SEDIMENT TRANSPORT:
AN APPRAISAL OF AVAILABLE METHODS
VOLUME 1
SUMMARY OF EXISTING THEORIES
By
W. R. WHITE B.Sc. Ph. D. C. Eng. MICE
H. MILLI Ingeniero Civil
A. D. CRABBE C. Eng . MICE
November 1973
Crown Copyright
Report No
INT 119
Hydraulics Research Station
Wallingford
Berkshire
England
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CONTENTS
Page
NOTATION
1
INTRODUCTION
BED LOAD EQUATION OF A SHIELDS
BED LOAD EQUATION OF A A KALINSKE
THE REGIME FORMULA OF C INGLIS
2
(1936)
(1947)
4
6
(1947)
BED LOAD EQUATION OF A MEYER-PETER AND R MULLER
BED LOAD EQUATION OF H A EINSTEIN
(1948)
10
(1950)
TOTAL LOAD FORMULA OF H A EINSTEIN
7
15
(1950)
BED LOAD EQUATION OF H A E INSTEIN AND C B BROWN
(1950)
17
TOTAL LOAD FORMULA OF A A BISHOP, D B S IMONS AND
E V RICHARD SON (1965)
18
BED LOAD E QUATION OF R A BAGNOLD
21
(1956)
TOTAL LOAD FORMULA OF R A BAGNOLD
(1966)
22
TOTAL LOAD FORMULA OF E M LAURSEN
(1958)
26
BED LOAD EQUATION OF J ROTTNER (1959)
28
(1963)
29
BED LOAD EQUATION O F M S YALIN
REGIME FORMULA O F T BLENCH
31
(1964)
TOTAL LOAD FORMULA OF F ENGELUND AND E HANSEN
TOTAL LOAD FORMULA OF W H GRAF
(1967)
35
(1968)
TOTAL LOAD FORMULA OF F TOFFALETI
35
(19681
TOTAL LOAD FORMULA OF P ACKERS AND W R WHITE
33
(1972)
41
44
REFERENCES
APPENDICES
1.
Evaluation of Y t w and v
c
49
2.
Solution of integrals used in the Einstein methods
51
3.
Th eori es not evaluated
53
CONTENTS
(Cont'd)
TABLE
1.
Summary of theoretical methods
NOTATION
a
�
O .4
Constant, 2. 45 Y /s
cr
a
Exponent in the concentration distribution e q uation
(Yalin)
(Toffaleti)
A*
Constant
A
Value of F
b
Stream breadth, surface width if not otherwise
(Einstein)
gr
at initial motion
(Ackers, White)
indicated
B*
Constant
C
Coefficient in the general function
C
Constant of proportionality in the concentration
x
C
xp
(Einstein)
distribution equation
(Ackers, White)
(Toffaleti)
Concentration as weight per unit volume in a given
size fraction, p, in the lower, middle and upper zones
of transport
C
Lp
Concentration as weight per unit volume in a grain
size fraction, p, in the lower zone of transport
(Toffaletil
CL
2
Constant of grain volume/constant of grain area
C
Temperature related parameter used in evaluating the
z
(Toffaleti)
middle zone exponent of the concentration distribution
d
Mean depth of flow
d'
Mean depth with respect to the grain
Sediment diameters
D*
Sieve size No 80
D
Dimensionless grain size (Ackers, White)
gr
e
b
(US Standard!.
Bed load transport efficiency
e
Suspended load transport efficiency
s
Efficiency of the sediment transport process (Ackers,
E
White)
F
F
Bed factor
b
bo
F
gr
(Blench)
Bed factor when there is no movement of bed particles
(Blench)
Sediment mobility (Ackers, White)
Fr
Froude Number
F
Side factor
S
(Blench)
g
Acceleration due to gravity
g
Bed load transport rate, dry weight per unit width
bt
g
st
per unit time
Total transport rate, dry weight per unit width per
unit time
G
Bed load transport rate, dry weight per unit time
2
G
Reynold's number for solids sheared in fluid
GF
p
Nucleus load, dry weight per unit width per unit time
for a given grain size range
1 ,1
1 2
Integrals
k
Correction factor fOr models
k
k
k
k
m
r
s
se
(Einstein)
Coefficient of bed roughness, grain and form (Strickler)
Coefficient of particle friction, plane bed (Strickler)
Grain roughness of the bed
Coefficient of total roughness, grain and form
(Strickler)
k
w
Coefficient of side wall roughness
K
Meander slope correction
log
Common logarithm (base 101
(Blench)
(Strickler)
ln
Naperian logarithm (base e)
m
Exponent in the general function
M
Sediment transport rate, mass per unit width per unit
(Ackers, White)
time
n
Transition exponent
n
Manning roughness coefficient
p
The proportion by weight of particles of size D
(Ackers, White)
or,
p
as a suffix, denoting association with a particular
size fraction
Pb
Fraction of bed material in a given range of grain
sizes
PB
Fraction of bed load in a given range of grain sizes
Pt
Fraction of the total load in a given range of grain
P,P
P
Factors which indicate the proportion of the bed
q
Water discharge per unit width
q
bt
q
st
sizes
taking the fluid shear
(Kalinske)
Bed load transport rate, submerged weight per unit
width
Total load transport rate, submerged weight per unit
width per unit time
Q
Water discharge
Q
s
That portion of Q whose energy is converted into
eddying close to the bed
R
Hydraulic radius
R'
Hydraulic radius ascribed to the grain
s
Specific gravity of sediment
*
s
Expression,.
S
Water surface or energy slope
o
(Y/Y
cr
-
l)
(YaUnl
A variable
t
t
g %
T
Coefficient of dynamic friction
A parameter including variables which are functions of
temperature
7
T
F
Temperature of water in degrees Fahrenheit
v
Flow velocity at a distance y* above the bed
v* I
Shear velocity related to grain
v*
Shear velocity
v*
c
Critical shear velocity
V
Mean velocity of flow
w
Fall velocity of the sediment particles
(T/p)
(Einstein)
�
Concentration by weight, ie weight of sediment/weight
x,x ,x
s b
of water . . Subscript s denotes suspended load,
b denotes bed load.
X
Characteristic grain size of a mixture
x
Parameter for transition (smooth to rough)
y
Pressure correction in transition
y*
Distance above the bed
Y
Dimensionless Mobility Number
Y
C
Z
(Shields)
Mobility number at threshold conditions
Bxponent relating to the suspended distribution
Z
Z
(smooth to rough)
v
p
Exponent relating to the velocity distribution
Exponent relating to sand distribution in the middle
zone of transport
A coefficient (Ackers, White and Bagnold)
Logarithmic functions
Specific weight of the fluid
p
Density of the fluid
Density of solids
\!
