior. With 

the relatively large grains used in the present experiments, the bed forms gradually progressed from dunes 

directly to antidunes. Bypassed completely were the 

transition from dunes to plane-bed stage and the planebed stage (at least at the greater depths), where the 

reversal of the S/V trend occurred in the Brooks and 

Vanoni-Brooks tests. As pointed out by Simons, 

Richardson, and Nordin (1965a), with coarse sand and 

shallow depths the dune stage may not terminate until 

the Froude number is so large that the subsequent bed 

form is antidunes instead of plane bed. In the absence 

of transition and plane-bed stages similar to those that 

occurred when finer grain sizes were used, no marked 

reduction in flow resistance occurred in the present 

runs, and the mean velocity increased consistently 

with increase in slope.

In summary, the measurements made in this investigation indicate that as soon as depth and any other 

variable (except flow resistance) were chosen, the values 

of all other variables were uniquely determined. (The 

value of flow resistance could also be ascertained because 

resistance is uniquely determined once D, S, and V are

determined.) The merit of depth as an independent 

variable in flume studies perhaps deserves emphasis. 

When depth is kept constant, the relations among 

variables appear to be much more clearly defined.

SURFACE VELOCITY

Surface velocities were measured only to provide an 

approximate check on the mean velocity. Inspection 

of the data, however, reveals some consistent relations 

involving surface velocity.

The relation between surface velocity (V,) and rate 

of sediment transport in the present experiments was 

surprising (fig. 11), because the influence of depth was 

virtually eliminated. Except in the yRS/'i plot, an 

elimination of the depth effect did not occur with the 

other measures of flow strength related to sediment 

transport (figs. 2-6). Therefore, in the present study, 

the easiest and quickest method of estimating sedimenttransport rate was a simple measurement of surface 

(float) velocity. Furthermore, the point scatter in 

figure 11 is small enough that any estimate of i using 

Vs would be reasonably accurate.

The variation between mean velocity ard surface 

velocity for the various depths investigated is shown 

in figure 12. The W/D ratio is used on tha, abscissa
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B17

I I

EXPLANATION 

Symbol

I I I I I I I 1 I I I II

o o

UJ Q0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 11. Surface velocity versus transport relation.

1.00

0-90 -

~l;-" o.so

0.60

Depth = 0.3 foot 

x x

Depth = 0.5 foot

Depth = 0.1 foot 

x x

6 

W 

D

10

FIGURE 12. VIV, relation for various depths.

although width was constant throughout. Average 

V/VS values as indicated by the line of best fit are 

0.85, 0.83, and 0.74 for depths of 0.5, 0.3, and 0.1 foot, 

respectively. The V/VS ratio is thus seen to decrease 

with depth by a very small increment, so that Vs departs farthest from V at a very shallow depth. For 

the present runs, use of the approximations would introduce possible V/VS errors of about 3.0, 12.0, and 7.0 

percent for the 0.5-, 0.3-, and 0.1-foot depths, respectively.

Figure 12, then, shows the proportionality that 

exists between Vs and V for a given depth (or W/D 

ratio). A fixed proportionality also exists between 

Vs and Q for a given depth, as might be expected (fig. 

ISA). Since Q varies directly with V for a constant 

depth and width, Q also varies directly with Vs.

The relation between surface velocity and slope is 

shown in figure 135. If the plot of slope as a function 

of mean velocity were transferred for comparison

purposes to figure 13B, it would be clear that the surface-velocity versus slope proportionality is eractly 

the same as that between mean velocity and slope, as 

would be expected. This relation is Vs ocS°A1 .

In summary, the surface velocity in this study was 

consistently related to all the other variables. Consequently, use of surface velocity may be an gid in 

determining the values of other variables, at least 

within the range of conditions examined here. Surface velocity is obviously easier to measure, and, for 

the present data, its relation to other variables is 

dependable. Vs alone can provide a reasonably good 

appraisal of both sediment-transport rate and mean 

flow velocity. When water depth is also known, Vs 

could be used to uniquely determine the slope and the 

discharge. Thus, such factors as power and shear also 

are uniquely related to surface velocity, for a given 

depth.
B18 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

> cc.

UJ 

UJ Q_

1 

0.1

I I I I I I I I

I I I I I I

0.5 1 

DISCHARGE, IN CUBIC FEET PER SECOND

B

i i i i i i

0.001 0.005 0.01 

SLOPE, IN FEET PER FOOT

FIGURE 13. Relations of surface velocity to slope and discharge.

0.3 

0.5

The surface-velocity relations may never be well 

defined for many natural-stream situations, owing to 

the complexities of these situations, nor for other 

laboratory investigations. However, sufficient information is not yet available to negate the possibility 

that reliable relations exist. In future laboratory and 

field studies, it may be worthwhile to measure surface 

velocity and see what relations exist between it and the 

other variables.

