GOVERNMENT
•
OF
THE
MINISTRY
DIRECTORATE
GENERAL
REPUBlIC
OF PUBLIC
OF
OF
INDONESIA
WORKS
WATER RESOURCES DEVELOPMENT
PROGRAMME Of ASSISTANCE fOR THE IMPROVEMENT
Of HYDROLOGIC DATA COLLECTION. PROCESSING
AND EVALUATION IN INDONESIA
BED- MATERIAL LDAD
(EINSTEIN'S
METHOOJ
by
M TRAVAGLIO
~
~~
SOCIETE CENTRALE
POUR L" EQUIPEMENT
DU TERRITOIRE
INIERNATIONAt
INTERNATIONAL
po
BANDUNG MARCH
1981
f
Bed-Materia1 Load
(Einstein's Methed)
by
M. TRAVAGLIO
Bandung, March 1981
Taole of Contents
Page
List of symbols •
r
Introduction
1
Einstein's Procedure
2
1.
Hydraulic.Calculations
2
1.1
Test Reach •
2
1.2
Surface Drag and Bedform Drag (or Bar Resistance)
3
1.3
Mean velocity
4
a.
Manning-Stricler's Equation
4
b.
Logarithmic Type Formula • • •
5
1.4
2.
Bed-Material Load Calculation .
2.1
3.
step by Step Procedure for Hydraulic Calculations
Rouse Equation for Vertical
Distribution of Suspended Matter
6
8
8
2. 2
Suspe nded Load Equation
la
2.3
Einstein's Bed-Load Formula
11
2.4
Bed-Material Load Equation ••
13
Example of Bed-Material Load Calculation
21
Concluding Remarks
Annex 1
Annex 2
Annex 3
Annex 4
·
·
·.
·
27
28
30
31
33
Annex 5
35
References
37
l
LIST OF SYMBOLS
A
cross-sectional area
d
diameter of particle. In a mixture
D
depth of flow
g
gravitational constant, mean value 9.81 rn/s
gs
bedload rate in weight per unit time and unit width
gss
suspended.load rate in weight per unit tirne and unit width
gst
GS
bed-material load rate in weight per unit time and unit width
bedload rate in weight per unit time
GSS
suspended load rate in weight per unit time
G
bed-material loadrate in weight per unit time
n
Manning roughness value
p
fraction of bed rnaterial in a given grain size
p
wetted perimeter
st
d
= d 50
or median diameter
2
3
water discharge (m /s)
hydraulic radius
S
channel slope
u
fluid velocity
~
A
=p
shear or friction velocity
v
settling velocity of particle
l
0
density of fluide For water at 20 C
kg/m
= 1000
density of particle.Usually taken as 2650 kg/m
3
3
when the actual value is unknown
1000 kgtm 3
3
partie le specifie weight. Taken usually as 2650 kgf/m
0
fluid specif~c weight. Water at 20 C
=
when actual value is unknown
o·
kinernatic viscosity of fluide For water at 20 C.
= 10
-2
2
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"'"
\0'0
=
y
R S
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 de termine 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"
of bed-material load.
is not taken into account when one speaks
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
1.
Figures number 2 ta number 9 are grouped fr.om page
15 to page 20.
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 related
H
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"o
b.
+ 't;'
(3)
0
in terms of friction velocities
=
the "prime",
1
,
(4)
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
y
35
and defined as
4
=
is the bed sediment size forwhich
RES
to the ratio
u
35% of the material is finer)
of the me an stream velocity, denoted u,
u"
(5 )
*
velocity due to the bar resistance denoted
to the friction
u;. This curve which has come to
be known as "bar resistance curve" is shown in fig. 3.
Nota:
1.3
Different bedform shapes are sketched in Annex 1
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' *
where
d
65
is
7.66
=
(RH )i/6
d
(6 )
65
the bed sediment size for which 65%
of the bed material is
finer.
The well-known Manning fonnula is defined as
u
=
1
n
R 3/2 51 / 2
H
(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
the hydraulic radius
i1i
n'
Be
and
~
n
and
the values when no bedform exists.