Kinematic viscosity
Specific weight of grain in fluid, g(p -p)
s
constant
(Einstein)
Hiding factor
e
Correction factor
w
Stream power per unit boundary area
T
Tractive shear
Tractive shear associated with sediment particle
Critical tractive shear to initiate motion
Thickness of the lamlnar sublayer
Laminar sublayer with respect to v*'
Intensity of shear on a particle
1jJ'
Intensity of shear on representative particle
Intensity of shear for individual grain sizes
Intensity of transport
Intensity of transport of individual grain sizes
Friction factor,
BgRS /V
o
2
Strip width, Simpson Rule
Sum of ....
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
SEDIMENT TRANSPORT:
AN APPRAISAL OF AVAILABLE METHODS
VOLUME 1
SUMMARY OF EXISTING THEORIES
INTRODUCTION
In 1972 Ackers and White
(Refs 24 and 25)
proposed a
new sediment transport theory and tested the theory against
flume data.
This data consisted of about 1000 measurements
with par�icle sizes in the range 0.04 < D (mm)
< 5 and
sediment specific gravities in the range 1.07 < s < 2.65.
Subsequent phases in this investigation have included
(i) the
acquisition of-about 270 field measurements of transport rates
from the literature and (ii)
the testing of all commonly used
sediment transport theories against this extended range of
data.
These two phases are the subject of the present report.
The report also includes a proposed modification to one of
the existing stochastic theories which significantly improves
its performance.
Volume 1 describes the sediment transport theories and
indicates in each case how the theories have been used in
the present investigation.
Where the theories include
graphical solutions, analytical equivalents have been worked
out to facilitate the use of a digital computer for the
analysis.
The limitations of the theories are indicated
wherever the original authors have given specific recommenda­
tions.
1
Volume 2 describes and classifies the data used for
comparative purposes.
It defines the criteria on which the
comparison between observed and calculated transport rates
are based and presents the results provided by the theories
described in Volume 1.
The problems of graded sediments are
discussed and illustrated using gravel river data.
A
modification to the Bishop, Simons and Richardson theory is
proposed together with suggestions for further refinements
which could be introduded at a later date.
The performance
of the various theories is compared and recommendations are
made regarding their usage.
BED LOAD EQUATION OF A SHIELDS (1936)
In 1936 A Shields, Ref
(1), gave the following formula
for the ped load transport rate:�
Y
s
S y
0
Where Y
Y
s
T
T
C
Q
=
10
T-T
C
:D'Y
•
s
=
tra.ctive shear
=
critical tractive shea.r
o
(1)
=
50
water discha.rge
= bed load transport rate,
G
,
= specific weight of solids in transport
=
Q
•
specific weight of the fluid
= mean diameter defined as D
D
S
=
2
G
dry weight per unit time
Energy gra.dient or surfa.ce slope
2
Substituting
Q
=
bdV
G
=
b g
=
2
v* /gd
T
=
pv*
g
bt
=
X Vdpg
b
1' IY
s
=
s-l
S
in e quation
0
bt
2
=
gpdS
o
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
(2)
(3)
(4)
(5)
( 6)
(7)
( 1) gives a concentration for the bed load, X ,
b
as:-
10 v*
(s-l)
2
4
Ddg
1 2 (
\
•
•
•
(8)
But defining the mobility number at the threshold conditions
as:-
10 v*
(s-l)
2
4
Ddg
2
(
1
•
•
•
•
.
•
(9)
( 10)
This equation is claimed to be valid at low transport
rates and under conditions where bed formations are relatively
flat .
Knowing v, v*, d, D, s and g sediment concentrations
are calculated as follows:-
1.
D
is computed from equation
gr
2.
y
( 1 6 1 ) , see page 4 0
is calculated from equation (A4) , see
cr
Appendix 1 , or from the original graphs of A. Shields
(Ref 1 ) .
3.
Substitution in equation
( 10) yields X .
( 10) is, of course, non-dimensional .
3
b
Equation
BED LOAD EQUATION OF A A KALINSKE (1947)
A A Kalinske
(2)
developed an equation for
(1947)
computing the bed load sediment transport rate .
He also
presented a method for adapting the equation to sand
It is as follows:-
mixtures.
•
where g
•
•
( ll )
=
bed load transport rate in a range o f grain
P, P
=
factors which indicate the proportion of the
D
=
btp
p
p
sizes , dry weight per unit width per unit time.
bed taking the fluid shear.
effective particle size ( diameter ) within a
given range of sizes.
The factor P was taken as 0.35 ( Ref. 2 ) and P was
p
evaluated using the expression
P
p
= P
substituting equation
relate g
btp
p/D
l:
(p
P
iD
(12)
p
)
•
=
•
(12)
in (11) and using (6) to
to the concentration by weight gives
•
where n
•
•
•
(13)
number of fractions into which the grading curve
is divided .
The value of '
c
was giVen by Kalinske as : "'""'
•
Hence , using ( 12 ) and substituting
4
'
0
=
pv*
2
•
•
(14)
TC
T
O
P ( s - l)g
2
v*
1
=
"3
p
.
•
.
( 15)
for rough. turbulent flow.
I n programming for the computer, the graph expressing
the function of T IT in the original paper of A A Kalinske
c 0
was brought to the fo l lowing analytical form:I f 0. 40 < T IT < 2. 50
c 0
. .. ( 1 6 )
0 , T IT < 0. 4
c 0
If
f
(: )
c
o
=
1O
( 0. 3 7 5 -T h )/o.945
c o
. . . ( 17)
A l l the equations are dimensiona lly homogeneous s o that arty
consistent set of units can be used and the calculation
proceeds as follows:1.
The representative grain size is chosen as the
mean diameter of each size range into which the
grading curve is divided.
The ratio T IT is evaluated using equation ( 15 )
c 0
with P = 0.3 5 for each fraction.
2.
3.
The function f (T IT ) is computed either with a
c 0
graph ( Ref 2 ) or by using equations ( 1 6 ) and ( 1 7).
4.
The total concentration for the who le bed material
load is computed from equation ( 1 3).
5
THE REGIME FORMULA OF C INGLIS (1947)
The regime theory rests on the principle of channel
(3)
self-adjustment .
According to E S Lindley
(1919) this
principle can be expressed as follows:-
"When an artificial
channel is used to convey silty water, both bed and banks
scour or filli
depth , gradient and width change until a
state of balance is attained , at which the channel is said
to be in regime" .
I n the discussion on the paper "Meanders and their
(4 , 5 )
bearing on river training" by C Ingli
1947, as an
�
answer to C M White, he suggested the following dimensionless
set , in which the effects of the sediment "charge" was
introduced into the Lacey regime equations of that time:
b
V
d
S
0
=
2 .6 7
=
0 . 7937 9
=
=
g
Q
l:1
1/3 v1/12
)/18
g
6
vl/3
X
W)4
(�
l/ 6
(DX w)
s
s·
=
. . . (18)
1/12
.. . (19)
v l/9 Q l/3 D l/ 6
0 . 4725
l/1 8
1/3
g
(X w)
s
. . . (20)
5/ 1 2
(DX w)
s
0 . 000547
6
6
v5/3 g 1/18 Q l/
. . . (21)
Algebraic manipulation o f equations
X
"
0. 5 6 2
( 1 8 ) and
(19) produces:-
•
•
.