COMPARISON WITH DATA OF GILBERT AND MURPHY

In this section the data from the present investigation are compared with the Gilbert-Murphy data 

(Gilbert, 1914) for their grade E, a comparable grain 

size (0.84-2.00 mm, nominal diameter 1.71 mm). With 

this grain size Gilbert and Murphy used flume widths 

of 0.66, 1.00 and 1.32 feet, and depths ranging from 

0.077 to 0.562 foot. In the present comparison, all 

their grade E data have been included regardless of 

widths and depths. Their results have been computed 

for unit flume width, and their transport rates (dry 

weight) have been converted to immersed weight by the

i * A u* 2.65 1.00^ , . UA relation immersed weight= ^-^z Xdry weight=

4.OO

0.624Xdry weight. Depth was not controlled in 

the Gilbert-Murphy experiments; transport rate and 

discharge were the independent variables. Most of 

their runs were made at the higher transport rates. 

The stream-power versus transport relation, on which 

the depth influence would be expected to appear, is 

shown in figure 14. The Gilbert-Murphy points are 

represented by dots, and the water depth is given 

beside each dot. For comparison, the lines for each 

depth from the present study are also included. In a 

general way the depth effect is discernible from the 

plotted points, and it is perhaps more noticeable at 

higher sediment-transport rates. It would be difficult, 

however, to sketch in lines of constant depth for the 

Gilbert-Murphy data. In other words, the GilbertMurphy points occupy the same band on th°. graph as 

do the points for the present data, but the points for the 

present data reveal the influence of depth much more 

clearly.

Because all the present runs plot on a single line if 

shear (yRS) is used as the measure of flow strength, a 

graph of the yRS/i relation was also prepared (fig. 15) . 

On this plot the point scatter of the Gilbert-Murphy 

data is so reduced that the depth can not be labeled 

beside each point, and symbols are used to represent 

depth ranges. The depth effect is eliminated, as in the 

present data, but there is slightly greater point scatter 

in this plot than in the plot of the present data (fig. 4) . 

The line of best fit for the present data is reproduced 

in figure 15; the agreement with the Gilbert-Murphy 

data is very good.

The relations among mean velocity, depth, slope, and 

resistance to flow in the two sets of experiments can be 

seen from a plot of V versus DS (fig. 16) ; DS is, of course 

a measure of shear. A straight line on log-log scales 

can be fitted to both sets of data, indicating the existence of a power function, similar to the Chezy equation 

of the form

Closer inspection shows that the two fitted lines duplicate each other. The fact that both sets of data are 

identical indicates that for the two investigations (a) 

the relation of velocity to shear is identical, and (b) the 

relation of flow resistance to velocity and shear is 

identical (resistances are the same) . The actual definition 

of resistance is irrelevant.

The value of the exponent b is about 0.42. For 

values of 6<^0.50, resistance to flow increases with increasing shear or mean velocity. The very slight 

increase of flow resistance with increasing shear or 

mean velocity, as indicated by the value of b, can be seen 

from the trend in the computed resistance (table 2).
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B19

TT

EXPLANATION

a

z

9 o.i

Present study

.0.45

Gilbert-Murphy grade E 

Number is wafer depth, in feet

0.37.. 0-39 

.0.56^ 0.19 

0.22 .0.11

.0.19 _

I I I I I I I

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 14. Comparison of stream power versus transport relations with data of Gilbert and Murphy.

 r~n 

EXPLANATION 

Gilbert-Murphy grade E 

x Depth<0.20 ft

? 0.20<depth<0.40 ft 

+ Depth>0.40 ft

o.i

0.01

Present study

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 15. Comparison of shear versus transport relation with data of Gilbert and Murphy.

The general agreement between part of the GilbertMurphy work and the present study is perhaps significant, for it indicates that some of the Gilbert-Murphy 

data are reproducible. Several details of experimental 

equipment and procedure differed in the two investigations, but, these differences apparently did not greatly 

affect the experimental results. For example, in the

Gilbert-Murphy work on stream traction using grade E

sand:

1. Two different flumes were used. One of these had 

a length of 24.5 feet from sediment infeed to sediment catchment (31.5 ft, overall), and the other 

had an overall length of 150 feet. (The present 

study employed a 52-foot-long flume.)
B20 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

ZQ

to o w

ZQ

Z 4 

>O

toO w

o«

EXPLANATION

Symbol Depth, 

in feet

A 0.1

o 0.3

D 0.5

Present study

Gilbert-Murphy grade E (all depths)

1

0.0001

x x x

I__II_L

0.001 

DS, IN FEET

FIGURE 16. Mean velocity versus shear (DS) relations compared with data of Gilbert and Murphy.

2. Various flume widths were used.

3. Depth was not controlled transport rate and discharge were the independent variables.

4. The flume was horizontal and was devoid of sand at 

the beginning of each run; that is, slope was not 

estimated and not preset.

5. The method of sediment infeed was different from 

the present study. Sand wa? dropped into the 

stream from above usually by hand (by dumping 

in a boxfull at timed intervals), but occasionally 

by a hopper (at low transport rates).

6. Transport rate was measured during a relatively 

short period.

7. Sediment-infeed rate did not always equal the collection rate.

Some of the above features might account for some 

point scatter in the Gilbert-Murphy data.

