THE PROBABILISTIC CHARACTERISTICS OF BED LOAD TRANSPORT IN ALLUVIAL CHANNELS

A TRESIS

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA

**BY**

AMREEK SINGH PAINTAL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Acknowledce~ents• • • List. of Ill~s-Ll.tions

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i i i List. Cf Syr.:bols### ..

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viiii..,I IN:,?CDUcrlOi'I

II REVI~,**O:? LITERATURE**

?orees Actin.: on a.Pc!.ct.icle

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1### s

5

Variat.ion of Forces ona. Particle _{9}

Conce?t. of Critical Condition • • • •

III ~~ALYTICAL CONSIDERA~IONS
**3ed** E~~etnent ~odel

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191121

Statist.ical Ana.lysis of Particle Movement

IV COLLEC7ION O? DATA • • •

12

ZY.;:leri::ental Set-Up

;·!a.ter'..zl Used

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v PRSSE:?i'ATIOfr*t:.::n*

ANAL!SIS OF DA:'A
~t...-.oance ::i'::ect of the 71une

**Ens=gy** Slo~e • • • • • • • • • • • • • • • • • •
300 Shear
46

### 49

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'IT ~esentation of DataCriticalShear Force Cor..cepts

cmrCLUSTIONS

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**53**

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59

3ib110 graphy. • • •

Photos ar~ **Figures • . ' •**

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ACKNOIflSDGMENTS

The author considers it. a fine privilco'.l t.o express his deep

aenseof grat.itude to his faculty adivser, Dr. Alvin G. Anderson,

Professor in Civil Engineering, University of Minnesota, for his

un-failing exhortation, encouragement, and many contributions during

the discussions of this investigation. The author takes this

oppor-tunity -to thank Professor Edward Sllberman, Director, St. Anthony

Falls Hydraulic Laboratory. for providing a rich experience in the process of working on various projects in the Laboratory.

Thanks are also due to those who were directly or indirectly

assoicated with the auth~during this investigation.

Financial assistance for the investigation was provided by the

Highway Research Board. This is gratefully acknOWledged.

Figure

1 Shields Diagra~for Critical Shear Stress--Variation

of Critical Shear Stress Hith~• • • • • • • •

*v*

2 ihite's Bed Embedment !·todel for the Evaluation of

Critical Shear Stress • • • • • • • • • • • • • • • • An exposed C:i.'ain SUbject. to Hydrodynamic Forces •

### 3

4 Bed Embedment Hodel

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5 Ge"'!lletrical Picture of Probability of ~Iovemcnt --Triple
**Integral - E(;r.** **33 • • • • • • • • • • •** **• • • .**

6 Variation of Probability of Move;nent p with Di!:I'~nsion

less Instantaneous Bed Shear *T..*

7 Variation of Probability of Hovement (mean) Po with

Dinensionlcss l1ean Bed Shear Stress *T*0* • • •

8 Geometrical Repres~ntationfor Elvaluating the Arrival of

Particles at a Certain Cross Section • • • • •

9 Variation of Bed Load Transport Pa....""allleter qs* with

Dimensionless Shear Stress *T*0* • • • • • • •

10 Size Distribution 01'Bed Material used 1.."1 the Present
**Investigation • • • • • • • • • • • • • • • • • • • •**

11 Velocity Profiles in the Vertical at Various Stations

Showing Growth of Boundary !ayer • • • • • • • • • •

12 The Bedand the Water Surface Profiles ".long the Lengt..h
**of the Flume** **•• • • • • • • • • • • • • • • • • • •**

13 Variat1o:l of BedIoadTransport *p,,+..,.*

### lls

with BedShearT 0 for the Various Bed !1a.terials • • • • • • • • •

14 Variation of Bed LoadTrar.sport Ra.te Para..neter qs* at

*Lo:l*Shear Values with *T*0* • • • • • • • •

15 ';a.:!"iation of Bed Load Transport Parameter lls* at high

~uearValucs wi~~ *TO*

** .•• . • • . • . • . • . •.*

### 16

Variatio" )f qs* withand B . . **l!4.** 60Y • • • •
11

*TO*

***

(Evaluation of Constants A
Figure

17 COMparison with other Bed Load Formulations • • 18 Critica1 Tractive Force as determined by Shieldst

**Method.** **• • • • • • • • • • • • • •** **• • • •**

### 19

Variation of Critical Shear Stress of Bed Mater1a.l d • • • • • • • •i l l

T

the coefficient. of lift the bed width

constants in bed load formula

Froude number of n"ou

~ a sediment particle

drag on the sediment particle

ma....imum dra.g on the sediment particle

constants in the bed load formula~area

With respect. to walls and bed; respect-ively

diamoter of sediment particles

diameter of sediment particle

### (50%

isfiner than this diameter size by weight) depth of nOIl

density function of instantaneous dimension-less shear

n~izeddensity function of mean dimen-siouless shear

random variable for particle's exposure

density function of particle step lengths *'f*

density function of travel length of a sedi-ment particle

mean density function of travel length of a sediment particle

distribution function of

### Y

rela.tive expoS'

acceleration due to g:t:avity

joint ciensity function of el' e_{2}, and e)
joint density funct10n of el' and e_{2}

### A,

### At. ,

_{~,}

_{"z,}

A),
Aw' ### ~

13_{B.}

Be
CL
d
d_{SO}

D
D
r
D_{max}

### e,

_{el' e}

_{2},

_{e)}El f

### (T.)

foe### '111)

f (y/ f (s) foeS) F (y) Fr g g(e_{l}, e 2, e) g(e;L' e

_{2 )}iv

### ,

N n p p q R r s density function of e 2 distribution function of EfUnctional relationship between el' e2, and

e3

Nikuradse' s sand grain roughness a number. an interval

lift force on a sediment particle number of grains

a number

Manning's ~gosity coefficient of walls and

bed, respectively probability of movement mean probability of movement

probability of an event not occurring probability of movement per second pressure difference

pressure

dis~h?xgeof water per unit width

### per

secondbed load transport per unit width per second

water discharge in cubic feet per sec~nd

hydraulic mean radius

hydraulic mean radius with respect to bed hydraulic mean radius with respect to walls hydraulic mean radius with respect to

rough-ness of grains

radius of a cylinder

distance traveled by a particle slope of the bed

8
e
·8
t
u
**u'**

### u

v'*fJ*P Ps 11 T TC T D energy slope

specific gravity of the sediment particles dummy variable

time of movement and rest (one cycle) time of rest of a sediment particle

time of movement of a. sediment particle

ratio of maximum drag and submerged weight local velocity

velocity fluctuation of u due to turbulence mean velocity of-fIou

shear velocity

velocity fluctuation in the vertical direct-ion due to turbulence

angle of escape of particle specific weight of water specific weight of sddiment mass density of water mass density of sediment kinematic viscosity of water packing coefficient due to White instantaneous bed shear

critical shear force for a sediment particle

dimensionless instantaneous bed shear

mean bed shear

dimensionless mean bed shear total bed shear

bed shear pertaining to bed 1rrf!gularities vi

Einstein •s bed load parameters dimensionless step length step length

characteristic fUnction

Gamma function (probability distribution

function)

### ,

The relatio~ship between the bed load trar.sport rate and the

~lo~ cha.-acteristics has received a great deal of attention by

re-searchers in se~ent transport, and a be~dldering number of

er.-pirical and seiJier.pi-"'"ical equat.Lonz have been prollosed. :'!ost of these bed load tra.'ls:Port relations.'lips are based on the concept of

incipient ~over.er.tthat is supposed to be governed by ce~~in

de-finable critical condatiens. However, tone present experiJr.cnts ha.ve shcxn that the motion of sediment rarticles on the bed is highly U-'lsteady and nonuniformly distri~tedover the bed area, so a dis-tinct condition for the initiation of motion does not exist.

