COEY

Determination of the 

Manning Coefficient From 

Measured Bed Roughness 

in Natural Channels

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1898-B

Prepared in cooperation with the 

California Department of 

Water Resources
Determination of the 

Manning Coefficient From 

Measured Bed Roughness 

in Natural Channels

By J. T. LIMERINOS

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1898-B

Prepared in cooperation with the 

California Department of 

Water Resources

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1970
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

Library of Congress Catalog Card No. GS 77-608592

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

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

Page 

Symbols - . ____ ____._________._..._______.....__...IV

Abstract.... _... __..._._....._..._.__..._____.__...__.________ Bl

Introduction.. _.___._ _.......__.___.__.._..____......._________ 3

Purpose and scope. .. _..___...._..______.._. ___-__.._ 4

Acknowledgments ..._._......_.,_._.____.___.._._______ 7

Previous investigations of channel resistance coefficients .__.._..._..__ 7 

Investigations involving relative smoothness_.._...._......_._.____ 8

Investigations involving velocity ratios-_ . ..._____.____ 11 

Conclusions.._.._ _. .___.._._____........_.__ ._......_______ 12

Collection and assemblage of data.......__.__.___..._........__._.__.._ 13

Determination of the Manning roughness coefficient- -___..._..._ 13 

Determination of characteristic streambed-particle dimensions.___. 19 

Determination of velocity ratios _.._._...__.____.__.._._..._._. 22 

Relation of roughness parameter to relative smoothness- . ___- 22 

Relative smoothness based on 84-percentile size of particles ._...__ 28 

Relative smoothness based on 50-percentile size of particles ___.... 29

Relative smoothness based on 16-percentile size of particles.__ _ 32 

Relative smoothness based on weighted particle size ._.._._ 32 

Discussion of results _.._._..___.._.._..__...._.._._..._..._..... 37

Relation of roughness parameter to velocity ratio _....._.__.__ 41

Summary.._.._._._.___.____...__._.__..______....______ 42

References cited. - _.....___........._.____.._.____... 46

ILLUSTRATIONS

Page

FIGURE 1. Index map of California showing location of study reaches ..B14 

2-3. Graphs showing size distribution of bed material 

2. Site 1, Austin Creek near Cazadero ....._..._._____ 20

3. Site 4, Kaweah River at Three Rivers. _____ 21 

4-7. Graphs showing relation of roughness parameter to 

relative smoothness based on 

4. 84-percentile size of minimum diameter of streambed

particles._._..._.._ ___. _.__.._.______ 30

m
IV CONTENTS

Page

FIGURES 4-7. Graphs showing relation of roughness parameter to 

relative smoothness based on Continued

5. 84-percentile size of intermediate diameter of

streambed particles .. _ -___......._._...B31

6. Weighted size of minimum diameter of

streambed particles . _ _._._ ___..._ 38

7. Weighted size of intermediate diameter of

streambed particles _.._ _ ._._..______ _.... 39 

8-9. Graphs showing relation of roughness parameter to 

8. Velocity ratio... _._ _ ._ ____.. 42

9. Logarithmic values of velocity ratio ____. 43

TABLES

Page

TABLE 1. Summary of basic data. ___ _ ___ _..._...B15 

2-4. Results of relating roughness parameter to relative 

smoothness based on 

2. 84-, 50-, and 16-percentile sizes of minimum diameter

of streambed particles _..._ _ 24

3. 84-, 50-, and 16-percentile sizes of intermediate

diameter of streambed particles _ _.._._ 26

4. Weighted size of streambed particles ___... 34

5. Comparison of results of correlations using d84 and dw...._____._ 41

SYMBOLS

a and 6 Constants dependent on boundary roughness 

A Cross-sectional area of the channel 

Ai Area of the ith subsection of the measured cross section 

C Chezy coefficient

d Characteristic size of a streambed particle 

d'16 Size of the minimum particle diameter that equals or exceeds

that of 16 percent of the streambed particles 

d'5Q Size of the minimum particle diameter that equals or exceeds

that of 50 percent of the streambed particles 

d'84 Size of the minimum particle diameter that equals or exceeds

that of 84 percent of the streambed particles 

d'w Weighted size of a minimum particle diameter computed as

O.ld'ie +0.8d'50+0.6d'84 

d"16 Size of the intermediate particle diameter that equals or exceeds

that of 16 percent of the streambed particles
CONTENTS v

d"50 Size of the intermediate particle diameter that equals or exceeds

that of 50 percent of the streambed particles 

d"84 Size of the intermediate particle diameter that equals or exceeds

that of 84 percent of the streambed particles 

d"w Weighted size of an intermediate particle diameter computed as

O.ld"16 +0.3d"50+0.6d"84 

E A coefficient (1.0 or 0.5) representing the proportion of velocity

head recovered

/ Darcy-Weisbach resistance factor 

F Fall, in feet, in water surface between the most upstream and

downstream cross sections of a reach

g Acceleration due to gravity, 32.16 feet per second per second 

k Effective roughness height (diameter of uniform sand grains

used in the Nikuradse experiments) 

k' Height of discrete roughness elements 

L Length of reach between cross sections 

log Common logarithm to base 10 

In Natural logarithm to base 2.718 .... 

m Number of particles within an arbitrary range of size 

m' A roughness factor such that 0.109 < m' < 3.17 

n Manning roughness coefficient 

P Wetted perimeter of a cross section

P6 Percent of bed material within an arbitrary range of size 

Q Total discharge at a cross section

A

R Hydraulic radius, 

S Energy slope

V Mean velocity, f- A

Vf Friction velocity, V

Vi Velocity in ith subsection of the measured cross section

v Velocity at a point in the vertical

w Top width of channel

x Ratio of velocity at 0.2y to O.Sy, where distances, y, are measured

	from water surface 

x Area-weighted ratio of velocities,

__

y Depth of flow

ym Mean depth, 

w

y0 Constant of integration, similar to y'

y' Thickness of the boundary layer within which the velocity is considered to be zero 

y" Vertical distance measured from the channel bottom to a point in

the vertical
STUDIES OF FLOW IN ALLUVIAL CHANNELS

DETERMINATION OF THE

MANNING COEFFICIENT FROM

MEASURED BED ROUGHNESS

IN NATURAL CHANNELS

By J. T. LIMERINOS

ABSTRACT

This report presents the results of a study to test the hypothesis that basic 

values of the Manning roughness coefficient of stream channels may be related 

to (1) some characteristic size of the streambed particles and to (2) the distribution of particle size. These two elements involving particle size 'can be 

combined into a single element by weighting characteristic particle sizes. 

The investigation was confined to channels with coarse bed material to avoid 

the complication of bed-form roughness that is associated with alluvial 

channels composed of fine bed material.

Fifty current-meter measurements of discharge and appropriate field surveys were made at 11 sites on California streams for the purpose of computing the roughness coefficient, n, by the Manning formula. The test sites were 

selected to give a wide range in average size of bed material, and the discharge measurements and surveys were made at such times as to provide data 

covering a suitable range in stream depth. The sites selected were relatively 

free of the extraneous flow-retarding effects associated with irregular channel 

conformation and streambank vegetation.

The characteristic bed-particle sizes used in the analyses were the 16,- 50- 

and 84-percentile sizes as obtained from a cumulative frequency distribution 

of the diameters of randomly sampled surficial bed material. Separate distributions were computed for the minimum and intermediate values of the 

three diameters of a particle. The minimum diameters of the streambed 

particles were used in the study because a particle at rest on the bed invariably 

has its minimum diameter in the vertical position; this diameter is, therefore, 

the most representative measure of roughness height. The intermediate diameter was also studied because this is the diameter most easily measurable 

either by sieve analysis or by photographic techniques and because it is 

the diameter that had been used in previous studies by other investigators. 

No significant difference in reliability was found between the results obtained 

using minimum diameters and those obtained using intermediate diameters.

Bl
B2 STUDIES OF FLOW IN ALLUVIAL CHANNELS

In analyzing the field data, the roughness parameter, -£ (where R is

/C/o

R

hydraulic radius), was related to relative smoothness, (where d is a chara

acteristic, or weighted characteristic, particle size). The parameter-^-, rather

/CTO

than n, was used because it is directly proportional to the square root of the 

Darcy-Weisbach friction factor, /, which is more widely used in theoretical

studies of hydraulic friction. If the transformation of -^7- to V / is made, /f 

the relations obtained in this study are of a form that is identical with that 

of the theoretical friction equation obtained by several investigators and that 

derived from field data by Leopold and Wolman (1957). The constants in the 

equation vary, of course, with the characteristic particle size used.

The relations best fitting the field data for this study were obtained by 

using either a characteristic particle diameter equal to the 84-percentile size 

(dg4, the size equal to, or exceeding, that of 84 percent of the streambed 

particles), or a diameter obtained by weighting three characteristic particle 

sizes (dw, the size obtained by assigning a weight of 0.1 to d16, a weight of 

0.3 to dgo, and a weight of 0.6 to d84 ). The use of d84 alone gave slightly 

better results than the use of dw, and, in addition, the use of d84 alone is 

attractive from a standpoint of simplicity. It is difficult, however, to rationalize 

the use of d84 alone because of the implication that the distribution of sizes 

is irrelevant, and it matters not at all whether 84 percent of the bed material 

is sand or whether it is large cobbles, as long as 16 percent of the material 

is of greater size. Consequently, the author recommends the use of dw rather 

than <J84 , although there was no unanimity of opinion on this recommendation 

among his colleagues who reviewed this paper. The reader is free to make 

his own choice between the use of d84 and dw.

As an adjunct to this study an attempt was made to relate the roughness

parameter,-^-, to the ratio of velocities at two-tenths depth (V02y) and at

/C7o

eight-tenths depth (VOSv ). These are the two depths at which velocity is 

usually observed when current-meter measurements of discharge are made. 