50 we have
u
By
combin~ng
=
l
n'
R'
3/2 si/2
{8}
H
{7} and {8} we get:
n'
~
{
n
}3/2
{9}
~
and by combining· {6} and {8} we get
d
n'
1/6
65
. {10}
24
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
Formula
Type
Einstein and Barbarossa chose the fo110wing equation
which was
derived from Nikuradse's experirnents by Keulegan.
where
k
U
2.3
u'
.*
k
12.27
log {
d
~
x
}
{11}
65
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
d
the roughness being in turn related to the ratio
T65 '
where ~ is the thick-
ness of the so-called laminar sublayer and is defined as
11.6 J)
(J,I
u~
In figure 2, the factor
x
kinematic viscosity of the fluid)
(12)
is given as a function of
Use of Manning-Strickler's formula is recommended when the grain roughness produces a hydraulically rough surface, i.e. when
r
d 65
about 5. Whereas use of a logarithmic formula when
d 65
~
~s
is more than
less than about
5 (see fig. 2).
In case direct measurements of velocities are made, a trial and error
procedure is used to determine
R'
H
and
x. The chosen values have not only
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
3.
cros~
2.1
Curve of
~ versus
2.2
Curve of
A
versus
2.3
Curve of
p
versus
section, that is,
D
A
Cross section area
D
D
Depth or stage
D
P
Wetted perime ter
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)
-
7.
Calculate y 35 fram equation (5 )
.
u
frcm fig. 3 then calculate un
ob ta~n-;;*
u *
Calculate
=~ +~
8.
Determine
5.
6.
and
RH
~
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 diffusiondispersion 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
vc
=
.2s.
_ E
(13)
s dy
where
v
is the settling velocity of the given particle and
at the height
y
above the bed.
v
c
the concentration
is given with fig. 4 as a function of
the particle diameter, the curve due to Rubey will roughly describe the sediment of most streams.
9
E
s
being a function of
diffusion coefficient
E
f3
m
@Em
=
s.
ft
(14)
factor is taken as unity. "Though experiments
decreases when both the diameter
concentration increase
observed in
so we have:
E
In most applications the
have shown that
which has been found to be proPQrtional to the
y
d
and the sediment
such changes are small in comparison with the changes
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
(15)
Assuming the Karman-Prandtl velocity law valid, that is,
2.3
u-u
max
-~
y
log 0
(16)
we finally get the so-called Rouse equation (see Annex 3 for the derivation
of this equation).
c
c
a
The quantity
~
"ku.
(17)
=
is often denoted
z.
It has been found that the dis-
crepancies observed between theoretical values of
experiments are chiefly .due to variations of the
unity as
wel~
as using for
do not seriously change the
v
and the ones based on
z
k
factor. So taking
~
the settling velocity in clear, still water
z
values. (See Annex 2).
as
10
D
- - - - -.... 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
a
, at a distance, a , above the bed is
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 the
ss
concentration over the part of the vertical concerned with suspended load,
say from
a
to
D.
=
[
(18)
cudy
This time, we use for the velocity distribution the following relation
due to Keulegan which relates the velocity not only to the depth
d
65
as well.
y
but to
11
u
2.3
\T
*
k
30.2 yx
l og'd
65
Substituting the Rouse equation (17) for
(19)
c
and cquation (19) for
u
into (18) we get: (see Annex 4 for the derivation)
_1..:1
gss -
k
c
a
u'
(20)
*
where
::
a
D
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, sol id
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
d
as the effective dia35
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" ,
intensity" ,
y
* ' are expressed respectively by:
~ * ' and "flow
12
l
(21)
P
r
~f
being the fraction
sentative diameter is
bed material in the given grain size whose repre-
d.
Y.
=
j
.
y
~og
~
10. 6
log 10.6 Xx
2
(fs P- r) _.RHd _S
d 65
(22)
'
X is defined as a characteristic grain size of the mixture computed as follows
d
O.7·7~
x
X
=
X
We recall that
d
if
65
;> 1.80b
x
d
~
~
laminar sublayer is equal to
if
(23)
c::: 1.80~
65
1.39
or
x
(23 ' )
11. 6 V
~= --ur•
Two correction factors are introduced namely ;
and
Y.
or "hiding" factor takes into account the fact that srnall particles seems
S
to hide between larger ones. Fig.
depicts the relation between.5 and the
5
ratio
d
.