It is a well-established fa.ct that the flow generates forces of li:ft and.drag on a solid ;:article resting on the bed of a streaul. The ~agnitude of tr£se ferces fluctuates with time, a. phenomenon strongly associa-::ec. ~iith turbuler.ee. ;~a.turally, t."le larger t."le

ex-posure of the particle, the c~eater is the intensity of forces. A

~icle a:';. the bed aoves ,lhen the hyd...""Odynat1i.e forces overco::e its submerged ;:eight. ':'he ?=oeability of movement of ... sediment 1J2.r'ticle

is governed by its exposure to flow, the position it happens to occupy

at the red and the fluctuati~nature of forces to which it is

sub-jected. The lIlOver::er.t of a !=3.-..-ticle is in qUick steps loo"ith inter-l:l1tte:lt periods of rest. Both the stell Lengtha and the rest periods are :-andon va-'"iables. 3ased on these concepts, an ana.lytical

expres-sion has been developed for t~e rate of bed load transport as i t is

gove-~edby the flow, the fl~id, and the sediment cr~teristics.

The experimental data have been obtained in a laboratory

flume in which lot-r shear values can be generated. At low shear

values in the proximity of so-called critical shear, the bed load

transport rate has a 16th power correlation With the shear stress,

which seems consistent With the concept of incipient shear. Along

with these data, the publ1shed data of Gilbert (11), the United

States Waterways Experiment Station (22), and Casey(131have been

The movement of sediment along the beds of alluvial rivers

has long been one of the most perplexing and challenging problems

to those tlho have tried to explain it. In the past few decades,

be-ccuse of rapid expansion of activity in the field of river valley development, the importance of this problem has been gravely felt. One of the most difficult problems encountered in alluvial channel hydraulics is the determination of rate of movement of bed material by nowing water.

The movement of bed load has been studied principally in

lab-oratory channels where suspension of sediment w.s neglected.

Re-searchers have tried to develop expressions to describe bed load

transport based on the analysis of experimental data. The earliest

concept regarding the pattern of movement was that the loose bed

slides in layers under the action of now above , The top layer of

the bed is set into motion by the tractive force or shear when this

force is larger than the force resisting the motion. The rate of

transport was found to be a function of the difference between these

two forces. These expressions were based on the concept of initial

movement that is supposed to be governed by a certain definable

crit-ical condition. The concept of a critical tractive force was

intro-duced by Shields (22]*, Kramer[16] , and others. Shields defined it

as the value of shear stress for zero sediment transport obtained

by extrapo!d.ting to zero a graph of observed sediment versus shear

stres~.

2

Kramer (l6J studied the problem of sediment movement in flumes

with the objective of selecting sediment for use in moveable - boundary

hydrauliC models. He defined three rates of I;lovement which he termed:

(1) weak movement, (2) medlull1 :novement, and (3) general movement. The Task Committee on Preparation of Sedimentation Manual (21] observes that Kramer's 'weak' movement corresponds most closely to

the beginning of movement. But the United States Waterways

Experi-ment Station (23) defines the critical tractive farce as the shear

stress that brings about the general movement of sediment particles

a.t the bed. I t has been proposed [2:3) that the shearing stress that

causes bed load movement of 1 lb/ft/hr be called the critical

tract-ive force. This definition is for sand--it cannot be used for gravel

beca.use of the fact that the movement of a few isolated stones may satisfy this requirement even when the movement is not at all general.

As such the definition of the critical condition is rather indefinite

qualitatively as well as quantihtively.

A.,examination of experinents on bed load movement reveals

that a distinct condition for the initiation of movement does not

exist. There is no single flow condition below which not a single

particle will move and above which all narticles of that size Will move. The observations show that t!.", motion of sediment grains at

the bed is unsteady and nonuniformly dis\.ributed over the bed , In

the proximity of the so-called ' critical condition' the movement of

particles occurs in small areas scattered over the bed. The

in-cidence of movement seems to be random in both time and space •. The boundaries of an alluvial stream are composed of discrete

particles. The distortion of flon field around a solid particle

invest-ig.ators have considered the nature of these forces which remove the

particle from its position. In many cases the shearing stress has

been considered to be the only disturbing force. The possibility

of a lift force was seldom considered. The force of lift is caused

by the velocity gradient in the flow and the possibillty of stagnation

pressure under the particle. For any given material there is no

single value of hydrodynamic force beloti which not a single particle

will move a.nd above which all the particles of the same size will

move. First, the forces acting on the particle are not constant,

but fluctuate about some average value, a phenomenon closely asso-ciated with turbulence, as suggested by Einstein (7] and Kallnske

[14]. Secondly, the forces tendirlz to move the particle depend on

the position which the particle heppens to occupy and how much

in-fluence its neighbors exert. Particles which project above the mean

bed level maybe expected to have a 101ler 'critical' force than those

imbedded in the sur£ace layers. If this critical force is exceeded

at any instant, the particle moves with a rolling or sliding motion. Once in motion, the chance of a particle's being redeposited is not the same at all points on the bed, but depends upon the local flow conditions and the amount of protection which the particle receives

trom its new neighbors. Mter being redeposited, further movement

depends on its nen location and the flow conditions around it, but

not on its previous history. Thus bed load movement may be considered

to be the motion of bed particles in quick steps With periods of

rest between.

In.this investigation, attempts have been made to derive

move-4

ment of particles has been assumed to be governed by the exposure

of the particles to the now and to the nuctuations of the

hydro-dynamic forces acting on them. The distance traveled by a particle

before coming to rest and the duration of the rest periods have been considered random variables.

An abundance of bed load transport data above the so-called critical now condition is available. so i t was felt unn;.:.:essary to

conduct more experiments in this region. The experimental data in

the proximity of so-called critical shear have been obtained in a

laboratory flume in which lO~l shear values could be generated.

The objectives of this study were (1) to make a thorough re-view of literature concerning the various aspects of bed load trans-port which mifiht be applicable to the proble&1, (2) to formulate an

analytical description of the problem. (3) to experimentally observe

the movement of the bed particles in the vicinity of the so-called

critical condition. and (4) to develop an expression for the bed load transport rate as it is governed by the now, the fluid, and the sediment characteristics.

A number of attempts--empir1cal and semi-empirical aa well as

analYtlcal--have been made to selve the numerous problems of alluvial

channel hydraulics. The works of major importance in the area of

bed load transportation are reviewed 1n brief 1n this chapter.

Forces Acting on a Particle.

The boundaries of an alluvial stream are composed of discrete particles which as a result of forces arising from the flon over them

can be set in motion. Surprisingly few writers ~ve engaged in

spe-culation about the nature of forces which remove particles from the bed, being satisfied with the concept of a shearing stress as the

only important quantity upon whicl ;0 base their analysis. The

shear-ing stress has been considered to be a steady force derivable by

applying the principles of statics to a prism of flowing fluid moving

at a constant velocity. The magnitude of this force 1s given by

where T .. sheadng stress in Ibs/sq. ft,

y .. density of fluid in Ibs/cu. ft,

Se .. slope of total energy line, R '"' hydraulic radius in ft.

