•GOVERNMENT OF THE REPUBlIC OF INDONESIA

MINISTRY OF PUBLIC WORKS

DIRECTORATE GENERAL OF WATER RESOURCES DEVELOPMENT

PROGRAMME Of ASSISTANCE fOR THE IMPROVEMENT

Of HYDROLOGIC DATA COLLECTION. PROCESSING

AND EVALUATION IN INDONESIA

BED- MATERIAL LDAD(EINSTEIN'S

by

METHOOJ

po ~ SOCIETE CENTRALE~~ POUR L" EQUIPEMENT

DU TERRITOIREINIERNATIONAt INTERNATIONAL

M TRAVAGLIO

BANDUNG MARCH 1981

f

Bed-Materia1 Load

(Einstein's Methed)

by

M. TRAVAGLIO

Bandung, March 1981

Taole of Contents

List of symbols •

Introduction

Einstein's Procedure

1. Hydraulic.Calculations

1.1 Test Reach •

1.2 Surface Drag and Bedform Drag (or Bar Resistance)

1.3 Mean velocity

a. Manning-Stricler's Equation

b. Logarithmic Type Formula • • •

1.4 step by Step Procedure for Hydraulic Calculations

Page

r

1

2

2

2

3

4

4

5

6

2. Bed-Material Load Calculation .

2.1 Rouse Equation for VerticalDistribution of Suspended Matter

2. 2 Suspended Load Equation

2.3 Einstein's Bed-Load Formula

2.4 Bed-Material Load Equation ••

8

8

la11

13

3. Example of Bed-Material Load Calculation

Concluding Remarks

Annex 1 ·Annex 2 ·Annex 3 · .Annex 4 ·Annex 5

References

21

27

28

30

31

33

35

37

l

LIST OF SYMBOLS

A

d

D

g

gs

gss

gstG

S

GSS

Gstn

p

p

cross-sectional area

diameter of particle. In a mixture d = d50 or median diameter

depth of flow

gravitational constant, mean value 9.81 rn/s 2

bedload rate in weight per unit time and unit width

suspended.load rate in weight per unit tirne and unit width

bed-material load rate in weight per unit time and unit width

bedload rate in weight per unit time

suspended load rate in weight per unit time

bed-material loadrate in weight per unit time

Manning roughness value

fraction of bed rnaterial in a given grain size

wetted perimeter

water discharge (m3/s)

hydraulic radius A~ = p

.; ...= 1000 kgtm3

32650 kgf/m

when the actual value is unknown

fluid specif~c weight. Water at 200 C

partiele specifie weight. Taken usually as

when actual value is unknown

channel slope

fluid velocity

shear or friction velocity

settling velocity of particle

density of fluide For water at 200 C l = 1000 kg/m3

3density of particle.Usually taken as 2650 kg/m

v

S

u

o·kinernatic viscosity of fluide For water at 20 C.

-2 2= 10 cm /s

shear stress or friction force per unit area exerted by

the fluid at a depth y above the bed

shear stress at the bottom 'Ï"'" = y R S\0'0 H or 1:

0= "(DS

other symbols are defined in due course in the following sections.

l

INTRODUCTION

The bed-material load is made up of only those particles consisting

of grain sizes represented in the bed.

In theory the bed-material load can be predicted with the hydraulic

knowledge of the stream J that is,

velocity

bed composition and configuration

shape of the measuring section

water temperature

concentration of fine sediment

Therefore the problem at issue is to determine the relationship

between the bed-material load and the prevailing hydraulic conditions such

a problem has proved to be a difficult task and is not yet completely solved.

50 far comparisons of measured and calculated bed-material loads

exhibit discrepancies which lead to think that first the problem o~ sediment

transport is not fully understood and second great care must be taken in

using bed-material load formulae.

As pointed out by GRAF (see references at the end) "Einstein's method

represents the most detailed and comprehensive treatment, from the point of

fluid mechanics, that is presently available". This method is described in

the following paragraphs.

. Nota We prefer the name "Bed-material load" to the name "Total load" since

the so-called "washload" is not taken into account when one speaks

of bed-material load.

2

EINSTEIN'S PROCEDURE

Introduction

The bed-materi~l load is divided in two parts according to the mode

of transport. In the immediate vicinity of the bed in the so-called bed

layer takes place the bedload whereas the suspended-load takes place above

the bed layer where the particle's weight is supported by the surrounding

fluid and thus the particles move with the flow at the same average velocity:

Some researchers think the division of the bed-material load in two

fractions is questionable. Actually such a division is rather artificial

particularly when it comes to define a zone of demarcation between bed-load

and suspended-load, nevertheless it is often convenient for the sake of

clarity to distinguish these two modes of transport.

Nota Figures number 2 ta number 9 are grouped fr.om page 15 to page 20.