BED FORMS

For sediment finer than 0.6 mm in median fall diameter, the usual sequence of bed forms produced in 

flume studies by increasing stream power was defined 

by Simons, Richardson, and Nordin (1965a) as ripples, 

ripples on dunes, dunes, transition, plane bed, antidunes, and chutes and pools. For mater al coarser 

than 0.6 mm they reported that the ripple and ripple-
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B21

on-dune stages are absent, and that the transition and 

plane-bed stages may or may not occur.

The types of bed configuration produced in the present study by increasing stream power were plane bed 

(at extremely low sediment-transport rates), dunes 

(at depths of 0.3 and 0.5 ft), and antidunes. A description of the bed configuration for each run is given in 

the section starting on p. B28. The initial flat-bed 

stage appeared to be relatively unstable, as extremely 

low crests (probably incipient dunes) occasionally 

formed at a few locations during the first run (D=0.3 

ft). At very low sediment-transport rates, definitive 

bed forms appeared. These first bed forms were 

different in runs at the 0.1-foot depth than in runs 

at greater depths.

At a depth of 0.1 foot, the run at lowest stream 

power produced an almost flat bed. With increase in 

flow strength (and hence in transport rate) a peculiar 

bed feature appeared that was a few grain diameters 

high. Such a bed form, herein called a meandering 

scar, began at the flume wall and curved toward the 

center of the channel downstream (fig. 17). It usually 

faded out completely in the middle of the channel. 

The adjacent bed feature, a mirror image of its neighbor, would begin from the opposite wall a short distance 

downstream from the first feature. (Similar features 

also were found under certain conditions after the 

water had been drained from the flume.) With 

further increase in flow strength the meandering scars 

disappeared except in the downstream part of the flume 

(run 4). At the same time (run 4) symmetrical 

antidunes of very low amplitude appeared throughout 

most of the flume, though present only in the center 

of the channel in a single row. The distinguishing 

feature of antidunes is an inphase relation between 

sand crests and water-surface crests. These initial 

antidunes had wavelengths of about 0.5 foot. In 

subsequent runs the meandering scars disappeared 

entirely and only antidunes were present. With 

continued increase in flow strength, two or possibly 

three rows of antidunes occurred side by side in the 

flume simultaneously. Adjacent lines of these bed 

forms had staggered sand crests and troughs, so that 

a checkerboard pattern formed. Some rhythmic periodicity occurred in the formation and disappearance

FIGUKE 17. Pattern of typical meandering scars.

of water surface irregularities (waves). At the highest 

flow strength examined at 0.1-foot depth, the antidunes 

were mostly flattened out, and so was the wate~ surface. This rather flat bed was only occasionally 

interrupted by a short train (3-10 ft long) of antidunes, 

moving downstream. All the antidunes traveled 

downstream.

At depths of 0.3 and 0.5 foot, dunes were the initial 

bed configuration (except in the first run at 0.3-f t depth, 

during which the bed remained flat). The distinguishing characteristic of dunes as opposed to antidunes is 

an out-of-phase relation between the sand crests and 

the water-surface crests. (Ripples are generally less 

than 1 ft in wavelength, and the present features had 

an average minimum wavelength of 1.5 ft). These 

dunes were noticeably asymmetrical from a side view; 

they were characterized by a long upstream side, a 

crest which varied from angular to somewhat rounded 

with increasing flow strength, and a relatively short 

and steep downstream face. At low transport rates 

the upstream face was extremely long, in comparison 

with the downstream face, and the downstream face 

seemed to be about at the angle of repose. The spacing 

of well-defined crests under these conditions was somewhat irregular downstream, and the crests were usually 

perpendicular to the flume walls and extended across 

the whole channel. As flow strength was increased, 

three particular changes were observed in the dunes: 

(a) the length of the upstream face progressiveT y decreased, and that of the slip face increased; (V) the 

angle of the slip face tended to decrease as the sand 

grains began to be propelled down the slip face rather 

than sliding down under their own weight; and (c) the 

angle between the dune crests and the flume walls 

changed though still extending from wall to wall, the 

crests began assuming irregular and curving fronts and 

reached the opposite wall 1 or 2 feet farther downstream 

from their origin. Furthermore, the water surface 

became increasingly irregular. (At D=Q.5 foct the 

maximum flow capacity of the flume was reached, 

while the dunes were beginning to be inclined to the 

walls and were showing some irregularity in crest 

directions.)

The characteristics just described probably represent 

a transition from dunes to antidunes. This trarsition 

evolved gradually; no tendency to develop a plare bed 

was observed. Clearly defined antidunes, as indicated 

by an inphase relation between bed and water-surface 

crests, were recognizable in the final three runs at the 

0.3-foot depth. For the complete series of runs at the 

0.3-foot depth the bed-form heights and rates of travel 

consistently increased with increase in flow strength, 

except, possibly, the travel rates of the antidunes. 

When antidunes began to form, the water surface be-
B22 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

came smoother than in the preceding runs and showed 

rather symmetrical waves. With further increase in 

flow strength these water-surface waves tended to break 

occasionally on the upstream side of the wave crest. 

In the final run the surface waves extended almost 

completely across the channel and broke with considerable turbulence just upstream from the crest. The 

antidunes (a) were almost symmetrical from a side 

view, (b) possessed well-rounded crests that extended 

across most or all of the channel and were perpendicular 

to the flume walls, and (c) moved downstream.