(1)

The possibility of lift forces acting on the particle has seldom

been considered. Lift forces My arise due to the existence of a

velocity gradient in the flow and the possibil1'tY of a stagnation

pressure existing under the grains. L1ft forces must contribute

7

pressure difference betwep"l the top and the bottom of the hemisphere was 2.85 times the lift. If this "scaling factor" is correct, the value of CL based on the work of Einstein and El-samni should be.O 624, a value slightly smaller than the value determined by Chepil.

The investigations of Chepil, and Einstein and El-samni were conducted with hemispheres glued to the bed of the channel. These hemispheres were considered ac idealised sediments. In contrast to hemispheres, the natural sedimerrt particles are irregular in three different ways: 1. The shape of individual particle Le irregular, 2. The size and the shape of the particles are different frolD one another, and

### J.

The location of the different particles with respect to theoretical wall is irregular. When a flat sediment particle is placed on the channel bed with its flat facesloping upward in the direction of flow, there is a force acting downwarde:tu.!

### te

deflection of stream lines. On the downstream side there is a tendency for lowered pz-es sure to form clue to the curvature of stream lines which causes an additional force acting d' .mwards. These forces keep the particle inplace. These forces cause the negative lift. Ent if the flat face of the particle is sloping down in the downstream direction, the direction of fo"'ce would be reversed, there would be an upward force on the lower side that would turn to overturn the sediment particle. Thus the lift may change its direction and magnitude depending on the positior. which the particle happens to occupy on the bed with respect to other particles. Einstein and El-samni measured the dynamic lift on the bed

particle by a similar method to that used in the studies with the hemispheres. The mean value of coefficient of lift obtained in the case

of natu::-al material agreed very well with the one obtained in case of hemispheres.

The coefficients of lift and drag :~0u1dbe different depending upon the position which the particle happens to occupy on the channel bed - the exposure of the particle itself and its neighbors to the now. For the sake of simplicity, it would be assumed that the ratio of forces of lift and drag is independent of

icle on the bed.

the relative position of the

pq.rt-It is possible to estilnate the ratio of lift and drag forces on a particle. The veloc~ty distribution on such a Loundary is given by:

= 2.5 loge

### -f

s+

### 8.5

(J)where u is the local velocity at a distance

*s'*

from the theoretical
wall (bed), k_{s}is Nikuradse's sand grain roughness, and 0* is the shear velocity. The coefficient of lift, ~, of a hemisphere lying on the bed is.C

### 68

with its velocity rnaesured at a distance### 0.35

particle diameter away from the bed. Therefore,Lift Drag 2

### 0.5

CL### A

p u = Ap### 0;

(4)The va1.ue of Nikuradse's sand grain roughness, k_{s'} may be taken
equal to the diameter of the particle composing the bed. Eq. 4

becomes

Lift

Drag

### =

1.207Thus based on the Prandtl-Karman velocity distribution law and experimental results, the ratio of lift and drag forces is 1.207.

As a result of the review presented above it may be concluded that a particle lying on a bed experiences the forces of lift and drag, which may be considered equal in magnitude.

9

Variation of Forces on a Particle

### .

The forces to which a particle is subjected are variable and

may fluctuate about their mean values. These fluctuations are caused

by two conditions: First, the small eddies which are generated

be-hind the particles cause the fluctuation of the hydrodynamic forces.

Second, the main flott strean may be turbulent and this will cause

the fluid velocity acting on any individual particle to vary

con-siderably ~Iith time.

Ka.l~.nske (14, 151demonstrated that the velocity fluctuations in rivers are distributed according to the normal probability

dis-tribution, the density function of which is given by

feu') • 1

### ~::z

2TTand

*r*

f(u') due ### =

1-00

where f(u') is the density function.

### ~

u'2 is the standarddevi-at10n of u', u' is the fluctuation of velocity about the mean and

eJPis the base of llaperian logarithms. The quantity r(u .) du'

in-dicates the prC1J?ortion of the time that the velocity fluctuation u'

lies between the magnitude of u' and u' +du ",

In the concept of monentum eXchange in turbulent flow, it has

been shoan that the unit shear at any point is equal to *P*

### U'V'1

thatls, the correlation of u' (the horizontal component) and v' (the

vertical component). Slnce the longitudinal component is in general

be taken as proportional to

### ~.

It may be assumed also that the fluctuations of the bed shear stress can be described statistically by the normal probability distribution whose density function is given by:= 1 exp (-(

_{(6)}

### •

Nhere T the 'I" the and*aO*= the a~2iT

local intensity of shear stress at any instant, mean intensity of shear stress,

standard deviation of T •

In

### 1959

Che?il and Siddowy (4) devised a strain gauge anemo_ meter for measuring turbulence near the bed.### e.r

using a system of strain gauges mounted on a sediment particle, a record of instantaneous values of lift and drag was obtained. Oscillogram obtained with this instrument indicated that both lift and drag are distributed statistically to a "some skew normal law". The ratio of maximum to mean lift and drag was found to be approximately 2.7. It seems that the frequency response of the instrument was not perfect to pick up very high values of instant-aneous drag and lift. Hence from a statistical point of view, the maximum lift and drag has no definit'3 limit.The maximum recorded pressure was ~ven by the expression

; +

### JJF2

and the ratio (P +### J./

p2 )/ P (where P is the mean pressure and.J p2 is the standard deviation of### p)

was termed as turbulence factor. It was found to have the value of approximately 2.7, which is the value used by~~te[24] and Kalinske(14) in their theoretical treatments of drag forces on a sediment particle. This would give the value of the standard deviation as 0.57### F.

Einstein and El-samni determined the value of standard deviation of turbulent fluctuations of 4p to be 0.5P. Therefore, i t may be concluded that the fluctuations of lift and11

ion whose standard deviation is 0.5 TO' The distribution is not strictly

normal as it is referred to have an lowe:- li'llit of zero. Concept of Critical Condition

In 1929 Jeffrey[121 presented a theoretical solution for the

problem of grain motion based upon the stabilit7

*ot*

a long cylender of
rlidius l' resting on the nat bed of a deep stream with its axis

perpen-dicular to the now. If the upward force acting on the cylender exceeds the submerged weight of the cylender, it will be lifted up into the now. For a sand and water complex, the result was expressed as follows:

### u2

### >

1.19 grwhere U is the free stream velocity and g is acceleration due to gravity.

This model, of course, gives the correct answer if there is sufficient

justification for describing the dynamic effects in a real fluid by

irrotational theory. Jeffrey's answer to this question is affirmative. He claimed that during the initial stage, when a particle is just dropped on the bed of a stream, the theory is applicable because the flow around the

particle has not yet been modified by viscosity. Even if the validity of

this argument is accepted, this model fails to recognise the erratic fashion

in which particles move on the bed. At tha same time, this theory assumes

that the now responsible for the forces induced on the particle is uniform and steady. Therefore, a particle that has once started moving would never stop at the bed anywhere. This is inconsistent with the form of motion actually observed.

