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Resistance to Flow in 

Alluvial Channels

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-J

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Resistance to Flow in 

Alluvial Channels

By D. B. SIMONS and E. V. RICHARDSON

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-J

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 Price 50 cents (paper cover)
CONTENTS

Abstract _ ________ _______

Introduction __________________________________

Experimental equipment and data collection __________

Forms of bed roughness and flow phenomena _________

Bed configuration without sediment movement ____

Ripples _________ ___________________

Dunes __________ ___________________

Plane bed with sediment movement ______________

Antidunes ________________________ 

Chutes and pools ____ _ _______

Regimes of flow in alluvial channels ____ _______

Lower flow regime __ _ _ _ _______

Upper flow regime __ _ ____________

Transition __ _ _ __________

Comparison of flume and field conditions _____________

Variables _ _____ _ __________________

Concentration of bed-material discharge _________

Fine-sediment concentration ____________________

Dependent variable __________________ 

Change of variables _____________________________

Shape factor for the reach and cross section ______

Seepage force _____ _ ___-_______

Resistance coefficient _ _ _____________

Slope ______________________________________

Depth _______________________

Page 

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Variables Continued

Resistance coefficient Continued

Size of bed material ______________________

Fall velocity _______-__________________

Apparent viscosity and density _____________

Gradation of bed material __________________

Prediction of form of bed roughness - _ ________

Velocity distribution ___ _ _ ______________

Plane bed with sediment movement _ _______

Roughness coefficients _____________________

Evaluation of resistance to flow ____________________

Evaluating resistance to flow by adjusting slope 

Evaluating resistance to flow by adjusting depth

of flow ______________ _____________

Adjusting depth to average alluvial grain

roughness 

Alluvial sand-bed roughness _________

Baffle and cube roughness __ _______

Adjusting depth to smooth boundary roughness __________________________________

Baffle and cube roughness __ _ _.

Gravel and cobble roughness ________

Alluvial sand-bed roughness ______ 

Determination of average velocity _______

Summary and conclusions >. _____ _ _______

Literature Cited _____________________ 

38

41

41

45

46

48

48

49

52

56

60

Page

FIGURE 1. Size-distribution curves for the sands used in the 8-foot-wide flume ________ J3

2. Size-distribution curves for the sands used in the 2-foot-wide flume __ 3

3. Idealized diagram of the forms of bed roughness in an alluvial channel _ __ 5

4. Photograph showing upstream view of ripple bed _ _ _____ 6

5. Photograph showing side view of ripple bed ___________ _________ 6

6. Graph showing variation of V with V* and D for flow over a ripple bed _____ 6

7. Photograph showing upstream view of dune bed ________ ___ :__ ____ 6

8. Photograph showing side view of the dune bed shown in figure 7____^_____ 7

9. Photograph showing upstream view of plane bed ________ _________ 8 

10. Graph showing comparison of resistance to flow for flow over a plane bed with

and without movement of the bed material _i__________________ 8 

11-15. Photographs showing:

11. Downstream view of antidune flow ________________________ 8

12. Side view of the antidune flow shown in figure 11 _______________ 9

13. Downstream view of chute-and-pool flow __________________ 10

14. Downstream view of the chute and breaking antidune shown in figure 13 10

15. Side view of the breaking antidune shown in figure 14 __________ 10

16. Graph showing relation of average velocity and stream power for runs using sand I 10

17. Plan view of alternate bar development in the large flume at large width-depth

ratio _________ _________________ '__:_______________ 13

18. Photograph showing large alternate bar in Marala Ravi Canal, Pakistan _____ 14

111
IV CONTENTS

Page

FIGURE 19-45. Graphs showing:

19. Change in Darcy-Weisbach / with slope, depth, and fall velocity of bed

material __________________ -_ J17

20. Relation of depth to discharge, Elkhorn River near Waterloo, Nebr. 19

21. Approximate relation of length of dunes to median fall velocity of bed

material, at constant depth ______ 20

22. Change in resistance to flow with temperature 21

23. Variation in resistance to flow with concentration of fine sediment 21

24. Apparent kinematic viscosity of water-bentonite dispersions 22

25. Variation of fall velocity with percentage of bentonite in water _ 22

26. Variation of fall velocity wjth temperature ________ ___ 23

27. Variation of resistance to flow with slope as a function of size distribution of bed material _____________ ____ _ 23

28. Relation of form of bed roughness to stream power and median fall

diameter of bed material _______________-___ 24

29. Relation of T<? V/RV*, V*, and regime of flow for laboratory and field

conditions __________________________-_-_-_ < 25

30. Typical vertical-velocity-distribution curves in alluvial channel _ 26

31. Variation of slope of vertical-velocity distribution and shear velocity for

plane bed with sediment movement _________ ___ 27

32. Relation between intercept of vertical-velocity distribution and shear

velocity __________________ _____ 28

33. Comparison of roughness heiglrt with dsr, size of bed material _ 29

34.: Comparison of equation C/Vg = 7.4 log (!?/£) with relation between 

Chezy discharge coefficient and depth for a plane bed with sediment 

movement _______ -___ 29

35. Comparison of resistance data from the 8-foot-wide flume with Einstein 

and Barbarosa's (1952) bar-resistance relation for rivers and with 

Einstein and Kalkanis' (1959) relation for flumes 30 

36-38. Relation between slope adjustment, shear stress, bed configuration, and 

depth:

36. Sand I ____________ __ 32

37. Sand II ___________________ ____ _ _ __ 33

38. Sand III ___________________________ _ 34 

39-41. Relation between slope adjustment, slope, bed configuration, and depth:

39. Sand I _________________________________- 35

40. Sand II _________________________________ 36

41. Sand III ___________________________________ 37

42. Relation between slope adjustment, shear stress, bed configuration, and

depth for sand IV ______________________________ 38

43. Relation between C*, shear stress, and depth for sand IV 39

44. Relation between C*, depth, and slope for dune-bed configuration ___ 39

45. Comparison of computed velocity and measured velocity _______ _ 40

46. Diagram showing typical flow pattern over two-dimensional ripple and dune

roughness elements _________'._________________________________________ 40

47. Sketch illustrating effective depth and effective velocity ________________ 41 

48-64. Graphs showing:

48. Effect of bed configuration on resistance to flow for sand I _______ 41 

49-54. Relation between depth adjustment, depth, and slope for:

49. Ripples and dunes (sand I) ____________^_______ 42

50. Ripples and dunes (sand II) ____________________ 42

51. Ripples and dunes (sand III) ____________________ 43

52. Dunes (sand IV) ___________________________ 43

53. Dunes (flume and field data, <&» = 0.15-0.25 mm) _______ 44

54. Dunes (flume and field data, dm = 0.25-0.35 mm) _______ 45 

55. Comparison "of computed velocity with measured velocity _________ 46 

1>6. Relation between discharge coefficient and Reynolds number for different 

' "' AD/D ratios __________________________________ 46

57. Relation between depth adjustment and depth with type of form roughness in a,rigid-boundary flume ________________________ 46

58. Relation between discharge coefficient, depth, and slope __________ 47 

;- 59. Relation between resistance coefficient, depth, and temperature _____ 47
CONTENTS

Page

FIGURE 48-64. Graphs showing Continued

60. Relation between depth adjustment and depth for baffle and cube roughness elements _________ J48

61. Resistance diagram relating C/Vg, R*, and AJ9/D for baffle roughness.. 49

62. Resistance diagram relating C/VV, R#, and AD/D for cube roughness.. 50

63. Relation of depth adjustment to depth for rock roughness in a flume 50

64. Resistance diagram relating C/Vg, R*, and &D/D for rock roughness

in a flume _________ 50 

65. Photograph showing typical San Luis Valley, Colo., canal with cobble roughness.. 51 

66-76. Graphs showing:

66. Relation of depth adjustment to depth for San Luis Valley canals with

cobble roughness ______________ 51

67. Resistance diagram relating C/Vg, R*, and AD/D for San Luis Valley

canals with cobble roughness 51 

68-71. Relation between depth adjustment, depth, and slope for:

68. Ripples _________________________________ 51

69. Dunes ___________________________________ 52

70. Plane bed (sand I) ____________________________ 52

71. Antidunes (sand II) __________________________ 52

72. Resistance diagram relating C/-\/~g, R*, and AD/D for ripples and dunes 53

73. Relation between hydraulic-radius adjustment, hydraulic radius, and

slope for Punjab canal data ______________ 54

74. Resistance diagram relating C/ \/g, R*, and &R/R for Punjab canal data 54

75. Relation between hydraulic-radius adjustment, hydraulic radius, and

slope for Pakistan canal data _____________ ___ 55

76. Resistance diagram relating C/-\/g, R*, and &R/R for Pakistan canal data 56

TABLES

TABLE 1. Comparison of the various characteristics of ripples and dunes with fall velocity of 

the bed material _ __________________________________________________

2. Data from a stable reach of the Rio Grande at similar discharges but different stream 

temperature _____________________________________________________

3. The slope A* and intercept B* of the velocity profiles for a plane bed with sediment 

movement _________________________________________________________

4. Values of B when £ = d->0 , and of £ when B = A ____________________________________

5. Equation for C* as a function of bed form and grain size ________-____________

6. Bed configurations and range in size of bed material for which figures 49-52 can be 

used to determine the A£) adjustment based on an average grain roughness _____

GLOSSARY OF TERMS

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Alluvial channel. A channel whose bed is composed of noncohesive sediment that has been or can be transported 

by the flow.

Antidunes. Bed forms of curved symmetrically shaped sand 

waves that may move upstream, remain stationary, or 

move downstream. They occur in trains that are inphase 

with and strongly interact with gravity water-surface 

waves. The water-surface waves have larger amplitudes 

than the coupled sand waves. At large Froude number, 

the waves generally move upstream and grow until they 

become unstable and break like surf (breaking antidunes). The agitation accompanying the breaking obliterates the antidunes, and the process of antidune initiation and growth is repeated. At smaller Froude numbers

the antidunes generally remain stationary and increase 

and then decrease in amplitude without breaking (standing waves).

Breaking antidune. Curved symmetrically shaped waves on 

the water surface and on the channel bottom that build 

up with time and then break like surf.

Bed configuration. A complex of bed forms covering the bed 

of an alluvial stream.

Bed form. A generic term used to denote any irregularity 

produced on the bed of an alluvial channel by flowing 

water and sediment.

Bed material. The material of which a streambed is composed.
CONTENTS

Bed-material discharge. Sediment discharge that consists of 

particles large enough to be found in appreciable quantities in the streambed.

Clay. Sediment finer than 0.004 mm (millimeters) regardless 

of mineralogical composition.

Chutes and pools. The flow phenomenon and bed configuration accompanying flows that occur at steep slopes and 

large bed-material discharges. The flow occurs at slopes 

steeper than for antidunes and consists of a series of 

pools in which the flow is tranquil, connected by steep 

chutes where the flow is rapid. A hydraulic jump forms 

at the downstream end of each chute where it enters the 

pool. The bed configuration consists of triangle-shaped 

elements with a steep upstream slope, a flat, almost horizontal back, and a gentle downstream slope. The chutes 

and pools move slowly upstream.

Dunes. Large bed forms having triangular profiles, a gentle 

upstream slope, and a steep downstream slope. They 

form in tranquil flow and, thus, are out of phase with 

any water-surface disturbance that they may produce. 

They travel slowly downstream as sand is moved across 

their comparatively gentle, upstream slopes and deposited 

on their steeper, downstream slopes. The downstream 

slopes are approximately equal to the angle of repose of 

the bed material. Dunes are smaller than sand bars but 

larger than ripples. They generally form at higher 

velocities and larger sediment discharges than do ripples, 

but at lower velocities and smaller sediment discharges 

than do antidunes. However, ripples form on the upstream slopes of dunes at lower velocities.

Fall diameter or standard fall diameter. The diameter of a 

sphere that has a specific gravity of 2.65 and 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.

Fine sediment. That part of the sediment discharge that consists of sediment so fine that it is about uniformly distributed in the vertical and is only an inappreciable fraction of the sediment in the streambed (referred to by 

some writers as washload). Its upper size limit at a 

particular time and cross section is a function of the 

flow as well as of the sediment particles.

Flow regime. A range of flows producing similar bed forms, 

resistance to flow, and mode of sediment transport.

Lower flow regime. A category for flows producing bed forms 

of ripples, ripples on dunes, or dunes. In this flow regime, 

flow is tranquil, water-surface undulations are out of 

phase with bed undulations, and resistance to flow is 

large.

Median diameter. The midpoint in the size distribution of 

sediment such that half the weight of the material is 

composed of particles larger than the median diameter 

and half is composed of particles smaller than the median 

diameter.

Plane bed. A bed form in which there are no irregularities 

larger in amplitude than a few grain diameters.

Ripples. Small triangular-shaped bed forms that are similar 

to dunes but have much smaller and more uniform amplitudes and lengths. Wave lengths are less than about 

2 feet, and heights are less than about 0.2 foot.

Sand. Sediment particles that have diameters between 0.062 

and 2.0 mm.

Sand bar. A dune-shaped bed form whose upstream surface 

is extremely long in relation to the geometry of the 

channel (length, 2-3 times the width of the channel). 

The bar may often protrude above the flow.

Sand waves. Crests and troughs (such as ripples, dunes, sand 

bars, antidunes, or standing waves) on the bed of an 

alluvial channel that are formed by the movement of the 

bed material.

Sediment. Fragmental material that originates from weathering of rock and is transported by, suspended in, or deposited by water or air.

Sediment concentration. The ratio of dry weight of sediment 

to total weight of the water-sediment mixture, expressed 

in parts per million.

Sediment discharge. The amount of sediment that is moved 

by water past a section in a unit of time.

Silt. Sediment particles whose diameters are between 0.004 

and 0.062 mm.

Standing waves. Curved symmetrically shaped waves on the 

water surface and on the channel bottom that are virtually stationary. When standing waves form, the water 

and bed surfaces are roughly parallel and inphase.

Suspended sediment. Sediment moving in suspension in a 

fluid as a result of turbulent currents and (or) colloidal 

suspension.

Transition. A category for flows that occur between the lower 

and upper flow regimes and produce bed forms ranging 

from those typical of the lower flow regime to those 

typical of the upper flow regime.

Upper flow regime. A category for flows producing bed forms 

of plane bed with sediment moving, standing waves, antidunes, or chutes and pools. In the upper flow regime, 

water-surface undulations are inphase with bed undulations, except in breaking antidune or chute and pool flow.
CONTENTS Vll

SYMBOLS

A Slope of the vy /^/gDS versus In y/% relation. (A = 1/X).

A* Slope of the % versus In y relation.........-.-.._..-_ ...........-------..... . . .. fps per .w unit

B Intercept of the vy /\/gDS versus In y/% relation.

/?* Intercept of the vy versus In y relation . ........ . . . - ... ...... .......... . . . fps

C Wave celerity equal to |/grL/2i. tanh ;

C = vfifD when L is large relative to D, and

C = j/£_ when L is small relative to D................. ...... _ ....-...-. fps

j 2ir

C* Correction term applied to the grain-roughness equation for a plane bed to obtain the

resistance coefficient for other bed forms (C = A/S'/S). 

C-r Concentration of bed-material discharge._.________________.____..._---_._------ ppm

C/ Concentration of fine sediment. -.--.-_..----._....-...._....-------.---------- ppm

C., Suspended-sediment concentration..........................._....._........---- ppm

The dimensionless Chezy discharge coefficient equivalent to

f = V/VgRS;

where \/gRS = V^gDS for two-dimensional flow. 

C'/\/g The dimensionless Chezy discharge coefficient for a channel having either grain roughness or hydraulically smooth boundary^ or for an equivalent channel having either

grain roughness or hydraulically smooth boundary. 

D Average depth of flow _____ .._...__.--_--.__-.._---------------.-.---.------- ft

D, The average depth of flow in a channel with ripple or dune roughness corrected for the

separation zones _______.__...__.__.___________-______.-------_---.------ ft

D' The average depth of flow a channel would have for the measured slope and discharge

VD. if only an average grain roughness affects the flow, D' = -^7'- ---------------- ft

D', The average depth of flow a channel would have for the measured slope and discharge

VD if the resistance to flow was that of a hydraulically smooth boundary, £)' = ^7,

where V is determined using the equation of Tracy and Lester (1961) --------- ft

AZ) Increase in average depth resulting from the form roughness and wave activity,

AD = D-D' orD-D'... .......... ..................................... ft

d Fall diameter of the bed material.. _________________________ ................... mm or ft

df,n Median fall diameter of the bed material ........_................._............. mm or ft

F Froude number_of the flow; the ratio of average velocity (V) to wave celerity (C).

F = VI V^gD when L is large relative to D. 

f, Seepage force caused by the water flowing through the streambed . ._ ....... Ib per sq ft

/ Darcy-Wiesbach resistance coefficient;

/ = 8

/' Resistance coefficient for a channel having grain roughness. 

A/ Increase in resistance coefficient caused by form roughness and wave activity; A/ =

/-/'. 

g Acceleration of gravity. .-.-.--.-....--.-_...................---....-.._.... ft per sec per sec

h Wave height from trough to crest. ...-.-.........-............-------.-_..-..-. ft

L Wave length (from crest to crest or trough to trough) of either water-surface waves

or sand waves ......__.._........._......_......._............__._...... _ ft

P The wetted perimeter, ..---.........---.-........_..........._.........._...... ft

Q Discharge of water-sediment mixture --_--------..-..---.--.-._--........,..... cu ft per sec

R Hydraulic radius (area/wetted perimeter). For wide channels R ^ D. ............. ft

R' Hydraulic radius a channel would have for the measured slope and discharge if the

only resistance to flow was grain roughness. _......................... ... ft

R',, Hydraulic radius a channel would have for the measured slope and discharge if the

resistance to flow was that of a hydraulically smooth boundary; equation of

Tracy and Lester (1961) - ._............._......._................_... . . ft

Aft Increase in hydraulic radius resulting from form roughness and wave activity;

Aft = R - R' or AH = R - R\. .................................. . .. ... ft

K Reynolds number; VD/v or VR/v.
Vlll CONTENTS

Units

K* Ratio of the Reynolds number to the Chezy discharge coefficient;

	R/(C/Vff) - V*D/v or y*fl/v.

S Slope of energy-grade line equal to water-surface slope with equilibrium flow.

S' The slope the channel would have for the measured depth and velocity if there was only

	grain roughness. 

AS Increase in slope required to compensate for the energy dissipated by form roughness

	and wave activity; AS = S S'. 

Sc Shape factor of the channel cross section. 

Sp Shape factor of the sediment particle.

Sr Shape factor for the reach of the stream; sinuosity and change in area with distance.

T Temperature _______________________________________________________________ °C or °F

t Time__________________'___________-_____----__-__-___---_-__..___--__-__--_- sec or min

V Average velocity based on continuity principle .........-^~........................ ft per sec

V The velocity of flow for a channel without energy dissipation caused by form roughness

	and antidune activity_______-----__-------_-----------------_-__---_----- ft per sec

Ve Velocity of flow when the area of the separation zones are subtracted from the area

	of flow; VcDc = VD..................................................... ft per sec

vy Velocity at the point y, where y is the distance from the bed__ .................... ft per sec

V* The shear velocity, V* = \A0/p = \/gDS or \/gRS. ___________.._________-._--- ft per sec

Y Specific weight of the water-sediment mixture.._________________________________ Ib per cu ft

YW Specific weight of water._____________________________________________________ Ib per cu ft

Y, Specific weight of the sediment particles._______________________________________ Ib per cu ft

AY Difference between the specific weights of sediment and water, ___________________ Ib per cu ft

K Von Karman's kappa.

(A Apparent dynamic viscosity of the water-sediment mixture _______________________ Ib-sec per sq ft

v Apparent kinematic viscosity of the water-sediment mixture_____--____--______--- sqft per sec

q Height of the roughness element, ___-__---____-___--__-_____________________-_ ft

p Mass density of the water-sediment mixture ____________________________________ Slug per cu ft

PI. Density of water.___________________________________________________________ Slug per cu ft

p., Density of sediment. ________________________________________________________ Slug per cu ft

a A measure of the gradation of the sediment distribution.

TO The shear stress on the bed. fDS or the average shear stress jRS. _________________ Ib per sq ft

w Fall velocity of sediment particles _____________________________________________ ft per sec

ENGLISH-METRIC CONVERSIONS

I'rinciiHil items h'nulish unit Factor Metric, unit

Length ___ ft _____________ 0.3048 _________m

30.48 __________cm 

304.8 ___________mm

Area ___________________sq ft ___________929.0 __.__________sq cm

.09290 ________sq m

Velocity ft per sec _______ 0.3048 ________m per soc

30.48 _________cm per sec

Discharge __ __ ______-_cu ft per sec (cfs) __ 0.02832 ________cu m per sec

2.832X104 ______cu cm per sec 

Acceleration of gravity ____ ft per sec per sec __ 0.3048 ________m per sec per sec

30.48 __________cm per sec per sec 

Force per unit area _________Ib per sq ft __-_,____478.8 ___________Dynes per sq cm

.4882 _________gm per sq cm 

4.882 __________kgm per sq m 

Dynamic viscosity __________Ib sec per sq ft ____ 0.04788 ________gm per cm-sec (Poise)

.4482 _________gm-sec per sq cm 

Kinematic viscosity _________sq ft per sec ______929,0 _________sq cm per sec (Stokes)

Specific weight ____________Ib per cu ft ______ 0.01602 ________gm per cu cm 

Mass density _____________slug per cu ft _____ 0.5154 _________gm per cu cm
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

RESISTANCE TO FLOW IN ALLUVIAL "CHANNELS

By D. B. SIMONS and E. V. RICHARDSON

AMSTKACT

Resistance to flow in alluvial channels is intimately related 

to the bed configurations. Flume and field studies have proved 

that the bed configuration in alluvial channels will be ripples, 

ripples on dunes, dunes, plane bed, antidunes, chutes and 

pools, or some combination thereof. These bed configurations 

are classified into either a lower or an upper flow regime, 

or into an intermediate, transition zone. The lower flow 

regime (ripples, ripples on dunes, and dunes) is characterized 

by small bed-material discharge and large resistance to 

flow (7.0 < C/Vg < 13.2, or 0.04 < / < 0.16). The upper 

flow regime (plane bed, antidunes, and chutes and pools) is 

characterized by large bed-material discharge and, except in 

chute and pool flow, low resistance to flow (0.02 < / < 0.07, 

or 10.7 < C/Vjf/ < 20). In the transition stage, the bed 

forms are extremely variable, ranging from dunes to antidunes. Knowledge of the relation between bed forms and 

associated flow conditions provides a means of relating depositional features to environmental conditions in both recent 

and ancient settings.