1. HYDRAULIC C.l\LCULATIONS

1.1 Test Reach

To calculate or measure the flow and the sediment transport in a

stream, a test reach has to be selected first, the following requirements

have to be fulfilled, the better they are the more reliable the results.

It should be sufficiently long to determine rather accurately

the slope of the channel

It should have a fairly uniform and stable channel geometry

with uniform flow conditions and bed material composition

It should have a minimum of outside effects such as strong

bends, islands, sills or excessive vegetation

No tributaries should join the river within ~r immediatly

above the test reach.

It is worth noting that the foregoing requirements are those usually

. sought-for to set up a gauging station.

3

1.2 Surface Drag and Bed-Form Drag (or Bar Resistance)

To take into account the contribution the bedforms make to the channel

roughness it was proposed that both the cross section area, denoted A, and

the hydraulic radius, denoted ~, be di~ided into two parts: one related to

the surface drag or grain roughness designated by A' and R' , the other relatedH

to the bedform drag designated by A" and R~ respectively.

In terms of hydraulic radii we have

= + R"H

It follows that both shear stress and friction velocity are in turn

divided since:

~=

=

=

=

Y(RH ~ R")S and (1)

(2)

so we have:

a. In terms of shear stresses

= 't" + 't;'o 0(3)

b. in terms of friction velocities

= (4)

the "prime", 1 , used in the notation pertains to the surface àrag whereas

the "double prime", fi , pertains to the bedform drag.

Einstein and Barbarossa derived a curve fram data of river measurements

which relates the "flow intensi ty" denoted y35 and defined as

4

=RES

is the bed sediment size forwhich

35% of the material is finer)

(5 )

to the frictionof the me an stream velocity, denoted u,to the ratiouu"*

velocity due to the bar resistance denoted u;. This curve which has come to

be known as "bar resistance curve" is shown in fig. 3.

Nota: Different bedform shapes are sketched in Annex 1

1.3 Mean Velocity

DePending on the surface roughness, Einstein and Barbarossa recommended

use of either the Manning-Strickles equation or a"logarithmic type formula.

a. Manning-Strickler's equation

Is defined as

u

u' *= 7.66 (RH )i/6

d65

(6 )

where d65

is the bed sediment size for which 65% of the bed material is

finer.

The well-known Manning fonnula is defined as

u = 1n

R 3/2 51/ 2H

(7)

5

Let us assume firstly the velocity would be the same with a fIat bed

and secondly the bedform would affect both the roughness coefficient n and

the hydraulic radius i1i Be n' and ~ the values when no bedform exists.

50 we have

l R' 3/2 si/2 {8}u = n' H

By combin~ng {7} and {8} we get:

n'n

{ ~ }3/2

~{9}

and by combining· {6} and {8} we get

n'

d 1/665

24. {10}

Equations {9} and (lO) enable to ascertain whether there is a bedform

drag or not and ta calculate ~ if need be. This is the case when direct

measurement were made of the mean velocities for examp1e at a permanent

gauging station.

b. Logarithrnic Type Formula

Einstein and Barbarossa chose the fo110wing equation which was

derived from Nikuradse's experirnents by Keulegan.

U

u '.*

2.3k

12.27 ~ xlog { }

d65

{11}

where k is the Prandtl - Von Karman coefficient equal to 0.4 for clear

fluid and, x , is a correction factor for the transition from hydraulically

(see AnneA 2 for a discussion about k) "rough to hydraulically srnooth surface,

6

d65the roughness being in turn related to the ratio T ' where ~ is the thick-

ness of the so-called laminar sublayer and is defined as

11.6 J)u~

(J,I kinematic viscosity of the fluid) (12)

In figure 2, the factor x is given as a function of

Use of Manning-Strickler's formula is recommended when the grain rough­

ness produces a hydraulically rough surface, i.e. when d65 is more than

d 65~about 5. Whereas use of a logarithmic formula when r ~s less than about

5 (see fig. 2).

In case direct measurements of velocities are made, a trial and error

x. The chosen values have not onlyprocedure is used to determine R' andH

to verify equation (11) but to verify both the 2 functions depicted by the

curves given in figures 2 and 3.

1.4 Step by Step Procedure for Hydraulic Calculations

Once· a test reach has been selected, the following informations are

needed.

l. Slope

2. Description of the cros~ section, that is,

2.1 Curve of~ versus D A Cross section area

2.2 Curve of A versus D D Depth or stage

2.3 Curve of p versus D P Wetted perimeter

3. Bed sediment distribution curve

7

The determination of the depth (or stage) - dischargerelation proceeds

as follows:

l. Select a value of ~2. Calculate u' and ~ through

*equations (2) and (12) respectively

3. Determine x frcm fig. 2

4. Calculate u through equation (6)

or equation (11)

5. Calculate y 35 fram equation (5 )-6. b . u frcm fig. 3 then calculate un and RHo ta~n-;;-

u * *7. Calculate ~ =~ +~

8. Determine A and D through the description of

the cross section

9. Calculate Q = u A

Remark

In flume experiments a side-wall correction is introduced to take into

account differences in roughness between the sand-coverad bed and the flume

walls. In most natural streams such a correction neednot be applied.

