Sediment Transport: an Appraisal of Available Methods

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Report No INT 119

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SEDIMENT TRANSPORT:

AN APPRAISAL OF AVAILABLE .f).1ETHODS

VOLUME 1 SUMMARY OF EXISTING THEORIES

By

W. R. WHITE B.Sc. Ph.D. C.Eng. MICE H. MILLI Ingeniero Civi1

A. D. CRABBE C.Eng. MICE

November 1973

Crown Copyright

Hydrau1ics Research Station Wa11ingford

Berkshire England

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CONTENTS Page NOTATION INTRODUCTION ₁

BED LOAD EQUATION OF A SHIELDS (1936) BED LOAD EQUATION OF A A KALINSKE (1947) THE REGIME FORMULA OF C INGLIS (1947)

2 4 6 BED LOAD EQUATION OF A MEYER-PETER AND R MULLER (1948) 7 BED LOAD EQUATION OF H A EINSTEIN (1950)

TOTAL LOAD FORMULA OF H A EINSTEIN (1950)

10 15 BED LOAD EQUATION OF H A EINSTEIN AND C B BROWN (1950) 17 TOTAL LOAD FORMULA OF A A BISHOP, D B SIMONS AND

E V RICHARDSON (1965) . 18

21 BED LOAD EQUATION OF R A BAGNOLD (19561

TOTAL LOAD FORMULA OF R A BAGNOLD (1966) TOTAL LOAD FORMULA OF E M LAURSEN (1958}

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BED LOAD EQUATION OF J ROTTNER (1959) BED LOAD EQUATION OF M S YALIN (1963) REGIME FORMULA OF T BLENCH (1964)

26 28 29 31 TOTAL LOAD FORMULA OF F ENGELUND AND E HANSEN (1967)

TOTAL LOAD FORMULA OF W H GRAF (1968}

33 35 TOTAL LOAD FORMULA OF F TOFFALETI (19681 35 TOTAL LOAD FORMULA OF PACKERS AND W R WHITE (1972) 41

REFERENCES 44

APPENDICES

1. Eva1uation of Y

c,

wand

v

49

2. Solution of integra1s used in the Einstein methods 51

TABLE

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CONTENTS (Cont'd) 1. Surnmaryof theoretica1 methods

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NOTATION

a Constant, 2.45 Y~ /sO.4 (Yalin)

cr . .

Exponent in the concentration distribution equation (Toffaleti)

a

Constant (Einstein)

b

Value of Fgr at initial motion (Ackers, White) Stream breadthf surface width if not otherwise indicated

A

B* Constant (Einstein)

C Coefficient in the general function (Ackers, White) C Constant of proportionality in the concentration

x

distribution equation (Toffaleti)

C Concentration as weight per unit volume in a given xp

size fractionf p, in the lower( middle and upper zones of transport

CLp Concentration as weight per unit volume in a grain size fraction( p( in the lower zone of transport

(Toffaletil

CL2 Constant of grain volume/constant of grain area (Toffaleti)

C

z

Temperature related parameter used in evaluating the middle zone exponent of the concentration distribution d Mean depth of flow

dl Mean depth with respect to the grain Sediment diameters

D* Sieve size No 80 (US Standardl.

D Dimensionless grain size (Ackersf White) gr

F gr Fr F s g GF P k m k r k s k se k w K log

Suspended 10ad transport efficiency

Efficiency of the sediment transport process (Ackers, White)

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Bed factor (B1ench)

Bed factor when there is no movement of bed partic1e-s (B1ench)

Sediment mobi1ity (Ackers, Whi.te) Froude Number

Side factor (B1ench)

Acce1eration due to gravity

Bed load transport rate, dry weight per unit width per unit time

Tota1 transport rate,.dry weight per unit width per unit time

Bed 10ad transport rate, dry weight per unit time Reyno1dJs number for solids.sheared in.f1uid

Nucleus 10ad, dry weight per unit width per unit time for a given grain size range

Integra1s (Einstein)

Correction'factor for mode1s

Coefficient of bed roughness, grain and form (Stric~ler) Coefficient of partic1e friction, p1ane bed (Strick1er) Grain roughness of the bed

Coefficient of tota1 roughness, grain and form (Strick1er)

Coeff!cient of side wa11 roughness (Strick1er) Meander slope correction (B1ench}

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ln m M n n p P,P P q Q R RI s s* S o

Naperian logarithrn (base e)

Exponent in the general function (Ackers, White)

Sediment transport rate, mass per unit width per unit time

Transition exponent (Ackers, White) Manning roughness coefficient

The proportion by weight of partieles of size Dor, p as a suffix, denoting association with a particular size fraction

Fraction of bed material in a given range of grain sizes

Fraction of bed load in a given range of grain sizes Fraction of the total load in a given range of grain sizes

Factors which indicate the proportion of the bed taking the fluid shear (Kalinske)

Water discharge per unit width

Bed load transport rate, submerged weight per unit width

Total load transport rate, submerged weight per unit width per unit time

Water discharge

That portion of Q whose energy is converted into eddying close to the bed

Hydraulic radius

Hydraulic radius ascribed to the grain Specific gravity of sediment

Expression( (Y/Y - 1) (Yalin} cr

y* Distance above the bed

Y Dimensionless Mobility Nurnber (Shields)

YC Mobility number at threshold conditions

Z Bxponent relating to the suspended distribution

Z Exponent relating to the velocity distribution v

Z Exponent relating to sand d!stribution in the middle p zone of transport

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t A variable tg ~o Coefficient of dynarnicfriction

T A parameter including variables which are functions of temperature

TF Temperature of water in degrees Fahrenheit

v Flow velocity at a distance y* above the bed

v*' Shear velocity related to grain (Einstein)

v* Shear velocity (TIp)

v*c Critical shear velocity

V Mean velocity of flow

w

Fall velocity of the sediment particles

X,xs,xb Concentration by weight, ie weight of sediment/weight of water .. Subscript s denotes suspended load,

b denotes bed load.

X Characteristic grain size of a mixture x Parameter for transition (smooth to rough)

y Pressure correction in transition (smooth to rough)

A coefficient (Ackersf White and Bagnold)

Logarithrnicfunctions

p

Specific weight of the fluid

Density of the fluid

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'Ys

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no~

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'[ 0 r c

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0

0'

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lIJ'lIJ*

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<1> <1>*

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À fj

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Kinematic viscosity

Specific weight of grain in fluid, g(ps-p) Constant (Einstein)

Hiding factor Correction factor

Stream power per unit boundary area Tractive shear

Tractive shear associated with sediment particle Critical tractive shear to initiate motion

Thickness of the lamïnar sublayer Laminar sublayer with respect to v*'

Intensity of shear on a particle

Intensity of shear on representative particle Intensity of shear for individual grairi sizes Intensity of transport

Intensity of transport of individual grain sizes Friction factor, 8gRSo/V2

strip width, Sirnpson Rule Surn of •••

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SEDIMENT TRANSPORT:

AN APPRAISAL OF AVAILABLE METaODS

VOLUME 1 SUMMARY OF EXISTING THEORIES

INTRODUCTION

In 1972 Ackers and White (Refs 24 and 25) proposed a new sediment transport theory and tested the theory against f1ume data. This data consisted of about 1000 measurements with par~ic1e sizes in the range 0.04 < D(mm) < 5 and

sediment specific gravities in the range 1.07 < s < 2.65. Subsequent phases in this investigation have inc1uded (i) the acquisition of 'about 270 field measurements of transport rates from the 1iterature and (ii) the testing of all commonly used sediment transport theories against this extended range of data. These two phases are the subject of the present report. The report also inc1udes a proposed modification to one of the existing stochastic theories which significantly improves its performance.

