Flume Studies Using 

Medium Sand 

(0.45 mm)

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-A

RECEIVED

MAY 2

Flume Studies Using 

Medium Sand 

(0.45 mm)

By D. B. SIMONS, E. V. RICHARDSON and M. L. ALBERTSON

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-A

A comprehensive study of 

fluvial hydraulics

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C.
CONTENTS

	Page 

Symbols and dimensions.-.-__________-_--_-_________-_______--__-__ vi

Glossary of terms_______-____________________-_-_-____-___---__---_ vu

Abstract__ ______________________________________________________ A-1

Introduction. _____________________________________________________ 1

History ______________________________________________________ 2

Importance of roughness in alluvial channels._____________________ 5

Research program _____________________________________________ 6

Theory of resistance to flow in alluvial channels_______________________ 7

Dependent and independent variables.___________________________ 7

The Pi-theorem.__..________._.___________________________ 10

Summary of theory.___________________________________________ 19

Experimental equipment and procedure._____________________________ 20

The flume.------_-----------_-_-_-___-_-_------------------_- 20

Alluvial bed material____--___-_-______-_-______-___-_____--_-__ 21

General procedure_________-______--_____-_-_-___-_-_-_-_------ 22

Data obtained_______-______-_______-___-_-_-_--_-___-__-_-__- 23

Water-surface slope._______-_______-___-__-___-_____-__-_-_ 23

Discharge _________-___-__-_____-___-_-_________-__-_--___ 24

Water temperature.---------------------.----------------- 24

Depth____-_____-_____--_____._____._____--____.______ 25

Mean velocity___________________________________________ 25

Velocity profiles._____--____-__-_______-_-____-_____----___ 26

Bed material-.__________________________________________ 26

Total sediment load______________________________________ 26

Suspended sediment samples-_______________________________ 27

Bed configuration________________________________________ 27

Presentation of data_______________________________________________ 28

Water-surface slope.___________________________________________ 28

Discharge.___________________________________________________ 28

Depth of flow_____________---____-________--__-_-_-_-_----_--- 32

Kinematic viscosity___.________-__-__--_________-___--__--__-_- 32

Mean velocity_______________________________________________ 32

Velocity profiles._______-____-____---_-____-___-__--____-_-__-_ 33

Bed material..___________________________-_-__-_-_-----_---_-- 34

Total sediment load__________________________________________ 35

Suspended sediment---_--_--_________--_-_--------------_--__- . 35

Bed configuration____________________________________________ 36

m
IV CONTENTS

	Page 

Observed flow phenomena._________________________________________ A-36

Bed forms in the tranquil-flow regime ____________________________ 39

Plane bed without movement _______________________________ 39

Ripples__-____-_-_________________________________________ 41

Dunes____________________________________________________ 42

Transition from dunes to rapid-flow forms ____________________ 47

Bed forms in the rapid-flow regime_______-_------____________--_ 49

Plane bed with movement._________________________________ 49

Standing waves ___________________________________________ 50

Antidunes ________________________________________________ 52

Analysis of data___________________________________________________ 58

Variation of velocity with depth.________________________________ 59

Variation of total sediment load concentration and Manning n______ 60

Variation of tractive force and total sediment load_________________ 61

Resistance to flow relations _____________________________________ 62

Forms of bed roughness related to size of bed material. ____________ 68

Conclusions._ _____________________________________________________ 73

Literature cited ___________________________________________________ 75

ILLUSTRATIONS

Page 

FIGURE 1. Schematic diagram of the flume___------__---___-____---- A-21

2. Size distribution curve of the bed material..___-_____---___ 22

3. Typical water-surface profiles_-__------_----------------- 29

4. Typical vertical velocity distribution curves at different stations across the flume.________________________________ 33

5. Major forms of bed roughness___----------------_-------- 37

6. View of bed with no sediment movement (run 14)___________ 39

7. View upstream of ripple configuration (run 9)______________ 41

8. View upstream of dune configuration (run 23)______________ 42

9. Flow over dune configuration similar to that shown in figure

10__________________________________________________ 43

10. View upstream of dune configuration (run 24)______________ 44

11. Three-dimensional flow over the sand bar configuration of

figure 12 (run 35)_______________-----_--__---__------- 48

12. View upstream of sand bar configuration (run 35)__________ 49

13. View upstream of a plane bed during rapid flow (run 26)___ 50

14. View downstream of standing wave during rapid flow (run 39) _ 51

15. View upstream of cross-laced configuration (run 27)_________ 51

16. Water surface before the antidune wave starts to build up

(run 42)_________________________________________-__ 52

17. View across the flow at an antidune wave that is forming____ 53

18. View upstream of an antidune wave at the point of breaking-- 53

19. Antidune wave after breaking (run 42)____________________ 54

20. View of bed configuration after antidune flow._____________ 54

21. Profiles of the actual water surfaces and bed configurations

(runs 44 and 41)______--_-_-_-_-_---_---------------- 56

22. Variation of velocity V with depth D___-------------_--_- 59

23. Variation of shear TO and concentration of total load CT----- 62

24. Variation of V r0/ V*(ATs)d with V#d/». ------------------- 63
CONTENTS V

Page 

FIGURE 25. Variation of (F*d/»0+5 with VrQ/V^(^ys) d and FT. ___._.__ A-64

26. Variation of V*d/v + 5 with [rT0/V*(AY.)d]°-*«/(T r/V?O) I/8 - 66

27. Variation of VdyD*j(VH^D)u* with [OSd/D^/VC^-D/")]2'3 -- 67

28. Criteria for bed roughness in alluvial channels____-_-------- 69

TABLES

Page 

TABLE 1. Summary of data. ________,_-_-__-__________-____-_------ A-30

2. Values for predicting bed form. _-_--__-_--_---_--_---__---- 70
	Symbols and Dimensions

A Area of flow cross section.__________-_-__-_.__

B Width of the flume__.___________--__-_-_-___

C Chezy coefficient of discharge in dimensionless

^ form which is equivalent to V/V*.

CD Drag coefficient for the particle. _ _ _ _ I ~J"_ _______

CT Concentration of total load.____-____---_-__-_

Cw Concentration of fine material.________________

d Median fall diameter of bed material.__________

d t Median fall diameter of total sediment load___-_

D Average depth of flow________________________

Fr Froude number (Vj-^gD) ______________________

g Acceleration of gravity.______________________

G Total load being transported-_________________

h Average height of bed roughness.______-___-_-_

K Any constant,________________-----_-------_

L Average distance between features of bed rough- 

	ness. 

n Manning coefficient of roughness ______________

rib Manning coefficient of bed roughness (Einstein).

P Wetted perimeter________________-_---_-.----

Q Discharge of water-sediment mixture____-__----

r Relative coefficient of correlation______________

R Hydraulic radius (A/P) ___________-__--__-_--

Re Reynolds number VD/v _____________-------_-

sfe Shape factor of the channel cross section. ______

sfp Shape factor of the sediment particle___________

sfr Shape factor for the reach of the stream, sinuosityS Slope of energy gradient equal to water surface

	slope in steady, uniform flow. 

t Time____________________________-------_-

T Temperature. _ ________________.___--_-_----_

V Average velocity based on continuity principal. _

Vp Average velocity based on velocity profiles______

l 7* Shear velocity which is ygDS, VWp- __---_-----

w Fall velocity of sediment particles. ____________

7 Specific weight of water._____-_________---_--

y, Specific weight of sediment________.____--__-_

AY Difference between specific weights of air and

	water. 

AY, Difference between specific weights of sediment

	and water. 

5' Thickness of laminar sublayer.________________

v Kinematic viscosity___--_-__-----------------

H Dynamic viscosity.-_____-_________-___------

Dimensions

L2

L

O

O

ppm

ppm

L

L

L

O

L/T2

F/T

L

O

L

Li/e

Li/e

L,

L»/T

O

L

O

O

0

O

O

T

O

L/T

L/T

L/T

L/T F/LS

F/U F/LS

F/L3

L

L2/T

Ft/L»

Units

ft2

ft

 

ft

ft

ft

ft/sec2

Ibs/sec

ft

ft

fti/e

ftl/6

ft

ftS/sec

ft

sec

0 C

ft /sec

ft/sec

ft/sec

ft /sec

lbs/ft3

lbs/ft3

lbs/ft»

lbs/ft 3

ft

ft2 /sec

lb-sec/ft2
CONTENTS VII

Dimensions Units 

P Density of water....-_-_____________-_---_-- Ft2/!4 Slug/ft3

P, Density of sediment... ______________________ Ft2/!4 Slug/ft3

Ap, Difference between density of sediment and Ft2/L4 Slug/ft3

water. 

o- Relative standard deviation of the size distribu- O

tion of the sediment. 

TO Tractive or shear force developed on the bed, F/L2 lbs/ft2

yDS. 

r e Critical tractive force associated with beginning F/L2 lbs/ft2

of bed movement.

GLOSSARY OF TERMS

Alluvial channel: A channel whose bed is composed of appreciable quantities 

of the sediments transported by the flow at a given discharge or greater.

Antidunes: Symmetrical sand and water-surface waves which are in phase, 

and which move upstream. The surface waves build up with time, becoming 

gradually steeper on their upstream sides until they break like surf and disappear. 

These waves usually develop, break, and reform in groups of two or more.

Bed material: The material of which a stream bed is composed.

Dune: A sand wave of approximately triangular cross section in a vertical 

plane in the direction of flow with gentle upstream slope and steep downstream 

slope. It travels downstream as a result of the movement of the sediment up 

the upstream slope and the deposition of part of this material on the downstream 

slope.

Equal transit rate (ETR): A method of sampling suspended sediment to obtain 

the mean concentration of the water-sediment mixture in the flume. By this 

method the depth integrating sampler (DH-48) is traversed through equally 

spaced verticals at an equal transit rate for each vertical.

Fall diameter: The diameter of a sphere that has a specific gravity of 2.65 and 

also has the same terminal uniform settling velocity as the particle (any specific 

gravity) when each is allowed to settle alone in quiescent distilled water of infinite 

extent and at a temperature of 24° C.

Median diameter: The midpoint in the size distribution of a sediment such 

that one-half of the weight of the material is composed of particles larger than 

the median diameter and the other one-half is composed of particles smaller 

than the median diameter.

Plane bed: A bed without elevations or depressions larger than the maximum 

size of the bed material.

Ripple: Small ridges and (or) crests, and troughs similar to dunes in shape, 

but smaller in magnitude, which have rather small width normal to the direction 

of flow.

Sand wave: A ridge (such as ripples, dunes, or symmetrical undulations) on 

the bed of an alluvial channel formed by the movement of the bed material.

Sediment: Fragmental material that originates from weathering of rocks and is 

transported by, suspended in, or deposited by water.

Sediment concentration: The ratio of dry weight of sediment to total weight of 

the water-sediment mixture, usually expressed in parts per million (ppm).

Standing waves: Symmetrical sand waves and water waves that are in phase 

and that gradually build up and just as gradually die down. Waves of this type 

are stationary, or essentially so, and usually develop in series and often reform, 

somewhat periodically, after disappearing.
VIII CONTENTS

Suspended load: The sediment moving in suspension in a fluid as a result of 

turbulent currents and (or) by colloidal suspension.

Total load: The total amount of sediment that is transported by water in a 

given length of time.

Fine material: That part of the total load composed of sizes not found in 

appreciable quantities in the bed material normally the silt and clay sizes 

(d<0.062 mm).
STUDIES OF FLOW IN ALLUVIAL CHANNELS

FLUME STUDIES USING MEDIUM SAND (0.45 mm)

By D. B. SIMONS, E. V. KICHARDSON, and M. L. ALBERTSON

ABSTRACT

The results pertaining to the progress during the first year of a comprehensive 

study of fluvial hydraulics, specifically roughness in alluvial channels, are presented.

The report is based on the data collected by using a recirculating rectangular 

flume of adjustable slope, 8 feet wide, 2 feet deep, and 150 feet long with an 

alluvial bed of sand approximately 0.7 foot deep. A typical river sand has been 

utilized. Its median diameter, d, is 0.45 mm and its relative standard deviation, 

a, is 1.60.

A total of 45 runs have been completed over a range of bed roughness forms 

extending from the plane bed with no movement to antidunes. In order to achieve 

this range, the discharge was varied from 2 to 21 cubic feet per second, the average 

velocity was varied from 0.5 to 7 feet per second, the average depth of flow was 

varied from 0.3 to 1.0 foot, and the slope of water surface was varied from 0.00014 

to 0.01. Other variables measured included: water temperature, bed roughness, 

suspended sediment load, and total sediment load.

Terms describing channel roughness were formulated and tested based on the 

data collected. The results indicate, as one possibility, that the Chezy coefficient 

of discharge in dimensionless form C/Vg> is a function of parameters involving the 

Froude number, viscosity of fluid, fall velocity, specific weight of the sediment, 

median diameter of the sediment particles and slope of the water surface. The 

various expressions presented were formulated on the fundamental concepts of 

fluid mechanics, dimensional analysis, and a detailed study of the variations of 

the variables measured.

In the two regimes of flow the following forms of bed roughness were observed.

For tranquil flow regime: plane bed without movement, ripples, dunes and 

transition from dunes to rapid flow forms.

For rapid flow regime: plane bed with movement, standing sand waves, and 

antidunes.

These forms of bed roughness are discussed and defined in various relationships.

Other data of both a laboratory and a field nature were combined with the flume 

data to develop a graphical relationship in which the form of bed roughness is 

related to size of bed material.

INTRODUCTION

The problem of defining roughness in alluvial channels dates back 

several centuries. A satisfactory solution of this problem has thus far 

eluded man. The principal reasons that only limited answers, which 

in some cases are of questionable value, have been developed are the

A-l

577668 O 61 2
A-2 STUDIES OF FLOW IN ALLUVIAL CHANNELS

broad scope of the problem and the multitude of variables influencing 

resistance to flow in alluvial channels. Alluvial channels are more 

complex than rigid channels because the form of the bed is a function 

of the flow and the bed material. That is, not only do the roughness 

elements resist the flow but they in turn are formed by the flow. In 

the experiments discussed in this report, the form of the bed was 

changed from plane to ripples, to dunes, and to antidunes by altering 

flow conditions. The resistance to the flow for each bed form is quite 

different. Thus far, most scientists working in this field have been 

limited by time, facilities, lack of instruments and fund. As a result 

only small parts of the complex problem have been thoroughly investigated. However, sufficient isolated groups of data have been collected 

and are being collected so that ultimately it should be possible to 

combine ideas and data resulting from these separate efforts to determine a general solution superior to any obtained thus far.

HISTORY

It is impossible to give proper credit to all who have made contributions of value in the field of fluvial hydraulics. Consequently, only a 

few of the more recent investigators whose work relates directly to 

resistance to flow in open channels will be cited.

One of the most thorough and complete groups of flume data ever 

collected resulted from a study by Gilbert (1914) which was sponsored 

by the U.S. Geological Survey and the University of California. He 

investigated the natural laws controlling the movement of material 

found on the beds of streams. The bed materials that he used consisted of sands and gravels obtained from typical California river 

beds. In all, more than one thousand tests were completed. These 

data have proved to be very valuable to the many investigators 

interested in flow in open channels.

Shields (1936) presented an analysis of the formation of roughness 

in stream beds. He proposed that the major parameters controlling 

bed undulations are r0 /Apd and V*d/v=11.6 d/5' 

in which

TO=critical drag force per unit area 

A/a=submerged density of the particles

d=median diameter of the particles

d'=thickness of the laminar sublayer 

V*= shear velocity 

v=kinematic viscosity

His parameters have been rather widely used in the analysis of problems involving drag and transportation of sediment.

The fundamental analyses that Einstein applied to sediment 

transportation (1950), to hydrodynamic forces on a rough wall
FLUME STUDIES USING MEDIUM SAND A~3

(Einstein and El-Sayed Ahamd El-Sammi, 1949), and to river channel 

roughness (Einstein and Barbarossa, 1951) have played an important 

role in stimulating interest and in providing a better understanding 

of the complexity of the laws of fluvial hydraulics and hydrodynamics.

Kalinske and Hsia (1945) presented the results of a study of the 

transportation of fine material by flowing water that is of importance 

because of the extremely small size of material used (d=Q.Qll mm). 

These data provide valuable information on how very fine material 

influences alluvial channel flow phenomenon.

The importance of fluvial morphology in the study of open channels 

was clearly defined by Lane (1955). An adequate understanding of 

this science is necessary to the understanding of stream forms and 

their behavior.

The hydraulic geometry of stream channels, as presented by Leopold 

and Maddock (1953), has contributed to a better understanding of 

the behavior of streams and rivers and verifies the existence of general 

laws that describe channel geometry. These laws are of the form 

developed for canals by Lindley, Lacey, Blench and others in India 

(Blench, 1957). That is, the wetted perimeter P, the hydraulic 

radius R, and the velocity V are all functions of the discharge Q.

Barton and Lin (1955) reported the results of a flume study done 

at Colorado State University, which was sponsored by the U.S. Army 

Corps of Engineers. The data presented therein were collected using 

a flume 80 feet long and 4 feet wide. The data, the collection procedures, and the analysis of the data are worthy of attention.

A study of roughness in alluvial channels by Ali and Albertson 

(1956) provides a qualitative expression defining roughness of the form

and relating these variables to the transition function

(7=Chezy coefficient 

Re=Reynolds number 

5 hydraulic radius 

c?=median diameter of bed material

Their work indicates that:

1. The laminar sublayer and Reynolds number are intimately related 

to the formation of ripples, dunes, and sand waves;
A-4 STUDIES OF FLOW IN ALLUVIAL CHANNELS

2. An increase in size of bed material decreases the influence of the 

laminar sublayer;

3. The normal sequence of the forms of bed roughness with increasing 

velocity is: no movement, ripples, dunes, sand waves, plane bed, 

and antidunes;

4. Other variables being held constant, the magnitude of bed roughness increases with a decrease in size of bed material within the 

noncohesive size range.