The two best known and most widely accepted results are those of

Shields (22] and White (241. By considering the disturbing forces to be

restricted to shear, each author derived an expression for a critical shear stress which if exceeded would cause motion. Shields (221 developed the relation:

(8)

where_{.} _{. c}*T* is the value of shear stress under which the motion of a
particle begins, 'Ysand 'Y *are* the specific weights of sediment
grains and water, respectively, d is the size of particle, U*=~Tc7p
is the so-called shear velocity, *P* is the mass density of fluid,
and 11 is its kinematic viscosity. Sh1.elds·s results in which he

de-termined the function f(U*d/lI) are sheen in **Fig. 1.** In his original

paper Shields did not showa.curve for the function, but indicated it

.by a shaded area, The curve shown in the figure wasintroduced by Rouse [20

### J.

ShieldsIs data were obtained from flume experimentsWith fully turbulent flow over artifica.lly flattened sediment beds. He found in experiments Wit.lt different bed stresses that were just above the critical, that small rates of bed load transport occurred. The fUnction betlieen the shear and bed load transport wasextrapolated in the direction of decreasing shear stress to the point where bed load transport was zero. The corresponding shear stress of this point

was called the critical shear stress.

White(241 determined the critical shear stress for several sediments under conditions of completely developed laminar flow,

lami-nar boundary layer flo.:, and. turbulent boundary layer flow. Figure 2 shows the embedment model. White argued that bed shear is transmitted from the fluid to the bed throUfjh the surface grains so that each Will

2

x:eceive a drag force equal to T ~1J where " i s a packing coef-ficient reflecting the spacing of the surface grains on the bed. A grain would

### :r..a:rt

to nove by rolling over a point of contact Wit.'1 a do:mstream particle. The point of contact 'IIould 11e on a line passing through the center of gravity of two particles at angle {j With the1)

vertical. The force causing movement is drag and the force

resist-ing movement is submerged ;leight.

~er(16) studied the entrainJaent problem in flumes with fully

developed velocity profiles With the object of selecting sediment for

use in movable bed hydraulic models. His definitions of rates of bed

load movement are as follo'lls:

1. 'None" refers to that condition where there is no sediment movement.

2. 'Weak' movement indicates that a. considerable number of

particles are in motion at isolated spots on the bed.

). 'Nedium' movement refers to that condition under lIhich quite

a large number of particles are in motion at many places on the bed.

4. 'General' means that condition in which there is an

appre-ciable amount of movement everywhere on the bed. The motion

is general in character.

The *U5:lES*(2)1 defined the critic:aJ. tractive force as that
trac-vive force which brings about. the general movement of bed material.
The state of general lJlo·.rement is obtained lIhen both the following
cc:nditions are satisfied:

1. The material in transportation is reasonably similar in

character to material composing the bed.

2. The rate of movement 1s equal or exceeds one pound per foot

width per hour.

The definition is applicable only to sand less than 0.6 IlIIlI in size.

This definition cannot be applied to gravel or small stone mi:xtures

due to the fact that the movement of only a few isolated stones might

### ,

movement was not general in character. For gravel mixtures the St. Anthcny Falls Hydraulic laboratory Ll) has defined critical tractive force as that force uhich removes J'J"; of the surface particles every hour. Assuch the concept of critical tractive force 1s only qual-itative in character and it has not been possible to define it 1n terms of bed load transport on a universal basis.

BedLoad Formulations

Several investigators have developed relationships for the bed load transport rate based on analytical evaluations of the behavior of sediment particles on the bed of an alluvial stream. Different types of models have been proposed, and the transport relationships were based on the magnitude of forces acting on the bed. The trans-port phenomenon has a high degree of randomness. The grains vary among themselves in shape and size as well as in their exposure to the nou. In addition, because of nuid turbulence the nOlf exhibits strong variations in time and space.

The first and the oldest approach to a bed load f~'rmulais based on the principles promulgated by DuBojs (quoted 11) (I?] ). This prin-ciple 1s based on the concept that the boundary shear force on the bed causes successive layers of sediment to slide one upon the other. The top layer of the bed is set into motion by the shear force when it becomes larger than the force resisting motion. The rate of bed load transport dete-"'lIlined experimentally wasfound to be a function of the difference of the two forces. No effort llaSaade to develop this theory to explain the actual mechanism of the interaction

be-tween the solid particles and the fio;: field. The theory assuned that the disturbin6 force of shear has an average constant value and does not vary ,11th time and space. I t i:mplies that all particles

### •

### 15

should start moving on the bed whenever the average shear is larger

than the so-called critical shear. This, of course, is contrary to

the well-established fact that motion near the bed takes place in the form of sudden movements by individual particles alternating with rather long periods of rest.

Recently two bed load equations based on stoehastic models,

which may be termed theoretical, have been proposed. Kalinske (14J

and Einstein

### t

7) both derived, from different sets of explicitassump-tions, tranSJlort relationships which allow a prediction of bed load to be made. They considered the stability of the individual particle

subjected to hydrodynamic forces that vary rapidly with time due to

nuid turbulence. Kalinske [14] trea.ted. these variations byassuming

that velocity nuctuations follow nomal distribution law. 'l'he number

of particles in motion was equated to the number of particles at the bed, wzU.ch iJaplies that rega.rdless of the transport rate the number

of particles in motion is constant. The particle velocity wasassumed

to be related to nuid velocity; i.e., the shear velocity. A

part-icle that has been set into motion would keep on movingwit~a

con-stant speed depending upon the now conditions. Observations by the

writer of grain motion in a laborato-ry nume showed these concepts

to be unrealistic. Based on these assumptions Kalinske [14) gave the

folloldng expression for the bed load transport:

is a function of T

### /r •

C 0

Einstein (7) postulated that a given particle size moves in a

con-tinuosly, but is deposited on the bed after a few steps. The mean length and the frequency of these steps are assumed to be functions

of ~rticle f:ize. This implies that the particles have a relatively

constant velocity of movement. The particle moves whenever the

hy-draulic lift exceeds its submerged weight. A normal distribution

function ,,-as assumed for the variation of hydraulic liftl this allows

infinitely high lift forces on the particle. The prol:ability of

move-ment p would then be simply the probability of lift exceeding the

submerged weight of the particle and is expressed as follows:

p =

### 1

1### ..In

dt (10)

lfhere " and & are u"'1iversal constants and 'It

### *

is a shear stressfunction. The end result of Einstein's analysis is the relationship

(lla)

where

### =

qs### ~g(s-l)

Fed 3/2and

The above relationship is based on the experimental cl.a.ta from

Zurich and the data by Gilbert. In 1950. Einstein (8) modified his

theory and redefined the bed load transport rate parameter in the

17

and gave the analy-t;ical result

where

### 4

and ET are constants and were evaluated frOI'\ theexper-imental data. The fUnctional relationship

### was

developed analy-t;icallyand based on"!he original assur.lptions.

Both Einstein (0) and Kal1nske tl4.) failed to consider the effec1

of the particle's exposure and the position it happens to occupy at

the bed on its novement. A particle which is protruding into the flow

is SUbjected to a larger force than one completely imbedrled in the

surface layer of the bed. Mercer (18) introduced the concept of

're-lative exposure' and developed a bed embedment model. The probability

of novenent., p, of a particle was evaluated by statistically

anaIyz-ing the randomness of rela.tive exposures of dti"ferent particles at

the bed. The she3J:i.ng stress \las considered as the ma.in disturbing

force. and the effect of turbulence wasignored.