(22)
in which w is the fall velocity o f a characteristic sediment
particle which is as sumed to be the grain having
the mean diameter (D
6
50
) of the bed material.
The above formulae are for quartz sand in water , and can be
used with any consistent set of unit s.
The calculation proceeds as follows : The fal l velocity is computed either using Rubey's
1.
equations (A5 )
( See Appendix I ) or from any experimental
curve, for the mean diameter D
50
•
The concentration is determined from equation (22) .
2.
BED LOAD EQUATION OF A MEYER-PETER AND R MULLER ( 1 9 4 8 )
Meyer-Peter and Muller
(6 )
( 1 9 4 8 ) determined, for the
bed load transport rate, an empirical relation which can be
written as follows:
Q
,,2.
Q
/
3
( ) 2
k
�
k
r
where q
bt
Q
k
k
D
(2:)
g
•
•
.
. = bed load transport rate, submerged weight per
s
se·
r
dS = 0. 047 Y D + 0. 2 5
o
s
1/3
(23)
unit width .
= that part of Q whose energy is converted into
,
eddying near the bed.
= coefficient of total roughne s s due to skin and
form roughness in the Strickl er formula.
=
=
coefficient of particle friction with plane bed
in the Strickler formula.
effective diameter of the sediment given by
D =
l:P
1
p . D
p
in which p is the fraction of bed
material in a size range D
P
7
.
Now ,
·
3
sv*
8"7"::"-"''-;�
( s -1)Vd;-g
=
[Q
s (
k
.
( 24)
(4) and ( 2 4), equation ( 2 3 ) reduces
Hence using equations
to the form:-
•
se
Q \ kr
)
3/2
0.047
-
2
v* /gD ( s - l)
J
3/2
•
.
•
•
. (25)
According to the authors of Reference ( 6 ) the ratio
determined for rectangular channels as fol lows:3/2
b k
w
=
3 /2
+ b
2d k
,se
.
( 2 6)
in which k is the side wal l roughnes s
w
k
and
se
is computed using the following expressions:k
m
=
R
V
2/3
s�
0
Vg '
l/ 6
v* R
k
=
k k b2/3
m w
=
k
3 2
se
/2
3/2 2/3
+ 2 dCk / _k
)]
[bk
w
m
•
•
•
•
•
.
!
When b
+
00
( 2 7)
(28)
equation ( 2 8 ) reduces t o ( 2 7) and for
can be taken as k
practica l purposes, natural rivers, k
se
m
b
when
� 15 .
It can be proved that in this case the error
d
in the computed value of k
is about 10 per cent .
For
se
typical prototype sizes equation ( 2 6 ) also reduces to unity
and Q
s
=
Q.
Mos t experimental f lumes are constructed of concrete,
glass or steel so that k can be estimated as:w
=
- -1
n
=
1
0.01
=
100
where n is the Manning coefficient .
The value o f k
r
is given by
8
•
•
•
( 29)
k
r
•
. . (30)
The friction factor A is obtained from the well-known
equation of Nikuradse in which it is expres s ed as a function
of the Reynoldsnumber and the relative roughness
1
_ _
IX
=
_
2 log
(
D
Q
0-",
9-;:;
..,...,-=.
-:;:1 4 .8 RQ
s
+
O . 6 2 75
v)
RV IX
In the region of fUlly developed turbulence k
calculated with:-
k
r
=
�
(metric
l/ 6 .
D
90
units)
r
•
•
.
•
•
•
(31)
can be
(32)
Except equations (27) , (2 8 ) , (29) , (30) and (32 ) which are
in m and m/s , all the equations are dimensionally homogeneous
so that any consistent set of units may be used with them.
The calculation proceeds as follows:
1.
The value of D , effective diameter of the sediment ,
is determined by D =
p D '
p
Ll
and the ratio Q /Q are determined
se
s
from equations (2 8 ) (with k = 100) and (2 6 ) respectively ,
w
when the sediment transport rate in experimental flume
2.
The values of k
is required .
(27) with R
In natural rivers k
=
d and Q
s
=
Q.
se
=
k using equation
m
The value of k is evaluated using equations (30) ,
r
(31) or (3 2 ) .
3.
4.
The concentration by weight is computed by using
equation (2 5 ) .
9
BED LOAD EQUATI ON OF H A EINSTEIN
The bed load relationship developed by H A Einstein is
derived using considerations of probabilities, (Ref 7) .
His
method, proposed initially in 1942, was considerably
modified in his later works dating from 1950 onwards.
In this method the bed load discharge is computed for
individual size fractions within the whole bed material .
This means that the size distribution of bed load is also
obtained .
The H A Einstein Bed-Load function is as follows : -
1
A* 4>*
+
A* 4>*
=
where
4>*
=
1)1*
=
1 -
PB
1
B *1)1* - In
_t 2 0
dt
e
f
1
ITI
q
-B * 1)1* -.
btP
ti
•
.
•
•
•
•
•
•
1
n
0
�
(s-l)D g
2
(1*
(v�)
(33)
-
Pb (Y D )3/2
s p
� Y
p
•
12
2
(34)
(3 5 )
=
fraction of the bed load of a given grain size,
=
fraction of the bed material of a given grain size,
4>*
=
intensity of transport for individual grain size,
v'*
=
shear velocity with respect to the grain,
S
=
"Hiding factor" o f grain in a mixture,
y
=
pressure correction in transition (smooth-rough)
where P
B
Pb
10
•
According to Einstein,
132
13;
=
[ (
lO
log
10. 6
10 . 6
2
t) J
. . . ( 3 6)
and B* = 0 . 143, A* = 43 . 50, n = 0 . 50 .
o
substituting from equation
( 2 4) the bed load concentra-
tion associated with a particular size fraction becomes : 3/2 >;;
p <p* ( s-l) >;; s D
g
b
P
= PBX =
X
Vd
bp
.
•
.
(37)
and the total concentration of bed load
X
b
= LP X
1 bp
. . . (38)
Einstein considered that the velocity distribution in
an open channel is described by the logarithmic formulae
based on V Karman' s similarity theorem with the constant as
(8)
proposed by Keulegan
.
He gave the mean velocity including
the transition between the rough and smooth boundaries as:
V =
5 . 7 5 log
v�
in which : -
(
12 . 27
)
d' t
"'i:"
. . . ( 39)
d' = mean depth with respect to the grain,
v � = shear velocity with respect to the grain ,
=
/l
the apparent roughnes s of the surface ,
/l =
where
k
x
s·
k
s
x
. . , ( 40)
= the roughness of the bed.
= function of k. /o'
s
6' = the thickness of the laminar sublayer at a
smooth wall , 1 1 . 6 v/v�.
f
In the original works R' is used rather than d'.
11
. . . ( 41)
The value of X in equation ( 3 6 ) can be calcul ated
according to Einstein as fol lows:X = 0. 77� if �/O'
>
1 . 80
X = 1.390 if �/O' < 1 . 80 or if there is a uniform
grain size
•
•
•
(42)
and the values o f the two correction factors in equation ( 3 5 )
are given by:-
i;
y
=
(i)
f
(D/x)
)
(
f E ( v* D/ v )
=
( i)
f E
( k / 0')
s
·
•
.