65
X
y
takes into account changes of the lift coefficient in
various roughness. Fig. 4 depicts the relation between
Y
mixtures with
and
d 65 •
S
Once ~.is deterrnined, we get ~. through figure 6 which depicts the
Einstein's bedload function, namely,
l
l
ff
r
J
1/7Y.(-2)
=
-1/7'r.(-2)
e
-t
2
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, Einstein
a
introduced the notion of bed-layer whose depth is equal to 2d and stated
that suspension is possible only above this layer.
ment in the bed layer
Assuming a bed-load move-
he derived the reference concentration at
2d
fram
the bed as (see Annex 5 for the derivation)
l
c
11.6
Introducing relation (25)
with
a
a
=
(25)
2d
into the suspended load equation (20)
we get
gss
=
gs
0.216
t
n
1:
z-l
A
30.2 Dx
d
65
(l-A) z
The bed-material load denoted gst
(~)z. dy + Az-l
Y
(l-A)~
(27)
into (27)
we obtain
(27' )
=
where
FE
=
ln
30.2 Dx
d
65
0.216·
Il
(l-Y)z lny d }26)
Y
Y
A
is given by
=
Substituting (26)
r
~
(28)
z-l
(l-~)
z
r
(!:.:lé) Z
A
E
dy
(29)
Y
"" r!=l
z-l
1
2
=
0.216'
(
(l-A ) Z
E
A
E
Y
).
Z
Iny dy
(30 )
14
The two integrals are not expressible in closed form in terms of
elementary functions.
and
for various
I
2
and
are graphically depicted in figures 8 and 9 respectively
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 watercourse, Einstein (1950) stresses the following points:
(1)
The length of a uniform reach should be such that the
slope
(2)
5
may be determined accurately;
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.
15
FIGURES
Fil. 3
Flow rcsisl~nce due 10 bedforms.
[Afra EI:-.;snIN
f!t
al. {/952}.J
16
~EDIMENTATION ENGINEERING
200
I.e ~;-..;.=--...:.l __~2.:...0.9'
0.8;
O.T ~;_: _., .
1
~1
0.&,
_
i).. . f . ~-1.-{ . ~., -r
i
'/
~ oS>
15
.;.:.4_.;;,"'1>
,;.;,"_.:;;;
.•:.-.:a~'....;;;ar;..,..;
•
_,
/_"
-_. _.. -
.... f - - - - -
."
---fT
"
1"
i
0.41 ~..,!--'o-+-.;__I_---_+...,.I-+-i---t--I~
1,,"";"'--1
~ O.3~_~.~~~;_;._~.~.:~.~:-j-Ë; E··~-~-;;B'~Ë·~1§ll'g~
O.Z:~§§~§§§
-5
4
2
3
l
0.8
0.6 0.5 0.4
0.3
VII"'~·." - - FIG. 4.-Fac:tor Y ln Einstein'. Bed Load Functlon (Einstein, 1950) ln Y.nn. of
. d."j
200.---,---------.. . ._ --.. . . . . .
1
•
.
1
·1
r-
"0
!
~
>
5
4
3
2
l 0.8 0.6
VI"" 01
•
AG. 5
d./X
-Fact«
~
0.4 0.3
\+•.---
0.2
0.1
ln Einstein'. Bed Load Functlon (Einstein, 1950) ln Y.nn. of
-t++++1ti 10
~
mtllmttttl!l1t:
1
j :
l,
I
IJ
.!
1
I
--f
'1
, 1t
1
5
.~
l
l '.
t
l ,1
1
:
i
rH-Wt++ltt-H-f'ld+H
1.0
Il H-H-I+HII++HtH:!
1 tIlH+1-"l+#q:++f+'ftH 1
l'
1
lil ;qll'
5 fi 7 89 0.1
10
18
. .e
~
~
~
~
~
10
~~
0.1
1000
l ' 45
11.0
I.()
1100
100
;
,
1.0
' : , ",Ii
Iii l'i!:!