Volume 1 describes the sediment transport theories and indicates in each case how the theories have been used in the present investigation. Where the theories include

graphical solutions, analytical equiva1ents have been worked out to facilitate the use of a digital computer for the

analysis. The limitations of the theories are indicated

wherever the original authors have given specific recommenda-tions.

Volume 2 describes and classifies the data used for comparative purposes. It defines the criteria on which the comparison between observed and calculated transport rates are based and presents the results provided by the theories described in Volume 1. The problems of graded sediments are discussed and illustrated using gravel river data. A

modification to the Bishop, Simons and Richardson theory is proposed together with suggestions for further refinements which could be introduced at a later date. The performance of the various theories is compared and recommendations are made regarding their usage.

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Ys _G 10 T-Tc S y2

Q

'

=

D Y_s 0

Where Y

=

specific weight of the fluid

'Ys

=

specific weight of solids in transport

T

=

tractive shear

TC

=

critical tractive shear

.... (1)

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BED LOAD EQUATION OF A SHIELDS (1936)

In 1936 A Shields( Ref (I), gave the following,formula for the ped load transport rate:~

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D

=

mean diameter defined as D50 Q

=

water discharge

G

=

bed load transport rate, dry weight per unit time S

=

Energy 9radient or surface slope

0

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2

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Substituting Q = bdV· •••(2) G

=

_{b gbt} • •• (3) S

₌

v. /gd2 •••(4) 0 2 gpdSo • •• (5) T

=

_pv*

=

gbt

=

XbVdpg •••(6) 'YsI>'

=

s-l • •• (7)

in equation (1) gives a concentration for the bed load, Xb,

as:-I

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4 10 v* 2 2 (1 -(s-l) Ddg \ • •• (8)

But defining the mobility number at the threshold conditions

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as:-2

=

V*c/(gD(s-l» • •• (9)

x

_b

=

4 10 v* ( 2 2 1 (s-l) Ddg y - v*2/g~(S-1») •••(10)

This equation is claimed to be valid at low transport rates and under conditions where bed formations are relatively flat. Knowing v, v*' d, D, s and g sediment concentrations are calculated as

follows:-l. D is computed from equation (161), see page 40 gr

2. y is calculated from equation (A4), see cr

Appendix 1, or from the original graphs of A. Shields (Ref 1).

3. (10)

Substitution in equation (10) yields Xb· is, of course, non-dimensional.

Equation

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BED LOAD EQUATION OF A A KALINSKE (1947)

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A A Ka1inske(2) (1947) deve10ped an equation for computing the bed load sediment transport rate. He a1so presented a method for adapting the equation to sand mixtures. It is as

fo11ows:-••• (11)

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where gbtp

=

bed load transport rate in a range of grain

sizes, dry weight per unit width per unit time.

pin p

=

p p p E(p!n ) p •...(12)

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P, P

=

factors which indicate the proportion of the

p

bed taking the f1uid shear.

D

=

effective partic1e size (diameter) within a p

given range of sizes.

The factor P was taken as 0.35 (Ref. 2) and P was p eva1uated using the expression

v*(s-l) Xb·

=

--.-=-:;--Vd n E1(p • fel Il ) ) C 0

P

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Substituting equation (12) in (11) and using (6) to

re1ate gbt to the concentration by weight gives p • •• (13)

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where n

=

number of fractions into which the grading curve

is divided.

The va1ue of lc was given by Ka1inske

as:-lc

=

~(ps-P)g Pp D

P ••• (14)

Hence, using (12) and substituting lo

₌

_pv*2

4

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=

1 . P(s-l)g •

"3

2 v* p _{• •• (15)} I;n1(p/D ) . P

for rough_turbulent flow.

In programming for the computer, the gra~h expressing the function of T IT. in the original paper of A A Kalinske

c 0

was brought to the fol10wing analytical form:-Jf ·0.40 < T.IT. < 2.50 c 0 • •• (16) If o ~ T.IT. < 0.4 c 0

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f

(:c)

=

10(0.375-T.c/T.o)/0.945 o ••. (17) All the equations are dimensionally homogeneous so that any consistent set of units can be used and the calculation proceeds as

follows:-1. Thè representative grain size is chosen as the mean diameter of each size range into which the

grading curve is divided.

2. The ratio T.IT. is evaluated using equation (15) c 0

with P = 0.35 for each fraction.

3. The function f(T. IT. ) is computed either with a c 0

graph (Ref 2) or by using equations (16) and (17). 4. The total concentration for the whole bed material load is computed from equation (13).

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5

THE REGIME FORMULA OF C INGLIS (1947)

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The regime theory rests on the principle of channe1 se1f-adjustment. According to E S Lind1ey(3) (1919) this principle can be expressed as fo11ows:- "When an artificia1 channe1 is used to convey si1ty water, both bed and banks scour or fi11; depth, gradient and width change unti1 a state of ba1ance is attained, at which the channe1 is said to be in regimeI~•

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In the discussion on the paper "Meanders and their bearing on river training" by C Ingli~(4,5) 1947, as an

answer to C M White, he suggested the fo11owing dimension1ess set, in which the effects of the sediment "charge" was

introduced into the Lacey regime equations of that time:

~ ~ X w)0\ b = 2.67 _{1/3Q 1/12 ( ;} 9 v . 7/1B Q1/6 (OX w)1/12 V = 0.7937 9: 1/36 s v d = 0.4725 v1/9 Q1/3 01/6 gl/1B(X w)1/3 s (OX w) 5/12 S = 0.000547 s 5/36 1/18 Q1/6 0 v 9 •.•(lB)

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•••(19) •••(20)

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•••(21)

A1gebraic rnanipu1ationof equations (lB) and (19)

produces:-X

=

0.562

s •••(22)

in which w is the fal1 velocity of a characteristic sediment partic1e which is assumed to be the grain having the mean diameter (D50) of the bed materia .

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The above formu,laeare for quartz sand in water, and can be used with any consistent set of units.

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The calculation proceeds ,as follows:~

1. The fall velocity is computed either using Rubey's equations (A5) (See Appendix I) or from any experimental curve,for the mean diameter 050•

2. The concentration is determined from equatlon (22).

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.BED LOAD EQUATION OF A MEYER-PETER ANO R MULLER (1948)

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Meyer-Peter and Muller(6) (1948) determined, for the bed load transport rate, an empirical relation which can be written as follows:

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Q

(k

)3/2 1/3 Y QS k:e dSo

=

0.047 'YsO + 0.25 (~) • q~~3 •••(23)

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where qbt = bed load transport ratet submerged weight per

unit width.

Qs

=

that part of Q whose energy is converted into

'eddying near the bed.

k = coefficient of total roughness due to skin and se·

form roughness in the Strickler formula.

k

=

coefficient of particle friction with plane bed

r

in the Strickler formula.

0

=

effective diameter of the sediment given by o_.

=

EP

₁ p .0 in which p is the fraction of bed

P

material in a size range D

.