8

2. BED-MATERIAL LOAD CALCULATION

The bed-material transport is calculated in its two modes, namely,

bed-load and suspended-load for each grain fraction of the bed at each

given flow depth.

The procedure used to compute the suspended-load is based on the

so-called Rouse equation which is in turn an application of the diffusion­

dispersion model.

The Einstein's bedload-function is used to calculate the bedload rate.

Sorne theoretical considerations are in place here to shed some light on the

procedure.

2.1 Rouse Equation for vertical Distribution of Suspended Matter

Let us consider particles of uniform shape, size and density in a two

dimensional, uniform,' turbulent flow.

Since the particle continuously settles with its settling velocity in

relation to the surrounding fluid an equilibrium suspension is possible only

if the flow provides a countermotion with an equal velocity. This.upward

movement is due to the turbulence of the flow, which turbulence results fram

eddies that are bei~g formed continuously and are swirling in an irregular

manner as they are carried along by the flow.

The diffusion-dispersion theory states that the settling rate due to

gravity per unit area is balanced by the upward movement due to diffusion.

This can be expressed by the following·equilibrium equation

where

vc = _ E .2s.s dy

(13)

v is the settling velocity of the given particle and c the concentration

at the height y above the bed. v is given with fig. 4 as a function of

the particle diameter, the curve due to Rubey will roughly describe the sedi­

ment of most streams.

Em

E being a function ofs

diffusion coefficient

y which has been found to be proPQrtional to the

so we have:

9

Es. = @E

m(14)

In most applications the ft factor is taken as unity. "Though experiments

have shown that f3 decreases when both the diameter d and the sediment

concentration increase such changes are small in comparison with the changes

observed in k.

Furthermore, the local shear stress, that is, the shear stress at the height

y above the bottan can be expressed as:

CE ~, m dy

Assuming the Karman-Prandtl velocity law valid, that is,

(15)

u-umax

2.3-~

ylog 0 (16)

we finally get the so-called Rouse equation (see Annex 3 for the derivation

of this equation).

cc

a= (17)

It has been found that the dis-The quantity ~ is often denoted z."ku.crepancies observed between theoretical values of z and the ones based on

experiments are chiefly .due to variations of the k factor. So taking ~ as

unity as wel~ as using for v the settling velocity in clear, still water

do not seriously change the z values. (See Annex 2).

D10

-----.... fla,,",

J

•

Figure l

50 relation (5) may be used to calculate the concentration, c , of a

given grain size whose diameter is, d , at a distance, y , above the bed

provided that the concentration, c , at a distance, a , above the bed isa

available. 5ee fig. 1.

2.2 5uspended Load Equation

To obtain the suspended load rate in weight per unit time and unit

width, denoted g , we have to integrate the product of the velocity and thess

concentration over the part of the vertical concerned with suspended load,

say from a to D.

= [ cudy (18)

This time, we use for the velocity distribution the following relation

due to Keulegan which relates the velocity not only to the depth y but to

d65 as well.

u\T

*

2.3k

l30.2 yx

og'­d

65(19)

11

Substituting the Rouse equation (17) for c and cquation (19) for u

into (18) we get: (see Annex 4 for the derivation)

_1..:1gss - k

where

ca

u'*

a::

D

(20)

According to equation (20) when y approaches zero the concentration becomes

infinite,obviously this is not true. In fact the sediment distribution does

not apply right at the bed because the concept of suspension, that is, solid

particles being continuously surrounded by the fluid fails and so the proclem

is to determine the thickness of the layer above which suspension is possible

and under which takes place the so-called bedload which is actually the source

of the suspended load.

2.3 Einstein's Bed-Load Formula

For mixtures with small size spread the total bedload transport of

the mixture can be determined directly by using d35

as the effective dia­

meter, that is the case when only the bulk rate is needed to predict scour

or deposition or when the suspended load is negligeable. The case was dealt

with in a previous note entitled "Bedload measurement and sampling."

A few more parameters come up when transport rates of each size fraction

have to he computed, mainly to take into account the fact that particles of

different sizes in a mixture have not the same behaviour as uniform bed

materials.

In that case, the "intensity of bed. load transport" , ~ * ' and "flow

intensity" , y * ' are expressed respectively by:

l

P(21)

12

r being the fraction ~f bed material in the given grain size whose repre­

sentative diameter is d.

Y.. 2

= j y ~og 10. 6 ~ (fs - r) _.d_log 10.6 Xx P RH S

d65 '

(22)

X is defined as a characteristic grain size of the mixture computed as follows

d d65X O.7·7~ if ;> 1.80b (23)orxx

X = 1.39 ~ ifd

65 c::: 1.80~ (23 ' )x

We recall that ~ laminar sublayer is equal to

~= 11. 6 V--ur-•

Two correction factors are introduced namely ; and Y.