The effect of introducing fine material, clay size, on transportation 

of bed load was investigated in a turbulence tank by Makarechian. 1 

He found that adding fines and holding the energy input constant 

does not dampen the turbulence appreciably, decreases the fall velocity 

of the sediment in transport; and increases the concentration at all 

points in the turbulent fluid.

Fine material may play an important role in channel stability, 

channel roughness, and sediment transport, since it causes an increase 

in viscosity and an increase in fluid density.

The analysis of sediment transportation by Bagnold (1956) is of 

importance to the study of roughness in alluvial channels, for, his 

concepts and equations provide a foundation from which predictions 

can be made regarding the influence of size and gradation of sediment on flow phenomenon. As the size of sediment is reduced, a 

size is reached where no additional energy input is required to transport additional quantities of this size and smaller. In fact, Bagnold's 

equations imply that the addition of very fine material may in effect 

impart energy to the flowing fluid. This concept is consistent with 

the results of the research in the turbulence tank done by Makarechian. 

It also compliments the current concepts proposed by many individuals working in the field of fluvial hydraulics concerning the influence 

of fine material on the behavior of fluvial streams.

A study of the theory of stable channels was completed by Simons 

and Albertson (1960). The field data presented therein should provide a valuable means of testing the results of laboratory studies of 

roughness in alluvial channels.

Liu (1957) presented a paper that explains accurately and in detail 

the mechanics of ripple formation. He concludes that ripples result 

primarily from the instability of the zone of large velocity gradient 

at the surface of the alluvial bed. He has formulated an experimental criterion that predicts the formation of ripples. That is,

i Makarechian, A. H., 1956, Effect of wash load on suspension of bed load: M.S. thesis, Colorado State 

University, Fort Collins, Colo.
FLUME STUDIES USING MEDIUM SAND A~5

the beginning of ripples is given by plotting experimental data in 

accordance with the functional relation.

y* =

w

where V* = shear velocity 

w = fall velocity 

v = kinematic viscosity 

d = median diameter of bed material

The results of using these same parameters to define other regimes 

of bed roughness were presented by Albertson, Simons, and Richardson 

(1958).

Maddock (written communication, 1957) points out some of the 

significant factors of the foregoing analysis of ripple formation presented by Liu (1957) and also the importance of Durand's (1953) 

study on the transport of sediment in closed conduits. In addition, 

Maddock proposes that:

1. Sediment in motion should not be divided into a bed load and a 

wash load;

2. V*, the shear velocity, is not a parameter of primary importance 

with respect to sediment movement;

3. Velocity distribution in alluvial channels is different from that in 

rigid boundary channels;

4. The shear velocity Reynolds number V*D/v is not directly related

to sediment movement.

Finally, Maddock proposes a method for analysis of roughness in 

channel utilizing the concept that C/Jg probably consists of additive 

terms.

IMPORTANCE OF ROUGHNESS IN ALLUVIAL CHANNELS

The importance of accurately describing the resistance to flow or 

roughness in alluvial channels becomes immediately apparent when 

it is realized that the accurate evaluation of channel roughness is 

essential to the solution of all problems associated either directly or 

indirectly with water flowing on alluvial material. A more precise 

knowledge of channel roughness would assist materially in the design 

of stable channels. For example, slopes could be accurately determined consistent with existing conditions. Furthermore, an understanding of the behavior and control of rivers and river systems 

could be developed, and, the influence of imposing changes such as 

cutoffs, increasing or decreasing the sediment load, contractions at
A-6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

bridges, and changing the characteristics of the sediment load could 

be predicted with greater precision.

In the design of hydraulic structures that are influenced by alluvial streams, or that influence the behavior of alluvial streams, an 

improved knowledge of channel roughness is a practical necessity. 

Here the applicability (or lack thereof) of data from the study of 

models to the prototype for flood routing, scour problems, backwater 

curve computations, and similar problems depends to a large extent 

on how accurately channel roughness can be estimated.

The accuracy with which discharge can be determined, excluding 

direct measurement, is a function of the accuracy with which channel roughness can be evaluated. Resistance to flow in alluvial channels varies between wide limits with discharge, temperature and size 

of bed material, and the magnitude and type of sediment load. As 

a result, a change in discharge may occur without a corresponding 

change in stage. The cost of flood control structures, bridges, and 

soil conservation methods is directly related to the accuracy with 

which floods can be measured.

These foregoing examples as well as many others, such as the influence of roughness on sediment transport, flow problems involving 

multiple roughness, all serve to illustrate the need for study and 

improvement of the knowledge of how and why channel roughness 

varies and how to evaluate its magnitude for any given set of 

conditions.

RESEARCH PROGRAM

In recognition of the need for a more basic and theoretical treatment 

of the science of fluvial hydraulics a research project was organized 

at Colorado State University in September 1956 to study the mechanics of water and sediment movement in alluvial channels. Since 

this is an extremely broad and complex field of study, it was decided 

to consider only one aspect of the total problem at a time, in order 

to minimize the diffusion of effort. The initial objective of this 

program, therefore, was the evaluation of bed roughness in alluvial 

channels, a subject of paramount importance, as already stated. 

This objective was further subdivided into a laboratory phase to be 

followed later by a field phase.

The project is a part of the research program of the Water Resources 

Division of the U.S. Geological Survey. P. C. Benedict and R. W. 

Carter assist with project planning and review of research developments. The study is under the supervision of S. K. Love, chief, 

Quality of Water Branch.

The laboratory studies of roughness in alluvial channels are being 

made in a recirculating laboratory flume 150 feet long, 8 feet wide and
FLUME STUDIES USING MEDIUM SAND A~7

2 feet deep with a sand bed. The flume is described in detail in the 

section, "Experimental equipment and procedure."

The investigation thus far done in the flume has involved :

1. Modification of the pumping system to meet the needs of the 

program;

2. Development of special techniques applicable to the measurement 

of the independent variables;

3. Selection and placement of a suitable sand to a depth of about 0.7 

feet on the bed of the laboratory flume;

4. Collection of data for the sequence of runs thus far completed to 

obtain information on discharge, velocity, velocity distribution, 

slope, temperature, depth, bed roughness, total sediment load 

and photographs.

THEORY OF RESISTANCE TO FLOW IN ALLUVIAL

CHANNELS

In a study of flow in alluvial channels, so many variables are 

involved that it is difficult to determine the fundamental relationship 

that exists between them. Fortunately, dimensional analysis can 

be employed so that the variables are arranged in dimensionless 

parameters to help reduce the number of variables, make the results 

applicable regardless of the system of units employed, and systematize 

the research and the analysis of data.

DEPENDENT AND INDEPENDENT VARIABLES

If the depth of flow D is selected as the single dependent variable 

for study, it will depend upon variables that fall into each of these 

four categories as follows:

Z)=</>[Geometry, flow, fluid, sediment] (1) 

or more specifically

D=(f>2 [B, sfc , sjr, Q, S, p, n, Ay, Cw, d, sjp> a, Ays] (2)

in which D is the only dependent variable and all the others are 

independent variables as defined in the list of symbols.

At this point it is important to be sure that each of the variables 

on the right side of equation 2 is actually independent and that each 

is necessary in determining the depth D. Frequently, in dimensional 

analysis, the mistake of inserting an extra dependent variable or 

extra independent variable is made. Such mistakes cause confusion 

that may be quite difficult to eliminate. Therefore, the initial 

selection of variables is worthy of considerable study before proceeding 

with the dimensional analysis.
A-8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

The variables that describe the g-eometry of the channel are B, 

sfc , and sfr . The width of the channel B enters the problem because 

the discharge Q is distributed over this width, the width may control 

to some extent the secondary flow, and there may be one type of flow 

along one side of the channel and a different type of flow along the 

other side. If the shape of the cross section sfc is varied, the depth 

of flow also varies to some extent. Although this variation may in 

some cases be relatively unimportant, the shape factor sfe should, 

nonetheless, be included. The shape factor for the reach sfr as 

measured by sinuosity may vary, affecting the depth, and should 

also be included. The flow variables are discharge Q and slope of the 

energy gradient S. The discharge Q is obviously of paramount importance because it can be varied at will, and such variation will 

cause the depth D also to vary. Since the slope can vary independently of the other variables and the depth of flow will then also vary, 

the slope S is both independent and of great importance. In establishing the depth of flow, no additional variables are needed to describe 

the geometry and the flow.

The variables that describe the fluid and influence the depth are 

the density p, the viscosity v, and the difference in specific weight 

Ay, across the interface of the lighter and heavier fluids at the surface. 

Density influences the depth because it expresses the mass effects 

and, therefore, is the heart of inertia, which plays a very important 

role in nearly all flow problems, aside from purely viscous flow (even 

here density seems to play an important part in describing completely 

the flow pattern, such as flow around tiny particles of sediment, for 

very small Keynolds numbers). Viscosity, quite obviously, has an 

influence because of its close association with boundary shear or drag 

and fall velocity.

The influence of Ay is not as obvious as that of viscosity. It is 

the fluid property that accounts for the influence of gravity. Actually, 

a stream is always submerged in an atmosphere of some kind. This 

atmosphere is usually air, but it might also be clear water with a 

stream of sediment-laden water (such as a density current in a reservoir) flowing under it, fresh water with a salt-water intrusion flowing 

under it, or warm water with a cold-water stream flowing under it. 

In each case it is the effective or submerged weight of the lower fluid 

that is important in reflecting the influence of gravity. If the atmosphere is air above water in an open channel, then AT is essentially 

the same as the 7 for the water. But if Ay is quite small, as for an 

interface of cold and warm fluids or for a salt water intrusion, then 

Ay must be considered as a factor. This is discussed in greater detail 

on page 9.
FLUME STUDIES USING MEDIUM SAND A-9

In brief, the fluid properties are very closely associated with motivating and retarding forces involved in open-channel flow. The 

effect of gravity, as reflected in Ay is to cause flow to take place; the 

viscosity is the important fluid property that resists flow causing 

shear; and inertia, as reflected in the density, p, is a property that 

resists any change in velocity (either in magnitude or direction). In 

steady, uniform flow the change in velocity is associated with turbulence and secondary circulation, which both play an important part 

in flow in open channels.

The properties of the sediment involved with flow in alluvial 

channels can best be described by the characteristic size d of the 

sediment in the bed, its characteristic size distribution as represented 

by a, its characteristic shape factor sfp , and the effective or submerged 

weight of the sediment.

The characteristic size d of the sediment is a subject of considerable 

discussion since it is intended that this size be the significant size. 

When there is a wide range of sizes, it is difficult to say whether the 

50 percent size or some size smaller or larger is most significant. 

Until further information is available, the 50 percent size will be 

assumed as most representative and significant. The relative standard deviation is the second moment of the size distribution about the 

mean size, relative to the mean size. It may be dimensionless or not, 

depending upon the method of determination. The selection of the 

characteristic shape factor, like the diameter d, is subject to debate. 

In addition, it is difficult to measure shape of the particle. An 

indirect method would be to use Ad, the difference between the 

median diameter, as determined from mechanical analysis, and the 

median fall diameter, as determined by sedimentation methods, or to 

use a ratio of the two diameters. The ratio has the advantage that 

it would vary from 1 for perfect spheres, whereas the Ad would vary 

from 0 for perfect spheres. The shape factor may not be very important with ordinary sediments in the sand size range, as it may 

approach a constant. However, sjp , d and Ays relate the interaction 

of the fluid and the sediment. Also d is involved as a grain roughness 

element, in addition to being related to the form of bed roughness, 

such as ripples and dunes. That is, with a plane bed condition the 

size of the sediment determines the roughness; with a dune bed 

condition the resistance to flow results from the bed configuration 

generated by the flow and d.

The use of Ays is another item that warrants discussion. Actually, 

Ays represents the effective specific weight of the particles of sediment, 

but it gives no clue whatsoever about either the density or the weight 

of a particle, except through gravitational acceleration, or the specific 

weight of the fluid. Basically, the question is which one is more

577668 O 61 3
A-10 STUDIES OF FLOW IN ALLUVIAL CHANNELS

important, the effective specific weight Ay, of a particle or the actual 

density ps of the particle. The Ays represents the net effect of gravity 

and hence the driving force causing downward motion, whereas ps 

reflects the inertia of a particle that resists any change in velocity 

(either in magnitude or direction or both). The inertia may be of 

particular importance in the association of the particle with the 

turbulence of the flow and the consequent continual changing of 

velocity (both in magnitude and direction). But if a difference 

(such as A7S =7S 7 or Aps=ps p) is used instead of simply ys or ps> 

then A7S is most significant.

The concentration of fine material Cw is a property of the fluidsediment mixture, which actually changes the effective properties of 

the fluid. Past research, both in the laboratory and in the field, 

shows that as Cw is increased from small to large values, the concentration of the bed material load in suspension is increased. Therefore, Cw is of considerable importance.

THE PI-THEOREM

Using D, Q, and p as the repeating variables and applying the 

Pi-theorem to equation 2 yields

, .

' sfc> sfr> QP Q2 n d ~wD2 =0

(3)

In equation 2 the fall velocity w which relates the fluid and the 

sediment has been substituted for sfp . The fall velocity depends on 

d, sjp , A7 S and /*, therefore it is not an independent variable. However, at this time it is the best measure of the interaction of the 

fluid and the sediment.

Although it is very helpful to use Q in determining equation 2, 

it is not usually included in the parameters involving ju, A7, w, and 

A7,. If these parameters are multiplied by (B/D) n in such a way 

that Q is divided by the area BD, the velocity V can be substituted, 

which then makes these dimensionless parameters more significant. 

Equation 3 then becomes

Q VDP V* n d_ _

o, > ) L> w, j-j> -J-7-: (4)

The parameters involving p, and A7 can now be recognized as the 

Reynolds number and the Froude number, respectively. Furthermore, since Ary/p^g for flow of water in open channels, the Froude 

number can be put in the form
FLUME STUDIES USING MEDIUM SAND A-ll

The last term in equation 4, and various other forms of it, have 

been much abused. Frequently, attempts have been made to call 

this term the Froude number of the particle when it is actually proportional to the drag coefficient, except that the depth D must be 

replaced by d, the velocity V must be replaced by the fall velocity w, 

and the inverse of the parameter must be used. Therefore, if the 

parameter is multiplied by D/d and (w/V)2 and then taken to the 

minus-one power there results the drag coefficient CD \

which can be used to replace the last term in equation 4. 

Equation 4 can now be written as

05 -jy */«, sfT , S, Re, Fr, Cw, -^ ^ a, <70J=0 (6)

In some cases it is desired to involve viscosity in a Reynolds number of the falling particle Rep instead of the Reynolds number of the 

flow Ref. This can be done for equation 6 by multiplying the dimensionless Reynolds number term by w/V and d/D which yields

VD w d wd >.

and equation 6 can be rewritten as

0e I jy sfc, sfr, S, Rep , Fr, Cw, p> ^ <r, CD \ Q (8)

Before equation 6 or 8 can be used satisfactorily, it is necessary 

to simplify them by reducing the num.ber of terms being considered. 

This must be done with considerable care to be sure that important 

variables are not eliminated.

Laboratory investigations of alluvial channels and most field channels may be approximated by a rectangular shape. If this is not true

in the field, it usually is due to one of the following three factors:

1 . The channel is so wide, relative to the depth of flow and the size

of the bed material, that two or more forms of bed roughness

are existing side by side in the same channel. For example,

there may be a deep part of the cross section and a shallow

part, or a dune-covered part and a plane part or a combination
A-12 STUDIES OF FLOW IN ALLUVIAL CHANNELS

of these. Under these conditions the cross section should be 

divided into as many parts as necessary and usually each part 

can be approximated by a rectangular shape.

A potential danger in assuming that the channel can be separated into parts is the possibility of transverse flow between 

parts. Such flow has not been investigated but is probably of 

secondary importance.

2. The bed of part of the cross section may have rock or clay exposed 

so the channel is not truly alluvial throughout. By assuming, 

however, that the bed is completely alluvial and that all parts 

of the cross section can be represented by rectangular shapes, 

the shape factor term sfc can be eliminated.

3. In the flume the sides of the channel are rigid and straight; consequently, the shape factor, sfr can be eliminated.

The width-depth ratio B/D is a term that is probably of secondary 

importance provided B/D>5. When B/D<^5, the resistance and 

other effects of the side walls or banks may become appreciable. The 

ratio may be of some importance for larger values, however, because 

of its relation to secondary flow and, possibly, its relation to threedimensional flow. The actual phenomenon of secondary flow, as well 

as its importance, is not at all well understood. However, preliminary studies now under way indicate that a series of pairs of rotating 

vertical cells are formed under certain conditions. The extent to 

which the ratio BfD is associated with these cells, and the importance 

of the cells has not yet been established. However, the importance 

may be assumed for the present to be secondary compared with the 

other variables involved.

The concentration of the fine material Cw is a variable that enters 

the problem only if sediment is introduced that has a size smaller than 

that found in appreciable quantities in the bed material. If such 

material is introduced then Cw must be considered. Otherwise 

Cw may be neglected. The extent of the influence of this fine material 

has not yet been determined. On the one hand, sediment of a size 

appreciably smaller than those sizes found in appreciable quantities 

in the bed material will effectively increase the density and viscosity 

of the fluid. Consequently, research in the turbulence tank and 

data from the Colorado River have shown the concentration of bedmaterial load to be appreciably increased with fine material present 

over that concentration with clear water.