100
sr·~Fc.
Wc.:,.\.11
~
~'5~/e...'!>
Wf11 t-r
1
.
!
1
l, '
1
1
il il::'
,
1 -': l',
.'1 1
'
10·) 10·'
10" 10'2
Il,I
"
1
1
! [j'II
;
'
'.:
"Ii
,
j'II'
1 i
i
I!;I!
1
i
! 1 j 1iilll
1
i "
1
,
i;Pi
1
1
1
1
:
",
'I:
il:!
1 : [1
QOl
FIG.
,..
7
!lli
l
i
0.1
. Seulini velocit)'
1 J.
~.
1
:!l
,
:,.,
Il'!
1
1111'1
ilili
1
1!lIi
1.0
10
1
' ',1::
1
.. ,
,,
;"
I! li;11
/
':
1
,
1! i
~ j
Il
1
l'OC
il
. ,
Il
'
1
; ,: :
1
Jill
~
••
.
te""l"er..""n:
j
,,'
Il Iii
11111,
l'
lili
100
',000
Groin sile. m m -
for quartz iraïna of various aizes according to Ruhey [lOt.
19
....
l
Fi,. 8
FunClion JI in terms of AI for values of
=.
[Afur E/:-;STEIS (/950).)
'
20
!
~: ::-~~':~}1??~;:·:i\:.p~:;:~~,,~\
;
~ . ,'." .. :1:,"~ .' ~:·~"~".N . .~~.~~
IOZrooo...
h--'.
~...
!
l' l
'i
!! Iii
,
~:
1
j
1
Il
il
r
1
'-'-;::-i .
!
- i _ ,-
!
iii:!I'
.,
:
~
----::-:••: ..: } :
l ,
-lilil:
N-U
-.~:
,
tO"~-~ê'3---~"~--~'.~~~:~-~.~;~'
'~:~~;~~~~~~~~~-~--i
~~
i
~--l--+---+--4
~---++t+-++';---I-++_":-_-";""'-+-+~";'+~
,1-:-7TI
l
,
Il!
Fil. 9
FunClion 1, in t~rms of A E 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
Distribution, mm
Average Grain Size
mm
d > 0.589
0.589
0.417
0.295
0.208
0.147
>d
>d
>d
>d
>d
> 0.417
> 0.295
> 0.208
> 0.147
Peroentage
2.4
0.495
17.8
0.351
40.2
0.248
32.0
0.175
5.8
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.
22
t.O
0.9
0.8
0.7
0.6
1
;-.
,, i
~ 0.5
..
::0.4
1
1
1
1
~
-ct,~-
r
1
1
,
l'
:
1
1
1
i
1
1
1
i
1
1
95
90
1
!
- -.--~
i
r-'
0'),
i
i
1
,
1
1 1
i
0.1
,
,
!
..,-----.;;;-
1
0.2
J
,
,
1
:1
,
1
,
1
I~
. , ,
1.
1
l'
!1
!
1
1
1
80 70 605040 30 20
10
10
Grain size distribution of bed material.
;
i,
j
,
1
'
Plrelnl finer
FI,_
1
i
1
il
5
2
23
Table 2
Hydraulic calculation for sample problem
.'
~
u"
103S"
1
2
3
*
0.0647
0.61
d 65 /
x
r
4
0.179
1.96
5
1.40
-u
Y35
ü/u:
u"
*
R}i
6
7
8
9
10
11
0.25
1.745
1.12
34
0.51
3
10 d 65 / x
0.379
.'
(i'ft)
m
(1)
m
mis
Values of
m
mis
mis
m
are assumed
~
friction velocity due to grain roughness
=
(3)
11.6
V
laminar sub layer.
u'
*
d 65 /
(5 )
x = fct (d /S"
65
(6)
d 65 / x
(7)
u
r
given with fig. 2
apparent roughness
12.27'
u~
5.75 log
d
d
(8)
y
35
=
(9) --lL- =
u"
*
fct
= (l/u~
(11) R"
,,2
=_u_
gS
(12)
=
~
35
~s
Y35
*
(10) u"
(kinematic viscosity) at 20 0 C
V = 10-2 cm2/ s = 10-6m2/ s •
(4)
-=
V
RH +
)
u
Correction factor for roughness
transition.
diamet~r
Rif
65
flow intensity with d 35 ,as representative diameter
given with fig. 3
friction velocity due to bedform drag
*
hydraulic radius due to bedform drag
RH
hydraulic radius
24
Table 2 (Continued)
P
A
0
Q
~
u.