S or "hiding" factor takes into

to hide between larger ones. Fig.

ratiod

65 .X

account the fact that srnall particles seems

5 depicts the relation between.5 and the

y takes into account changes of the lift coefficient in

various roughness. Fig. 4 depicts the relation between Y

mixtures with

and d65 •

SOnce ~.is deterrnined, we get ~. through figure 6 which depicts the

Einstein's bedload function, namely,

ll

ff=

r1/7Y.(-2)

J-1/7'r.(-2)

2-t

e dt 43.5~ *=1+43. S9i. (24)

J3

2.4 Bed-Material Load Equation

For a given vertical, it is logical to think that the summation of the

bed-load and the suspended load leads to the determination of the bed-material

load. In order to relate the concentration c to the bed-load, Einsteinaintroduced the notion of bed-layer whose depth is equal to 2d and stated

that suspension is possible only above this layer. Assuming a bed-load move­

ment in the bed layer he derived the reference concentration at 2d fram

the bed as (see Annex 5 for the derivation)

l11.6

ca with a = 2d (25)

Introducing relation (25) into the suspended load equation (20)

we get

gs tn30.2 Dx

z-l 1: (~)z. dy z-l r (l-Y)z lny dY}26)A + A

gss = 0.216 d65 (l-A) z Y (l-A)~ A Y

The bed-material load denoted gst is given by

= (27)

Substituting (26) into (27) we obtain

= (27' )

where ln30.2 Dx

(28)FE = d65

z-l r (!:.:lé) Z~

(29)Il 0.216· z dy(l-~) A Y

E

z-l

12

0.216' "" r!=l ZIny dy (30 )= ( ).

(l-A ) Z A YE E

14

The two integrals are not expressible in closed form in terms of

elementary functions.

and

for various

I2

are graphically depicted in figures 8 and 9 respectively

and z values.

Equation (27') gives a stream's capacity as to how much bed material

load it can transport under uniform.and steady flow conditions; washload is

not included in Equation (27'). In applying the methodfor a particular water­

course, Einstein (1950) stresses the following points:

(1) The length of a uniform reach should be such that the

slope 5 may be determined accurately;

(2) the channel geometry, the sediment composition, and

aIl other factors influencing the roughness velue n,

such as vegetation, etc., should be uniform, so that

an average representative cross section may be selected.

50 Einstein's (1950) method of computing the bed-material load 15

elegant and allows the calculation without measuring either the suspended

or the bedload matter.

FIGURES

15

Fil. 3 Flow rcsisl~nce due 10 bedforms. [Afra EI:-.;snIN f!t al. {/952}.J

16

200 ~EDIMENTATIONENGINEERING

2 l 0.8 0.6 0.5 0.4 0.3

VII"'~·." ---

I.e ~;-..;.=--...:.l__~2.:...- ,;.;,"_.:;;;.•:.-.:a~'....;;;ar;..,..;•.;.:.4_.;;,"'1>

0.9' _, 10.8; /_" _

O.T ~;_:_., . '/ i i).. . f . ~--

0.&, ~ 1 -_. _.. - ." 1.-{ . ~., -roS> "~ ....f----- ---fT 1" i

15 0.41 ~..,!--'o-+-.;__I_---_+...,.I-+-i---t--I~1,,"";"'--1

~ O.3~_~.~~~;__;.. ~.~.:~.~:-j-Ë; E··~-~-;;B'~Ë·~1§ll'g~O.Z:~§§~§§§

-5 4 3

FIG. 4.-Fac:tor Y ln Einstein'. Bed Load Functlon (Einstein, 1950) ln Y.nn. of. d."j

200.---,---------......_--............

•·1

1 .1

r-"0!~>

0.15 4 3 2 l 0.8 0.6 0.4 0.3 0.2

VI"" 01 \+ •.--­-Fact« ~ ln Einstein'. Bed Load Functlon (Einstein, 1950) ln Y.nn. of• AG. 5

d./X

rH-Wt++ltt-H-f'ld+H 1.0

5 fi 7 89 0.110

l

1--f 5

.~

-t++++1ti 10~

mtllmttttl!l1t :

j :l, '1

IJ

I1

, 1t

. !t l '.Il ,1

1 1 : i

Il H-H-I+HII++HtH:!

1 tIlH+1-"l+#q:++f+'ftH1

l' li ;qll1 l '

, 1 . ' '.: 1,

1 -': l', 1 ': ! 1 l, ' "Ii 1 ' ',1:: 1 ! i ~ j 1

'1 1 Il,I1

! [j'II 1 1 il il::' 1

1 j 1iilll Il il10·) 10·'

.. ' !