On the other hand, if the fine material introduced is just barely 

smaller than the bed material its influence is not known. It may 

increase the concentration of bed-material load, or it may decrease 

the concentration of the bed-material load if it acts to replace in part
FLUME STUDIES USING MEDIUM SAND A~13

the larger sizes. Fine material can be neglected only if there is none 

present.

The relative standard deviation a of the size distribution of the 

bed material is a factor that is of considerable importance under certain conditions and, apparently, of much less or no importance under 

certain other conditions. In the unsteady process of scour, Thomas 2 

has shown that the rate of sediment scour may be doubled by cutting 

the standard deviation in half. Liu (1957), however, has shown that 

ripples are initiated in sediments of the same median diameter, but of 

varying standard deviation, at about the same flow conditions. Since 

it is likely that ripple formation is more closely related to steady flow 

in alluvial channels than scour, the assumption can be made as a 

first approximation that the relative standard deviation v is of secondary importance.

The use of the Froude number as a parameter has caused considerable controversy. Fundamentally, it expresses the ratio of the forces 

of inertia to the forces of gravity. It is important when the flow is 

accelerating either locally dF/d^O or convectively dF/da^O. In 

other words, it can be important when the free surface flow involves 

unsteady flow or nonuniform flow. Unsteady flow exists when the 

velocity at a point is changing with time dF/d^O as is reflected by 

increasing and decreasing discharge and velocity during a runoff event. 

Usually, if the rise and fall take place over a relatively long period of 

time the flow can be approximated as steady that is dF/d^O. 

Nonuniform flow exists in most natural channels because of local expansion or contraction of the cross section. In uniform channels 

such as a laboratory flume or especially straight reaches of artificial 

channels, however, the flow can be assumed to be uniform. Nonuniformity in these cases may be due to backwater curves, surface 

waves or sand waves. It should be remembered that since velocity 

is a vector quantity, it has both direction and magnitude. Therefore, 

the flow is nonuniform if neither direction or magnitude changes 

with distance in the direction of flow.

Although there is little question that the Froude number enters the 

problem if surface waves are present, there is considerable controversy over the importance of the Froude number if sands waves are 

present on the bed without water waves at the surface. For example, 

as the flow passes over the crest of a dune, it is accelerated by the action of gravity, and the contact load of the sediment climbs the dune 

to the crest, beyond which it drops by gravitational action into the 

trough. Some scientists argue that gravitational action is inex_

2 Thomas, E. K., 1953, Scour in a gravel bed: M.S. Thesis, Colorado State University, Fort Collins, 

Colo.
A-14 STUDIES OF FLOW IN ALLUVIAL CHANNELS

orably involved in this process and hence the Froude number is important. Conversely, other scientists argue that essentially the 

same accelerations of flow can take place in a closed conduit, owing 

to irregularities of the boundary where the driving force is due to a 

pressure gradient, rather than a sloping water surface or an energy 

gradient due to gravity. They further argue that in each case a gradient of piezometric head P/y-{-Z is the driving force, regardless of 

the cause. Finally, they point out that data taken from a closed 

conduit flowing full with an alluvial bed have given no conclusive 

evidence to prove that Froude number enters the problem. Also, 

Rouse (1950) states that with a completely confined flowing liquid 

gravitational acceleration does not effect the flow pattern, and, hence, 

there is no Froude number effect. When the Froude number has 

been used, in some cases it may be a modified coefficient of drag.

The Froude number Fr can be combined with the slope S to obtain 

the Chezy discharge coefficient C/-Jg. In so doing, either the slope or 

the Froude number apparently can be replaced by Cf-Jg". Because 

the Froude number is so intimately related to the flow with water 

surface waves present, it is retained and the slope is dropped when 

C/-Jg is included.

The need for some form of the Reynolds number to reflect the influence of viscosity has clearly been demonstrated by Ali and Albertson (1956), and by Liu (1957) to be important if the full significance of the phenomena associated with flow in alluvial channels is to 

be understood. Apparently, this can be accomplished by using the 

Reynolds number of the flow Ref= VD/v the Reynolds number of the 

particle Rep=wd/v, or the shear velocity Reynolds number Res= 

V*d/v=ll.6d/d', depending upon the particular need. The variable 

8', is the thickness of the laminar sublayer.

The relative depth of flow D/d has been found to be important in 

rigid-boundary fluid mechanics as an expression of relative roughness. 

It is logical, therefore, that for certain conditions of flow in alluvial 

channels it would also be important. In addition to expressing the 

roughness of a plane bed relative to the depth of the flow, it also relates the size of the bed material to the scale of the flow system. 

Therefore, D/d is retained, at least for the present.

The relative fall velocity wfV is important in relating the fall 

velocity of the bed material to the velocity of the flow system. This 

parameter includes the influence of shape of the sediment particles 

and, together with the size ratio D/d and the drag coefficient CD, may 

make it possible to compare the data taken with sediments of various 

specific weights in fluids of various specific weights.
FLUME STUDIES USING MEDIUM SAND A-15

In view of the foregoing discussions of the variables in equation 8, 

the following equations can be written

<h[S, Re, Fr, D/d, w/V, CD]=0 (9) 

or

0io[CVV0, Re, Fr, D/d, w/V, CD]=0 (10) 

Since

C/^[g) = = ====- (11) 

equation 10 can be written as

0i2 TT' ^ e i Fr> D/d, w/V, CD =0 (12)

which now permits using Res =V*d and V* if so desired. Hence

v w

0i3 Pl^ Res, Fr, ^, Z*, CJ=0 (13)

I % ^^ -i.l

or 

and

0,4 tf/V0, Re, Fr, , *, ^=0 (U)

HS' fi^'7'»' C°J=0 (15)

in which Re can be any one of the three Reynolds numbers.

The drag coefficient CD can be modified by combining it with 

V*/w to yield T0/^ysd which is a parameter frequently used, for example, by Shields and Einstein. This gives

"=0 (16)

Various combinations of these variables can be used to satisfy the 

particular problem in question. All are equally valid theoretically 

but certain ones may produce better and more significant relationships than others.

The foregoing equations may satisfactorily introduce all of the 

dimensionless parameters necessary to analyze correctly the problem 

of resistance to flow in alluvial channels. On the other hand it may 

be that even more significant groups of parameters can be established 

That is, it may be desirable to evaluate all possible Pi-terms and then 

select n k of them for the final relationship in which n is the total 

number of variables, and k is the number of physical dimensions, 3 

in this case. All of the possible Pi-terms can be evaluated by the
A-16 STUDIES OF FLOW IN ALLUVIAL CHANNELS

same procedure used to introduce the Reynolds number for the 

particle Eev and the coefficient of drag for the particle CD or, as 

another possibility, various repeating variables can be selected and 

Pi-terms evaluated until all terms have been used as repeating variables. This procedure yields all possible terms in their most simple 

form.

The total number N of possible Pi-terms can be determined, according to Van Driest (1946), by the equation

(k+l)\(N-k-i)\

where n and k are defined as in the foregoing paragraph. To illustrate (three physical dimensions being involved^ &=3) 5 variables 

yield 5 possible Pi-terms, 6 variables yield 15 possible Pi-terms and 

7 variables yield 35 possible Pi-terms, howevei, not more than n k 

of these Pi-terms should be included in any one functional relationship.

As a final possibility, not introduced thus far, there also exists the 

possibility of adding still more dimensionless Pi-terms by considering 

new terms formulated by taking sums and (or) differences of existing 

terms. Such an approach could be justified if indicated by a mathematical model or even by simple logical reasoning.

Other parameters of interest, some dimensional and others nondimensional (the latter group being formulated as indicated in the 

foregoing paragraphs), that will be utilized in the analysis of data as 

follows:

CT is the concentration of total load in ppm. 

G is the total sediment load.

C

-1= is the Chezy discharge coefficient in dimensionless form.

n is the Manning coefficient of roughness.

TO is the shear or tractive force exerted on the bed by the

flowing mixture of water and sediment. 

F* is the shear velocity.

-j=p is the ratio of Fr2/Re which is a ratio of viscous forces to 

gravitational forces.

-jj is the product of -^r and S.

-^ is the Reynolds number in terms of the shear velocity and 

mean diameter of the bed material. This parameter is 

proportional to the ratio of particle size to the thickness 

of the laminar sublayer. Specifically, V#d/v=ll.Q d/5'.
FLUME STUDIES USING MEDIUM SAND A-17

- is a mobility number of the sediment particle. That is, 

the ratio of the force producing motion and the force resisting motion of the particle.

 ^-5 is the Shields parameter (1936). It is used to indicate 

Ts the beginning of movement on the bed.

^r- -jr^-j is the product of the Shields parameter and the velocity

ratio T^r- ^*

^r ~r~ ri is the product of the preceding parameter and -=r' V* Ays U r ror- p

These parameters can be introduced into equations 9 or 10.

To determine the dimensionless parameters that are most important 

in describing C/^fg in equation 6, or subsequent equations that evolve 

from substitutions, an equation involving (n k) of the dimensionless 

Pi-terms is selected. Such simplification (like that employed to 

obtain equations 9 and 10) must be done with considerable care to be 

sure that important variables are not eliminated. Regardless of the 

apparent importance or unimportance of the variable at present, it 

must be remembered that at a later time the variable may prove to be 

of greater importance than it was initially thought to be.

The various combinations of dimensional and dimensionless parameters deemed significant on the basis of existing concepts, a study 

of the data collected, and dimensional analysis are related graphically 

and in many cases by equations, in such a way as to emphasize what 

variables are of prime importance for evaluating alluvial channel 

roughness.

Ali and Albertson (1956) simplified (as a first approximation) the 

dimensional analysis of resistance to flow to the point where three 

parameters remained.

(17)

While this is undoubtedly an over simplification of the problem, it 

serves as a basis from which a first approximation to bed roughness 

can be made and it emphasizes the effect of Reynolds number. This 

relationship may be slightly improved, using Laurson's (1958) concept, for the flume data by selecting D/h as an index of roughness in 

preference to d/D although, considering field application, d/D can be 

determined more easily than D/h since h is very difficult to measure. 

The height of roughness h equals d for a plain bed, and height of the 

ripple, dune, or other form of bed configuration when they exist. In 

addition, A- is a function of the same variables as the roughness and,

577668 O 61 4
A-18 STUDIES OF FLOW IN ALLUVIAL CHANNELS

thus, is not an independent variable, as directly as is d. However, 

insight into the problem can be gained by considering the equation

(18)

It is noteworthy that the variables in equation 18 bear a close resemblance to the variables employed in the diagrams for pipe roughness.

If the methods used to describe resistance to flow in alluvial channels 

are to be improved, more variables must be considered in the analysis 

than are considered in equation 17 or equation 18. That is, it seems 

quite certain that Cf^fg must vary to a certain extent with more of the 

geometric characteristics, fluid properties, and sediment characteristics than are indicated in these equations. This is borne out by 

some of the relations for Cf-Jg developed in the Analysis of Data. 

For example, in the following equation

, CD, ReP , (19)

most variables have been included and yet accuracy is still inadequate. 

Another approach that might further improve the concepts of 

channel roughness is that Cf-Jg is an additive term. This approach 

seems logical, especially since total resistance to flow probably consists of surface drag, which is dependent upon the viscous effects as 

reflected in the Reynolds number; the effect of the sand grains as 

roughness (form drag) ; and the effect of ripples and (or) dunes as a 

form of roughness (form drag) . That is, in equation form

C

-p=022 [surface drag, form drag due to grain roughness, 

V0

form drag due to ripple and (or) dune roughness]

The grain roughness is of decreasing importance as the magnitude of 

the ripples and dunes increases. The drag due to ripples and dunes is 

usually of minor importance at the beginning of movement, but it

increases in importance (TJ< 10 j as the size of the ripples and dunes

increases until the' rapid flow is established. At this point the sand 

waves develop a plane or a symmetrical pattern of roughness, as shown 

in figure 23, and there is very little separation and, consequently, only 

a small amount of form drag. However, there is dissipation of energy 

in antidune flow from the breaking of the wave. This should decrease 

the discharge coefficient C/-Jg.
FLUME STUDIES USING MEDIUM SAND A~19

Another factor that may be important, particularly with the rapid 

flow regime, is that the concentration of sediment being transported is 

very large on and near the bed, relative to that being transported as 

suspended load. This is the effect of the density gradient, which 

Barton and Lin (1955) have related to a special form of the Froude 

number. Its effect, which is of unknown magnitude, would be to 

reduce resistance to flow, perhaps in somewhat the same manner as for 

plug flow in pipes.

In order to describe the types of resistance, considering the foregoing discussion, it would be necessary to know which combination of 

variables expresses the grain effect and which expresses ripple and dune 

effect. This accentuates the need for a mathematical model to serve 

as a guide. Such an equation could help bridge the gap between the 

empirical treatment and theory. Currently no adequate mathematical treatment has been developed. With progress in experimental research and an improved general understanding of fluid flow 

phenomena, it is possible that a mathematical model will be developed 

describing alluvial channel roughness. As an example of correlations 

involving C/-Jg as an additive term, refer to figure 26. In this case

c vd ATs<f (21) ran

which of the terms in the foregoing equation are related to each type 

of roughness is still a matter of conjecture. On the basis of the flume 

data thus far collected, this expression describes Cf-Jg more adequately 

than any of the preceding relationships.

SUMMARY OF THEORY

Although roughness in alluvial channels has never been described 

accurately over the complete range of operating conditions, it is quite 

apparent that in the final analysis C/Jg, for this flume experiment, 

must be expressed in such a way that the effect of all the important 

variables in equation 6 are included over the complete range of operating conditions.

As a final step, assuming that a satisfactory laboratory solution can 

and will be obtained, it will be necessary to adapt the results to field 

conditions. This implies that appropriate scale factors must be 

determined and ultimately the effect of the bank material; the shape 

of the channel, which is probably a function of the bank and bed 

material; the bank vegetation; the wind; and the seepage forces should 

be included, even though it may be virtually impossible to describe 

some of the effects for example, the effect of vegetation on resistance 

to flow in a precise quantitative manner.
A-20 STUDIES OF FLOW IN ALLUVIAL CHANNELS

EXPERIMENTAL EQUIPMENT AND PROCEDURE

The establishment of relationships that exist between the resistance 

to flow in an alluvial channel and the characteristics of the flow and 

the sediment requires laboratory experiments in which:

1. Uniform steady flow exists. This is only possible in a statistical 

sense for with a rough movable boundary the velocity is changing 

both in magnitude and in direction with time and distance. 

However, just as turbulent flow, with velocity fluctuations, may 

be considered steady and uniform in a statistical sense, so may 

flow in alluvial channels. Similarly in terms of scour, the sum 

of the depositional forces must balance the sum of the aggrading 

forces. This also is only possible in a statistical sense as the 

bed at a cross section is continually changing form and the total 

load fluctuates with time. However, considering a long time 

average, neither the bed nor the load is changing with time. 

Possibly equilibrium flow is a more suitable term than steady 

uniform flow.

2. The variables that describe the flow and sediment characteristics 

must be measured accurately. The interval of time required to 

measure precisely a given variable depends upon the variable 

involved and how it varies with time. To illustrate, a sediment 

sample should be collected over a relatively long interval of time 

in order to average its variation with time. Conversely, the 

point velocities in a vertical should be measured quickly before 

the bed condition, and consequently, the form of the velocity 

profile has had an opportunity to change appreciably.

In addition, the scope of the investigation must simulate the conditions found in nature. That is:

1. Various sizes and gradations of bed material that are found in the 

field must be used, subject to the limitations of the flume.

2. Slope, depth, and water discharge must be varied enough to cover 

the complete range of field conditions.

3. The effect of temperature variation must be investigated.

Thus far, 45 runs have been completed using one size of bed material 

with slope, depth, and water discharge varied from run to run. The 

slope was varied from 0.00014 to 0.01 foot per foot, the discharge 

was varied from 2 to 21 cubic feet per second, and the flow conditions 

ranged from no sediment movement to antidune flow. This is, to our 

knowledge, the first set of data, excluding Gilbert's, that covers this

range of conditions.

THE FLUME

The runs were made in a tilting recirculating flume 150 feet long, 

8 feet wide, and 2 feet deep as shown in figure 1. The flume is of
FLUME STUDIES USING MEDIUM SAND A-21

b,b2

EXPLANATION 

Pumping units 

Orifices

Head box and diffuser 

Baffles and screens 

Flume 8 by 2 by 150 ft 

Tailgate

Total-load sampler 

Tail box 

Jacks supporting the flume

t Storage sump

FIGURE 1. Schematic diagram of the flume.

wood construction resting on steel I-beams. The slope of the flume 

was changed and adjusted by mechanical jacks placed every 10 feet 

along the flume. Any slope could be set between the limits of 0 to 

0.013 foot per foot. The flume is constructed so that the watersediment mixture that leaves the flume is returned to the flume 

entrance by either a 7 cubic feet per second or 14 cubic feet per second 

centrifical pump (or both). This procedure of recirculating the watersediment mixture aids in obtaining equilibrium conditions, and in 

simulating conditions in an infinitely long channel, because the sediment load and size distribution are varied by the flow, rather than by 

mechanical traps and shakers.

Flow enters the flume through a vertical manifold diffusor, Fiala 3 

constructed in the head box. It then flows through a %-inch wood 

lattice and a %-inch mesh screen. The diffusor and screens effectively 

distribute the water-sediment mixture and produce a small-scale 

turbulence that is quickly dissipated. The boundary layer is fully 

developed for all flow conditions beyond approximately 25 feet downstream from the entrance. This was determined by inspection of the 

water surface, the bed configuration, and from calculations pertaining 

to the theoretical boundary layer.

The backwater curve and depth are controlled by a system of 

vertical notches in which slats are raised or lowered. This method 

gives precise control of the backwater curve and depth.