P

7

Now,

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•.• (24)

Hence using equations (4) and (24), equation (23) reduces t.o the

form:-I

3 [ Q k 3/2 ] 3/2

=

8 sv

*

___.ê. ( se ) _

o.

047 (s-l)Vdg Q \kr v*2/gD(s-1)

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••. (25)

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Qs

According to the authors of Reference (6) the ratio

0-

is determined for rectangular channels as

follows:-b k3/2 w

=

2d k3/2 + b .se ·•. (26)

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in which k is the side wall roughness

w

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and k is computed using the following

expressions:-se k m

v

=

=

R2/3 _So~. _v* Rl/6 Vg~ • •• (2 7)

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k se ••• (28)

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When b + 00 equation (281 reduces to (27) and for

practical purposes( natural rivers, k can be taken as k

b se m

when d ~ 15. It can be proved that in this case the error in the computed value of k is about 10 per cent. Por

se

typical prototype sizes equation (26) also reduces to unity and Qs

=

Q.

Most experimental flumes are constructed of concrete, glass or steel so that k can be estimated

as:-w

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k w 1 1

= - = ----

,

= 100 n 0..01 • •• (29)

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where n is the Manning coefficient.

The value of k is given by r

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k r ••• (30)

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The friction factor À is obtained from the well-known

equation of Nikuradse in which it is e~pressed as a function of the Reynolàsnurnber and the relative roughness

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l_

= -

2 log (1~9~ ~Q + 0.6275\») •••(31)

Ir

.

s

RVIr

In the region of fully developed turbulence k can be

r calculated

with:-I

kr

= ~

(metric units) Dl,6. 90 ••• (32)

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Except equations (27), (28), (29), (30) and (32) which are in mand mis, all the equations are dimensionally homogeneous

50 that any consistent set of units may be used with them. The calculation proceeds as follows:

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1. The value of D, effective diameter of the sediment, is deterrnined by D

=

L

l

p Dp'

2. The values of k and the ratio Q

IQ

are deterrnined

se s

from equations (28) (with kw

=

100) and (26) respectively, when the sediment transport rate in experimental flume

is required. In natural rivers k

=

km using equation se

(27) with R = d and Q = Q. s

3. The value of k is evaluated using equations (30),

r (3l) or (32)..

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4. The concentration by weight is computed by using

equation (25).

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BED LOAD EQUATION OF H A EINSTEIN

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The bed load re1ationship deve10ped by H A Einstein is

derived using considerations of probabi1ities, (Ref 7). His

method, proposed initia11y in 1942, was considerab1y modified in his later works dating from 1950 onwards.

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In this method the bed load discharge is computed for individua1 size fractions within the who1e bed material. This means that the size distribution of bed load is a1so obtained.

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The H A Einstein Bed-Load function is as

fo11ows:-1 A*<p* B*W*-

In

1

f

_{_t2 0} 1 _{+ A~<p*}

=

1 - e dt /TI ₁ -B W-* W-*

n

0 where PB ~ <p* qbtP

=

(Y D )3/2 Pb s p

e

(s....l)D g W'k

=

_{~p Y 2} 12 (v~)2 p*

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• •• (33)

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• •• (34)

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• •• (35)

where PB

=

fraction of the bed load of a given grain size,

Pb

=

fraction of the bed material of a given grain size,

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<p*

=

intensity of transport for individua1 grain size,

v!

=

shear velocity with respect to the grain,

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~

=

"Hiding factor" of grain in a mixture,

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y

=

pressure correction in transition (smooth-rough).

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According to Einstein,

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~2 [

lO(

10.6 J2 ~: =. log 10.6

*)

•••(36)

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and B* = 0.143, A* = 43.50, "0 = 0.50.

Substituting from equation (24) the bed 10ad

concentra-I

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tion associated with a particu1ar size fraction

becomes:--

~ s D3/2 g~ Pb~* (s 1)

X_bp

=

P X_B

=

_Vd E •••(37) and the tota1 concentratiön of bed 10ad

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Xb

=

EP X

1 bp • •• (3 S)

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Einstein considered that the velocity distribution in an open channe1 is described by the 10garithmic formu1ae based on V Karman's simi1arity theorem with the constant as proposed by Keu1egan(S). He gave the mean velocity inc1uding the transition between the rough and smooth boundaries as:

~

.

=

5.75 log (12.27

~,)t

...

(39)

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in

which:-d'

=

mean depth with respect to the grain, v!

=

shear velocity with respect to the grain,

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/::, = the apparent roughness of the surface,

k

where /::,=- s

x

k = the roughness of the bed. s •••(40)

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x

=

function of k./0' s

6'

=

the thickness of the 1aminar sub1ayer at a

smooth wa11, 11.6 v/v!. •••(41) t In the origina1 works R' is used rather than d'.

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The value of X in equàtion (36) can be calculated according to Einstein as

follows:-I

X

=

0.77~ if

~/o'

> 1.80 grain size • •• (42)

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X

=

1.390 if

~/o'

< 1.80 or if there is a uniform

and the values of the two correction factors in equation (35) are given

by:-ç;

=

f (i) (D/X) f (ii)(v*D/v) y

=

f(iii) (k /0') s •••(43)

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•••(44)

H A Ein~tein presented graphs for the evaluation of x,

ç;

and y. The hiding factor,

ç;,

was introduced to define the influence of the mutual interference between the

particles of different sizes with respect to the buoyancy forces. The original paper of Einstein(4) gave the factor

ç;

as a function of D/X only. In his later works carried out with Ning Chien(9) in 1954 he introduced a second correction factor, this being a function of v*D/v. Equation (43)

represents the more recent concept.

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The Einstein method includes a procedure for computing mean velocity, obtaining v*" from a bar resistance curve by trial and error and hence v*' since v* is known. The v*' value enables x to be calculated and the mean velocity can then be determined. Howevert in the present exercise,

measured mean velocities were available and the following

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procedure was adopted for determining

(v

*

J ) 2

=

gd'S0

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•••(45)

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2 gdS v*

=

₀ (v*I )2 Hence d'

₌

d

.

2 v* •••(46) •••(47)

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12

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Substituting equations (47')and (40) into (39) gives (V!)2

=

5.75 log ( 12 •27d. 2 °65 v*

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v

•••(48)

(ks has been taken as 065) •

The va1ue of x as a function of °65/

0'

is eva1uated from either the graph given by Einstein or the fo110wing set of equations:-If _°65/0' < 0.4 ; x

=

1.70 log °65/0' + 1.90 If 0.4

,

_°65/0' < 2.35; x

=

1.615 - 1.544 (log °65/0,)1.6 If 2.35 , D65

/O'

< 10 x

₌

0.926(10g D /6'_1)2.43 + 1.00 65 If 10 < D65

/O'

x

=

1.00. ••• (49)

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Vanoni and Brooks (Ref 23) give a direct graphica1 method and the reader interested in this approach shou1d refer to this work. In programming for computer the graphs given by Einstein for determining ~, y and

e =

f(iii) (v*O/v) from equations (43) and (44) were expressed ana1ytica11y as fo110ws:-1. Hiding factor, ~

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°

If 0.10 ,

X

< 0.73; If 0.73

,

D < 1.30; X If 1.30 <

n

X with:-v*O If

--

,

3.50 v

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~

=

(i)","2.385 0.70

e

~

(~)-O.