Bed forms and the resulting resistance to flow vary with 

the slope of the energy grade line, depth of flow, fall velocity, 

physical size and gradation of the bed material, seepage force, 

and shape and sinuosity of the channel. In the transition 

stage antecedent form of the bed roughness influences the 

bed configuration. These independent variables are interrelated in their effect on the bed form, so that isolation of 

the effect of any one variable on resistance to flow is very 

difficult. However, the flume experiments and field studies 

were used to qualitatively and, in some tests, quantitatively 

determine the effect of each variable on the mechanics of 

flow. These studies confirmed the essential role of depth, 

slope, and fall velocity of the bed material in determining 

resistance to flow. The studies also proved that changes in 

fall velocity with such changes in fluid properties as can 

occur in natural rivers will substantially alter resistance to 

flow. Thus, although depth, slope, and fall velocity are primary factors in determining resistance to flow, the effects of 

the other variables are not inconsequential.

Fine sediment dispersed in water has a definite effect on 

the viscosity and specific weight of the fluid mixture, and 

thus on fall velocity, bed configuration, resistance to flow, 

and sediment transport. Tests made with a Stormer viscosimeter in water whose temperature was 24 °C showed that 

the apparent kinematic viscosity of an aqueous dispersion 

of bentonite (10 percent bentonite by weight) was 8.75 times 

greater than that of pure water.

Methods are presented for estimating the resistance to 

flow and the average velocity of the flow for both analysis 

and design purposes. The methods depend on knowledge of,

or methods for, estimating the bed configuration for given 

fluid, flow, bed material, and channel characteristics. A relation between stream power, median diameter of the bed material, and bed configuration was developed, from the field and 

laboratory data, for estimating the bed configuration. The 

methods of determining the resistance to flow are based on 

adjusting the measured depth or measured slope to an equivalent depth or slope in a channel having grain roughness or 

hydraulically smooth rigid-boundary roughness. The methods 

are applicable to most types of roughness (rigid or alluvial 

boundary) in all types of channels, flumes, canals, and rivers. 

Utilizing the foregoing concepts of the mechanics of flow 

in alluvial channels gives valuable insight into the difficult 

problem of modeling the behavior of alluvial channels and 

the concepts assist in the development of a better understanding and control of the dominant factors affecting the 

geomorphology of streams and rivers. These concepts can be 

utilized in designing stable irrigation canals constructed in 

alluvium and in determining the discharge that occurred in 

a natural channel during a runoff event if only residual 

information is available.

INTRODUCTION

Many significant aspects of resistance to flow in 

open channels are still only vaguely understood after 

nearly two centuries of study. In contrast to knowledge about resistance to flow through closed conduits, 

the lack of progress is a result of the problems created by the free surface and unsymmetrical cross 

section, the large variety of roughness elements, and 

the lack of any concerted sustained study since that 

of Darcy and Bazin in 1865. Other problems exist 

because many open channels have a movable or 

alluvial boundary rather than a fixed or rigid 

boundary.

An alluvial boundary is formed in cohesive or noncohesive materials that have been and can be transported by the stream. The noncohesive material 

generally consists of silt (0.004-0.062 mm (millimeter) ), sand (0.062-2.0 mm), gravel (2.0-64 mm), 

or cobbles (64-256 mm), or any combination of 

these; silt, however, normally is not present in 

appreciable quantities in streams having larger bed 

materials.

The alluvial boundary, as shown by Gilbert (1914), 

can be molded into many bed configurations by the

Jl
J2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

flow, and changes in resistance to flow result. In 

addition, experiments of Vanoni (1946), Elata and 

Ippen (1961), Bagnold (1954), and Simons, Richardson, and Hauschild (1963), showed that the fluid 

properties and turbulent characteristics of the flow 

are changed by the moving sediment. The movable 

alluvial boundary thus adds another dimension to 

the resistance-to-flow problem and increase its complexity ; not only the fluid, flow, and boundary characteristics but also the sediment characteristics must 

be included in the study of alluvial channels. At 

some discharges the material forming the boundary 

may not move, and the alluvial channel behaves as 

a rigid channel. Therefore, the flow conditions under 

which the material forming the boundary begins to 

move must be known.

To determine resistance to flow and bed-material 

discharge in alluvial channels, a systematic study 

was undertaken by the U.S. Geological Survey. Part 

of the investigation is being conducted at Colorado 

State University, where the study is limited to steady 

equilibrium flow over sand-bed material, although 

some qualitative preliminary studies have been made 

for unsteady flow (Simons and Richardson, 1962b). 

The median fall diameter of the bed material ranged 

from 0.19 mm to 0.93 mm. Each bed material was 

investigated from beginning of motion of the individual sand particles to antidune flow. Depths of 

flow ranged from 0.3 foot to 1.0 foot, velocities from 

less than 1 fps (foot per second) to more than 8 fps, 

and discharges from 3 cfs (cubic feet per second) 

to 22 cfs. Special studies have been made to determine the effect of changes in viscosity, either real or 

apparent, and the effect of the gradation of the bed 

material on the bed configuration, resistance to 

flow, and bed-material transport.

This report describes the bed configurations which 

form in an alluvial channel and the corresponding 

resistances to flow, as measured by the DarcyWeisbach discharge coefficient or by the related 

Darcy-Weisbach resistance coefficient. These bed 

forms are classified into lower flow regime, transition, or upper flow regime. Knowledge of why the 

bed configurations form and the mechanics of their 

formation is not known. Thus, no perfect method 

is known for predicting the bed forms that will exist 

in an alluvial channel for particular flow conditions. 

However, a qualitative method for predicting the bed 

form that appears to have merit' is presented. The 

variables that influence the form of the bed configuration and resistance to flow are quantitatively 

and, in some instances, qualitatively discussed. Some 

practical aspects of the effect of bed configuration

or change in bed configuration on flow in rivers and 

artificial waterways are described, and equations 

and methods for predicting resistance to flow and 

velocity of flow are developed on the basis of current fluid mechanics theory and a study of the collected data. The equations, developed from a study 

of the flume data, apply to flow in natural alluvial 

rivers and in artificial waterways. The equations 

can also be applied to flow in rigid-boundary open 

channels, as well they should be if there is any theoretical justification for them. With a method available for predicting the bed form and the resistance 

to flow, procedures are described for designing 

stable water-conveyance channels to carry a known 

water discharge or to determine water discharge 

in a channel when slope, bed-material, size, cross 

section, and water viscosity are known.

Only a brief description of the sands, equipment, 

and procedure used in the data collection is given. 

A more detailed description of the procedure is given 

together with the basic data in a companion publication by Guy, Simons, and Richardson (1965).

The authors acknowledge the advice, assistance, 

and encouragement given by Messrs. L. B. Leopold, 

P. C. Benedict, R. W. Carter, Thomas Haddock, Jr., 

F. C. Ames, W. L. Haushild, D. W. Hubbell, W. W. 

Sayre, F. M. Chang, H. P. Guy, and C. F. Nordin, Jr., 

of the Geological Survey, during the study and in 

the preparation of this report. We also thank Messrs. 

M. L. Albertson and A. R. Chamberlain of Colorado 

State University for their assistance during the 

investigation. '

EXPERIMENTAL EQUIPMENT AND DATA COLLECTION

The basic studies were conducted in a tilting recirculating flume 150 feet long, 8 feet wide, and 2 feet 

deep. The slope in the flume could be varied from 

0 to 0.013 foot per foot, and the discharge, from 2 

cfs to 22 cfs. A smaller tilting recirculating flume 

60 feet long, 2 feet wide, and 2.5 feet deep was used 

for special studies. Slope in the smaller flume could be 

varied from 0 to 0.025 foot per foot, and discharge, 

from 0.5 cfs to 8 cfs. The special studies in the 

2-foot-wide flume were made to determine the effect 

of viscosity (Simons and others, 1963; Hubbell and 

Al-Shaikh Ali, 1961), density of the bed material 

(W. H. Haushild and R. K. Fahnestock, written 

comm., 1962), and gradation of the bed material 

(Daranandana, 1962) on flow in alluvial channels.

The size distribution of bed materials used in the 

8-foot-wide flume studies are given in figure 1. Those 

for materials used in the 2-foot-wide flume studies 

are given in figure 2. The size distribution is in
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J3

3.0

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IV fall diameter^.

III

IV (sieve di

0.1 99.9

FICUKE 1. Size-distribution curves for the sands used in the 8-foot-wide

flume.

terms of the fall diameter (Colby and Christensen, 

1956) unless specified otherwise. The curves shown 

in figures 1 and 2 are the average curves for a group 

of runs, determined from an averaging of the size 

analyses of 10 grab samples of the bed material 

per run.

Sand I (median diameter 0.19 mm) was pure 

quartz sand obtained from a decomposed sandstone 

deposit near Denver, Colo. Sands II (median diameter about 0.28 mrn) were almost pure quartz obtained from the bed of the Elkhorn River near 

Waterloo, Nebr. The difference between the size 

distributions of sands IIj (0.28 mm) and IL (0.27 

mm) resulted from screening the coarse sand fraction from sand llt . The difference between sands 

IL. and 1I;( (0.32 mm) resulted from natural changes 

during the experiments. Sands III (median diameter about 0.45 mm), consisting of a mixture of 

feldspar and quartz with some mica, were obtained

Fir

1.0 10 20 30 40 50 60 70 80 90 95 99 99.9 99.99 

PERCENTAGE OF MATERIAL FINER THAN INDICATED SIZE

i.'iir: '2. Size-distribution curves for the sands used in the 2-foot-wide 

flume'.

by screening the gravel from a sand deposit of the 

Cache la Poudre River at Fort Collins, Colo. The 

differences in the size distribution among sands IIIi 

(0.45 mm), III, (0.47 mm), and Ills, (0.54 mm) 

were caused by natural changes of the bed material 

as they were used in the experiments. Sand IV (0.93 

mm) was a natural river sand, mostly quartz, obtained by screening the material coarser than 5 mm 

from the bed material of the North Platte River 

near Scottsbluff, Nebr. Bed material V, for which 

both the sieve size (0.68 mm) and the fall-diameter 

size (0.36 mm) distributions are given, was a lightweight aggregate (expanded illite) marketed by the 

Ideal Cement Co. under the trade name Idealite. 

Sands VI (0.33 mm) were almost pure silica obtained from the Black Hills Silica Corp., Hill City, 

S. Dak. Sand VIX passed through a U.S. standard 

No. 40 sieve and was retained on a No. 60 sieve, 

whereas sand VL was prepared by mixing six sands 

in predetermined proportions to obtain a graded 

sand having the same median diameter as sand VIi. 

The sands were placed in the flumes to a depth of 

about 0.7 foot.

The general procedure for each run was to recirculate a given discharge of the water-sediment mixture until equilibrium flow conditions were established. Equilibrium flow is defined as flow which has 

established a bed configuration and slope consistent 

with the fluid, flow, and bed-material characteristics 

over the entire length of the flume, neglecting entrance- and exit-affected reaches that is, the timeaverage water-surface slope of the flow is essentially 

constant and parallel to the time-average bed surface, and the time-average concentration of the bedmaterial discharge is constant. Equilibrium flow 

should not be confused with steady uniform flow; 

in equilibrium flow, velocity may vary at a point or 

from point to point. Steady uniform flow, as classically defined (dV/dt = 0, dV/dx = 0), does not 

occur in an alluvial channel unless the bed is plane 

and the flow is steady.

After equilibrium flow was established, average 

water-surface slope (S), discharge-of the watersediment mixture (Q), water temperature (T), 

depth (D), velocity distribution in the vertical (v,,), 

total sediment concentration (Cr), and the'geometry 

of bed configuration (length, L\ height, h), and 

shape) were determined.

Water-surface slope was measured with a Lory 

point gage and a precision level by determining 

water-surface elevations at definite intervals along 

the flume (every 5 ft in the 8-ft-wide flume; every 

2 ft in the 2-ft-wide flume). Average water-surface
J4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

slope was obtained by averaging several individual 

slope determinations made during a run. Discharges 

were measured with calibrated orifice meters located 

in the return-flow pipes. Water temperature was 

measured to the nearest 0.1 °C with a mercury thermometer. Depth was determined by subtracting 

mean bed elevation from mean water-surface elevation. Velocity distribution in a vertical was measured with a calibrated Prandtl pitot tube; however, 

the mean velocity of the cross section was computed 

from the discharge and cross-sectional-area data, 

V = Q/A, Total sediment concentration was measured by traversing the outflow nappe at the end of 

the flume with a width-depth integrating sampler. 

Suspended-sediment concentration was measured 

with a depth-integrating sampler. Bed configuration 

was measured with the Lory point gage and, in later 

runs, with a newly developed sonic depth sounder 

(Richardson and others, 1961). Photographs of the 

bed and water surfaces for all runs were taken with 

a still camera; normal and time-lapse sequences of 

the flow of water and sediment were photographed 

with a 16-mm movie camera.

Many of the flow phenomena observed are recorded in the Geological Survey film "Flow in Alluvial Channels." The filming was done by the Survey 

staff, assisted by personnel from Bandolier Films, 

Albuquerque, N. Mex., in the laboratory at Colorado 

State University and in the field near Albuquerque, 

on the Rio Grande.

A complete documentation and description of all 

basic data collected from 1956 to 1961 by the Geological survey at Colorado State University is included in a data report, Professional Paper 462-L.

FORMS OF BED ROUGHNESS AND FLOW PHENOMENA

The bed configurations (roughness elements) that 

may form in an alluvial channel are plane bed without sediment movement, ripples, ripples on dunes, 

dunes, plane bed with sediment movement, antidunes, and chutes and pools. These bed configurations are listed in their order of occurrence with 

increasing values of stream power (VyDS) for bed 

materials having dun less than 0.6 mm. For bed materials coarser than 0.6 mm, dunes form instead of 

ripples after beginning of motion at small values of 

stream power. The typical form of each bed configuration is shown in figure 3.

These bed-roughness elements are not mutually 

exclusive occurrences in time and space in a flume 

or a natural river. They may form side by side in 

a cross section or reach of a natural stream, giving

a multiple roughness; or they may form in sequence, 

in time, producing variable roughness.

Multiple roughness is spacially related to variation in shear stress (yDS), stream power (VyDS), 

or alluvial bed material. The greater the width-depth 

ratio of a stream or flume, the greater is the probability of a spacial variation in shear stress, stream 

power, or bed material. Thus, the occurrence of 

multiple roughness is closely related to the widthdepth ratio of the stream or flume. Multiple roughness in these experiments occurred when the widthdepth ratio was greater than 20.

Variable roughness is related to changes in shear 

stress, stream power, or reaction of bed material to 

a given stream power. An example of the effect of 

changing shear stress or stream power is the change 

in bed form which occurs with changes in depth 

during a runoff event; this action has frequently 

been observed in natural rivers. An example of the 

effect of the bed-material-stream-power relation is 

the change in bed form that occurs with change in 

the viscosity of the fluid as the temperature or concentration of fine sediment varies. It should be noted 

that a transition occurs between the dune-bed and 

the plane bed; either bed configuration may occur 

for the same value of stream power.

In the following sections, bed configurations and 

their associated flow phenomena are described in the 

order of their occurrence with increasing stream 

power.

BED CONFIGURATION WITHOUT 

SEDIMENT MOVEMENT

If the bed material of a stream moves at one discharge but not at a smaller discharge, the bed configuration at the smaller discharge will be a remnant 

of the bed configuration formed when sediment was 

moving. The problems are in knowing when beginning of motion of the bed material occurs (Kramer, 

1935; Shields, 1936; and White, 1940) and which 

bed form may develop. The bed configuration after 

the beginning of motion may be any of the preceding 

ones, depending on flow and bed material. Prior to 

the beginning of motion, the problem of resistance 

to flow is one of rigid-boundary hydraulics. After 

beginning of motion the channel is alluvial, and the 

problem related to defining resistance to flow under 

the alluvial condition is the main subject of this 

report.

Plane bed without movement was studied to determine the shear stress for the beginning of motion 

and the bed configuration that would form after 

beginning of motion. A plane bed for this study was
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J5

/Water surface Water surface.

£ Plane bed

A Typical ripple pattern

F Antidune standing waves

6 Dunes and superposed ripples

, Boil

C Dunes

G Antidune breaking wave

0 Washed-out dunes or transition H Chute and pool

E 3. Forms of bed roughness in an alluvial channel.

obtained artifically by screeding the bed. Within the 

accuracy of visual observation, Shields' relation for 

the beginning of motion was adequate. After beginning of motion, the plane bed changed to ripples for 

sand smaller than 0.5 mm, and to dunes for 0.93-mm 

sand. This was contrary to Liu's (1957) findings; 

he reported a plane bed for a range of shear stresses 

after beginning of motion. Knoroz (1959) reported 

that a plane bed did not persist after the beginning 

of motion and that ripples do not form for coarser 

sands. Resistance to flow is small for a plane bed 

without semiment movement. In the flume runs, 

values of C/^g ranged from 15 to 20.

RIPPLES

Ripples are small triangle-shaped elements having 

gentle upstream slopes and steep downstream slopes. 

In this study they ranged from 0.4 foot to 2 feet in 

length and from 0.02 foot to 0.2 foot in height and

were narrow normal to the direction of flow. (See 

figs. 4 and 5.) Resistance to flow is__large, and the 

resulting discharge coefficient (C/Vg) ranged from 

7 to 12. As depth increased, resistance to flow due 

to roughness decreased. (See fig. 6.) Thus, there 

is a relative roughness effect produced by the ripple 

bed. Figure 6 shows that resistance to flow is independent of sand size when the bed configuration is 

one of ripples. This is because the ripple shape is 

independent of sand size and the effect of grain 

roughness is small relative to the form roughness. 

Knoroz observed that the length of the separation 

zone when the bed form was either ripples or dunes 

was about 10 times the height of the ripple or the 

dune. The separation zone downstream from a ripple 

causes very little, if any, disturbance on the water 

surface, and the flow contains little suspended bed 

material. The water is clear enough that the bed 

configuration can be photographed through the run-
J6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

FICURE 4. Upstream view of ripple bed. Bed material is sand II; slope = 

0.00108, depth = 0.57 foot, discharge = 7.2 cfs, C/VT= 11.2.

Flf.URE 5. Side view of ripple bed. Bed material is sand II; slope = 

0.00023, depth = 1.01 feet, discharge = 7.7 cfs, C/V7T= 11.1.

ning water. The bed-material discharge is small, 

ranging from 10 to 200 ppm.

DUNES

When the shear stress or the stream power is 

gradually increased for flow over a bed having ripples or, if the bed material is coarser than 0.6 mm, 

over a plane bed, a rate of bed-material transport, 

a magnitude of velocity, and a degree of turbulence 

will soon be achieved that cause sand waves called 

dunes to form. At smaller shear-stress values, the 

dunes will have ripples superposed on their backs. 

(See fig. 7.) These ripples will disappear at larger 

shear values, particularly if coarse sands (e£50 > 0.4 

mm) are involved.

Dunes are large triangle-shaped elements similar 

to ripples. (See figs. 7 and 8.) Their lengths range 

from 2 feet to many feet, depending on the scale of

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SHEAR VELOCITY (~Vf =TfjfRS) , IN FEET PER SECOND

FICUHE 6. Variation of V, V*, and D for flow over a vipple bed.

the flow system. Dunes formed in the large flume 

used in this study ranged from 2 feet to 10 feet in 

length and from 0.2 foot to 1 foot in height, whereas 

those in the Mississippi River described by Carey 

and Keller (1957) were several hundred feet long 

and as much as 40 feet high. The maximum amplitude to which dunes can develop is approximately the 

average depth. Hence, in contrast with ripples, the 

amplitude of dunes can increase with increasing 

depth, so that the relative roughness can remain 

essentially constant or even increase with increasing 

depth of flow.

 ISB 7. Upstream view of dune bed. Bed material is sand_II; slope = 

0.00167, depth = 0.94 foot, discharge == 15.6 cfs, C/ V» = 9.3.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J7

FICUKK 8.--Side view of the dune bed shown in figure 7. Note the bottomset, foreset, and top-set layering of the sand.

Field observations by the authors indicated that 

dunes form in any channel, irrespective of the size 

of bed material, if the stream power is sufficiently 

large to cause general transport of the bed material 

without exceeding a Froude number of unity. Also, 

the length and shape of the dunes are functions of 

the fall velocity of the bed material; dune height 

did not appear to be a function of fall velocity of 

the bed material. The length of the dunes increased 

and the angle of the upstream and downstream faces 

decreased as fall velocity decreased. Dunes formed 

of fine sand (dr>(} < 0.4 mm) were longer and less 

angular than those formed of coarser sand.

Resistance to flow caused by dunes is large, but 

not as large as that caused by ripples formed of finer 

sand and at shallow depth. For dunes, the discharge 

coefficient C/Vg varied from 8 to 12. Resistance to 

flow increased with an increase in depth for coarser 

sands (dr>n > 0.3 mm) and decreased with an increase in depth for finer sands (d-)0 < 0.3 mm). Also, 

when dunes were more than 10 times longer than 

they were high, the grain roughness on the back of 

the dunes influenced resistance to flow. Knoroz 

(1959) observed that resistance to flow for dunes 

depended on the grain roughness, in addition to the 

form roughness, whereas, that for ripples did not 

depend on grain roughness.

Dunes cause large separation zones in the flow. 

These zones, in turn, cause large boils to form on the 

surface of the stream. Measurements of flow velocities within the zone of separation showed that velocities in the upstream direction existed that were j ?. 

the average stream velocity. Boundary shear stress 

was sometimes sufficient to cause formation of ripples oriented in a direction opposite to that of the

primary flow in the channel. With dunes, as with 

any tranquil flow over an obstruction, the water 

surface is always out of phase with the bed surface. 

The flow accelerates over the crest of dunes and 

decelerates over the trough, thereby contracting- the 

flow over the crest and expanding it over the trough. 

Some investigators do not agree that there is a 

difference between ripple- and dune-bed configurations. Vanoni, Brooks, and Kennedy (1961), for 

example, saw little reason for distinguishing between 

ripples and dunes because the mechanisms by which 

they are formed and by which they move are similar. 