ALLUVIAL BED MATERIAL

The flume was filled to a depth of 0.7 feet with a natural river sand. 

Sand from the Cache La Poudre River was obtained from a Corns' Fiala, G. R., 1957. Study of manifold stilling basins, M.S. thesis, Colorado State University, Fort 

Collins, Colo.
A-22 STUDIES OF FLOW IN ALLUVIAL CHANNELS

mercial sand and gravel company. The sand was modified by washing 

and screening to meet specifications. Although it was supposedly 

free of particles larger than one-fourth inch, a few particles of this size 

were observed although they were seldom sampled. The sand, 

principally quartz and feldspar, also contained a small amount of 

mica. The micaceous particles have a large sieve diameter, but a 

small fall diameter. That is, they have a smaller settling velocity 

than a sand particle of the same sieve diameter. The median fall 

diameter of the bed material is 0.45 mm, based on sample analysis by 

the visual accumulation apparatus. The size distribution curve for 

the bed material is given in figure 2.

1.0 

0.9 

0.8 

£0.7

LLJ

tiO.6 

§0.5

jo,

Ta

cc 0.3

LLJ HLLJ

5 0.2 

0.15

0.01 

(

/

O

/^

/

/

^

/

^

^~/

/'

S^

/

^

</cn-

<r =

<s

0.45mm _ 

1.60

).l 0.5 1.0 5.0 10 30 50 70 90 95 9

PERCENT OF MATERIAL FINER 

FIGURE 2. Size distribution curve of the bed material.

The size analysis of the bed material and sediment load were made 

using the visual accumulation apparatus cited in the foregoing paragraph. It was developed by the Inter-Agency Sediment Project at 

Minneapolis, Minn. The theory and application of this apparatus 

were presented in detail by Colby and Christensen (1956).

GENERAL PROCEDURE

The general procedure followed for each run involved recirculating 

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

given slope until equilibrium conditions were established.

Only in a general sense were slopes selected. In any flow system 

where discharge, depth, and slope can be varied, only two of the 

three variables can be considered as independent. In a natural 

stream the discharge and slope are normally independent with depth 

dependent. In the flume the discharge was independent, slope was
FLUME STUDIES USING MEDIUM SAND A~23

independent within limits, and depth was dependent. The slope was 

preset at the beginning of a run by adjusting the tailgate. Such 

adjustment indirectly influences the depth as a dependent variable. 

Generally, and especially at the flatter slopes (0.00014 to 0.0006), the 

slope of the water surface was adjusted parallel with the bed. With 

the development of bed configuration, the slope and depth adjusted 

to the new condition of bed roughness. Thus, for these experiments, 

the slope and depth are a function of Q and the roughness that develops for that regime of flow. The nonuniformity of flow caused by a 

change in bed roughness was eliminated by continuing the run until 

the bed slope and the slope of the water surface again became parallel 

to each other by natural adjustment of the sand bed.

For tranquil flow the mechanics of establishing the desired slope at 

the beginning of each run were to set first an Ml backwater curve, 

and then alternately open the tailgate and measure the water surface 

slope until the water surface was gradually adjusted to the desired 

slope. For the flatter slopes, (0.00014 to 0.0006^ the bed was carefully 

screeded to the desired slope before the run. With steeper slopes, 

since the increased sand movement assisted in obtaining equilibrium 

conditions quickly, the bed was only raked to remove the dunes from 

the previous run before starting the new run. In the rapid flow 

regime the slope was changed by altering the control at the lower 

end of the flume. The movement of sediment quickly established 

equilibrium conditions.

Equilibrium flow was considered established if the same bed configuration was established for the full length of the flume, excluding 

the Section influenced by entrance conditions and the water surface

slope remained essentially constant with respect to time ~jx7=0-

The time required to establish equilibrium conditions varied with 

the slope and the discharge. Some runs with flat slopes required 3 

and 4 days to achieve equilibrium, whereas, at steeper slopes equilibrium was established in 2 or 3 hours. Every run involved continuous 

flume operation until it was completed. Whether or not equilibrium 

conditions were established, in the final analysis, rested on the measured data and the judgment of the experimenter. To insure the 

achievement of equilibrium conditions most of the experiments were 

continued longer than the required time as indicated by measurements.

DATA OBTAINED

WATER-SURFACE SLOPE

Water-surface elevations were measured every 5 feet along the flume 

using a precision level and a Lory point gage. These elevations were 

plotted on graph paper and an average slope determined. Average
A-24 STUDIES OF FLOW IN ALLUVIAL CHANNELS

slope of the water surface was determined periodically throughout 

each run. These slopes were used to establish the water surface slope 

at equilibrium. The final slope that is used in the analysis of the 

data was computed by the method of least squares. To measure 

slopes for those runs in which antidunes formed was difficult because 

of the rapidly changing water surfaces. Water-surface elevations for 

these runs were determined by averaging the water-surface elevations 

taken at an upstream and downstream cross section over a period of 

time. In addition, the slopes were also determined in the conventional 

way by taking more time and not measuring the water-surface elevation while antidunes were breaking.

DISCHARGE

To obtain the desired range in discharge, two pumps were used. 

One, a 12-inch pump with a maximum discharge capacity of 7.98 

cubic feet per second and the other, a 15-inch pump with a maximum 

discharge capacity of 13.38 cubic feet per second. The 12-inch pump 

was used for runs 1-18, for which Q<^8 cubic feet per second. For 

the runs 19-45 inclusive, the 15-inch pump was used for all discharges 

up to 13.34 cubic feet per second. For Q>13.34 cubic feet per second the 12-inch pump was used at its maximum capacity and the 

discharge from the 15-inch pump was varied to obtain the desired 

total discharge. Carefully calibrated orifice meters and water-air 

manometers, read to the nearest one-hundredth of a foot, were used 

to measure the discharge. The orifice meter for the 12-inch pump is 

made of stainless steel, is circular, and has a diameter of opening of 

10.137 inches. It is located in a vertical section of pipe 10 diameters 

downstream from the pump and 6 diameters downstream from a 

right angle bend. The orifice meter for the 15-inch pump consists of 

a vertical side contraction with stainless steel edge. It is located in 

a horizontal section of pipe, 11 diameters downstream from a 45° 

bend, (fig. 1).

Both orifices were calibrated in place. The orifice meter for the 

small pump was calibrated using a calibrated Cipolletti weir. The 

orifice meter for the larger pump was calibrated volumetrically.

Water discharge was measured four to five times each day during 

a run. The average discharge for a run is the mean of these periodic 

discharge determinations.

WATER TEMPERATURE

Water temperature was measured with a mercury thermometer to 

the nearest 0.5° C each time the water discharge was determined. 

The effective temperature recorded for each run is the average of the 

temperature measurements made after equilibrium conditions were 

reached.
FLUME STUDIES USING MEDIUM SAND A~25 

DEPTH

Depth was determined by three methods. Two of the methods 

involved the use of the level and point gage, whereas the third method 

used discharge and average velocity measurements and the continuity 

equation. Depth was measured directly by the first two methods 

after the run was completed and the flume drained.

Using the first method, the elevation of the bed was measured 

along the centerline of the flume at 2- to 5-foot intervals, depending 

on the bed roughness. From these bed elevations, the average elevation of the reach was computed. The difference between the average 

bed elevation and the average water-surface elevation is the depth.

Using the second method a reach in the channel, from 2 to 20 feet 

long, between stations 80 and 105, was carefully screeded. The 

length of reach used depended on the bed roughness. After screeding, the bed was inundated with water to help attain natural compaction of the bed material. The mean elevation was determined at 

the center of this screeded section by averaging the elevations for 25 

to 60 points located symmetrically in the reach. This average elevation was then subtracted from the corresponding water-surface elevation to determine the average depth.

The third method was to divide the mean water-sediment discharge 

by the average velocity of the flow as determined from the velocity 

profiles. That is, D=A/S=Q/8VV for a nominal width of channel 

of 8 feet.

The depths as determined by methods 1 and 2 are in good agreement. 

Five of the runs with the roughest bed configuration had differences in 

depth of about 0.04 foot. All the other runs had differences less than 

0.02 foot. However, the differences between depths determined by 

averaging bed and water surface elevations (methods 1 and 2) and 

depths calculated from the velocity profiles and discharge measurements (method 3) were quite large. Some of these differences exceed 

0.10 foot. The large difference between depths computed by method 

3 and methods 1 and 2 indicate that the velocity profiles obtained in 

the cross section were not adequate to determine the true average 

velocity.

MEAN VELOCITY

The mean velocity was computed from the velocity profiles, and the 

continuity principal, using the depth as computed by methods one or 

two. In the first case the mean velocity in each of three velocity 

profiles was calculated and then averaged to determine the mean velocity in the cross section. It was this velocity that was used to determine depth by the third method in the foregoing paragraph. In the 

second case, the average velocity is V= Q/8D as determined from the 

continuity equation.

577668 O 61 5
A-26 STUDIES OF FLOW IN ALLUVIAL CHANNELS

VELOCITY PROFILES

Velocity profiles were obtained by measuring point velocities in the 

selected verticals with a calibrated pitot tube using a water-air manometer. The pitot tube was calibrated in a towing tank. For slower 

velocities a tilting manometer which gave a 1X10 magnification of 

head was used, and for the faster velocities a vertical manometer was 

used. After equilibrium conditions were established, profiles were 

taken at 3 to 5 verticals in a cross section with 4 to 12 velocity points 

in each vertical. The cross sections where velocities were measured 

were always located between stations 95 and 105.

Velocity profile data at a cross section were determined as rapidly 

as the pitot tube could stabilize. However, in a run with a dune bed, 

the dunes would often change the lower part of the profile considerably, 

especially if measuring was started over a trough and then the crest of 

a dune moved in where the trough had been before the measurements 

had been completed. In the regime of flow with antidunes it was 

virtually impossible to obtain velocity profiles in that part of the flow 

where the antidune existed. However, velocities above the crest and 

in the trough of antidunes were determined with the pitot tube, and 

surface velocities were measured by timing floats through the crests 

of the antidunes and to the side of the antidunes.

BED MATERIAL

Except for runs 13, 14, 16, 17, and 18, at least four samples of the 

bed material were taken from the bed at random for each run. The 

size distribution of each sample of bed material was determined using 

the visual accumulation tube (VA tube). The VA tube determines 

the fall diameters.

TOTAL SEDIMENT LOAD

Total load was determined periodically, after equilibrium conditions 

were established, by using a width-depth integrating total load sampler. A water-sediment sample weighing from 100 to 160 pounds 

was taken 4 or 5 times during a run. These samples were analyzed 

and the results were combined to determine total load. The time 

during which the samples were collected varied from half an hour to 3 

hours.

From these samples of the total load, the concentration was determined on a dry-weight basis and the size distribution was determined 

in terms of fall diameter, using the VA tube. When the amount of 

sediment in the samples was small, the samples were composited for 

the size analysis. When the amount of sediment was large, the sam-
FLUME STUDIES USING MEDIUM SAND A~27

pies were combined and then split until reduced to the amount of 

material that could be analyzed. This sometimes meant four or five 

splittings. Duplicate analysis of separate portions of the splittings 

were run through the VA tube apparatus and averaged to establish 

the size distribution of the total load.

SUSPENDED-SEDIMENT SAMPLES

Suspended-sediment samples were collected at the same cross 

section as the velocity profile data using the DH-48 hand sampler. 

Five samples were obtained for each run, using the equal transit rate 

(ETR) method of sampling. Where the depth was shallow (less than 

0.5 foot), the sampler was rocked forward to sample a larger portion 

of the flow. Some runs were not sampled and for runs 1-6 and 8-18 

the suspended load was too small to measure. Concentration and 

size distribution were determined for the suspended sediment using 

the same procedure as for total load.

BED CONFIGURATION

The height and length of the ripples, dunes, and antidunes were 

measured by the point gage. The bed configuration was measured 

along the center line of the flume over a reach that varied from 20 to 

80 feet in length, depending upon the roughness pattern. That is, 

as the magnitude of roughness increased, the length of reach for which 

bed configuration was measured was increased.

The bed configuration was always measured after the flume had 

been drained. However, when standing waves and (or) antidunes 

existed, the bed configuration was also measured during the run. 

The elevation of the crest and trough of the water surface with 

standing waves and (or) antidunes and the corresponding elevation 

of the bed was measured. Because the bed is extremely firm with 

this type of flow, the elevation of the bed was accurately determined.

To determine the average length and height of the bed configurations, 

the bed profile was plotted. The height was computed by summing 

the vertical distances between the crests and troughs and dividing by 

the number of these distances. The wave length was computed by 

dividing the overall flume length, from crest to crest or trough to 

trough, over which the bed configuration had been measured by the 

number of dunes or ripples in that length.

In addition to measuring the bed configuration, photographs of 

the bed were taken for all runs. The velocity of the dune was 

determined for some of the runs by observing the time of travel of 

the crest of the dune with respect to a fixed point at the glass window.
A-28 STUDIES OF FLOW IN ALLUVIAL CHANNELS

PRESENTATION OF DATA

In the foregoing section, descriptions are given of the method of 

measuring and (or) collecting the data needed to describe the channel 

geometry and the characteristics of the flow and sediment as they 

relate to channel roughness. In this section these data will be 

discussed with respect to their range, variation, and accuracy of 

measurement.

The basic variables that were recorded for each run are given in 

table 1 and a discussion of the basic variables follows.

WATER-SURFACE SLOPE

The final water-surface slope based on water-surface elevations 

measured along the flume and the corresponding coefficient of 

correlation are given for each run in table 1. The variation of 

magnitude in slope for the runs ranges from 0.00014 to 0.0101.

The precision with which the water-surface slope determinations 

were measured is excellent. This is illustrated graphically in figure 3 

by plotting water-surface elevations versus distance along the flume 

for typical runs. The bed roughness for these five runs range from a 

plane bed with no movement of bed material to antidunes. Note 

the relatively small amount of scatter about the least-square lines.

DISCHARGE

The average discharge of the water-sediment mixture for each run 

is given in table 1. The tabulated discharge is the mean of the 

periodic discharge measurements recorded during a run. The maximum deviation from the average discharge for any one run, as 

determined from the periodic discharge measurements, is approximately 4 percent. However, in most of the runs the maximum 

deviation from the mean was less than 2 percent. The magnitude 

of the discharge ranges from 1.84 to 21.62 cubic feet per second.

The discharge calibration curves for both pumps are well defined, 

although there is more scatter for the small circular orifice than for 

the side-contracted orifice. This scatter results from the method of 

calibration. The maximum deviation of any calibration point from 

the rating curve in the range of discharge used is 4 percent for the 

circular and 0.7 percent for the side-contracted orifice.

The maximum change in discharge for each 0.01 foot change in the 

manometer reading for the entire range of discharge was 2 percent 

of the discharge for either of the orifices. However, for discharges 

greater than 2.7 cubic feet per second for the 12-inch pump and 6.0 

cubic feet per second for the 15-inch pump, the change in discharge 

was less than 1 percent.
FLUME STUDIES USING MEDIUM SAND A-29

0.94

0.93

0.90

0.88

0.86

0.80

W 0.70

0.60

0.50

0.90

0.70

0.50

1.20

1.00

0.80

0.60

0.40

20 30 40 50 60 70 80 90 100 110 

DISTANCE ALONG THE FLUME, IN FEET

120

FIGURE 3. Typical water-surface profiles.
j^ j<I p CO C* 00 ' i i H- i O fc^lH-COOOOicOO^COOCntOt-'COtOi 

P

Oi CO "^ Ci l 5^tOQOOOOCOOOO5O5OiOOCJil 'OSOih^Ci ^" ^ ^' ^ ^OOtOh^JOlOl 

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p ......... 

P . . . . !"*. 4 I" . h- . t

p _ t f f f 1 J- 1 t- J 1 H- - 

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tO^-^«OCC-qrf^s^sS>J?^>^^ CO-^OOO5CC^r 00 Cn O 00 rf!>. tO CD «O 00 00 O CO H-CO 00-^ 00-^-^ OO 00 

O -^CnOcO^ OOOOs!^.OiOOCO l W<MO»-^H-SiOiOOOOOOrf>.-^lO-^cDCC^COOOJ--

^.p ^^ ^-N. 

00^"^^"' H-'^'^^^^'^ 1 J- tO h^. O W^"' CJiOOCOh-'OOCOOCOM-^QSCnCJiCCOO^D l^-cDOtOOOOOCOOiQ 

tO CCJ-"^OiOO OiCCOCCCOOOrf^H-rf^i^J-'rf^OOCriCCtO-^OO-^tOCTiCOcOOOOOCn 

H- tO tO OS CO-^ CJi 00 tO rf^ CO lOCn 00 J-i-^ CC CC CJi Oi Oi 00 tO CJi 

OCO tOtO CO O5 tO - 1 rf Qihf^-*-IOQiCriWOOO5O5OCriO*<ri 't^-O l ^COJ 'CDI 

tOOOOOOtO^OiCCfcOJ-'rf!>.J-'OOtOJ O l OOOOOOOOOOOOOOOOCTiOOOOOh-OOJrf* OS Oi CO tOCn -^ i-- tO CO Cn O 00 

OOrf*. tO CCOOiH-OOStO OOH-OOO 

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v tO-^ ^ICO OOOO OiCOCnCT>COtOCnOO^OOCnOO^-^» tCh-rf>.Cn COCO 0000-^rf^-tOO 

OOCnJ-'O-4COOtOtOcOOiCO»-'COOiOOOCOOi»-'l i tOh-'OiOiOtOCOCOtOOiO-^ 

OO OO»-'OOOOsS l J-'COCOrf*.COtOCnCOtOO»-*otOO>- OOH-OOJ-'J-'OH-OtOO 

Oirf^ OH-H-'J-'OiOitOj-'OiOiOOOOOiCTiOi'^rOascO-^^- OOcOOOO-^tOrf*-COCOOiCOCn 

COtOJ-'j-'i itOn-h-itO»-'j-*j-'tOi 'tOH-'i 'tOh-'i-n-n-i ^^ tOCOCOCntOCOCTiCntOCn 

^l" piCO^prf^rf^jt^Cnrf^jt^piptOtO^Wrf^ptOCn^r ^tO^ppppOH-'WlO^it-' 

tc en "' x <r oo-<r-<rj-'co»-'cncnoiOOO5Cooi '-^cotooj-'j-'to -^cooi'<rtoj-'OO-^Oioo^. 