The Meyer-Peter (19] formula. flaS developed at the Hydraulic

Ia.b-atory in Zl.U'ich, Sldtzerland, and has been used quite extensively ~

Europe. Using the Froude lal.. of similarity as a guide. it was

pub-lished as folious:

*p*

The effects of.sediment density and the separation of the form

re-s1stance from -the total resistance were also considered. In the above

2/3 1/2 *2/3*

### •

A and B are constants, and qb is the sediment transport rate

measured as dry weight per unit width per unit time. The total

re-s~nce was divided into grain resistance and :form resistance by

holding _{Rb constant and dividing the energy slope into a part }

de-pendent on grain resistance ~• and a part dependent upon :form

re-sistance sb".

Chien

### t61

has shown that eq. l2a. can be mod1£ied to expressthe :following relationship between q~ and *T*0* - - the same parameters

introduced by Einstein (8

### J

in his bed load fUnction311.

qs* .. (4 *TO*

### * -

0.188) (12b)The bed load equa.tions o:f Einstein (8) • Meyer-Peter (19) • and Kall.nske (l4] have been plotted in Fig. 15. All t.l'" equations are

sim-ilarat *lOll* transport rates, but at high transport rates they diverge

### ,

III. IINALYTICAL CONSIDERATIONS

When water f10lfS over a flat surface of loose sediment particles, a. fey grains here and there move with the water because of the forces

exertedby the flow. The moving grains are sUbject to a weight force

which opposes their movement and a tangential force which tends to

maintain the forward motion. There are also shear stresses between

the grains in motion and those forming the stationary boundary, the

fluid between them taking part in shearing. Figure:3 shows an

assem-blage of discrete particles forming the bed of a channel in which the

water is flolling. In general there will exist near the bed a Dean

velocity profile u

### =

fey). The velocity profile represents thetem-poral mean velocity at each point in the vertical and shows the

de-crease in velocity as the bed is approached. Superimposed upon this

temporal llIean velocity maybe turbui.,,:lt f'luctuations. These temporal mean fluctuations mayor may not cause local instantaneous modifi-"

cations of velocity field. As the flow passes the particle, the

streamlines are deflected upward around the particle. The more

ex-posed particles shed eddies and a wake is formed dOl!!lstream. The

size of the wake depends upon the size, the shape of the particle,

and the point of separat~onof the boundary layer formed on the pa.."'t-icle. The point of separation is dependent on the particle shap<:! and the local Reynolds nUlllber of the

### no!"!.

Tha hydrodynamic forces exerted on the particle maybe resolved

into two components--one parallel to the direction of aean flow, called

thedrag, and the other normal to it, called the lift. The drag is composed of a surface d....-ag and a fom drag and is caused by the

pres-sure difference in front and behind the particle. The point of

### ,

the particle is not fixed and depends on the relative magnitude of

the lift and the drag forces. which in turn depends on the local

Reynolds number. the shape, and the position that the particle happens to occupy at the bed.

The lift is similar to the form drag and is the resultant of

the pressure difference above and below the particle. On the upper

side the pressure is reduced below the static pressure by virtue of the curvature of the streamlines and the increased velocity around the

particle. Underneath the particle the interstitial velocity is

re-latively very small, and the pressure approaches static pressure.

Both the lift and the drag are fluctuating quantities in magnitUde.

point of application, and direction due to the turbulence. In fact,

it is conceivable that in certain instances one or both might be

re-versed in directior..

Opposing these hydrodj'Ilarnic forces is the submerged weight of the particle plus any constraining force caused by contact with the

other particles in the bed. The submerged weight depends upon the

size, the shape, and the density of the p&-""1;icle. Consequently, the

force system that d~tcrminesthe motion of the particles is a complex

:f\lnction of time, space, and distribution of physical properties of the system.

These two sets of forces--the hydxodyna.mic and the resisting forces--tend to dislodge the particle in one of the following ways.

The possible 1nltial motions of the particle are limited by the

part-icle's contact with its neighbors. The type of motion requiring least

hydrodynar.dc force is a rolling motion with the particle pivoting about its point of contact with its immediate downstream neighbor.

21 contact is greater than the moment of submerged weight about the same point, the particle will be rolled from its initial position to some point dOlmstrea.m, Hhere the combination of forces. including its own momentUIII, is such that i t is more stable. If the lift force at any

instant becomes larger than the submerged weight. the particle will

be li£ted bodily £'rom the bed. The drag force acting on the particle

may also tend to move it dotmstream. so that its motion will be in

the nature of a hop. A simpler type of motion which is easier to

analyze and which requires only a slightly higher force is a linear translation of the particle parallel to the plane of contact with its immediate dOlll1stream neighbor particle.

Because of the possible variation of the forces generated by the water flOWing about the typical particle, the instant of movement

is indeterminate. Considering the assemblage of particles as a whole,

particles will moveWhenever and ~rhereverthe hydrodynamic forces

over-come the forces resisting motion.

The funda.l!lental variables associated with low rates of transport would be the variation in the exposure of the particles and the var-iation of the hydrodynan1c forces caused by turbulence.

Bed Embedment Hodel

Figure 4 shows a longitudinal section of a bed consisting of

spherical particles of uniform size d. These particles are supposed

to be randol:l1y oriented relative to a mean bed level, which is defined

as a line drawn tangent to the bottom of those particles which have

the g:reatest exposure to the flow. The relative exposure "e" is the projection of a particle above the mean bed line measured

exposure to nOlI in com:P<lXison With its neighbors. U 1s subjected

to higher velocities, and hence to larger forces. This particle also

has a smaller angle of escape. Therefore it is most likely to 1II0ve.

\/hen this particle having the maximum exposure is removed, the

par-ticle underneath becomes the least exposed parpar-ticle on the surface.

Us upper edge projects a little above the bed level. The probability

of movement of this particle will be very small. Therefore, the

pro-bability of movement should be based on the statistical analysis of

the particle exposures, as suggested by Mercer t18 ). In other words,

the probability of a randomly selected surface particle having a given

relative exposure should be determined.

The distribution function of particle exposure E should be a

continuous fUnction, as no single value has a non-zero probability.

U has a derivative at each point. The derivative of the distribution

function ~(e) is called the probability density function of the

distribution and is denoted by g( e) :

gee) c

### -S

(G(e»### =

d P.rob. (E ~ e)de

### de

dC(e) c g(e)de (14)

This differential of probablli.ty (E

### <.

e) can be used, for small de,to approximate an increment in probability

### <E (,

e) correspondingto an increment in e from e to e +de:

g(e)de = P.rob. (E ~ e +de) - P.rob. (El~ e)

Onc of the properties of a function nhich is differentiable, is that it is the integral of its derivative.

(16)

### "

From. this it follows that

*f*

*ez*

Probab. (0

### <

E~_{92 )..}g(e)de

o

The determination of the probability that a particle would move requires some assUl'llptions about the method of dislodgement and the

forces causing it. The forces acting on a particle are:

.1.. SUbmerged weight, ti, of the particle, acting vertically

dowmmrd;

2. the fluid drag D

r acting in the direction of flow; and

### 3.

the hydraulic 11ft L on the particle acting upward normalto the flow.