With antidunes the water-surface waves and sand 

waves were not always well developed throughout the 

flume length; rather, any given segment of the flume 

would possess well-developed waves during one period, 

and poorly developed ones during another period. 

Two or three segments of well-formed waves existed in 

the flume at any one time; the length of time such a 

train of well-developed waves persisted ranged from a 

fraction of a minute to several minutes.

Because of the variations in appearance and measurements of bed forms, the minimum of information which 

will adequately describe bed conditions must include a 

description of the features (rather than a single name) 

in addition to some measurements of bed-form heights, 

wave lengths, and rates of travel. The characteristics 

of the water surface must also be noted, especially the 

phase relation between bed-form crests and watersurface crests. The recording of this information 

should be a standard practice in flume studies.

During the experiments it became evident that the 

description and measurement of bed features after the 

water has drained from a flume should be undertaken 

only with great caution. No significant difference was 

found in dune wave lengths during the run and after 

the run, in those runs for which both measurements 

were taken. However, bed features existing after the 

water was drained were frequently quite unlike those 

that existed during the run. Some examples of feature 

changes are the following: (a) All antidunes were 

obliterated as the water left the flume; (b) the general 

bed-form appearance as well as the heights were often 

modified as the water drained; (c) whenever the depth 

was relatively shallow or the bed was somewhat flat 

during the run, a peculiar type of bed feature (the 

meandering scars of fig. 17) appeared as the water 

receded. In many runs the peculiar features did not 

form until the flume was drained. No sound explanation can be offered for the formation of these features, 

but the writer suspects that they are associated with 

deflection of the water by the fixed walls of the flume.

INTERRELATIONS BETWEEN CHARACTERISTICS

The rate of bed-form movement (c), average bedform length (1), and average height (fi) are given in 

table 4. Heights of well-developed antidunes were 

generally fairly constant for any given run. The range 

in dune heights during runs under given flow conditions 

was small in some runs and several fold in others. 

The determination of dune lengths was occasionally a 

subjective procedure. As reported in othe^ studies, 

dunes under certain conditions are continually dying 

out, reforming, and overtaking other dunes. In many 

studies the dunes do not extend very far laterally, and 

the crests frequently trend diagonally downstream. 

In the present study, extremely low dune crests which 

were a few grain diameters high were disregarded. 

Such incipient bed forms were more common at low 

sediment-transport rates. In spite of occasional subjectivity in obtaining the values, they should be fairly 

representative of the prevailing bed forms.

No well-defined relation was obtained betveen bedform height and length. The antidunes at D 0.1

TABLE 4. Bed-form measurements 

[No data for runs 1-3]

Run

'4. ________

1 5.. .......

} 6. ........ 1 7

1 8__._______

1 9. .........

10__________

ll._._ -_

12 _______

13 _______

14 

15 ____

16_____.____

17__. ______

18 ________

19 _______

20 _______

21..........

22 _______

23 ______

24 ___

25 ______

26 _______

27 ______

28 ______

29 ____

30 _____

31 ________

32 ____

33 _____

34 ________

35 _____

36 _____

37 _____

Height (ft)

0 021 

042 

083 

053 

073

. 021

042 

042 

053 

063 

042 

053 

053 

053 

063 

073 

104 

167 

125 

167 

250 

250

083 

042 

104 

063 

042 

063 

167 

167 

167

Length (ft)

During run

0 5 

5 

6

7 

1 0 

6

8.9 

5.8 

4.8 

4.3 

3.6 

3. 5

1. 8 

1. 9 

1.3 

1. 5 

1. 5 

1.6 

1.6

5.2 

3.9 

3.7 

2.8 

2.9 

2.5 

2. 0 

2. 2 

1.8

Dried bed

2.3 

2.4 

2. 0 

1. 9 

2. 0 

1.9 

1. 9

4. 6 

4.6 

3. 5 

3. 0

2.7

2.2 

1.8 

1.8

Velocity 

(ft per sec)

C.l-ft depth

0. 0200 

.0216 

.0300 

. 0683 

. 0683

0.8-ft depth

. 0007

. 0017 

. 0027 

. 0015 

. 0035 

. 0050 

. 0027 

. 0075 

.0117 

. 0158 

. 0267 

. 0167 

. 0250 

. 0384 

. 0083 

. 0400 

. 0350

C.5-ft depth

. 0012 

. 0007 

. 0017 

. 0018 

. 0022 

. 0020 

. 0072 

. 0063 

. 0217

Antidunes.
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B23

foot tended to increase in height as their length increased, but dunes increased in height as length decreased. Height and length relations alone could not 

distinguish dunes occurring at a depth of 0.3 foot from 

those occurring at a depth of 0.5 foot.

The data further indicate that bed-form height increased as rate of travel increased. The relation 

between either height or wavelength and bed-form 

velocity serves to distinguish the runs roughly on the 

basis of flow depth. Although the point scatter precludes a precise relation, figure 18 suggests that for a 

given dune height the rate of movement downstream 

is slower at a greater depth. The wavelength versus 

bed-form velocity relations (fig. 19) show that as dune 

velocities increased the wavelengths gradually decreased, apparently reaching a limiting minimum value 

of about 1.5 foot. The wavelengths of the antidunes 

at Z?=0.1 foot generally increased as bed-form velocity 

increased.