Twenty-seven sets of such velocity data, obtained at seven sites, were available from the discharge measurements made in connection with this study 

of the effect of streambed-particle size. If a suitable relation were found, the 

myriad of current-meter measurements of discharge in U.S. Geological Survey 

files could be used to compile an extensive library of n values for streams in 

the United States. Unfortunately, the range in magnitude of the 27 values of

-^7- was small, and therefore the small standard error of estimate obtained /£%

by correlating -^7- with the velocity ratio had little significance by itself. #%

The correlation coefficient, 0.41, was low, and it was therefore concluded that, 

on the basis of the available data, the velocity ratio is an unsatisfactory 

index of the roughness parameter.
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B3

INTRODUCTION

The Chezy equation with the empirically determined resistance 

coefficient n, ascribed to Manning (1891), has been used by many 

engineers in design of waterways and in other hydraulic applications. Under steady-state, uniform-flow conditions, satisfactory 

results can be obtained by the equation

S*

n 

where

t* V=mean velocity, -- ,

radius, -^ ,

S=energy slope,

w=Manning roughness coefficient,

Q= total discharge at a cross section,

A cross-sectional area of the channel, and

P=wetted perimeter.1

Equation 1, commonly referred to as the "Manning formula," 

has been used extensively as an indirect method for computing 

flows in natural channels. Although this equation was theorized 

and developed for uniform-flow conditions, adjustments for 

assumed energy losses attributable to an expanding or contracting 

river reach permit its use in channels with gradually varied flow. 

These assumptions have led to relatively reliable results when used 

within the range of verified channel-roughness data.

Confident selection of values of the Manning roughness coefficient, n, usually requires considerable experience. Tables of computed n values for various channel conditions, and illustrations 

and stereoscopic color slides of channels for which n has been 

verified, are available to help the practicing engineer select an 

appropriate n value. Barnes (1967) has presented color photographs and descriptive data for 50 stream channels, including the 

average values of the Manning roughness coefficient for each 

channel. These aids show various channel conditions and the combined effect of the several factors that influence channel resistance. 

However, the lack of complete similarity in channel conditions and 

geometry from stream to stream makes it difficult to estimate 

channel roughness from illustrations and stereoscopic slides.

1 In this paper a uniform set of symbols has been adopted. Symbols in formulas of other authors 

have been changed where necessary for consistency, but the definitions remain the same.

397-414 O - 70 - 2
B4 STUDIES OF FLOW IN ALLUVIAL CHANNELS

In streams with relatively stable boundaries, the total resistance 

to flow or retardation of velocity results from the interaction of 

many elements. Among them are particle size of streambed material, bank irregularity, vegetation, channel alinement, bed confiiguration, channel obstructions, converging or diverging streamlines, sediment load, and surface waves. With our present knowledge the quantitative effect of most of these factors is not determinable and must be estimated subjectively.

In many channels streambed particle size exerts a major influence on the flow. Fortunately, the quantitative effect of this 

factor can be determined more accurately than most of the other 

parameters involved in resistance to flow. Engineering knowledge 

and confidence would be increased if an objective relation between 

Manning's n and streambed-particle size could be shown. An objectively obtained basic value of n that is, the value of n for a 

straight uniform channel in the materials involved could then be 

adjusted for any of the other flow-retarding effects mentioned in 

the preceding paragraph, in the manner described by Chow (1959, 

p. 101-123). Some of the numerous theoretical studies and laboratory and field experiments made in attempts to solve the problem 

of determining basic values of flow resistance are noted in the 

next section of this report.

The investigation here described was conducted to provide an 

improved method for estimating the Manning roughness coefficient 

used in indirect measurement of high-water discharges at gaging 

stations operated by the U.S. Geological Survey. Rating curves 

showing the stage-discharge relation at those stations are normally defined by velocity-meter measurements of low- and mediumstage flows. Similar measurement of the maximum or peak discharge is often not possible because of impassable roads, destroyed 

measuring structures, or excessive flood drift. Extension of the 

rating curves to include flood flows are therefore commonly made 

on the basis of some type of indirect measurement of maximum 

discharge such as the slope-area method, and in those methods 

evaluation of the Manning roughness coefficient is a critical factor.

PURPOSE AND SCOPE

The primary purpose of this study was to define the relation 

between basic values of the Manning roughness coefficient, n, and 

an index of streambed-particle size and size distribution. The index 

used for particle size was a characteristic particle size obtained 

from a cumulative frequency distribution of the diameters of randomly sampled surficial bed material that is, the size corresponding to a selected percentile of the frequency distribution. By
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B5

weighting selected characteristic particle sizes, a single index was 

obtained that combined both size and size distribution. The retarding effect of a particle of given size is relative; it will have 

greater retarding effect on shallow flow than on deep flow. Therefore, instead of directly using the particle-size index, d, it was

R combined with the hydraulic radius, R, in the term,-,-, to provide

M 

an index of relative smoothness. The parameter -=^-, rather than

n, was used in this study because it is directly proportional to the 

square root of the Darcy-Weisbach friction factor, /, which is 

more widely used in theoretical studies of hydraulic friction. Thus,

n R the relation sought was one between -==7- and -3-. This course of K a

action was consistent with that taken by the more recent investigators of friction in open channels.

A lack of agreement exists in the results previously obtained 

in laboratory and field studies of open-channel friction. The theoretical and laboratory studies have been helpful to the investigator 

in the field because they define the types of equations to be expected in the relation of the roughness coefficient to hydraulic 

radius and bed-particle size. The constants in the laboratoryderived relations cannot be corroborated in the field because they 

are predicated on the condition of a channel uniformly covered by 

roughness particles of uniform height, a condition that is never 

found in natural channels. The field studies themselves do not 

give results that are entirely consistent, and it is probable that 

the inconsistencies are due primarily to the inadequacy or incompleteness of the data used. Where data in sufficient volume were 

available for analysis, they usually consisted of information not 

specifically collected for the purpose of the study and were incomplete ; where complete data were available, they were often too few 

for the purpose. This study should be free of those shortcomings.

Fifty current-meter measurements of discharge and appropriate field surveys were made at 11 sites on California streams for 

the purpose of computing the roughness coefficient, n, by the 

Manning formula. The investigation was confined to channels 

with coarse bed material to avoid the complication of bed-form 

roughness that is associated with alluvial channels composed of 

fine bed material. The test sites were selected to give a wide range 

in average size of bed material, and the discharges and surveys 

were made at such times as to provide data covering a suitable 

range in stream depth.

All discharges tested were in the tranquil range, their Froude
B6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

numbers being less than unity. The sites were selected to be as 

free as possible of flow-retarding effects that are not associated 

with particle size.

Separate analyses involving characteristic particle size were 

made for the minimum and intermediate values of the three diameters of a particle. The minimum diameter of the streambed particles was used in the study because a particle at rest on the bed 

invariably has its minimum diameter in the vertical position; that 

diameter is, therefore the most representative measure of roughness height. The intermediate diameter of the streambed particles 

was studied because that is the diameter most easily measurable 

either by sieve analysis or by photographic techniques and is the 

diameter that had been used in previous studies by other investigators.

The spacing and spacing pattern of streambed particles are 

also important elements in the retardation of flow by bed particles. 

However, it is extremely difficult, if not impossible, to obtain a 

quantitative index for those elements in a natural channel, and 

they, consequently, were not considered in this study. Because 

they were eliminated from consideration, despite their importance, 

there was little point in considering other factors of probably 

lesser importance, such as Froude number; width-depth ratio; 

sediment transport; and statistics, other than characteristic sizes, 

of the distribution of particle size. It was, therefore impossible to 

isolate completely the effect of particle size and size distribution on 

flow retardation, and, consequently, deviations from the relation of

 =r- to -= are to be expected. It is not likely that the determina- K dw

tion of n values for natural channels will ever be an exact science. 

Bed roughness is known to affect the vertical-velocity distribution (distribution of horizontal velocities in a vertical plane). 

When current-meter measurements of discharge are made, velocities are usually observed at two-tenths depth (Vo.2v ) and at eighttenths depth (Fo.sy), and those velocities may be considered to be 

characteristic, or index, values of the vertical-velocity distribution. 

Twenty-seven sets of such velocity data obtained at seven sites 

were available from discharge measurements made in connection 

with the study of the effect of streambed particle size. As an adjunct to this study, those data were used in an attempt to relate

the roughness parameter, -~r to the ratio, T7°' 2y-. The values of w* yo.sv 

VQ.ZV and FO.SJ/ for each discharge measurement were weighted with

respect to subsection area in the measured cross section to provide
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B7

a single velocity ratio for each discharge measurement. If a suitable relation were found, the myriad of current-meter measurements of discharge in U.S. Geological Survey files could be used 

to compile an extensive library of n values for streams in the 

United States.

ACKNOWLEDGMENTS

This study was authorized by a cooperative agreement between 

the U.S. Geological Survey and the State of California, Department of Water Resources. This report was prepared by the Geological Survey under the general supervision of Walter Hofmann and 

R. S. Lord, successive district chiefs of the Water Resources 

Division, California District, and under the immediate supervision 

of L. E. Young, chief of the Menlo Park subdistrict office.

Technical guidance was given by S. E. Rantz, research hydrologist, who offered valuable suggestions and criticism during the 

study. Lucile Coleman assisted in the computation and preparation of the data.

PREVIOUS INVESTIGATIONS OF CHANNEL RESISTANCE

COEFFICIENTS

A review of the literature on flow-resistance investigations in 

open channels has made it apparent that little agreement exists on 

a practical method of objectively evaluating the roughness coefficient. However, Carter and others (1963) have prepared a report 

on friction factors in open channels. Their comprehensive report 

summarizes the numerous investigations made of this subject.

The findings or results from several well-known studies of the 

relation of the roughness coefficient to the size of streambed 

particles or to the velocity distribution are included in this report 

because they provide valuable background for the study.

Investigation of fixed-bed, open-channel resistance reportedly 

begins with the unpublished work of Antoine Chezy. According to 

Herschel (1897), Chezy inferred that channel resistance varied 

directly with the wetted perimeter and the square of the velocity, 

and inversely with the cross-sectional area and the hydraulic 

slope. Thus,

V2P

resistance cc r-, (2)

or, as expressed in the Chezy equation,

V=C V&S (3) 

where

C=a resistance coefficient (Chezy coefficient).
B8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

Experiments by Darcy (1854) and Bazin (1865) on artificially roughened surfaces showed that

C'= . (metric units) (4)

where

C"=a resistance coefficient, and

a and ^constants dependent on boundary roughness. 