OCn OJ-'-^Ol'OcOCOCncOCOOOOCn-qCOWCOO-qJ-'cOCn OiCOOOOO-^-^OCO-^COO 

>£* toco co co t i H-^ j ' co to 

1 t- tO H- tO J-» h-' ' tO-<T O Oi COrf*.OOCO-^ 

CO j*». CO 0^ ^ 1 H^tOCnpd^tOCn^^tOH^^cOOdlOtOH^^M^CO^^^^COpO^CTippOcDcOHW^d ^2 S^OtdU 0 

g t3* § 5* S 5" G "3" S 

* a ppppopp®^p® 1 ^r m pp®. m ppppppp. : o66g'p^"b'^rpg'^:g*Q'g

s

OS~V saianis .ID 'iviAmiv
32......

45......

44......

42......

43......

.65580

.86200

.89800

.98600

14.96

5.58

10.83

13.43

21.42

1.140

1.113

1.098

1.082

1.125

.372

.279

.283

.313

.433

5.027

2.500

4.783

5.363

6.184

4.65

3.36

4.76

4.97

18.0

18.9

19.4

20.0

18.5

1.35

1.33

1.74

1.30

1.57

.213

.215

.218

.220

.214

1.67

1.59

1.57

1.72

1.28

1.84

1.61

1.60

1.65

1.80

.246

.236

.233

.262

.180

618

963

1510

1140

1 Computed on basis of average median diameter (0.00148 ft) and water temperature for the run.

2 A is the median diameter of the bed material when a plain bed or antidune regime exists.

s 3-dimensional flow.

4 No sample.

6 No size analysis.

«Dunes not measured.
A-32 STUDIES OF FLOW IN ALLUVIAL CHANNELS

DEPTH OF FLOW

The average depth recorded in table 1 was obtained by the 

screeding method described in the preceding section (method 2). This 

procedure gives better results with fewer bed elevation determinations 

than are required when working with the undisturbed bed. That is, 

relatively speaking, the average elevation of a plane can be described 

more accurately with fewer readings than are required to describe 

the average elevation of an active stream bed when there are dunes 

and potholes that are sometimes of the same order of magnitude as 

the depth. Screeding the bed was done with care since any disturbance, such as that caused by walking on the bed, changed the degree 

of compaction of the bed material, and hence its average elevation 

and the average depth. To offset this danger, the bed was first 

carefully screeded without increasing the compaction and then flooded 

to achieve natural conditions before taking elevation measurements.

Another possible source of error always present is the degree of 

compaction of the bed material, which varies with the form of the 

bed roughness. For instance, a bed with dunes and potholes is fairly 

loose prior to draining the flume as compared to its conditioning 

after drainage. This problem of changing density could be avoided, 

as well as the problem of the erosion of the roughness pattern caused 

by draining the flume, if a suitable means of measuring bed elevation 

during the run could be found.

KINEMATIC VISCOSITY

The kinematic viscosities are given in table 1 and are based on the 

temperature of the water during the runs. In the runs, the temperature varied from 9° C to 20° C, which resulted in a range in kinematic 

viscosity of 1.082 to 1.46X10"5 square feet per second.

The recorded temperature for each run is the average of the water 

temperatures obtained after equilibrium was established. The 

maximum deviation from the recorded temperature during any 

individual run was approximately 1° C. This magnitude of temperature fluctuation changes the kinematic viscosity about 3 percent.

MEAN VELOCITY

The mean velocity V, used in the computation of parameters was 

obtained by dividing the discharge by the product of the width of the 

flume times the average depth. The velocity thus accumulates the 

errors inherent in depth and discharge calculations. The errors in 

discharge and depth may be additive or compensating in their effect 

on the velocity. The velocity, as determined by the continuity 

principal, is given in table 1. Also included in table 1 is the mean 

velocity VP as determined from the velocity profiles.
FLUME STUDIES USING MEDIUM SAND

VELOCITY PROFILES

A-33

Typical velocity profiles are given in figure 4. This figure illustrates the form of the distribution curves of the velocities on semilogarithmic paper for a plane bed with no movement, run 14, for a 

bed covered with ripples, run 9, for a bed covered with a fully formed 

dune pattern, runs 21 and 24, for a plane bed with Fr>l, run 26, 

and for antidune flow, run 44. The stations given in the figure locate 

the velocity profiles in the transverse direction. The zero station in 

the traverse direction was at the right edge of the flume.

Velocity profiles in an alluvial stream are continually changing in 

form. For example, the lower part of the velocity profile is considerably different when taken at the crest of the dune than when 

taken in the trough where a separation zone exists. In the separation 

zone, the direction of flow is reversed. For flow with standing waves

oo o oo o oo o oo o oo o o

VELOCITY, IN FEET PER SECOND

FIGURE 4. Typical vertical velocity distribution curves at different stations across the flume. ' 

577668 O 61 6
A-34 STUDIES OP FLOW IN ALLUVIAL CHANNELS

and antidunes, the velocity profile in the vertical is considerably 

different through the trough than through the crest. In addition, 

the velocity of this type of flow at any point changes so rapidly that 

the profile is constantly changing.

BED MATERIAL

The median fall diameter d, the nondimensional standard deviation 

a, and the fall velocity w of the median diameter of the bed material 

as determined from the samples for each run are given in table 1.

The median fall diameter of the bed material d for each run was 

determined from size distribution curves plotted on log-probability 

paper. The standard deviation a of the bed material for each run 

was calculated from the equation

^DE+T] (24)

in which

di6 is the size of the material for which 16 percent is smaller. 

<?84 is the size of the material for which 84 percent is smaller.

These sizes were obtained directly from the size distribution curves 

for each run.

The median fall diameter of the bed material used to calculate 

parameters was obtained by averaging all of the data for size analysis and plotting the results to obtain the composite size distribution curve given in figure 2. This was done because the probability 

is small that any single sample or small group of samples will give 

the size distribution of a natural unsorted sand. The more samples 

that are analyzed the greater the probability of obtaining the actual 

size distribution. The possibility that the characteristics of the bed 

material change with time was investigated by taking the size distribution, as determined from samples from the first 18 runs, and comparing it with the size distribution for the last 26 runs. There is 

very little difference between the two, which is an indication that 

the sand size was not changing as a result of abrasion and there is no 

appreciable selective sorting of certain sediment sizes.

Using the data shown in figure 2, the median fall diameter, as determined from all the samples, is 0.45 mm or 0.00148 foot, and the 

relative standard deviation is 1.60. Statistically there are 99 chances 

in 100 that this median diameter is within 4 percent of the true median 

diameter of the bed material.

The fall velocity of the bed material, which is given in table 1, is 

based on the average median fall diameter (0.45 mm), shown in 

figure 2, and the temperature of the water for the run in question.
FLUME STUDIES USING MEDIUM SAND A~35

This fall velocity was determined using the graph on page 41 of InterAgency Report No. 4 (1941).

TOTAL SEDIMENT LOAD

The median fall diameter, the standard deviation, the fall velocity 

of the median diameter, the concentration of the total load, and the 

computed total sediment load for each run are given in table 1. The 

methods used to determine size, relative standard deviation, and fall 

velocity were similar to those used to establish these values for bed 

material. The total load was computed by multiplying the concentration by the discharge and a constant to convert to the units of pounds 

per second per foot of width.

The median diameter of the particles in the total load varied from 

0.000361 to 0.00174 foot. In certain runs the median diameter d for 

the total load was slightly larger than the median diameter of the particles in the bed. This may have been due to segregation of the bed material and possibly error in sampling and analysis. In other runs the 

d for the total load was smaller than the d for the bed material. To a 

limited extent, this may be due to some sampling error, but it is 

largely caused by the transport capacity being proportionately smaller 

for certain sizes than their proportionate availability in the bed. 

Beyond a certain limiting bed shear, however, the d for the total load 

should be equal to the d for the bed material.

The concentration of the total load varied from 0 to 15,100 ppm 

which gave a total load variation from 0 to 1.28 pounds per second 

per foot.

Samples had to be taken over a time long enough to average the 

effects of storage in the dunes and the resulting fluctuations in concentration passing the samplers. A short sampling time resulted in 

concentration data that were usually either too high or too low, and 

size distributions that were either coarser or finer than they should 

have been. This may explain the smaller total load and finer size 

distributions for runs 20-24, 39, 40, and the coarser size distribution 

for runs 6 and 10. However, the sampling times for run 20 were long 

enough for the dunes to move a distance equal to four times their 

length, and for the dunes corresponding to the other runs to move a 

distance at least equal to their length. This indicates that the 

sampling time was long enough and that the finer size distribution 

resulted from storage of certain sizes of bed material in the dune. 

However, larger samples collected over a longer time should be 

obtained with the rougher beds.

SUSPENDED SEDIMENT

The concentration of suspended sediment varied from 0 to 14,500 

ppm. There were some concentrations of suspended sediment that
A-36 STUDIES OF FLOW IN ALLUVIAL CHANNELS

were larger than the concentrations of total load. This is a physical 

impossibility considering average values and illustrates the inadequacy 

of the method of sampling. The greatest difficulty occurred when the 

bed was extremely rough and (or) the depth of flow was quite shallow, 

and it is attributed to positioning the nozzle of the hand sampler too 

close to the bed. Suspended sediment is not included in the tabulation 

of concentrations.

BED CONFIGURATION

The geometry of the bed is described by the spacing and height of 

the sand waves, the ratio of length to height of the sand waves, the 

ratio of depth of flow to height of sand waves, and the form of the 

bed configuration for each run, which are all given in table 1.

It is difficult to measure the bed configuration accurately, as in the 

measurement of depth, because the accuracy depends upon the effects 

on the original configuration of stopping the flow and draining the 

flume, and upon the number of bed configuration measurements taken.

It was necessary to exercise care in stopping a run so that a wave 

would not be created that would alter the bed roughness as the wave 

traveled up the flume, and in draining the flume so that the roughness 

pattern would not be eroded. In the tranquil flow regime it was 

possible to stop the flow and drain the flume without appreciably 

altering the bed configuration. However, with steeper slopes (rapid 

flow regime) this was not possible.

OBSERVED FLOW PHENOMENA

The form of bed roughness in alluvial channels is a function of the 

sediment characteristics and the characteristics of the flow. That is, 

the bed configuration may be changed by changing either discharge 

(which effects the depth), slope, temperature, or the median diameter 

or size distribution of the bed material. In the flume experiments the 

bed material was not changed and the slope, temperature, and depth 

were varied. Under these conditions the following forms of bed 

roughness were observed:

Tranquil-flow regime:

1. Plane bed without movement,

2. Ripples,

3. Dunes,

4. Transition from dunes to rapid-flow forms.

Rapid-flow regime:

5. Plane bed with movement,

6. Standing sand waves,

7. Antidunes.

The major forms of bed roughness are sketched in figure 5.
FLUME STUDIES USING MEDIUM SAND 

_______________^ Water surface

A-37

TYPICAL RIPPLE PATTERN

TYPICAL DUNE PATTERN

TYPICAL STANDING WAVES

TYPICAL ANTIDUNE PATTERN 

FIGURE 5 Major forms of bed roughness.

Slope and size of bed material are the dominant parameters that 

det rmU what form of bed configuration will "ccur although^

material is close to the critical slope. That is, lor a given
A-38 STUDIES OF FLOW IN ALLUVIAL CHANNELS

material with a small slope and the dune bed form of roughness, it is 

impossible to create the standing wave form of roughness by changing 

depth or temperature. With a given depth and temperature, all bed 

forms can be created by changing slope when using a sand size bed 

material. This last statement has to be qualified because threedimensional flow will occur in shallow depths and with it occur 

multiple bed roughness forms. It is possible'to have a size of bed 

material that will prevent formation of some of the bed forms regardless of the depth, slope or water temperature. Figure 28 in the 

section Analysis of Data illustrates the effects of temperature, slope, 

depth, and size of material on the forms of bed roughness.

For a given size of bed material the change from one bed form to 

another is not necessarily abrupt. Neither does it occur at the same 

slope (depth and temperature held constant) nor the same depth 

(slope and temperature held constant), if the change in bed form is 

reversed. That is, the change of bed form from ripples to dunes may 

occur at a different slope (depth and temperature held constant) 

than the reversed change from dunes to ripples. This gradual change 

and (or) hysteresis effect result in a transition from one bed form to 

another. This transition of bed forms is of major importance when 

the flow changes from tranquil to rapid, or from rapid to tranquil, 

that is, when the bed form changed from dunes to standing sand 

waves with standing water waves or to a plane bed with a plane water 

surface, or when it changes from standing sand waves to dunes. The 

change in flow regime from tranquil to rapid or from rapid to tranquil 

may, if conditions are right, occur in a natural river. That is, if the 

slope of the energy grade line is close to the critical slope, a change in 

the stage of the river could result in a change in the flow regime. 

Tranquil flow exists at low stage and rapid flow exists at high stage. 

With a change in regime there would be a break in the discharge 

rating curve for that stream. Because there is a hysteresis in the 

transition from rapid flow to tranquil or from tranquil flow to rapid, 

the point at which the break in the discharge rating curve would occur 

depends on whether the stage is rising or falling and also on the rate 

of change.

It is important to note that the velocity of the flow in an alluvial 

channel is not zero at the bed. The velocity of the water through the 

bed is slow but important, especially when considering seepage force. 

The velocity of this flow in the bed is highest in the dune bed form, 

since the porosity of the material is high, and smallest in the rapid 

flow regime, where the bed material is closely packed. In the flume, 

as a result of the effects of the floor, the velocity of the flow through 

the bed was probably lower than in most natural streams, and consequently, the seepage force was lower. In a natural stream the velocity
FLUME STUDIES USING MEDIUM SAND A-39

in addition to being greater in the direction of flow, has vertical components in the bed resulting from inflow or outflow. This inflow or 

outflow, which depends on the water table of the surrounding area, 

causes large seepage forces. These forces may have a considerable 

effect on the form of bed roughness and transport of sediment.

BED FORMS IN THE TRANQUIL-FLOW REGIME

PLANE BED WITHOUT MOVEMENT

The plane bed with no movement of bed material was soft and easily 

disturbed. Figure 6 is a photograph of a plane bed. This bed was 

photographed through the water during the run. The word "plane"

n w

FIGURE 6. View of bed with no sediment movement. Water-surface slope S=0.000147 (run 14).

is used to emphasize that the surface of the bed was not hydraulically 

smooth. That is, for a smooth, rigid hydraulic boundary the magnitude of d/8' must be such that d/5'<0.25. The minimum d/d' for the 

flume runs was 0.40. Obviously it is possible to obtain a hydraulically 

smooth boundary without bed movement by decreasing the slope or 

the depth. However, obtaining a smooth bed or a plane bed without 

sediment movement in itself has little practical importance. The 

important thing to consider is the point at which movement of bed 

material will begin as the slope or depth, or both, are increased. It 

is interesting to note that the point where movement begins, for this 

size of bed material, is not the point where a transition from the hy-
A-40 STUDIES OF FLOW IN ALLUVIAL CHANNELS

draulically smooth boundary to the hydraulically rough boundary 

occurs according to rigid boundary hydraulics. That is, the thickness 

of the laminar sublayer must be decreased beyond that point corresponding to the transition point before movement begins. It must be 

remembered that the bed material utilized is nonuniform and computations are based on a median diameter of the material.

Movement of bed material began with a slope of 0.00015 when the 

depth was increased (by increasing Q) from 0.61 to 0.71 feet. The 

d/d' value at the beginning of movement of the bed material was 

between 0.48 and 0.53. The Manning n for the two runs without 

movement was 0.015, and C/-\/g varied between 14.00 and 15.0. 

These statements and others that follow can be verified by referring 

to table 1.

When d/8' approached 0.53, movement of the fine material began 

and ripples started to form on the bed. The ripples formed at the upstream end of the flume and at any small depression or ridge on the 

bed, and then continued to form in a downstream direction below 

these points of minor disturbance, until the bed was covered. As 

ripples formed on the bed, the slope and depth in the flume increased 

from 0.00015 and 0.71 foot to 0.00023 and 0.80 foot respectively. 

The Manning n increased from 0.015 to 0.022 and Cf-\fg decreased 

from 15.0 to 10.3.

The change from a plane bed, with no sediment movement, to a bed 

with ripples occurred at a definite depth (slope and temperature held 

constant). As stated in the proceeding section, the change occurred 

when the depth was changed from 0.61 to 0.71 foot with a slope of 

0.00015. For the bed material utilized in these experiments ripples 

started to form when movement began. Other investigators have reported movement of material on a plane bed without the formation of 

ripples. Whether this movement without ripples is natural or the 

result of the experimental procedure and equipment is unknown. For 

example, movement without ripples may result from using a sorted 

bed material whereas this bed material was unsorted.