12
13
14
15
lG
17
0.99
0.U83
1.02
94
94.3
164
10 3 X
y
0"-
\10 9 10.6)
'=<
mis
m
2
m
m
m
3
m
Is
18
0.249
19
O.GO
20
1.024
21
1.003
m
Jg~S
friction velocity
(13)
u.
=
(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
(18)
X
:::
·d
0.77 -2.i
x
if
or X
=
1.39 i'
if
\'later discharge
d
. (20)'
(22)
y
log
:::
> 1.80
x
65
<
10.6 XX
d
65
Characteristic grain size
1.80
given with fig. 4
:::
0<.
65
x
d
(19)
2
Pressure correction term
PE
22
11. 72
25
Table 3
R'
Bed material load. calculations for sample problem
Gs
8
9
diX
§
y*
~*
2
3
4
5
6
7
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
1
0.61
gs
p
3
10 d
H
ZG s
10
.
m
m
kg/m-sec kg/sec
kg/sec
(1)
RH
(2)
d
taken fram fig. 10 and Table 1
grain size diameter
(3)
p
taken fram table 1
fraction of bed material whose
diameter is d
(4)
..s!.
(5)
5
=
fct (d/X)
(6)
Y*
=
j
(7)
X
§. =
G
s
=
y [lOg 10.61 2 (
,0<
fct (Y*)
P
(9)
given "in fig .' 5
!p. '(s
Pgs
d
~ s - r) (RHs)
r
given in fig. 6
Jt ~ f r;. p
hiding factor
flow intensity on individual
grain size
intensity of transport for individual grain size
bedload rate in weight per unit
time and width for a size fraction
bedload rate ln weight per unit time for a size fraction
for the entire cross-section
bedload rate in weight per unit timefor all size
fractions 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)
103 A
E
11
v
z
Il
- I2
P I +I +1
E 1 2
12
13
14
15
16
gst
G
st
17
18
~Gst
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
22.64
0.396
37.362
180.887
kg/rn-sec kg/sec
kg/3ec
10.0
mis
2d
D
(11)
~ =
( 12)
v
=
fct(d)
(13)
z
=
v
0.4 u~
( 14)
Il
=
f(~,
z)
given with fig. 8
(15)
I
=
f(~,
z)
given with fig. 9
(16)
P I +I +1
E 1 2
(17)
gst
(18)
G
st
2
=
ratio of bed layer to water depth
given with fig. 7
gs(P I +I +1)
E l
2
bed rnaterial rate in weight per unit time
and width for a size fraction
Pg
bed rnaterial rate in weight per unit time
for a'size fraction for the entire crosssection
st
. (P : wetted perimeter)
(19)
Z.
Sett1ing velocity
bed rnateria1 rate in weight per unit time
for aIl size fractions for the entire
cross section
Gst
Obviously
the digits (given by using a calculator) after the decima1
point in colurnn 19 are not significant/at best
figures is
the number of significant
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'
*
in calculating
instead of
tion may seem
z
in the suspended load equa-
E ' upon
m
which the equation is based is likely to depend on the total shear stress
1;o
inapprop~iate b~cause
and not only on
the diffusion coefficient
\:' , let alone that taking 0.4 for
0
k
is also
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
the supply
were
i~if
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
load (i.e. the factor
k
wash~
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
R'
and
H
CI~ssificatjon
l/CJ(>5) und
5"10:-;5 et
of bedforms
~nd
Bed lIlureriul
clJnc<'ntratilJlIs.