CHARACTERISTICS RELATED TO FLUID FLOW STRENGTH

When antidunes at depths of 0.1 and 0.3 foot became 

clearly recognizable, the Froude number, representing 

the ratio of inertial to gravity forces, was about the 

same for both depths (F«0.85). The value of the 

fluid flow strength was considerable greater, however, 

at a depth of 0.3 foot at the onset of formation of antidunes. The antidunes at the two depths had somewhat

different characteristics; at the 0.1-foot depth they were 

much smaller in size, more symmetrical in all directions, 

and usually did not exist near tlie flume walls. Also, 

trains formed side by side in runs at a depth of 0.1 foot, 

but not at a depth of 0.3 foot.

Prior to the antidune stage, dunes existed at depths of 

0.3 and 0.5 foot but not at a depth of 0.1 foot. Furthermore, the dunes at the two greater depths were rather 

similar in appearance. This similarity, in contrast to 

the complete absence of dunes at the shallowest depth, 

seems rather suggestive of a depth effect on type of bed 

form. Unless wall drag or some other factor had a 

great influence on the bed forms which occurred in these 

experiments, it appears that the depth of flow caused 

substantial differences between bed forms occurring at 

the shallowest depth and bed forms of the two deeper 

depths. Apparently, somewhere between the 0.1 arid 

0.3-foot depths a critical depth existed below which 

the bed features, especially dunes, were greatly inhibited 

in their formation by shallow depth. Thus, flow depth 

may place certain limitations on bed-form characteristics. The existence of these limitations, such as on 

maximum attainable heights of bed forms, has been 

suggested by various investigators.

Height

In figure 20 the average height of the bed forms is 

plotted against mean velocity, surface velocity, shear,

I i I i i i i i i i

EXPLANATION

Symbol Depth, 

in feet

0.1

0.3

0.5

^. 0.1

D D o 0-;

0.01 I I I I I I I I I I I I I I I I I I

0.001 0.01 

TRAVEL VELOCITY, IN FEET PER SECOND

FIGURE 18. Variation in bed-form height with travel velocity.

o.i
B24 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

10

\

\

EXPLANATION

Symbol Depth, 

in feet

A 0.1

o 0.3

 0.3 (after draining water)

D 0.5

 0.5 (after draining water)

I I I

0.001 0.01 

TRAVEL VELOCITY, IN FEET PER SECOND

FIGURE 19. Variation in bed-form wavelength with travel velocity.

o.i

and power. (The constant 7 is dropped in the computation of shear and power. The fitted lines do not include 

the antidunes at D=Q.l ft.) The heights of the bed 

features increased progressively with increase in flow 

strength. Dune heights grew with about the 3.2, 3.2, 

1.2, and 0.9 powers of mean velocity, surface velocity, 

shear (either definition) , and power, respectively. The 

antidunes at D=Q.l foot differed from the bed forms 

at greater flow depths whose heights were greater for a 

given flow strength. In spite of the point scatter, 

there is no clear indication (except possibly with the 

Vg plot) that the heights of the dunes occurring at the 

0.3-foot depth were significantly different from those 

at the 0.5-foot depth.

The relation of mean wavelength to each of the 

measures of flow strength is shown in figure 21. The 

antidunes that formed at the shallow flow depth had 

short intervals and are separated from the other bed 

forms. Also, the lengths of these antidunes increased, 

whereas those of the dunes decreased, as flow strength 

increased. The curves indicate that a fair estimate of 

flow strength could be obtained from a determination 

of mean dune wavelength. The antidunes that formed 

at a depth of 0.3 foot (the last few runs) did not show 

any significant change in wavelength with increase in 

fluid flow strength. For a given flow strength, a change

in depth from 0.3 to 0.5 foot apparently die1 not affect 

the mean dune wavelengths.

Velocity

The relation of rate of bed-form travel to the various 

measures of flow strength is presented in figure 22. 

The velocity of the bed features increased with greater 

flow strength. A slight increase in flow strength caused 

a relatively large increase in rate of bed-form movement. 

The rate of bed-form travel varied approximately with 

V7 - 2, Vs7 - 2, (RS) 2 - 6, (DS) 2 - 2 , and (QS) 1 - 5 . Liu found in 

certain flume experiments using finer sand that bedform velocity is proportional to Vs.

The depth effect can be distinguished on the DS and 

QS diagrams. As indicated earlier on the sedimenttransport diagrams (figs. 2, 3), the amount cf sediment 

transported, at a given DS or QS value, increased as the 

water depth decreased, within the range of flow conditions studied. Figure 22 is consistent with this 

trend; it shows that the bed forms moved downstream 

faster as depth decreased when DS and QS were the 

measures of flow strength.

As can be seen in the diagrams, the lines for the 

various depths are farthest apart on the power plot. 