Changing from ripples to the plane bed by altering flow conditions 

is a physical impossibility unless movement of sediment occurs without the formation of ripples. By changing depth and (or) slope it is 

possible to stop the movement of sediment. However, the ripples 

would still exist although they may not be as angular in form as when 

sediment is moving. To stop movement with ripples on the bed, the 

depth and (or) slope (temperature constant) would have to be reduced 

to smaller values than that which would stop movement with a plane 

bed. If d/8' is a criteria for the change from plane bed to ripples or 

ripples to plane bed, d/d' would have to be reduced to less than 0.48 

to stop movement of sediment with a ripple bed configuration.
FLUME STUDIES USING MEDIUM SAND

RIPPUES

A-41

The form of the ripples is illustrated in figure 5, and a photograph of 

typical ripple formation is shown in figure 7. The height of the ripples 

was less than 0.10 foot and their longitudinal spacing was less than 

2.0 feet. Their length-height ratio, Lfh, varied from 10 to 20. The 

ratio of depth of flow to ripple height ranged from 4 to 24; Manning n 

ranged from 0.019 to 0.027; and C/^fg ranged from 7.8 to 12.4. The 

upper limit of n-values, and the lower limit of C/-\fg, in general, were 

associated with the shallower depths.

There was little or no suspended sediment movement with the 

ripple bed form. The water was sufficiently clear so that the bed was 

visible at all times. The total sediment load ranged from 1 ppm to 101 

ppm and the sediment moved in more or less continuous contact with 

the bed, rolling up over the crest of the ripples and coming to rest on 

its forward slope. The sediment on the forward slope did not move 

again until it was exposed in the upstream part of the ripple as the 

ripple migrated downstream. Thus the movement of the sediment 

particles takes place in steps. The length of the step and time interval 

between steps depends on the length of the ripple and the ripple 

velocity.

Beds with ripples were soft, but not as soft as those with dunes. The 

water surface was very smooth with little visible difference from the 

water surface for a plane bed without movement. The separation zone

FIGTJKE 7. View upstream of ripple configuration. Water-surface slope S=0.000403 (run 9).
A-42 STUDIES OF FLOW IN ALLUVIAL CHANNELS

downstream from the crest of the ripple is such that there was little 

evidence of jet impingement on the downstream ripple. The reverse 

flow in the trough moved only the finest mica particles and the turbulence was small at the interface between the main current and the 

separation zone.

As the slope and (or) depth were increased beyond a certain limit, 

the ripples were modified and the appearance of the bed changed. 

Compare the ripples shown in figure 7 with the dunes in figure 8.

FIGURE 8. View upstream of dune configuration. Water-surface slope S= 0.00247 (run 23).

The bed changed from the ripple pattern to a dune pattern when 

the slope and (or) depth were changed such that d/dr was approximately 1.0. The Froude number was approximately 0.28. This 

change was abrupt and the dunes established themselves over the full 

length of the flume in a matter of hours. To change from dunes to 

ripples (Q=constant) it was necessary to decrease the slope and 

increase depth from those values at which ripples changed to dunes. 

Because of the reduced transport capacity caused by decreasing slope 

and increasing depth, considerable time is required for the flow to 

convert a dune configuration to ripples.

DUNES

The form of the dunes is illustrated in figure 5 and a photograph 

of typical dunes is shown in figures 8 and 10. In this sequence of 

runs, dunes ranged in average height from 0.15 to 0.52 feet and in
FLUME STUDIES USING MEDIUM SAND A-43

length from 4.0 to 7.5 feet. The length-to-height ratio, L(h, varied 

from 14 to 28. The ratio of depth of flow to dime height, D/h, ranged 

from 1.5 to 5.0. The Froude number varied from 0.38 to 0.60 with 

0.60 fixing the upper limit of the dune bed form. The Manning n 

varied from 0.018 to 0.032, although from observations of the bed, 

much larger values of Manning n would be expected (figs. 8, 9, and 

10). The reason for these smaller n-values is that in a flume this 

large, as in a natural stream, the dunes form a pattern that allows part 

of the flow to sideslip or bypass the dune. This sideslippage or 

meandering of the flow around the dunes is very obvious when a smaller 

discharge is run over a dune bed formed by a larger discharge.

Dunes move downstream as a result of sediment toppling over the 

crest and accumulating on the downstream face of the dune. The 

larger the amplitude of the dune the more sediment is stored in the 

dune and the lower the dune velocity. Smaller dunes with their 

higher velocities overtake the larger dunes. This results in a much 

larger dune and decreases the dune velocity. Velocity of the dunes 

varied from 0.03 to 0.70 feet per minute. The higher dune velocities 

were associated with the steeper slopes and shallower depths. Conversely, the larger dunes were associated with flatter slopes and deeper 

depths.

The L/h ratios for dunes arid ripples are similar. The ratios ranged 

from 10 to 24 for ripples and 14 to 28 for dunes. This indicates that

FIGURE 9. Flow over dune configuration similar to that shown in figure 10.
A-44 STUDIES OF FLOW IN ALLUVIAL CHANNELS

I

FIGURE 10. View upstream of dune configuration. Water-surface slope S=0.00289 (run 24).

there is a correlation between dune height and dune length for a given 

sediment. That is, if the height of the dune or ripple increases with 

change in slope or depth, the length also increases. The variation in 

the L(h ratios may be the result of the randornness of the dune configuration, making it difficult to obtain true average length or height. 

Some of the variation in the L/h ratios may result from including the 

separation zone in the measured lengths of the dune. That is, the 

true length of the dune may extend from the end of the separation 

zone to the downstream crest. The measured length as used, however, was from crest to crest. The magnitude of the separation zone 

increases as slope increases.

The ratio of depth of flow to dune height, or relative roughness D(h, 

was different for ripples from that for dunes. Ripples had a larger 

D/h ratio than dunes. This parameter may help explain why the n 

values were smaller for ripples than for dunes. Also for the ripple and 

dune bed form, the larger n values occurred with the smaller D/h 

values.

It seems from the flume data that maximum dune height, during 

two-dimensional flow, is limited by the depth of flow. The dune will 

only grow until its crest is within a certain distance from the water 

surface, unless three-dimensional flow exists. Also, with a given 

slope and continuously increasing depth, the dune height may reach 

some maximum value for that size of sediment and water temperature,
FLUME STUDIES USING MEDIUM SAND A~45

and increasing the depth bej^ond this limiting depth will not increase 

the dune height. If this is true, then in order to predict accurately 

the resistance to flow in an alluvial channel, the point where dune 

height becomes independent of depth must be determined. This 

point may be a function of the width of the channel. If dune height 

becomes independent of depth, the w-values may decrease with stage. 

That is, h/D, the relative roughness, will decrease.

The bed with dunes is very soft and fluid, and there is considerable 

segregation of the bed material. The upstream slope and crest of the 

dune, which has a slope of approximately 2 to 4 degrees, consists of the 

fine and coarse material moving downstream. This slope of the upstream face is practically the same as that for the upstream face of the 

ripples. On the upstream face of the dune, at the lower slopes, ripples 

form. At the higher slopes these ripples on the dunes are not apparent. The downstream face, which has a slope between 31 and 39 

degrees, consists of the coarser fraction of the bed material. This 

coarser fraction is the material that can not be swept into suspension 

at the crest but topples over the crest and avalanches down the foreslope. It is this avalanching that results in the soft nature of a dune 

bed. The material is deposited by the force of gravity and has relatively large voids. If it were deposited by the dynamic force of the 

fluid it would be packed into place, with a relatively small voids ratio. 

Bagnold (1941, p. 238-240) observed and explained the same phenomena in wind blown dunes. The trough of the dune contains a 

lajTer one eighth to 2 inches thick of the fine material from the bed. 

Part of the fines in the trough comes from the sediment that is swept 

from the crest and part conies from the lower part of the upstream face 

of the downstream dune, where the main current that overrides the 

separation zone impinges on the bed (fig. 5). As the water-surface 

slope is increased over the dune form of bed roughness, the toe of the 

downstream dune is more actively blasted and eaten away by the 

water with high velocity that impinges on it. This is an additional 

source of fine material that is deposited in the trough by the reverse 

flow in the separation zone. Velocities as high as 1.3 feet per second in 

the upstream direction were measured in the separation zone.

The slope of the downstream face of the dune, which varied from 31 ° 

to 39° with an average of 34°, was clearly defined by lines formed by 

the dark colored mica particles as the dune moved downstream (fig. 5). 

The angle of repose in air for dry noncohesive sand 1 mm in diameter 

(95 percent of the bed material in the flume was finer than 1 mm) 

varies from 29° to 32° depending on angularity of the particles 

(Simons and Albertson, 1960). That is, well rounded 1 mm particles 

have an angle of repose of 29° and very angular particles have an 

angle of 32°. It is probable that the angle of the forward face of the
A-46 STUDIES OF FLOW IN ALLUVIAL CHANNELS

dune is increased above the angle of repose of dry sand in air as a result 

of the increase in pressure in the separation zone on the foreplane of 

the dune and as a result of the reverse flow in the trough moving up the 

face of the dune. This is verified by the fact thai, as the velocity of 

flow was reduced to zero in the flume, the face of the crest of the dune 

slumped, reducing the angle of inclination of the face.

With the dune form of bed roughness the water surface is uneven 

and turbulent (see fig. 9). Over the crest of the dune, the water 

surface is lower than over the trough. This is the result of the acceleration and deceleration of the flow as it contracts over the crest and 

expands over the trough. This was illustrated for rigid boundaries 

by House (1946 p. 135 and 139). The degree of roughness (turbulence) of the water surface increases with an increase of bed roughness. 

The roughest water surface occurred where there were large watersurface boils. These boils stood approximately one-tenth of a foot 

above the surrounding water surface. However, the roughness of 

the water surface was not great enough to affect the determination of 

the water-surface slope (fig. 3). The turbulence created at the interface between the main flow and the flow in the separation zone dissipated considerable energy, and normally the disturbance caused by 

the interference was visible on the water surface downstream. In 

the dune bed form, the suspended sediment concentration, the intensity of the turbulence, the relative roughness, and the velocity of 

the reversed flow in the trough all increased with increasing slope. 

Along with the dunes, potholes formed that had a depth equal to the 

height of the dunes. These, as well as the dunes, caused boils on the 

water surface, which were evident downstream from the dunes. The 

potholes and boils moved downstream in front of the dunes at the 

same velocity as the dunes. There were normally from 10 to 20 potholes and boils evident in the full length of the flume for this type of 

bed roughness. The potholes appeared to develop as a result of the 

increased strength of the secondary circulation in the separation zone. 

The combination of the velocity in the direction of flow and the secondary circulation causes a spiral circulation similar to that in a whirlwind. This rotating motion scours out additional material at a point 

in the separation zone and produces a pothole. The existence of 

dunes and potholes is indicated visibly by the formation of very 

strong boils that transport considerable suspended sediment upward 

to the water surface.

This segregation of the material by the formation and movement of 

dunes means that a large number of bed material samples must be 

obtained and analyzed before the size distribution can be accurately 

established. For this flume material, 60 samples were needed before 

another sample would not alter the median diameter by a significant
FLUME STUDIES USING MEDIUM SAND A~47

amount. This segregation should have some value, though, because 

in a natural stream where Fr<0.6 it might be possible to determine 

the characteristics of the bed roughness by taking core samples of the 

bed, conditions permitting.

TRANSITION FROM DUNES TO RAPID-FLOW FORMS

The change from dunes to rapid flow forms is complex and the form 

of the bed roughness is erratic. With small changes in depth and (or) 

slope the bed configuration may change from a form typical of dunes 

to a form typical of rapid flow or some combination of the two. The 

transition occurs when the depth and (or) slope are changed to give 

d/S'>2 and 0.6<Fr<1.0.

There is a definite break in the Froude number when changing from 

dunes to rapid flow forms. For flow with dunes the maximum Froude 

number was 0.60. Obviously, the minimum Froude number is 1.0 for 

rapid flow. Runs with a Froude number between 0.6 and 1.0 

showed a multiple roughness. That is, the bed form was in between 

dunes and a plane bed and consisted of washed out dunes and sand 

bars.

It is logical that the upper limit of the Froude number, for the dune 

bed form, is considerably less than 1.0. The change in roughness, and 

consequently the change in resistance to flow, and the dissipation of 

energy are large when changing from the plane bed or undulating bed 

during rapid flow to dimes, whereas the change in energy for changes 

in the Froude number in the vicinity of Fr=l is small. Thus, the 

velocity and depth, when the bed form changes, are changed considerably, resulting in a low Froude number. Conversely, the change from 

dunes to rapid flow forms results in smaller loss of energy because of 

the reduced roughness, resulting in higher velocities, low depths and 

Froude numbers> 1.0.

There is a hysteresis effect in the change of bed roughness from dunes 

to rapid flow forms and back to dunes. The value of slope and (or) 

depth for the change depends on the bed configuration prior to the 

change. If the bed is covered with dunes, a slope of 0.0035 may be 

required before rapid flow will occur and the dune configuration is 

destroyed. If the bed is plane, the slope may be decreased to 0.0025 

before the flow will change from rapid to tranquil and dunes will form. 

This hysteresis effect may be attributed to the change in energy with 

a change in flow regime from tranquil to rapid, or from rapid to 

tranquil. That is, a change from potential to kinetic energy with a 

change from tranquil to rapid flow or change from kinetic to potential 

energy with a change from rapid to tranquil flow. This change in 

energy results from the large change in roughness associated with the 

change in flow from tranquil to rapid or from rapid to tranquil. The
A-48 STUDIES OF FLOW IN ALLUVIAL CHANNELS

Manning n values for dunes range from 0.018 to 0.032, whereas, with 

rapid flow, Manning n values range from 0.010 to 0.015.

The fact that, for this bed material, a typical dune pattern did not 

occur with a Froude number equal to or greater than 0.60 is important, 

especially if it holds true for other bed materials. Runs 40 and 29 

had Froude numbers equal to or slightly greater than 0.60. Their 

bed forms were washed-out dunes with rather large L/h ratios. The 

Manning n values were 0.018 and 0.021 respectively, which are small 

when compared with n values for dunes, but are higher than rapidflow n values. It seems that the magnitude of Manning n increases 

as the percent of the area of the bed that is covered with dunes decreases.

Three other runs (30, 35, 36) with 0.6<Fr<1.0 had long, low bars 

diagonal to the flow. These bars were 20 to 30 feet in length with 

L/h ratios between 30 and 50.

Runs 29, 30, 35, and 36 had distinctly different types of flow occurring side by side (fig. 11 and 12). This three-dimensional flow was 

probably caused by using discharges that were too small with respect 

to slope. This phenomena was also observed where the antidune bed 

forms existed and the discharge was decreased beyond a given value 

(run 45). For three-dimensional flow the n values ranged from 

0.014 to 0.023, whereas the n values for standing waves was approximately 0.012.

FIGURE 11. Three-dimensional flow over the sandbar configuration of figure 12 (run 35).
FLUME STUDIES USING MEDIUM SAND A-49

FIGURE 12. View upstream of sandbar configuration. Water-surface slope S= 0.00494 (run 35).

Three-dimensional flow probably results because discharge and 

slope were controlled in the experiments, and with the steeper slopes 

the small discharges set up conditions favorable for three-dimensional 

flow. Other experimenters may have observed this flow condition 

because of the limited discharge capacity of their flume system and 

the resultant large width-to-depth ratio that develops as slope is 

increased. In runs with the dune bed form, the smallest discharge 

was sufficient to insure two-dimensional flow. Presumably, however, 

by further decreasing the discharge, three-dimensional flow could also 

occur in runs with the dune bed form.

Runs 29, 30, 35, and 36 were discounted in analysis of data because 

they were three-dimensional.

BED FORMS IN THE RAPID-FLOW REGIME

PLANE BED WITH MOVEMENT

In runs with rapid flow C^K>1.0), three forms of sand bed surfaces 

and water surfaces were observed: plane bed and water surface, 

standing sand and water waves, and antidunes.

A completely plane bed with a plane water surface over the full 

length of the flume was only produced once in all the runs (fig. 13). 

It was anticipated from a study of the existing literature that this 

would be the most common type of bed configuration between dunes 

and antidunes. Most experimenters have recorded a large number
A-50 STUDIES OF FLOW IN ALLUVIAL CHANNELS

FIGURE 13. View upstream of a plane bed during rapid flow. Water-surface slope S= 0.00366 (run 26)

of runs with plane beds, particularly when they worked with short 

flumes. In many of the runs in this study, there was a plane bed 

surface and plane water surface for the first 60 to 70 feet of the flume, 

but beyond that point standing waves developed. Laursen (1958), 

in his experiment using a flume 105 feet long and 3 feet wide, also 

reported difficulty in obtaining a plane bed. Perhaps plane beds 

result from using flumes with insufficient length for standing waves 

or antidunes to form.

For the plane bed of figure 13 the Manning n value is extremely 

small (n=0.0078); the total sediment load is large (4,580 parts per 

million); the Froude number is 1.6; and the d/8' value is only 2.1. 

This Froude number is the next to largest recorded even though the 

slope for this run was not as great as for the runs in the antidune 

regime. The sand bed was very firm.

STANDING WAVES

The water surface consisted of symmetrical standing waves of low 

amplitude, (fig. 14). The standing waves formed and gradually 

disappeared but unlike antidunes they had no tendency to break or 

migrate upstream. The forms of bed roughness observed with standing waves, in the order of increasing slope, were a diagonal dune 

pattern cross-laced like a shoe string (fig. 15), a plane bed, (fig. 13),
FLUME STUDIES USING MEDIUM SAND A-51

FIGURE 14. View downstream of standing wave during rapid flow. Water-surface slope <S= 0.00364

(run 39).

FIGURE 15. View of cross-laced configuration looking upstream. Water-surface slope S= 0.00435 (run 27).
A-52 STUDIES OF FLOW IN ALLUVIAL CHANNELS

and a symmetrical, undulating sand wave similar in form to those 

observed in the antidune regime.