Additional experiments by Bazin (1897) resulted in a similar 

relation for English units of measure,

C'= 157' 6 , (English units) (5)

1+-?= 

V# 

where m'=a roughness factor, such that 0.109 <ra'<3.17.

Manning experimentally determined that the Chezy coefficient 

varied as the sixth root of the hydraulic radius, and in metric 

units,

7?% C= . (6)

n

According to Rouse and Ince (1957), Hagen in 1821 and 

Gauckler in 1868 had made similar determinations of the Chezy 

coefficient.

Substitution of the value for C from equation 6 into the Chezy 

equation 3 results in the widely used equation, ascribed to Manning, 

which in English units is

,, 1.486/2*5* ,-v V "=. . MI n v 

Consequently, the equation for discharge is

n

Strickler (1923) derived the following equation for n for streambeds composed of cobbles and small boulders,

nr=0.039<25o% (8)

where d50=median size of streambed particles, referred to the 

intermediate diameter.

INVESTIGATIONS INVOLVING RELATIVE SMOOTHNESS

An equation describing the vertical distribution of velocities in 

turbulent flow was derived by Prandtl (1926), and verified by 

Von Karman (1930). This equation, known as the Prandtl-Von 

Karman universal velocity-distribution law, is

v=2.5Vf ln, (9)
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B9

where

vr^velocity at a point in the vertical,

Vf= friction velocity, V ffRS, (10) 

ln=natural logarithm to base 2.718 . . .,

y"= vertical distance measured from channel bottom to a point 

in the vertical,

k 2/0=constant of integration, -^r ,

ol)

^acceleration due to gravity, 32.16 feet per second per second, and

fc=roughness height.

Equation 9 indicates that turbulent-flow velocity varies logarithmically in the vertical with the ratio of distance from the channel 

bottom to roughness height. The roughness height, k, is the effective height of the roughness elements. Linear dimensions of roughness elements may vary and yet produce the same effective roughness because of difference in shape and position or orientation. 

Vanoni (1941) observed a close relation between observed and 

theoretical velocity distributions. 

The Keulegan (1938) equation,

F=V> (6.25+5.75 log |), (11)

was derived from equation 9 and the Nikuradse (1933) experiments which used pipes on whose inner surface sand grains had 

been cemented. Note that common logarithms to base 10 are used 

in equation 11, whereas natural logarithms to base 2.718 are used 

in equation 9.

By combining equations 3, 10 and 11, the Chezy C can be expressed as a function of the hydraulic radius and effective roughness height, k. The resulting equation is

^-. (12)

The term "relative smoothness" has been applied to the ratio

R k

-j- ; the inverse ratio, -~- , is known as relative roughness.

Expressed in English units, the relation between the Chezy C 

and the Manning n is given by the equation

n 

Chow (1959) combined equations 12 and 13 to obtain the following

TO comparable relation between a roughness parameter, -^7 , and
BIO STUDIES OF FLOW IN ALLUVIAL CHANNELS

relative smoothness

n * flA\ 

-DiT = D- ( 14 >

21.9 log 12.2^

Robinson and Albertson (1952) developed a relation between a 

resistance parameter, -7=, and the ratio of depth of flow to height

of discrete regularly spaced roughness elements. Their experimental data conformed well to the equation

C=26.651og (15)

where

&'=height of discrete roughness elements and 

2/=depth of flow.

Boyer (1954), in a theoretical analysis and a study of several 

streams in northwestern United States, concluded that for fully 

developed turbulence in very wide rivers, 

n 0.105

where

ym=the mean depth in the channel cross section, or ; 

w=top width of channel; and 

,_A

The term y' is identical with the term y0 in equation 9 and represents the thickness of the boundary layer within which the velocity 

is considered to be zero.

Leopold and Wolman (1957) developed an empirical relation 

between a particle-size parameter and the resistance factor, /. 

Their equation is

4==1.0+2.0 log -A. (17)

V/ a^ 

where

,__ Sgn* _,, 1 _ 0.0926

and d84=ihe particle size, referred to intermediate diameter, that 

equals or exceeds that of 84 percent of the streambed 

particles.
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS Bll

The resistance factor, /, was derived by Weisbach (1845) for flow 

in pipes. Related studies by Darcy (1854) resulted in / being 

subsequently designated as the Darcy-Weisbach resistance factor. 

By combining equations 17 and 18, the equation of Leopold and 

Wolman can be put in the form,

n ____0.0926 , 1Q .

 pu. S~- \*-9)

R* 1.0+2.0 log A<*84

The Vennard (1961) equation for flow in pipes,

4=1.14+2.0 log pipediameter (20) 

V/ k

was developed from the Prandtl-Von Karman equation 9 and the 

Nikuradse experiments on rough pipes. By combining equations 

18 and 20, we obtain equation 21, which again relates the rough-

/yi

ness parameter,- ^, to a measure of the relative smoothness. 

Equation 21 is

n 0.0926

1.14+2.0log diameter (21)

The similarity in form of equations 17 and 20, or 19 and 21, is 

striking. Furthermore, if we take the right-hand side of Chow's 

equation 14, multiply numerator and denominator by 0.0926, and 

substitute for log 12.2 its equivalent value, 1.086, we get another 

similar equation,

n 0.0926

2.2+2.03 log ^-

(14a)

INVESTIGATIONS INVOLVING VELOCITY RATIOS

At this point we turn our discussion to a summary of investigations relating the Manning n to x, the ratio of velocities at the 

0.2 and 0.8 depths.

The Prandtl-Von Karman equation 9 and the results of the 

Nikuradse experiments on turbulent flow in pipes were combined 

by Chow in the following equation for vertical-velocity distribution over rough surfaces,

0=5.757, log . (22)

Equation 22 was used by Chow to show the Manning coefficient, 

n, for a wide, rough channel to be a function of mean depth and 

the ratio of velocity at 0.2 depth to the velocity at 0.8 depth. This 

is seen in the equation,

397-414 O - 70 - 3
B12 STUDIES OF FLOW IN ALLUVIAL CHANNELS

~ 6.78 (z+0.95) * 

Kolupaila (1930) used data from 34 discharge measurements 

at eight stations on the Nemunas River in Lithuania to substantiate his theoretical determination for the relationship between the 

roughness coefficient and the velocity distribution. His analysis 

resulted in the equation,

%=0.240 log x. (24)

The Boyer theoretical analysis of the Manning coefficient as a 

function of the velocity distribution resulted in the equation,

« 

Boyer also graphically related the roughness parameter, 77- , to 2/ffi 

the velocity ratio, T °- 2y , for average values of three measurements

"o.8j/

at each of seven streams. He concluded that there was insufficient 

data to define precisely a curve for the relation between the roughness parameter and the velocity distribution.

Toebes (1955), using the Boyer equation 25, determined an 

exceptionally good relation between the roughness parameter and 

the velocity ratio using eight discharge measurements at one 

gaging station.

Graf (1966) related the friction parameter to the velocity distribution for several ratios of water-surface width to mean depth 

in the cross section. The Graf equation, in English units, is

n 0-26 ( .

< 6.25+5.75 |VO-rc*-l-88) _log / 

[_\ l x ) \ w 

where w=top width of channel.

CONCLUSIONS

The results of theoretical studies cited, relating the roughness

ti/ parameter, - r , to relative smoothness, are not entirely consistent. it 

The results are basically similar in that they all show an inverse 

relation between the roughness parameter and relative smoothness, but there is variation in the form of the relation, as in 

equations 16 and 21. Field corroboration of the theoretical relations 

has usually been hampered by the use of data that were inadequate 

in quality because they were originally collected for other purposes, 

or because the data were insufficient in quantity. Even with adequate data, complete agreement between field results and those 

derived from theoretical studies or laboratory experiments is not
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B13

to be expected. Two reasons for this are: (1) The theoretical and 

laboratory-derived relations are predicated on the condition of 

uniform size and distribution of channel roughness elements a 

condition that never exists in natural channels, and (2) complete 

freedom from the many velocity-retarding influences other than 

the size distribution of streambed particles cannot be achieved in 

natural streams. The agreement in the form of relation derived 

from field data by Leopold and Wolman (eq 19) and the theoretical 

equations presented by Vennard for flow in pipes (eq 21) and by 

Chow (eq 14a) is, therefore, noteworthy. It might be mentioned 

at. this point that relations of similar form evolved from the study 

being reported here.

Little agreement is found among the results of previous investigations relating the Manning n, or some function of n, to the 

ratios of velocities at the 0.2 and 0.8 depths.

COLLECTION AND ASSEMBLAGE OF DATA

The 11 study reaches used in this investigation were located 

near U.S. Geological Survey stream-gaging stations because facilities for measuring discharge by current meter were available at 

the stations. The test sites, shown in figure 1, are given with 

identifying number and drainage area in table 1. Seven sites are 

on streams in north coastal California, and the other four sites are 

on streams draining the west slopes of the Sierra Nevada.

Fifty current-meter measurements of discharge and appropriate 

field surveys were made for the purpose of computing the roughness coefficient, n, by the Manning formula. The test sites were 

selected to give a wide range in average size of bed material, and 

the discharge measurements and surveys were made at such times 

as to provide data covering a suitable range in stream depth. The 

maximum discharge used at the various sites was that equivalent 

to a flood of about 1-year recurrence interval, but the ratios of 

depth to vertical height of streambed particles of median size 

reached values as high as 300. Because values of Manning's n not

iyt

to be confused with the roughness parameter, -^r show little

XV 

change with ratios greater than 30, the stream depths used in this 

study cover an adequate range.

