An analysis of relationships between flow conditions and statistical measures of bed configurations in straight and curved alluvial channels

":b

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-...bwu,...

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t.~ ~o ..\4I"Cft ~

\rac.d..

~~~

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~\~:\I.YSIS OF r,ELATIOXSRI?S EE'2:\\'ESN F:'G'..! COXDITIOXS .\.'0 STATISTICAL ~fEASt;"R2S OF EED CO~FIGITl!..'l':ï:CXS

rx

STRAIGHT AI\'D CURVED ALLUVli2. CHA:;;~E!.S

/

by

lm .:-\s tr.~)':-.C::

s :. è:s.s~:-~atioi1 s ubcrictcd i:l p~::-ti.~l fulfill~2:- ..t:of che

:~çti.i::i:=en~s fo'!: tlle G2g::-ee 0= Docco r of :.J~iluso:lhy ~...~t...e... ~ .., .... d T" • .. • ._,.._

_. _lj .iJepz..rtr.12~t 01: r:c crian i.c s an dyûra\:.L:tcs in th~ Gr~~u~te Col12g~ of The Uni.vcxs Lcy of l-:J"';.:a .' June:...19ó3 i I

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The reliability of predictors for friction factors and rates or

scdi~ent transport in alluvial channels is open to question when they

are applied to sinuous ch armeLs', Bed+Eo rm geo:üctry in a curved

chan-nel and a straight flume which are subject to the same nominal flO"tv

conditions ~is investigated by ~tatistical analysis of records of

. _"'j1 .

stream"bed profiles .

_L=

_{_}

A~~~correlation, s.pe_ctral_density and p~oba~_ili~y

density_functions of a process defined by the bed elevation as a iunc-_

tion of the distance along the chaili~el,or as a function of elapsed

tiQe at a fixed point of the channel are computed by digital computer.

-:

Comparison of the statistical descriptors obtained from the curved

channel and from the straight flume permitf a quantitative evaluation

of the mar kad differences b etwe en bed geometry in curved and str-ai.ghr;

channe Ls •

.

,

"

The total rat e of sed iment trailsport in t he curvcd channe 1

is-approximately 15 times as much as that oii a straight flu..-newh ï.ch is

subject to nominally identical f Low conditions . This difference

in-creases "'vitaincrease in Froude numbe r , At the same time t he overall

cean vlater surface s Lope in the curved charme L is comparable to the

water surface slope in the straight flu~e.

~

It is shown that bed+f r LcrLon factors_I in alluvial bed s can be

cletermined either in terms of flow conditions or in terms of tne size

of the bed forT.ls. The statistical approach ~ibed in-th~e.:-;t:_

•

permits pract~cal and relatively sirnple methods to be used for

obtain-ing charactcristic heights and lengths of the bed foros in terms of

~

...'nemo._{... -}~en~s·o....~ the _{spec ra}t 1 d_ens~~y_~ _{rune ~on.}~ t'

These charactcristic

1 2

..

.

""

• 1 straiocht fluwe. ~~ ,~ I A': .'-:r~..

It '-is demon'strated that characteristic dImens Lons of the bed

. 'can be obtained from statïonary as weLl as from nonstationary

;..:.::T.lS

Ie records.

';.lOp

Comparison between time and space spectra permit?_:"evaluation of

-:'~,:

ri?ple celerity. The resulting re lationship shows t haf smal I ripples

faster than large ones and th at the celerity of ripples increases

!

.

v

_·

,_,,-:-

.

'v'

I

~ith increasing f Lowvelocity •. These results are confLrmed by results

.

_I

;o;tained from time-lapse photography and are suggested for use in

re-;lating time and space domains.

_/

)

.8

:phenomenon • Sugge s t Lcns for future s t udy ~ --1is teà---.'_

-Abstract approved:

,

..

\ I ! , dissertation supervisor Assist~t Professor

title and depart~ent

May

2, 1968

date .'---.

..

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AN Al\ALYSIS OF RELATIO~SHIPS BETI.JEEN FLCH CONDITIONS

I

_~" AND STATISTICAL_{L~STRAIGh"'T AND CUR\o'ED ALLL"'VIAL CHAI\T~lELS}NEASURES OF BED CO:NFIGURATIONS

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_{David Squarer}by

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f

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A dissertation suomitted in partial fulfHlment of the

requirements for the degree oi Doctor of Philosophy

in rhe Department of :fechanics and Hydraulics

" in the Graduate College of ::...

The'Univers ity of Iowa

s

"

"

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J\;ne, 1968

<,

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•

Chairman: Assistant Professor E~T.ett

M

.

O'Loughlin

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Graduate College The Ua Lvexsi.t'y of Towa Iowa City) Lowa

CERTIFICATE OF APPROVAL

PH.D

.

DISSERTATIO~

I

This is to certify that the Ph.D. dissertation of

David Squarer

with a major in Necharlics and Hyàraulics

has been ap proved by the ExaraLnLng Corr:.:nitteeas

satisfactory ior the'dissertatioa requirement for

Ph.D. degree at the'convocation of June, 1968.

rhe .

i

!

!

Dissertation CO~aittee:

i;_F~7V

f~

0

.z~

?-

~~

Chairman I S .,' Hember

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,j /"'1

./~

/"; •• 1:':._...

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ACKNcr.{LEDGHE~'TS

I

Thc writer ,,,ishes to express his gratitude to Professor Emmecr }f.

I

·O'!..o\;ghlin for his constant advice and for critically r-evLewLng rhe

I

t:..lnu5cript; to Professor J. R. Glover, whose help

in

solving rrumer oua

'\

instr~~entation problems made the study possible at all.

I

A~d stiSpecialmulatingthanksdiscuaressions,due toandNr.to

C.

ProfessorFarell for

J.

many usefulF. Kennedy for'suggestionshis·

I

at tentLon and encouragement.· T~e wrLter is also grateful to

~

rr.

F. A.

lLochcr for' some stLmuLat.Lng diseuss ions; to }fr. T. Pandit for ass

ist-I

ance in carrying out part of the measurements of bed configurutions; "

.

.

"

·

1

co Hre. S. Annambhotla for permitting use of part of his unpub1ished

.éata; and to Nr. Dale

C.

Harris and tho staff of the shop for the

con-I

struetion of the experimental equi?me:1t .

. Th anks are also due to the H.

-r-:.

Keek Laboratory of Hydraulics and

I

\'!ater Resources Divis ion of Eng Lneer Ing , California Institute of Tech-

s

I

nology for permitting the use of their "Dual Channe

--...

l

...

Stream Honitor".

Last but not least.deep appreeiation is due to my wife ior her

11

encouragement and support in every respect.

This work was 'vholly supported by the United States Department oi

I

tne Intcrior as authorized under the Hater Resources Act of 1964,

Public Law

83-379.