692

=

1 •20 """"ë"'""' t;

=

-8-1.00 ••• (50) v*O (0.544 - log

---v-)

e =

10 0.412 13

V*D If --_\) > 3.50

I

I

e

=

1.00 .•. (51)

I

I

2. Pressure correction in transition (smooth-rough)

D D 1.187 If ~ , 0.47 _Y

=(0~5)

ó' D65 ( D65 .)2 If 0.47 < , 1.70; 10-2.23 log ~-0.0492 -0.083

T

y

=

D65

o

s(D65) -0.378 If 1.70 <

T

,

3.15; Y

=

.

0'

If 3.15 < D65 _{'" 5.00;} 0.525

T

y

=

I

I

I

I

If uniform grain sizei y

=

1.00 .•. (52)

I

I

I

I

All the equations are dimensiona11y homogeneous so that any

consistent set of units may be used. The va1ue of the

integra1 in equation (33) was eva1uated by expanding the

integranà in Mac1aurin's series and integrating term by term

(see Appendix II).

The ca1cu1ation proceeds as fo110ws:~

1. From the granu10metric curve D65 is taken as the

skin roughness kso

2. The friction velocity v~ is ca1cu1ated either using

the method described previously or the direct graphica1

method given in Ref (23).

3. The thickness of the 1aminar sub1ayer

0

is obtained

from equation (41).

I

I

I

4. The correction factor x is evaluated either from

the ~raph given by Einstein or from the equation set

(49)•

5. The apparent roughnes$ ~ is obtained using equation

(40) •

I

I

I

14

I

I

I

I

I

I

6. From the ratio

A/6'

the value of X is calculated using the set equation (42).

I

7. The pressu~e correct ion factor is determined using the set of equations (52).

8. The ratio p2/~*2 is calculated from equation (36).

I

9. The representative grain sizes are chosen as the

mean diameter of each size fraction.

I

10. The "hiding factor" ~ of each fraction is

deter-mined from the sets of equations (50) and (51).

I

11. For each grain slze fraction the parameter ~* is

calculated using equation (35).

I

I

I

I

I

12. The intensity of transport,

4>*,

for individual

grain size is evaluated from equation (31) (see Appendix II) .

13. The concentration by weight for each fraction and the total concentration for the whole bed material are then computed using equations (371 and (38).

I

I

I

TOTAL LOAD FORMULA OF H A EINSTEIN (1950)

In his 1950 paper (Ref 7) Einstein also gave a method of determining the total sediment load, excluding wash load. The total load in a parttcular grain size range was given

as:-I

I

I

I

I

••• ($ 3)

in which PT

=

fraction of the total load in a given grain size

gst

=

totalload, dry weight per unit width per unit time.

15

Z-l 1 Z Il 0.216 A

J (

l~t)

dt •••(54)

=

(l-A)Z A Z-l 1 Z 12

=

0.216 A

!

_(1~t) In t dt ••.(55) (l-A)Z A

where A

₌

ratio of bed layer thickness to water depth

A

₌

2D/d ••• (56) Z

₌

2.5w/v*

,

•••(57) 1 ( 30.2) _{•••(58)} p

₌

0.434 log b./d

I

I

I

I

I

I

I

I

The transport rate is related to the concentration by weight using equation (24) and making these substitutions in equation (53) we obtain

• •• (59)

I

I

and the ~otal bed material load for all the fractions is given by

x

=

r

P X 5 1 sp • •• (60)

I

I

I

The two integra1s in equations (54} and (55) were so1ved by numerical integration using Simpson's Rule (see Appendix

II) •

The calculation proceeds as

follows:-I

I

1. The sett1ing velocity is calculated from equation (A5) using the mean diameter for each fraction of the bed sample (see Appendix I).

2. The values of Zand A are calcu1ated using equations (57) and (56) for each fraction.

3. The integra1s 11 and 12, equations (54) and (55) are eva1uated by numerical integration (see Appendix II) or by the original graph of H A Einstein (Ref 7).

I

16

I

I

I

I

I

I

I

_{4. The value of P is determined from equation (SB).}

I

5. Introducing the values of Xbp from equation (37)

in equation (59) the concentration, X , is obtained,

sp

and from equation (60) the concentration for all the fractions.

I

I

I

_{BED LOAD EQUATION OF H A EINSTEIN ANO C B BROWN (1950)}

I

I

I

I

I

The bed load equation of Einstein-Brown is a

modifica-tion developed by H Rouse, M C Boyer and E M Laursen of the

formula of H A Einstein. It was présented by C B Brown (li)1

in 1950 and appears to be based on empirical consideration~. The equations can be written as

follows:-where

i

₌

_{f (~)} F <I> gbt

=

3/2 p (5-1)~ 03/2 g _{5 50} 2 v* l/W

=

{s-1)050 g 2 ~ ~ F

= (;

+

36 \)₃

)

( 36

v

2 ) - g D~O{S-l)

..

g DSO(s ....l) •••(6l) • •• (6 2)

I

• •• (6 3)

I

I

• •• (Ei

4 )

I

I

I

The relationship between <1>, F and W was given by the authors

of Reference ll·in a semi-graphical form. When l/W exceeds

0.1 it becomes <I>/F

=

40·(l/w)3. To programme the relationship

when l/W is'less than 0.1 the function was approximated by

an exponential equation

I

I

I

m -4.0 + (23.695 + 16.925 log

i)~

.x..

=

10 F ••• (65) 17

I

I

Combining the above equations and relating the bed load transport to the concentration by weight we obtain the

following predictive

equations:-I

I

2 v* If D ( 1) ;.0.1 g 50

5-I

If gD50(s-1) < 0.1 ; Xbt

=

_Vd . <P _{••• (6 6)}

I

I

where <p is determined from equation (65)

I

For the above expressions the quantity F appears in the

Rubey equation (see Appendix' I) and is defined as a

dimensionless function of fall velocity.

I

The calculation proceeds as follows:~

1. The dimensionless function of the fall velocity is calculated using equation (64).

I

2. The concentration of bed load transport by weight

is evaluated using the equations (62) to (66) inclusive.

I

I

I

TOTAL LOAD FORMULA OF A A BISHOP, D B SIMONS

AND E Y RICHARDSON {1965}

I

The quantities A* and B* which appear in the relation

(33) were assumed by H A Einstein as constants. Later works of A A Bishop , D B Simon's and E V Richardson (Ref 10)

revealed that the relation (33) can be improved if A* and B* are treated as variabIe quantities.

I

I

I

The authors of Reference (10) reasoning that "the

instantaneous variations in the lift forces may lift some

I

18

I

I

I

I

I

I

_{particles from the bed into suspension whereas other particles}

will be moved only within the bed layer", concluded that the

relationship ~. - $. should be concerned with the total bed

material sediment transport rate rather than with the bed

load transport rate. Working on this basis they found a

relation between A. and B. and the grain size 0.