The possible in!tial movements of a particle are liJnited by the

par-tic.1.e's contact with neighboring particles. Every particle is supposed

to be in contact with a.t least two other particles--one upstream and

one downstrea.ll. Its points of contact with these two particles lie

in the plane of the figure. The plane along which the particle is

supposed to escape is nomal to the plane of the figure and ta~ntial

to the particle and its point of contact with 1ts downstream neighbor. The condition favoring escape exists, then, when the follonng force inequa11ty is satisfied:

D_{r}

### >

(it - L) tan### 8

where (1 is the angle of escape.

the exposure of particles. 'fhe hydrodynamic forces acting on the

par-ticle have been as suraed to depend upon the amount of exposure (Mercer

### us) )

which the particle has in comparison with the upstream particle. Assuming a linear relationship between the forces and the particleex-posure, the following expression can be written:

### Pr ..

Dmax (e_{2}- e

1)

L .. L_{max (e2 - e1)}

where Dmax and Lmax are maximum drag and maximum lift on a cOTolpletely

exposed particle _{(e 2}

### =

1, el .. 0, e) .. 0) at any instant in time.

The angle of escape of particle 2,

### a ,

maybe shown to begeo-metricaly related to the exposure of particles 2 and 3by the expression

The inequality 18 may be expressed in terms of particle exposure

by substituting the values of L, D, and tan

### a

The forces of lift and drag on the particle have been assumed to be

equal in magnitude. Denoting Dmax/W .. T, the inequality 19 may

### or

### 25

~ 1 -(e2 - eJP (e2 - ell e2 - eJ### >

### 0

T### [

### '(1

*-C.. -*

### '3)'

### J

(20) -I + e2 - eJ### H

_{(el' e2, eJ)}

_{<}

C (21)
where

### H

represents the functional relationship and### C

is a constant.No~ the density function g(e2) may be expressed as:

g(e

_{z)}

### =

*J*

*f*

g (el. e2' eJ) del de
J

H(el,eZ,e

J)

### >

C0.-,el~1

O~e3~1

where g(el' e2' ElJ) is the joint. density funct.ion of the pa-"'ticlEl
expoSl:res el' e_{2•} ann.e3. The :particle exposures have values ranging

between zero and one. For a function to be a valid probabilit.y density

function it should satisfy the follOWing relationship:

*t*

*j*

o 0

(23)

The above relation can be solved to give,

Substituting the value of _{g(el' e2, e}_{J}) -- joint density function
--in Eq.22.

g(e2) is the density function of those exposures of particle 2 when

it is at the brink of movelJent. Therefore, the probability of move:nent.pI

where e2¥' is given by the follo«ing expression assuming e

l and e

### J

equal to zero in the inequality 20:

### y(l -

### e2~

(e2¥' -I-

### V(

1 - e2*2### )J

The physical interpretation of e2* is that it is the minimum

ex-psoure for particle 2 to escape when e

1 and eJ• i.e., the

ex-psoures of its neighboring particles, are zero. For particle 2 to

be in an incipient state the inequality may be put in the follo«ing form:

### 27

If el' e_{2,} and e_{3} are represented by three axes ofa
three-dim-ensional domain, the expression represents the equation of a surface.
The values of particle exposures lie betlleen zero and one, and thus

the three-dimensional domain is a cube: 0 ~ e

l ~ 1, 0~ e2~ 1, and

### o

~ e### 3

~ 1. The surface and the three-dimensional domain are shown geometrically in Fig.### 5.

The volume of the domain being a cube isunity, Thevolume contained between the surfaces H(e_{1}, e_{2}, 'e,) .. C,
e_{1} .. 0, e

### 3

= 0, and e2 .. 1 represents the probability of movementof particle 2. A.ll values of el' e

2, and e

### 3

that pertain to the movement of particle 2 lie in this region of the domain. The triple integral Eq. 2E evaluates the volume of this region.If the r~lativeexposures of particles 1 and 2 are known, the exposure o! particle

### :3

would have a certain maximum value in accordance with Fq. 20 for the dislodgement of particle 2. This maximum value of e:y is e;iv~nby:1

### ~]

where

D

### T ::

:axNow the probability that particle 2 will move is the probability that 83 will lie between zero and 83*':

(30)

### ,

This probability is actually the probability density that particle 2

will move out if e_{l} and e_{2} are fixed and e_{3} is random. Now, if

the exposure of particle 2 is fixed, there is a maximum value of the

exposure of particle 1 for the movement of particle 2. This maximum

value of el' e_{l}

### *,

is given by the folloWing expression:***

*..J(*

1 - *el.)*

el .. e - 6.,.-;

2 T ( e_{2} + (1 - e_{2} *J )*

(31)

The above equation is obtained by putting e

3 equal to zero in Eq. 20.

That particle 2 will be dislodged is the probability that e

l

will lie between zero and e

l* ! This probability is the probability density i f e

2 is fixed and el and e

### 3

are random._ 1 _

*2Vf!*

+_1__

### 2*

(32)

For movement, the particle 2 should have a certa~.n minimum exposure

to the now. This minimum exposure e2* is g1ven by the *J'*ollowing

~-~

### -~---29

2 "(1 -

### e~)

e2* ..

The rane;e of values of _{e2 would be between e2* and one for 'the}

dislodgement of particle Z. Now the probab1.lity that. particle 2 will

have the given exposure for the movement iSI

### "'[1 - L -

### .1

e3• +### L

### a.

2]### :3

2T:3 2T _ 1### [11

### 4T

### T::

-1 TT ] [ 2 2### V(

2) +Sin e~ - _ + 1 1 - 62* + 62* 1 - e2* 2### lIT

- 1_{16}

(4 e_
c.--- 1 [ (2 T - 1) Sinh-1 (IT-I) **-J**

(ZT - 1)2 +1 ]
4'**-J**

### "JZ

Tt1

+

### 4'f2

T2 {(2T e2* - 1) Sinh-l (21 e2* -l)yzr e2* - 1) 2 +l}### +

1 1 1 [ Sinh-l (2T - 1) - Sinh-l (2T e2* - 1)] (33)### 8V2

T2The probability of movement of' particle 2 has been computed

### :trom

theknoim probabilities associated with individual particle exposures. Each of' these particles has exposures independent of the others.

The parameter T is obtained by dividing the maximum drag (the

drag experienced by a completely exposed particle, i.e., el C 0,

e3 .. 0, and e2

### =

1) by the submerged weight.T .. Dmax -W-Tr 2

### 4d

T### .. .2

2 T .. ()4)The functional relationship for p with T" ( ..

### j

T) is plotted in### Fie.

### 6.

T.. is the dimensionless instantaneous bed shear.In order to incOI'lJOrate the effect of' time variation on pro-bability of' movement, the variation of' the bed shear should be

con-Sidered. The fluctuations of the dimensionless bedshear can be

des-cribed statisticallyby a. normal distribution function, the density

### 31

where *a* is the standard deviation and has been taken equal to 0.5 T

O

### *

and T 0. is the dimensionless meanbed shear intensity. The

male-imum value of instantaneous shear to which a particle is subjected

has been found to be

### 2.5

times the ave~e. Statistically, the valuesof T 0* range between0 &00 , The density funct.Ion g1ven by Eq.