They are closer together on the DS diagram, and the 

plots of bed-form velocity versus RS, V, and Vs practically eliminate the effect of depth.
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B25

0.2

£ 0.1

0.05

0.02

I I I I I I I I 

EXPLANATION

 Symbol Depth, 

in feet

I I I I I I I

0.0002 0.001

I I I I I 1IIII 'I L

0.001 0.01

DS QS

0.2

0.1

0.05

0.02

0.0002

MEAN VELOCITY, IN 

FEET PER SECOND

SURFACE VELOCITY, 

IN FEET PER SECOND

I I I I I I I

0.001 0.002

RS

FIGURE 20. Bed-form height versus measures of flow strength. (Lines exclude D = 0.l ft.)

CHARACTERISTICS RELATED TO SEDIMENT-TRANSPORT

RATES

The general trend of bed-form measurements with 

increase in transport rate can be seen in table 4. The 

bed-form length versus transport relation is virtually 

the same as that shown by the plots of wavelength 

versus flow strength. Bed-form height increased consistently with increase in transport rate (fig. 23). The 

point scatter precludes, and indeed the accuracy of the 

height values does not warrant, an attempt to fit an 

extremely accurate line to the data. The approximate 

relation for the bed forms at. depths of 0.3 and 0.5 

foot is A,°ci°-40 . The same remarks also apply to the

bed-form velocity versus transport diagram (fig. 24), 

where the general relation is about coci0 - 76 . Bed-form 

heights, wavelengths, and velocities are frequently not 

constant from one bed form to the next, within a given 

run. In both the h/i and c/i diagrams there is a faint 

suggestion that if more accurate means of measurement 

were available it might be possible to draw a separate 

line for each flow depth.

The bedload equation proposed by Simons, Richardson and Nordin (1965b), based on ripple and (or) dune 

heights and rates of travel, was tested on the present 

data. The authors cited used the equation for sand 

ranging from 0.19 to 0.93 mm in median fall diameter.
B26 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

i r i i I i 1111 i r

0.3 (after draining water)

0.5

0.5 (after draining water)

I I I I I I I I I

MEAN VELOCITY,

IN FEET 

PER SECOND

SURFACE 

VELOCITY,

IN FEET 

PER SECOND

FIGURE 21. Bed-form wavelengths versus measures of flow strength.

o.oi

0.001 Sy

EXPLANATION 

Depth, 

in feet 

0.1

0.3 

0.5

I I

I I I I II I

234 3 4

MEAN VELOCITY, IN SURFACE VELOCITY, 

FEET PER SECOND IN FEET PER SECOND

0.0005 0.001 

RS

0.0005 0.001 

DS

0.001 0.01

QS

FIGURE 22. Bed-form velocity versus measures of flow strength.

Their data were obtained in the laboratory in a large 

recirculating flume. A basic assumption in the derivation of the equation was that the bed forms are triangular from a side view. This was true for only some of

the bed forms of the present experiments. Since triangular bed forms are restricted to a very limited range 

within the spectrum of bed configurations, it would 

seem that bedload equations which depend on the
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B27

EXPLANATION

0.1

Symbol Depth, 

in feet

0.1

0.3

0.5

0.01 _LL

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 23. Relation of bed-form height to sediment-transport rate.

I T TT

o.oi

0.001

EXPLANATION

Symbol Depth, 

in feet

A 0.1 

o 0.3 

a 0.5

0.001 0.01 0.1 

SEDIMENT-TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGURE 24. Variation in bed-form velocity with sediment-transport rate.
B28 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

existence of an ideally shaped bed form will have a 

rather limited application. The equation is

where

<2& =volume rate of sediment transport (ft3 per

sec per ft width), 

A = porosity of sand bed, 

c=average bed-form velocity (ft per sec), 

h=average bed-form height (ft), and 

Ci=& constant of integration, considered equal to 

zero as long as the bed is entirely covered 

with ripples or dunes.

Volume rate of sediment transport can be changed 

to weight rate by dividing by the measured unit 

(specific) weight of the quartz sediment (91.5 Ibs per 

cu ft). This dry weight in turn can be changed to

j w *v. i ^ i, 2.66-1.00. ,., immersed weight, the latter being ^~RR~~X(dry

u. DO

weigh t) = 0.624 X (dry weight). By making these 

changes and using the measured porosity value of 

O.445, the above equation can be reduced to the form

 ic =15.8 ch,

where ic is the computed sediment-transport rate in 

pounds (immersed weight) per second per foot of channel width. (The constant C\ in the original equation 

can be taken as zero.)

Although the equation was not intended to apply to 

antidunes, those runs which produced antidunes were 

included in the present computations merely out of 

curiosity. A comparison of all the computed versus 

measured bedloads is shown in figure 25. The amount 

of point scatter is probably not excessive in view of the 

uncertainties involved in measuring bed-form heights 

and rates of travel.

CONCLUSIONS

For the sediment, flow conditions, and flume of the 

present study, the following conclusions were drawn:

1. Scatter of points in the sediment-transport diagrams 

is vastly reduced and existing relations are clarified 

by keeping water depth constant for a series of 

runs that is, by using depth as a third variable.

2. If shear, expressed as y RS, is used as the measure of 

flow strength in computing the sediment-transport 

rate, all the runs can be described by a single curve 

in which shear is uniquely related to transport 

rate.