•
•
•
(43)
( 44 )
H A Eins tein presented graphs for the evaluation of x,
I; and y .
The hiding factor,
1;, was introduced to define
the influence of the mutual interference between the
particles of different sizes with respect to the buoyancy
(4)
forces.
The original paper of Einstein
gave the factor
I; as a function of D/X only.
In his later works carried out
(9)
in 1954 he introduced a second correction
with Ning Chien
factor, this being a function of v* D /v .
represents the more recent concept .
Equation ( 4 3 )
The Einstein method includes a procedure for computing
mean velocity, obtaining v* " from a bar resistance curve by
trial and error and hence v* ' since v* is known .
The v* '
value enables x.to be calculated and the mean velocity can
then be determined .
However, in the present exercise,
measured mean velocities were available and the following
procedure was adopted for determining
( v* ' )
v*
Hence
2
= gd' S
2 =
gdS
d' = d .
o
•
o
·
( v* ' )
v"
2
12
. . (45 )
. . ( 46 )
2
•
•
•
( 47)
substituting equations (47 ) and (40) into
v
= 5 . 7 5 log
(k
s
(;
( v�)
1 . 2 7d •
65
v*
( 39) gives
2
• •• (48 )
2
has been taken as D 6 ) .
5
The value of x as a function of D 6 /0' is evaluated
5
from either the graph given by Einstein or the fol lowing set
of equations:-
,
D 6 /0' < 0 . 4 .
5
If
If 0 . 4
;;
If 2 . 3 5
;;
D 6 /0' < 2 . 3 5 ;
5
D 6 /0' < 10
5
If 10
x = 1 . 70 log D
6 5 /0' + 1 .90
1.6
x = 1 . 6 1 5 - 1 . 544 ( log D /0,)
65
2 43
x = 0 . 92 6 ( log D /0'_1 ) .
+ 1 . 00
65
x = 1 . 00 .
·
•• (49)
Vanoni and Brooks ( Ref 2 3 ) give a direct graphical method
and the reader interested in this approach should refer to
this work .
In programming for computer the graphs given by
(iii)
e
= f
( v* D/ v ) from
Einstein for determining S, y and
equations
( 43 ) and
fol lows:1.
(44) were expres sed analytically as
Hiding factor , s
If 0.10
;; �
If 0 . 7 3 ;;
< 0 . 73;
D <
1.30,
X
s = 0.70
s = 1 . 20
with:v* D
v
e
(�t
O . 6 92
. e .
1 . 00
s = -e
X
--
t2•3 8 5
-
D
If 1 . 30 <
If
(i
(
• . . ( 50 )
0 • 544 - log
;;
3 . 50
e
= 10
13
0 . 41 2
--V)
v* D
If
2.
V*D
\!
--
>
e
3 . 50
= 1 . 00
. . . (51)
Pressure correction in transition ( smooth-rough)
If
D
22
cl"
y =
.; 0 . 4 7
If 0 . 4 7 <
I f 1 . 70 <
If 3 . 15 <
D
65
T
D
65
T
D
65
T
.; 1 . 70 ;
C� )
D
5
f
Y = 10
1 . 187
(
-2 . 2 3 log
G� )
5
.; 3 . 1 5 ;
y = 0.8
.; 5 . 00;
y = 0 . 525
D
65
-0 . 0 4 9 2
�
Y
-0 . 08 3
-0. 3 7 8
I f uniform grain size; y = 1 . 00
. . . (52)
All the equations are dimensionally homogeneous so that any
consistent set of units may be used .
The value of the
integral in equation ( 3 3) was evaluated by expanding the
integrancr in Maclaurin's series and integrating term by term
(see Appendix Ill.
The calculation proceeds as follows : �
From the granulometric curve D
1.
s kin roughnes s k '
s
The friction velocity
2.
y�
65
is taken a s the
is calculated either using
the method des cribed previously or the direct graphical
method given in Ref (23).
The thickness of the laminar sublayer 0 is obtained
3.
from equation ( 4 1 ) .
The correction factor x is evaluated either from
4.
the graph given by Einstein or from the equation set
(4 9 )
•
The apparent roughness � is obtained using equation
5.
(40 )
•
14
From the ratio 1!./6"
6.
the value of X is calculated
using the set equation ( 4 2 ) .
7.
The pre ssure correction factor is determined us ing
the set of equations
(52) .
8.
2
2
The ratio p /p * i s calculated from equation ( 3 6 ) .
9.
The representative grain sizes are chosen as the
mean diameter of each s i z e fraction .
The " hiding factor" � of each fraction i s deter­
10 .
mined from the sets of equations ( 5 0)
and ( 5 1 ) .
For each grain s i z e fraction the parameter � * i s
11.
cal culated us ing equation ( 3 5 ) .
The intensity of transport , $ * , for individual
12.
grain s i z e i s evaluated from equation ( 3 1 )
Il)
( see Appendix
.
The concentration by weight for each fraction and
13 .
the total concentration for the whole bed material are
then computed using equations
( 3 7) and ( 3 8 ) .
TOTAL LOAD FORMULA OF H A EINSTEIN ( 1 9 50)
In his 1950 paper (Ref 7) Einstein a l so gave a method
of determining the total sediment load , excluding wash load .
The total load in a particular grain s i z e range was given
as : •
in which
•
.
(53)
PT = fraction o f the total load in a given grain
size
g
st
=
total load, dry weight p e r unit width per
unit time .
15
1
1
where A
=
1
2
=
=
0 . 216
0 . 216
A
( l_A)
A
1
Z-l
Z
)
�
)
Z
dt
. . . (54)
ln t dt
. . . (55)
A
1
Z-l
( l_A)
f(
�
l t
Z
f(
l t
Z
A
ratio of bed layer thicknes s to water depth
A
=
2D/d
Z
=
2 . 5w/v*
p
=
. . . (56)
,
1
1og
0 . 434
3Q . 2
( Md
)
. . . (57)
. . . (58 )
The transport rate i s related to the concentration by
weight using equation ( 2 4 )
and making these substitutions in
equation ( 5 3 ) we obtain
. . . (59)
and the �otal bed material load for all the fractions i s
given by
X
s
=
l: P X
sp
1
•
•
•
(60)
The two integrals i n equations ( 5 4 ) and ( 5 5 ) were solved
by numerical integration using Simpson' s Rule ( see Appendix
Il) .
The calculation proceeds as fol lows : 1.
The settling velocity i s calculated from equation
(A5 ) u s ing the mean diameter for each fraction of the
bed sample ( see Appendix I ) .
2.
The values of Z and A are calculated using
equations
( 5 7 ) and ( 5 6 ) for each fraction .
The integrals 1
and 1 , equations ( 5 4 ) and ( 5 5 )
2
1
are evaluated by numerical integration ( see Appendix 11)
3.
or by the original graph of H A Einstein
16
(Ref 7 ) .