I

I

I

The experiments reported do not make clear the influence

of such parameters as specific gravity and temperature. The

graph of this relation is expressed here in the following

I

I

analytical :eorm: If _°50 < lmm _B*

₌

0.0375 + 0.1054 °50(mm) If °50 ) lmm ; B*

=

0.143 If °50 < 1.12 mm A*

=

(°50 (mm)0.2

+

lY

If D50 ) 1.12 mm _A*

=

43.5 •••(67)

I

I

I

The method outlined by Bishop, Simons and Richardson

involves:-I

I

(i) The computation of $' as in the Einstein method

but, in order to utilise the measured mean velo-

_,

cities available from experiments, v. was computed

by the method presented by Vanoni and Brooks (Ref

23). $' is then given by

$'

=

035(s-1)g/(v*,)2 •••(68)

(ii) The use of expression (33) in the following

form:-.1

I

I

I

=

1 ••• (69)

I

I

I

I

19

I

I

I

in which ~ is given by ••• (70)

I

I

~ =

(y D )3/2

=

s 50

and A* and B* are given by the set of equations (67). Thus the values of PB and Pb are assumed equal to unity and the concentration of the total load is given by equation (37) in which Pb

=

l.O.

According to the authors of Reference (10) the relati. on-ship described by equation (69) is valid only when ripples and dunes exist. In the transition reg.ime and for later stages of motion systematic errors develop and the relation-ships have been modified to fit the data empirically.

I

I

I

I

In the original works the relationships between ~, ~I and D50 are given graphically.

The calculation proceeds as

follows:-I

I

1. The value of v* is computed using either equations (45) to (48) or the graphs presented in Reference (23).

I

I

2. The shear intensity factor ~I is determined using equation (68).

I

3. The values of A* and B* are obtained from the set of equations (67).

4. The intensity of transport for the total bed

material load is evaluated using either equations (69) and (70) or the graphs given by Bishop, Simons and Richardson in Reference (10).

I

5. The concentra~ion is computed from equation (37) putting Pb

=

1 and Dp

=

D50.

I

I

I

I

20

I

I

I

I

I

I

_{BED LOAD EQUATION OF R ~ BAGNOLD (1956)}

I

I

bed load transport rate.R A Bagnold(12) (1956) gave a formula to express the. It was based on principles of energy and,may be written as

follows:-I

~ qbt p aY~(Y-Yc) (y . D) 3/2 = s •••(71)

I

I

in which y

₌

_{Shield's mobility parameter given by} 2

Y

=

v* /(gD(s-l»

Y

c =

The value of Y at initial motion

qbt·

=

bed load transport rate,'submerged weight pe'r

unit width

a

₌

8.5eb/t 1jJ _{•••(72)}

g 0

eb

=

bed load"transport efficiency tg1jJo·

=

coefficient of dynamic friction.

I

I

I

I

Substituting the Shield's parameter into equation (71) and noting that the bed load transport rate is related to the concentration by weight through

I

I

qbt

=

x

bVdpg(s-l)s • •• (73)

I

we obtain 3

XI>

=

~s:~)v:g

[1 - v.2/(:~(S-1)J

• •• (74)

I

I

I

where D is taken as D = ED P in which p is the proportion by

p

weight of particles in a size range D •

P

R A Bagnold related eb/t ljJ to.particle size and

g 0

presented the functional relationships graphically. In

programming for computer these relationships were introduced into equation (72) in the following forrn:~

I

I

I

I

If D < 0.5 mm a

=

8.50 D (mm)

I

I

If D ~ 0.5 mm a

=

4.25 ••• (75)

I

This theory has the following

restrictions:-(i) The ratio eb/t

W

is given for sand and water.

g 0

It 15 thus only applicable where Ys

=

1.65. It is claimed to be valid only for the earlier stages of the movement, say Y , 0.4.

I

I

(ii)

(iii) In his derivation R A Bagnold assumed a rough plane bed (without sand waves) and rough

turbulent conditions.

I

I

The calculation proceeds as

follows:-I

1. The value of the effective diameter of the sediment forming the bed of the channel is determined from

D

=

El

pDp.

2. The value of a is determined from the set of equations (75).

I

I

3. The dimensionless critical tractive force Y

c

is calculated either from the equation A4 (see Appendix I) or from the original graphs presented by Shields (Ref 1).

I

4. The concentration by weight is evaluated using equation (74).

I

I

TOTAL LOAD FORMULA OF R A BAGNOLD (1966)

I

I

RA Bagnold(13) has developed a theory to express the total load of bed material which is based on similar

principles to the bed load equation.

I

I

I

22

I

I

I

I

I

_{The tota1 load equation can be written}

as:-I

--w • •• (76)

I

I

where qst

=

'tota1 load (bed material) , submerged weight per unit width per unit time

w

=

stream power per unit boundary area

I

=

bed load transport efficiency

I

e

=

suspended load transport efficiency

s

tgwo.

=

coefficient of dynamic friction w

=

effective fa11.ve1ocity

I

I

I

The stream power, w, is related to the hydraulic properties as follows:-and substituting pgQS 0 pgdVS w·

=

= b 0 2 gdS v* = ₀ pVv*2 w

=

·•• (77)

I

••. (78)

I

I

The transport rate for the total load (bed material) is related to the concentration Xs by

(s-l) q

=

X Vdpg'~~~ st s s • •• (79) Substituting (78) and (79) in (76)

I

I

X

=

s· 2 v* s ( eb

l

-+ gd(s-l) t

W

g 0 e (l-e

)~J

s b w • •• (80)

I

In his later works R A Bagno1d re1ates t

W

to a

g 0

parameter c1ose1y analogous to the'f1uid Reynolds number

I

I

t 1./1

=

f{G2} ••• (8l) g 0 2 2 G2 s D v* where

=

2 ••• (82) 14 _v

I

I

23

I

I

I

R A Bagnold gave graphs to determine eb and t ~ but g 0

these have been brought to analytical form as

follows:-I

(i) _tg~O

If G2 < 150 _tg~o

=

0.75

If 150 , G2 < 6000 _tg~o

=

-0.236 log G2 + 1.250

If G2 > 6000 _tg~o

=

0.374 ••• (83) The theory is only claimed to be applicable at high

transport rates, the applicable range being defined by the

I

I

I

expression G2 0.65 3 >

_""14"

.

s(s-l)D 9:

.

_tg~o ••• (84) 2 v

(ii) eb (Only for 'Y

=

1.65 and 0.3 ~ V (mis) ~ 3.0) s

If D < 0.06(mm) _eb

=

-0.012 log 3.28V + 0.150 If 0.06 , D(mm) _< 0.2 _eb

₌

-0.013 log 3.28V + 0.145 If 0.2 , D(mm) _< 0.7 ; _eb

=

-0.016 log 3.28V + 0.139 If 0.7 < D(mm) ; _eb

₌

-0.028 log 3.28V + 0.135 where V

₌

mean velocity of flow (m/s)

D

₌

effective particle diameter.

I

I

I

I

I

I

I

In order to bring his transport formula into a form that can be used for practical purposes, Bagnold assumed, ignoring certain uncertainties, th at the suspension efficiency e has

s the universal constant value 0.015 for fully developed suspension by turbulent'shear flow, and took for the

numerical coefficient in the second term of equation (76) the round figure of

1

0.015

=

0.01.

I

I

I

I

24

I

I

I

I

I

I

I

Equation (76) thus

becomes:-I

2 v* s [eb Xs

=

gd(s-l) ~ + g 0

vl

0.01

;ij

• •• (86)

The quantities D and ware taken as

follows:-I

of the whole bed material • •• (87)

I

I

w

=

LP P

w

1 p

L1

p

of the suspended material ••• (88)

I

Or

L1

p w

w

= ~

P of the whole bed material ••• (89)

L1

P

I

I

I

All equations are dimensionless except the set (85) where the units are mis. The total load theory has the

following

restrictions:-I

.