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t.C;OF TABLEST •

.

. .

LIST OF FIGURES

I

IST OF SYl-IBOLS

..

TABLE OF CO~"TENTS

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H

A

PTER

INTRODUCTION • • •

I.

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.

.

THEORY AND REVIE.ivOF THE LITERATURE

Il.

.

. .

General Considerations ••.••

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,6•.._L

A~..J.

Nomenclature for an Er~dible. Bed Geometry •

'~~~- Occurrence of Bed Conf1gurat1ons •••••

.( ·--;-I~I'... Resistance to Flow Over an Erodible Bed •••••

=_~(""'-""..Lc...

.

.

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lIL ANALYS IS OF THE PROBLEM

. . .

. .

.

.

Conclusions Drawn From Chapter 11

Effect of Meandering • • • •

Quantitative Definition of Bed Configurations

,Method of Analysis • • • • • • • • • • • • • • •

Particle Size .•••••••••••••••

Dimensional Analysis •.•••••••••

Incorporation of the Bend Loss Coefficient Into

Bed Friction Factor Computations

Application of Statistical Methods to the Field

of Sediment Transportation • • • • • • • • • •

General Nathematical Techniques in Spectral Analysis:

General Definitions; Nean; Autocovariance •

Stationarity and Ergodicity • • • • • • • •

Power Spectrum • • • • • • • • • • • • • •

Estimation of the Statistical Quantities

Theoretical Moaels for Power Spectrum and Auto- .

. corre lat ion • • • • • • • • • • • • • • • •

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IV.

_{EXPERïPffiNTALAPPARATUS AND PROCEDURES}

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_{. . . .}

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_{. .}

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Flumes

.

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.Instrumentation. • • •

Experimental Procedure

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...

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iv 1 '·t

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Page, vi vii x I

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4

5

6 11 20,

"

20 20 20 21 21 22 27 s

,

32

34

34

38

43 46

55

57

57

58

62

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~i'TER

V.

METHOD OF ANALYZI

N

G THE EXPERIMENTAL

.

RESULTS •

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PRESENTATION AND DISCUSSIO

N

OF RESULTS •

• •••

V •

Page

68

78

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Summary of the Presented Results

. • • • • • • •

Qualitative Discussion of Bed Geo

m

etry in Curved

Channel and in Straight Flume Based Upon Spectra

Probability Functions •••

Rate of Sediment Transport

•••

Water Surf

a

ce Slopes

• • • •

Characteristic Wave Height and Wave Leugth

Bed

Pr LetIon

Factor • • • • • • •

Celerity of Bed For

m

s • • • • • •

Theoretical Model for Spectra

78

.

'

82 86

87

89 90 97 100 101

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...

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VII.

CONCLUSIO

N

S

104.

I

LIST OF REFERENCES

TABLES

.

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. . . .

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.

113107

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FIGURES

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• • • • • • 139 s î

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LIST OF TABLES

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lc

_Page

1.

Resu1ts of Exper~ents

in Straight F1ume

. .

.

.

₁₁₄

2.

Resu1ts of Exper~ents

in Curved Channel

_{• 117}

3.

Results of Experiments in Curved Channel

Sections . • • • • • •

• • • •

.

.

. .

135

Transverse

.-,

• • • • • • -e

4.

Miscellaneous Results from Straight Flume Experiments

_{•• 136}

5.

Celerity of Ripples (F

=

0.25, d

=

0.48 ft.).

r

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.

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• 138 >. --I J

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Figure

8.

9.

1 LIST OF FIGURES Page

la. Ge ome try of Meandering Channel ₁₄₀

lb.

Photograph of Meandering Channel ₁₄₁

2.

Plot of Bed Elevation in Meandering Channel (F

=

0.35,

d

=

0.383 ft., r

=

29.92 ft.,

e

=

0 - 0.l43n) r ₁₄₂

3.

Photograph of Sampling- Station ₁₄₃

4.

_{Bed Shear Contours in Curved Channel (after C. L. Yen}

(30», and Location of Locally Stationary Sample Records

for F = 0.30, d

=

0.383 ft.

r .

Procedure for Analyzing a Single Sample Record

₇₀

144

5.

6.

_{Photographs of Sand lvaves in (a) Curved Channel,}

Fr

=

0.35, d = 0.383 ft. (Tef t ) and Fr

=

0.40,

d

=

0.383 ft. (right);· (b ) Straight Flume, F = 0.50,

d

=

0.383 ft. . r

)

145

7.

_{Photographs of Bed Topography in, (a) Curved Channel,}

Fr

=

0.35,-d = 0.383 ft.; (b ) Fl;:

=

0.40, d

=

0.383 ft.;

(c) Straight Flume, Fr = 0.35, d

=

0.48 ft. . 1~6

₁₄₇

(a) Comparison Becween Spectral Density Functions Computed

By Four D~fferent Spectral Windows; (b) Autocorrelation

Function in Straight Flume Computed from Space Sample

Record ~

" s

148

(a) Effect of Number of Class Intervals on Probability

Density Function; (b) Probability Distribution Function

in Straight Fl~~e Computed from Space Sample Record;

(c) Probability Density Function in Straight

FillineCom-puted from'S,ace S~~ple Record; (cl) Probability

Dis-tribution Function in Straight Flume Computed from

Space Sample Record; (e) Probability Density Function

in Straight Reach of Curved Channel C~mputed from

Space Sample Record; (f) Probabiiity Distribution

Function in Straight Reach of Curved Channel Computed

from Space Sample Record

149

150

151

vii

16. (a) Spectral Density Function in Straight Flume Computed from Space Sample Record; (b) Autocorrelation Function in Straight Flume Computed from Space Sample Record

17. (a) Spectral Density Function in Straight Reach of Curved Channel Computed from Space Sa~ple Record; (b) Auto-correlation Function in Straight Reach of Curved Channel Computed from Space Sample Record

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1 Page (a) Spectral Density Function in Straight Flume Computed

from Time Sample Record; (b) Autocorrelation Function in Straight Flume Computed From Time Sample Record

152

u

.

Spectral Density Function in Straight Flume Computed trom Space Sample Record Obtained Af ter Time Sample Record; (b) Autocorrelation Function in Straight

Flume Computed from Space Sample Record Obtained Af ter Time Sample Record

153

12

.

(a) Spectral Density Function in Straight Reach of Curved

Channel Computed from Time Sample Record; (b) Autocorrela-tion FuncAutocorrela-tion in Straight Reach of Curved Channel Com-puted from Time Sample Record

(a) Spectral Density Function in Straight Reach of Curved Channel Computed from Space Sample Record; (b) Auto-correlation Function in Stra Lght; Reach of Curved Ch anne L Computed from Space Sample Record.