The value of P i s determined from equation ( 5 8 ) .
4.
5.
from equation ( 3 7 )
Introducing the values of X
bp
in equation ( 5 9 ) the concentration , X
, i s obtained ,
sp
and from equation ( 60 ) the concentration for all the
fraction s .
BED LOAD EQUATION OF H A EINSTEIN AND C B BROWN ( 1 9 50 )
The bed load equation of Einste in-Brown i s a modi fica­
tion developed by H Rou s e , M C Boyer and E M Laursen of the
(ll)
formula of H A Einstein .
It was presented by C B Brown
i n 1 9 50 and appears to be based on empirical considerations .
The equations can be written as follows : -
<p
where
=
1/1jJ =
g
3/2
p
( s -1 ) D
s
(s-l)
50
k
•
•
•
•
•
•
•
.
•
•
•
•
2
g
•
( 61)
(62)
(63)
(64)
The relationship between <p, F and 1jJ was given by the authors
of Reference 11 in a semi-graphical form .
When 1/1jJ exceeds
3
To programme the relationship
0.1 it become s <P/F = 40.( 1/1jJ)
•
when 1/1jJ i s' l e s s than 0 . 1 the function was approximated by
an exponential equation
�
:t:.
F
=
10
-4 . 0
+
( 2 3 . 6 9 5 + 1 6 . 9 2 5 log
1 k
�)
2
•
17
.
•
(65)
Combining the above equations and relating the bed load
transport to the concentration by weight we obtain the
fol lowing predictive equation s : -
If
gD
50
( s- 1 )
<
0.1
i
X
bt
=
Vd
• q,
. . . (66)
where q, i s determined from equation ( 6 5 )
For the above expre s s ions the quantity F appears i n the
Rubey equation ( see Appendix' I ) ' and is defined as a
dimensionless function of f a l l velocity .
The calculation proceeds as follows : 1.
The dimensionless function o f the fall velocity i s
calculated using equation
(6 4 ) .
The concentration o f bed load transport by weight
4'
i s evaluated using the equations ( 6 2 ) to ( 6 6 )
inclusive .
TOTAL LOAD FORMULA OF A A BISHOP , D B SIMONS
AND E V RICHARD SON ( 1 9 6 5 )
The quantities A * and B * which appear in the relation
( 3 3 ) were as sumed by H A Einstein as constant s .
Later works
'
of A A B i s hop , D B S imon s and E V Richardson (Ref 10)
revealed that the relation ( 3 3 ) can be improved if A* and B*
are treated as variable quantitie s .
The authors of Reference
( 10 ) reasoning that " the
instantaneous variations in the l i ft forces may l i ft some
18
particles from the bed into suspension whereas other particles
will be moved only within the bed layer " , concluded that the
relationship �* - ** should be concerned with the total bed
material sediment transport rate rather than with the bed
load transport rate .
Working on this basis they found a
relation between A* and B* and the grain s i z e D .
The experiments reported do not make clear the influence
of such parameters as specific gravity and temperature .
The
graph of this relation is expressed here in the fol lowing
analytical form :
< I mm
If 0
50
B*
=
i B*
=
< 1 . 1 2 mm
If 0
50
A*
=
1 . 1 2 mm
A*
=
If 0
If 0
50
50
;. 1 mm
;.
0 . 03 7 5
+ 0 . 1054
0
50 (mm)
0 . 14 3
( 050 ( mm )
0.2
43.5
+ l
Y
•
•
•
(67)
The method outlined by Bishop , Simons and Richardson
involve s : (i)
The computation o f * ' a s in the Einstein method
but , in order to uti l i se the meas ured mean velocities available from experiments , v*
,
was computed
by the method presented by Vanoni and Brooks
23) .
(Ref
*' i s then given by
*' = D
3S
(S- 1 ) g/ (v*,)
( i i ) The u s e of expre s s ion ( 3 3 )
=
1
2
•
•
(68)
in the fol lowing form : -
•
19
•
•
•
(69)
in which $ i s given by
$
=
(y D ) 3 / 2
s 50
=
•
•
.
(70)
and A* and B * are given by the set o f equations
Thus the values o f P and P are assumed
B
b
equal to unity and the concentration of the total
( 67 ) .
load i s given by equation ( 3 7 ) in which P = 1 . 0 .
b
According to the authors o f Reference ( 10 ) the rel ation­
ship described by equation ( 6 9 )
and dunes exi s t .
is valid only when ripples
In the transition regime and for later
stages of motion systematic errors develop and the rel ation­
ships have been mod ified to fit the data empirically.
In
and D
50
the original works the relationships between $, W'
are g iven graphically .
The calculation proceeds as follows : 1.
The value of v*
,
i s computed using either equations
( 4 5 ) to ( 4 8 ) or the graphs pre sented in Reference ( 2 3 ) .
2.
The shear intensity factor W' i s determined using
equation ( 6 8 ) .
3.
The values of A* and B* are obtained from the set
of equations
4.
(67 ) .
The intensity of transport for the total bed
material load i s evaluated u s ing either equations
(69)
and ( 70 ) or the graphs given by Bi shop, Simons and
Richardson in Reference
5.
( 10 ) .
The concentration i s computed from equat ion ( 3 7 )
putting P = 1 and D
p
b
=
D
50
20
.
BED LOAD EQUATION OF R A BAGNOLD ( 1 9 5 6 )
R A Bagnold
(12)
( 1 956 ) gave a formula to express the
bed load transport rate .
It was based on principles o f
energy and,may b e written as follows:-
l:2
q
bt P
�
= ClY (Y-Ye )
3 2
(I' D ) /
s
Y =
in which
•
(71)
=
V
/ / ( gD (S-l»
Ye
=
The value of Y at initial motion
bt
=
bed load transport rate , submerged weight per
Cl
=
unit width
8.5e /t 1jJ
b g 0
•
e
= bed load transport e ff i c i ency
,"
=
b
t
•
Shield's mobi lity parameter given by
Y
q
•
g'l'o·
•
•
(72 )
coeffic ient of dynamic friction.
Substituting the Shield's parameter into equation ( 7 1 ) and
noting that the bed load transport rate i s related to the
concentration by weight through
we obtain
Cl
v*
3
•
•
•
•
•
•
s
( s -l ) Vdg
(73 )
(74 )
where D i s taken a s D = ED
P in which p i s the proportion by
p
weight o f particles in a s i z e range D
P
•
R A Bagnold related e /t 1jJ 0 to particle s i z e and
b g
presented the functional relationships graphi cal ly .
In
programming for computer these relationships were introduced
into equation ( 7 2 )
in the following form : -
2l
0 . 5 mm
Cl.
=
8 . 50 D
If D � 0 . 5 mm
Cl.
=
4 . 25
If D
<
(mm)
•
•
•
(75)
This theory has the fol lowing restrictions : (i)
The ratio e /t W i s given for sand and water .
b g o
It i s thus only applicable where Y = 1 . 65 .
s
(ii)
I t i s claimed to be valid only for the earlier
stages o f the movement , say Y � 0 . 4 .