The following factors, however, indicate that there 

are major differences, in addition to the difference 

in size, between the two bed configurations:

1. The effects of a change in depth on resistance to 

flow are opposite: an increase in depth causes 

a decrease in the resistance to flow for flow 

over a ripple bed, but an increase in depth 

causes an increase in the resistance to flow for 

flow over a dune bed when the bed material is 

coarser than 0.3 mm, and a decrease in the 

resistance to flow when the bed material is 

finer than 0.3 mm.

2. Ripples do not form if the median diameter of 

the bed material is larger than 0.6 mm.

3. Resistance to flow caused by ripples is independent of the grain size of the bed material, 

whereas that caused by dunes is dependent on 

the grain size.

This question might seem academic if it were not 

for the fact that ripple beds are dominant in flume 

investigations and dune beds are dominant in the 

field. Taylor and Brooks (1962) pointed out that 

if there is a fundamental difference between ripples 

and dunes, the problems of roughness analysis and 

of modeling alluvial channels cannot be resolved by 

small-scale laboratory studies.

PLANE BED WITH SEDIMENT MOVEMENT

A plane bed is a bed without elevations or depressions larger than the largest grains of the bed material (fig. 9). The resistance to flow for flow over a 

plane bed results largely from grain roughness and 

C/Vg is large, ranging from 14 to 23. Grain roughness is not of the usual type, however, because grains 

roll, hop, and slide along the bed. For flow over a 

plane bed with sediment movement, the resistance 

to flow is slightly less than that for flow over a static 

plane bed, which is essentially an artificial rigidboundary condition that exists after screeding when 

stream power is insufficient to cause significant
J8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

transport of the bed material. The difference in 

resistance to flow between that on a plane bed with 

sediment moving and that on a static plane is shown 

in figure 10. This lower resistance to flow for flow 

over a plane bed with motion has been observed by 

many experimenters and was attributed by Vanoni 

and Nonicos (1960) to dampening of the turbulence 

by the suspended sediment. Elata and Ippen (1961) 

indicated that the structure of the turbulence was 

not damped by the suspended bed material but that 

the structure was changed by the movement of the 

sediment at the boundary.

The magnitude of the stream power (r0V) at 

which the dunes or transition roughness changes to 

the plane bed depends mainly on the fall velocity of

;UI;E 9. Upstream view of plane bed. Bed material is sand II ; slope =

0.00153, depth = 0.60 foot, discharge 14.9 cfs, C/V<T= 18.1.

21

, 20

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17 

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DEPTH, IN FEET

FtOURE 10. Comparison of resistance to flow for flow over a plane V>od 

with and without movement of the bed material. (Sam! II.)

the bed material. Dunes of fine sand (low fall velocity) are washed out at lower values of stream 

power than are dunes of coarser sand. Consequently, 

in the flume experiments, where the depths were 

shallow (0.5-1.0 ft), the plane bed formed with 

finer sands at smaller slopes than it did with coarser 

sands; the result was smaller velocities and smaller 

Froude numbers (F). The plane bed with the fine 

sand occurred at slopes such that the Froude number, based on average depth and average velocity, 

was as small as 0.3, and it existed until the Froude 

number increased to about 0.8. If coarse sands are 

involved, larger slopes are required to effect the 

change from transition to the plane bed; the result 

is larger velocities and larger Froude numbers. 

Hence, in a flume containing fine sand, the plane-bed 

condition commonly exists after the transition and 

persists over a wide range of Froude numbers 

(0.3 < F < 0.8). If the sand is coarse and the 

depth is shallow, however, transition may not terminate until the Froude number is so large that the 

subsequent bed form may be antidunes rather than 

plane bed. In natural streams, because of their 

greater depths, the change from transition to plane 

bed may occur at a much lower Froude number than 

in the flumes.

ANTIDUNES

Antidunes form as a series or train of inphase 

(coupled) symmetrical sand and water waves. (See 

figs. 11 and 12.) The height and length of these 

waves depend on the scale of the flow system and the 

characteristics of the fluid and the bed material. In

ir.ruE 11. Downstream view of antidune flow. Bed material is^ sand n, 

slope = 0.00o9, depth = QM foot, discharge = 21.2 cfs, C/Vff = 14.6.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J9

FIC.UIIE 12. Side view of the antklune flow shown in figure 11.

the 8-foot-wide flume, the height of the sand waves 

(from the bottom of the trough to the top of the 

crest) ranged from 0.03 foot to 0.5 foot, and the 

height of the water waves was 1.5-2 times the height 

of the sand waves. The length of the waves, from 

crest to crest or trough to trough, ranged from 5 to 

10 feet. In natural streams, such as the Rio Grande 

or the Colorado River, much larger antidunes form. 

In these streams surface waves 2-3 feet high and 

10-20 feet long have been observed.

Antidunes do not exist as a continuous train of 

waves that never change shape; rather, they form 

as trains of "waves that gradually build up from a 

plane bed and a plane water surface. The waves may 

grow in height until they become unstable and break 

like the sea surf, or they may gradually subside. The 

former have been called breaking antidunes, or antidunes ; and the latter, standing waves. As the antidunes form and increase in height, they may move 

upstream, downstream, or remain stationary. Their 

upstream movement led Gilbert (1914) to name them 

antidunes.

Resistance to flow due to antidunes depends on 

how often the antidunes form, the area of the reach 

that they occupy, and the violence and frequency of 

their breaking. If the antidunes do not break, resistance to_flow is about the same as for a plane bed, 

and C/ V g ranges from 14 to 23. The acceleration 

and deceleration of the flow through the nonbreaking 

antidunes (frequently called standing waves) causes 

resistance to flow to be slightly more than that for 

flow over a plane bed. If many antidunes break, 

resistance to flow can be very large because the 

breaking waves dissipate a considerable amount of 

energy. With breaking waves, C/ Vgr may range 

from 10 to 20.

With antidune flow, the fact that the water and 

bed surfaces are inphase is a positive indication that 

the flow is rapid (F > 1). With dunes, the fact that 

the water surface is out of phase with the bed surface is a positive indication that the flow is tranquil 

(F < 1). In both instances the Froude number, 

which is based on the ratio of the mean velocity of 

the flow to the velocity of a gravity wave (C), must 

take into account the wave length (L) . That is,

C- = i- tanh -^- and cannot be approximated

by c- = V#Z?, as is often done in open-channel flow, 

unless L/D > 10. Also, the mean velocity must be 

for the flow in the section of the stream where the 

antidunes are forming and not for the entire cross 

section of the stream.

Kennedy (1961) made a detailed study of antidune 

flow. His conclusions were:

1. The wave length of antidunes is given by 

L = 2irV2 /g.

2. Where a limited range of depth and velocity 

exists, antidunes will form only if an initial 

surface wave is introduced.

3. The Froude number (V/C) for the occurrence 

of antidunes decreases as the depth of flow 

increases or as the size of the sand grains 

decreases.

4. Two-dimensional waves (those that occupy the 

full width of the flume) break when the ratio 

of wave height to wave length reaches a value 

of approximately 0.14. This value of 0.14 

agrees with the theoretical value for deepwater waves.

From Kennedy's first conclusion it is obvious that 

the wave length is independent of depth of flow and, 

hence, the Froude number. However, Kennedy 

(1963) revised his first conclusion and stated instead 

that only the minimum length is given by 2<irV2 /g 

and that, for each bed material, there is a characteristic wave length dependent on velocity and depth.

CHUTES AND POOLS

At very steep slopes, alluvial-channel flow changes 

to what has been called chutes and pools. In the 

large, 8-foot-wide flume, this type of flow and bed 

configuration occurred only in runs using sands I 

and II. Chute-and-pool flow could not be attained 

when coarser sands were used because the steeper 

slope required could not be set up in the flume. This 

type of flow consisted of a long chute (10-30 ft) in 

which the flow accelerated rapidly, a hydraulic jump 

at the end of the chute, and then a long pool (10-30
J10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

ft) in which the flow was tranquil but was accelerating. (See figs. 13, 14, and 15.) The chutes and 

pools moved upstream at velocities of about 1-2 fpm 

(feet per minute). The elevation of the sand bed 

varied within wide limits, although at no time was 

the flume jloor exposed. Resistance to flow was large, 

and C/ Vg ranged from 9 to 16. This type of flow 

was frequently accompanied by a decrease in the 

mean velocity of the flow in the flume even though 

there was an increase in stream power (fig. 16).

FIGURE 13. Downstream view of chute-and-pool flow. Note the pool in 

the foreground and the breaking wave in the background, at the downstream end of the chute. Bed material is sand II ; slope = 0.0095, 

depth = 0.45 foot, discharge = 15.4 cfs, C/ \/(j = 11.5.

FIGURE 14. Downstream view of chute (foreground) and breaking antidune shown in figure 13. Standing waves are terminating in the breaking antidune.

FiouitE 15. Side view of the breaking antidune illustrated in figure 14. 

Note the large quantity of bed material in suspension.

REGIMES OF FLOW IN ALLUVIAL CHANNELS

The flow in alluvial channels is divided into two 

flow regimes with a transition zone between (Simons 

and Richardson, 1963) . These two flow regimes are

01234

AVERAGE VELOCITY (V). IN FEET PER SECOND

FIGURE 16. Relation of average velocity to stream power for runs with 

sand I as the bed material.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS Jll

characterized by similarities in the shape of the bed 

configuration, mode of sediment transport, process 

of energy dissipation, and phase relation between 

the bed and water surfaces. The two regimes and 

their associated bed configurations, discussed in the 

following sections, are listed as follows:

Classification of bed forms into flow regimes

Lower flow regime (smallest values of shear, yDS, or stream 

power, VyDS) :

1. Ripples

2. Dunes with ripples superposed

3. Dunes

Transition zone (bed roughness ranges from dunes to plane 

bed or antidunes).

Upper flow regime (largest values of shear, yDS, or stream 

power, VyDS):

1. Plane bed

2. Antidunes

a. Standing waves 

b. Breaking antidunes

3. Chutes and pools

LOWER FLOW REGIME

In the lower flow regime, resistance to flow is large 

and sediment transport is small. The bed form is 

either ripples or dunes or some combination of the 

two. The water-surface undulations are out of phase 

with the bed surface, and there is a relatively large 

separation zone downstream from the crest of each 

ripple or dune. Total resistance to flow is the result 

partly of grain roughness and dominantly roughness 

of form. The most common mode of bed-material 

transport is for the individual grains to move up the 

back of the ripple or dune and avalanche down its 

face. After coming to rest on the downstream face 

of the ripple or dune, the particles remain there until 

exposed by the downstream movement of the dunes; 

then the cycle of moving up the back of the dune, 

avalanching, and storage is repeated. Thus, most 

movement of the bed-material particles is in steps, 

of the order of magnitude of a ripple or dune length, 

separated by rest periods dependent on ripple or 

dune height and velocity. The velocity of the downstream movement of the ripples or dunes depends 

on their height and the velocity of the grains moving 

up their backs. In flume studies, the ripple bed is 

the configuration usually formed in the lower flow 

regime; in natural streams and rivers, dunes or 

dunes with ripples superposed are the dominant bed 

forms in the lower flow regime.

UPPER FLOW REGIME

In the upper flow regime, resistance to flow is 

small and sediment transport is large. The usual

bed forms are plane bed or antidunes. The water 

surface is inphase with the bed surface except when 

an antidune breaks, and normally the fluid does not 

separate from the boundary. A small separation zone 

may exist downstream from the crest of an antidune 

prior to breaking. Resistance to flow is the result 

of grain roughness with the grains moving, of wave 

formation and subsidence, and of energy dissipation 

when the antidunes break. The mode of sediment 

transport is for the individual grains to roll almost 

continuously downstream in sheets one or two grain 

diameters thick; when antidunes break, however, 

much bed material is briefly suspended, then movement stops temporarily and there is some storage 

of the particles in the bed.

TRANSITION

The bed configuration in the transition zone is 

erratic. It may range from that typical of the lower 

flow regime to that typical of the upper flow regime, 

depending mainly on antecedent conditions. If the 

bed configuration is dunes, the depth or slope can be 

increased to values more consistent with those of the 

upper flow regime without changing the bed form; 

or, conversely, if the bed is plane, depth and slope 

can be decreased to values more consistent with 

those of the lower flow regime without changing the 

bed form. Often in the transition from the lower 

to the upper flow regime, the dunes will decrease in 

amplitude and increase in length .before the bed becomes plane (washed-out dunes). Resistance to flow 

and sediment transport also have the same variability as the bed configuration in the transition. It 

was the transition zone, which unfortunately covers 

a wide range of shear values, that Brooks (1958) 

was investigating when he concluded that a singlevalued function does not exist between velocity or 

sediment transport and the shear stress on the bed.

In many instances when the flow conditions are 

such that the bed form is in the transition zone, the 

bed configuration will oscillate between dunes and 

plane bed. This phenomenon can be explained by the 

changes in resistance to flow and, consequently, the 

changes in depth and slope as the bed form changes. 

Resistance to flow is small for flow over a plane bed; 

so the shear stress on the bed decreases to a value 

that is incompatible with the shear stress for a plane 

bed but compatible with the shear stress for a dune 

bed, and the bed form changes to dunes. The increase 

in resistance to flow causes an increase in the shear 

stress on the bed so that the dunes wash out and the 

bed becomes plane, and the cycle continues.
J12

COMPARISON OF FLUME AND FIELD CONDITIONS

The preceding comments were based on observations of bed configurations and flow phenomena that 

occurred in the flume experiments and in natural 

rivers (field conditions) and are equally true for 

both situations. However, there are major differences between flume and field conditions. In the usual 

flumes only a limited range of depth and discharge 

can be investigated, but slope and velocity can be 

varied within a wide range. In the field, a larger 

range of depth and discharge is common, but slope 

of a particular channel reach is virtually constant. 

Consequently, the variation in shear stress (yDS) 

and stream power (VyDS) is principally the result 

of slope variation in flume studies and of depth variation in a particular natural stream. The walls of a 

flume are nonerodible, and, hence, the width is constant; but in most alluvial channels the width depends on flow arid bank conditions. Larger Froude 

numbers (V/VgD) can be achieved in flume studies 

than will occur in most natural alluvial channels 

because natural banks cannot withstand prolonged 

high-velocity flow without eroding. This erosion increases the cross-sectional area and this causes a 

reduction in the average velocity and the Froude 

number. Rarely does a Froude number, based on 

average velocity and depth, exceed unity for any 

extended time period in a natural stream with erodible banks. In the field, where the slope of the energy 

grade line is constant, the Froude number is also 

constant unless there is a change in the resistance

to flow (F =

V

Flow is more nearly two dimensional in flumes 

than in natural streams. However, the main current 

meanders from side to side in a large flume, as it does 

in the field, and bars of small amplitude but large 

area develop in an alternating pattern adjacent to 

the walls of the flume. (It is on these bars that the 

bed forms shown in fig. 3 superpose themselves.) 

If very large width-depth ratios are maintained in 

the flume by keeping depth of flow shallow, these 

bars may grow to the water surface. Figure 17 

shows the bars that formed in the 8-foot-wide flume 

when the width-depth ratio was about 22, the slope 

was 0.0020, the total discharge was 4.0 cfs, and the 

bed material had a median fall diameter of 0.19 mm.

In the field it is even more obvious that the flow 

meanders between the parallel banks of a straight 

channel, and the alternate bars which form opposite 

the apex of the meanders are easier to distinguish. 

As in a flume, if the depth is decreased the alternate

bars increase in amplitude until they are close to 

the water surface, or even exposed. (See fig. 18.) 

In fact, scour in the main current adjacent to a large 

bar may cause the water surface there to drop 

slightly so that the top of the bar is exposed. This 

has been observed in the Rio Grande (R. K. Fahnestock, oral commun., 1962) and in other natural 

channels. If the banks are not stable, erosion occurs 

where the high-velocity water impinges, and deposition occurs on the opposite bank. The ultimate development is a meandering stream if other factors such 

as slope, discharge, and size of bed material are 

compatible (Leopold and Maddock, 1953).

VARIABLES

Resistance to flow in alluvial channels is complicated by the large number of variables. It is further 

complicated by the interdependency, either real or 

apparent, of the variables. In fact, some variables 

may be altered or even determined by the flow, and 

changes in flow conditions may change the role of a 

dependent variable into that of an independent one. 

It is difficult, especially in field studies, to differentiate between the independent and dependent 

variables.

The slope of the energy grade line of an alluvial 

stream illustrates the changing role of a variable 

and the difficulty of selecting the dependent variable. 

If a stream is in equilibrium with its environment, 

slope is an independent variable. In such a stream 

the average slope over a period of years has adjusted 

so that the flow is capable of transporting only the 

amount of sediment supplied at the upper end of the 

stream and by the tributaries. If ^or some reason 

a tributary or upstream reach supplied a larger or 

smaller quantity of sediment than the stream was 

capable of transporting, the slope would change and 

would be dependent on the amount of sediment supplied. Clearly, it is difficult to determine if a stream 

is in equilibrium. Lane (1955) and Leopold and 

Langbein (1962) pointed out that a stream reacts 

quickly to any change and that the slope of the 

stream, although it may never reach true equilibrium 

(geologically, it never does), will approach its final 

equilibrium value in a short time. If during a runoff 

event in a natural stream the water and sediment 

discharges do not change rapidly, the slope of the 

water surface and the energy gradient for a reach 

of the stream also will not change significantly. If 

the discharge does change rapidly (surge), so that 

there is an appreciable difference in discharge between the ends of a relatively short reach, the slope
RESISTANCE TO PLOW IN ALLUVIAL CHANNELS J13

LONGITUDINAL STATION, IN FEET

70 80 R 90 100

i____i r** i____L_

130

I

140

A'

x

t -2 2

50

A' C'

A A' B B' C C' 

FICUKE 17. Plan view of alternate bar development in the 8-foot-wide flume at a larjre width-depth ratio. Contour intervals 0.05 and 0.10 foot.

of the water surface and the energy gradient will 

vary.

We have given much thought to the significant 

variables in an alluvial channel, as have others 

(Rouse, 1959; Vanoni, 1946; Brooks, 1958; Einstein, 

1950; and Bagnold, 1956, to mention a few), and 

have presented our ideas in several publications 

(Simons and others, 1961; Simons and Richardson, 

1962a; and Richardson and others, 1962) publications which have provoked stimulating discussions. 

In the following sections the variables affecting 

resistance to flow are listed and their dependency 

or independency is discussed. Also discussed are 

some conditions whereby a dependent variable may

become an independent one; which variables may 

be eliminated as a first approximation to simplify 

the problem; and, in general terms, the effect or 

importance of a variable on resistance to flow. The 

measure of resistance to flow used in these discussions (and in this report) will be either the Chezy 

discharge coefficient (C/Vgr) or the Darcy-Weisbach 

resistance coefficient (/ = 8/(C/Vgr) 2 ). 

Under the restraint of equilibrium flow,

dCr _ dCT _ dV dV

dt dX dt dX

the variables that determine resistance to flow are:

0 [V, D, S, p, [L, g, d, a, p s, Sf, Si/, Sr, / ] = 0, (1)
J14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

where

CT = concentration of bed-material discharge,

V = velocity,

D = depth,

S = slope of the energy grade line,

p = density of water-sediment mixture,

JJL = apparent dynamic viscosity of the watersediment mixture,

g = gravitational constant,

d = representative fall diameter of the bed 

material,

a = measure of the size distribution of the bed 

material,

p s = density of sediment,

Sp = shape factor of the particles,

SR = shape factor of the reach of the stream,

Sc = shape factor of the cross section of the 

stream, and

fs = seepage force in the bed of the stream.

FlGUKB 18. Large alternate bar in Manila Ravi Canal, Pakistan.

CONCENTRATION OF BED-MATERIAL DISCHARGE

The concentration of bed-material discharge (Cr), 

which some investigators have suggested is a variable, has been left out of equation 1. Admittedly, the 

concentration of bed material affects the fluid properties by increasing the apparent viscosity and the 

density of the water-sediment mixture (Einstein, 

1906; Bingham, 1922; Bagnold, 1956; and Elata and 

Ippen, 1961). The effect of the sediment on viscosity 

(P.) and density (/») in any resistance to flow relation is accounted for by using their values for the 

water-sediment mixture instead of their values for 

pure water (Elata and Ippen, 1961; Simons, and 

others, 1963). Also, as Vanoni and Nomicos (1960), 

Vanoni and Brooks (1957), and Elata and Ippen 

(1961) demonstrated, the presence of sediment in 

the flow causes a change in the turbulence characteristics and in the velocity distribution. However, 

Elata and Ippen found that the effects of the particles on the resistance to flow, by the change in 

turbulence and velocity distribution, can be accounted for by using the apparent jviscosity of the 

mixture to evaluate R in the C/Vg versus R diagram. Furthermore, both Vanoni and Brooks (1957) 

and Elata and Ippen (1961) found that the effect 

of the sediment concentration on resistance to flow 

was small. Elata and Ippen found about a 5-percent 

difference in C/ Vg between clear-water runs and 

runs in which as much as 27 percent by volume of 

the water-sediment mixture was neutrally buoyant 

granular material in suspension.

A more fundamental reason for not including the 

concentration of bed-material discharge in equation 

1 is that, for equilibrium flow, it is a dependent variable. If, however, the flow is not in equilibrium 

that is, the concentration of bed material entering 

the flume or reach of a stream is larger or smaller 

than the concentration leaving the concentration is 

an independent variable and should be included as a 

variable. Needless to say, for nonequilibrium flow the 

concentration of bed-material discharge would have 

to be known, but determining the concentration is 

almost an impossibility.

FINE-SEDIMENT CONCENTRATION

Fine sediment is that part of the total Sediment 

discharge that is not found in appreciable quantities 

in the bed material. If much fine sediment is in suspension, its effect on the viscosity of the watersediment mixture should be taken into account 

(Simons and others, 1963). The effect of fine sediment on resistance to flow is a result of its effect on 

the apparent viscosity and the density of the watersediment mixture; so the fine-sediment factor is not 

included in the list of variables, but it is included in 

the viscosity and density terms and is discussed in 

a later section.

The fundamental differences between fine sediment 

and bed material are in the availability of the sediment, the source of the sediment, and the capacity 

of the flow to carry the sediment. Fine sediment is 

not available in appreciable quantities from the bed, 

and it generally is not transported in concentrations 

that approach the fine-sediment transport capacity 

of the stream; bed material, however, is always available and is transported at the capacity of the stream. 