155

154

13.

(a) Spectral Density Functions in Straight Flume Cornputed from Three Different Sample Records; (b) Autocorrelation Function in Straight Flume Computed from Three Different Sample Records ; . !

156

.,' l... _

15. (a) Spectral Density Function in Straight Flume Computed from Space Sample Record; (b) Autocorrelation Function in Straight Flume Computed from Space Sample Record

157

158

:

s

159

I

18. Curved Channel Computed(a) Spectral Density Functionfrom Space Samplein Straight Reach ofRecord; (b) Autocorrelation Function in Straight Reach of Curved Channel Computed fl-om Space Sample Record

19. (a) Spectral Density Function in Straight Reach of Curved Channel Computed from Space Sample Record; (b) Auto-correlation Function in Straight Reach of Curved Channel Computed from Space Sample Record

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160

161

viii

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2

0.

(a) Spectral Density Function in Straight Reach of Curved

Channel Computed from Space Sample Record; (b)

Auto-correlation Function in Straight Reach of Curved Channel

Computed from Space Sample Record.

Page

162

163

164

165

l6ó

167

168

169

170

171

172

173

I 1 ! ! j

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s

21.

(a) Spectral Density Function in Straig'lt Reach of Curved

Channel Computed from Space Sample Record; (b)

Auto-corre1ation Function in Straight Reach of Curved Channel

Computed from Space Sample Record

22.

(a) Spectra1 Density Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

23.

(a) S~ectral Density Functicn in Curved Channel"Computed

fro~ Space Sample Record; (b) Autocorrelation Function

in Curved Channel Computed from Space Sample Record

24.

(a) Spectral Density Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

25.

(a) Spectral Density Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

26.

(a) Spectral Density Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

27. (a) Spectra1 Density Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

28. (a) Spectral Density Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

29. (a) Spectra1 Dens~ty Function in Curved Channel Computed

from Space Sample Record; (b) Autocorrelation Function in

Curved Channel Computed from Space Sample Record

30. l/Jfi:versus rb/ox for straight Flume Data

31. Re1ation Between Peaks of Spectra of Time and Space Sample

Records and Effects of Flow Velocity on Ripple Celerity

...

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LIST OF SYMBOLS

I

=

constants

I

,- total bed area over which A is me asur'ed

s

=

the sum of upstream projected area of thè roughness elements

or the horizontal projection of the lee faces of the bed

I

'

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forms

lt." = wetted area of the_,I 'wall

"

=

constant ,

-

,

a

= channel width

l' = frequency band width

e

b = constant

t

~

'

~

,

:

·

_J

I

,

I

c.C

I,C"

=

cons tants

I

C(l,l+~)

=

autocovariance function for process with non-zero mean

!

;

!

i

I

r s

I

e

=

constant c

₌

concentration of sediment s

D representative particle size, usually equals D50

D.

_1.

₌

weighting function

_--...

d

₌

mean depth of flow

r: [

J

=

expected value or average value

e

₌

_{roughriess concentration}

..

r

₌

_{a functional relationship}

F(f)

₌

_{cumulative power spectrum}

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Fr:

u/lid

= Froude number

I

x

I

I

_·t

I

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1 ...

_ Darcy-Heisbach friction factorj' frequency or wave number

_ Darcy-l-leisbachfriction factor associated with the channel

bed

a Darcy-Weisbach gráin roughness friction factor

=

Darcy-Weisbach form drag friction factor

=

equivalent 10ss coeffieient for a bend

t

.

1/~

=

Nyquist frequeney

r.

I

t

=

Darcy-Heisbach friction factor associated with the wa11

-ti

C (f)

=

physica11y rea1izab1e one sided power spectra! density

. x function

I

ë

(f,A) = an average va1ue of G (f,l) for nonstationary process

x

x

I

I

s

,S{>:) h

·

1

h

I

I

h"

b

I

h lot

I

j=êl d

I

I

Kil

'b

ie f

I

I

= "raw" es t imate of G (f) x

=

acce1eration due to gravity

= any rea1 sing1e-va1ued continuous function of x(l)

=

bed form height or height of roughness e1ements

= mean wave height

= head 10ss associated with fb

.= he ad 1055 associated with fl

b

j

j

= head 10ss associated with fb

=

head 10ss per bend

s

= head 10ss associated with f

w

= dimension1ess lag distance

=·number of c1ass intervals

. 1 .. 1 1 1 ff . f f

an equ~va ent oca oss eoe ~eient or b

=

an equivalent localloss coefficient for f'

'b

= an equivalent localloss coefIicient for f"

b

=

an equivalent localloss coefficient for f

I

I

I

I

I

I

I

I

I

I

,

I

I

I

I

1

..

=

an'equivalent localloss coefficient for fL

=

an equivalent localloss coefficient for f

w

k = wave number

=

equivalent sand roughness

..

..

L = wave 1ength or bed form leng th

1

=

distance along a channel

leer_{.e e}.

e

=

length of a bend along the center line

M

=

maximum number of lags

Nom.

= ith moment of the spectral density function

1

N

=

slope of the side walls or total number of data points or

total number of sample functions

N.

1.

=

number of data points which fall inside each of the K c Las s

intervals

= _{number of zero crossings per unit 1ength}

.r

l

..

.

~~...

= _{number of degrees of freedom}

N O. n

P(x)

=

cumulative probability distribution function

Pr ob [

]

probability

p(x) = probability density function

Pw

=

wetted perimeter of the wall

Qs

=

total sed iment discharge

~

R = hydraulic radius

s

'

1

~

or rb = hydraulic radius associated with the bed

I

I

I

I

Reynolds number . ~

R

_x(À,l)

=

Autocorrelation function

₌

_{Autocovariance function when ~ =0} -'.

x

r _{= radial coordinate}

r

c

=

radius of bend center 1ine

r

P

=

radius of a pipe

I

,.

J • 0x

T

e

e

n U

u

_c u n

I

I

I

1 .)

..

..

-

slope of energy 1ine

• encrgy slope associated with grain roughness

- energy slopc associated with form roughness

J ({)-dF(f)/df = two sided power spectra1 density function

•

I

I

I

I

I

·

1

\I X ~

,.