FllJ'" regilll<'
L•.m .:r regiml:
Be"J"rm
. PP"'
100-1.200
dun~s
Washed·,lul
dunes
: Plane b<:tfs
Antidunes
Upp.:r rcgimc:
ChutC'i anJ
.\tud~
SI'10:-.s "t ul.
lJI
s~dil/li!nr
T.r~uI
transport
rlJughll<'ss
Io-~OO
Rippk'S
Rippll:S ,ln
Dun.:s
Transilion
inform~tion (ufrer
other
al, (/966 JI
Discrl:ll:
sleps
200-2.000
; Variable
1.000-3.000
2,<>00-6.000
2,000 2.000 -
. Form
roughncss
predominales
. R(}/'3hn~ss ..
.("\
,
:;l
7.8-1~.4
7.0-13.~
7.0-:0.0
16.3-:0
Conlinuous
pools
Grain
10.8 ·:0
roughnl:SS
9A-10.i
pr.:dominalcs :
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
=l
~
classification is as follows:
tranquil (streaming) flow
lower regime
critical flow
transition regime
rapid (shooting) flow
upper reg ime
29
Annex l
(Continued)
Sketches of various bedforms are shown in the following figure •
..,..
Ct'l Plane Dea
_-----c.:!!.~':.
lD:
::>unes ... fft flCDles
..
':~::___----
5yD@fOOStd
C9a,'
lc) Dunes
..,..
.
~-
/.\""'tI',
~
Poo'
(d) WO$h~d-ouT dunes or tranSITion
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 flow
pattern. When the discharge is least, the distortion of the flow
pattern is greatest; as witness the meandering of natural streams
at low flows. As the discharge increases and hence the sediment
transport along the bed also increases, the distortion of the flow
pattern becomes less and less because the alinement of flow becomes
progressively straighter. Consequently, one rnay expect that the
additional 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
1 O , . . - - - - - - - - - " ' r " ' - -.....
lÇ~=_~~
O~===
0.9 > - - - - - - - - - f - - - - ' f - i
------+
-C.4 ------+---,...
0.6~,
O~'-·-
0.8 !---------t--~t-.
0.71----y
03 ----~-_,,I--
0.6,........-----
15 0.51----------t-4-.-J~o 4 :---------+---+-~
02 ~---+___"+--____.
y
[)
O' _
~.O
0.3 r----------J'--~---; 0.C8-~~-j~~~~~~
0.2'-':,--------j,-~--__I
0.06-=
1
0-0295 ft: o.c~
D·".2?: •• !
~
0.C4.: o'
; - - - - s' ·)0025 1
1--.::::-.
.....J1 O.O~ l
'.
1.0
2.;)
3.0
40
'0
2C
30
4.0
-s.o.ooas
Veloe' ~ y
\'.~:\u:-"I
Il.
~cs
.. !IC-C, ~.,
J.
'C\
VelOl:llY protil~." fvr ,lear-\\;lIer and >.:Jiment-Iaden I1v\\ .. [Afra
,'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
and
wash-Ioad is the predomlnant forro of transport.
50
load is made up of particles finer than the bed mate rial ones
31
Annex
3
Derivation of the Rouse Equation
We have the following set of equations
vc
- Es ~.
dy
=
1: y =
Es
t
0
du
dy
Em
Equilibrium equation
(2)
E
m
diffusion coefficient in the diffusion theory
=
~ Em
(3)
~
=
OSD
(4)
Bottom shear stress, often simply called shear stress
(5)
Ratio of the local shear stress to the bottom
shear stress
(6)
Shear stress velocity or friction velocity
(7)
Karman-Von Prandtl law
'Ç'
D-y
~-=
l'V
D
\"0
ft
=
u...
(1)
u-umax
2.3
Y
- 1 09k
D
u ...
constant
Let's take the derivatille in equation (7) noting that 2.3 logL= ln:L
D
D
we get
du
dy
=
(8)
ky
Let's' express
equation (5)
(D-Y)
D
't'
0
1i in
we get :
(9)
terms of
~o
in equation (2) by means of
32
Annex 3 (Continued)
Substituting equation (8) into equation (9) and expressing
terms of
u*
/'\,.