### 35

has been normalized. In its final form it can be put as follows:

(36)

The average value of the probability of movement, Po' 1s given by:

P f_{o} (T* )dT

### *

Onsimpllfications Eq. (32) 1s reduced to the form

(J8) _t2! 2 e dt 1

### __

### JCo

### pe

Po .. '2 TT*-pr*

where t is a dUIDrny variable and is equal to (T* -TO*) / *C1.* The

above integral maybe evaluated with the help of FiZ. 6 and standard

)2

of movement with dimensionless bed shean 'T . ' At lower values of_{o} T . 'P

0 , 0

approached the variation. with the fourth power of To•• With the increase in fluid shear Le. 'To.' the probability of movement increases. According

to Fig 7 the fraction of particles that are in motion is less than 0.68 even for very high values of shear. This limiting value of probability is inherent in the model chosen and does not neccessarily represents any limit in the real phenomenon. It may be speculated that as the number of particles in motion increases, collision between moving particles may become more frequent and their effects cannot be ignor~. The particle collision give ri~e to an additional shear resistance due to the momentum transferred by the particles. This shear resistance would be in addition to the forces considered in this investigation. Therefore, the present model does not cover this situation.

Once the particle is set into motion, the hydrodynamic forces of lift and drag vary throughout its path of movement both in magni-tude and direction. The probability of the particle's coming to rest is not equal at all the points on the bed. It depends upon the amount of shielding that the particle gets from its new neighbors.Once the particle is redepo-sited it may move again when it is re-exposed to the flow due to the move-ment of its neighbors. Thus the movemove-ment of sedimove-ment particle consists of steps and periods of rest. The distance traveled by the particle before coming to rest - the step length - depends upon the angle of escape of particle and the characteristics of the flow, the fluid and the sediment.

Statistical Analysis of Particle movement

Consider a particle at rest at a point after which it makes a movement to another point. Consider the particle's coming to rest as an event. The distribution of the random interval between two consecutive events is determined as follows:

### JJ

Let

### Pr

be the probability that the particle does not cometo rest between y and y

### +

h. This probability is independent ofevents ~ccuringoutside of the interval

I y

### t

h Jand is uniform along the axis of movement. It depends only upon h.

Thus it is only a function of a single v.ariable hand is designated

### Pr

(h). consider two consecutive intervals 1 and J between thepoints y and y+h: and y+h and y4h+k, respectively. The

pro-babllity that the particle does not Come to rest in (y, y+h+k). the

union of two adjoint intervals I and J maybe written in the

follow-1ngfom:

Probabilities: Pr(h +k) '"Pr (h).Pr(k)

In order to write the above expression the independence between the

events in rand J has been considered. This relation which holds

for every value of hand k is a distinctive feature of exponential

fUnctions. It means that:

*t*

(40)

1I1thA> 0 and 0<1)~l

-0

Let . Ei and EH l be two points where the particle rests-consecutive

steps. The distance Y betlleen _{Ei and Ei +1 is the length of the}

Probability of no events

Prob (Y> y).. in (Yo. Yo + Y)

(41)

Let F(y) be the distribution function of Y.

F (y) = P.rob (0

### <

Y ~ y) c l-Prob (Y### >

y)The density function of the interval _{(Ei• EH l). i.e., Y,} is:

f (y) c

### s,

F(y) 1 e*-r*

*{Po>"*

dy .. Po>" (43)

The random distance (interval) between two successive points on the axis of particle movement l!l2.y be taken as following the negative

ex-ponential distribution. It satisfies the following conditions:

1. The probability of making a zero displacement 1s finite.

2. The probability of making a travel (displacement) of

in-finite length is zero.

### 3.

The function exists and is finite at all points betweenzero and inf1nity•

and 4. The area between the function and axis of y is unity.

The mean step length is PoA. where Po is the probability of

movement and >.. is a constant and is the mean length of the step when

the probability of movement is one.

The density function of the step length, Y, has been taken

### 35

successive steps of the particlE" Yn. _{Yn+1• Yn+2, ••• are }

independ-ent of each other. The total distance trave1ed by a particle between

the ll0ints Ezt and E"n+k is the SUlll of k independent steps. of random length.

### •

A Now Ezt+1### (Em.

E_{m+l ) ..}Y

_{m for}m" (n, n+l. • •• n+k) ~+k

**•••**

These steps are independent lengths. Each of them follows negative

exponential distribution. Finally the total distance,s, trave1ed by

a particle is the sum of k independent random lengths with the same

distribution function Y l'

The characteristic function of the negative exponential distribution is given by:

*.r*

*00*

it

tp(t)

### DJ_

e Y' f (y) d.yo

The characteristic function of the sum of independent random variables possesses a11 i.Ill!JOrt&nt factorization property. The characteristic function of any finite sum of independent random variables is the product of their respective characteristic functions.

The characteristic function of s:
*.. 'P _{n (t)}* 'Pn+l (t)

**...**

*'" ['P(t)]*

k (46)
where <pn(t) is the characteristic function of Y_{n}• Therefore. the

characteristic function of s:

1

This ls the characteristic function of Gamma. (k) distribution. The

density function of s is given by:

:Let us consider the particle on a. channel bed that has probability of

movement Po depending upon the nowand nuid characteristics. Suppose

that this particle has been set into motion at a certain polnt A.

It undergoes a displAcement Yl and stops at a. poin't El' The density

function of displacement (s a

### Y)

ls:*-slAP*

e 0

It aay again aove and can have another displacement _{12,} and thus the

de-sity function of distance s (a Y

l +Y2) is:
and so on,
1
AP_{o}
-sAp
e 0 (...!L)
'APo

### 1!

E2 E

### ~n

_{+ 1}

A _{El} I ,n

2 n n + 1

Probability of movement 1 _{Po} _{Po} _{Po} _{Po}

2 n-1 n n n +1

Probability of staying 0 I-po Po-Po Po - Po Po - Po

Density Funotion of

Considering the density function of distance traveled by a particle as a random variable, let us now find the mean density function of f(s): E (f(s» .. (l-p )....L e-si.. Po +P (l-p) 1 e -s

*lA*

Po
o A Po 0 0 AP(j
(p_{o}n-l

_{-}p

_{0}n) e-s/~p

_{..0}

**+ •••••**

**Cl**

### -

1### .

APOTherefore, the mean density function of distance s traveled by a

par-ticle has the following form:

### ..,

### ....

_{s::}S :t Gl s:: 0

**l-**

x _{'f9~}

**c**Flow Fig.

### 8

*(SO)*

39

Figure 8 shows the cross section and the rectangle with unit Width and semi-infinite length where all particles start their move-ment ~hich together form the number of particles N in motion. The number of particles per second that cross1:."16Une labeled "section"

must be equal to the number of particles starting to travel a variable

length in one second that is long enough 'to cross the line. The

nua-ber of particles per unit width beginning to move in one second fron

the differential area, which is at a distance x from the line, is

given by (Einstein (7)):

in which Ps is the probability that the particle will sta-..-t moving

in one second, d is the d1alneter ofthe particle, and ~d2 is the

area that the grain covers at the bed. The probability that·a

par-ticle will cross the line is designated by Px(s ~x}, It is given

by:

The total number of particles beginning movementthat are large enough

to reach or cross the line per unit width:

**_ 5",£**

PsPo A

A. has the dimensions of length and may be termed the mean distance traveled by a. particle i f the probability of movement is unity.