The standing water waves formed when Froude number was 

approximately 1.0. The standing water waves with the diagonal 

cross-laced sand bars shown in figure 5 formed with this lower Froude 

number. The standing water waves on a plane bed formed at a 

Froude number of about 1.1, and standing water waves on a bed 

with undulating sand waves formed when 1.2<^Fr<^1.5. The Manning n for the standing sand wave bed form is relatively small, ranging 

from 0.012 to 0.015. The water surface waves are 1.5 times as high 

as the corresponding sand waves. Concurrent measurements of the 

sand bed waves and the water-surface waves are illustrated in figure 21.

ANTIDUNES

When the Froude number, computed on the basis of average velocity 

and average depth, was greater than 1.5 and d/8' was greater than 3.0, 

antidunes formed.

Antidunes are defined as a train of symmetrical sand waves, inphase 

with a corresponding train of symmetrical water-surface waves. 

Both trains of waves move upstream and grow in height until they 

break (figs. 16-20), resulting in a cyclical fluctuation of the waterFIGUKE 16. Water surface before antidune wave starts to build up. View is upstream. Water-surface

slope S= 0.00986 (run 42).
FLUME STUDIES USING MEDIUM SAND A-53

FIGURE 17. View across the flow at an antidune wave that is forming.

FIGURE 18. View upstream of an antidune wave at point of breaking.
A-54 STUDIES OF FLOW IN ALLUVIAL CHANNELS

FIGURE 19. Antidune wave after breaking (run 42).

FIGURE 20. View of bed configuration after antidune flow. The pattern has been somewhat altered

by draining the flume.
FLUME STUDIES USING MEDIUM SAND A~55

surface waves and the sand waves. The waves build up from a 

plane bed with a plane water surface. They grow and move upstream 

until one or two of the waves become unstable and break as shown 

in figures 18 and 19. Normally, when one wave breaks, other waves 

break immediately thereafter for a distance of one or two waves 

upstream and 4 or 5 waves downstream. That is, depending on discharge and slope, from 1 to 8 waves at a spacing of 1 to 6 feet usually 

break within a short time interval. This chain reaction that follows 

the breaking of the first wave is apparently triggered by the action 

of the first wave that breaks.

In making concurrent measurements of the bed and water surfaces 

in the troughs and crests of antidunes, it was observed that antidunes 

become unstable and break when the water surface in the trough of 

the wave train is approximately the same elevation as the crest of the 

downstream bed wave (fig. 21). The measurements also indicated 

that the water waves are 1.7 times larger in amplitude than the 

corresponding sand waves. The total height of the water-surface 

waves from trough to crest was from 1.0 to 1.5 times the average 

depth. When the waves started to break, their heights were about 

twice the average depth. The breaking of the waves is very spectacular. There is considerable turbulence, dissipation of energy, and 

mixing of the flow. When they break they sound like the surf in the 

ocean and there is probably some resemblance, in that the ocean 

wave, as it climbs the sloping beach, also builds up to an unstable 

height before breaking.

When the antidunes are building up, the bed is very firm and there 

is no separation between the flow and bed. However, when the antidune breaks, the crest of the sand wave becomes soft and fluid and 

seems almost to explode. When the water waves break, the flow is 

very turbulent; there is separation, and considerable sediment is 

thrown into suspension. Normally, most of the sediment load is 

moved as contact load. Total load concentration is very high, 

ranging from 4,240 to 15,000 ppm for these runs. It appears that the 

movement of the antidunes upstream results from scour on the downstream side of one sand wave and deposition on the upstream side of 

the adjacent wave and consequently, the wave moves upstream.

Except for run 45, which was considered three-dimensional, there 

was only one train of waves in the cross section. The train of waves 

was located off center at about the left %-point in the cross section. 

In subsequent runs, however, two trains of waves symmetrically 

located in the flume were observed several times. The wave train was 

not continuous throughout the entire length of the flume. That is, 

a train of antidunes would build up and break in one 20- to 40-foot 

length of flume and then repeat in another 20- to 40-foot reach, or
A-56 STUDIES OF FLOW IN ALLUVIAL CHANNELS

1.20

1.00

0.80

0.60

0.40

0.20

0.00

-0.20

Water s jrface RUN 41 

STANDING WAVE

85 87 91 93 95 97 99 

STATION ALONG FLUME

101 103 105 107

FIGURE 21. Profiles of the actual water surface and bed configuration for runs 44 and 41. Note: arrows 

indicating the point in run 44 where the elevation of the crest of the bed surface is greater than the elevation 

of the trough of the water surface.

they would build up and break in two or three discontinuous lengths of 

the flume. The building up and breaking, in the different reaches of 

the flume, might be in phase or out of phase.

The period of time it took for one antidune wave to build up and 

break, and the number of antidune waves built up at one time varied 

with the slope and the discharge. During run 32 only one train of 

six to eight antidune waves would build up and break at a given time. 

This train of waves, with a total length of about 40 feet, would build
FLUME STUDIES USING MEDIUM SAND A~57

up in, for example, the lower section, then next in the middle or upper 

sections, and again in the original position. Normally, about two 

complete cycles occurred every hour (a cycle includes starting with a 

plane surface, the building up of waves, the breaking of the waves, and 

the return to a plane surface). For other runs with steeper slopes 

there were three or four reaches in the flume where the train of waves 

built up and broke in phase and out of phase, almost continuously.

When the antidunes break, a considerable amount of water is stored 

in the flume. Observations at the glass-walled section of the flume 

show that when an antidune breaks, the water arid sediment in the 

wave crest ceases to move or moves upstream until the breaking wave 

vanishes and then normal flow is restored. At the higher slopes (run 

43), when three or four series of antidune waves in the train would 

build up and break simultaneously, so much water was stored in the 

flume that the pumps would surge. In the extreme case, antidune 

waves would build up and break in phase with the surging of the 

pumps. This resulted in a fluctuating discharge and an increase in 

antidune activity. So much water was stored in the full length of the 

flume as a result of the more violent antidune action, that the tailwater level in the tailbox dropped until the pumps lost their prime. 

To eliminate this degree of surging of the pumps, the level of the water 

in the sump was raised and more water was continuously added to the 

sump to replace that stored in the flume. When the antidunes were 

not breaking, there was a surplus of water, which was discharged 

through an overflow at the top of the sump.

The storage of water caused by the breaking of the antidune waves 

probably explains the surging discharge observed in alluvial streams 

with steep slopes. That is, the antidunes store and release water in 

the upper reaches of the stream in a random, haphazard manner, but 

as the flow travels downstream this storage and release of storage 

causes the antidunes to break in a more systematic pattern until 

surges develop that cause the antidunes to form and break at regular 

time intervals. Some field examples of surging flow are Muddy Creek 

near Pavillion, Wyo., and Medano Creek in the Great Sand Dune 

National Monument, Colo.

Antidune flow, in terms of fluid mechanics, is very efficient. That is, 

the Manning n ranges from 0.010 to 0.013. Even though the breaking 

of the antidunes dissipates considerable energy, the length of flume 

that this breaking occupies is small in comparison with the total flume 

length and the period of time the breaking waves exists is small in 

comparison with the total time for one antidune cycle. This explains 

the low n values. It was observed that as the antidune activity increased with an increase in slope there was a larger dissipation of 

energy with a decrease in discharge coefficient C/-Jg and an increase in
A-58 STUDIES OF FLOW IN ALLUVIAL CHANNELS

Manning n. Surface velocity measurements, obtained by timing 

floats along a centerline of a train of antidune waves (when they were 

not breaking) and along a nearby parallel line where the water surface 

was smooth, showed that the surface velocity was greater in the train 

of waves. The velocity in the trough of the antidune is considerably 

greater than that in the crest. It is possible that the antidunes break 

when the velocity and depth over the crest decrease because of increasing wave height until a Froude number less than 1.0 results.

The standing wave and the antidune wave may be compared to the 

flow regimes of rigid boundary hydraulics illustrated by Rouse (1946). 

The crisscrossed bed pattern (figs. 15 and 20) that is observed with 

both standing wave and antidune flow resembles the shock waves 

that form with rapid flow (Fr^>l.O) in rigid boundaries.

ANALYSIS OF DATA

The basic data (table 1, p. 30) have been utilized as dictated by the 

investigations of others, by various concepts (many of which are 

partly intuitive) and the dimensional analysis of the channel roughness 

problem to obtain the relationships presented in this section. It 

should also be borne in mind that these relations, excluding figure 28, 

are based on only one group of flume data, and that the size and 

gradation of bed material remained constant except as altered by 

miscellaneous sorting, which may or may not be of significance. 

That is, d and a are constant, and w varied only because of temperature 

changes. The relations presented in this chapter are given in a logical 

sequence that can be subdivided into four major phases.

1. Relations of a simple nature that illustrate the interrelationship of the variables, 

geometric variables, the form of bed roughness, and the regimes of flow.

2. Relations of a more complex nature that include the variation of the Manning 

n, tractive force and total sediment load concentration.

3. Relations that involve roughness in alluvial channels. These relations vary 

from simple to complex.

4. A plot that utilizes data from both the laboratory and the field to expand the 

usefulness of the foregoing concepts and to accentuate the existence of two 

regimes of flow and six major forms of bed roughness.

Appreciable insight of the scope of the data collected, and into 

the interrelationships of flow to geometric variables can be gained by 

considering a few simple dimensional plots.

VARIATION OP VELOCITY WITH DEPTH

The variation of velocity with depth and their relations to the six 

major forms of bed roughness are shown in figure 22. In this and 

subsequent plots the following notation is used to signify the two 

regimes of flow and six forms of bed roughness: Tranquil flow PI, 

is plane bed and water surface with no bed movement; R, is ripples;
FLUME STUDIES USING MEDIUM SAND A-59

8.0

7.0

6.0

5.0

4.0

8 3.0

2.0

1.0

"AA0

Plane bed

or 

antidunes

Fr= 1.30

 37

FORMS OF BED ROUGHNESS 

TRANQUIL FLOW 

6 Plane 

O Ripples 

® Dunes 

O Transition

RAPID FLOW

A Plane

 Standing waves

A Antidunes 

Plus sign indicates 

three-dimensional flow

Plane bed

or 

standing waves

Transition 

Critical velocity curve

0.2 0.4 0.6 0.8 

DEPTH,IN FEET

FIGURE 22. Variation of velocity, V, with depth, D.

D, is dunes; T, is transition from dunes to rapid flow forms. Rapid 

flow: P, is plane bed with a plane water surface; W, is standing 

sand waves; A, is antidunes.

Isoclines of Froude numbers have been superposed on the figure to 

illustrate the wide scope of the data. Velocities range from those 

for no movement of bed material to magnitudes far in excess of the 

critical velocity. That is, V varies from 0.65 to 6.2 feet per second. 

Note that the transition roughness includes those runs that have a 

bed roughness somewhere between dunes and roughnesses associated
A-60 STUDIES OF FLOW IN ALLUVIAL CHANNELS

with rapid flow. Runs of this type plot just below the critical velocity 

curve. The lines dividing the plot into forms of bed roughness are 

based on the observed roughness and the Froude number, Fr.

A similar relation, that of D to Q, can be plotted using slope and 

forms of bed rougness as the other variables. As in the preceding 

figure, the transition runs will plot just below the critical depth curve.

VARIATION OP TOTAL SEDIMENT-LOAD CONCENTRATION AND

MANNING ft

Values of Manning n range from 0.008 to 0.032. These n values 

have been corrected for side effect in accordance with the procedure 

presented by Einstein and Barbarassa (1951). The magnitude of 

nb increases with increasing V until the regime of flow shifts from the 

tranquil regime to the rapid flow regime where the n's reduce suddenly 

to approximately 0.012. Undoubtedly, a better relationship of the 

foregoing type could be obtained by relating Manning n or some other 

measure of channel roughness with suspended sediment load, except 

that in shallow depths it is difficult to sample suspended sediment 

load with significant accuracy. More often than is desirable, when 

taking the suspended sediment sample, part of the bed material load 

and in some cases even a part of a ripple or dune may be intercepted, 

particularly when the velocity is relatively large.

By relating the concentration CT and the Manning n to the regimes 

of flow these significant conclusions can be reached.

1. As the flow is increased and the bed changes from a plane bed with 

no movement to one with ripples, the Manning n increases from 

about n 0.015 to w=0.025 with no significant discontinuities.

2. As dunes form a marked increase in Manning n occurs, from 

#,=0.025 to n=0.033. This increase in n is caused by the 

large size of the dunes at a spacing conducive to maximum 

roughness.

3. In the transition from tranquil flow to rapid flow, the resistance 

to flow is also in transition. It shifts with a small change in 

depth and slope from high resistance, almost as high as for the 

dunes, to low resistance, which is not quite as low as for rapid 

flow. The amount of the resistance to the flow is dependent on 

the area of the bed covered by dunes.

4. In the rapid flow regime there is a significant decrease in Manning 

n. This large decrease in resistance to flow can be attributed to 

the change from dunes to symmetrical sand waves. Dunes have 

a large separation zone with large form drag, whereas the 

symmetrical sand wave has no separation zone and has only the 

form drag resulting from the particle. Possibly some of this 

decrease in Manning n can be attributed to the movement of 

large quantities of sediment as bed load. The turbulence
FLUME STUDIES USING MEDIUM SAND A~61

created is insufficient to hold large quantities of sediment in 

suspension. Consequently, the sediment load that was carried 

in suspension when dunes formed now is carried near the bed. 

This heavy concentration, in turn, markedly changes the 

properties of the fluid-sediment mixture, which means that the 

fluid is no longer homogeneous. A sort of stratified flow results 

that inhibits mixing, and the effective bed roughness is extremely 

reduced. The flow has some similarity to plug flow in pipes. 

For the single run with n=0.0078 the bed was plane, and the 

resistance caused by the standing waves was not in effect. 

Under these conditions the similarity to plug flow is even more 

pronounced.

5. The largest concentration of total load for standing wave roughness 

occur just before these waves begin to move upstream as antidunes. With antidunes the CT values are even larger. The 

same large concentration of sediment exists near the bed as 

under standing wave conditions, except for the short time while 

the antidunes are breaking, and hence the Manning n remains 

small, 0.011<rc<0.015.

VARIATION OF TRACTIVE FORCE AND TOTAL SEDIMENT LOAD

Figure 23 is obtained by plotting the tractive force TO , which has 

been calculated by the equation

r0=yDS (25)

with the total sediment load concentration in parts per million. 

Although there is appreciable scatter, a trend definitely exists. That 

is, the magnitude of total load CT increases with increasing magnitude 

of tractive force TO . Each run in figure 23 has been labeled in accordance with its form of bed roughness. Using this notation the regimes 

of flow and forms of bed roughness have been indicated. There is a 

definite tendency, in this case, for the tranquil flow runs to separate 

from the rapid flow runs because of the large tractive force associated 

with runs that have large dunes on the bed.

The deviation of the runs that form dunes from the general trend 

is a result of an increase in shear TO , due to the large size of the dunes, 

without a corresponding increase in concentration CT . It is significant 

that a sudden decrease in TO occurs at about 0^=2,000 parts per million 

as the bed changes from dunes to plane bed or standing waves. This 

is caused by a reduction in roughness, which also decreases the 

ability of the flow to transport sediment. Consequently, CT is cut 

in half from run 40 to run 36, beyond which point CT, in the transition 

from dunes to rapid flow bed forms, steadily increases. No such 

discontinuity occurs at the change from ripples to dunes or at the
A-62 STUDIES OF FLOW IN ALLUVIAL CHANNELS

3f

Pl9

0.001

FORMS OF BED ROUGHNESS 

TRANQUIL FLOW

Plus sign indicates 

three-dimensionalfjow

10 100 1000 10,000

CT, IN PARTS PER MILLION 

FIGURE 23. Variation of shear ra and concentration of total load CT, in parts per million.

change from standing waves to antidunes. There is, however, 

considerable scatter in the values for the rapid flow regime, which 

shows that another variable is needed to describe the phenomenon 

more completely.

RESISTENCE TO FLOW RELATIONS

Working more directly toward the evaluation of variable bed 

roughness in alluvial channels, and using (7/V<7 as the index of roughness, the applicability of the following equation, suggested by Ali 

and Albertson (1956), was investigated.

(26)

Certain trends were apparent involving these three parameters but 

there was excessive scatter. It may be that the collection of additional data will improve this condition. Certainly the data presented by Ah and Albertson show that the foregoing functional 

relationship has merit and warrants additional consideration.

Applying an approach suggested by Liu (1957), the relationship 

between V*d/v and VTO /V* (&ys)d, for ripple formation, was investigated for both regimes of flow (fig. 24) by using form of bed roughness 

as the third variable. 

It is important to note how precisely the equation

(27)
FLUME STUDIES USING MEDIUM SAND A-63

describes all forms of bed roughness. These data plot as a curve on 

log-log paper.

The foregoing curve is of such a nature that it can be modified by 

adding a constant to the ordinate term to yield two lines that are

10

Ts

'**

<]

bl

1.0

o.ii .

>

/

/

/»

/

u

)

X

ffi %

o

D

J

 "//

^J A

jr £ @

»

FORMS OF BED ROUGHNESS

TRANQUIL FLOW

© Plane

O Ripples

© Dunes

Q Transition

RAPID FLOW

A Plane

 Standing waves

A Antidunes

Plus sign indicates

three-dimensional flow

1.0 10 . 10

Vm d d/ 

 1 1 fi /f '

FIGURE 24 Variation of VT«/V*(&y,)d with Vtd,.
A-64 STUDIES OF FLOW IN ALLUVIAL CHANNELS

essentially straight, one line representing runs for 7*V>1, the other 

line representing runs for FK<!. These lines are shown in figure 25. 