", , .. , . ' 1

1 ;" ; , j'II' 1 i i " 1 , 1 Il ,10" 10'2 / I! li;11 i I!;I! 1 i i;Pi 1 1

;,: :~ Jill1 •• j

: ", :!l , :,., . ,,'1 'I: il:! Il'! 1 1111'1 Il Iii l'

1 : [1 !lli l i1 ilili 1 1!lIi 11111, lili

, ':, ",Ii

Iii l'i!:!

~. ~e

~ ..~ ~~~

10 1000

;

1.0 100

0.1 l' 45 I.() 11.0 100 1100

sr·~Fc.Wc.:,.\.11 ~

~'5~/e...'!>

Wf11 t-r

te""l"er..""n:

l'OC

18

,..

FIG. 7

QOl 0.1 1 J. 1.0 10 100 ',000Groin sile. mm-

. Seulini velocit)' ~. for quartz iraïna of various aizes according to Ruhey [lOt.

Fi,. 8

....

FunClion JI in terms of AI for values of =. [Afur E/:-;STEIS (/950).)

l '

19

20

~: ::-~~':~}1??~;:·:i\:.p~:;:~~,,~\ ;IOZrooo... ~ . ,'." .. :1:,"~ .' ~:·~"~".N . .~~.~~

h--'. ~... '-'-;::-i .

! l' l 'i

! !! Iii, ~: 1

1 j 1 Il

r il

! - i _ ,- . , l , ~----::-:••: ..:}: N-U! iii:!I' : -lilil: -.~:

tO"~-~ê'3---~"~--~'.~~~:~-~.~;~''~:~~;~~~~~~~~~-~--i,~~ i

~--l--+---+--4 ~---++t+-++';---I-++_":-_-";""'-+-+~";'+~,1-:-7TI l , Il!

Fil. 9 FunClion 1, in t~rms of AE for values of :. [Afler EI:-;STEl:-O (/950,.]

(I:l i~ .. ,G. t";",~ )

21

3. EXAMPLE OF BED-MATERIAL LOAn CALCULATION

(After GRAF'Hydraulics of Sediment Transport*p.222)

A test reach, representative of the watercourse to be investigated,

has been selected. It was concluded that thé channel can be represented by

a trapezoidal cross section with bank slopes of 1:1 and a bottom width of

91.45 m. The channel slope was determined and given by S = 0.0007.

Five samples, taken down to a depth of approximately 2 ft, were.collected

to obtain information on the grain size distribution of the entire wetted

perimeter. The average values of the five samples are given in table 1.

Table 1

Grain Size Average Grain SizeDistribution, mm mm Peroentage

d > 0.589 2.4

0.589 > d > 0.417 0.495 17.8

0.417 > d > 0.295 0.351 40.2

0.295 > d > 0.208 0.248 32.0

0.208 > d > 0.147 0.175 5.8

0.147 > d 1.8

The average grain size is the geometric mean between the upper and the

lower limits of each division, i.e. 0.495 "0.589 x 0.417 .

The grain size distribution curve is given in fig. 10.

Description of cross section is given in fig. Il.

Hydraulic calculations are presented in Table 2 and bed material load

in table 3. The table heading, its meaning and caleulation are explained

with footnotes.

t.O0.90.80.70.6

~ 0.5

::0.4..

0.2

0.1

1,

, i 1

1 1 1 J 1 !;-. 1 , 1

, i ~,

1,

i ;, !

r -ct,~- --.--~r-' 1 : , i ji 1 1

,1 1

,..,-----.;;;-

I~1

1 0'),1 l' ,

i 1 : 1. '1. , ,

i,

1

1 1

il11 l' !

1i 1

!111 1

1

i 1 1

95 90 80 70 605040 30 20 10 5 2Plrelnl finer

22

FI,_ 10 Grain size distribution of bed material.

23

Table 2 Hydraulic calculation for sample problem

.'

103S" 3 -Y35 ü/u: R}i~ u" d65/r

x 10 d 65/ xu u"

* *

1 2 3 4 5 6 7 8 9 10 11

0.61 0.0647 0.179 1.96 1.40 0.25 1.745 1.12 34 0.51 0.379

.'

(i'ft)

m mis m m mis mis m

(1) Values of ~ are assumed

friction velocity due to grain roughness

(3) = 11.6 Vu'*

laminar sub layer. V (kinematic viscosity) at 200 C

V = 10-2cm2/s = 10-6m2/s •

(4) d65 / r(5 ) x = fct (d65 /S" given with fig. 2

(6) d65 / xapparent roughness diamet~r

Correction factor for roughnesstransition.

flow intensity with d35 ,as representative diameter

12.27' Rifd

65d

35

~s

5.75 log-u = u~

y 35 =(8)

(7)

(9) --lL- = fct Y35u"*

(10) u" = (l/u~ ) u* *

,,2(11) R" =_u_

gS

(12)~ = RH + RH

given with fig. 3

friction velocity due to bedform drag

hydraulic radius due to bedform drag

hydraulic radius

Table 2 (Continued)