x

'"

_~x

(x)

. Yo z(l) Cl ~

r

r

_s 6. Ö f,(f-f)

I

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I

..estimate of the standard deviation

._ length of the tangent reach of the channel or wave period

or temperature

.. time

= student t distriH~tion with n degrees of freedom

::mean flow velocity

c critical velocity for incipient mot ion

.l

.. set of original discrete .sample record with non-zero mean

..V't'/p

=

VgRS =: shear velocity

=

fall velocity of partieles

ee bed elevation

=

estimator to the mean

~ stochastic process

=

normal depth of flow s

c standarized random variabie

.. constant or level of significance

- constant

.. specific weight of water

.. specific we~ht of sediment

=

measure of spectrum width

m·lag distance between 10cal transport rate and local velocity.

=: Dirac delta function

I

l'

c

I

,

1

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• normalized standard error

_ function of the wave numher

- angular coordinate in direction of mean flow

_ 90° - central angie of bend

e

c total length of sample record

.,.lag distanee

.,.spaeing between suecessive data points

adynamie viseosity

c mean value (or expected value or ave rage value)

=

kinematie viseosity

_.~

=

mass-density of fluid

=

mass-density of sediment

= geometrie standard deviation of particle size

=

standard devintion of wave height

.= varianee of x(l)

..

.,.shear stress

= critical shear stress

=

a funetional relationship or a true value of Bny function

s

=

est~ator to the function ~

=

ehi-Square variabie with n degrees of freedom

=

mean square value of x(l)

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CHAPTE RI. TNTRODUCl' TON

•

the problcm of undc r standLng the mccb an Lcs of flow over ailuvial

bed. has long attracted the'nttentlon of lU:llly i.nvestigators. ",ho have

tried to attack it in many different ways.

Of great concern h.:ls been thc pr.ohlclll of "t"\?sistancc to flow over al1uvial beds, which is closely r e l nto.I to thc dctorrni.nntLon of the

't~ge-discharge relationship id' a rivcr.

In spite of the vast knov l edge gn i ned during the last decade from

intens ive research on the mec han ics of f l ow over movabLe b cd s , a

defi-n(te solution to this complicated problcm is not yet available.

,

It is interesting to note thnt Du Hu?t

(1734-1809),

two centuries

ago had listed many of the unknown Ltems whLch at that time needed

further study, and as of today are a lrnost entirely evaluated. In a

s!milar manner Rouse and Ince

(1)

have LncLudcd among four typical,ly

hydraulic problems which are little subject to solution by me.:lsurements

alone, "the statistical ev a Luat;ion of sur f ac e roughness and its c f fcc t

Upon houndary resistance."

It is perhaps this same concept, "stati.stical evaluation" ",hich

has beep the stwnbling block to ste ady pr ogres s nncl to more pr cc Lse

•

evalu-'ltion of pr ob lems categor Lzcd un.Ie r t.he ge~era 1 subject of "s cd i-ment tr<'lnsportation." Gener:tl1y sp~.,kinr,. any oh;.erved data

rcpresent-lng a physicnl phenomenon cnn be hro:lJly cl.,ssj[i~d as heing eithcr

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1 relationship - or nondeterministic. A rigid body suspended

."--,,ciea

...

~

...

-f~ed foundation by a linear spring, for example, produces

physi-rr:tfl

.

4 .

4.1 doStawhieh are determinist ie. However, there are many other phys

i-~1 phcnomena which produce nondeterministic data. Such are, for

exam-_ electrical output of a noise generator, the height of waves in a

t·'.

! cd sea, various meteorological phen.omena, etc. Nondeterministie

(.~ us

l~t4 are random in character and must be described in terms of

proba-11

tt

y

statements and statistical averages rather than by explicit

c~u3tions. The decision as to whether or not physical data are

deter-cwistic or random is usually based upon one's ability to reproduce the

d4ta by controlled exper~ents.

A glance through the vast and invaluable amount of data collected

fronl experiments involving flow over alluvial beds during the past :

century, ean lead to thè conelusion that some aspects of the process, such as local bed elevation, are indeed nondeterministic, and henee should be measured and analyzed as random data.

In trying to Improve existing predictors involved wLt h fLow over

covable beds it was the writer's belief that perhaps the statistical approach could better and more accurately define the problem.

_--

_.

_~

Con-seqüently this approach has been adopted.in designing, performing and analyzing the experiments presented in this report. The material pre_ -sented in this study involves a detailed

..

description of the statistical

..

method of analysis, which has been used to dèscribe the bed eonfig-urations. At the same time, the physieal reasoning supporting the need for applying statistical rnethcds is elaborated. Needless to say, such an approach could not be adopted, say, twentr years ago when

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modern high precision equipment such as high speed computers we re not

fully incorporated in various aspects of applied research.

The v~rious motives which led to attacking the problem from the

statistical point of view are described in Chapters 11 and 111. Also.

described in these chapters are the already known predictors suggested

by other researchers, some of which are in current use. In order to

~valuate the reliability of existing expressions for resistance

coeffi-cients and sediment discharge in alluvial channels, this study involves

experiments which were conducted simultaneously in a straight laboratory

flume and in a laboratory meandering channel.

This study by no means completes the task of yielding definite

and final formulations for resistance coefficients and sedinent

dis-charge in alluvial streams. However , hopefully by applying techniques

similar to those described herein, especially in the early st~ges of

designing the data acquisition procedure, the complicated problem of

flow over alluvial beds will eventually be solved.

:_o

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:-:_-.::_::._<;:..

CHAPTER 11. THEORY AXD REVIEW OF.THE LITERATURE

-

-

-

_

-

-

-

-__ . - ._-__: -c_:-: --r.":.

_-

_- --.

-

-_-'"

- ".IC'.

-pO_:"

.··..GeneraColnsiderations· :.

Various.aspec_ts_.~~~h.e problem of ·flow o_yer movab Le beds have

en attacked by numerous ..investigators -e - OnLy Lntoe last-twodecades'

vc a f~f researchers_{~--- - -}succeeded_{-- .} ._-'.in predicting_{_. - -} stage-discharge reIa~

ons, tqe.hest known of all beipg the Eins~ei~-Barbarossa bar

resist-ce curv~ê_published in .1950 (2). .The considerable_.~ uncertainties and·

'pers al1Ècliscus s ions •

Not until l~5S_,.\vhen~rook~.J3) cont r Ibut ed to the understanding. '.'

_;.

the mec~a~~cs of ~ treams wi~~ movab Ie bed~,.could any significant '.

short review à~ p~~~:Ln~n~ studie~~\l!i!L.be~m~qe 'Vlit}:l.theaim of

point-out the necessity for applying statistical methods as areasonabie

1 for approaching the problem.