(i) It is not claimed to be valid at the early stages 2

of movement, say v* /gD(s-l)) < 0.4,

(ii) there is no evidence to support the theory at depths below 150 mm,

I

I

(iii)eb values are related to sand transport in water, Ys = 1.65.

The computation proceeds as

follows:-I

1. The effective grain diameter and fall velocities

are computed from the grading curve of the bed material using equations (87), (88) and (89) and A5 (see Appendix

I) .

I

I

I

I

I

2. The bed load transport efficiency is evaluated either from the original graph of R A Bagnold (Ref 13) or by using the set of equations (85).

25

3. The va1ue of G2 is determined from equation (82).

I

I

5. Concentrations are determined from equation (86).

I

I

4. The coefficient of dynamic friction t ~ is

g 0

eva1uated using equations (83) and (84) or the graphs inRef (13).

I

THE TOTAL LOAD FORMULA OF E M LAURSEN (1958)

I

I

The method of calculating tota1 10ad proposed by

E M Laursen (Ref 14) rests on empirica1 relationships. It predicts both the quantity and composition of the sediment in transport.

I

I

I

The basic equation can be written:

-••• (90)

where LoI

=

tractive shear associated with the sediment grains,

LcI

=

critica1 shear associated with the grains.

2

'D

)1/3

LoI

=

₅₈pV ( 50_d •••(91)

I

I

I

L I

=

C •••(92)

I

I

Substituting from equations (91) and (92) into (90)

yie1ds:-P

(D )7/6

2

D

)1/3

=

0.01~· P ~ (58YC

6

g(S-l)( ~O .

L

1 P

1) .

f(:*)

p ••• (93)

I

26

I

I

I

I

I

I

I

I

The value of Y

c

was related by Laursen to Dp/O as

follows:-I

I

I

If D

/ó

> 0.1 Y

c

= 0.04 P If 0.1 ~ D

/0

> 0 ..03

.

Y

c

= 0.08 P I If 0.03 > D

/0

Y

c

= 0.16 P

where Ó = thickness of .the larninarsublayer

0

= 11.6 v/v*

•••(94)

•••(95)

I

Laursen gave the function of v*/w in equation (93) in a

p

graphical form, the analytical equivalent of which can be

written:-I

I

I

v* _2v* v 0.97 10g-+0.85 log -+1.20 3.0;f(w*) = 10 wp wp p ;~:*)

C*)"30

20 = 5.6 -w P P v* _2v* 3 (v*) 3.16 log --0.57 log -+0.413 10 w w 10 ;f

w

= P P P •••(96)

I

I

I

suggested by Laursen.The p in equation (91) was not in the original equationIt has been introduced in this report

so that the equation becomes dimensionally homogeneous and

Laursen's coefficient has been changed accordingly. The

Laursen method, as it stands at present, is only applicable to natural sediments with a specific gravity of 2.65.

I

I

The calculation proceeds as follows.~

·

1

1. Representative grain sizes D are chosen as the

p mean size of each size fraction, p.

I

I

I

27

I

I

2. The thickness of the laminar sublayer is computed

using equation (95).

I

3. The values of Y

c

for each fraction, p, are determined using the set of equations (94).

4. The values of f(v*/w ) are determined for each

p

size fraction using the set of equations (96), the fall velocity being determined using equation A5 (see

Appendix I).

I

·

1

5. The total concentration is evaluated using equation (93). This represents the sum of the concentrations of the individual size fractions.

I

I

v )0.253 10.8 (w* P

I

The first equation in set (96)

I

gives the bed load transport rate, according to Laursen, 50

long as conditions are within the range 10-2 ,v*/w < 103

p

I

This seems to indicate that if v*/w < 0.3 the transport

p

of material is solely a bed process.

I

I

I

THE BED LOAD EQUATION OF J ROTTNER (1959)

I

I

I

J Rottner (Ref l5) has developed an equation to express the bed load transport .rate in terms of the fluid and

sediment properties which is based on dimensional considera-tions with certain empirical coefficients introduced to satisfy available data. The main interest in Rottner's work is that he carried out a systematic investigation into the effect of the geometrical ratio d/D.

I

I

28

I

I

I

I

I

The expression given by Rottner

is:-I

M

[[

D )2/3

]

.

=

0.667( ~O + 0.14. '\ . ; (s-l) (gd)

I

I

D

2/3

3

-0.

778(

~O)

J

• •• (97)

. where M

=

~ediment transport rate, mass per unit width

per unit time,

I

I

D50

=

mean grain s.izediameter.

Substituting. M

=

XVdp • •• (9 B)

I

I

and rearranging equation (97)

yields:-x

= S(S-lt~(9d)~[[0.66jD~0)2/3 + 0.14J V '\ (gd(s-l))~

I

I

I

I

2/3 ]3

_ O.77B(D~O)

• •• (99)

The above equation is dimensionally homogeneous and any

consistent set of units can be used. In his derivation,

wall effects were ignored and according to Rottner the

theory "may not be 'applicable to uses where only small

quantities of material are being conveyed".

The calculation of concentrations is achieved'by di~~ct

substitution in equation (99).

I

I

THE BED LOAD FORMULAOF M S YALIN (1963)

I

I

I

M S Yalin developed a bed load equation based on a

theoretical analysis of saltating particles. The final

expression

is:-I

I

29

qbt 0.635S*[1- 1n (1 + as*)

J

=

_as* y Ov*_s v;/(gO(s-l» where s*

₌

- 1 Y

c

Y ~ 2.45 C a

₌

_--0:4

s

I

I

•••(100)

I

•••(101)

I

I

••• (102) Ov* s Xb

=

0.635 Vd s* [1 ....1n(1_as*+ as*)] ••• (103)

I

I

Re1ating the bed load transport rate to the concentration using equation (24), equation (100) becomes

This equation is dimensiona11y homogeneous. The

formu1a is restricted (i) to p1ane bed conditions, (ii) to fu11y deve10ped turbulent flow and (iii) large depth/diameter ratios.

I

I

I

Equation (103) was deve10ped to app1y to material of uniform.grain size. When the method is app1ied to a graded sediment Ya1in suggested that "an effective or typica1" grain size shou1d be used. He was not specific on this point and the mean diameter (0501 has been used in the present ana1ysis.

I

I

The ca1cu1ation proceeds as

fo11ows~-I

1. The mean diameter (050) is determined from the grading curve of the bed material.

I

2. Shie1ds critica1 Mobi1ity number ca1cu1ated using the set of equations Shie1ds graphica1 representation (Ref

(Ycl is

(A4) or from

1) •

I

3. The va1ues of s* and a are computed from equations

(101) and (10~)..