That is, fine-sediment concentration is dependent 

upon the availability of the sediment to the stream ; 

and bed-material concentration is dependent upon 

the ability of the stream to transport a particular
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J15

size in a conglomeration of sizes and, hence, upon 

the variables of equation 1. Fine sediment that is 

in suspension at one discharge or in one stream may 

be bed material at another discharge or in another 

stream. Generally the fine sediment is uniformly 

distributed in the stream cross section.

DEPENDENT VARIABLE

The velocity (V) was selected as the dependent 

variable in the present study, and the other variables 

were assumed to be independent. If the study had 

had other objectives, any one of the other variables 

could have been selected as the dependent variable, 

if the conditions of the experiment and the research 

objectives were properly treated. For example, the 

size distribution of the bed material could have been 

the dependent variable; then, the experiment would 

have been designed to determine the change in size 

distribution of the bed material with changes in V, 

D, S, and the other variables.

To study resistance to flow in an alluvial channel, 

either V, D, or S may be selected as the dependent 

variable. In the present study velocity was selected 

because in a natural stream it is generally the unknown quantity and the other variables are measured 

quantities. In our experiments with given bed materials, the slope was varied and the resultant depth 

measured for different values of discharge. The 

velocity was then computed from the discharge and 

depth. Bed-material discharge and bed configuration 

were also measured, and either could be substituted 

in equation 1 for one of the other variables. Whether 

they should be included depends on the objectives of 

the study and the availability of data both in the 

laboratory and in the field. If the objective of the investigation is sediment transport or bed configuration under equilibrium conditions, then obviously the 

concentration or the bed configuration should replace 

one of the other variables. Including either one or 

both as independent variables in a resistance to flow 

study, however, might make the analysis more difficult by obscuring an essential relationship. Another 

reason for not including them is that they are often 

unknown quantities in natural streams, although by 

visual observation of the water surface or of the bed 

configuration and stratifications in the bed material 

after a runoff event, the bed configuration can usually 

be estimated. Later in this report, bed configuration 

is used as an independent variable in determining 

resistance to flow.

CHANGE OF VARIABLES

In equation 1, fluid viscosity, particle shape, and 

density can be eliminated if the fall velocity of the

bed material is included in the. list of variables, 

giving equation 2:

V = <MAS, 9, 9, d, to, S«,X/.f *];. (2) 

where fall velocity, w, depends on several variables,

(.) = <t> [d, p., p, g, Sp, (L, a]. (3)

Viscosity can be eliminated because flow over an 

alluvial bed after the beginning of motion is fully 

developed, rough, turbulent flow. The laminar sublayer, since it is very much smaller than the grain 

diameters, should not exist. Resistance to flow for 

this type of flow is independent of the flow Reynolds 

number (R) and the viscosity. The changes in 

resistance to flow over sand beds which have been 

observed with changes in viscosity (Straub, 1954; 

Hubbell, 1956; Hubbell and Al-Shaikh, All,, 1961; 

Vanoni and Brooks, 1957) are the result of changes 

in the fall velocity of the bed material, which cause 

a corresponding change in the bed configuration 

(Hubbell and Ali, 1961; Simons and others, 1963; 

and Simons and Richardson, 1963).

The density and shape factors of the particles were 

included in equation 1 with the size of the bed material and the viscosity to account for the^ interaction 

of the fluid and the bed material. To have kept them 

in equation 2 with fall velocity would have resulted 

in a redundancy. Furthermore, it has frequently 

been demonstrated that fall velocity is more representative of the interaction of the fluid and the bed 

material than any other single variable.

The size and gradation of the bed material cannot 

be replaced by the fall velocity because, in addition 

to determining the fall velocity, they determine the 

grain roughness of the bed. Gradation parameters 

based on the physical sizes of the bed material and 

on fall velocities may be necessary. This is true 

because of the nonlinearity of the relation between

the coefficient of particle drag versus

The shape factor for the reach (Sri) was included 

in equations 1 and 2 to focus attention on the energy 

losses resulting from the nonuniformity of the flow 

in a natural stream caused by the bends and the 

nonuniformity 'of the banks. Study of these losses 

in natural channels has long been neglected, although, as Ippen and Drinker (1962) stated, there 

is no net excess energy loss in bends with straight 

approach and exit reaches of sufficient length. In 

straight flumes with parallel walls, the shape factor 

of the reach is a constant and can be disregarded.
J16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

The shape factor for the cross section (Sc ) is a 

constant for the flume and can be disregarded. However, as is explained in the following paragraph, a 

shape factor for the cross section is needed when 

data from flumes of different widths are compared. 

Under these conditions, flow phenomena, bed configuration, and resistance to flow vary with the width 

of the flume.

Simons, Richardson, and Haushild (1963) noted 

that:

1. Ripples observed in the 8-foot-wide flume with 

sand III did not occur with the same sand in 

the 2-foot-wide flume.

2. Dune fronts in the 8-foot-wide flume seldom were 

continuous from one wall to the other, whereas 

those in the 2-foot-wide flume were continuous.

3. Antidunes in the 8-foot-wide flume could be threedimensional and could occur in more than one 

row, whereas those in the 2-foot-wide flume 

were always two-dimensional and extended 

from wall to wall.

Hence, the resistance to flow was different in the 

two flumes for the same conditions (depth, slope, 

fall velocity, and so on). Vanoni and Brooks (1957) 

found a difference in results for their 10.5-inch-wide 

and 33.3-inch-wide flumes which could not be accounted for by a sidewall correction.

SEEPAGE FORCE

The seepage force (Fs ), which occurs whenever 

there is inflow or outflow through the bed material 

and banks Of a channel in permeable alluvium, may 

have a significant affect on bed configuration and 

resistance to flow (Simons and Richardson, 1962b). 

The inflow or outflow through the interface between 

water and the bed and the bank depends on the 

difference in pressure across the interface and the 

permeability of the bed material. For flow in a 

flume with a rigid bed and walls, the pressure difference and the permeability of the bed material 

depend only on the variables in equation 1; thus, the 

seepage force is a dependent variable and can be 

eliminated from equation 2. For flow in a natural 

channel, however, the pressure difference will depend 

on the location of the water table in the alluvium 

and also on the variables in equation 1; now, seepage 

force is an independent variable insofar as resistance 

to flow is concerned.

If there is inflow, the seepage force acts to reduce 

the effective weight of the sand and, consequently, 

the stability of the bed material. If there is outflow, 

the seepage force acts in the direction of gravity and 

increases the effective weight of the sand and the

stability of the bed material. As a direct result of 

changing the effective weight, the seepage forces can 

influence the form of bed roughness and the resistance to flow for a given channel slope, channel shape, 

bed material, and discharge. For example, under 

shallow flow a bed material with median diameter 

of 0.5 mm will be molded into the following forms 

as shear stress is increased: Ripples, dunes, transition, standing sand and water waves, and antidunes. 

If this same material was subjected to a seepage 

force that reduced its effective weight to a value 

consistent with that of medium sand (median diameter, d = 0.3 mm), the forms of bed roughness would 

be ripples, dunes, transition, plane bed, and antidunes for the same range of flow conditions. The 

reason for this difference in the forms of bed roughness is the reduction in effective weight and increased mobility of the bed material.

A rather common field condition is outflow from 

the channel during the rising stage; this process 

builds up bank storage and increases the stability 

of the bed and bank material. In the falling stage, 

the situation is reversed; inflow to the channel reduces the effective weight and stability of the bed 

and bank material and influences the form of bed 

roughness and the resistance to flow.

RESISTANCE COEFFICIENT

With the proposed simplifications and by grouping 

V, D, S, p, and g into the Darcy-Weisbach resistance 

coefficient or into the Chezy discharge coefficient, 

equation 2 becomes: 

SgRS , 0

V* >, d, co, a, g, p]; (4)

or the bed configuration can be substituted for the 

dimensionless resistance coefficient to give:

bed configuration = $ [S, D, d, w, a, g, p]. (4a) 

The variables were not grouped into dimensionless 

parameters to present more clearly the essential role 

of each. Because of the interdependency of slope, 

depth, bed-material characteristics, bed configuration, and resistance to flow, it is difficult to isolate 

the effect of any one variable. For example, an increase in fall velocity may increase resistance to 

flow at one slope and decrease it at another slope. 

The relation of the variables in equations 4 and 4a 

to the bed configuration and resistance to flow are 

discussed in the following sections.

SLOl'E

The effect of slope on bed configuration and resistance to flow for two depths and four bed materials 

is shown in figure 19. With constant depth and in-
RESISTANCE COEFFICIENT (f)

O S3

o B

cn p 

O S

_ T3

3 o 3 

o -»

* C

-V

\

;-^5.

IA

e w IAP a

fo «

T3

n
J18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

creasing slope, the bed configurations for the bed 

material having a median fall velocity equal to or 

less than 0.22 fps (sand III) were (1) an artificial 

plane bed without sediment movement, (2) ripples, 

(3) ripples on dunes, (4) dunes, (5) transition 

(where the bed configuration was either dunes, plane 

bed, or a combination of them), (6) plane bed with 

sediment movement, and (7) antidunes. For the bed 

material having a median fall velocity of 0.48 fps 

(sand IV), the bed forms were the same but ripples 

did not form.

Resistance to flow is a function of slope even when 

the class of bed configurations given on page J4 does 

not change. For example:

1. With shallow depths and the ripple-bed configuration, resistance to flow increased with an increase in slope; at greater depths, resistance 

to flow did not change. At greater depths, ripples gave way to dunes; but with an increase 

in slope at the shallow depths, dunes did not 

form. The shallow depth appeared to inhibit 

the formation of dunes.

2. With the dune-bed configuration, an increase in 

slope increased resistance to flow for bed material having fall velocities greater than 0.20 fps. 

For those bed materials having fall velocities 

less than 0.20 fps, an increase in slope decreased resistance to flow at shallow depths 

and slightly increased resistance to flow at 

greater depths.

3. The slope (where the transition from flow in the 

lower flow regime to the upper flow regime 

occurred) depended on the fall velocity of the 

bed material and the depth of flow. Constant 

depth required an increase in slope with an 

increase in fall velocity for the occurrence of 

the transition. With fall velocity constant, an 

increase in slope and a decrease in depth were 

necessary for the occurrence of the transition.

4. With antidunes, resistances to flow increased 

with increasing slope; the magnitude of the 

increase of resistance to flow with an increase 

in slope depended on the fall velocity of the 

bed material. The smaller the fall velocity, the 

larger the increase in resistance to flow with 

an increase in slope.

DEPTH

The effect of depth on bed form and resistance to 

flow is not well defined because only a limited range 

in depths have been studied. However, just as 

changes in slope can change the bed configuration 

when depth and size of bed material are constant,

so will changes in depth change the bed configuration when slope and bed material are constant. With 

a constant slope and bed material, an increase in 

depth can change a plane bed without movement to 

ripples, and a ripple-bed configuration to dunes, as is 

illustrated in figure 19. Although not clearly shown 

in figure 19, flume studies (Simons and Richardson, 

1962b) and field studies (Colby, 1960; Bawdy, 1961; 

and Culbertson and Bawdy, 1964) have demonstrated that an increase in depth, without varying 

slope and bed material, may cause a dune bed to 

change to a plane bed or antidunes, and that a decrease in depth may cause a plane bed or antidunes 

to change to a dune-bed configuration. Figure 20 

shows a typical break in a depth-discharge relation 

caused by a change in bed form from dunes to plane 

bed or from plane bed to dunes (Beckman and Furness, 1962).

Limiting the variables to discharge and depth, 

resistance to flow will also vary with depth even 

when the bed configurations given on page J4 do not 

change. When the bed configuration is plane bed, 

-either with or without sediment movement, there is 

a decrease in resistance to flow with an increase in 

depth that is, a relative roughness effect. When 

the bed configuration is ripples, because their form 

is independent of depth, there is also a relative roughness effect resistance to flow decreases with an 

increase in depth, as shown in figures 6 and 19. 

When the bed configuration is dune bed, although 

dunes increase in size when depth is increased, resistance to flow increases with an increase in depth only 

for sands coarser than 0.3 mm; for sands finer than 

0.3 mm, resistance to flow decreases with an increase 

in depth. This effect is the result of a decrease in 

angularity of the dunes as they increase in size. 

Field studies indicate that at large depths, resistance 

to flow may decrease with an increase in depth even 

when the bed material is composed of sands coarser 

than 0.3 mm. This is true even though the size of 

the dunes continues to increase with an increase in 

depth. The anomalism may be explained by the facts 

that (1) the long dunes with mobile grain roughness 

on their backs compensate for the form effects of the 

separation zones downstream from the crests, and 

(2) at large depths the dunes, although large, may 

not cause appreciable nonuniformity of the flow. In 

the flume studies, dunes of the coarser sands created 

much acceleration and deceleration of the flow. 

Studies are needed to determine the conditions where 

resistance to flow for flow over a dune bed ceases to 

increase with an increase in depth.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J19

10

u! 4

LJ

EXPLANATION

A 

Plane bed

Dune

O 

Transition

500 1000 2000 4000 6000

DISCHARGE (Q), IN CUBIC FEET PER SECOND

FIGURE 20. Relation of depth to discharge for Elkhorn River near Waterloo, Nebr.

10,000

When the bed configuration is antidunes, resistance to flow increases with an increase in depth to 

some maximum value, then decreases as depth is 

increased further. This increase or decrease in flow 

resistance is directly related to changes in length, 

amplitude, and activity of the antidunes as depth 

is increased.

SIZK OF BED MATERIAL,

The effects of the physical size of the bed material 

on resistance to flow are (1) its influence on the fall 

velocity, which is a measure of the interaction of the 

fluid and the particle in the formation of the bed 

configurations, (2) its effect as grain roughness, and 

(3) its effect on the turbulent structure and the 

velocity field of the flow.

The physical size of the bed material, as measured 

by the fall diameter (Colby and Christensen, 1956) 

or by sieve diameter, is a primary factor in determining fall velocity, although only in relation to the 

other variables in equation 3. Use of the fall diameter instead of the sieve diameter is advantageous 

because the shape factor and density of the particle 

can be eliminated as variables. That is, if only the 

fall diameter is known, the fall velocity of the particle in any fluid at any temperature can be computed ; whereas, to make the same computation when 

the sieve diameter is known, knowledge of the shape 

factor and density of the particle are also required.

The physical size of the bed material is the chief 

factor in determining the friction factor for the 

plane-bed condition and for antidunes when they are 

not actively breaking. The additional dissipation 

of energy when antidunes are present is caused by 

the formation and breaking of the waves, which increases with an increase in energy input or a decrease in the fall velocity of the bed material.

The physical size of the bed material for a dunebed configuration also has an effect on resistance to 

flow. The flow of fluid over the back of dunes is 

affected by grain roughness, although the dissipation 

of energy by the form roughness is the major factor. 

The form of the dunes is related to the fall velocity 

of the bed material.

FAT.L, VELOCITY

Fall velocity is the primary variable that determines the interaction between the bed material and 

the fluid. For a given depth and slope, it determines 

the bed form that will occur, the actual dimensions
J20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

TABLE 1. Comparison of the various characteristics of ripples and dunes with fall velocity of the bed material

Sand

No.

I_.._.

IIIII. 

IV....

d60

(mm)

0.19

0.27, 0.28

0.45, 0.47

0.93

Fall Velocity 

(fps)

0.064-.073

.10 -.12

.20 -.22

.48 -.49

Ripples

Depth

(ft)

1.01

.96

.81

---

c/V7

13.1

11.1

11.2

...

L

(ft)

0.63

.95

1.13

---

h 

(ft)

0.04

.04

.10

---

Dunes

Depth

(ft)

0.91

.93

.81

1.02

c/VJ

12.0

10.6

9.6

10.8

L

(ft)

8.4

7.6

5.9

4.7

h 

(ft)

0.30

.30

.30

.21

of the bed form (table 1), and, except for the contribution of the grain roughness, the resistance to flow.

The significance of fall velocity of the bed material on the spacing or length of dunes for various 

sizes of bed material at virtually constant depth is 

shown in figure 21. The dunes are not only shorter 

in length but are also much more angular when the 

fall velocity of the bed material is relatively large. 

For example, with sand IV (fall velocity « 0.48 fps) 

the average length of dunes was 6.5 feet as compared 

to 11.6 feet for sand I (fall velocity « 0.068 fps).

The differences in resistance to flow for the four 

sands in figure 19 are primarily the result of differences in fall velocity. The differences in resistance 

to flow when the bed form is plane bed, however, are 

caused by the differences in grain roughness for the 

different diameters of bed material. Nevertheless, 

the magnitude of depth and slope at the beginning 

of motion (when a plane bed without motion changes 

to ripples or dunes) and that when a dune bed 

changes to a plane bed with motion or antidunes 

depends primarily on the fall velocity. An increase 

in fall velocity requires an increase in the product 

of depth times slope (that is, shear stress) for the 

change from static plane bed to ripples or dunes, or 

the change from dunes to plane bed and antidunes.

Fall velocity is the critical factor in determining 

(1) whether ripples or dunes will form after the 

beginning of motion, (2) the shear stress at the 

beginning of motion when ripples or dunes begin to 

form from a plane bed, and (3) the shear stress at 

which ripples change to dunes. Ripples will not form 

in material having a fall diameter larger than 0.6 

mm or a fall velocity higher than about 0.22 fps; 

there is some evidence, however, that the size of the 

bed material is a factor in determining whether 

ripples form. The magnitude of the shear stress 

required for the formation of ripples increases with 

an increase in fall velocity.

When the bed form is dunes, the slower the median 

fall velocity of the bed material is, the longer and 

less angular the dunes will be and, also, the smaller

6 7 8 9 10 11 

AVERAGE LENGTH OF DUNES, IN FEET

FIGURE 21. Approximate relation of length of dunes to a median fall 

velocity of the bed material, at constant depth.

the range in shear stress or stream power will be 

within which a dune bed occurs.

For bed material having a median fall velocity 

equal to or greater than 0.2 fps, the range of stream 

power within which dunes occur is large and that 

within which a plane bed occurs is small. In fact, 

whether or not a plane bed does occur between dunes 

and standing waves is intimately related to fall 

velocity and depth. If the depth is shallow and the 

fall velocity of the bed material is fairly high, the 

bed configuration may change from transition to 

standing waves without the development of a plane 

bed.

The magnitude of the shear required for the formation of antidunes and the amount of antidune 

activity (formation and breaking of the antidunes) 

that will result from a given shear depends on the 

fall velocity. With a decrease in fall velocity there 

is a corresponding decrease in the magnitude of the 

shear required for the formation of the antidunes 

and for an increase in antidune activity. The in-
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J21

crease in antidune activity with a decrease in fall 

velocity causes an increase in resistance to flow.

Special flume studies verified that the changes in 

bed form and resistance to flow are primarily the 

result of changes in the fall velocity. These special 

studies were:

1. The fall velocity of a given bed material was 

varied by changing the viscosity of the watersediment mixture. Viscosity was varied by 

changing either the water temperature or the 

concentration of fine sediment in the watersediment mixture. A change in fall velocity 

caused the bed configuration and resistance to 

flow to change (Hubbell and Al-Shaikh Ali, 

1961; Simons and others, 1963). A decrease 

in fall velocity resulting from an increase in 

fluid viscosity caused resistance to flow to decrease for a dune bed and to increase for the 

antidune bed. (See figs. 22 and 23.) Thus, by

0.11

0.10

0.09

0.08

0.07

g 0.06

.bJ

o

I

0.05

0.04

0.03

0.02

0.01

A\

N

«Xv

1

O.

X<^ 

X

0

N ^~

N v

o^

\^A^>_

 x 

\

*S

rf

p-""""

^~«^ 

^ """ 

X 

\

\

 ^

,'/' 

,''/

/

7--^-

x. \

\

^ ~^r^V 

/<Ss /^- """"

" \

EXPLAf

^ ^.

"x

X 

N

X

/ /

/ .