3. Mean velocity, stream power, shear expressed as 

7 DS, and a regime bed factor are uniquely related 

to sediment-transport rate only for a constant 

water depth. Two corollaries of this conclusion 

are:

(a) If depth is permitted to vary, the transport rate 

increases as depth decreases for a given value of 

any of these four measures of flow strength.

(b) The critical or threshold flow strength required 

to initiate sediment transport, as expressed by 

these four measures, varies with water depth, 

rather than being uniquely determined by the 

sediment characteristics and bed slope.

4. The interrelationships between discharge, slope, 

and sediment-transport rate are uniquely defined 

only for a constant water depth.

5. If water depth is known, surface velocity is the only 

other measurement needed to determine the 

values of all other major variables. Surface 

velocity may be very useful in this respect because 

it is so easily measured.

6. The Gilbert-Murphy data for a comparable grain 

size are in substantial agreement with the present 

data.

7. A critical water depth somewhere between 0.1 and 

0.3 foot exists in the formation of bed forms. At 

depths shallower than this critical depth, the bed 

features have distinctly different characteristics 

than do features formed at greater depths.

DESCRIPTION OF BED CONFIGURATION, BY RUN

The following are partial descriptions of the bed configurations 

that formed during each run (numbered). The depth of water 

for each set of runs is indicated by centerhead.

0.1-foot depth

1. Flat bed.

2. Almost flat bed, although some barely perceptible features 

were present. These features were meandering scars 

which began at the wall and curved diagonally outward 

in a downstream direction; they generally fadod out completely in the middle of the channel. The;r points of 

origin alternated systematically from wall to wall. The 

amplitudes (heights) of the features were on the order of 

a few grain diameters. There was a downstream distance 

of about 2 feet from the start of one scar to thn start of its 

neighbor at the opposite wall.

3. Same as previous run.

4. Bed almost flat, although two different types of bed features

were faintly discernible:

(a) Symmetrical antidunes down much of the flume 

length but not present near the walls Crests of 

these bed forms corresponded to crests of watersurface crests immediately above. Amplitudes of 

antidunes were too small to measure accurately; 

wavelength estimated to be 0.5 foot.

NOTE: These features disappeared PS the water 

was drained from the flume at the completion of the 

run.
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE B29

EXPLANATION

Symbol Depth, 

in feet

0.1 

0.3 

0.5

0.1

O o

0.01

ine of perfect agreement

W 0.001 -

0.001 0.01 0.1 

MEASURED TRANSPORT RATE, IN POUNDS (IMMERSED WEIGHT) PER SECOND PER FOOT WIDTH

FIGUEK 25. Comparison of computed versus measured transport rates.

i.o

(b) Meandering scars, like those formed in the previous 

runs, were present, but only downstream. Heights 

of scars were as much as a quarter of an inch at 

flume walls but decreased toward the center of the 

channel, where the bed was almost flat. Downstream distance between successive scar inceptions 

was about 1 foot.

NOTE: After the water drained from the flume, 

features of the latter type were found throughout 

most of the flume. Rather than fading out in the 

middle of the channel, each one curved from its 

point of inception, at a wall, downstream and over 

to the opposite wall, at which point a similar feature 

began. Downstream distance between these two 

inception points averaged 1.5 feet. 

5. Rather symmetrical antidunes (crests were directly beneath

the crests of the ripply water surface). As the water

drained from the flume, these bed forms disappeared, and

meandering scars appeared.

6. Symmetrical antidunes like those in the previous run, but 

formed in two rows in a longitudinal direction rather than 

in a single row. An alternating pattern existed whereby 

one sand crest was closer to one flume wall and the next 

crest downstream was closer to the opposite wall. The 

distribution of sand crests and troughs was so regul*r that 

it resembled a checkerboard pattern. The water surface 

was correspondingly rippled in two rows, with neigl boring 

crests staggered relative to each other, in a downstream 

direction. The amplitude of these water-surface waves 

was higher than that of the bed forms. The distance from 

one sand crest to the downstream range of it? offset 

neighbor (parallel to flow) was about 0.35 foot.

All these bed features were smoothed out and disappeared as the water drained from the flume.

7. Same as preceding run.

8. Antidunes (sand crests directly beneath water-surface 

crests) in a single row; moved downstream in groups or 

trains. A single sand crest sometimes extended across
B30 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

the channel width but at other times occupied only about 

0.4 foot in the middle of the channel. At times the bed 

forms were almost obliterated (latter phenomenon was 

possibly more common in the first 10 ft and in the last 

10 ft of the flume). All bed features were regularly spaced 

throughout the flume length. The water surface was 

ripply but irregular.

9. Mostly flat bed; an occasional train of antidunes generally

about 3 feet long but sometimes as much as 10 feet long.

Water surface mostly rather smooth except where ripply

above antidunes; inphase relation between water and sand

crests, where such crests existed.

O.S-foot depth

10. Flat bed. Smooth water surface. Very little sediment 

movement.

11. Almost flat bed; one or two widely spaced bed features 

(dunes). Water surface smooth.

12. Almost flat bed; an occasional small dune.

13. Dunes, with fronts higher in center of channel than near 

walls. Wavelengths of dunes increased systematically 

in downstream direction. When bed-form observations 

were made, five dunes were present in the flume.