I

I

I

4. Equation (103) gives the bed load concentration.

30

I

I

I

I

I

I

THE REGIME FORMULA OF T BLENCH (1964)

I

I

Based on the data and regime princip1es described by

C Ing1is and G Lacey, T B1ench (Ref 17) gave the fo11owing

three basic

equations:-I

I

V2/d

=

Fb V3/b

=

F s ••. (104) •.• (105)

I

I

v

2 3.63 (1 +

~~~x)

(

~b )\ ••• (106) gdS·

=

K 0

where _Fb

=

bed factor

F

=

side factor

s

b

=

mean breadth

d

=

mean depth

K

=

meander slope correction.

I

I

A1gebraic manipu1ation of equations (104) and (106)

yie1ds:-I

F 11/12 3.63 g b\ q1/12 S b 0 ••• (107) 105X

=

1 _{+ 233} Kv"

Substituting q

=

Vd and v*

=

(gdSo) ~ then gives

F 11/12 _{3.63 b\} V1/12 2 b v* ••• (108)

=

. ~ _d11/12 105X 1 _{+ 233} K v

I

I

I

I

B1ench suggested tne fo11owing va1ues for K

K

₌

1.25 for a straight reach

K

₌

2.00 for areach with we11 deve10ped meandering

without braiding

K

=

3.00 for a braided reach

I

I

I

I

I

31

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K

=

4.00 for an extremely braided reach

The bed factor (Fb) is related to the "zero bed factor" (Fbo) by the

expression:-Fb

=

Fbo(l + 0.12XI05) •••(109)

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for values of X less than about 10-4• The "zero bed factor" is evaluated as

follows:-I

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If 0 ~ 2(mm) ; _Fbo

=

1.9 10(mm} If

o

> 2(mm) _{i Fbo}

=

o

•58 w7011/24(v70/ )11/72v or _Fbo

₌

7.3 0~(V70/v}1/6 These apply

for:-.

)~

s

=

2.65 and Fbo ~ 38 (~ •••(110)

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•••(111) •••(112)

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•••(113)

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If the calculated value of Fbo using equations (111) or (112) does not Sjtisf

Y

equation (113) then the value of Fbo is takenoas 38(~ 1/12.

The units are confusing. In equation (110) 0 is in mm, in equation (111) w is in cm/s and in equation (112) 0 is in ft. The answer for Fbo is always in ft/s! w70 is the fall velocity of the median sand size (050) in water at 70 deg F and v70 is the kinematic viscosity at this temperature. The limits of applicability of equations (110 to (112) are not given by Blench in a precise way.

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Substituting (109) in (108) 3.63 v*2 b~ V1/12 K d11/12 ~ F11/12 v bo

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•••(114)

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According to Blench the left hand side of equation (114) can be approximated as

follows:-32

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If 10-5 < X ~ 10-4 If 10-4 < X ~ 0.002 LHS

=

19.91 X\ LHS

=

293 537 X13/24 ••• (lIS) ••• (116)

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Blench interpreted X as the concentration of "that part of the total which is not suspended".

The equations have the fo11owing

restrictions:-I

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(i) Equation (106) was established for low flow condit.i.ons.

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(ii) Equation (109) is va1id on1y for sand, not for coarse material. The va1ue of the constant 0.12 in the same equation applies to subcritica1 flow conditions and .concentrations 1ess than 10-4• (iii) All the equations are un1ikely to app1y if bid

falls below about 4 or depth be10w about 0.4 m.

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E~cept for Fbo the equations are dimensional1y

homogeneous. Fbo' equations (110) to (113)t is in ft/s2• The calculation proceeds as

fol1ows:-I

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1. The bed factor Fb is ca1culated using equations (110) to (113).

2. The concentration X is determined by trial and error using equations (1141 to (116).

TOTAL LOAD FORMULA OF F ENGELUND AND E HANSEN (1967)

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F Engelund and E Hansen (Ref 18) developed an equation to express the total 1óad (bed material) as

follows.-I

. ~ <p = 0.1 y5/2 4 . ••• (117) where À

=

8gdS o _{••. (118)}

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33

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4> gst = P (s-l)~ g3/2 03/2 s 50 2 v* y ₌ 050 (s-l) g v* = IgdSo

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••• (119)

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•••(120)

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•••(121)

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Substituting from (118), (119), (120) and (121) into (117) yie1ds:-0.05 p g~ V2 O~ y3/2 s 50 gst = (S-l)~ But, as before, gst·= X V d p g 0.05 s V v*3 Hence, X = 2 2 dg 050(s-1)

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•••(122)

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•••(123)

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•••(124)

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In bheir origina1 paper Enge1und and Hansen gave a graphica1 solution which faci1itated the determination of both the water flow parameters and the sediment transport rates. In the present exercise mean ve10cities were

avai1ab1e and were not computed by the Enge1und and Hansen method.

Equation (124) is dimensiona11y homogeneous and the sed~ent transport concentration can be ca1cu1ated direct1y from this equation using any consistent set of units.

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THE TOTAL LOAD FORMULA OF W H GRAF (1968)

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W H Graf (Ref 20) proposed a re1ationship to express the tota1 transport rate (bed material) in both open and c10sed conduits as f

o11pws:-I

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~ =

10.39(~)-2.52 •••(125)

or •••(126)

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where ~ and ~ have the same meaning as in the Einsteip-Brown equations.

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Using equation (123) and re-arranging, equation (126) becomes

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•••(127)

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W H Graf was not specific about the interpretation of the sediment diameter, D. However, he used tota1 load data to derive his formu1a taking the mean diameter as the

relevant size. Hence~ in this report D has been assumed equa1 to the D50 size of the bed material.

The ca1cu1ation proceeds direct1y from equation (127) using artyconsistent set of units.

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THE TOTAL LOAD FORMULA OF F TOFFALETI (1968)

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The tota1 load formu1a (bed material) presented by F Toffa1eti (Refs 21 and 22) is based on the concepts of H A Einstein (Ref 7) with certain, main1y empirica1,

modifications. There are three ma in differences from the Einstein methode

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3~

..

(i) The velocity profile in the vertical is represented by the

relationship:-I

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v

z

V (dY*)

v

=

(1 + Z ) v

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•••(128)

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where where Z

=

0 •.1198 + 0.00048 TF v •••(129)

TF

=

water temperature in degrees fahrenheit

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v

=

flow velocity at a distanc~ y* above

the bed.

(ii) The three Einstein correction factors ~2/~*2,

~/e

and y are reduced,to two

viz:-I

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•••(130)

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•..(131),

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in which D65 must be expressed in feet.

(iii)The re1ationship ~* versus ~* was assumed between the level of y*

=

2D and y*

=

d/11.24 rather than between y*

=

0 and y*

=

2D as in the Einstein

theory.

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The levels y* equa1s 2D, d/11.24 and d/2.5 are used by Toffa1eti as discontinuities within the sand concentration distribution. In each of the three zones the sand concentra-tion is defined by the expression

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(

y*)-azp

c

_xp

=

_.

c

_x

-

_d •••(132)

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where C and a are constants for each zone.

x

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•••(133)

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36

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CZ

=

_132.2__ (260.67 - 0.667 TF)

=

8.095 - 0.0207 TF •••(134)

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If Z (equation (133» < Z (equation (129», Z must

p v p

be arbitrari1y taken

as:-I

Z

=

1.5 Z ••• (135)

P v

F Toffa1eti gives his ~* versus ~* re1ationship

as:-I

~*

=

17.17/~* 5/3 ••.(136) T Al k D 104 where _~*

₌

•.• (137)

v

2 and T

₌

g(s-l) CL2 (0.09158 + 0.0000028 TF) •.• (138)

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Equation (136) can be re-written making substitutions from equations (137) and (34) as fo11ows:~

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3.68 P g3/2 s(s-l)~ D*3/2 Pb 10-6 GF

=

---~~~--~~----P TA kD 5/3

[v~ ]

•••(139)

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where GF

=

sand fraction load within the depth range p

y*

=

2D and y*

=

d/11.24 for a particu1ar grain size range, dry weight per unit width per unit time.

D*

=

mean size of Sieve No 80 U.S. Standard (0.177mm·= 0.00058 ft).

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For natura1 sand rivers s

=

2.65 and in Imperia1 Units (tons/day/ft width) equation (139) reduces

to:-I

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•••(140).

(tons/day/ft width)

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which is the origina1 re1ationship proposed by F Toffaleti. However, equations (136) and (139), are the more fundamental non-dimensiona1 re1ationships. It does not fo11ow, though,

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37

that the theory is necessarily applicable to all types of sediment since the functions Al and k were determined for sand on1y.

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Using the above concepts, the sediment transport rates were ca1cu1ated by Toffa1eti for each depth zone as

fo11ows:-(i) LOWER ZONE (2D < y* , d/11.24) 0.756Z -Z gSLp

=

Pb CLp(1+Zv)Vd p v. NL

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•••(141)

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where ) 1+Z -0.756Z 1+Z -0.756Z (d v P \11.24 - (2D) v P NL

=