// -*'

f

~^-~y- /

/

X __

x^

"\

y*

NATION

'->i

/ 

. /

 ><

/" 

/

 =*-'

~^>A

^X 

\A

P

/ 

/ 

/ 

/

/ 

/

"^e 

/Q

~~~ "- 

--A

^^.-D

O Ripples O Transition A Standing waves 

 Dunes having ripples Plane A Breaking antidunes

© Dunes

i i i i i

10 15 20 25 30 

TEMPERATURE, IN DEGREES CELSIUS

35

FIGURE 22. Change in resistance to flow with temperature. Connected 

points represent paired runs during which temperature was the only 

independent variable (Hubbell and Al-Shaikh Ali, 1961).

2.

decreasing fall velocity, a dune bed could be 

changed to a plane bed or a plane bed could 

be changed to antidunes.

The density of sand V was 1.87, the median fall 

diameter was 0.36 mm, and the median sieve 

diameter was 0.68 mm. (See fig. 2.) The bed 

configurations and the magnitude of the shear 

stress for the various bed configurations and 

for a change in bed configurations corresponded to those for a bed material with the 

same fall velocity but not for material with the 

same sieve diameter. In fact, ripples formed 

with this bed material, which is expected for 

a 0.36-mm sand but not for a 0.68-mm sand. 

In addition to the flume experiments, observations 

of natural streams have shown that the bed con0.12 

( 

0.10

0.08 

0.06

)

R

0

jn 8,

_^

Q=3.8cfs

 V

Run 15 Q=7.0cfs_

0.04

0.06

0.04

Run If

Ri

,Q=

n 17

ii^

L 

Q=7.9cfs

1

O 

Dune

10 102 103 10«

Cf, IN PARTS PER MILLION

EXPLANATION

D : 

Transition dune Transition plane Standing wave

A 

Antidune

FIGURE 23. Variation in resistance to flow (/) with concentration of fine 

sediment (Cf). Each run number represents a sequence in which 

concentration of fine sediment was the only independent variable.
J22 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

figuration and resistance to flow will change with 

changes in fall velocity when the discharge and bed 

material are constant (Hubbell and others, 1956). 

For example, the Loup River near Dunning, Nebr., 

has bed roughness in the form of dunes in the summer when the stream fluid is warm and less viscous 

but has a nearly plane bed during the cold winter 

months. Similarly, two sets of data collected by R. K. 

Fahnestock (written commun., 1962) on a stable 

reach of the Rio Grande at similar discharges show 

that when the water was cold, the bed of the stream 

was plane, the resistance to flow was small, the depth 

was relatively shallow, and the velocity was large; 

but when the water was warm, the bed roughness was 

dunes, the resistance to flow was large, the depth was 

deeper, and the velocity was smaller. (See table 2.)

TABLE 2. Data from a stable reach of the Rio Grande at 

similar discharges but different temperatures

Temperature <°F)

50°... .........

80°............

Velocity 

(fps)

4.25

2.53

Depth

(ft)

2.45

3.66

Slope

0.00049

.00053

Bed

material 

(dio)

0.24

.24

Bed

roughness

Plane

C/Vg

21.7

10.2

APPARENT VISCOSITY AND DENSITY

The effect of the various factors given in equation 

3 on fall velocity .are "well known. However, the 

effects of fine sediment in suspension on fluid viscosity and fall velocity are less well known (Simons 

and others, 1963). Figure 24 shows the effect of fine

10 20 30 40 

TEMPERATURE, IN DEGREES CELSIUS

50

sediment (bentonite) on the apparent kinematic 

viscosity of the mixture. The magnitude of the effect 

of fine sediment on viscosity is large and depends on 

the chemical makeup of the fine sediment.

In addition to changing the viscosity, fine sediment 

suspended in water increases the mass density (?) 

and, consequently, the specific weight (y) of the mixture. The specific weight (y) of a sediment-water 

mixture can be computed from the relation, (Simons 

and others, 1963)

T =

T, (5)

FIGURE 24. Apparent kinematic viscosity of water-bentoriite dispersions.

yw = specific weight of the water, about 62.4 

Ib per cu ft (pounds per cubic foot);

7s = specific weight of the sediment, about 

164.5 Ib per cu ft; and

Cs = concentration (in percent by weight) of 

the suspended sediment.

A sediment-water mixture, where Cs = 10 percent, 

has a specific weight (y) of about 66.6 Ib per cu ft, 

and any change in y affects the boundary shear stress 

and the stream power.

Changes in the fall velocity of a particle caused 

by changes in the viscosity and the fluid density are 

evident in figure 25. For comparative purposes, the

i

0.8 

0.6

0.4

EXPLANATION

co 

Fall velocity

Computed CO

O 

co from VA tube analysis

.2 4 6 8 

PERCENTAGE OF BENTONITE, BY WEIGHT

10

FIGURE 25. Variation of fall velocity with .percent bentonite in water.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J23

effect of temperature on fall velocity is given in 

figure 26. The details of the determination of fall 

velocity by computation and by visual accumulation 

(VA) tube analysis are given in the report of Simons 

and others (1963).

GRADATION OF BED MATERIAL

The effect of gradation of the bed material on flow 

in alluvial channels was investigated by Daranandana (1962) under the supervision of the authors 

and the sponsorship of the U.S. Geological Survey. 

His study was conducted in the 2-foot by 60-foot 

flume using two quartz sands with the same d50

10

. 10 ^ . 20 30

' WATER TEMPERATURE, IN DEGREES CELSIUS

FIGURE 26. Variation of fall velocity with temperature.

40

(sands VI in fig. 2; dso = 0.33 mm). Sand Vli had 

a uniform gradation coefficient (a) of 1.27, and sand 

VI2 was graded with a = 2.07, where

_ 1 ["^60 . ~ 2 (6)

In this study, depth and water temperature were 

kept constant, and the change in bed configuration, 

resistance to flow, and sediment transport were observed and measured for various slopes and discharges of the water-sediment mixture.

The effect of the gradation of the toed material on 

bed form and resistance to flow is shown in figure 27. 

The resistance to flow for flow over a plane bed with-

'0.10 

0.08 

0.06 

0.04 

0.02

0.01 

O.C

i

J]

i

 t

t

J 

1

i

1

\

/

^.

.

.(

''

;

i

>

e

Uniform sa

/-"

Graded sar 

/"^

fj

nd

\

d '

~ ~

k

\

)2 0.04 "6.06 0.08 0.1 0.2

EXPLAN

O 

Plane

Q

K- D

a

S

^

i 

p 

/

eptr

&

Ri

D

0

rra

e

lan

\nt 

i 0

/

PP

im 

ris

e

di 

50

y 

J.

A

(

36 

(

es

1

3S 

[

tic

[

D6

I

n

c

**

f

T

3 

ds

a 

 

i

n

3

ds

9

3S

0.

^

X

ON

05

r

E

a

> ^

-S"

^

0.4 0.6 0.8 1.0

SLOPE (S x io 2;

FIGURE 27. rVariation of resistance ,to flow with slope as a function of 

the size distribution of the bed material.

out motion was slightly larger for the uniform sand, 

whereas that on a plane bed with motion was nearly 

the same for the two sands. Resistance to flow was 

much larger for the uniform sand on beds having 

ripples, dunes, or antidunes. The transition from a 

dune bed to plane bed occurred over a narrower 

range of shear values for the uniform sand than for 

the graded sand.

Visual observation of the bed configuration provided the reason why resistance to flow was different 

for the two sands, but it did not show why the bed 

configurations were different. In the studies 'using 

the graded bed material, however, the'continual sorting and remixing which took place suggested that 

the representative fall velocity arid gradation also 

varied. In essence, then, the characteristics of the 

bed material were continually changing.
J24 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

For flow over a plane bed without movement, the 

finer particles in the graded sand filled the voids, so 

that the boundary was smoother than the plane bed 

formed in graded sand. On the plane bed with movement, however, the effect of sediment on the turbulent structure of the flow probably had a counterinfluence on resistance to flow that compensated for 

any filling of the large voids by fine said, and resistance to flow was similar for the two sands.

The average height and length of ripples in runs 

using uniform sand were larger than those in runs 

using graded sand, which increased resistance to 

flow. Ripples formed of uniform sand averaged 0.032 

foot in height and 0.86 foot in length; those formed 

of graded sand averaged 0.025 foot in height and 

0.46 foot in length.

Dunes of the uniform sands had ripples superposed 

on them, whereas those of the graded sand did not. 

Although the dunes of the uniform sand were 

smaller, the effect of the ripples on their back increased resistance to flow. Dune length and height 

were 3.33 feet and 0.095 foot, respectively, in the 

uniform sand and 4.00 feet and 0.14 foot in the 

graded sand.. Also, the longer dunes, crest to crest, 

decreased resistance to flow for the graded sand.

The greater resistance to flow in runs using the 

uniform sands cannot be explained by the differences 

in antidune length and amplitude. The average antidune lengths and heights were greater for the uniform sands (5.05 ft and 0.15 ft, respectively) than 

for the graded sand (4.3 ft and 0.085 ft, respectively). These changes do not adequately explain 

the large differences in resistance to flow.

Daranandana's investigation proved that there is 

a gradation effect on resistance to flow and emphasized that natural river sands should be used in flume 

experiments if the results are to be extrapolated to 

the field. The importance of using natural river 

sands in flume experiments is further emphasized 

by Blench's (1952) conclusion that alluvial bed material is always adjusted by the flow so that its distribution conforms to a fairly definite normal distribution law.

PREDICTION OF FORM OF BED ROUGHNESS

A completely satisfactory method for predicting 

form of bed roughness has not been developed. Various methods of predicting form roughness (Albertson and others, 1958; Simons and Richardson, 1963; 

and Garde, 1959) have been proposed, but none have 

been suitable for both laboratory and field conditions. 

A simple relation (fig. 28) was developed by Simons 

and Richardson (1964) to relate stream power,

0.001

0.2 0.3 0.4 0.5 0.6 0.7 0.8 

MEDIAN FALL DIAMETER, IN MILLIMETERS

FIGURE 28. Relation of form of bed roughness to stream power and 

median fall diameter of the bed material.

median fall velocity of bed material, and form roughness. This relation gives an indication of the form 

of bed roughness one can anticipate if the depth, 

slope, velocity, and fall diameter of bed material are 

known. Flume data were utilized to establish the 

boundaries separating plane bed and ripples, ripples 

and dunes for all sizes of bed material, and dunes 

and transition for the 0.93-mm bed material. The 

lines dividing dunes and transition and dividing 

transition and upper regime are based on the following field data: (1) Elkhorn River, near Waterloo, 

Nebr. (Beckman and Furness, 1962), (2) Rio 

Grande 20 miles above El Paso, Tex. (see table 2), 

(3) Middle Loup River at Dunning, Nebr. (Hubbell 

and Matejka, 1959), (4) Rio Grande at Cochiti, near 

Bernalillo, and at Angostura heading, N. Mex. (Culbertson and Dawdy, 1964), (5) Punjab canal data
J25

(Simons, 1957), and (6) Harza Engineering Co., 

International (1963) canal data. If just the flume 

data were used, the dividing line between dunes and 

transition would occur at about 10 percent less 

stream power than the field data indicates. Figure 

28 shows that the range of stream power in which 

dunes occur becomes smaller with decreasing fall 

diameter of bed material. Thus, a small change in

stream power can change the bed form and resistance 

to flow when the median fall diameter of the bed 

material is small. . Using additional alluvial canal and river data, 

Tipton and Kalmbach, Inc. (written commun., 1963) 

further validated the stream-power relation for predicting form of bed roughness. They also suggested 

using figure 29 to illustrate the large number of

0.001

0.00001, 0.1 0.2 0.3 0.4

V*.IN FEET PER SECOND

FIGURE 20. Relation of r»~ V/!&/*, V*, and regime of flow for laboratory and field conditions. The data are from the Rio Grande (Culbertson and 

Dawdy, 1964), Elkhorn River (Beckman and Furness, 1962), Pakistan canals (Harza Engineering Co., Internat., 1963), and the 8-foot-wide flume.
J26 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

upper regime flows that have been observed in large, 

irrigation canals that have fine-sand beds.

Knowledge of what the bed form is or may be 

under given fluid, flow, and sediment conditions is 

ihiportarit in designing channels that will neither 

erode nor fill up with sediment (stable channel), 

determining the stage-discharge relation for natural 

channels, and estimating the water and sediment 

discharge of a stream. For example, in stablechannel, design, if dunes with a large resistance to 

flow are anticipated, then a small average velocity, 

fairly large depth, and steep slope must exist for a 

given discharge. With fine bed material, a small 

change in stream power can change the bed form, 

and the channel may not function as designed. The 

bed form may not be dunes as anticipated but may 

be transitional or plane and result in small resistance 

to flow, high average velocity, relatively shallow 

depth, and an unstable channel having high transport 

capacity.

The Marala-Ravi Canal in Pakistan is an example. 

This canal has bed material with a median diameter 

of approximately 0.16 mm and was designed to carry 

21,000 cfs. When the canal was put in operation, the 

slope steepened from 1:10,000 to 1:8,000 because 

of excess sediment from the river. This resulted in 

a combination of slope, depth, and velocity which 

gave a stream power of 0.71. A stream power of 

this magnitude was sufficient to cause development 

of a plane bed and cause channel instability, large 

changes in channel geometry, and a bed-material 

discharge of 132 tons per day per foot of width.

VELOCITY DISTRIBUTION

The velocity distribution in a wide alluvial channel 

is as complex as the bed forms that occur. With 

dunes and antidunes, the velocity distribution is constantly changing with time and space. The variation 

of the velocity distribution in the vertical with these 

bed forms is so great that a detailed statistical study 

would be required to describe them. This would require many more profiles, covering a longer period 

of time over a larger area of the flume, than are 

available. Typical velocity profiles for ripples, dunes, 

and plane-bed configurations are given in figure 30.

With both plane bed and ripples, the velocity profiles are constant with time and space outside a zone 

near the boundary roughness. (See fig. 30.) In the 

following section the velocity distribution for the 

plane bed with sediment moving is analyzed. From 

this analysis an equation for the mean velocity and 

the roughness coefficient for the plane bed with sediment movement is evolved.

PLANE BED WITH SEDIMENT MOVEMENT

The velocity distribution in the vertical for twodimensional flow over a plane bed with sediment 

movement can be described by

vv = A* Iny + £*, (7)

where A* and B# are the slope and the intercept 

of the vy versus Iny plot. The values of A* and B# 

for the various runs are given in table 3 along with 

other pertinent information for the run.

VELOCITY fy^/IN^EET PER SECOND

FIGURE 30. Typical vertical-velocity-distribution curves .for an alluvial channej.-.Data are from .the runs-in the 8-foot-wide flume with'sand III'an 

 i ' the bed material'.'' Station refers'.to the normal clistance: from the flume wall. - < .- " i r . . 
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J27

TABLE 3. Slope (A*: ) and intercept (B#) of the velocity 

profiles for a plane bed with sediment movement

Sand

I. .....

II ......

Ill ......

IV .......

rfeo

0.19

.27

.28 

45

47

Q3

Run

10

15

16

46

22 

25 

28 

26

60 

61 

71 

72 

70 

100 

27

32

24

V*

0.167

.169

.190

.199

.172 

.215 

.201 

.200

.261 

.264 

.234 

.238 

.249 

.360 

.095

.103 

.104

A*

0.561

.595

.610

.608

.548 

.589 

.545 

.700

1.005 

.782 

.826 

.813 

.590 

1.170 

.261

.244 

.261

B*

4 22

4.50

4.92

4.74

3.91 

4.89 

4.69 

f\ OQ

6.05 

5.90 

5.43 

5.20 

5.21 

7.94

9 (19

2.30 

2.30

K

0.30

on

.29 

.34 

.34

.27

.22 

.31 

.27 

.28 

.41 

.29 

.34

.40 

.38

CT

2480

2000

2750

1670

1540 

2710 

2760 

4580

3290 

3390 

5250 

5680 

6310 

8440

26 

1

Velocity varies with In y, so it is natural to assume 

that the velocity distribution will have a form similar 

to that of the Von Karman-Prandtl velocity-distribution equation for flow near hydraulically rough 

boundaries:

 A In + B. (8)

With flow over a rigid boundary, A (A = l/k, where 

k is Von Karman's kappa) has the constant value 

2.5, B varies with the roughness and form of the 

cross section, £ is some roughness height, and Vt 

is the shear velocity ( VgDS). Experiments with 

flow over an alluvial bed have revealed that A is a 

variable and can exceed 2.5 (Vanoni, 1946; Ismail, 

1952; Vanoni and Brooks, 1957). The difference in 

A, comparing flow between movable alluvial boundaries and rigid boundaries, was attributed by these 

investigators to a damping effect on the turbulence 

by the suspended sediment. Laursen and Lin (1952), 

in a discussion of Ismail's paper, disputed this view 

and held that the change in A resulted from the 

change in bed roughness. Vanoni and Brooks (1957) 

showed a good correlation between the change in A 

and the ratio of the power to suspend the sediment 

in a thin layer near the bed (AyCg o> Ay) and stream 

power to overcome friction (r0V). Elata and Ippen 

(1961) used neutrally buoyant particles to show that 

suspended sediment causes an increases in A. Also, 

by measuring turbulence they found that the change 

in A was not the result of damping the turbulence 

but of a change in its structure. Elata and Ippen's 

investigation indicated that the size and density of 

the bed material, in addition to its concentration near 

the boundary, affected A. Thus, the size and concentration of the bed material moving in suspension and 

in1 contact with an alluvial bed fairly conclusively 

affect A, but so also might changes in bed form. 

Also, the relative importance of the three factors 

(suspended sediment, contact load, or bed form)

affecting A is not known.

To determine A and B, equation 7 is equated to 

equation 8, from which

and

A V~ *

v *

B = * + A In §.

V 0,

(9)

(10)

Plotting A* versus V* (fig. 31), A, which is the

EXPLANATION

Sand

O A 

I III

X 

II IV

 I

FIGURE 31. Variation of the slope of the vertical-velocity distribution (A*) 

with shear velocity (V*) for a plane bed with sediment movement.

slope of the line, does not vary systematically with 

concentration, size of bed material, or any other 

variable. Thus, it appears that the slope of an 

average line through the plotted data represents A 

for the plane-bed data for sands I, II, and III. The 

three points representing sand IV are for plane-bed 

runs in which shear values were slightly larger than 

were required for beginning of motion, and A for 

this sand should be about the same as that in runs 

in rigid-boundary channels. The capabilities of the 

flume system were inadequate to obtain a plane bed 

with high rates of sediment movement (after the 

dune-bed configuration) with sand IV. The average 

value of A for sands I, II, and III in a channel with 

a moving boundary was 3.20, and that for sand IV 

was 2.64 ; these values are equivalent to kappa values 

of 0.31 and 0.38, respectively.

The absence of 'a systematic variation of A with 

sand size for flow over a plane bed having a moving 

boundary can be explained by the nature of the experiments. In experiments' conducted under equilibrium plane-bed conditions, the concentration of 

suspended sediment and the movement of the contact 

bed-material discharge depend on the magnitude of 

the shear velocity and the size of the bed material.
J28 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Similarly, A*, the slope of the vy versus In y plot, 

depends on the existence of the plane bed, the concentration of the suspended sediment, the magnitude 

of the contact bed-material discharge, and thus on 

the shear velocity and the size of the bed material. 

If a 1:1 correspondence exists between the shear 

velocity for the plane bed and the effect of the shear 

velocity on the slope of the velocity profile irrespecA tive of the size of the bed material, then A = ^-* " * 

should be a constant. That is, the increase in shear

velocity required by larger size bed material for the 

formation of a plane bed resulted in a compensating 

increase in the slope of the vy versus In y relation 

so that A remained constant.

To determine B and £ in equation 10, B* (the 

intercept of the plot of vv versus In y) is plotted 

versus V* in figure 32. Either B or £ can be selected 

and the other solved with the aid of figure 32. Except

TABLE 4. Values of B when £ = d and of £ when-B = A.

[Calculated values of B assume e = d.-,o, and those of £ assume B == A.~\

Sand

I 

II...

III...

</50

mm

0.19

0.27, 0.28

0.45, 0.47

ft

0.000613

.00090

.00151

dss 

(ft)

0.00078

.0015

.003

£*/v* 

25.9

23.3

21.9

B

(fps)

2.3

.9

1.1

f

ft

0.00083

.0018

.0029

mm

0.25

.55

.88

0.8

0.6

0.4

0.2

EXPLANATION

Sand

O A 

I III

0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32

FIGURE 32. Relation between the intercept of the vertical-velocity distribution (B*) and shear velocity (V*).

for sand IV, the slope M of the B* versus V* relation 

depends on the size of the bed material. The relation between B* and V* for sand IV at low rates of 

sediment movement is not the same as for high rates. 

Consequently, the slope of the B* versus V* relation 

is considered to be a function of the size of the bed 

material for all sand sizes.

For each bed material having a large range in sand 

sizes, there are many possible values of the roughness size (£) and, thus, many combinations of B and 

|. From the original concept of equation 8, B should 

be a constant for a given roughness and channel 

shape. Also, B should have such a value that £ should 

be a size which always occurs with the same frequency in the bed material. That is, | should equal 

the,.$5o, ^65> doo* or some other size in the bed material.

When the d50 size of the bed material was substituted 

for £ in equation 10, B was not a constant; but when 

the value of B in equation 8 was assumed to be equal 

to A, then £ was approximately equal to the ds5 size 

(table 4). Using B equal to A and integrating equation 8 gives a simple mean-velocity relation

V = V, [A in(D/5) + (B - A)]. (11)

However, because B varies with the shape of the cross 

section as well as with the grain roughness, other values 

of B or of £ may have to be selected for other cross 

sections. With B = A, equation 8 becomes

vy/V* = 3.2 [foifo/$) + 1] (12) 

and

- = 3.2 " * = 7.4 log (13)

where

(in eq 12) = shear velocity for the

position where the velocity distribution 

in the vertical was measured,

(in eq 11 and 13) = mean shear 

velocity for the flume, and

£ = d8b .

The different methods of calculating V* are necessary because in the velocity-distribution equations 

(eq 7, 8, and 12) the flow is two-dimensional and 

the distribution is governed by the local shear 

(y.DS). In the mean-velocity equations (eq 11 and 

13) the flow is three-dimensional and the mean 

velocity is governed by the mean shear (yRS).

ROUGHNESS COEFFICIENTS

V _ C'

The ",-. . measure of the roughness coefficient, f/ ~/-, v * V g

given by equation 13 was evolved from a study of 

the velocity distribution normal to the bed under the 

following assumptions:

1. The velocity profiles would conform to the Von 

Karman-Prandtl equation (eq 8).
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J29

2.

3.

4.

The coefficients A and B in this equation are

equal. 

The shear stress for the velocity-distribution

equation is equal to yDS', thus, V* = VgDS

in equations 8 and 12.

The shear stress for the integrated equation is 

equal to yRS; thus, V* = ^gRS in equations 

11 and 13.

For equation 13 to be valid, the values of £ (calculated using V and v'gRS) should be approximately 

equal to the values of £ calculated from the velocitydistribution equation (eq 12) using vy and ^/gDS. 

As shown in figure 33, the values of $ from equations 12 and 13 compare very well, and their height 

is approximately equal to that of the D85 size of the 

bed material. To further check equation 13, it was 

plotted on; the relation between C'/^/lj and D for 

the plane-bed runs (fig. 34). Equation 13 fits the 

experimental points well and may be used as the 

basic velocity equation for the plane-bed runs with 

sediment movement. However, even though equation 