DETERMINATION OF THE MANNING ROUGHNESS COEFFICIENT

The most important criteria in selecting a site for study was 

that it be relatively free from velocity-retarding influences other 

than those associated with the size and size distribution of streambed particles. The spacing and spacing pattern of roughness ele-
B14 STUDIES OF FLOW IN ALLUVIAL CHANNELS

EXPLANATION

Test reach location

1 . Austin Creek near Cazadero

2. Cache Creek at Yolo

3. Middle Fork Eel River below 

Black Butte River, near Covelo

4. Kaweah River at Three Rivers

5. Kings River below North Fork

6. Merced River at Clarks bridge, 

near Yosemite

7. Merced River at Happy Isles 

bridge, near Yosemite

8. Outlet Creek near Longvale

9. Middle Fork Smith River at 

Gasquet

10. Van Duzen River near Bridgeville

11. Van Duzen River near Dinsmores

FIGURE 1. Index map of California showing location of study reaches.

ments have an effect, of course, but they were given no consideration in site selection. It was expected that the use of 11 sites would 

randomize the effect of natural particle spacing on the velocity. 

Sites that had an appreciable percentage of particles smaller than 

about three-quarters of an inch in diameter were eliminated from 

consideration to avoid the complication of having bed forms at a 

site ripple, dune, or plane bed that varied with discharge.

In other words, a basic value of the Manning coefficient was 

sought that is, one that could be related to streambed particle
TABLE 1. Summary of basic da

No.

1

1

2

3

4

5

6

7

Stream and 

location

2

Austin Creek near

Cazadero.

Cache Creek at

Yolo.

Middle Fork Eel

River below Black

Butte River, near

Covelo.

Kaweah River at

Three Rivers.

Kings River below

North Fork.

Merced River at

Clarks bridge

near Yosemite.

Merced River at

Happy Isles

bridge, near

Yosemite.

Drainage 

area 

(sq mi)

3

63.1

1,138

367

418

1,342

181

Length of 

reach 

(ft)

4

288.0

290.5

295.0

303.0

279.0

309.5

372.0

531.0

533.0

536.0

248.0

256.0

248.0

254.0

248.0

180.0

200.0

220.0

216.5

221.0

220.0

221.0

Discharge 

(cfs)

5

5,050

1,370

853

672

2,180

944

277

9,000

1,350

1,050

869

405

3,690

3,660

3,200

2,440

1,340

1,840

1,650

983

1,170

666

622

1,950

1,950

1,990

1,840

1,650

1,340

1,170

983

666

622

211

Hydraulic 

radius 

(ft)

6

5.90

2.70

2.01

1.88

4.25

2.62

1.02

4.88

1.54

2.14

1.94

1.45

3.70

3.56

3.49

3.40

3.62

3.60

3.48

3.18

2.96

2.68

2.63

4.38

3.93

3.74

3.28

3.08

3.03

2.89

2.68

2.47

2.25

1.58

Manning - 

n

7

0.036

.036

.036

.038

.020

.022

.023

.035

.043

.071

.067

.083

.064

.059

.064

.066

.044

.035

.036

.052

.050

.068

.064

.060

.068

.067

.060

.058

.065

.066

.070

.087

.074

.100

Statis

Minimum diame

16 per- 50 pe

centile, centi

89 

0.016 0.0

.005 .0

.011 .0

.094 .

.11 .

.15 .

.29 .
TABLE 1. Summary of basic data Co

Stati

No.

1

8

9

10

11

0 . , Drainage 

Stream and area 

location (sqmi)

2 3

Outlet Creek near 161

Longvale.

Middle Fork Smith 130

River at

Gasquet.

Van Duzen River 216

near Bridgeville.

Van Duzen River 85.1

near Dinsmores.

Length of 

reach 

(ft)

4

243.5

242.0

137.5

242.0

250.0

241.0

243.0

239.5

241.0

253.0

254.0

256.0

304.0

313.0

313.5

Discharge 

(cfs)

5

15,200

4,420

5,640

1,200

1,210

1,610

1,130

542

348

3,000

1,950

1,570

1,840

5,310

3,570

2,510

Hydraulic 

radius 

(ft)

6

10.9

7.38

6.00

3.97

3.89

3.35

3.20

3.06

2.57

3.92

3.32

3.01

2.76

5.27

4.52

3.60

Manning

TC

7

.033

.036

.035

.029

.028

.028

.025

.036

.038

.042

.044

.047

.039

.088

.107

.098

Minimum diam

16 percentile,

8

1 0.015

2 .019

.13

.088

.066

50 pe

centi

d'o

9 

1 0.

2 

1 Particle-size distribution applies to discharge measurements made before Dec. 22,1964.

2 Particle-size distribution applies to discharge measurements made after Dec. 22,1964.
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B17

size alone. Therefore, the following additional criteria were used 

in site selection:

1. Straight channel alinement with little increase in width in the 

downstream direction.

2. Minimal vegetation on the banks and in the channel.

3. Stable banks and bed, both devoid of major irregularities.

4. Relatively wide stream of simple trapezoidal shape that will

contain the entire discharge without overflow. 

The first criterion concerning expansion in width in the downstream direction was used because of the uncertainty regarding the 

magnitude of energy losses attributable to channel expansion. The 

last criterion, concerning trapezoidal cross section without overflow plain, was used to avoid the complications associated with (1) 

a cross section of complex shape that required subdivision in 

computing Manning's n, and (2) differing values of n in the main 

channel and overflow plain.

After selecting a site, cross sections were established at the 

upstream and downstream ends of the study reach. At all but one 

site, a third or intermediate cross section was established, dividing 

each study reach into two subreaches. The exception was site 3, 

Middle Fork Eel River below Black Butte River near Covelo, where 

the length of reach was not much greater than the width. Table 

1 gives the length of each reach.

The following description of the fieldwork and computational 

procedures is brief because the details are known to most hydraulic 

engineers that are concerned with open-channel flow. For detailed 

discussions the reader is referred to reports by Benson and 

Dalrymple (1967) and Dalrymple and Benson (1967). Those reports are concerned primarily with the determination of discharge 

when values of n are known or have been subjectively estimated. 

In this study we will start with a measured value of discharge and 

solve the equations for Manning's n, but otherwise the procedures 

used in this study follow those given in the two reports.

Stakes for referencing water-surface elevations were installed 

at the ends of each cross section, and the reach and cross sections 

were surveyed. When a current-meter measurement of discharge 

was made, concurrent elevations of the water surface were 

obtained at the ends of each cross section. Those elevations and the 

measured discharge were used with cross-section dimensions from 

the original survey to compute n by means of the Manning equation. The simple Manning equation, shown as equation 7 in this 

report, is used only for uniform flow that is, flow in a channel 

whose cross-sectional area does not vary within the reach. A
B18 STUDIES OF FLOW IN ALLUVIAL CHANNELS

modification of the equation for use with gradually varied flow is 

shown below as equations 27 and 28 for unit (unsubdivided) 

cross sections.

The equation for a reach with two cross sections is

(27)

l. 2

The equation for a reach with three cross sections is

(28) 

where

subscripts denote cross-section number in downstream order

A=cross-sectional area, in square feet;

E=a coefficient (1.0 or 0.5) representing the proportion of

velocity head recovered ; 

Fmfall, in feet, in water surface between the most upstream

and downstream cross sections of a reach ; 

L= distance, in feet, between cross sections ; and

The value of the velocity-head coefficient of energy recovery, E, 

in the equations depends on whether the subreach is hydraulically 

contracting or expanding. Because all cross sections were of 

simple geometric shape and therefore required no subdivision, it 

was assumed that no adjustment to the velocity head was required 

for the effect of variation of point velocities within the cross 

section. Consequently, whether a subreach was hydraulically contracting or expanding depended entirely on the relative size of the 

upstream and downstream cross-sectional areas. If the area of 

the upstream cross section was equal to, or greater than, that of 

the downstream section, E was assumed to equal 1.0; if the 

upstream area was smaller than the downstream area, E was 

assumed equal to 0.5. The 50 measured discharges at the 11 sites 

and the corresponding computed values of the Manning coefficient, 

n, are given in table 1. Also given in table 1 are values of the 

hydraulic radius, R, each of which represents the average of R 

values for the two or three cross sections in the reach of channel 

studied.
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B19

DETERMINATION OF CHARACTERISTIC STREAMBED-PARTICLE

DIMENSIONS

/yt

Because the roughness parameter -=^7- was to be related to

characteristic bed-particle size, it was necessary at each site to 

(1) sample the bed material objectively, (2) obtain a cumulative 

frequency distribution of the dimensions of the particles, and 

(3) select sizes for study that corresponded to some predetermined 

percentiles of the distribution.

The sampling procedures used was patterned after the Wolman 

(1954) method. In each reach a grid system was established in the 

transverse and longitudinal direction of the channel to obtain 100 

intersections. The streambed particle at each intersection was 

measured along the implied maximum, intermediate, and minimum 

diameters. The ease with which the Wolman method is performed 

suggests measuring more than 100 particles for more reliable size 

distributions of the bed material. However, Wolman made a statistical test of the median diameter using nine samples of 100 

pebbles and found that one sample of 100 pebbles would result in 

a value within 12 percent of the median of the 900 pebbles, about 

two-thirds of the time. The samples of 100 particles were, therefore, considered adequate for this study.

The sampling of bed material was done during periods of low 

flow. Because of a change in channel conditions that occurred 

during the flood of December 22, 1964, two sets of bed samples 

were obtained at site 8, Outlet Creek near Longvale. The samples 

obtained before December 1964 supplied data for use with discharge measurements, cross-section measurements, and watersurface profiles obtained before the flood; the samples obtained 

after December 1964 were used similarly with measurements and 

surveys made after the flood.

Individual cumulative frequency-distribution curves were fitted 

graphically for the maximum, intermediate, and minimum diameters of each sample of 100 particles. The data were plotted on 

logarithmic normal probability paper. Plotting positions for points 

on each curve were cumulative values of P6 , computed by use of 

the formula,

where

Pb=: percent of bed material within an arbitrary range of size,

and 

m=rnumber of particles within that arbitrary range of size.