In contrast to an invariant roughness in a channel with a fixed

d. the roughness in a channel with an erodible bed is extremely'

iable due to the format ion of different types of bed features on

<,

erodible channel bed. Furthermore, a close interaction exists

..

,t'.:eethe erodiblen channel bed ~:ld the flow over it, and vice versa.

The fact that the friction factor depends on the geometry of the

fOrtrQ generated on the channel's bed, while the latter depends in

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factor, furthcr complicates the problem. Other major complexities

involved are: the chance of having a suspension in the water, which is

otherwise treated as clear; the plan geometry of the river, which is

closely related to the secondary currents; an a Ll.uvLaI bed wh Lch is

posslbly cohesive; and the'deterillinationof the independent variables.

The last factor has been clarified by Brooks (3) and has gerierally

been accepted by other investigators.

Nomenclature for an Erodible Bed Geometry

Unfortunately, discussions related to the general subject 'of

sedi-ment transportation should as yet includ~ definitions of the different

terms used. If such definitions are absent, one might obviously be

confused as to what; the autho;rmeant by writing "ripples»" "dunes,"

l

"bars," etc. It is'particularly important to use in this report a

ciear and familiar nomenclature for two reasons. Firstly, most of the

known bed configurations have been observed in various phases of this

study. Secondly, it is the primary interest of this study to describe

quantitatively the various bed configurations wh Lch until now had meceIy

descriptive name s , This in turn will hopefully allow for a bet ter f

orm-ulation of the friction factor.

To avoid any unnecessary confusion, the nomenclature describêà by

the Task Force on Bed Forms in Alluvial Channels of the Co~~ittee on

Sedimentation (4), has be~ adopted.

An

endeavour is made to name

each of the observed bed forms in accordance with the detailed

-~":~Occurrence of- Bed Configurations

rne-flume experiments~whieh have-been performed in connection with pr~sent s tudy , and which _are described in detail in Chapt ez s IV and

I

•t are rather limited in their nature. 'Obv.Lous'- --, -,Ly ,-,.norc_- -all'"- rhe- - -independ-, __

Ggt vatiables could be varied over the entire range of interest, in the

,1i;:tite(l,period of time over w_hich t.he-study has been conduct:ed.

Con-,,cqucntly, ,?_n~~_9ne

_

~i.~~

_

.9t

,be,d_se~~mel'!_~has bee~ used in ~oth the

$::raight flume and the meandering channel. 'l\vq di~,~~rent_ ~~pt:~_s of

, .

-{ter..r were studied in the'~s1:-räi-grï~f Lumë-atn:l":-only'-one-in~-t~~__

~':lrve~~~n-"

W,ith,~_coIls~a_nt depth of flow and various discharges , 'differen_!:,

..

.

--.~ -::~-.:-_-

-

-

-"'roude numbe r s have been stud ied .

_: - -!- - - - .

studied in the straight flume and only three in the meandering ~ha?~~l. In order to relate the nar r ow range of 'exper Lments"reported in- ~

. . .~.

study to the more genera 1s ituat ion, it is pèrt inent to zev Lew

I

- - _-

-rhat whLch is already known from other studies- 'conc'ernirig-'the-'occurrence

....:...-

-

-

-

_

of bed forms.

-

-

-'-

-- - ~- ~- . _.

The variety of bed conf Igur arLons referre-d to 'iil"':the

I

one shoul.d be ab Le to c1assify bed forms.-by-inü-:(cating "certafn ïimits"

. , .

_----

-

-

~- - r

ThLs has been the sûbject -of many studies-, --the,',

~

-

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of flOtv conditions •

-

-

--

-

-

-

-

-

---

-

--

..

-~

-

.; .

~ost prominent of which are those of Simpns and Richardson (5) and (10),

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ii~

ê

Ó)

-

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:

'Alb~~t~~~~ S{mons"arid -Riëhárdson -(8),

~og<l.rdi (9), and Kennedy (U).

I

It is weU és t ab Lf.shed by no_:;"that -the -~cèurr~n~è of bed -fîorms

;>rogresses in the fo l Lowfng ~rder whê~

~hë

.;

veï6èlt·y-

6f

flow is- gradually

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i:lcreased: In th~ "1ower fl~~v'regunei'- af ter rhe initiation of partiele ripples, dunes having ripples superimposed on their upstream

I

sLopes (or bars in a meandering channeLj , and dune s \vill occur consecu-tivcly. Thcreafter the "transition regime" \..rillprevail. With an in-creas e in the velocity of f l.ow to the "upper f Lot...regime", flat bed

and antidunes will occur in sequence. A shift from one bed

configura-tion to another, using the sarr.efluid, at the same temperature, the

same bed material, and the same depth of flot..r,requires a change in t1:e

Froude number of the flow.

In order to mark the limiting flow conditions for a certain bed

configuration, coilected data have been plotted two dimensionally by

different researchers. Liu (7), for cxample, has used a

llr

/w versus

~D/y

plot, where u~

=0r7P

= !gRS, and D

=

representative particle

sizc,

V

= kinematic viscosity of the fluid, w

=

particle fall velocity,

~= shear stress on the bed, R

=

hydraulic radius, g

=

acceleration of

_

;f.

gravity, S = slope of energy line, P = dcnsity of the'fluid. It is

H

l

interes ting to note that (u,:!w)2is proportLonaL to -C-/6'1D, where 6'1 "

.~_.,

':

'

1

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'I

=

"'s - gp is the difference be twe en the specLf'Lc weight of the

!

partieles and that of the fluid. Thus , plotting

'l7:

hv

versus

urDI)J

aüounts to plotting ~'l6"'Dversus

UrD1V'

which is none other than the

weIl k~~~n Shields diagram (12). Albertson, Simons and Richardson (8).

-

...._

have extended Liuls work by using different partiele sizes and differeat

Froude numbers and incorporating the friction factor in their final

re-sults.

..

Of particular interest is the observation cade by Albertson et al.

,..,

.

that for D

<

1.7

mmo

all bed configurations may be formed, depending

on the fLow conditions. If the particle size D is greater than 1.7 mra.,

..

>

5.0 mm.., dune s cannot be forrned, and the only possible bed forms

this part ic Ie s ize are 11trans ition11 and 11antidunes.11

Classification of bed configurations on a~/67D versus

F

plot,

r

\.-rc F =

u/lid

is the Froude number , d

=

mean dept.h of fLow and

v ••" r

:'..mean f Low velocity has been attempted by Garde and Albertson (6).

~~cir observations could be divided approximately into three main

:-.:r.ions,0.1

<

Fr

<

0.4 to 0.6, 0.4 to 0.6

<

Fr ~ 1..0 to 1.5 and

i.