13 is valid in form, it may not be the correct equation for other conditions because it was developed 

for an idealized situation and neglects many factors 

such as the shape factor for the reach and the cross 

section.

EVALUATION OF RESISTANCE TO FLOW

Various methods have been suggested for determining average velocity and resistance factors for

1.6

1.4

S i b

O

bo

0.6

0.2 z

EXP

Sand

I 

II

! in

;. IV

°/

LANAT 

O

X'

x' if

jf x

ION 

£2

O' - 

X' 

A'

&

/

/

/

i

/

/

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1. 

rf 85 , tN MILLIMETERS

FIGURE 33. Comparison of the roughness height J£) with the dan size of 

the bed material. £r was calculated from C/Vff = 7.4 log (Z>/£) ; and 

£., from va/V« = 3.2 [Zn (y/£) + 1].

Ill 

§

24

22

20

18

16

14

EXPLANATION

Sand

O A 

I III

0.1 0.5 

DEPTH (D), IN FEET

FIGURE 34. Comparison of the equation C/Vff = 7.4 log (£>/£) with the 

relation between the Chezy discharge coefficient and depth for plane 

bed with sediment movement.

flow in alluvial channels. The Manning and Chezy 

equations developed for channels having rigid boundaries, the various regime equations (Inglis, 1948), 

and Einstein and Barbarosa's (1952) treatment of 

alluvial river-channel roughness have all been used 

to estimate channel resistance and average velocity.

The data for the 8-foot-wide flume follow those 

plotted in Einstein and Barbarosa's (1952) river 

curve, which relates V/ Vg&RS to &yd/yR'S, reasonably well within the range of Ayd/yR'S values for 

which ripples and dunes occur. However, for both 

large and small values of &yd/yR'S, the data departs 

systematically and radically from the proposed 

curve. Therefore, some modification is required 

when Einstein and Barbarosa's relation is applied 

in the upper flow regime. (See fig. 35.)

Resistance to flow in alluvial channels is the result 

of one of the following processes or a combination 

of them:

1. Surface resistance (grain roughness). Where 

surface resistance occurs, the flow does not 

separate from the macroboundary but does 

separate from the grains, or microroughness. 

This type of resistance occurs on a plane bed, 

on the back of dunes, and in antidune flow.

2. Form resistance. Where form resistance occurs, 

the flow separates from the macroboundary. 

The result is a pressure reduction in the separation zone (form drag) and the generation of 

large-scale eddies; both processes dissipate
J30 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

70

30

20

10

EXPLANATION

Sand

0 V 

I III

 

IV

Curve 1 /

0.5 0.7 -345 10 20 30

FIGURE 35. Comparison of resistance data from the 8-foot-wide flume to 

Einstein and Barbarosa's (1952) bar-resistance relation for rivers 

(cinvve 1) and with Einstein and Kalkanis' (1959) relation for flumes 

(curve 2).

, energy. This type of resistance occurs with 

ripples, with dunes, and, to a limited extent, 

with antidunes.

3. Acceleration and deceleration of the flow (nonuniform flow). With the growth and subsidence of the inphase water- and bed-surface 

waves, acceleration and deceleration of flow 

occurs. This is a source of energy dissipation 

in addition to that caused by the grain roughness. This type of energy dissipation occurs 

with antidune flow, although with dunes there 

is also some acceleration and deceleration of the 

out-of-phase flow over the roughness elements.

4. Breaking waves. The inphase water- and bedsurface waves reach an instability point and 

break. The eddies and turbulence generated by 

the breaking waves dissipate energy of the 

same order of magnitude as does a hydraulic 

jump with a Froude number -slightly larger 

than unity. Fortunately, a breaking wave, except in chute-and-pool flow, occupies a limited 

amount of time and space and causes a rather 

small increase in resistance to flow. This type 

of energy dissipation occurs with antidunes 

and with chutes and pools.

It is very difficult to separate the processes or to 

mathematically describe the forms of bed roughness 

which occur on the bed of a sand channel. The 

spacing, amplitude, and shape of these roughness 

elements are affected by many interrelated variables. 

Because of this, it is very difficult to write a generalized function to determine resistance to flow. 

A generalized function may not exist, because (1)

more than one resistance to flow may occur for a 

given slope, depth, and bed material; (2) hysteresis 

exists in the change in bed configuration so that the 

bed configuration and resistance to flow depend on 

preceding flow conditions; and (3) the bed configuration will oscillate between a dune bed and a plane 

bed for a given bed material at certain slopes and 

discharges. The problem is further complicated by 

three-dimensional flow, varying depth, varying bank 

roughness, and nonuniformity of flow in alluvial 

channels. The net result is varying bed configuration from point to point on the stream bed, both 

longitudinally and laterally, and, consequently, varying resistance to flow. Hence, average resistance to 

flow is the result of the combined effects of many 

different roughness elements and the turbulence they 

create. In a straight sand channel it is not uncommon to find a plane bed under the main current of 

the stream, transition roughness adjacent to the bed 

plane, and dunes near the banks. Meanders and bars 

also complicate the problem; meanders are present 

to some extent even in straight reaches of sand-bed 

streams.

Various methods of treating resistance to flow and 

of determining the average velocity of flow are presented in the following sections. The methods are 

based either on adjusting the measured slope to 

compensate for the increase in energy loss caused 

by the different bed configurations, or by adjusting 

depth or hydraulic radius to compensate for the 

energy loss and the increase in flow cross section 

caused by the form roughness. The latter method 

results from the fact that there is no flow through 

part of the measured depth. Two methods of depth 

adjustment are discussed: (1) by adjusting the 

depth to the equivalent depth for a plane bed with 

grain roughness and, (2) by adjusting the depth to 

an equivalent depth for a smooth boundary as defined by the equation of Tracy and Lester (1961). 

These methods depend on knowledge of the bed configuration. If the bed configuration is not known, 

.an estimation is made of what it was or may be, and 

the velocity is computed using one of the foregoing 

methods. With the computed velocity, shear stress, 

and bed-material size, figure 28 is used to determine 

if the selected bed configuration would occur. If the 

selected bed configuration would not occur, another 

trial is made and the velocity is recomputed.

EVALUATING RESISTANCE TO FLOW 

BY ADJUSTING SLOPE

Equation 13 can be used to calculate resistance to 

flow and average velocity for flow over a plane bed 

upon which there is sediment movement. Therefore,
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J31

it is proposed that an adjustment term be applied to 

the plane-bed equation to obtain the resistance to 

flow for the ripple, dune, antidune, or chute and pool 

bed configurations. This adjustment term would take 

into account the increase in resistance to flow caused 

by the form roughness, wave acceleration and deceleration, and wave breaking (all these effects will be 

referred to hereafter as form roughness effects). 

The resistance to flow is in terms' of /' and A/, where 

/' is the resistance to flow due to grain roughness 

and A/ is the increase in resistance to flow caused 

by various form roughness effects. This implies that

/ = /' + A/. (14) 

From equation 14, the total-resistance factor is

/ = 8(gR/V*)(S' + AS) (15) 

or

/ = S(gRS'/V*)(l + AS') = 8(gRS'/V*)(S/S'),

where S' is the slope the channel would have for the 

same depth and velocity if the resistance to flow was 

only grain roughness, and AS is the increase in slope 

required to compensate for all other types of energy 

dissipation. From equation 14,

S = S' + AS. (16) 

In terms of1 the Chezy discharge coefficient, C/ Vg =

where

I = 8/ccy Vfj* = g/cy Vj)*(s/sr ), (17)

C7 Vg = 7.4 log D/l

or

C/ Vg = C'/ Vg = (7.4 log

(18)

,, (19)

and C* is the correction factor applied to equation 

18 to obtain the discharge coefficient for bed forms 

other than a plane bed. V * is based on the hydraulic 

radius R and not the depth D.

To determine C* it is first necessary to determine 

AS, the increase in slope of the energy grade line 

which results from the form roughness. The increment of slope (AS) should be a function of the bed 

form, shear stress on the bed, and size of bed material. Therefore, AS = S S' was calculated for all 

the data and was plotted versus the shear stress for 

each sand with bed form and depth as additional 

variables. For .sands finer than sand IV, a lijiear 

relation existed between the logarithms of AS and 

the shear stress (depth and bed form as additional 

variables). :($ee figs. 36-38.) Thus,

AS = KDM (20)

where K, M, and N are constants that are functions 

of the bed form and sand size. N is the slope of the 

AS versus TO line for the different bed forms, and 

K and M are determined from

(21)

AS by plotting 7 as determined from ^ for each depth TO

versus that depth on log paper and determining the 

slope M of the line and the intercept K.

The equations expressing AS as a function of depth 

and shear are given in the preceding three figures. 

These figures indicate that, given bed roughness of 

ripples and dunes, AS is independent of depth of 

flow for sands I and II and that it has a minor effect 

for sand III if the hydraulic radius is approximately 

equal to the depth that is, when TO = yRS ~ yDS. 

Therefore, AS was plotted against the slope. (See 

figs. 39, 40, and 41.) The correction factor C* for 

each sand for the plane-bed equation was determined 

from these figures and is given in table 5.

The flume data indicate that AS and C* are independent of depth for the ripple- and dune-bed configurations in the finer sands. This requires further 

verification because of the small range in depth 

investigated. To determine if AS was still independent of depth for the dune-bed configuration at 

large depths, field data from the Elkhorn River 

(Beckman and Furness, 1962), which was the source 

for sand II, were studied. Runs having fully developed dunes (indicated by the depth-discharge relation in fig. 20) were plotted in figure 40. As indicated 

in figure 40, AS was also independent of depth for 

the Elkhorn River, where the depth ranged from 

1.4 feet to 5 feet. This tends to confirm that AS is 

independent of depth for dunes. However, additional 

field and laboratory studies are needed to conclusively 

determine whether or not C* is independent of depth 

for ripples and dunes in fine sands.

TABLE 5. Equation for C* as a function of bed form and 

grain size

Bed form

Plane bed-....

Antidunes....

I 

(0.19 mm)

Vl - 1.5S- M

Vl - 1.2S-°»

1

Jl - 34S-»

1 £> '

Grain size

II

(0.27 mm)

Vl - .62

Vl - .26/S- 1 *

1

Vl - TOOS'-s

III

(0.45 mm)

 \ /I 9 7Q.2

Vl 2.6D- 8S- 2

1

V I - 2.9 X 10 4S»

Z>2-5,
J32 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0.08

0.06

0.04

0.02

0.01

0.008

0.004 0.008 0.02 0.04 0.06 0.1 0.2 

SHEAR STRESS (Tn ='VDS).\N POUNDS PER SQUARE FOOT

FIGURE 36. Relation between slope adjustment (AS), shear stress (TO), bed configuration, and depth (D) for sand I.

Numeral by the point is the depth.

The lack of variation of AS with depth is in conflict with prior statements concerning the effect of 

depth of flow on resistance to flow and with, the 

changes in resistance to flow shown in figures 5 and 

19. The only explanation is that the effect of depth 

is taken care of by the D/£ term in equation 18 from 

which S' is determined.

With sand IV, AS was not linearly related to the 

shear stress. (See fig. 42.) Therefore, C* was computed for each run and plotted against the shear 

stress as shown in figure 43. With the dune-bed 

configuration, C* was inversely related to the shear 

stress. An increase in shear stress, either by an 

increase in slope or an increase in depth, caused a 

decrease in C* (that is, an increase in roughness). 

The variation of C* with depth and slope in the 

lower flow regime is given in figure 44. When the

bed form was in transition from the lower to upper 

flow regimes, C* was affected by depth and the shear 

stress. Also, there was a systematic change in C* 

with depth and shear stress in the transition which 

was contrary to the results of runs using finer sands. 

C* was inversely related to slope and directly 

related to slope and directly related to depth when 

flow was in the upper regime. An increase in slope 

resulted in a decrease in C* (an increase in resistance to flow), and an increase in depth resulted in 

an increase in C* (a decrease in resistance to flow). 

The effect of depth and slope on C* has been qualitatively verified by field observations. Antidime 

activity and resistance to flow in natural alluvial 

channels will decrease with an increase in depth 

when slope is constant, and they will increase with 

an increase in slope when depth is constant. Studies
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J33

1.0 

0.8

0.6 

0.4

0.2

0.1

EXPLANATION

0 

Ripple

Dune

0

Transition

A 

Plane

A 

Antidune

D

Dune, Elkhorn River 

data (Beckman and 

Furness, 1962)

0.04

0.59

0.01

AS-^OJ 7"0 

Ai> " D

0.004

0.90

A

1.00

A

0.001

0.96 

1.0

0.74

9

0.002 0.004 0.01 0.04 0.1 

SHEAR STRESS (TQ = JDS), IN POUNDS PER SQUARE FOOT

0.4

FIGURE 37. Relation between slope adjustment (AS), shear stress (TO), bed configuration, and depth (D) for sand II. Numeral by the point

is the depth.
J34 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0.08

0.06

0.04

0.02

0.01

0.008

0.01 0.02 0.04 0.06 0.08 0.1

SHEAR STRESS (TQ = JDS), IN POUNDS PER SQUARE FOOT

0.2

FIGURE 38. Relation between slope adjustment (AS), shear stress (TO), bed configuration, and depth (D) for sand III. Numeral by the point is the

depth.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J35

A S is independent 

of D for ripples 

and dunes

0.08

0.06

0.04

0.02

0.01

0.015 0.02 0.04 0.06 0.08 0.1

SLOPE

0.4 0.6 0.8 1.0

FIGURE 39. Relation between slope adjustment (AS), slope (S), bed configuration, and depth (D) for sand I. Numeral by the point is the depth.
J36 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

0.8

0.6

0.4

0.2

0.1

0.08

0.06

0.04

EXPLANATION

O 

Ripple

 

Dune

9 

Transition

A 

Plane

A 

Antidune

D

Dune, Elkhorn 

River data 

(Beckman and 

Furness, 1962)

0.94*

0.59 A

(jl.61 Q0.82 

^0.91

Antidunes 

AS=700S 2' 5

0.02

fjO.84

0.01

0.02 0.04 0.06 0.08 0.1 0.2

SLOPE fSX 10 2)

0.4 0.6 0.8 1.0

FIGURE; 40. Relation between slope adjustment (AS), slope (S), bed configuration, and depth (D) for sand II. Numeral by the point is the depth.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J37

0.08

0.06

0.04

0.02

0.01

0.008

0.015 0.02 0.6 0.8 1.0

SLOPE (SX1Q 2) 

FinURB 41. Relation between slope adjustment (AS), slope (S), bed configuration, and depth (£>) for sand III. Numeral by the point is the depth.
J38 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0.002

0.001

0.01 0.02 0.04 0.06 0.1 0.2 0.4

SHEAR STRESS (T0 = JDS), IN POUNDS PER SQUARE FOOT

FIGURE 42. Relation between slope adjustment (AS), shear stress (TO), 

bed configuration, and depth (D) for sand IV. Numeral by the point 

is the depth.

in the field and laboratory are needed to quantitatively determine the relation between C#, slope, and 

depth for different sand sizes.

Using equation 19 and the values of C* from either 

table 5 or figure 43, the velocity was computed for 

all the flume runs. The computed velocity and the 

measured velocity are compared in figure 45.

In summary:

1. The discharge coefficient C'/^g for a plane bed 

can be determined for the flume runs by

= 7.4 log (D/^), (13)

in which £ is the rf85 size of the sand in the flume 

(may vary for other streams), because £ will 

be a function of the form of the cross section. 

The plane-bed equation can be multiplied by a 

parameter C* to obtain the discharge coefficient 

for the other bed configurations.

3. The parameter C* depends on the bed form, sand 

size, slope, and depth. 

A. In the lower flow regime:

a. The coefficient C* for the finer sands 

(d50 < 0.4 mm) was independent of 

depth and decreased (an increase in 

resistance to flow) with an increase 

 in slope. The magnitude of the decrease in C* with an increase in 

slope depended on the sand size and 

whether the bed configuration was 

ripples or dunes.

b. For the coarser sands (d > 0.4 mm), 

C* was dependent on the depth and 

the slope and decreased with an increase in depth and (or) an increase 

in slope. The coarser the sand, the 

greater was the effect of depth on 

the value of C*. 

B. In the upper flow regime:

a. The value of C* for sands finer than 

0.93 mm decreased with an increase 

in slope and increased with an increase in depth. The magnitude of 

the change depended on the sand 

size. 

C. In the transition zone:

a. The value of C* was undetermined for 

all but the coarser sands. With the 

0.93-mm sand, C* increased with an 

increase in slope or a decrease in 

depth.

4. The value of C* can be determined from equations for sands I, II, and III (table 5), and 

from figure 43 for sand IV.

EVALUATING RESISTANCE TO FLOW 

BY ADJUSTING DEPTH OF FLOW

Resistance to flow or mean velocity can be determined by adjusting either slope or depth of flow. 

The depth adjustment takes into account the increase 

in energy dissipation resulting from the form roughness and the possible error in depth measurement 

resulting from inclusion of the separation zones 

downstream from ripples and dunes in the total area 

of flow.

The measured depth of flow includes part of the 

separation zones downstream from ripples and 

dunes. This part does not actually convey the fluid 

downstream. The roughness elements and their associated separation zones were observed, measured, 

and photographed. Together with these measurements, much attention was given to the flow pattern
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J39

1.1

1.0

0.9

1.03 A F0.53

0.8

0.7

0.6

EXPLANATION

Dune

3 

Transition

A 

Plane

A 

Antidune

0.56\

,Z)=0.6

,0.68

\

0.603

0.51

,£>=0.9

0.5

1.11*

0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 

SHEAR STRESS (T0 = 7#SJ,IN POUNDS PER SQUARE FOOT

0.20 0.30 0.40

FIOUHB 43. Relation between form roughness correction (C»), shear stress ( T<>), and depth (D) for sand IV. Numeral by the point is the depth.

1.1

0.8 1.1 

DEPTH (D), IN FEET

1.2

FIGURE 44. Relation between form roughness correction (C«), depth (D), 

and slope (S) for dune-bed configuration in sand IV. Numeral by the 

point is the slope X 10 2 .

over the individual roughness elements. For example, downstream from the crest of each ripple and 

dune is a separation zone in which a part of the 

fluid rotates about a horizontal axis which passes 

through the mass of fluid within the zone of separation, as is shown in figure 46.

By careful measurement, it was determined that 

the length of these separation zones conformed reasonably well with the length of separation zones 

observed downstream from baffles placed in a wind 

tunnel (Nagabhushanaiah, 1961). These lengths 

were approximately 10-12 times the amplitude of 

the roughness element measured from trough to 

crest. A study of these separation zones suggested 

that average velocity based on Q/A and average 

depth based on the average distance from the water 

surface to the bed surface are perhaps misleading. 

An effective depth (Dc ) may be defined as the average depth (D) reduced to compensate for the zones
J40 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

234 

MEASURED VELOCITY (V),\N FEET PER SECOND

FIGURE 45. Comparison of the computed velocity and the measured velocity.

Water surface

Separation zones - 

DunesFIGURE 46. Typical flow pattern -over two-dimensional ripple and dune 

roughness elements.

of separation, and an effective velocity may be defined correspondingly by applying the principle of 

continuity. The concept of an effective velocity and

depth is further clarified by figure 47, from which it 

can be seen that the effective depth,

J.+J. + D.+ ........+D.,

n

^ '

and the effective velocity, Vc = q/De .

The effective depth and velocity given in equation 

22 can be approximated in the flumes by mapping 

the form roughness. This is done using sonic equipment and ejecting dye to define the limits of the 

separation zone. Knowing effective depth and velocity, the average velocity in the channel can be 

determined from

V = (23)
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J41

V'=q/D t

Water surface 

V

FIGURE 47. Sketch showing effective depth (Do) and effective velocity 

(V = q/De).

This procedure is extremely difficult and is generally 

impossible to use. In addition, effective depth and 

velocity do not consider the increase in energy dissipation resulting from the form roughness. Therefore, in the next sections an adjusted depth (D') 

and velocity (V) are defined that take into account 

both correction factors. They are also relatively 

simple to use to evaluate resistance to flow and 

velocity.

ADJUSTING DEPTH TO AVERAGE ALLUVIAL, 

GRAIN ROUGHNESS

The resistance to flow and the average velocity 

are evaluated by adjusting the measured depth (D) 

to the depth of flow of an equivalent channel with 

the same slope, discharge, and average grain roughness. That is,

V'D' = VD (24) 

and

D' = D - AD,

depth of the channel would have for the

same slope and discharge if the resistance

to flow was the result of an average grain

roughness,

increase in depth resulting from the form

roughness,

mean velocity of the equivalent plane-bed

channel,

where

D' =

D = 

V =

V = C'l V0 V^S (25) 

To obtain D' it is first necessary to define the 

average grain roughness. This is done by arbitrarily 

using the average discharge coefficient (eq 25) obtained for each sand in the plane-bed runs made in 

the 8-foot-wide flume.

The magnitude of C'/ ^ g is a function of the 

median diameter of the bed material:

C'A/g Median diameter of sand (mm)

20.60 _______________________ 0.19

18.28 ______________________ .28

17.10 _______________________ .45

16.25 _______________________ .93

As the tabulation shows, C7 ^g is largest for the 

finest bed material, because resistance to flow for 

the plane bed is a function of the magnitude of the 

grain roughness.

These C'/V0 values ignore the effect of relative 

roughness (depth range from 0.4 ft to 1.0 ft) and 

the difference in resistance to flow between staticsand-grain roughness and moving-sand-grain roughness. Average C'l^fg values rather than values obtained from figure 34 are used for the grain roughness because both the relative roughness and the 

moving-sand-grain effects for the plane-bed condition will be accounted for by the adjustment of the 

measured depth to the adjusted depth (D').

ALLUVIAL SAND-BED ROUGHNESS

Development of method

To determine the adjusted depth (D') for an alluvial sand-bed roughness, consider the points plotted 

in figure 48 for sand I. The points for ripples and 

dunes fall below and to the right of those for the 

relation of average grain roughness. Similar figures 

can be drawn for the other sands. The adjusted 

depth (D') can be determined by finding the difference necessary to adjust a run with form roughness 

to the average grain-roughness line. This difference, 

AD, between the measured depth (D) and the adjusted depth (D') is a measure of the effect of the 

form roughness on resistance to flow. For the plane 

bed, AD is small; for ripples and dunes, AD is relatively large; for antidunes, AD is relatively small; 

and in the transition zone, AD varies from large to 

relatively small values. As was indicated previously, 

AD should be a function of the depth, slope, size of

a

Q

§< 

O

UJ (/>

EC 

UJ

0.

K 3

UJ 

UJ

u.

Z

E 2

>

o 

o -1 ,

UJ !

>

EXPLAfNATION

O 

Ripple

«

Dune

O

Transition

A

Plane

/

/A n

b c

4

*s s

o 

o ^

0° "

<$

$r

'fsf

Y

O

)°

s'

S

jf

'Q

3

9

,

* 0

s

t^r

f

0.24

SHEAR VELOCITY (V#). IN FEET PER SECOND

FIGURE 48. Effect of bed configuration on resistance to flow for sand I.
J42 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

the bed material, and the bed configuration. If a 

functional relation can be determined for AD, then 

knowing the average depth of flow (D), the adjusted 

depth (D'} can be determined. Knowing D', the adjusted velocity (V) can be calculated from equation 