In h i s derivation R A Bagnold as sumed a rough
(iii)
plane bed (without s and wave s ) and rough
turbulent conditions .
The calculation proceeds a s fol lows : The value of the effective diameter o f the sediment
1.
forming the bed of the channel is determined from
D
2.
=
1:1
pD '
p
The value of
equations
3.
Cl.
i s determined from the set of
(75) .
The dimen sionless critical tractive force Ye i s
calculated either from the equation A 4
( se e Appendix I )
or f rom the original graphs presented by Shields
4.
(Ref 1 )
The concentration by weight is evaluated using
equation
(74) .
TOTAL LOAD FORMULA OF R A BAGNOLD
R A Bagnold
(13)
(1966 )
has developed a theory to express the
total load of bed material which is based on similar
principles to the bed load equation .
22
•
The total load equation can be written as:=
where q
st
w
e
b
=
e
b
tijJ
0
g
+ e (l
s
v
e ) b w
-
•
(76)
unit width per unit time
=
stream power per unit boundary area
=
bed load transport efficiency
suspended load transport effic iency
t 1JJ
g o
=
coeffi cient of dynamic friction
w
=
effective fall,velocity
s
.
'total load (bed material ) , submerged weight per
=
e
·
w,
The stream power ,
is related to the hydraulic properties
as fol lows:w·
=
pgQS
0
b
and substituting
v*
2
w
=
pgdVS
=
gdS
=
pVv*
o
.. . (77)
o
2
. . . (78)
The transport rate for the total load (bed materi a l ) i s
related to the concentration X
q
st
=
s
by
)
�-� l�
� (s
X Vdp g
s
s
.. . (79)
substituting ( 7 8 ) and ( 7 9 ) in
X
s
=
v*
2
r e
s
gd ( s - l )
b
l tijJ
g
0
+
e
s
(76 )
( l-e
)�J
b w
. . . ( 80 )
In h i s later works R A Bagnold relates t * 0 to a
g
parameter closely analogous to the fluid Reynolds number
t *
g 0
where
G
2
=
=
2
f {G }
s D
2
14 v
23
v*
2
•
•
•
•
.
•
(81)
2
(82)
R A Bagnold gave graphs to determine e
and t �0 but
g
b
these have been brought to analytical form as follows : (i)
\1�0
If
G
I f 150 " G
If
G
2
< 150
2
< 6000
2
>
;
6000
t �
g o
=
0. 75
t �
g o
=
-0. 2 3 6 log G
2
+ 1. 250
t � = 0 . 374
g o
•
•
•
(83)
The theory i s only c l aimed to be applicable at high
transport rates , the applicable range being defined by the
expre s sion
G
(ii) e
b
2
>
3
0 . 65 . s ( s - l ) D g
--r4
2
•
v
t �
g o
•
•
•
(84)
(Only for 'Y = 1. 65 and 0. 3 " V (m/ s ) " 3 . 0 )
s
D < 0. 06 (mm)
e
If 0. 06 " D ( mm) < 0 . 2
e
< 0. 7
e
If
I f 0. 2
"
If 0 . 7
< D (mm)
D (mm)
e
b
b
b
b
=
-0. 012 log 3. 2 8V
+
0 . 150
=
-0. 013 log 3 . 2 8 V + 0. 1 4 5
=
-0. 016 log 3 . 2 8V
=
-0 . 0 2 8 log 3. 2 8V + 0 . 1 3 5
+
0. 1 3 9
where V = mean velocity o f flow (m/ s )
D
=
effective particle diameter .
In order to bring his transport formul a into a form that
can be used for practical purposes , Bagnold assumed , ignoring
certain uncertaintie s , that the suspension effi ciency e
the universal constant value 0.015 for fully developed
s
suspension by turbulent ' shear flow , and took for the
numerical coe fficient in the second term of equation
2
the round figure of 3 0 . 015 = 0. 01.
24
(76)
has
Equation ( 7 6 ) thus become s : -
0 . 01
vi
wJ
. . . (86)
The quantities D and w are taken as fol lows : -
D
=
EP
pD
��
E
l
p
of the whole bed material
of the suspended material
w =
w = �
Or
E
l p wP
E
lP
•
.
•
(87)
. . . (88)
of the whole bed material . .
.
(89)
All equations are dimensionless except the set ( 85 )
where the units are m/so
The total load theory has the
fol lowing restrictions : (i )
It i s not claimed to be valid at the early stages
2
< 0.4,
of movement, say v* /gD ( s -l»
( i i ) there i s no evidence to support the theory at
depths below 150 mm ,
( i ii ) e
'Y
b
s
values are related t o sand transport i n water ,
=
1 . 65 .
The computation proceeds as follows : 1.
The effective grain diameter and fall ve locities
are computed from the grading curve of the bed material
using equations ( 8 7 ) ,
( 8 8 ) and ( 8 9 ) and AS
(see Appendix
I) .
2.
The bed load transport efficiency i s evaluated
either from the original graph of R A Bagnold ( Re f 1 3 )
or by using the set of equations ( 85 ) .
25
3. The value of G2 is determined from equation (82).
4. The coefficient of dynamic friction tgW is
evaluated using equations (83) and (84) or the graphs
in Ref (13).
5. Concentrations are determined from equation (86).
0
THE TOTAL LOAD rORMULA or E
M
LAURSEN (1958)
The method of calculating total load proposed by
E M Laursen (Ref 14) rests on empirical relationships. It
predicts both the quantity and composition of the sediment
in transport.
The basic equation can be written:x
st.
where , o '
=
=
c
, ' =
,o '
,c '
=
0.01 �
(90)
LL1
tractive shear associated with the sediment
grains,
critical shear associated with the grains.
2 'D 1/3
pV ( 50
... (91)
58 \ d )
• . . (92)
·
.
•
substituting from equations (91) and (92) into (90) yields:p D 7/6
2
D O 1 /3
Xst = 0.01 '\' p ( cf) (58Y � g (s-l)( � ) - 1 ) • fC*)
cP
p
Ll
(93)
•
26
•
•
The value of
follows:If DP!O
If 0.1
If 0.03
Ye
>
:.
>
was related by Laursen to Dp/O as
0.1
DP!O
DP!O
>
Ye
0.03
Ye
;
Ye
=
=
=
0.04
0.08
0.16
•
•
.
(94)
thickness of the larninar sublayer
.. (95)
11.6 v/v*
0
Laursen gave the function of v* /wp in equation (93) in a
graphical form, the analytical equivalent of which can be
written:0.253
*
*
0.3;fC ) = 10.SC )
p
P
V*
log2-+1.
0.97
wp 20
= 10
If
where
<5
=
=
If
If 20
·
20 ifC* )
=
P
=
V* 2.30
5.6(wp )
v*
v*
2
0.57
log
-+0.