24

103X \109 10.6)2

~ u. 0 A P Q y PE0"-'=<

12 13 14 15 lG 17 18 19 20 21 22

0.99 0.U83 1.02 94 94.3 164 0.249 O.GO 1.024 1.003 11. 72

2 3m mis m m m m Is m

(13) u. = Jg~S friction velocity

(14) D ::: fct(i1:I) given with fig. 11 Depth

(15) A ::: fct (0) given with fig. 11 Cross Section Area

(16) P ::: fct(O) given with fig. 11 Wetted perimeter

(17) Q ::z uA \'later discharge

·d d65

(18) X ::: 0.77 -2.i if > 1.80 Characteristic grain sizex x

1.39 i' ifd

65 < 1.80or X =x

(19) y ::: given with fig. 4 Pressure correction term

. (20)' 0<. 10.6 XXlog

d 65

(22) :::

25

Table 3 Bed material load. calculations for sample problem

R' 103d p diX § y* ~* gs G ZGsH s

1 2 3 4 5 6 7 8 9 10

0.61 0.495 0.1.78 1.99 1.00 1.15 6.7 0.140 13.202 13.202

0.351 0.402 1.25 1.01 0.82 9.6 0.271 25.555 38.757

0.248 0.320 1.00 1.13 0.65 12.2 0.160 15.088 54.637

0.175 0.058 0.70 1.60 0.65 12.2 0.018 1.697 56.334

.m m kg/m-sec kg/sec kg/sec

(1) RH

(2) d taken fram fig. 10 and Table 1

(3) p taken fram table 1

(4) ..s!.X

(5) 5 = fct (d/X) given "in fig .' 5

(6) Y* j y [lOg 10.61 2 ( ~ s - r) d= (RHs)

,0< r

grain size diameter

fraction of bed material whosediameter is d

hiding factor

flow intensity on individualgrain size

(7) §. = fct (Y*) given in fig. 6

P !p. '(s Jt ~ f r;. pintensity of transport for indi­vidual grain size

bedload rate in weight per unittime and width for a size fraction

(9) G = Pgss bedload rate ln weight per unit time for a size fraction

for the entire cross-section

bedload rate in weight per unit timefor all sizefractions for entire cross-section

50 according to Einstein's procedure the bedload rate is in the region

of 56 kg/s.

26

Table 3 (Continued)

103A v z Il - I 2P

EI

1+I2+1 gst G

st ~GstE

11 12 13 14 15 16 17 18 19

0.97 0.063 2.43 0.15 0.95 1.760 0.246 23.198 23.198

0.61 0.045 1. 74 0.27 1.80 2.36 0.640 60.352 83.550

0.49 0.035 1.35 0.51 3.00 3.98 0.636 59.975 143.525

0.34 0.022 0.85 2.70 10.0 22.64 0.396 37.362 180.887

mis kg/rn-sec kg/sec kg/3ec

(11)2d

ratio of bed layer depth~ = to waterD

(12) v = fct(d) given with fig. 7 Sett1ing velocity

v(13) z = 0.4 u~

(14) Il = f(~, z) given with fig. 8

(15) I 2 = f(~, z) given with fig. 9

(16) PE

I1+I 2+1

(17) gst gs(PEIl+I2+1)

(18) G = Pgstst

. (P : wetted perimeter)

(19) Z. Gst

bed rnaterial rate in weight per unit timeand width for a size fraction

bed rnaterial rate in weight per unit timefor a'size fraction for the entire cross­section

bed rnateria1 rate in weight per unit timefor aIl size fractions for the entirecross section

Obviously the digits (given by using a calculator) after the decima1

point in colurnn 19 are not significant/at best the number of significant

figures is 3.

50 according to Einstein's pxocedure the·bed rnaterial 10ad rate is in

the reqion of 180 kg/s.

27

CONCLUDING REMARKS

Several items in Einstein's method were questioned. For instance

to use u'*

instead of in calculating z in the suspended load equa-

tion may seem inapprop~iate b~cause the diffusion coefficient Em' upon

which the equation is based is likely to depend on the total shear stress

1; and not only on \:' , let alone that taking 0.4 for k is alsoo 0

questionable.

Anyway any method has its own limitations and is at best for the

time being a mere estimate even though aIl pertinent variables are taken

into account to set it up as it is the case in the Einstein's method.

In the foregoing chapt~rs it was assumed that at any time the sedi­

ment bed could afford a continuous and full availability of its particles

to be transported under any likely hydraulic conditions, if not/that i~if

the supply were partially exhausted the stream would obviously transport

less material and a bed material load equation which is supposed to give the

maximum capacity (load capacity) would fail.