25 and the mean velocity (7) can be determined 

from the continuity equation, equation 24.

A method of estimating AD was established by 

computing it for the ripple and dune runs for each 

bed material studied in the 8-foot-wide flume. To 

compute AD, equation 24 was combined with equation 25 to obtain

AD fo=

-12/3

(26)

The computed AD's were plotted against depth, with 

slope as a third variable for each bed material and 

bed configuration. (See figs. 49-52.) These plots 

verified that there was a relation between the depth 

adjustment (AD), depth, and bed form for each of 

the bed materials. The relation between AD and D 

is not a function of the slope for the ripple bed but 

is a function of slope for the dune bed formed in

0.40

0.30

0.20

LJ 0.10

HI

u.

Z

5 0.00

0.50 0.60 0.70 0.80 0.90 

DEPTH (D), IN FEET

1.00

0.10

0.00

0.50 0.60 0.70 0.80 0.90 

DEPTH (D), IN FEET

1.00 1.10

FIGURE 49. Relation between depth adjustment (AD), depth (D), and 

slope (S X 10 4 ) for ripples (upper part) and dunes (lower part) ; 

sand I.

0.40

0.30

0.20

0.10

0.00

12.60

7.3 0

0.40 0.50 0.60 0.70 0.80 

DEPTH (D), IN FEET

0.90 1.00

0.40 0.50 0.60 0.70 0.80 0.90 

DEPTH I'D;, IN FEET

1.00 1.10

FIGURE 50. Relation between depth adjustment (A£>), depth (D), and 

slope (S X 10 4 ) for ripples (upper part) and dunes (lower part); 

sand II.

the coarser (sand IV) bed material. The AD versus 

D relation is a function of slope for the dune-bed 

configuration for all sands if field data are considered. With a significant relation between AD, D, 

and S for the different bed forms and bed materials, 

it is now possible to determine the average velocity 

by adjusting the depth of flow to an equivalent 

plane-bed depth.

This method of determining average velocity can 

be summarized as follows. If the depth, the median 

fall diameter of the bed material, and the bed configuration are known, AD can be estimated; then, 

knowing D, the term D' = D AD can be computed. 

Next, if the slope of the energy gradient is known, 

the average plane-bed shear velocity, V* = V gD'S, 

can be evaluated, and the value of V can be computed using equation 25,

V = C'/ v^ Vj&S. (25) 

From the continuity relation,

V = V'D'/D (24)

the average velocity can be computed. The size of 

bed material and the corresponding text figure to be 

used in determining AD for other bed materials with
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J43

E o.o

0.4 0.5 0.6 0.7 0.8 

DEPTH (D), IN FEET

0.9 1.0

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 

DEPTH (D),\N FEET

FIGURE 51. Relation between depth adjustment (AD), depth (D), and 

slope (S X 10'') for ripples (upper part) and dunes (lower part) ; 

sand III.

1-0.4

0.5 0.6 0.7 0.8 0.9 1.0 

DEPTH (D), IN FEET

1.1

FIGURE 52. Relation between depth adjustment (AD), depth (D), and 

slope (S X 10M for dunes ; sand IV.

either the ripple- or dune-bed configuration are given 

in table 6. Similar relations can also be developed 

for transition or antidune roughness.

Using the data for canals with a dune bed presented by Simons (1957) and by Harza Engineering 

Co., International (1963) and the average grainroughness relations developed using data from the 

8-foot-wide flume, values of AD, D, and S for various 

size ranges of bed material were plotted to give 

figures 53 and 54. For field conditions, in which

TABLE 6. Bed configurations and range in size of bed material 

for which figures 49-52 can be used to determine the AD 

adjustment based on an average grain roughness

Range of median fall 

diameters of the bed 

Bed material for which each 

Figure configuration figure can be used (mm)

49................

50. ...............

51................

52

Ripples ., ..........

Dunes.... ...... ...

Ripples __ ___......

Dunes ....... .....

Ripples .... ---..

Dunes.. ....... .. .

Dunes

0.14-0.25 

0.25-0.35 

0.35-0.55 

0.9 -1.0

depths are greater, slope is a major variable in the 

AD, D, and S relations. For flume conditions, in 

which depths are shallower, slope is not a major 

variable. However, the flume data are compatible 

with the field data. The usefulness of such relations 

requires further discussion.

Applications

Velocity and discharge are determined by using 

figures 53 and 54 and making an initial estimate of 

form roughness and then using the other known 

variables, D, S, and c£50, computing the average 

velocity (V). As a check, it is necessary to enter 

figure 28 with the stream power, based on computed 

velocity, and median fall diameter of bed material 

(dm) to see if, in fact, the assumed bed roughness 

will actually occur. This procedure must be repeated 

until agreement exists.

In the design of stable channels, the water discharge is known; the form of bed roughness is selected (usually dunes for sand-bed channels) ; the 

size of the bed material of the proposed channel is 

estimated by studying the river characteristics, the 

diversion conditions, and the natural material in 

which the channel is to be constructed; and an estimate is made of the bed-material discharge from the 

river to the canal. A depth of flow is selected from 

D versus Q or R versus Q relations based on stablechannel data. These are the regime-type relations 

described by Simons (1957) and Simons and Albertson (1963). A figure, such as 53 or 54, that is appropriate for the anticipated size of bed material in the 

canal and form of bed roughness can be entered with 

the selected depth of flow. A limited choice of slope 

is available. If the canal must carry a relatively 

large bed-material discharge, select a steep slope. 

The depth of flow and the tentative slope fixes the 

depth adjustment (AD). The average velocity is 

computed using equations 24 and 25. Stream power 

is then computed. Using the d50 of the bed material, 

determine if the assumed bed roughness agrees with
J44 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

4.0

3.o

2.0

1.0

EXPLANATION

(Harza Engineering Co., 

International, 1963)

O 

(Simons 1957 data)

X

Flume data, sand 1

3.30 f 

2.80 C

2.20 o 

2.10°

y $ A $A

X*

O3.00

O2.70 

O3.40

02.70

^ O2.90

jra.io 01.70

1.18»

4

 1.08

 1.21

0.961

 0.885 

 l.OO

' 0.854

1.52

1.61 9 1.75

01.59

1.970

1.11 *

 1.14

1.04

00.893

1.00(

0.869

0.885

10 12

DEPTH (D), IN FEET

FIGURE 53. Relation between the depth adjustment (AI>), depth (D), and slope (S X 10') for dunes (flume data and field data; dM ranges from

0.15 to

that given in figure 28. If there is disagreement, the 

geometry of the channel must be modified until there 

is agreement. When this phase of the design is completed, the bed-material-discharge capacity of the 

canal is computed using some appropriate procedure, 

such as that proposed by Colby (1964). If the bedmaterial discharge differs from the required capacity 

based on river and canal conditions, another revision 

is made in the design, such as a change in depth 

and (or) slope, to increase or decrease the bedmaterial discharge to as near the required value as 

possible without changing the bed form. In many 

channels, conditions for stability can only be achieved 

by including exclusion and (or) ejection structures 

to control the quantity of bed material entering the 

canal from the river. Finally, by dividing the known 

discharge by the selected depth and computed velocity, the average width of canal is calculated.

If the depth relations and figure 28 are used to 

design stable channels to convey a given discharge, 

it will be necessary to avoid designs which have a 

combination of stream power and median fall diameter that plots close to the dividing line between 

dunes and upper regime (usually plane bed) flow. 

In this region a slight error, a change in fall velocity

0.25 mm).

of bed material, or just the inherent error in figure 

28 may be sufficient to give an altogether different 

form roughness than is desired. This can be a very 

serious error because the change in resistance to 

flow between plane bed and dunes is generally on the 

order of 100 percent. If one designs a channel anticipating dunes as the bed roughness (C/ V g value 

of about 12) but a plane bed develops ( '/ ^/~g value 

of about 20), the velocity in the channel would be 

almost double that anticipated. Because the depth 

would be fairly great, it would be much more difficult to divert flow from the channel, maintain stable 

banks in the channel, and supply bed material at a 

rate equal to the channel's new transport capacity.

The accuracy of predicting average velocity by 

use of the foregoing method for both laboratory and 

field conditions is indicated in figure 55. The average 

error is about 5 percent for the laboratory data and 

about 10 percent for the field data.

To show the usefulness of_this approach, resistance diagrams relating C/^/g and R for different 

values of &D/D or &R/R can be developed. In these 

diagrams the ratio &D/D or &R/R is similar to the 

measure of relative roughness £/D or £/R in the 

resistance diagram for pipes. A typical relation
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J45

4.0

u 3.0

Ld

U.

Z

$

^*

5 2 -°

^

HD 

 i

Q

I

h0.

^ 1 0 Q

0

EXPLANATION

(Harza Engineering Co.,

International, 1963)

O

(Simons 1957 data)

X

Flume data, sand II

3.1

X* 

XX*

x x

O 3.30

2.00 O

30 O2.80 2 ' 00 ^ ' /^*\ o nn

_. \-/ j.uu /"> o on Oo QJ^-^U

2.00°

1.70°

O 2 ' 30 1.40

1 60 1 50 (R1 - 60

O OO O 1 - 50 

O2.20 \> ^^ ,-) ^ Ovi /ir>

O1.50 L4°

0L8°

oL7 °

Ql-30

 1.32

^2.86

 2.00

1.82 1.82

 1.49

3.33

^1.30 n

Q1.20

24 6 8 10 12

DEPTH (D), IN FEET

FIC.UKE 54. Relation between the depth adjustment (A£>), depth (D), and slope (S X 10') for dunes (flume data and field data; d:M ranges from

0.25 to 0.35 mm).

between C/ Vgr, R, and bD/D for sand II is shown 

in figure 56. The &D/D ratio is a function of bed 

configuration and size of the bed material, so that 

a general equation cannot be developed as in studies 

of channels with rigid boundaries. However, this 

procedure can be applied to rigid-boundary problems.

BAFFLE AND CUBE ROUGHNESS

The applicability of the foregoing concept to rigid 

channels having baffles or cubes as the roughness 

elements was investigated to test the generality of 

the method. The baffles and cubes, like dunes, cause 

zones of separation, turbulence, changes in velocity 

distribution, and considerable dissipation of energy. 

A constant resistance coefficient for an average 

grain roughness can be assumed, and the measured 

depth in the channel containing baffles and cubes 

can be adjusted to an equivalent depth in a channel 

having the same slope and discharge. Then, as in 

studies of alluvial channels, a relation between the 

depth adjustment AD and D can be determined. With 

this relation and the assumed resistance coefficient, 

the velocity can be determined for flow over a 

boundary having baffle and cube roughness.

If the average grain roughness for a channel containing baffles and cubes is the same as for the 

0.19-mm sand, 

then

or

and

C7 Vg = VI VgD'S = 18.91 

D' = yV(18.91 20S)

(27)

AD = D - D' = D - V2 /(18.912flrS). (28)

The depth adjustment (A/)) is an indirect measure 

of the total effect of the form roughness in question. 

If for a given form roughness, values of AZ) are 

plotted against average depth (D), a unique relation results for each specific type of roughness. (See 

fig. 57.) Then, for this type of roughness, if the 

depth (D) and slope (S) are known, one can evaluate 

AZ), D', VgD'S, V', and finally, the average velocity,

V =

V'D'

D ' The whole scheme is shown in figure 57,

which is based on data from studies by Sayre and 

Albertson (1963) and Koloseus and Davidian
J46 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

1 2 3 

MEASURED VELOCITY (V), IN FEET PER SECOND

FIGURE 55. Comparison of the computed velocity with the measured 

velocity. The velocity was computed by adjusting depth to the equivalent depth for a channel having average grain roughness. Roughness 

is ripples and dunes, and the data were from the 8-foot-wide flume, 

Punjab canals (Simons, 1957), and Pakistan canals (Harza Engineering Co., Internat., 1963).

10

11

12

J

o0- 35

,0.39

^,0.35

g-29

O 

0.28

O 

0.27

30.3C 

°0

) 

28 

0.32

AD =Q 35

0.34 C

Q0.32 0.32 O OQ 32 

0.32

0.30 

^0.29

Oo.29 O028 

Q0.28

D°'28 Oa260°'27

^0.25

°0.35

^-=0.30

°oo8-

D

0.4 0.6 0.8 1.0 

RX10 5

2.0 3.0

FIGURE 56. Relation between discharge coefficient (C/^/g), and Reynolds 

number (R) for different AD/D ratios. Bed configuration is ripples 

and dunes in sand II.

(1961). The upper six lines in the AD versus D 

relation are for Sayre's AI, Bit A2, B2 , Ci and C2 

roughnesses formed by baffles at different spacings. 

The lower four curves are for Koloseus and 

Davidian's A = 1/8, A = 1/32, A = 1/128, and 

A = 1/512 cube roughnesses.

0.2 0.4 0.6 

DEPTH (D),\N FEET

0.8 1.0

FICUKE 57. Relation between depth adjustment (AD) and depth (D) 

for the type of form roughness in the rigid-boundary flume. £3 refers 

to Sayre and Albertson's (1963) roughness coefficient, and |.i refers to 

Koloseus and Davidian's (1961) roughness coefficient.

The general equation for the family of lines in the 

AD versus D relation in figure 57 is

AD = MD + b, (29)

and the general solution for C/ v'g for these roughness elements is

- 18.91

I 3/2

or

'D' D MD 63/'2 / V = -^r- = 18.91 - ^ - VgDS, (30)

D D

where

V = the average velocity,

D = the average depth,

M = the slope of the lines in the A£) versus

D relation, 

b = a function of the Y-intercept in the AD

versus D relation, and 

S = the slope of the energy-grade line.

ADJUSTING DEPTH TO SMOOTH BOUNDARY ROUGHNESS

Depth adjustments were determined by adjusting 

depth and velocity to an equivalent average grain 

roughness. The same effect is accomplished by correcting data for hydraulically rough channels, both 

rigid and alluvial, to fit the rigid-channel hydraulically smooth boundary relation presented by Tracy 

and Lester (1961). That is, .

R

5.38, (31)
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J47

where

C"/ Vlj = Chezy discharge coefficient for hydraulically smooth rigid open channels, 

and

R = Reynolds number, VR/v, which becomes 

VD/v for channels where sidewall effect 

can be neglected.

Equation 31 indicates that the discharge coefficient 

is a function of depth, slope, and temperature.

where

D' s = equivalent smooth-channel depth for the

same slope (S) and discharge, and 

T = temperature of the flow.

If both discharge and temperature are constant, a 

a relation between C'/Vg', D's, and S can be developed. (See fig. 58.) A quantitative relation can be 

written which expresses C'/ Vg versus D's and, holding the slope (S) and the unit discharge (q) constant, figure 59 is obtained. Using the results shown 

in figures 58 and 59, the smooth-boundary resistance 

(C'/Vg) given by equation 31 can be expressed as

C'/ V~g~ = t(D' 8 , S, T) =

[

D' S1/ 3 1 

1 °8 8.B2-lfl51og r + 4- 182] (33)

To develop a general method of adjusting the 

hydraulic radius of a particular run to the smoothboundary relation, first assume that the change in 

the wetted perimeter (P) is incorporated into the

32

30

28

22

20

18

r=60°F

0.1 0.2 0.4 0.6 0.8 1.0 

DEPTH (D'), IN FEET

2.0 4.0

FIGURE 58. Relation between discharge coefficient (C'/\/f/), depth (D'), 

and slope (S). Unit discharge ((/) and temperature (T) are constant.

32

30

28

26

24

22

20

18

S=10

0.1 0.2 0.4 0.6 0.8 1.0 2.0 

DEPTH (D1), IN FEET

4.0

FIGURE 59. Relation between discharge coefficient (C'/Vjy), depth (D'), 

and temperature (T). Unit discharge (q) and slope (S) are constant.

adjusted hydraulic radius. Then, from the continuity 

equation,

Q = AV = A'V, 

RPV = R'P'V

RV = R'^V = R'.V, 

V'R'

(34)

and for constant temperature,

V'R

The Chezy relation for smooth or equivalent 

boundary flow is

V = C'/ Vg VgWJ. (35) 

Solving equations 34 and 35 simultaneously for R'8 ,

VR 1 , 2/3 ^

The correction to the hydraulic radius (AJ?) required 

to plot a particular run on the smooth boundary 

relation is

A/v =/v R = R '_? __ h/3

"//Vg VgS] '

(37)

or for channel conditions such that R

VD AD = D -

J'/Vg VgS

2/3

2/3 = _

.C'/VaVjS. 1 ' (38)

Values of &R or AD can be computed using either 

equation 37 or 38, and from these values &R versus 

R or AD versus D relations can be developed.

For a particular roughness it appears that using 

A/2 versus R or AD versus D relations to adjust R
J48 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

or D to R's or D's, and using equations 33, 34, and 

35, the average velocity can be estimated. This procedure will be discussed in detail after the required 

relations are developed.

An alternative for determining the average velocity is suggested by referring to the resistance 

diagram in figure 56, which relates C/Vg, R, and 

AD/D. If in such a resistance diagram a new 

parameter,

or (39)

is adopted, a new resistance diagram relating 

C/Vffi V*D/v, and AD/D can be developed. It is 

then possible to use the AD versus D relation and 

the new resistance diagram to establish average 

velocity for a given boundary roughness. This concept will also be tested and discussed in more detail.

BAFFI.E AND CUBE ROUGHNESS

Using the data of Sayre and Albertson (1961) 

and Koloseus and Davidian (1961), in which the

bed roughness and the width of channel are large 

so that R « D, the value of AD for each run for 

each roughness was computed using equation 38.

The relations between AD and D for each of the 

families of roughness patterns studied are shown 

in figure 60. Resistance diagrams relating C/ V<y, 

R.,. = VJ.D/V, and AD/D were prepared as outlined 

in the preceding section for both Sayre and Albertson's (1961) and Koloseus and Davidian's (1961) 

data. These diagrams are presented in figures 61 

and 62, respectively.

As with the AD versus D relations, where AD was 

based on plane-bed roughness, the results based on 

the smooth bed are excellent.

GRAVEL AND COBBI.E ROUGHNESS

The flume data of Kharrufa (1962) and equation 

38 were used to compute AD. The relations between 

AD and D for Kharrufa's type B and H rock roughnesses, which have diameters of 4 and 2 inches, 

respectively, are given in figure 63. Also, a resistance diagram was developed using the flume rock

0.6

0.5

0.4

, =0.0467

0.3

0.2

=0.0198

0.1

-^X=0.0078, 3/16- and 3/8-in. cubes

0.4 0.5 

DEPTH (D), IN FEET

FIGURE 60. Relation between the depth adjustment (AZ>) and depth (D) for baffle and cube roughness elements. £3 refers to Sayre and Albertson's 

roughness coefficient, and f.t refers to Koloseus and Davidian's (1961) roughness coefficient.
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J49

OQ.47

00.53

.0.66

J0.71

0 0.62 

°0.62

0.53

.A D/D=0.45

0.4s|

c°-7i

00.7l

A D/D-0.70 

,0.70

A D/D=0.60

A D/D=0.55

0.2 0.4 0.6

R_xi<rVR

0.8 1.0 2.0

FIGUHE 61. Resistance diagram relating C/\/g, R*, and AD/Z? for the 

baffle roughness (Sayre and Albertson, 1963).

roughness for a limited number of these runs. (See 

fig. 64.)

Study of natural cobble roughness was possible 

using data from Lane and Carlson's (1953) examination of irrigation canals in the San Luis Valley, 

Colo. (fig. 65). The resultant roughness, a paved 

cobble bed, resulted from the interaction of the bed 

material and the flow. Values of AD for these canals 

were determined using equation 38. The canals were 

divided into three groups on the basis of the median 

size of the bed material at the streambed-water 

interface. The bed material in one group had a 

median diameter of about 3 inches, that in another 

group, about 2 inches, and that in the last group, 

about iy% inches. The AD versus D relations for 

each of these groups of canals are given in figure 66. 

The corresponding resistance diagram is given in 

figure 67. As with the rigid-boundary roughnesses 

previously considered, AD is independent of slope; 

but in canals it can remain independent of slope only 

as long as the bed material is stable. If a large discharge was turned into any of these canals, a boundary shear could develop that would cause movement 

of bed material. The bed material would ultimately 

become a little coarser through transportation of 

some of the smaller particles and movement of larger 

roughness elements into new positions. The previous 

AD versus D relation may no longer hold after such 

a flow condition, because a new boundary condition 

is developed by the flow. Also, a dune-bed configuration may develop that would be residual at lower 

discharge.

ALLUVIAL SAND-BED ROUGHNESS

Sand-bed roughness is much more complex than 

the baffle, cube, and rock roughnesses. Form roughness is a function of flow. A small change in any 

variable affecting the form of bed roughness will 

cause a change in the bed roughness and the resistance to flow. However, &R versus R or AD versus D 

relations can be developed for the bed form shown 

in figure 3.

For the flume data, D R, and the relations are 

in terms of AD and D. _

As can be seen in equation 32, C7 ^g and, hence, 

AD, for a constant (q) are a function of the slope 

and depth of the channel. Also, AD is more strongly 

dependent on slope when the bed form is dunes in 

the coarser sands. (See figs. 68 and 69.)

In contrast to the previous study for sand bed 

roughness where AD was determined by correcting 

to an average sand-grain roughness, in this section 

AD is determined by correcting to the hydraulically 

smooth, rigid-boundary conditions, and it will not 

be zero for the plane bed. The AD versus D relation 

for plane-bed runs using sand I is given in figure 70. 

A similar relation using slope as the abscissa for 

plotting antidune flow is given in figure 71. Similar 

relations for the other forms of bed roughness and 

other bed materials can be developed.

Although the resistance diagram for the flume 

data could be more precise if separate diagrams 

were made for each form roughness and size of bed 

material, only one diagram has beenjprepared. (See 

fig. 72.) This diagram relates C/Vg, R, and AD/D 

for all the ripple and dune flume data. To show the 

possibilities of using this procedure to estimate the 

average velocity in alluvial channels, the foregoing 

analysis was applied to the Punjab canal data reported by Simons (1957) and the more recently collected Pakistan canal data reported by Harza Engineering Co., International (1963). Relations based 

on these data were prepared in terms of the hydraulic
DEPTH ADJUSTMENT ('A £>;, IN FEET

DISCHARGE COEFFICIEN

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DISCHARGE COEFFICIENT (CVY0~)
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J51

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O 

0.47 0.48 °0.49 

0.49 0 30.49

0.3 0.4 0.6 0.8 1.0 2.0 3.0

FIGURE 67. Resistance diagram relating C/Vff. R*. and AZJ/Z? for the 

San Luis Valley, Colo., canals with cobble roughness.

FIGURE 65. Typical San Luis Valley, Colo., canal with cobble roughness.

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DEPTH (D), IN FEET

66. Relation between depth adjustment (AD) and depth (D) 

;he San Luis Valley, Colo., canals with cobble roughness (Lane and 

son, 1953). The third variable is median diameter of the bed matein inches.

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DEPTH (D), IN FEET

68. Relation between depth adjustment (AD), depth (D), and 

(S X 10 4 ) for the ripple-bed configuration in sands I, II, and III.
J52 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

0.3 0.4 0.5 0.6 .0.7 0.8 0.9 1.0 1.1 1.2 

DEPTH (D), IN FEET

FIGURE 69. Relation between depth adjustment (AD), depth (D), and 

slope (S X 104 ) for the dune-bed configuration in sands I, II, III, 

and IV.

(&D) ADJUSTMENT DEPTH

FEET IN

p co 

i-> k D*U>

Sand I

1.6 ° o1 .0

15.6. 11.2

0-..S

0.2 0.3 0.4 0.5 0.6 0.7 

DEPTH (D). IN FEET

0.8 1.0

FIGURE 70. Relation between depth adjustment (AD), depth (D), and 

slope (S X 10'1 ) for a plane bed in sand I.

0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

SLOPE fSXIO 2,)

FIGURE 71. Relation between depth adjustment (AD), depth (D), and 

slope (S X 10 2 ) for antidune flow in sand II.

radius, although the relations would be equally valid 

if expressed in terms of depth and a depth correction.

Figure 73 presents the AjR versus R relation for 

the Punjab canals, which, according to the relationship given in figure 28, have a dune-bed roughness. 

The slope in the canals is a very significant third 

variable. The accompanying resistance relation for 

the Punjab canals is presented in figure 74.

Following an identical procedure, the &R versus R 