~
in
o
by means of equation (6) we get :
D - y
D
)
u*
=
Em
(10)
ky
Combining equation (3) and (10)
Es
can be expressed by
(11)
Substituting equation (11) into equation (1) and separating the
variables we get :
~ =
v
---
c
Ddy
(12)
y(D -y
Let us assume that the concentration of suspended sediment at a
point
is
a
=
c . Then integrating (12) from
a
j
_~D...;;d:.l.Y__
Ya
y(l~
-
a
[ln
L
y)
D-y
v
=
a
The quant±ty
v
~
\Jku*
[a(D-Y)]
y (D-a)
Pku*
is. often ca11ed
y
(-.:L..)] Y
and taking the antilogarithms
c
c
to
z.
a
we get :
-:L- log a (D-y)
~ku*
Y (D-a)
33
Annex
4
Derivation of the Suspended Load Equation
We have the following three relations
5.75 log 30.2 xy
d
65
c [a
y
(D-Y)]
y (D-a)
=
ca
=
gss
Substituting
gss
5.75 u* c a
c u
dy
(3)
into (3) we get
j: [a <D-YU
30.2 x
y(D-a)
d'65
~
a
D
=-
=
respectively.
dy
+
r
a
ra (D_Y)]Z
y (D-a)
10
9
y
(~f
(l-~J
l-A
• Y
E
dyJ
(5)
-D
u=:L
D
then we have
D du
and the new limits of integration are
=D
Z
then we have
Let us take as new variable
y
(1)
65
(2)
a (D-Y)] Z
Y(D-a)
dy
d
Z
and (2)
Let us introduce
l
ln 30.2 xy
y Y
(1)
~oq
1
0.4
u = Ae: and
u = 1
for
y = a
and
(4)
34
Annex 4 (Continued)
Consequently we get
J
D Ga (D-y) ]
y(D-a)
a
J
D [a (D-yil
y(D-a)
~
Z
dy
A
D(_E_)Z
(l-u)
u
a
Substituting (6) and (7) into (4)
)%
(6)
E
log Y dy
or taking the Naperian
and
l-A
~Og
30.2 Dx
d
65
Z
J
Z
log u du + log D ~ (l-u
u)
du
l
J
(7)
we finally get:
l-u Z
( - ) log u
(-l-u)z du
u
u
(8)
logarithms
30.2 Dx
d
65
du
.
+
r
~
(l-u)z ln u
u
(8
1
)
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, u
B
' is related to the shear stress velocity by
50 assuming that the particles in the sublayer move with an average
veloci ty equal to
Ua' the bed.. load per unit wid th
as the product of the concentration
50
c
a
may be considered
g5
and the discharge per unit width,
we can write:
and with
a
=
9s
::
9s
=
2 d
C
c
a
a
a u
or
B
a 11.6 u.
we get
gs
Ca
=
(1)
Il. 6 u. 2d
Let us resume the suspended load equation
[Annex 4, equation (8 '
(l~Y)Z dy + J~ (l~y)Z
i]
ln y dy
J
(2)
which may be rewritten as follows:
l
- - u Oc
0.4 *
a
+
~
z-l
rI
(l-A )z JA
E
E
(l-y)zln y dyJ
y
(3)
36
Annex 5 (Continued)
Substituting (1) into (3) and noting that
a
~=D
g
ss
2d
=0
a
= 2d
and consequently
we get
1
1
2d
0.4
Il.6
o
= - u. D -
gs
[ .•• ] = 0.216
[-
Finally we get for the bed-material 10ad
gg (P E Il + I 2 + 1)
gs + g9S
gst
where
PE
Il
""
=
ln (30.2 Dx)
d
65
0.216
~
z-l
(l_~)z
\:z-l
12
=
0.216
(l_~)Z
[
(l-Y) z dy
y
J:
(l-Y) z Iny dy
y
...
-]
(4)
37
REFERENCES
Hydraulics of Sediment Transport. GRAF, W. H., MacGraw Hill.
Sedimentation Engineering. American Society of Civil Engineers.
River Sedimentation. EINSTEIN, H. A.
Hydrology{VEN TE CHOW~
in Handbook of Applied
These books are available at the DPMA library.