3.16 log 413
w
w
10
p
P
(96)
The p in equation (91) was not in the original equation
suggested by Laursen. It has been introduced in this report
so that the equation becomes dimensionally homogeneous and
Laursen's coefficient has been changed accordingly. The
Laursen method, as it stands at present, is only applicable
to natural sediments with a specific gravity of 2.65.
The calculation proceeds as follows.1. Representative grain sizes Dp are chosen as the
mean size of each size fraction, p.
·
27
.
•
The thickness of the laminar sublayer is computed
using equation (95).
3.
The values of Ye for each fraction, p, are
determined using the set of equations ( 9 4 ) .
4.
The values of f (v* /wp) are determined for each
size fraction using the set of equations ( 9 6 ) , the fall
velocity being determined using equation A5 (see
Appendix I).
5. The total concentration is evaluated using
equation ( 9 3 ) . This represents the sum of the
concentrations of the individual size fractions.
The first equation in set ( 9 6 )
v* )0 . 253
10. 8 (
wP
2.
gives the bed load transport rate, according to Laursen, so
long as conditions are within the range
-2
3
10
� v /w
* p < 10
This seems to indicate that if v * /wp < 0 . 3 the transport
of material is solely a bed process.
THE
BED
LOAD EQUATION OF
J
ROTTNER
( 1 959 )
Rottner (Ref 15) has developed an equation to express
the bed load transport ,rate in terms of the fluid and
sediment properties which is based on dimensional considera­
tions with certain empirical coefficients introduced to satisfy
available data. The main interest in Rottner's work is that
he carried out a systematic investigation into the effect of
the geometrical ratio diD.
J
28
The expression given by Rottner is:2/3
D
M
0
_��== = [[ 0 . 6 6 7\ � )
� �
O.14J
(s-l)
(gd)
p/gd3 (s-l)
2/3 3
D
O
-0 . 7 7 8 ( � )
J
+
•
•
•
(9 7 )
sediment transport rate, mass per unit width
per unit time,
mean grain size diameter.
D
50
(98)
M XVdp
substituting
and rearranging equation (97 ) yields:where
M
=
=
=
•
o
'
14J
•
•
V
(gd (S-l» �
•
•
•
(99)
The above equation is dimensionally homogeneous and any
consistent set of units can be used. In his derivation,
wall effects were ignored and according to Rottner the
theory "may not be applicable to uses where only small
quantities of material are being conveyed".
The calculation of concentrations is achieved- by direct
substitution in equation (99).
THE
BED
LOAD FORMULA OF M S YAL IN (1963)
M S Yalin developed a bed load equation based on a
theoretical analysis of saltating particles. The final
expression is:-:29
qbt
'YsDV*
where
s*
=
=
0.635 s * [ 1 - In (las+* as* ) ]
V; / (gD(s-l»
-1
Ye
·
.
•
•
•
.
(100)
(101)
Ye12
2 . 45
a
( 102 )
"""""0:4
s
Relating the bed load transport rate to the concentration
using equation ( 2 4 ) , equation (100) becomes
DV s
In (l + as*) ]
X b 0.635 *Vd
( 103)
as*
This equation is dimensionally homogeneous. The
formula is restricted (i) to plane bed conditions, (ii) to
fully developed turbulent flow and (iiil large depth/diameter
ratios.
Equation (103) was developed to apply to material of
uniform.grain size. When the method is applied to a graded
sediment Yalin suggested that "an effective or typical"
grain size should be used. He was not specific on this
pOint and the mean diameter (D501 has been used in the
present analysis.
The calculation proceeds as follows�1. The mean diameter (D50) is determined from the
grading curve of the bed material.
2.
Shields critical Mobility number (Ye) is
calculated using the set of equations (A4) or from
Shields graphical representation (Ref 1)
3. The values of s* and a are computed from equations
(101) and (102).
Equation (103) gives the bed load concentration.
4.
=
=
•
•
•
·
•
.
_
•
30
THE REGIME FORMULA OF T BLENCH (1964)
Based on the data and regime principles described by
C Inglis and G Lacey, T Blench (Ref 1 7) gave the following
three basic equations:(104)
V2 / d Fb
V 3/b Fs
(105)
V2 3.63 1 + x b
(106)
gdSo K ( ��� ) ( � t
where Fb bed factor
Fs side factor
b mean breadth
d mean depth
meander slope correction.
K
Algebraic manipulation of equations (104) and (106)
yields:Fb11/12 3.63 g bli ql/12 S
(107)
);
5x
'
10
Kv
1 + 233
=
•
•
.
=
•
•
•
=
•
•
.
•
•
.
=
=
=
=
=
0
=
Substituting
Vd and v* (gdSol � then gives
3.63 bli vl/12 v *2
� d11/l2
K v
q=
=
Fb11/12
(108)
5x
10
l, + 233
Blench suggested t�e following values for K
K
1.25 for a straight reach
2.00 for a reach with well developed meandering
K
without braiding
3.00 for a braided reach
K
=
=
=
=
31
•
•
•
4.00 for an extremely braided reach
The bed factor (Fb) is related to the "zero bed factor"
(Fbo) by the expression:Fb Fbo (l 0. 12X105)
(109)
for values of X less than about 10-4. The "zero bed factor"
is evaluated as fo11ows:If D ,;; 2 ( mm) i Fbo 1. 9 ID (mm)
(110)
11/72
/24
If D 2 ( mm ) Fbo o 5B w11
(111)
70 (\)70/\) )
or Fbo 7.3 D� (\)70/v) 1/6
(112)
These apply for:K
=
=
+
=
=
>
•
=
.
•
.
•
•
•
•
•
•
•
•
•
(113)
2. 65 and Fbo � 3B (�)�
If the calculated value of Fbo using equations (111) or
(112) does not S tisfY equation (113) then the value of Fbo
is taken 'as 3B(dD 1/12 .
The units are confusing. In equation (110) D is in mm ,
in equation (111) w is in cm/s and in equation (112) D is in
ft. The answer for Fbo is always in ft/s! w70 is the fall
velocity of the median sand size (D50) in water at 70 deg F
and v70 is the kinematic viscosity at this temperature. The
limits of applicability of equations (110 to (112) are not
given by Blench in a precise way.
Substituting (109) in (lOB)
3.63 v*2 b� v1/12
(114)
11/12 v% F11/l2
K d
bo
s
=
•
•
•
•
•
•
j
According to Blench the left hand side of equation (114)
can be approximated as fo11ows:32
(115)
If 10-5 < X � 10-4 LHS 19.91 Xli
(116)
If 10-4 < X 0.002 LHS = 293 537 X 13/24
Blench interpreted X as the concentration of "that part of the
total which is not suspended".
The equations have the following restrictions:(i) Equation (106) was established for low flow
conditions.
(ii) Equation (109) is valid only for sand, not for
coarse material. The value of the constant 0.12
in the same equation applies to subcritical flow
conditions and .concentrations less than 10-4.
(iii) All the equations are unlikely to apply if bid
falls below about 4 or depth below about 0.4 m.