397-414 O - 70 - 4
B20 STUDIES OF FLOW IN ALLUVIAL CHANNELS

10 20 30 40 50 60 70 80 90 95 98 

PERCENTAGE OF BED MATERIAL FINER THAN INDICATED SIZE

99 99.5

FIGURE 2. Size distribution of bed material, site 1, Austin Creek 

near Cazadero.

The distributions of bed-material sizes in seven of the eleven 

reaches sampled were approximately logarithmic normal that is, 

the data plotted on straight lines as illustrated in figure 2, which 

shows the distribution for site 1, Austin Creek near Cazadero. 

The distribution for the remaining four reaches were skewed; 

that is, lines fitting the plotted data were curvilinear as illustrated
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B21

u. 1

z 0.9

. 0.8

< 0.7

Q±

w 0.6

<

S 0.5

§' 0.4

o

iv 0.3

o 0.2

0.1 

0.09 

0.08 

0.07 

0.06 

0.05 -

0.04

EXPLANATION

Maximum diameter

A

Intermediate diameter

0 A

10 20 30 40 50 60 70 80 90 95 98 99 

PERCENTAGE OF BED MATERIAL FINER THAN INDICATED SIZE

FIGURE 3. Size distribution of bed material, site 4, Kaweah River 

at Three Rivers.

in figure 3, which shows the distribution for site 4, Kaweah River 

at Three Rivers.

The curves in general were not well defined for sizes that did 

not lie between the 16 and 84 percentiles, and, consequently, data 

outside those percentiles were not used for analysis. The 16 percentile represents a size whose logarithm is one standard deviation
B22 STUDIES OF FLOW IN ALLUVIAL CHANNELS

smaller than the mean of the logarithms of the sizes in the array; 

the 84 percentile represents a size whose logarithm is one standard 

deviation greater than the mean. The three characteristic dimensions of streambed particles selected for study were those corresponding to the 16 percentile, 50 percentile (median of the logarithms), and 84 percentile. Those sizes are given in columns 8-13 

of table 1 for the minimum and intermediate diameters. Values for 

the maximum diameter of bed material are not tabulated because 

they were not used in the correlations with the roughness parameter that are discussed in this report (p. B23). The maximum diameter is the least effective of the three diameters as a measure of 

roughness because bed particles usually orient themselves so that 

the maximum diameter or axis of each particle is parallel to the 

streamlines.

DETERMINATION OF VELOCITY RATIOS

It was mentioned above that as an adjunct to this study, the 

roughness parameter would be related to the ratio of velocity at 

two-tenths of the depth below the water surface (V0_ 2y ) and to 

velocity at eight-tenths of the depth below the water surface 

(Vo.8y)> To obtain a single composite ratio, ~x, for each discharge 

measurement, the following formula was used,

 __ Vo.2y _ ~T7 

VQ.ZV 

where

Ai=area of ith subsection of the cross section measured, and

Vi=velocity in the itln. subsection.

Velocity ratios could not be computed for all the current-meter 

measurements of discharge because, for some of the low-flow 

measurements, the depths in some subsections were too shallow to 

permit observation of velocities at the 0.2 and 0.8 depths. In those 

subsections, velocities were observed only at the 0.6 depth to give 

the mean velocities in the subsections. At three of the sites the 

high-flow discharge-measurement section lay outside the reach 

used for determining n, and the measured velocities therefore could 

not be used in a correlation with the roughness parameter. Consequently, usable velocity ratios could be obtained for only 27 of 

the 50 current-meter measurements of discharge; the 27 ratios are 

given in the last column of table 1.

RELATION OF ROUGHNESS PARAMETER TO 

RELATIVE SMOOTHNESS

The basic data assembled in table 1 are pertinent to the study

n?

of the relation of the roughness parameter, -^7, to the relative

/t/6
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B23

7? smoothness, -=-, where d is a characteristic size of the streambed

particles. It has already been established that the largest of the 

three diameters of the bed material will not be used in the study, 

but that separate analyses will be made using the intermediate 

and minimum diameter. Reasons for using those two diameters 

were given on page B6. It has also been established (p. B 00) that 

such factors as Froude number and width-depth ratio will not be 

considered in the analyses, and they therefore have not been given 

in table 1.

In the correlations of roughness parameter to relative smoothness that follow, individual characteristic particle sizes are first 

used, without regard for any other characteristics of the distributions from which those individual sizes were selected. Thus, in the 

first three sets of correlations, d will have values corresponding to 

the 84-, 50-, and 16-percentile sizes of bed particles. In the correlations that are next made, consideration is given to the parent 

distributions of size of bed material, and the three individual 

characteristic sizes will be weighted and combined into a value of 

d representing a single composite size.

In making the correlations discussed in the preceding paragraph 

it was found that equations expressing the relations were of a 

form similar to those presented by Chow (eq 14a), Leopold and 

Wolman (eq 19), and by Vennard (eq 21) in which

n 0.0926

R

(31)

where a and b are constants.

Because the procedure used was common to all the correlations,

it will be described at this time. For each discharge measurement,

-tt was computed and tabulated in tables 2 and 3. Values of the

roughness parameter, -~7, were then converted to corresponding

values of 1/V / by means of equation 18, which states that

_ O 0926 _ 

1/V /= . (The computation of values of 1/V / is merely an

IV

intermediate step in the correlation procedure, and therefore 

values of 1/vTare not given in the tables.) Values of 1/V / versus

r>

-=- were then plotted on semilogarithmic graph paper, the values

of I/ V7 being plotted as the dependent variable on the vertical 

natural-scale ordinate. The best fit to the plotted points of each
TABLE 2. Results of relating roughness para/meter to relative smoothness, 

minimum diameter of streambed pa

Relative smoothness

Stream 

No. and

location

R

d'84

R

d'50

R

#16

n~R%

Based on 84-percen

particle size

Computed Percenta

from deviation 

equation computed 

33 observed v

1 2

1 Austin Creek near

Cazadero.

2 Cache Creek at Yolo 

3 Middle Fork Eel River

below Black Butte

River, near Covelo.

4 Kaweah River at Three

Rivers.

5 Kings River below

North Fork.

6 Merced River at Clarks

bridge, near Yosemite.

7 Merced River at Happy

Isles bridge, near

Yosemite.

3

72.0

32.9

24.5

22.9

118

72.8

28.3

23.2

7.33

2.33

2.10

1.58

4.81

4.62

4.53

4.42

6.83

6.79

6.57

6.00

5.58

5.06

4.96

3.98

3.58

3.40

2.97

2.80

2.75

4

164

75.0

55.8

52.2

327

202

78.5

102

32.1

6.11

5.51

4.14

10.6

10.2

10.0

9.71

12.9

12.9

12.4

11.4

10.6

9.57

9.39

7.68

6.91

6.56

5.74

5.40

5.32

5

369

169

126

118

850

524

204

444

140

22.8

20.5

15.4

33.6

32.4

31.7

30.9

24.1

24.0

23.2

21.2

19.7

17.9

17.5

15.1

13.6

12.9

11.3

10.6

10.4

6

0.027

.031

.032

.034

.016

.019

.023

.028

.040

.062

.060

.078

.052

.048

.052

.054

.036

.028

.029

.043

.042

.058

.054

.046

.054

.054

.048

.047

0.54

7

0.021

.024

.026

.027

.019

.021

.025

.027

.037

.062

.066

.080

.044

.044

.045

.045

.038

.038

.039

.040

.041

.043

.043

.047

.050

.051

.054

.056

.057

8

 22.2

 22.6

 18.8

 20.6

+18.

+10.5

+8.7

 3.6

 7.5

0

+10.

+2.

 15.4

 8.3

 13.5

 16.7

+5.6

+35.7

+34.5

 7.0

 2.4

 25.9

 20.4

+2.

 7.4

 5.6

+12.

+19.

+5.
Outlet Creek near 

Longvale.

Middle Fork Smith 

River at Gasquet.

10 Van Duzen River near 

Bridgeville.

2.63

2.44

2.25

2.05

1.44

57.4

35.1

28.6

20.9

20.5

16.0

15.2

16.1

13.5

5.16

4.37

3.96

5.41

5.07

4.70

4.33

3.95

2.77

202

117

95.2

73.5

72.0

53.2

50.8

56.7

47.6

12.2

10.4

9.41

12.5

9.97

9.24

8.52

7.76

5.45

727

388

316

265

259

176

168

204

171

30.2

25.5

23.2

31.4

.055

.059

.075

.065

.093

.023

.026

.026

.023

.022

.025

.021

.040

.032

.033

.036

.039

.033

.058

.060

.063

.067

.086

.022

.024

.025

.027

.027

.029

.030

.029

.031

.042

.045

.047

.042

+ 5.5 

+1.7

 16.0 

+3.1

 7.5

 4.3

 7.7

 3.8 

+17.4 

+22.7 

+16.0 

+42.9

 27.5

 3.1

+27.3 

+25.0 

+20.5

+27.3

11 Van Duzen River near 

Dinsmores.

3.51 

3.01 

2.40

17.0 

14.6 

11.6

79.8 

68.5 

54.5

.074 

.083 

.079

.050 

.054 

.061

Summary of correlation statit

Coeffi 

Stam

cient of correlation

n 

Jard error of estimate, in units of _-r___._ _ __ ... .... _ .... ____________

0.89 

.0087

 32.4 

 34.9 

 22.8

sties

Mean (algebraic) of percentage differences - 

Standard deviation of percentage differences..

 0.05 

±18.6
TABLE 3. Results of relating roughness parameter to relative smoothness, b

intermediate diameter of streambed pa

Relative smoothness

Stream

No. and

location

1 2

1 Austin Creek near

Cazadero.

2 Cache Creek at Yolo. ...

3 Middle Fork Eel River

below Black Butte

River, near Covelo.

4 Kaweah River at Three

Rivers.

5 Kings River below

North Fork.

6 Merced River at Clarks

bridge, near Yosemite.

7 Merced River at Happy

Isles bridge, near

Yosemite.