TransposJ.llg

(54)

Noli', assuming that ~, the mean step length, is equal to hod,

### ho

being dimensionless,

PoPsA.o

"'---I-po

*(55)*

In order to make the right-hand side dimensionless, Ps must be

multi-plied by a given time tol to must be a characteristic time which

in someway is representative of the behavior of a particle in the

fluid under the given flo\! conditions. In fact, to is the time

through which the forces act on the particle and the' particle moves

from its position of rest to occupy a new position of deposit. By

this definition to may be llritten as

(56)

where t l is thepart of the time to during which the particle is at

rest (between two consecutive excursions) and t_{2} is the part during

which the particle is in motion. When the particle is subjected to

hydrodynatlic forces, the particle vibrates and tries to free itself

from the surrou.,ding particles before a. new journey. At low shear values

the movies have sholm that in general, for coarse particles and steady

mean fiotls, t

_{z}

(time of travel) is much smaller than t
l (tiJ:le of

rest). !heefore, for allpractical purposes it maybe assumed that

to:= t_{l"} This time should be dependent upon the flow and sediment
characf.oristics. The s1mplest time pa.raIIIeter having the right

di-41..

mensions is the time required to move a particle through a distance d

w1th the shear velocity U*• The time through which the hydrodynamic

forces should act on apartlcle to move it,would also be proportional

to the probability of movement, Po' of the particle. Aparticle ha.ving

a large value of Po would require less time than a particle having

a smaller value of Po' Considering these arguments, the follottl.ng

expression may be writtg~:

where A_{3} is the constant of proportionality.

### As

transposing

Po .. Ps x t

Substituting the value of Ps in Eq.(55) yields:

### P 3 A

*U*

HAd'" 0 o.

1

### ~(l-po)d

(58)Not.ing that qs" NYS"2d3 mere qs is sediment transport rate and

### Azd

### 3

_{is the volume of}

_{a}

_{particle}

1.2

### >0

### ~

### p/

AlA) d I-po

Two sides can be grouped into two dimens10nless groups:

### t

T~Calling the left-hand side, qs*' the dillIensionless sediment transport,

the above equation 1s reduced to the form

where A ..

### At

_{oIA1Aj}and Func ('&) represents the relationship

1/2

### *

bei...csn T 0* po'J/(l-po) andT 0*.

Because Eq.

### J8

wasderived for spherical particles, anothercon-stant

### B

is introduced to account the variation in particle's shape. Nowequation maybe put in the following form:

q .. AFunc(BT )

s. Olf (60)

The :function has been determined analytically with B .. 1. Tho variation

of q with T (when A;; B:= 1) is shown in Fig. 9. The parameter

S* ~

qs. is similar to Einstein's bed load function and T~ is the

IV. COLIECTION OF DATA

The science of sediment transport relies on the analysis of

obser..red- data. This is particularly true owing to the complexity of

the natural phenomena and the fact that a purely analytical approach

would lead nowhere. As there is available an abundance of flume data

on bed. loadtransport above the so-called 'critica1 condition'. it. seemed

unnecessary ;;0 conduct more experiments in this region. An

eXJleri-Illental progran wasu-"ldertaken to collect bed load data below and in the proximity of tre so-called 'critica1 condition.'

Collection of Data

The fiu,":Ie data from previous studies which are utilized in the

present investigation are those of Gilbert [13] • Casey (13). and the

United States Waterways Experiment Station [131. The details are given

in Table

### r,

Experir:!ental Set-U'D

All the present bed load experiments were conducted in a tilting

f1U1'1e. The water was taken from the Mississippi River through the

labora1tory supply system to the entrance of the flume stil1ing chamber. Flume:

The flume itself. illustrated in Photo 1. is a steel structure

with painted side walls forming a hydraulica.1ly smooth surfa.ce. The

value of Manning's rugosity coofficient for the inside surface was found

'to be 0.00923. The flume is rectangular in cross section • Its

approx-1.ma.to inside dimensions are: length. 50 it; width. J ft; and depth.

1.25 rt. I t is supportedby two I-beams which in turn are supported

Type of Spec1fio Classification

S. No. Designation of Material Material Gravity of Mixture Particles Mean Size

lIlJ1l.

1 Gilbert river

### 2.69

un1-granular sUb-e.n6\1lar### .305

sand

### •.506

11### "

sub-rounded 1.71### 3.17

### 4.93

7.011### 2

U.S. Waterways Experi- river### 2.65

un1-granular sub-angular### .691

ment Stat10n sand angular

### .976

sub-rounded

### 4.07

tosub-an-gular

Casey sand 2.70 unl-grenular sUb-angular 2.45

to rounded

45

screws. Through an electrically operated arrangement of jack screws

can be ma.nipulated to set the flUJ;1e to any desired slope.

The water was admitted from the main channel through a l2-in.

pipe controlled by a hydraulic valve. This_pipe is provided with a

Pitot cylinder at its entrance section to measure the flow rate in terms

of deflection of a differential manometer. This Pitot cylinder was

ealibrated by means of weighing tanks. The pipe diseharges water into

the inlet tank of the flume. This tank is equipped with baffles which

effeetive1y still the flow before it is allowed to enter the main portion of the flume.

The elevation of the water surface in the flume is controlled by

manipulating a tail gate hinged to the bottom of the flume a.t the

dosn-stream end. After being discharged over the tail gate, the water flowed

to the Mississippi river througha channel under the floor.

The position of the water surface and the bed wasmeasured with

point ga\18es which could be read to 0.001 ft by means of Vernier scales.

The point gauge ~iaS mounted on a carriage whieh moved along the flume

on level tracks. These tracks, one on each side of the flume, are

supported on steel framework resting on the floor. An adjustment screw

a.t each support. made a.ccurate leveling of the track possible. For

mea-suring the position of tIE bed a 2-10. -diameter circular plate was attached

to the pointer.

The measti:rement of the rate of movement of sand and gravel under

the test was made possible by the sand trap provided at the downstream

end of the flume. This trap consists of a suddenly enlarged section

about

### 5

inches deeper than the bed. This trap was found to beMaterial Used

The purpose of these experiments llaS to obtain bed load transport

data at low shear values in the proximity of the so-called critical

tract.ive force. In order to calculate shear accurately the depth of

flow and slope should not be of a small order of magnitude. The

sedi-Clent transport rate shotl.1d also be measurable. These condit10ns ilIIposed

a lower ~1mit on the particle size that could be used. Based on these

req'.:.lrements. three uniform materials and two milCtures were selected.

Before being tested

### m

the flume a small representative sampleof the material was subjected to a complete size and shape analysis.

The size analysis was conducted in a sieve shaker. For shape analysis

all three axes of the particle were measured with a micrometer screw

gauge in order to calculate the sphericity coefficient. The results of

size and shape ana.lysis are sumrrarized in Table 11. Figure 10 shows

the sieve analysis results graphically. Procedure of Exnerimentation

Before placement in the flume each o~ the sediments was thoroughly

washed to remove all traces of silt, clay, and other extraneo\ls material.

The flume was adjusted to the desired slope by regulation of jack screws.

The carr1a.ge trackwa.s checked by a level at 2-£t intervals along the rails.

All experiments were made with a uniform layer of sediment on

1lhe bed of about

### J

or### 5

inches thickness. The gravel surface was moldedto the exact slope b:r means of a vertical template so constructed as to

slide along the side rails of the flume. At the upper and lower ends

of the flume, wooden sills hadpreviously been placed to the same depth.