Note that a constant equal to 5 was assumed. That is, figure 25 

relates (V*df)v+5 to V T0/y* (Ays)d. These two lines are combined 

into a single line in figure 26 by noting that the Froude number can

10

1

J

y^

#^

9&v°

s 

^^

B

e

*r^f 4

^

v$^

r?\

^

'%£*

FORMS OF BED ROUGHNESS 

TRANQUIL FLOW 

e Plane 

O Ripples 

® Dunes 

« Transition

RAPID FLOW 

A Plane 

 Standing waves 

A Antidunes 

Plus sign indicates 

three-dimensional flow

).l 1.0 10.0 1C

FIOUEE 25. Variation of (F»d/i/)+5 with Vro/F,(A-y.)<f and Fr.

be used as a third variable to evaluate a common intercept. The 

magnitude of this intercept (I) is.

V V1/8

and the equation of the resultant line becomes

1/8

(28)

The relation between (V*d/v)+ 5 and

is given in figure 26. The foregoing equation modified to conform 

with this figure becomes

0.465 0.13

(29)
FLUME STUDIES USING MEDIUM SAND A~65 

Expressing this equation in the form of the Chezy equation

^=0.0049 +5 (Fr)°M (31)

and

The velocity term in the Froude number can be factored out and 

combined with the dependent variable for convenience.

All six forms of bed roughness are indicated in figure 26. Excluding 

some overlap of the runs having dune bedforms and the runs having 

transition bedforms, the forms of bed roughness plot in well-defined 

groups.

Equation 9 results from the dimensional analysis of the problem of 

channel roughness.

(9)

Considering for the present that because d, w and CD are interrelated, 

CD and w may be of secondary importance and may be eliminated at 

least temporarily from equation 9. Then, the foregoing expression 

reduces to

' (32)

in which Fr has been squared arbitrarily. Taking the liberty of 

combining the remaining parameter as follows as a first approximation,

FT^ . rs0"|

Re L^J (33)

a plot can be made using ^ =-^9=<t> T=T with V*D/v as the third Re yD2 [_D J

variable. A family of parallel lines of constant V*D/v values can be 

established on log-log paper. This family of lines can be compressed 

into a narrower band by finding the equation relating the intercepts 

and values of V*D/v and replotting according to the indicated 

relationship. That is,

TT r-/-o_7N3M x \-n

(34) yD*

In doing so it is noted that Fr can be used as a third variable to yield 

a family of parallel lines, which can be reduced to a single line by
-66

90 

80 

70

60 

50

40

 v 

f 30 

&

20

15

10 

0

STUDIES OF FLOW IN ALLUVIAL CHANNELS

Eqi

/

Y

latio

/

X«

A+

\

/

I

4 

©

© /

/ 0

^ ©

O

. X

yX,

~^y

VA>

/

FORMS OF BED ROUGHNESS

TRANQUIL FLOW

9 Plane 

O Ripples 

<8 Dunes 

O Transition

RAPID FLOW

A Plane 

 Standing waves 

A Antidunes

Plus sign indicates 

three-dimensional flow

I

8 0.9 1.0 1.5 2 34

FIGURE 26. Variation of Vtd/v +5 with [Wo/F,

again solving for the general intercept equation and substituting it 

in equation 34. Following this procedure the equation

V J

(35)
FLUME STUDIES USING MEDIUM SAND A-67

ixirr4

6

4

2

IxlO'5

6

4

2

ixio~6

Equation :

/

vn /

r s f3

7" 2/(vfr)/4 T 0 ' /^\

£

®/

s

O/

n x^Pj / ©

/

^n

#

^

/

/TBTi

e J

r

9

^

A*X i

\^V

/4

^

FORMS OF BED ROUGHNESS

TRANQUIL FLOW

O Plane

O Ripples

ffi Dunes

9 Transition

RAPID FLOW

A Plane

 Standing waves

A Antidunes

Plus sign indicates

three-dimensional flow

I

ixio- i x io~5

FIGURE 27. Variation of VnlvD*/(V/-JgDW* with [(SW/.D) 8/4/(V».D/«')F*.

was obtained. This relation is presented in figure 27. Solving for V,

I

f T\. Al*t S N9/Q / nt}fl \ */ & / t*t \ */ « _____

O ^ ' V V / V ^^/ /

and

n 1 / j\w / *, \2/3

(37) ^_l 01/6 ri 2/3 l^£ \ [ "

^_^6 U ( J ^TS

where CD, the drag coefficient of the particle, is equal to
A-68 STUDIES OF FLOW IN ALLUVIAL CHANNELS

FORMS OF BED ROUGHNESS RELATED TO SIZE OF BED MATERIAL

As suggested in the introduction, a study of Liu's (1957) curve, 

which predicts when ripples form, led the writers to speculate on the 

possibility of discovering similar curves for other forms of bed roughness. As has been illustrated in the preceding figures, various parameters and (or) combinations of parameters apparently make it possible 

to define forms of bed roughness. In order to relate the foregoing 

concept to recent publications, the equation

was selected to test the concept. Therefore, figure 28 and table 2 

were prepared using data from several sources, which are cited. In 

this figure it is clearly evident that, in general, dividing lines can be 

drawn to separate the forms of bed roughness. Furthermore, Liu's 

curve is extended more than one cycle to the left by means of the data 

of Kalinske and Hsia (1945), and it is extended more than one cycle 

to the right by the data of Lane and Carlson (1953). In this figure 

a temperature of 20°C was used to determine the fall velocity and 

kinematic viscosity where temperature was not available. The 

scatter among the points from Laursen (1958), Brooks (1957), Barton 

and Lin (1955) and the basic flume data is due in large part to the 

temperature variation. Lines of constant temperature can be drawn 

for these data at an angle of 45°. It is also noteworthy that for almost 

any transition line, a change in temperature for a given size of bed 

material can result in a change in the form of bed roughness.

The following are observations regarding figure 28 and table 2 

that may have significance.

1 . Kalinske and Hsia (1945) reported ripple formation, although they 

did not record the conditions under which movement began 

and the conditions under which ripples were initiated. The 

minimum value of d/8' for their data is approximately at the 

same point where the Colebrook-White curve (Rouse, 1946) 

connects with the smooth boundary curve. To the left of this 

point the boundary is completely smooth, no influence of the 

overhead turbulence is felt at the boundary, and the laminar 

sublayer is a completely protective blanket. This fact makes it 

doubtful that ripples form under these conditions. In any event 

the authors have been unable to locate any data for smaller 

values of d/d'. The instability of fine noncohesive materials 

may make it difficult to determine definitely when movement 

begins and ripples start to form.
o i <?~

rteS

FIGTJKE 28. Criteria for bed roughness in alluv
TABLE 2. Values for predicting bed fo

[Froude number, Fr, concentration of total load, CT', slope for predic

Reference

Kalinske-Hsia (1945).... __ -

Brooks (1956) __ -

Barton-Lin (1955) _ _______ .....

Gilbert (1914). . . 

Lane-Carlson 1 (1953).. ....... _ ....

Particle 

diameter 

(mm)

0.011

.10

.137

.18

.45 

1.506 

1.71 

3.17 

4.94 

42.0

Point of transi

Plane bed to ripples

FT 

(value)

0.15

CT 

(ppm)

1

S 

(percent)

0.023

Ripples to dunes

FT 

(value)

0.30

.25 

.20 

.30

CT 

(ppm)

17,000

70 

20 

90

S 

(percent)

0.025

.070 

.044 

.057

1 Complete stability of bed, no transition involved.
FLUME STUDIES USING MEDIUM SAND A~71

The curves describing the forms of bed roughness are rather clearly 

defined and are nearly parallel for d/8'<^10. In figure 28 it is 

apparent that Liu's curve is not parallel to the foregoing family 

of curves in this range, whereas the Shield's curve turns up at 

the left end more nearly in accordance with them. Based on the 

rather definite and systematic orientation of the regime curves 

and Shield's curve, it is possible that Liu's curve should be 

lowered in the vicinity of d/S'=l. This results in a curve (fig. 

28) that more nearly conforms to the curves defining the other 

regimes of flow. Such a minor adjustment might also be 

justified by a careful study of the effect of size distribution of the 

sediment.

For increasing values of d/S' (the laminar sublayer becoming 

relatively thinner and thinner) the influence of viscosity becomes 

less and less for each type of bed formation. Ripples do not 

form beyond d/d'^10 and dunes cease to exist beyond e?/5'«40. 

Apparently, the data are sufficiently complete to permit these 

approximate conclusions regarding both ripples and dunes. 

For bed formations of the rapid flow regime, however, the data 

are not complete enough for any conclusions to be drawn about 

whether the standing waves and antidunes ceases to exist beyond 

some large value of d/d'.

Because of the large surface waves and irregularities in rapid flow 

regime, the Froude number no doubt plays a very important 

part. No attempt has been made as yet to relate the Froude 

number beyond that done in table 2 and as illustrated in figure 

28.

Although the two variables V*/w and V*d/v, as shown by figure 28, 

are of primary importance, there is still need for a significant 

third variable that will correlate the data in a quantitative 

manner as now is done in a qualitative way by the terms, ripples, 

dunes, transition, plane bed, standing waves, and antidunes. 

The Froude number is significant, as shown in table 2, in defining 

ripples and dunes, but there is considerable question whether 

it will prove to be as significant for these forms of bed roughness 

as for the forms in the rapid flow regime. The drag coefficient 

CD is essentially a constant for each size of bed material, and, 

hence, is not a logical third variable, except perhaps in establishing the points where the forms of roughness, such as ripples 

and dunes, may cease to exist. The remaining variables that 

might be significant are: the slope S, the size of bed material 

d relative to the depth of flow D, the concentration of total load 

CT, the Chezy discharge coefficient Cj-Jg, and the parameter 

used by Shields, T0/d&ys. Which of these would be most
A-72 STUDIES OF FLOW IN ALLUVIAL CHANNELS

significant depends, at least to some extent, upon the use of the 

plot.

6. Figure 28 shows that as d/8' increases, the existence of two forms 

of the bed roughness are eliminated. When d/d'^>7 and sediment is moving, the possibility of the first condition (movement 

without bed waves) is eliminated. When d/5'>10, ripples will 

no longer form, and when d/5'!>40, dunes may not exist. For 

sediments with a specific gravity of 2.65, figure 28 shows that 

the changes in form of bed roughness are related approximately 

to the size of sediment, the limiting size for the existence of 

ripples, and dunes being approximately 2.5 mm, and 8 mm respectively. In the rapid flow regime beyond d/d'=4Q, several 

bed configurations are possible, such as a plane bed and a bed 

with symmetrical undulations of various sizes. The limiting 

conditions for the standing wave and antidune forms of bed 

roughness are not established. Although the factors that produce these limits are not known, they are no doubt related at 

least in part to the influence of viscosity as expressed by d/d'. 

However, the limits may also be related to gravitational forces 

and the forces creating the sand waves, the drag coefficient, the 

size of the sand waves, the size of bed material, and the Froude 

number. Perhaps both ripples and dunes disappear when the 

grain roughness approaches the same order of magnitude as the 

size of the dunes and ripples.

7. Within the tranquil flow regime the form of bed roughness is unstable when 0.4<Fr<O-0- Typical dunes cease to exist when 

0.4<^/V<^0.6, depending on the characteristics of the bed material. The flow for which (0.4 to 0.6)<Fr<].0 is in the transition zone between dunes and rapid flow forms. The proximity 

of the transition zone to critical flow conditions Fr=l probably 

accounts for the instability cited.

8. In the rapid flow regime following bed conditions occur:

a. A plane bed forms with no visible sand waves of any kind 

and with a very smooth water surface. These flume 

studies showed that this condition is unstable and is difficult to reproduce, particularly as the length of flume is increased beyond 75 feet.

b. Symmetrical standing waves of low amplitude form and 

gradually disappear. Unlike antidunes these water surface and bed waves have no tendency to migrate upstream 

or to break and shear off. The bed may be plane, have a 

diagonal dune pattern crisscrossed like shoestrings or have 

symmetrical undulations.
FLUME STUDIES USING MEDIUM SAND A~73

c. As d/d' increases beyond the value for the standing wave form 

of bed roughness, the formerly stationary undulations of 

the bed which are rather similar to sine waves (figure 21), 

grow larger until antidunes form and travel upstream. At 

increased values of d/d', the formation of antidunes depends on the size of the bed material, and, as both d/d' 

and V*/w are increased the formation of antidunes is 

mathematically related to the Froude number. As pointed 

out by Gilbert (1914), the antidunes occur in coarse material of a size even greater than that for which data is 

available.

CONCLUSIONS

Several forms of bed roughness are found in alluvial channels, the 

major ones being: tranquil-flow regime, which includes plane bed 

before movement of bed material begins, ripples, dunes, and transition from dunes to rapid flow forms; and the rapid-flow regime, which 

includes plane bed with a plane water surface, symmetrical standing 

sand waves, and antidunes.

In the tranquil flow regime C/-<fg varied from 14.0 for the plane 

bed without movement to 7.4 for dunes. For ripples, C/ifg, varied 

from 7.8 to 12.4, the larger C/^g values occurring with the deeper 

depths and smaller relative roughness values h/D. For dunes (7/V0 

varied from 7.4 to 12.8, and again larger Cj^g values were associated 

with small h/D.

Smaller values of Cj^g probably would have occurred but for the 

fact that in a flume this large, as in a natural stream, the form of 

bed roughness allows part of the flow to by-pass the dune.

In the rapid flow regime C\^g varied from 13.9 to 27.0. The 

smaller Cj^g values occurred with the smaller Froude numbers. 

The largest C\Jg value occurred for a plane bed and plane water 

surface. Standing sand waves in general had larger values of Cl-Jg 

than antidunes. This results from the dissipation of energy by the 

breaking waves. C/i/~g would be reduced further for antidunes but 

for the fact that the period of time and length of flume occupied by 

the breaking waves is small in comparison with the total time for a 

complete antidune cycle and total length of flume.

In the transition from tranquil to rapid flow (7/V0 varied from 10.0 

to 14.1. The magnitude of (7/V0 increases as the percent of the area 

of the bed that is covered with dunes decreases. Also, there is a 

hysteresis during the change from tranquil to rapid flow that depends 

on the form of the bed prior to the change. That is, with discharge 

constant, a larger slope is required to change from tranquil flow to 

rapid flow than is required to change from rapid flow to tranquil flow.
A-74 STUDIES OF FLOW IN ALLUVIAL CHANNELS

This change in flow regime from tranquil to rapid or vice versa may, 

if conditions are right, occur in a natural stream. That is, if the slope 

of the energy grade line is close to the critical slope, a change in stage 

results in a change in regime. When this occurs there is a break in 

the discharge rating curve. Because there is a hysteresis in the change, 

the stage where the break occurs varies depending on whether the 

stage is rising or falling and the rate of change of discharge with time.

The usefulness of the various equations for C/^fg developed in this 

report is severely limited by the range of data upon which they are 

based. As the influences of different bed materials, and a wider range 

of temperature conditions are determined, as a result of proposed 

future work, these expressions will undoubtedly need revisions, which 

will make them much more useful to the engineering profession because of the greater range of conditions considered.

In the series of runs upon which this report is based, the antidunes 

developed when Fr>1.3. This may or may not be the case for other 

bed materials. For values of 1 .0<^Fr<^ 1.3, water surface waves formed 

which rarely broke in true antidune fashion, and the sand waves 

underlying the water waves stayed in essentially the same position. 

There was a very little tendency for these waves to move in either 

the upstream or downstream direction. At values of Fr>1.3±, the 

water waves formed and broke in a rather systematic manner and the 

sand waves moved in the upstream direction. The sand waves associated with the water waves were approximately the same shape as the 

water waves, but on a reduced scale. The sand waves were about 

one-half the amplitude of the water waves. There was a tendency 

for the magnitude of the sand and water waves to increase in the 

downstream direction along the flume, particularly for those runs 

having 1.0<Fr<1.3±. The magnitude of the sand and water waves 

increased with depth. Breaking of the water waves and the sand 

waves occurred when the elevation of the crest of a sand wave was 

approximately at the same elevation as the water surface in an adjacent trough. It is apparent that antidunes reforming and breaking 

result in the surging flow observed in some natural streams.

There is a definite break in the Froude number when the bed form 

is changing from dune beds to rapid-flow bed forms. Dunes do not 

occur with this bed material when the Froude number is greater than 

0.6. Rapid flow starts when Fr>1.0. Flow with a Froude number 

between 0.6 and 1.0 had multiple forms of bed roughness.

Observations of the forms of bed roughness, of the way they develop, 

and of the way Jthey^-Haeve indicate that it may be possible to determine 

the magnitude and type of bed roughness (particularly dunes) by 

checking the manner in which the bed material is segregated within 

the beds of small and intermittent streams. This method cannot be
FLUME STUDIES USING MEDIUM SAND A~75

employed if the bed material is perfectly uniform and is not contaminated with any foreign matter, but this is a laboratory situation. 

Also, the appearance of the water surface seems to provide an excellent 

means of estimating the form of bed roughness.

LITERATURE CITED

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channels and some physiographic implications: U.S. Geol. Survey Prof. 

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Liu, H. K., 1957, Mechanics of sediment-ripple formation: Am. Soc. Civil 

Engineers Jour., v. 83, no. HY2.
A-76 STUDIES OF FLOW IN ALLUVIAL CHANNELS

Shields, A., 1936, Anwendung der Ahnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebbewegung: (Application of similarity principles 

and turbulence research to bed-load movement): p. 42 Mitt, (der Preussischen Versuchsanstalt f tir Wasserbau und Schiff bau, Heft 26, Berlin).

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