o

to 1.5

<

Fr

<

4.0. In the first reg ion bed configurations consisted

Co

:

ripples and dunes. In the second , the transition regLon was identi

-ficd, and in the third only antidunes formed. The authors have

fur-:.!1.::rspeculated that the transition be twaen ripples and dune s is as we Tl,

rbe transition from viscosity effects beLng of major importance and g

rav-itational effects of minor importance, to viscosity effects being of

.e.

~inor importance and gravitational effects of major importance.

Bogardi (9), in an interesting study, distinguished betHeen the var

-lOUS bed configurations by plotting flume data on a plot or gD/u~2 versus

D

.

By using the particle size diameter as an independent variabie, he

cade it immediately apparent that the reported results are in'complete

ol~~eementwith those of A1bertson et al. (8). That is,

~

whi.l,e D :::::

D ~ 1.7 mmo is

toe upper limit for the formation of ripples, _{5.0 mmo} is the

U??er limit for the formation of dunes. Bogardi has gone a step fur

-:~èr by incórporating in his plot the suspended sediment concentration

..

:U~ction developed by Laursen (13). This presumably shoulà permit one

teevaluate the sedi~ent load associated with a particular bed

config-U~é.ition,once the flow condition and the partiele size are known ,

SL~ons and Richardson (10) also have chosen the partiele size

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to be the Lndependcnr parameter, wh i Le the stream powe r , (:'U was chosen

---

~

--

--

~--

-as the other variabie. Knmvledge of the f Low conditions and the

partic-ular size permits the determination of bed configurations.

A somewhat; different approach to the field of sediment

transporta-tion as a whoIe \>1asundertaken by Kennedy (11). Kennedy has studied

the mechanLcs of dunes and antidunes by means of potential flm., theory.

I

The bed configuration was assumed sinusoidal, on account of the

separa-tion pocket wh Lch occurs usually in the lee of a ripple or a dune , and

'the potential flow over such a bed geometry was solved.

Kennedy was

first to

(

introduce the concept of the so called lag distance 8, i.e., th~~

)

,

distance by which the local sedLüen~ transport rate lags the local

veloc-ity at the bed. The lag distance concept was introduced to account for

the real fluid phenomena wh L'Le still being consistent and compatible

with the potential analysis.

_By

_{using the}

lag distance as one of the

independent va4iables, an expression for the local sediment transport

could be estaqlished. _{This in turn allmved the celerity of the bed}

'features and their amplitudes to be predicted.

The occurrence of

dif-ferent configurations were determined by Kennedy from his theoretical

solution, in terms of byo independent variables j

=

8/d and

F •

---.---- --- -- ---- __ -_~-- _'. _ .r_

Hore-Froude number was expresseà as a function of d. 0, and the wave

..u~ber k = 2~/L, in ~hich L is the wave length. _{Upon considering the}

;,>articular case for wh.i.chJ)

«

à. or j - 0, which obvious ly should be

re?resentative of most practical flow phenomena, one ean compare the

~neoretical study wLth experimental r esu Lts on a plot, of F versus kd ,

, " r

'Lnso dotng , good agreement be twaan the theory and results of various

e:-:;;criments was found by Kennedy for F

>'

0.8. that is when bed

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ç·C"urations are antiduncs. Hot...ever, for 0.2

<

F

<

0.8 a

consider-~~.lo r

I

=

2

~

d/L

for

F

<

0.5.

r

.Io)lc scatter of the data around the theoretical curve given by j

=

0

,,'.1& observcd. The scatter for F_r

<

0.5, i.e., when ripples and dunes

I

.-_r·_·...Vo'l:L·1, is very sioonificant and may lead one to the conc1usion that is no dependence of Froude number on kd

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~ attcmpt to improve Kennedy's theory was madeby Reynolds (14).

~yr.o1ds has introduced the continuity· and momenturu.equations including

I

i r~sistance coefficient, instead of the velocity potential used by

I

tc:nncdy, and which obviously could not have been associated with any

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rherc:sistance coefficient.Lr o....n merit, they do not representAlthough Reynolds'any significantresults aremodificationinteresting forto .

.

1

i\ennedy's result from the practical point of view, and as Reynolds

hirn-sel f concluded "further analysis of bed waves bas ed on' one dLmeris Lona L

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-hydr'au Li.c s is unlikely to be fruî.tful." It is interesting to note that

"

I

_e~~_vanneàyhas gone a step_{Iuat;e the ratio} h/L of the bed fofurther with rms, where hhis results

=

and appliedheight of them tothe bed

~OLm (15). The ratio h/L may be written aft er some simplifications and

·

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1

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_{;;SSU:ili)tionsas}

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.,

~.,which F .is a functional relationship and U is the critical velocity

c

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:or incipient motion.

..

This reViet'1 of the occurrence of bed configurations t...ill be

con-I

~ludeà with the introduction of the concept of areal concentration of

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,

roughness elements e, th at is, the number of individual roughness

~4eNents per unit area. This roughness co~centration should be

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m_a..::Y_b_e__e_n_v_i_S_~_"_o_n_e_d_a_s__t_h_e_u_t_i_l_i_z_a_t_i_o_n_o_f_t_'_o_e_b_e_d__c_o_n_f_~...;·g:::....u_r_a_t_~_"_o_n_s__e_x...;p:...e_r_i_-· o'_

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m~ntioned in this respect, because it is a necessary geometric parameter

whLch is required to describe correct1y the considered rough surface.

Indeed, this fact was recognized by Sch1ichting (16), who ascertained

that the usua1 types of roughness cou1d not be geometrically described

by a single dimension such as a roughness-element height, but th at a

roughness density must a1so be specified., Obviously, in"the case of

an erodible bed, the roughness elements are the various bed forms which

are generated on the bed.

"Resistance to Flmv Over an Erodible Bed

Generally speaking, the desired stage-discharge relationship must

be based upon know l.edge of the friction factor of the f Low •

The ultimate goal of this phase of the study is not a final

determina-,I

tion of the stage-discharge re1ationship, because of the lack or a

broad spectrlli~of data. Yet, the study has been conducted in this

direction in order to help o ther s predict such arelation. The

ulti-mate purpose of this study from the practical application point of vie,'l

enced in the experiments described in Chapters IV, V, and VI for

evalu-ating toe res istance to f1m". It is now evident that a short revie"l

of previous works concerning friction factors for flmv in sand-bed

chan-néls is in order. A brief review will be made only of that research

•

wh i.ch..lill be referred to in a later stage of this report. These are

zhe worles by Tay Lor' and Brooks (22), 0 'Loughlin and MacDonald (17),

Rouse (18), Hwang (19), Vanoni and ~~ang (20), and Alam, Cheyer and

tive roughness. Rence

fb

can be determined fr om Equations 2~ 3, 4a and

11

4b. Instead of

USL'b

~~e above analysis ior evaluating the friction

11

11

11

11

11

11

11

·

11

"

11

11

.