relation for the Pakistan canals is presented in 

figure 75, and the resistance diagram is given in 

figure 76.

Similar relations were tested for field conditions 

using river-channel data. Very good results were 

obtained for channels having the same pattern 

throughout that is, dunes, plane bed, or antidunes. 

However, if multiple roughness patterns and other 

complicating factors occur, such as a braided stream, 

the accuracy of the method is reduced.

DETERMINATION OF AVERAGE VELOCITY

The preceding relations and equations offer two 

methods of estimating average velocity by using the 

appropriate AD versus D or AJ£ versus R relations, 

equations, and resistance diagrams. The procedures 

are outlined in the following paragraphs.

It is assumed that the depth or hydraulic radius 

is known or can be determined from D versus Q and 

R versus Q relations (Simons 1957), that the slope 

is known or can be assumed on the basis of limits 

set by the AD versus D relations, and that the form 

of roughness is known (rigid or alluvial). If the 

boundary is alluvial, the median fall diameter of the 

bed material d50 must also be known. For many 

channels only an estimate of dgo is possible.
DISCHARGE COEFFICIENT

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J54 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

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HYDRAULIC RADIUS (R), IN FEET

FIGUKE 73. Relation between the hydraulic-radius adjustment (AR), hydraulic radius (R), and slope (S X 104 ) for Punjab canals (Simons, 1957).

R»X10~4 (R»= Vf R/v) 

FIGURE 74. Resistance diagram relating C/\/g, R», and AR/R for Punjab canals (Simons, 1957).
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J55

789 

HYDRAULIC RADIUS (R), IN FEET

FIGURE 75. Relation between the hydraulic-radius adjustment (AR), hydraulic radius (R), and slope (S X 10 4 ) for Pakistan canals (Harza Engi>

neering Co., Internal., 1963).

In the first method, if the depth (D) or hydraulic 

radius (R), the slope (S), and the temperature (T) 

are known or assumed, &R or AD can be determined 

from the appropriate relation. Then C'/^/g can be 

computed using equation 31 or 33. Knowing C'/ ^g, 

equation 35 can be used to compute V, and V determined from equation 34.

The second method of determining V, in which 

knowledge of D or R and of S is assumed, is as 

follows:

1. Read AD or &R directly from the appropriate 

AD versus D or A/2 versus R relation, in which 

slope may be a significant third variable.

2. Having determined a depth adjustment and 

knowing the corresponding values of S, D or 

R, form roughness, and size of bed material 

when dealing with alluvial channels: 

A. Compute AD/D or

B. Compute R* = V^D/v; __

C. Enter the appropriate C/ ^/g, R* relation in 

which AD/D or &R/R is the third variable, and read the value of C/^/g corresponding to the values of R* and of &R/R 

or AD/D.

3. Compute the average velocity from the relation 

V = C/V g VgRS, using D = R when appropriate.

If determining the average velocity in a sand-bed 

channel, for which it has been necessary to assume 

a form of bed roughness prior to the determination 

of AD or A/?, it is also necessary to compute the 

stream power, TO V. Then, with d50, enter figure 28 

to determine whether the preselected roughness will 

occur. To conform with the selected bed roughness, 

it may be necessary to make another trial computation using a different slope and (or) depth. For the 

design of a stable sand-bed canal, the bed-material
J56 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

n°- 37

0.36 o.3700 a37 Qo.38

FIGURE 76. Resistance diagram relating C/Vg, R*. and AB/jR for Pakistan canals (Harza Engineering Co., Internat., 1963).

discharge is computed and compared with the desired 

bed-material discharge. If they are not approximately equal, the design should be modified by 

changing slope and (or) depth without changing 

the form of bed roughness.

SUMMARY AND CONCLUSIONS

The forms of bed roughness and resistance to flow 

in alluvial channels are functions of many interrelated variables, which are sensitive to changes in 

each other. In flume experiments and in natural 

rivers, the bed forms that occur (depending on the 

flow, fluid, geometry, and sediment characteristics) 

are ripples, ripples on dunes, dunes, plane bed, antidunes, and chutes and pools. These bed forms are 

classified into a lower or an upper flow regime or 

into an intermediate transition zone. Classification 

is based on similarity of bed form, mode of sediment 

transport, and magnitude of resistance to flow. This 

classification and the range in resistance to flow, as 

measured by / or C/VJ, associated with each bed 

roughness are:

Lower flow regime

1. Ripples (0.052 < / < 0.13, or 7.8 < C/Vg< 12.4)

2. Dunes (0.042 < / < 0.16, or 7.0 < C/ Vff < 13.2)

Upper flow regime

1. Plane bed (0.02 < /0.03, or 16.3 < C/Vg < 20)

2. Antidunes _

Standing waves (0.02 < / < 0.035, or 15.1 <C/Vg< 20) 

Breaking waves (0.03 <f< 0.07, or 10.8 < C/ Vg_ < 16.3)

3. Chutes and pools (0.07 < / < 0.09, or 9.4 <C/Vg < 10.7)

The bed roughness and flow regime are not mutually exclusive occurrences in time and space. They 

may occur side by side in a cross section (multiple 

roughness) or one after another in time (variable 

roughness).

Resistance to flow in the lower flow regime is relatively large and results mainly from form roughness 

but also from grain roughness. Resistance to flow 

in the upper flow regime is relatively small except 

in chute-and-pool flow and is caused by the grain 

roughness, wave formation and subsidence, and the 

breaking of waves. Resistance to flow for a plane
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J57

bed is less when the bed material is moving than 

when the bed material is not moving.

The bed form and resistance to flow are extremely 

variable in the transition between the two flow 

regimes, ranging from those typical of the lower 

flow regime to those typical of the upper flow regime. 

The bed form in the transition (in addition to being 

dependent on flow, fluid, geometry, and bed-material 

characteristics) depends on the antecedent form of 

the bed. That is, if the bed configuration is dunes, 

depth and (or) slope can be increased, without 

changing the bed form, to values more consistent 

for the upper flow regime; or, if the bed is plane, 

depth and (or) slope can be decreased, without a 

change in bed form, to values more consistent with 

the lower flow regime. At certain depths and slopes, 

other variables being constant, the bed form in a 

flume or in a reach of a river will oscillate between 

dunes and either plane bed or antidunes. This oscillation to and from dunes is caused by the change in 

resistance to flow and, consequently, the change in 

depth and slope with a change in bed form. On a 

plane bed, resistance to flow is small, so that the 

shear stress on the bed decreases to a value that is 

incompatible for a plane bed, and a dune bed forms. 

The increases in resistance to flow and shear stress 

resulting from the dune bed cause the dunes to be 

replaced by a plane bed, and the oscillations continue.

Ripples are not small-scale dunes; although the 

two are similar in shape, they differ in size. Ripples 

range from 0.4 foot to 2.0 feet in length and from 

0.02 foot to 0.2 foot in amplitude; dunes range from 

2.0 feet to more than 100 feet in length and from 

0.2 foot to more than 10 feet in amplitude. Resistance to flow for flow over ripples is independent of 

grain size and decreases with an increase in depth. 

Resistance to flow for flow over dunes is dependent 

on grain size and may either increase or decrease 

with an increase in depth. Ripples will not form in 

bed material having a median fall diameter coarser 

than 0.6 mm, but dunes may form in any size noncohesive bed material. Ripples appear to move 

mostly in one horizontal plane because of their small 

amplitude, but they actually move in many horizontal planes because they have many amplitudes. 

Knowing these fundamental differences between ripples and dunes is extremely important, because it 

explains the difficulty of extrapolating the results of 

small-scale laboratory studies to natural streams. 

Ripples are the dominant bed form in the lower flow 

regime in small flumes, whereas dunes are the dominant bed form in the lower flow regime in the field.

The variables that govern the form of bed roughness and, thus, resistance to flow under the restraint 

of equilibrium flow,

FaCr = aCr = W ^ W = 0]

L dt dx dt dx _]'

are slope of the energy grade line; depth of flow; 

fall velocity, physical size, and gradation of the bed 

material; concentration of fine sediment; seepage 

force; and shape and sinuosity of the channel. Concentration of the bed-material discharge is not included as a variable because the definition of equilibrium flow eliminates it as an independent variable. 

However, bed-material discharge can be substituted 

for resistance to flow or bed form if the purpose of 

the investigation is to evaluate sediment transport.

The independent variables in alluvial channels are 

often interdependent in their effect on bed form, 

resistance to flow, and each other. That is, a change 

in one of the variables not only affects resistance to 

flow but may interact and affect one or more of the 

other independent variables. Owing to this interaction it is impossible to completely isolate the effect 

of an independent variable. The following conclusions resulted from the flume experiments and field 

studies.

1. With only slope and discharge varying, and 

with the original slope being the slope at which 

motion begins, an increase in slope will produce, in 

order, the bed forms listed on page Jll. Even if bed 

form does not change, resistance to flow will still 

vary with slope. The magnitude and direction of the 

change in resistance to flow with a change in slope 

will depend on the bed form and the other independent variables.

2. With only depth and discharge varying, a 

change in depth can cause a change in bed form. The 

change occurring with an increase in depth may be 

from ripples to dunes, dunes to plane bed, or plane 

bed to antidunes. With 'a large increase in depth, the 

bed form can change from dunes to plane bed or 

even to antidunes. The order will be reversed by 

decreasing depth. If the bed configuration does not 

change with a change in depth, resistance to flow 

will change. With either ripples or a plane bed as 

the bed form, an increase in depth will cause a decrease in resistance to flow as a result of changing 

relative roughness. With dunes as the bed form, 

an increase in depth will cause an increase in resistance to flow if the median diameter of the bed material is larger than 0.3 mm, because the increase in 

the length and amplitude of dunes caused by the
J58 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

increase in depth is more rapid than the decrease in 

relative roughness. With dunes as the bed form in 

bed material having a median diameter of less than 

0.3 mm, an increase in depth either will not change 

or will cause a decrease in the resistance to flow. 

This is because the length of the dunes increases 

more than the amplitude, and because the angularity 

of the dunes decreases with increasing depth. If the 

bed form is antidunes, an increase in depth will cause 

the antidunes to increase in length, amplitude, and 

activity until a maximum is reached, and then further increases in depth will cause the antidune characteristics to decrease. These changes in the antidunes with an increase in depth results in resistance 

to flow increasing to some maximum value and then 

decreasing.

3. Fall velocity of the sediment is the primary 

variable that determines the interaction between the 

bed material and the fluid. The experiments proved 

that for a given depth and slope, fall velocity is more 

important than physical size in determining the bed 

forms that will occur, the dimensions of the bed 

form, and the intensity of any flow phenomenon 

associated with the bed form. If both depth and 

slope are constant, an increase in fall velocity will 

cause a decrease in the length and an increase in the 

angularity of the dunes; thus, resistance to flow will 

be increased. If the bed form is antidunes, a decrease 

in fall velocity will cause the antidunes to increase 

in occurrence and activity, thereby increasing resistance to flow. A decrease in fall velocity can cause 

a plane bed to change to antidunes, whereas an 

increase in fall velocity can cause a plane bed to 

change to dunes. For either change (plane bed to 

antidunes or plane bed to dunes), resistance to flow 

will increase, although the increase is much larger 

when a change from plane bed to dunes occurs.

Fall velocity, in addition to depending on the physical size of bed material, depends on the viscosity of 

the fluid and the density and shape of the particle. 

The effect of the fluid viscosity on fall velocity, and 

thus on resistance to flow, is very important because 

within the range of changes in temperature or finesediment concentration that occur naturally in 

streams, the bed form may range from dunes to antidunes. With an increase in viscosity the fall velocity 

decreases and a dune bed may become plane, decreasing resistance to flow. 'This same change in 

viscosity and fall velocity may cause a plane bed to 

change to antidunes or, if antidunes are already 

present, may increase the antidune activity, thus 

causing a corresponding increase in resistance to 

flow. Although density and shape of the particles

are factors in determining fall velocity, their effect 

on resistance to flow in natural streams is small 

because their variation in nature is small. However, 

varying density and shape of the bed material is a 

common technique employed when attempting to 

model natural river behavior.

4. The physical size of the bed material is a fundamental variable in determining fall velocity and the 

bed configuration that will form for a given shear 

stress or stream power. Physical size is also important because grain roughness occurs to a certain 

degree with all bed forms. Whether resistance to 

flow will increase or decrease with an increase or 

decrease in physical size of the bed material depends 

on the bed form that will occur with the associated 

fall velocity of the sand grains and the relative 

importance of the dissipation of energy by grain 

roughness and form roughness.

5. Fine sediment dispersed in water affects the 

specific weight and apparent viscosity of the mixture. The apparent viscosity of a fine-sediment dispersion in water depends on the concentration of 

the fine sediment; the chemical and physical properties of the fine sediment; the amount and type of 

any base added as a dispersing agent; and, with some 

fine sediments at certain concentrations, the temperature of the fluid. The ratio of the apparent 

viscosity of an aqueous dispersion of bentonite, as 

measured with a Stormer viscometer, to the viscosity 

of water is independent of the temperature at concentrations of less than about 5 percent by weight 

but is dependent on the temperature at concentrations greater than about 5 percent. At 40 °C the 

apparent viscosity of an aqueous dispersion of 10 

percent (by weight) bentonite is 11 times the viscosity of water.

The specific weight of the aqueous dispersions of 

fine sediment increases in accordance with the 

amount and density of the fine material into the flow.

The change in fluid properties that results from 

the presence of fine sediment has a definite effect 

upon the fall velocity of the bed material and, thus, 

on resistance to flow. For example, at 24 °C the fall 

velocity of a 0.47-mm median fall diameter sand in 

an aqueous dispersion of bentonite is decreased by 

65 percent. This decrease of fall velocity is equivalent to the difference between the fall velocities of a 

0.47-mm and a 0.24-mm sand particle in water at 

24 °C. With fine sediment in the flow, the bed configuration, resistance to flow, and transport of bed 

material for the 0.47-mm sand have the attributes
RESISTANCE TO FLOW IN ALLUVIAL CHANNELS J59

of a 0.24-mm sand at the same slope, depth, and 

discharge.

6. Gradation of the bed material has a major 

effect on the resistance to flow. Resistance to flow 

is larger for the bed configuration formed in a uniform sand than in a graded sand.

7. Seepage force resulting from the flow of water 

into or out of the bed of a stream can increase or 

decrease the effective weight and, thus, the mobility 

of the bed material. The change in mobility can 

cause a change in the bed form and resistance to 

flow.

8. The shape of the cross section and the sinuosity 

of the flow affect the distribution of the shear stress 

or stream power on the bed and thus affect the bed 

configuration and resistance to flow. These changes 

in bed configuration and resistance to flow were 

observed even in laboratory flumes which had only 

slightly different widths. In natural rivers the shape 

of the cross section and the sinuosity of the flow may 

cause antidunes to form in one part of the channel 

and dunes or other features to form adjacent to the 

antidunes. The dunes formed in deeper parts of the 

cross section of a natural stream may be much larger 

than those formed in the shallow parts. These variations in bed form across a stream (called multiple 

roughness) considerably affect resistance to flow.

The velocity distribution in an alluvial channel 

is as complex as the bed forms that occur there. 

Except over a plane bed, the velocity distribution 

is variable, and many measurements are needed to 

determine its average form. Over a plane bed, however, the velocity distribution in the vertical can be 

described by vv/V* = 3.2 [MD/£)+1], and the 

mean velocity by V/V* = 3.2 In (D/£) . For flume 

data, the roughness coefficient (£) is the <£g5 size of 

bed material. The value of von Karman's kappa in 

the equations for plane-bed velocity distribution is 

0.31; it does not vary systematically with concentration, size of bed material, or any other variable 

when computed using flume data.

Knowledge of the bed configuration that will form 

for a given slope, depth, viscosity, bed-material size 

distribution, and shape of channel would greatly 

simplify the determination of resistance to flow and 

sediment transport. A satisfactory method or relation between the independent variables and the bed 

configuration has not been developed. However, a 

relation between stream power (involving the velocity of flow) and the median fall diameter of the 

bed material separates the different bed configurations for both the laboratory and field data.

Because of the large range of bed forms that may 

occur in an alluvial channel, the large variation of 

resistance to flow among the different bed forms, 

and the large number of interrelated independent 

variables affecting bed form, it has been impossible 

to write a generalized function to predict resistance 

to flow or the velocity of flow. However, if the bed 

configuration is known or can be determined by 

trial-and-error computation, there are methods for 

determining resistance to flow and average velocity. 

The methods are based on adjustment of measured 

slope to compensate for the increase in energy loss 

caused by the form roughness or on adjustment of 

depth or hydraulic radius to compensate for the 

energy loss and increase in cross-sectional area 

caused by the form roughness. The magnitude of 

depth or slope adjustment varies with the bed configuration. Therefore, the methods for determining 

resistance to flow and average velocity depend on 

knowledge of the bed configuration.

In the slope adjustment method, the plane-bed 

equation (eq 13) developed from the study of the 

velocity profile is multiplied by a parameter C* to 

obtain the discharge coefficient C/ ^g for the flow. 

The parameter C* depends on bed configuration, 

sand size, slope, and depth. Equations and graphs 

were developed for determining (7* when these 

variables are known.

In the depth adjustment method, two methods 

were used to determine the depth adjustment (AD) 

required to compensate for energy losses and crosssectional area losses. In one method the depth of 

flow is adjusted to the depth of an equivalent plane 

bed having the same average grain roughness as 

defined by the plane-bed conditions studied in the 

8-foot-wide flume. In the other method the depth 

or hydraulic radius of flow is corrected to the depth 

or hydraulic radius of an equivalent channel with 

a hydraulically smooth boundary as defined by the 

equation of Tracy and Lester (1961). The increase 

in depth resulting from the form roughness systematically varies with depth, slope, and bed form for 

the different sand sizes; so it is possible to develop 

relations for predicting AD. If AD is known, the 

resistance to flow and average velocity can be determined. The depth correction (AD) or the hydraulic 

radius correction (A/2) is analogous to the protuberance height (£) referred to in resistance diagrams 

for pipes, and the ratio of AD/D or &R/R is a 

measure of the relative roughness. Resistance o!iagrams relating the discharge coefficient (C/Vgr), 

the Reynolds number, and the measure of relative 

roughness (AD/D) are developed for both methods
J60 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

of depth correction and for both rigid and alluvial 

channels. Also, a very useful relation is obtained by 

using the ratio of the Reynolds number to the discharge coefficient instead of the Reynolds number 

as a parameter in the resistance diagrams. (See 

figs. 75, 77, 79.) This procedure eliminates a trialand-error solution for the velocity.

By correcting the plane-bed equation with the 

correction factor (C*) or by computing the depth 

correction (AD) and working it into a resistance 

relation, a more meaningful method of estimating 

average velocity for alluvial channels is developed. 

This method can be extended to all types of roughness in all types of channels, both for analysis and 

design. The methods are verified using laboratory 

and field data for both rigid and alluvial channels.

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shear: Royal Soc. [London] Philos. Trans., ser. A, v. 225, 

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Beckman, E. W., and Furness, L. W., 1962, Flow characteristics of Elkhorn River near Waterloo, Nebraska: U.S. 

Geol. Survey Water-Supply Paper 1498-B, 34 p.

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McGraw Hill Book Co., 440 p.

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Brooks, N. H., 1958, Mechanics of streams with movable beds 

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

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characteristics and hydraulic variables, Middle Rio 

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Paper 1498-F, 74 p.

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of gradation of bed material on flow phenomena in 

alluvial channels: Fort Collins, Colo., Colorado State 

Univ., Dept. Civil Eng., Ph. D. dissertation.

Dawdy, D. R., 1961, Depth-discharge relations of alluvial 

streams discontinuous rating curves: U.S. Geol. Survey 

Water-Supply Paper 1498-C, 16 p.

Einstein, A., 1906, Eine Neue Bestimmung der Molekular 

dimensionen: Ann. Physik., Leipsig., v. 19, p. 289.

Einstein, H. A., 1950, The bed load function for sediment 

transportation in open channel flows: U.S. Dept. Agriculture Tech. Bull. 1026, 70 p.

Einstein, H. A., and Barbarossa, N. L., 1952, River channel 

roughness: Am. Soc. Civil Engineers Trans., v. 117, p. 

1121-1146.

Einstein, H. A., and Kalkanis, G., 1959, Project report on sand 

deposit in canals: Berkeley, Calif., Univ. California Inst. 

Engineering Research, Ser. 93, Issue 6, 62 p.

Elata, C., and Ippen, A. T., 1961, The dynamics of openchannel flow with suspensions of neutrally buoyant particles: Cambridge, Mass., Massachusetts Inst. Technology, 

Dept. Civil and Sanitary Eng. Tech. Rept. 45.

Garde, R. J., 1959, Total sediment transport in alluvial channels: Fort Collins, Colo., Colorado State Univ., Dept. 

Civil Eng., Ph. D. dissertation.

Gilbert, G. K., 1914, The transport of debris by running water: 

U.S. Geol. Survey Prof. Paper 86, 263 p.

Guy, H. P., Simons, D. B., and Richardson, E. V., 1966, Summary of alluvial channel data from flume experiments, 

1956-1961. U.S. Geol. Survey Prof. Paper 462-1, 96 p.

Harza Engineering Co., International, 1963, West Pakistan 

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U.S. GOVERNMENT PRINTING OFFICE: 1966 O 215-323