~~~~--~~-=----~~~p~---1 + Z - 0.756 Z v P

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.••(142)

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(ii) MIDDLE ZONE (d/11.24 < y* ' d/2.5) 0.756Z -Z P v . gSMp

=

Pb CLp(1 + Zv)Vd .• NM •.•(143)

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where

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( d ~. 244Zp

[f

d )1+Zv-Zp _. N \11.24) \2.5 M

=

1 + Z - Z v p ( d )l+Zv-Zp] 11.24

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••• (144)

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(iii)UPPER ZONE (d/2.5 < y* ~ d) 0.756Z -Z gsuP

=

Pb CLp(l + Zv)Vd P v. NU .••(145) where

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( ) 0.244Z ( )0.5Z [ l+z -1 5Z )l+Z -l.5Z ] d P d P v· P {d v P N 11.24

'2.5

d

-\'2.5

.

U= l+Z -Z V P .•.(146)

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38

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_{These functions derive from the}

expression:-I

C_xp V dy* ••.(147)

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The bed load discharge is given

by.-I

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0.756Z -Z l+Z -O.756Z - P C (l+Z )Vd p v(20) v P ,- b Lp ,v • • •(148)

The value of CLp is computed solving equations (14t) and (142) with

• ... (149)

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F Toffaleti suggests a check on the concentration at the level y* = 20 with the intention of, quite arbitrarily,

avoiding unrealistically high values. The suggested maximum value is 100 lb/ft3 or 1.6 tons/m3,

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_{i.e. •••(150)}

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If condition (150) is not satisfied Toffaleti suggests making

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C = 100(lb/ft3) = Lp (C) ,

P

y*=20 1.6 (tons/m3) (Cp)y*=20 •••(151)

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The tatal sediment transport rate (bed material) size fraction, p, is giyen by

in a

and the total sediment transport rate (bed materiaI) for the whole bed material

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....(153)

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39

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For computationa1 purposes equations (130) and (131) were brought to the fo11owing ana1ytica1

forms:-I

Denoting

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•••(154)

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If 0.13 < PAM , 0.50 If 0.50 < PAM , 0.67 Al

=

10 PAM-1•487 Al

=

43 PAMO.6142

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If 0.67 < PAM , 0.725 ; Al

=

185 PAM4•20 If 0.725 < PAM ,1.25 ; Al

=

49

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If 1.25 < PAM ~.10 24 PAM2.79 •••(155)

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Denoting FAC

=

•••(156)

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If FAC

,

0.25 _i k

₌

1 If 0.25 < FAC ~ 0.35 k

₌

5.37 FAC1•248 If 0.35 < FAC

,

2.00 _i k

₌

0.50 FAC-1.1 •••(157) If the product Alk < 16,.it must be taken arbitrari1y as A k

=

16. _v*

,

is computed as in the Einstein Theory.

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The computations proceed as

fo11ows:-I

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1. The shear velocity with respect to the grain, v*', is computed as described herein for the Einstein

methods using the re1ationships of Vanoni and Brooks (Ref 23).

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2. The correction factors Al and k are determined using the sets of equations (155) and (157) or the origina1 plots of Toffaleti, (Ref.22).

3. The exponent Zv is computed using equation (129).

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,

4. The exponent Z is evaluated for each fraction from p

equations (133), (134) and (135), the fal1 velocity being derived using equation A4 (see Appendix 1)

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5. The nucleus load GF for each grain size fraction

p

is eva1uated using equation (139).

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40

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6. The va1ue of the concentration CL is determined

. p

from equation (149) and condition (150).

7. The bed load discharge and the suspension load within each zone and for each grain size fraction are

computed using equations (141) to (151).

8. The tota1 sediment transport rate (bed material) is then determined from equations (152) and (153).

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TOTAL LOAD FORMULA OF PACKERS AND W R WHITE (1972)

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The general function of PAckers and W R White (Refs 24 and 25) is one of the most recent formu1ae for the eva1uation of the tota1 sediment transport rate. It is

based on physica1 considerations and on dimensiona1 ana1ysis. The various coefficients were derived using a wide range of f1ume data. The general function

is:-I

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G

=

C [FÁr ....·lJm gr where G (sediment transport)

=

Xd

C.)n

gr sD V •••(158) •••(159)

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n v* [ F (mobi1ity)

=

---gr y'~g-=D"""'(~s-...."""l"""} 132

v

J ...

1-n (160) 10d log D

m, C, A and n are.given in terms of Dgr' the dimension-1ess partic1e size.

= (9:(5-1)

j

l

3 Dgr D

l

\)2

J

•••(161)

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41 I

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For coarse sediments (D > 60) these four coefficients gr are as fo110ws:-n

=

0.0 A

=

0.170 m = 1.50 C

=

0.025 •••(162)

For the transitiona1 sizes (60 ~ D ~ 1):-gr n

₌

1 - 0.56 log D gr A

=

0.23 + 0.14

ro-

_gr

f)

9.66 ~ ~ m

₌

+ 1.34 D

r/

gr

log C

=

2.86 log D - (log D )~ - 3.53

gr gr •••(163)

The partic1e mobi1ity, F , and the dimension1ess grain gr

size, Dgr' express the square root of the ratio shear forces/ immersed weight and the cube root of the ratio immersed

weight/viscous forces respective1y. Hence D describes the gr

inf1uence of viscous forces on the sediment transport phenomenon. The sediment transport parameter, Ggr, is defined

as:-=

Shear forces . E Ggr Immersed weight

where E is an efficiency factor "pased on the work done in moving the materia1 per unit time and the tota1

stream power".

The coefficients n and A have physica1 meaning. The

·coefficientn re1ates to the transition.zone of partic1e sizei n = 1 för fine sediments (D = 1) where corre1ation

gr

is with tota1 shearf v*l n ~ 0 for coarse

~ 60) where corre1ation of F and G is with

.. gr gr

and d/D being the representative variab1es. of F . and G gr gr sediments (D gr grain shear, V 42

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