11

·

11

11

11

11

·

·

11

12

Aftcr BrooksI find ing that the bed frict"ion factor, fT, is a

o

uniquc function when expressed in terms of velocity and depth of flo:.]

as the independent variables , a vast amount of reliable data were

collected, analyzed, and reported by many researchers.

A very interesting approach wh Lch is susceptible to an im:nediate

practical application was described by Taylor and Broaks. _{Consider the}

Darcy-t{eisbach bed friction factor

.

d~;

é=

\

J

L

.

\

8gR.S

o (1)

in wh Lch ~ is the hydraulic rad ius associated w:ith the bed and t he

corresponding friction factor related to the grain resistance is

f.' =

b (2)

in which SI is the energy sLope associated \vitn grain r'oughne s c ,

If

f"

=

b (3)

in whi.ch fb mld S" are t he Darcy-Heisbacn friction factor and the s Lope

associated wï.t h the bed configurations respectively, then

..

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S

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S'

+ S"

_(4a) fb

=

f'

+

f:'

8g~S

-

...

_(4b) b 0 U2 -,

the value of

fb

may be estLmated from a gr aph of rhe Nfku ads e data

la-13

..

factor associated with the bed configurations, Taylor and Brooks (22)

suggcstcd the usc of the critcrion

..'

s

S' f

,

s

tr=-s

(5)

.

,~,I •

=

_f'

In their analysis, they used the following definitions:

f'

=

8gRS' U2 .~;~

f"

₌

8gRS"

₍₆₎ / U2

f

₌

f'

_{+ f"}

I

r,

.

.

.

e, -:

s

= S'

+

S"

in which R is the hydraulic radius.

!

1

Equation 5 may be visualized as a s ImpLe me asur e of the energy

dissipation arising out of the bed irregularities in an alluvial s~re~~

as related to that expected in clear water flow over a flat bed.

Fur-rhermore, a.ccording to the authors, f/f' might be a us efuI descripti.on

of the bed configuration, that is, f/f!_,~ I for a flat bed, and

tt

z:

>

1

for other bed configurations. In the situation where slispended load

alters the tur~ulence structure, f/f' may have values less than one.

The aut hors have, presented a graph of U versus d in wh Lch f/f' is a

third parameter, and propos cd to use this para~eter for the dual

pu~-poses of clarifying terminology and predicting the resistancc_

I

..

)

•

)

-

1

;scussed many aspeets of th is eo~p1ieated problcm. Of partieular

,rtsnce to this study is his analysis of rough boundaries in tcrms

t

~

?

~

-_

I

.,

.

: _~·ou~

_.,

hncss eoneentration _-- e,_- and the 10ss coefficient assoeiatcd with

I

._.'_:;,;;_{th~ Iowa};me1bends._InstituteBoth of these_{of Hydrau1ics}pr obLems_Researchhave been eontinuously_{(IIHR) during the} studied_{past avo}

I

The last of the investigations on the r ougjmes s conc entr-at.Lon

I

~::.:'!cts has been presentedpointed out, for the present,by OlLough1inth~_~at_ureand MaeDonald (17).

=

.

the .rougbnes s _As Rouseeffects

I

'!.s still_best seen in terras of _the_ s Lze , sha~e _ar~~s?~c~ng_o_f_c3:_rtificial.

/

I)~óhness eie:nents. Using the expe r Irnerrta I resu1ts of Seh1ichting (16)

I

~.<!~ other invest igations performed at the IIHR, Rous e expressed the

.

.

----

_

-

-

I

rietion fac-tor for fully daveLoped roughness of he Lght;-r~_ by

~

I

1

= A log Alhe •

R

C

!f

for 0

<

e

<

0.15 (7a)

:'Ivh ich A and Care constants ref1ceting the shape of the cross

sec-J

.

~()n, and A1 is a fun'ction.' of both -the shape and the arrangement of the

-

·1

.)~ghness e lements, the latter effect being negligib1e at; Low

eoncen-:.!tions. Equation To. ean be written as

1

R

=

A

log he

+

Vi

Cl _(7b)

11

-

~hieh Cl is a constant for a given type and pattern of roughness

I

_{Equation 7 is based}

..

_{on the assumption that the equivalent roughnes_~_}

I

varies in direet proportion to the concentration when e

<

0.15.

I

,-= high_concent.ra t Lons , sir.1ilar e quat Lons cou Ld be derived on1y for

- roughness c Lements (spheres) studied by Sch Lf.chc Lng (16) and for

I

I

I

15

s and grains studied by O'Loughlin and HacDonald (17), viz.

-.

-:-1 = A 1 R

+

C"

. og h(l-e)

Vi

for 0.3

<

e

<

1 (8)

in whi.ch C" is a constant for a given type and pattern of roughness

elements.

It should be pointed out that

t~

~

as.~umption of_~ir~~t proporti

on-~1~_~y_È~_tFe~!!___t__resist.<ill1e _t_l~~_a~~_E~,:oncentration_~._ _, or__~~~~l::_'_l__~s~~L_~nd_

e, where ks is the equivalent sand zoughnes s , Ls_

~0t:

_

__

al:!a~s__va l Ld , and

should be verified-in every case. Indeed, it may even happen tnat k

In

s

wouLd depend upon the relative roughness y Ih as weLl., in whLch y =

0_0

normal de pth of fLow • In th is case t.he usefu1ness of the equfva l.ent

roughness concept is greatly diminished. This point has been discusscd

by the wr it.er in more detail eLsewhere (23).

-Because this study is concerned with meandering as weIl as wLth

straight channels, it is pertinent to mentioil in th is respect toe 10ss

-

_

. . ---.

-_

çoe!_ijC!_ieIlt_assoçLaced .with a bend. Rouse (18) and Hayat (29)" -in one

of -a fe\v studies of this nature, presented the los_s_coefficient_of_ one

of_~__êe_ri~~__Qi_21~~-eroèible channel bends. In their study t_h~_lo~s_

coe ffIc Lent; was found to depend upon the Fzoude number and the

depth-width ratio d/B.

An

interesting point stressed by Rouse (18), is the

fact that the loss coefficient of a bend for subcritical fLow depends

Upon the Froude numb!r. Consequently, in applying conclusions derived

fr om one bend (in the laboratory) to another (prototype), both changes

in çross sectLon (d/B) and the Froude number should he considered.

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I

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I

I

..._!,;:i:1gand its resu1ts are sOffie\vhat·siraLl.ar in character to .those

-

...

.

:..:::~,;":>y Vûnoni and Hwang (20). The fina1 result given by.~lam~ al.