R

Based on 84-percentil

 ._ .-., particle size

R

d"84

3

47.2

21.6

16.1

15.0

68.5

42.3

16.5

14.4

4.53

1.56

1.41

1.06

3.27

3.15

3.09

3.01

5.10

5.07

4.90

4.48

4.17

3.77

3.70

2.50

2.25

2.14

1.87

1.76

1.73

B

d"50

4

105

48.2

35.9

33.6

177

109

42.5

62.6

19.7

4.12

3.71

2.79

6.98

6.72

6.58

6.42

9.05'

9.00

8.70

7.95

7.40

6.70

6.58

5.28

4.75

4.51

3.94

3.71

3.65

B

o"ie

5

227

104

77.3

72.3

472

291

113

271

85.6

14.3

12.9

9.67

19.5

18.7

18.4

17.9

16.5

16.4

15.8

14.5

13.5

12.2

12.0

11.0

9.85

9.35

8.18

7.70

7.58

n Computed

~^

6

0.027

.031

.032

.034

.016

.019

.023

.028

.040

.062

.060

.078

.052

.048

.052

.054

.036

.028

.029

.043

.042

.058

.054

.046

.054

.054

.048

.047

.054

from

equation

34

7

0.021

.024

.026

.026

.019

.021

.026

.027

.037

.060

.064

.077

.042

.043

.043

.044

.036

.036

.036

.038

.039

.040

.040

.047

.050

.051

.054

.056

.057

Percentag

deviation o

computed fr

observed val

8

 22.2

 22.6

 18.8

 23.5

+18.8

+10.5

+13.0

 3.6

 7.5

 3.2

+6.7

 1.3

 19.2

 10.4

 17.3

 18.5

0

+28.6

+24.1

 11.6

 7.1

 31.0

 25.9

+2.2

 7.4

 5.6

+12.5

+19.1

+5.6
Outlet Creek near 

Longvale.

Middle Fork Smith 

River at Gasquet.

10 Van Duzen River near 

Bridgeville.

1.65

1.53

1.41

I.29 

.90

30.3

20.6

16.7

II.0 

10.8 

9.31 

8.89 

8.50 

7.14

3.24

2.74

2.49

3.07

3.48

3.23

2.98

2.71

1.90

109

73.8

60.0

39.7

38.9

33.5

32.0

30.6

25.7

7.69

6.51

5.90

7.46

7.22

6.70

6.18

5.62

3.95

419

231

188

153

150

105

100

118

98.8

18.7

15.8

14.3

18.4

.055

.059

.075

.065

.093

.023

.026

.026

.023

.022

.025

.021

.040

.032

.033

.036

.039

.033

.058

.061

.064

.067

.087

.022

.024

.026

.029

.029

.030

.030

.031

.032

.042

.045

.047

.043

+5.5 

+3.4

 14.7 

+3.1

 6.5

 4.3

 7.7

0

+26.1 

+31.8 

+20.0 

+42.9

 22.5 

0

+27.3 

+25.0 

+20.5

+30.3

11 Van Duzen River near 2.15 10.5 53.8 

Dinsmores. 1.84 9.04 46.1 

1.47 7.20 36.7

.074 

.083 

.079

.051 

.055 

.062

 31.1 

 33.7 

 21.5

Summary of correlation statistics

Coefficient of correlation _ . _____ ._ __ ____ . ___ __. __ _

Standard deviation of percentage differences. . __ _______ 

 

.

- 

......_...._.

0.88 

.0087

 0.43 

±19.0
B28 STUDIES OF FLOW IN ALLUVIAL CHANNELS

set of data was a straight line that had an equation of the form,

l/V7=a+61og^. (32)

Equation 32 was then transformed to equation 31 by means of the

 ti/ relation between 1/V / and -grr, given by equation 18. In all cor- K 

relations except those involving only the 16-percentile size of bed 

material, the constant b was found to differ only insignificantly 

from 2.0, the coefficient in equations 14a, 19, and 21. Where applicable, therefore, the theoretical values of 2.0 was used as the

R coefficient of log -=-.

For each of the relations obtained, the standard error of estimate 

and coefficient of correlation were computed. The standard error

% of estimate is in units of -^7- and, therefore, has very limited K/6

significance to the engineer who is interested in the percentages by 

which computed values of n differ from the observed values. Therefore, percentage differences were computed for each of measured 

discharges. The percentage differences associated with each of the 

relations were found to be distributed about a mean value that 

differed only negligibly from zero percent. Therefore, the standard 

deviation of those percentage differences was used as an index of 

error, because it showed the approximate range within which the 

percentage differences for two-thirds of the measurements lay. 

The standard deviation of the percentage differences was the basis 

on which the equations were compared with respect to their adequacy for predicting values of n from particle size and hydraulic 

radius.

RELATIVE SMOOTHNESS BASED ON 84-PERCENTILE SIZE OF PARTICLES

The roughness parameter was first related to relative smoothness based on the 84-percentile size of particles. It is well known 

that the larger elements of streambed roughness in a channel have 

a disproportionately large retarding influence on the velocity, and 

it was expected that a highly significant relation would be found. 

That expectation was realized.

The relation using the minimum diameter (d'8±) is described by 

the equation,

n 0.0926 "pu p~ \oo)

K 0.76+2.0 log -£-

«84

and has a standard deviation of percentage differences of ±18.6 

percent. The data used in the analysis are given in table 2, which
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B29

tl/ also shows a comparison between observed values of -J^T- and those

fC/G

computed by equation 33. The relation is shown graphically in 

figure 4.

The relation using the intermediate diameter (d"84 ) is described 

by the equation,

n 0.0926

1.16+2.0 log

" 84

and has a standard deviation of percentage differences of ±19.0 

percent. The data used and a comparison between observed and

M

computed values of -=r- are given in table 3. The relation is shown JK 

graphically in figure 5. The similarity between equation 34 and 

equation 19 of Leopold and Wolman, also based on the intermediate 

diameter, is remarkable.

RELATIVE SMOOTHNESS BASED ON 50-PERCENTILE SIZE OF PARTICLES

The roughness parameter was next related to relative smoothness based on the 50-percentile size of particles. It was expected 

that the 50-percentile size would not correlate so well as did the 

84-percentile size, and that proved to be the case.

The relation using the minimum diameter (d'50 ) is described by 

the equation,

n _ 0.0926 , 3g .

2.0 log -57 

and has a standard deviation of percentage differences of ±22.7 

percent.

The relation using the intermediate diameter (d"50 ) is described 

by the equation,

n 0.0926 

0.35+2.0 log

, Q ^ (36)

50

and has a standard deviation of percentage differences of ±22.4 

percent.

The data used in the correlations and a comparison between

7? observed and computed values of -=r- are given in table 2 for the

K 

analysis using minimum diameter, and in table 3 for the analysis 

using intermediate diameter. The curves of relation for the minimum and intermediate diameters are not given, but they 

closely resembled those in figures 4 and 5, and the pattern of 

deviations of the plotted points was similar to that for the 84- 

percentile size.
B30 STUDIES OF FLOW IN ALLUVIAL CHANNELS

0.2

S 0.1

< 0.09

< 0.08

<o °-07

" 0.06

I 0-05

§ 0.04

	0.03

3 456789 10 20 

RELATIVE SMOOTHNESS

30 40 50

EXPLANATION 

River test reach

Austin Creek near Cazadero

e 

Kaweah River at Three Rivers

Van Duzen River near Dinsmores

o

Middle Fork Smith River at Gasquet

e

Merced River at Clarks bridge 

near Yosemite

x 

Cache Creek at Yolo

Outlet Creek near Longvale

o

Kings River below North Fork

a

Van Duzen River near Bridgeville

o

Merced River at Happy Isles 

bridge near Yosemite

o

Middle Fork Eel River below 

Black Butte River

- 6.0

- 5.0

- 4.0

- 3.0

1.5 

1.3 

1.1 

1.0 

0.9 

0.8

0.7 

0.6 

0.5

i- 0.3

100

FIGURE 4. Relation of roughness parameter to relative smoothness based on 

84-percentile size of minimum diameter of streambed particles.
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B31

< o.i

< 0.09 °- 0.08

2 °-07

| 0.06

§ 0.05

3 4 5 6 7 8 9 10 20 30 40 50 

RELATIVE SMOOTHNESS (^-

\ dM

EXPLANATION 

River test reach

Austin Creek near Cazadero

e 

Kaweah River at Three Rivers

Van Duzen River near Dinsmores

D

Middle Fork Smith River at Gasquet

e

Merced River at Clarks bridge 

near Yosemite

x 

Cache Creek at Yolo

Outlet Creek near Longvale

o

Kings River below North Fork

a

Van Duzen River near Bridgeville

c

Merced River at Happy Isles 

bridge near Yosemite

o

Middle Fork Eel River below 

Black Butte River

6.0 

5.0

4.0 

3.0

- 2.0

1.5 

1.3 

1.1 

l.O 

0.9 

0.8 

0.7 

0.6

- 0.5

- 0.4

- 0.3

- 0.2

FIGURE 5. Relation of roughness parameter to relative smoothness based on 

84-percentile size of intermediate diameter of streambed particles.
B32 STUDIES OF FLOW IN ALLUVIAL CHANNELS

RELATIVE SMOOTHNESS BASED ON 16-PERCENTILE SIZE OF PARTICLES

The roughness parameter was also related to relative smoothness based on the 16-percentile size of particles. From comparison 

of the correlations previously made using 84- and 50-percentile 

sizes, it was virtually a certainty that inferior correlations would 

result from using the 16-percentile size of particles. Nevertheless, 

correlations using the 16-percentile size were made in the belief 

that the results obtained might be helpful in determining methods 

for weighting particle sizes for the correlations to be described 

in the next section of this report.

The relation using the minimum diameter (d'i6 ) is described by 

the equation,

n - 0-0926 " r P~ 

R 0.25+1.33 log -£-

and has a standard deviation of percentage differences of ±25.5 

percent.

The relation using the intermediate diameter (d"i6 ) is described 

by the equation

n _ 0.0926 (3g

0.10+1.60 log -IJa le

and has a standard deviation of percentage differences of ±25.7 

percent.