14. Dunes, many of which were poorly defined. Fronts generally perpendicular to flow direction across whole channel 

width, and higher in channel center than near walls.

15. Dunes. Wavelengths irregularly spaced.

16. Dunes, with fronts normal to flow and wavelengths not 

consistent in downstream direction.

17. Dunes, irregularly spaced and possibly tending to have 

longer wave lengths downstream.

18. Dunes, generally with well-defined fronts and perpendicular 

to flow. Wavelengths not consistent throughout flume.

19. Same as previous run. Wide range of wavelength 

magnitudes.

20. Same as previous run.

21. Dunes, with crests often curving irregularly across the 

channel rather than normal to flow. Fronts continually 

grew higher, receded, and moved downstream at various 

rates. Wavelengths were longer and more inconsistent 

in the downstream half of the flume length, but were 

regularly spaced in the upstream half of the flume.

22. Dunes, with very irregular crests that generally curved 

from one wall downstream to the opposite wall. Small 

parts of the dunes were normal to the flow direction near 

walls. Bed forms and water surface became increasingly 

rougher downstream. Dune lengths more variable in 

downstream part of flume.

23. Dunes, with very irregular and poorly defined crests, particularly upstream. Dune heights and wavelengths varying considerably in magnitude. Possible tendency for 

dune lengths to cluster, so that reaches of short wavelengths alternated with reaches of longer wavelengths.

24. Dunes, very irregular in cross-sectional and top views but 

rather evenly distributed throughout flume length.

25. Same as previous run. Water surface very rough and 

irregular.

26. Antidunes, with sand crests directly beneath water-surface 

crests. Wavelengths rather constant throughout flume 

length. Bed-form crests more rounded and water surface 

smoother than in the runs immediately preceding this. 

NOTE: Bed forms were considerably smoothed out 

after water drained from flume and bore little, if any, 

resemblance to the features that were present during the 

run.

27. Antidunes, with very well rounded crests; consistent in 

wavelength. As water drained, bed forms were considerably smoothed out, and many meandering scars (run 4b) 

appeared.

28. Antidunes, rather evenly spaced along the flume and rather 

symmetrical in form from a side view. Crests very well 

rounded. Water surface consisted of rough vraves whose 

crests were inphase with the dune crests. These water 

waves were distinctive because they broke on the upstream 

side of the wave crest or peak.

O.S-foot depth

29. Dunes, whose fronts were almost normal to Hew direction 

except near flume walls. Water surface smooth.

30. Dunes, irregularly spaced in downstream direction. Water 

surface rather smooth; very minor undulations.

31. Dunes, with very well defined angular crests th".t extended 

completely across flume width, normal to flow. Water 

surface rather smooth.

32. Dunes, with crests nearly perpendicular to flow from wall to 

wall. Crestline slightly irregular. Downstream dunes 

regularly spaced; upstream dunes not as regularly spaced 

and generally closer together.

33. Dunes, with well-defined crests normal to flow; dunes irregularly spaced relative to one another. So^ne possible 

tendencies of clustering of similar dune lengths. Minor 

undulations in water surface.

34. Dunes, with well-defined angular crests nornal to flow 

direction; not very regularly spaced in downstream direction. Rather smooth water surface, with very minor 

undulations about 1 foot long.

35. Same as previous run, except less variation in dnne lengths.

36. Dunes, with crests sometimes curving downstrerm from one 

wall to the other and at other times perpendicular to the 

flow direction. Possible tendency for clustering of similar 

dune lengths. Water surface had undulations about 1 

inch high and 1 foot long. No discernible relation between 

water-surface crests and bed-form crests.

37. Dunes, some with crests straight acorss chanrel (possibly 

more common upstream), and others with crests curving 

downstream from wall to wall (mostly in downstream hall 

of flume). Dunes tended to be regularly spaced along 

flume. Undulating water surface, with troughs directly 

over dune crests.

REFERENCES

Brooks, N. H., 1954, Laboratory studies of the mechanics of 

streams flowing over a movable bed of fine sane1 : Pasadena, 

California Inst. Technology Ph. D. thesis, 248 p.

 1958, Mechanics of streams with movable beds of fine 

sand: Am. Soc. Civil Engineers Trans., v. 123, p. 526-549.

Colby, B. B,., 1964a, Practical computations of bed-material 

discharge: Am. Soc. Civil Engineers Proc., v. 90, no. HY2, 

p. 217-246

 1964b, Discharge of sands and mean-veloc: ty relationships in sand-bed streams: U.S. Geol. Survey Prof. Paper 

462-A, 47 p.

Colby, B. R., and Scott, C. H., 1965, Effects of water temperature on the discharge of bed material: U.S. Geol. Survey 

Prof. Paper 462-G, 25 p.

Einstein, H. A., 1942, Formulas for the transportation of bed 

load: Am. Soc. Civil Engineers Trans., no. 107, p. 561-577.
FLUME EXPERIMENTS ON THE TRANSPORT OF A COARSE SAND B31

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U.S. GOVERNMENT PRINTING OFFICE : 1967 O-248-117