... c:....c farm

ló

(9)

.' .;.;rivation of this re1ationship is based on dimensional cons Lder a

-I· ...· , 1 1 h

h ~ ,2 nd ,....,1 then

$ufficient Y arge suc t at l.ann.--r:-""" ,

. and on the follOiving intermed Late relations: If 21rd/L

>

2.70

- ..-- ..>,

h.'

- =

_L (10)

...::.~.):-.10 is based on Kennedy' s t.hco retLcaL study (11) and (15) and

•:.':<: i.:-:vûlves all the as s umpt.Lons made therein.

from the definition of the friction fact~r, it folloHs'that

I __ 1 -.'

'

.s

V

re

e.

-

\

~(ll) TI c

~l

.

ft

vr;

_p

rt-e_

s: ~ LA.::: C-~ .» (12) ,

,

j _j .' 1

.

-j

I ~.

.. ;-..;:ly rough bo

.

undar Les , AléU"11et

_-

al. have assumed

I

~e noted that the important para~eter e (rough~ess

..:.:' .~s~i.ssing f r omEquation 12 for unknown reasons •

'Che re1ation

..

concentr

a-'t" =c.

D c .l I ~ r r_;.

-

,

.- r

t

(13)

..as been propos ed by the ASCETask CO:nrö'.itteeon Preparation of a

*

.

"'

,

--',·-=;::.::.:ion~{anual (24), one obtains fro;n.E~~ations ll, 12, and 13

I

'ji 1 I 1

1

I

!

I

I

I

17

(14)

Substitution of Equation 14 into Equation 10 yields,

(15)

Equation 9 may then be obtained from dimensional analysis co

nsidera-tion by making use of Equation 15 and assuming

2 ;' L

=

21(] g~ (ló) or otherwise, (17)

-... ',.

It is also assumed as an initial condition in the above derivation,

that bed undulations are present, hence one canr.ot use·this result to

predict friction factor for the flat bed regime. The final result has

been presented in the fOLm of a plot, b~sed on many àata~ in which

fb

is t~e 'o~~inate and ~ the abscis sa. Also appearing on the plot are

U U

curves of constant

r:gD

and constant Vg~ • The Alam et al. result is

araeriabLe to immediate practical application on condï.t Lon , howevez , thaz

one keeps in mind all the assumptions made in its derivation.

•

'._i .The analysis prese~ted by Vanoni and Thvang (19), (20) is very

sinilar to the analysis made by Rouse for fixed roughness elereents (18).

Indecd, thc final result of Vanoni and Th~ang is presenteà exactly in

thc fo~ of equation (7). However, it has indeed been applicd first

I

I

I

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I

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I

·

1

I

I

I

I

I

I

I

..

t

₁

3

f:

[

:

"

I·

ï

·

I'

t

··

I I·

,

.

r

_i

t

I· i. ! I· I t I I I· I

!"

t: i

:

,

t: f· t·

I

I

.

i

i

r

they chemically stabilized a ccrtain' bed configuration cr~'::'ê:::'~l

\

ccrtain fLow, conditions , over whi ch meas ureraent s of ve l.cci ty é.::ë ~-::1~'__ sure distributions as weTl as friction factors were made. Ass'---:::2.:.;~~.

isolatcd-roughness fLow, that is, no interference be tween 5UCê:SS::"'2 :.=;.::

....

irregularities, the roughness co?centration e is expresseè ~y ~hG authors according to Schlichting's definition as the ratio of thë: ~~~

of the upstream projected areas of the roughness el~ments to tte :lo~~

area

A

,

e

=

A s

A r: (18j

Because the lee faces of ripples and dunes stand approxL~ate:l ~_

the ang Le of repose of the bed sedLment....vhich varies Lî tt.Le, the ~-::C=~

A was taken by the authors as the horizontal projection of the 1(;(:

s

,

,. ~ r i

,

.

faces of the ripples or dunes. This ia turn was evaluated by dLrvc t,

photographic measurement. For hydrodynamically rough beds, wh i.ch usu -ally occur in most practical cases of fLow over an erodLb Le bed,

friction factor has been expressed l:>y_Yanoniand. lk-langas

f" == f" (~

A

s

\ ::::

f.b"(_h~ e)

b bh'

A/

\!

(

19)

Incorporating the relation be tween k /h and e,

à

s-

presented by Rc,:.;:;e s

(

18

),

Equation

19

yields

f

" ::::

f"

b b

..

(0

.

(2

ei

.

By as sumfng. k s (21) h J '.

19

..

..

ror 10\-1values of e (0

<

e

<

0.15), Equatiori 20 may be written

f"

=

f" (~\

_p

~

h~)

(22)

It should be kept in mind that the limitations of Equation 22 involve

all the restrietions imposed on Equation 21, some of whieh have a1ready

been mentioned. Expressing the f=ietion factor in terms of Equation 22

or Equations 7 and 8 gives much less scattering then is usually

encoun-tcrcd in studies of this nature, and henee appears to be quite

prom-ising.

A revic\-1of the pzob Lem of the resistance to f Low over an erociible

bed shou Ld not be concluded \áthout at least mentióning the name s of ~

I

Einstein anà Barbarossa (2), Engelund (25) and (26), and Garde and

Raju

(2

7

)

, (

28

),

all of which have eontributed to the progress made

in this complicated problem.

"

-

I

,

-

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--

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--

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1

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20

CHAPTER 111. A.~ALYSIS OF THE PROBLEt1

Conclusions Dra~-m FrornCharter 11

The g_eneral outlines of the present studr weze decided upon after

a caref~l examination of the material presented in èhapter 11. It is

thus proper to point out the conclusions drawn from Chapter 11, and

w"hich"\Vereused as guidelines in pursuing the study.

Effect of Meandèring

Very little is lmmvn as of today about; the plan geometry ef f ects

-uponthé resistancè to fLow over erodible beds. Because all ex i.st Lng

~-expressions for friction factors do not account for pla~ geo~etry

effects,-ït·hàs been suggested that a cOr:lparativestudy be conducted

in straight and meandering channeLs unde r the same nomf.riLa f Lo» con

-ditions • It is bel'ieved that an Lndirect evaluation of meandering

e~fects upon the friction factor may be assessed through comparative

measurem~nts of both the total sediment discharge and the bed-fom can

--

...~

-figurations, which deter;nine the resistance to flow.

Oua~titative Definition of Bed Configuratians

Ide:1tif:lcationof bed configurations Ln terms of f Low conditions

is required for e~aluation of the friction factors. Qualitative classi

-fications in terms of Fraude n~~ber and partiele size have bee~ sug

-gested by various researchers. Quantitative classification has been