Last but not least, wherever washload plays an essential role the bed

material equations.are merely helpful for the understanding of the problem

but cannot give correct results since not only such equations are of no help

to de termine the washload rat~ but the parameters used to derive them are

most likely to undergo drastic changes due to the very presence of the wash~

load (i.e. the factor k which is no longer equals to 0.4 when heavy sediment

laden flows are considered).

28

Annex l

The following table shows that in the lower regime the values of

RH are likely to be high as the form roughness predominates whereas in

the upper regime when grain roughness predominates RH is often negligeable

and R'H

CI~ssificatjon of bedforms ~nd other inform~tion (ufrer SI'10:-.s "t ul.l/CJ(>5) und 5"10:-;5 et al, (/966 JI

Bed lIlureriul .\tud~ lJIclJnc<'ntratilJlIs. s~dil/li!nr T.r~uI .R(}/'3hn~ss ..

FllJ'" regilll<' Be"J"rm . PP"' transport rlJughll<'ss .("\ :;, l

Rippk'S Io-~OO . Form 7.8-1~.4

L•.m.:r regiml:Rippll:S ,ln 100-1.200 Discrl:ll:

roughncssdun~s sleps

predominalesDun.:s 200-2.000 7.0-13.~

Transilion Washed·,lul 1.000-3.000 ; Variable 7.0-:0.0dunes

: Plane b<:tfs 2,<>00-6.000Grain

16.3-:0Antidunes 2,000 - Conlinuous roughnl:SS

10.8 ·:0Upp.:r rcgimc:

ChutC'i anJ 2.000 - pr.:dominalcs :9A-10.i

pools

A useful flow regime criterion is the Froude number denoted NF

and defined·as :

_u_

JgO

•where u is the stream mean velocity and 5 the mean depth over the entire

cross-section.

A~ classification is as follows:

= l

tranquil (streaming) flow

critical flow

rapid (shooting) flow

lower regime

transition regime

upper reg ime

29

Annex l (Continued)

Sketches of various bedforms are shown in the following figure •

..,..

_-----c.:!!.~':. ':~::___----

Ct'l Plane Dea

lD: ::>unes ... fft flCDles 5yD@fOOStd

C9a,'..

lc) Dunes

..,..

(d) WO$h~d-ouT dunes or tranSITion

. ~-

~/.\""'-tI',

Poo' Ct'u!e '

(/Il ChuTes and POOlS

Idcalized bcdrorms in alluvial channcls. [Afte, SIMO~S ~I al. (196/).)

It is worth noting that should the bedforrn change for the same depth

(or stage) bath .the velocity and the water discharge would in turn do,

sornetirnes discontinuous rating curves or rating curves with loops may be .

interpreted in this way.

To explain the fact that in the upper regime the depth-discharge

relation is reasonably stable we will quote Einstein and al.

The effect of irregularities (bedforrns) is to distort the flowpattern. When the discharge is least, the distortion of the flowpattern is greatest; as witness the meandering of natural streamsat low flows. As the discharge increases and hence the sedimenttransport along the bed also increases, the distortion of the flowpattern becomes less and less because the alinement of flow becomesprogressively straighter. Consequently, one rnay expect that theadditional friction loss, u~ , dirninishes as the discharge increases.

30

Annex 2

variations of k

The value of k is approximately 0.4 for clear fluids, but it has

been observed to diminish to as low as 0.2 in flows with high concentration

of suspended material. The following figure shows that the logarithmic

velocity distribution law holds true but with different values of k

according to the mean concentration.

THE SUSPENOEO LOAO

lÇ~=_~~O~===0.6~,------+O~'-·---C.4 ------+---,...03----~-_,,I--

1O,..---------"'r"'--.....

0.9>---------f----'f-i

0.8!---------t--~t-.

0.71----­0.6,........-----

y 02 ~---+___"+--____.15 0.51----------t-4-.-J~- y

[)o 4 :---------+---+-~ O' _ ~.O

0.3r----------J'--~---; 0.C8-~~-j~~~~~~0.2'-':,--------j,-~--__I 0.06-= 1

0-0295 ft: o.c~ D·".2?: •• !-s.o.ooas ~ 0.C4.: o' ;---- s' ·)0025 1

1-- -.::::-. .....J1 O.O~ l '.

1.0 2.;) 3.0 40 '0 2 C 30 4.0Veloe' ~ y Il. ~cs .. !IC-C, ~., J. 'C\

VelOl:llY protil~." fvr ,lear-\\;lIer and >.:Jiment-Iaden I1v\\ .. [Afra\'.~:\u:-"I ,'1 ul, (/Y6UI.)

It has been suggested that a reduction of k means that mixing is

less effective and that the presence of sediment suppresses or damps the

turbulence.

Anyhow drastic changes may arise in the veloèity distribution when

high concentrations take place but in that case it is likely that the bulk

of the ~otal load is made up of particles finer than the bed material ones

and 50 wash-Ioad is the predomlnant forro of transport.