Again, the data used in the correlations and a comparison be7? tween observed and computed values of -=^- are given in table 2

for the analysis using minimum diameter, and in table 3 for the 

analysis using intermediate diameter. The curves of relation for 

the minimum and intermediate diameters are not given, but they 

closely resembled those in figures 4 and 5, and the pattern of 

deviations of the plotted points was similar to that for the 84- 

percentile size.

RELATIVE SMOOTHNESS BASED ON WEIGHTED PARTICLE SIZE

The three preceding relations using individual characteristic 

particle sizes ignore all other characteristics of the distributions 

from which those individual sizes were selected. To give consideration to the entire cumulative frequency curve of particle sizes 

between the 84- and 16-percentiles, the 84-, 50-, and 16-percentile 

sizes were weighted and combined into a single composite size, dw. 

Other methods of considering the cumulative frequency curve 

might have been used for example, the three percentile sizes 

might have been used in a multiple correlation, or the slope of the
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B33

cumulative distribution curve might have been used by including 

an additional term in the equation for either of the ratios, -^ or

~-. It was thought, however, that the most desirable course of

action would be to maintain the theoretical form of the equation, 

and, consequently, a single weighted composite size was used. Two 

methods of weighting suggested themselves.

The most obvious method was to weight each of the three percentile sizes in inverse proportion to the square of the individual 

standard errors of estimate given in tables 2 and 3. When this was 

done, almost identical results were obtained for both the minimum 

and intermediate diameters. After rounding each weighting factor 

to a single significant figure, the resulting weights were as follows:

Characteristic size Weighting factor

84 percentile - -_ _ 0.6

50 percentile- -- .3

16 percentile .1

Actually, the weighting factors obtained were in the order of 

6, 3, 2, but the weighting factor of the 16-percentile size was reduced from 2 to 1, because the smaller weighting factor would have

7? resulted had the theoretical coefficient of 2.0 for log = been used

in equations 37 and 38. Particle sizes weighted as shown above are 

given in table 4 for both minimum (d'w ) and intermediate (d"w ) 

diameters. Each value represents a composite size for the sampling 

site, and each was obtained by multiplying d84 by 0.6, d50 by 0.3, 

and die by 0.1, and then adding the three products.

The second weighting method tried was based on rank correlation. All values of the roughness parameter, -^r , were ranked in

XV'6

order of magnitude. A set of composite values of dw , was then 

computed. (The values of d'w or d"w are such a set, using the previously mentioned weightings of 0.6, 0.3, and 0.1 for the three particle sizes.) The next step was to compute the relative smoothness,

R

, for each of the weighted particle sizes; the results of that d

R computation are given in table 4. Because -=- varies inversely with

U"U>

n R

, the values of -= were ranked in inverse order of magnitude. RVe

R

Differences between the rank of each value of -= and the rank

aw

/yt

of corresponding values of -=rr were squared, and the squared
TABLE 4. Results of relating roughness parameter to relative smoothness, 

Relative smoothness

Stream 

No. and 

location

1 2

1 Austin Creek near Cazadero _ _. ___ _____

2 Cache Creek at Yolo... ...... ___ .... ....... .

3 Middle Fork Eel River below

Black Butte River, near Covelo.

4 Kaweab River at Three Rivers.-. ~ __ __.

5 Kings River below North Fork.. _ . _____

6 Merced River at Clarks bridge,

near Yosemite.

7 Merced River at Happy Isles bridge,

near Yosemite.

Minimum 

diameter 

R

*«

3

95.2

43.5

32.4

30.3

163

101

39.2

34.9

11.0

3.19

2.88

2.16

6.38

6.14

6.02

5.86

8.62

8.57

8.29

7.57

7.05

6.38

6.26

5.09

4.58

4.35

3.80

3.58

3.52

Intermediate 

diameter 

R

<*"_

4

62.8

28.7

21.4

20.0

94.4

58.2

22.7

21.2

6.70

2.16

1.95

1.46

4.30

4.14

4.06

3.95

6.35

6.32

6.11

5.58

5.19

4.70

4.61

3.27

2.94

2.79

2.44

2.30

2.26

Observed 

n

K\/Q

5

0.027

.031

.032

.034

.016

.019

.023

.028

.040

.062

.060

.078

.052

.048

.052

.054

.036

.028

.029

.043

.042

.058

.054

.046

.054

.054

.048

.047

.054
8 Outlet Creek near Longvale-._. _ . ___ .._

9 Middle Fork Smith River at Gasquet_...--

10 Van Duzen River near Bridgeville _. ___

11 Van Duzen River near Dinsmores _____

3.36

3.12

2.87

2.62

1.84

83.8

49.2

40.0

30.5

29.9

22.3

21.3

23.5

19.8

7.00

5.93

5.38

7.26

5.27

4.52

3.60

2.16

2.00

1.84

1.68

1.18

43.6

29.5

24.0

15.9

15.6

13.4

12.8

12.2

10.3

4.36

3.69

3.34

4.12

3.23

2.77

2.21

.055

.059

.075

.065

.093

.023

.026

.026

.023

.022

.025

.021

.040

.032

.033

.036

.039

.033

.074

.083

.079

Summary of correlation statistics

Coefficient of correlation

n 

Standard error of estimate, in units of rr - ____ - -

«% 

Mean (algebraic) of percentage differences.... _. .-

Standard deviation of percentage differences. ....... . ....
B36 STUDIES OF FLOW IN ALLUVIAL CHANNELS

differences were added to give a sum of squares for the particular 

weighting system used. The process was repeated for other arbitrarily selected weightings of the three particle sizes. The weighting 

system that gave the smallest sum of squares was considered to be 

the one, of all the weightings tested, that would correlate best with

vt

 =7-. The logic behind this test is easily understood. In a perfect,

xv 

vt

or near perfect, correlation, the highest values of -~r would be tv 

K

associated with the lowest value of -= ; the second highest value of

aw

n R

would be associated with the second lowest value of -r ; and

so on. The better the correlation, the smaller the differences in 

inverse rank, and the smaller the sum of squares of those differences.

The three arbitrarily selected weightings, shown below, were 

tested.

Characteristic size Weighting factors tested

84 percentile - -- 0.5 0.6 0.7

50 percentile . . .4 .3 .2

16 percentile ___ .1 .1 .1

For the minimum diameter, the weighting of 0.6, 0.3, and 0.1 gave 

the smallest sum of squares; for the intermediate diameter that 

weighting gave the second smallest sum of squares, the smallest 

being associated with a weighting of 0.7, 0.2, and 0.1.

The results of the two weighting methods were then examined. 

Both methods showed the weighting of 0.6, 0.3, and 0.1 to be best 

for the minimum diameter, d'w. For the intermediate diameter, 

d"w, the rank correlation method showed the weighting of 0.7, 0.2, 

and 0.1 to be best, whereas the method based on the standard error 

of estimate indicated an optimum weighting of 0.6, 0.3, and 0.1 

From working with the figures it was evident that there would be 

little difference in the results of final correlations regardless 

of which of those two weightings was used. In the interest of 

consistency, therefore, a weighting of 0.6, 0.3, and 0.1 was used 

for both d',0 and d"w.

n R The relation found between -^7 and -^- , when minimum K"6 a w

diameter was used, is described by the equation, 

n _ 0.0926 BIT rr- i«yj

R 0.50+2.0 log -£- rt w

The standard deviation of percentages differences is ±19.1 per-
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B37

cent. The data used in the analysis are given in table 4, which also

fYL

shows a comparison between observed values of -=^7- and those

computed by equation 39. The relation is shown graphically in 

figure 6.

The relation using the intermediate diameter (d"w ) is described 

by the equation,

n 0.0926 p% = W 

RV° 0.90+2.0 log -

and has a standard deviation of percentage differences of ±19.5 

percent. The data used and a comparison between observed and

fit computed values of -=^7- are given in table 4. The relation is shown

/£76

graphically in figure 7.

DISCUSSION OF RESULTS

fL In general, the observed values of -=j- for each individual site

/T,7°

had a fairly consistent plotting pattern with respect to all curves 

relating the roughness parameter to relative smoothness. The

77 observed values of -=r- for four sites plotted consistently above K 

the curves, values for four other sites plotted below the curves, 

and values for the remaining three sites straddled the curves. That 

consistent pattern suggests that systematic factors other than size 

of bed material affected the roughness coefficients at the sites, and

77 and that a curve of relation for basic values of -^r that is, values 

based on size of bed material alone should perhaps be drawn on

fn

the basis of the lowest plotted values of - T- . However, the 11

/[TO

selected stream sites all gave the appearance of being relatively 

free of extraneous roughness effects, and an experienced hydraulic 

engineer viewing those sites would select n values on the basis of 

bedparticle size alone. If the curves of relation are to be of practical value to engineers in selecting n values in the field, it is therefore necessary to assume, erroneously or not, that extraneous 

roughness effects were negligible at all 11 sites and that the scatter 

of all plotted points about the curves is random.

The possibility that some of the systematic scatter might be due 

to the positioning of the coarser bed material in individual reaches 

was also considered. However, no relation was apparent; for

m

example, of the four sites whose values of -=j- plotted above the
B38 STUDIES OF FLOW IN ALLUVIAL CHANNELS

0.02 -

3 4 5 6 7 8 9 10 20 30 40 50 

RELATIVE SMOOTHNESS (^~) 

EXPLANATION

River test reach

 

Austin Creek near Cazadero

» 

Kaweah River at Three Rivers

A

Van Duzen River near Dinsmores

D

Middle Fork Smith River at Gasquet

e

Merced River at Clarks bridge 

near Yosemite

x 

Cache Creek at Yolo

Outlet Creek near Longvale

o

Kings River below North Fork

B

Van Duzen River near Bridgeville

o

Merced River at Happy Isles 

bridge near Yosemite

o

Middle Fork Eel River below 

Black Butte River

- 0.4

- 0.3

- 0.2

FIGURE 6. Relation of roughness parameter to relative smoothness based on 

weighted size of minimum diameter of streambed particles.
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B39

0.5 

0.4

0.2

0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

4 5 6 7 8 910 20 

RELATIVE SMOOTHNESS f -

EXPLANATION 

River test reach

Austin Creek near Cazadero

e 

Kaweah River at Three Rivers

Van Duzen River near Dinsmores

a

Middle Fork Smith River at Gasquet

e

Merced River at Clarks bridge 

near Yosemite

x 

Cache Creek at Yolo

Outlet Creek near Longvale

o

Kings River below North Fork

Van Duzen River near Bridgeville

e

Merced River at Happy Isles 

bridge near Yosemite

o

Middle Fork Eel River below 

Black Butte River

30 40 50

6.0 

5.0

4.0 

3.0

1.5 

1.3

l.l 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5

- 0.4

- 0.3

- 0.2

FIGURE 7. Relation of roughness parameter to relative smoothness based on 

weighted size of intermediate diameter of streambed particles.
B40 STUDIES OF FLOW IN ALLUVIAL CHANNELS

curves of relation, one site had its coarser material evenly distributed through the reach, two sites had their coarser material concentrated along the centerline or thalweg of the channel, and one 

site had its coarser material concentrated in the shallower water 

near the right bank.

