•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