31

Annex 3

Derivation of the Rouse Equation

We have the following set of equations

(7) Karman-Von Prandtl law

(1) Equilibrium equation

(6) Shear stress velocity or friction velocity

(4) Bottom shear stress, often simply called shear stress

E diffusion coefficient in the diffusion theorym

~ constant

(5) Ratio of the local shear stress to the bottomshear stress

(2)

(3)

vc = - E ~.s dy

1: duy = Em dy

Es = ~ Em

t = OSD0

'Ç' D-y~-=l'V D\"0

u... = ftu-u

2.3 1 Ymax-- 09-

u ... k D

Let's take the derivatille in equation (7) noting that 2.3 logL= ln:LD D

we get

dudy

= ky (8)

Let's' express 1i in terms of ~o in equation (2) by means of

equation (5) we get :

(D-Y) 't'D 0

(9)

Annex 3 (Continued)

/'\,.

Substituting equation (8) into equation (9) and expressing ~ ino

terms of u* by means of equation (6) we get :

32

D - yD ) u*

Em= ky (10)

Combining equation (3) and (10) Es can be expressed by

(11)

Substituting equation (11) into equation (1) and separating the

variables we get :

~ =c

v--- Ddyy(D -y (12)

Let us assume that the concentration of suspended sediment at a

point a is c . Then integrating (12) from a toa y we get :

= j Ya _~D...;;d:.l.Y__y(l~ - y)

[ln (-.:L..)] YL D-y a

-:L- log a (D-y)~ku* Y (D-a)

and taking the antilogarithms

cc

a

v

= [a(D-Y)] Pku*y (D-a)

vThe quant±ty ~

\Jku*is. often ca11ed z.

Annex 4

Derivation of the Suspended Load Equation

We have the following three relations

33

5.75 log 30.2 xyd 65

10.4

ln 30.2 xyd 65

(1)

cy = [a (D-Y)] Z

ca y (D-a)(2)

gss = c u dyy Y

(3)

Substituting (1) and (2) into (3) we get

5.75 u* ~oq 30.2 x j: [a <D-YU Z dy + r ra (D_Y)]Z 10 dyJ (4)gss c ya d'65 y(D-a) y (D-a) 9a

introducea then we haveLet us ~

=-D

l a (D-Y)] Z (~f (l-~J (5)Y(D-a) l-A • YE -D

Let us take as new variable

dy = D du

u=:LD

then we have

and the new limits of integration are u = Ae: and u = 1 for y = a and

y = D respectively.

34

Annex 4 (Continued)

Consequently we get

JD Ga (D-y) ] Zy(D-a) dy

a

AD(_E_)Z

l-AE

and (6)

JD [a (D-yil ~ log Y dyy(D-a)

a

(l-u) Z log u duu Jl Jl-u Z

+ log D ~ (u) du (7)

Substituting (6) and (7) into (4) we finally get:

)% ~Og 30.2 Dxd

65(l-u)z- duu

l-u Z(-) log u

u(8)

or taking the Naperian logarithms

(8 1)(l-u)z ln u

udu + r. ~

30.2 Dxd65

35

Annex 5

Derivation of the Be~-Material Load Equation

Einstein foun& that in the so-called laminar sub-layer whose depth is

the bottom velocity, uB

' is related to the shear stress velocity by

50 assuming that the particles in the sublayer move with an average

velocity equal to Ua' the bed.. load per unit width g 5 may be considered

as the product of the concentration c and the discharge per unit width,a

50 we can write:

9 s:: C a u

Ba

9 s= c a 11.6 u.

a

and with a = 2 d we get

gsCa =

Il. 6 u. 2d

or

(1)

Let us resume the suspended load equation [Annex 4, equation (8 ' i]

(l~Y)Z dy + J~ (l~y)Z ln y dyJ (2)

which may be rewritten as follows:

l--u Oc0.4 * a

z-l

+ ~ rI (l-y)zln y dyJ(l-A )z JA y

E E

(3)

Annex 5 (Continued)

Substituting (1) into (3) and noting that a = 2d and consequently

a 2d~=D = 0 we get

36

1 1g =- u. D­ss 0.4 Il.6

2do [ .••] = gs

0.216 [- ... -] (4)

Finally we get for the bed-material 10ad

gst gs + g9S gg (PE Il + I 2 + 1)

where

PE "" ln (30.2 Dx)d

65

z-l

[0.216~ (l-Y) z dyIl =(l_~)z y

\:z-l J: (l-Y) z Iny dy1 2 = 0.216(l_~)Z

y

REFERENCES

Hydraulics of Sediment Transport. GRAF, W. H., MacGraw Hill.

Sedimentation Engineering. American Society of Civil Engineers.

River Sedimentation. EINSTEIN, H. A. in Handbook of AppliedHydrology{VEN TE CHOW~

These books are available at the DPMA library.

37