The relations of -=7- to -j- that best fit the field data were ob- R* d

tained using either of the two characteristic particle sizes, d84 

or dw. From the standpoint of best fit, there was negligible difference between using intermediate or minimum diameter, but 

because of the relative ease of obtaining measurements of intermediate diameter either by sieving or by photographic methods 

it is expected that the relations using intermediate diameter will 

have more popular appeal.

It is not surprising that the larger particle sizes (d84 ) were 

shown to have greater effect on flow retardation then the smaller 

particle sizes, because the influence of the larger particles extends over a relatively greater volume of channel. Furthermore, 

the smaller particles are often in the wake of the turbulence created 

by the larger particles. Leopold and others (1964) have stated 

that a single flow-resistance parameter involving bed-particle size 

should use a size larger than d50 . Finer particles in the channel 

also have a lesser retarding effect during high flow because at that 

time they may move as suspended or bedload material, although 

this fine material is commonly replaced by the deposition of material of similar size when the stage of the stream falls. Large 

quantities of fine material in suspension may also reduce the retardation effect by dampening turbulence, but in this study the 

sites were so selected that they were relatively free of bed material 

that was as fine as sand.

The results obtained by the use of d84 and dw are summarized in 

table 5 which shows that the equations using d8± gave slightly 

better results than those using dw. The failure of the use of dw to 

improve the results obtained using d84 is attributed to the fact that 

the pattern of scatter of points about the curves was consistent, 

regardless of the characteristic particle size used. Usually, it is 

found that, if the pattern of scatter varies in individual correlations involving each of several elements of similar type, an improved correlation results if a weighted composite value of the 

individual elements is used.

The slightly better results obtained by using rf84, combined with 

the fact that the computations using d84 are somewhat simpler than 

those using dw, makes the use of d84 attractive. It is difficult, how-
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B41

TABLE 5. Comparison of results of correlations using dg4 and dw

Minimum diameter, d' Intermediate diameter, d"

Correlation -, , ,. ,. 0.0926 0.0926 0.0926 0.0926 equation^. - =-

0.76+2.0 log 0.50+2.0 log -p 1.16+2.0 log -j 0.90+2.0 log -y 

"'84 d' »"84 " IK

Coefficient of 

correlation . _ 0.89 0.87 0.88 0.87

Standard error of 

estimate, in units of

.0087 .0092 .0087 .0092

Mean percentage 

difference without

Standard deviation of 

percentage differences . ±18.6

15.6 15.5 16.1

±19.1 ±19.0 ±19.5

ever, to rationalize the use of <284 because of implication that the 

distribution of sizes is irrelevant, and it matters not at all whether 

84 percent of the bed material is sand or whether it is large cobbles, 

as long as 16 percent of the material is of greater size. Consequently, the author recommends the use of dw rather than <284, 

although there was no unanimity of opinion on this recommendation among his colleagues who reviewed this paper. The reader is 

free to make his own choice between the use of d%± and dw.

RELATION OF ROUGHNESS PARAMETER TO 

VELOCITY RATIO

The twenty-seven velocity ratios (x) given in the last column of 

table 1 were correlated with corresponding values of the roughness

/ % \ _ 

parameter (-=7- . It will be recalled that x is the area-weighted V R I

ratio of velocities at the 0.2 and 0.8 depths. Figure 8 shows the 

roughness parameter plotted against natural values of of; figure 9 

shows the roughness parameter plotted against logarithmic values 

of ~x. Equations of the straight lines best fitting the plotted points 

are shown in figures 8 and 9. The two correlations had identical

vt

standard errors of estimate, 0.011 in units of-^, and identical

correlation coefficients, 0.41. Although the magnitude of the 

standard error of estimate is on the order of those found in the 

previously described relations using relative smoothness, the correlation using velocity ratios is poor. This seeming anomaly re-

*i 

fleets the fact that the 27 values of -=7-, for which velocity ratios
B42 STUDIES OF FLOW IN ALLUVIAL CHANNELS

0.10

0.09

0.08

t

0.06

0.05

0.04

0.03

0.02

0.01

K = 0.0267 x- 0.0018

1.1 1.2 1.3 1.4 _ 1.5 1.6 1.7 

VELOCITY RATIO x

EXPLANATION

River test reach

Austin Creek near Cazadero

o

Kings River below North Fork

a

Van Duzen River near Bridgeville

e

Merced River at Clarks bridge 

near Yosemite

x 

Cache Creek at Yolo

Outlet Creek near Longvale

D

Middle Fork Smith River at Gasquet 

FIGURE 8. Relation of roughness parameter to velocity ratio.

were available, have a very much smaller range in magnitude than 

the 50 values used in the studies involving relative smoothness. 

On the basis of the available data, it was concluded that the 

velocity ratio is an unsatisfactory index of the roughness parameter.

SUMMARY

Fifty current-meter measurements of discharge at 11 sites on 

California streams were use to relate the roughness parameter,
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B43

o.io 0.15 0.20 0.25

LOG ~x 

EXPLANATION

River test reach

 

Austin Creek near Cazadero

o

Kings River below North Fork

o

Van Duzen River near Bridgeville

e

Merced River at Clarks bridge 

near Yosemite

x 

Cache Creek at Yolo

Outlet Creek near Longvale

a

Middle Fork Smith River at Gasquet

FIGURE 9. Relation of roughness parameter to logarithmic values of

velocity ratio.

Mt T>

K DJT , to relative smoothness, -j-, where d is a characteristic bed- a

particle size obtained from a cumulative frequency distribution of 

the diameters of randomly sampled surficial bed material. Separate 

distributions were computed for the minimum and intermediate 

values of the three diameters of the particles measured. The investigation was confined to channels with coarse bed material to 

avoid the complication of bed-form roughness that is associated
B44 STUDIES OF FLOW IN ALLUVIAL CHANNELS

with alluvial channels composed of fine bed material. Because basic 

values of n were sought those attributable solely to the size and 

size distribution of bed material the sites selected for study 

were relatively free of the extraneous flow-retarding effects associated with irregular channel conformation and streambank vegetation.

Three characteristic bed-particle sizes were investigated the 84 

percentile (d84 ), the 50 percentile (d50 ), and the 16 percentile (e^e). 

In addition, a weighted composite size (du,) was tested, using a 

weight of 0.6 for the 84-percentile size, a weight of 0.3 for the 50- 

percentile size, and a weight of 0.1 for the 16-percentile size. Best 

results were obtained using either dS4 or dw, but it made little 

difference in the reliability of the results whether the study was 

confined to the minimum or to the intermediate diameter. Results 

obtained using d8 ± and dl(. for both minimum, and intermediate 

diameters are summarized in table 5. Because of the relative ease 

of obtaining measurements of the intermediate diameter either 

by sieving or by photographic methods use of the intermediate 

diameter will probably have more appeal. The relations obtained

between ^7- and ^ were of the form R* d

n 0.0926

, O A 1 a+2.0 log -r

the only difference in the equations being in the values of the constant a, which varied with size (d84 or dw ) and diameter (minimum 

or intermediate) used. The above equation is virtually identical 

with the theoretical equations presented by Chow (1959) and 

Vennard (1961) and that derived from field data by Leopold and 

Wolman (1957).

The standard deviation of the percentage differences between

*YL

observed and computed values of - r- was about 19 percent when

c?84 was used, and only slightly more than 19 percent when dw was 

used. The slightly better results obtained by using rf84, combined 

with the fact that the computations using d8 ± are somewhat simpler 

than those using du., makes the use of d8± attractive. It is difficult, 

however, to rationalize the use of d8± because of the implication 

that the distribution of sizes is irrelevant, and it matters not at all 

whether 84 percent of the bed material is sand or whether it is 

large cobbles, as long as 16 percent of the material is of greater 

size. Consequently, the author recommends the use of dw rather 

than d84 , although there was no unanimity of opinion on this
MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B45

recommendation among his colleagues who reviewed this paper. 

The reader is free to make his own choice between the use of d^ 

and dw.

As an adjunct to this study an attempt was made to relate the

ti roughness parameter, -=7-, to the ratio of velocities at two-tenths K/Q

depth (V0 .2j/) and at eight-tenths depth (Vo.8v ~). These are the two 

depths at which velocity is usually observed when current-meter 

measurements of discharge are made. Twenty-seven sets of such 

velocity data, obtained at eight sites, were available from the 

discharge measurements made in connection with this study of 

the effect of streambed particle size. Unfortunately, the range in

/M

magnitude of the 27 values of -^77 was small, and therefore the RM

ti small standard error of estimate obtained by correlating -=7-, with K/&

the velocity ratio had little significance by itself. The correlation 

coefficient, 0.41, was low, and it was therefore concluded that on 

the basis of the available data the velocity ratio is an unsatisfactory index of the roughness parameter.
B46 STUDIES OF FLOW IN ALLUVIAL CHANNELS 

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MANNING COEFFICIENT FROM BED ROUGHNESS IN CHANNELS B47

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