Except for Fbo the equations are dimensionally
homogeneous. Fbo' equations (110) to (113), is in ft/s2.
The calculation proceeds as follows:1. The bed factor Fb is calculated using equations
(110) to (113).
2. The concentration X is determined by trial and
error using equations (l14l to (116).
=
�
•
•
•
•
•
•
TOTAL LOAD FORMULA OF F ENGELUND AND E HANSEN (1967)
F Engelund and E Hansen (Ref lB) developed an equation
to express the total 16ad (bed material) as follows:.2: � . = 0.1 y5/2
(117)
4
where
A
=
BgdSo
33
•
•
•
•
•
•
(l1B)
=
y
v*
=
=
gst
3/2 03/2
P s (s-l)� g
50
v*2
g 050 (s-l)
IgdSo
•
•
•
•
•
•
•
•
•
(119)
(120)
(121)
Substituting from (118), (119), (120) and (121) into (117)
yields:J,; y3/2
0.05 P s gJ,; V2 D50
(122)
gst·
(s-l)�
=
But, as before,
gst·
Hence,
X
=
=
X
Vd
P
g
0.05 s V v*3
dg2 D50 (s-1)2
•
•
•
•
•
•
•
•
.
(123)
(124)
In t!heir original paper Engelund and Hansen gave a
graphical solution which facilitated the determination of
both the water flow parameters and the sediment transport
rates. In the present exercise mean velocities were
available and were not computed by the Engelund and Hansen
method .
Equation (124) is dimensionally homogeneous and the
sediment transport concentration can be calculated directly
from this equation using any consistent set of units.
THE TOTAL LOAD FORMULA OF W H GRAF
W H Graf ( Re f
20)
(196B)
proposed a relationship to express
the total transport rate (bed material )
in both open and
closed conduits as follpws : �
=
10.39 (�)-2 .52
10.39
or
where � and
�
[
(S-l
v*
�gD -2.52
J
•
•
•
•
•
•
(125)
(126)
have the same meaning as in the Einstein-Brown
equations.
Us ing equation
(123)
and re-arranging , equation
(126)
becomes
•
•
.
(127)
-
W H Graf was not spec i f i c about the interpretation o f
the sediment diameter , D .
However , h e u s e d total load data
to derive h i s formula taking the mean diameter as the
relevant s i z e .
equal to the
Hence, in this report D has been assumed
D50
s i z e of the bed materia l .
The calculation proceeds directly from equation
(127)
using any consis tent set of units.
THE TOTAL LOAD FORMULA OF F TOFFALETI
(196B)
The total load formula (bed material ) presented by
F Toffaleti
( Refs
H A Einstein ( Re f
modifications .
21
7)
and
22)
i s based on the concepts o f
with certain , mainly empirical ,
There are three main differences from the
Einstein method .
35
(i)
The velocity pro f i l e in the vertical i s represented
by the re lationship:-
v
•
.
•
•
.
(128)
=
0 . 1198
T
F
=
water temperature i n degrees fahrenheit
v
=
flow velocity at a distance y* above
where Z
where
•
v·
0 . 0004 8 T
F
+
(129)
the bed.
2
2
( i i ) The three Einstein correction factors � /�* , �/e
and y are reduced. to two viz : -
i n which D
•
.
•
·
•
.
( 1 30 )
(131)
must be expressed in feet .
65
( i i i ) The relationship � * versus � * was assumed between
the level of y* = 2D and y*
=
d/ll. 2 4 rather than
between y* = 0 and y* = 2D as in the Einstein
theory .
The leve l s y* equals 2 D , d/ll . 2 4 and d/2 . 5 are used by
Toffaleti as discontinuities within the sand concentration
distribution .
In each of the three zones the sand concentra­
tion is defined by the expres s ion
c
where C
x
xp
=
c
. x
(- )
y*
-aZ
p
d
·
•
•
•
•
•
(132)
and a are constants for each zone.
Z
p
=
w
p
V
36
(133)
1_ (260.67 - 0.667 T ) 8.095 - 0.0207 T
(134)
_32.2
p
p
If Z p (equation (133» < Zv (equation (129» , Z p must
be arbitrarily taken as:Z
1.5 Zv
(135)
P
C
Z
=
=
=
•
•
.
•
•
.
P Toffaleti gives his <P * versus ** relationship as:(136)
<P * 17.17 /** 5/3
T Al k D 104
where 1)i *
(137)
V2
(138)
and T g (s-l) CL2 (0.00158 0.0000028 T )
p
Equation (136) can be re- written making substitutions
from equations (137) and (34) as follows , =
·
=
.
• . •
=
GPP
•
+
=
•
•
•
•
.
.
(139)
sand fraction load within the depth range
y * 2D and y * = d/l1. 24 for a particular grain
size range, dry weight per unit width per unit
time.
D* mean size of Sieve No 80 u . s .. Standard
(0.177 mm 0.00058 ft)
Por natural sand rivers s 2.65 and in Imperial Units
(tons/day/ft width) equation (139) reduces to:0.600 Pb
GPP --------�
(l40) .
�=
TA kD 5/3
(tons/day/ft width)
(0. 00�58V2)
which is the original relationship proposed by P Toffaleti.
However , equations (136) and (139), are the more fundamental
non-dimensional relationships. It does not follow, though,
where GPp
=
=
=
-
•
=
=
•
37
•
•
that the theory is necessarily applicable to all types of
sediment since the functions Al and k were determined for
sand only.
Using the above concepts, the sediment transport rates
were calculated by Toffaleti for each depth zone as follows:(i) LOWER ZONE (2D < y* ( djll.24)
0.756Zp-Z v
C
(l+Z
(141)
)Vd
NL
Pb Lp v
where
l+Z -0. 756Z P
l+Z v-0.756Zp
( d ) v
(2D
)
\1l.24
P
��
-�----------NL = ����--�
(142)
+�
Z v--�
ZP
1�
- 0.756
(ii) MIDDLE ZONE (d/ll. 24 < y* ( d/2. 5)
O.756Z p-Z v .
(1
C
(14 3)
)Vd
+
Z
. NM
Pb Lp
v
where
( d \O.244Zp [( d )l+Zv- Zp ( d )l+Zv-ZpJ
\2.5
11.24
11. 24 )
1 + Zv - Zp
(144)
(iii)UPPER ZONE (d/2.5 < y* ( d)
O.756Z P -Z v
· .. (145)
. NU
gsuP = Pb CLp (l + Z v)Vd
where
( d )O.244ZP( d )O.5ZP [dl+Z v-1 · 5Zp- i d )l+Z v-1.5Zp]
2.5
1l.24
\2. 5
1 + Z v - Zp
(146)
•
•
•
•
•
•
.
.
•
,
•
•
•
.
•
_
_
=
=
38
•
These functions derive from the expres s ion : -
C
xp
V dy*
•
•
•
( 14 7 )
The bed load d i s charge i s given by : -
( l+Z ) Vd
- P C
v
.
b LP
The value o f C
( 1 4 2 ) with
g
SLP
Lp
0. 7 5 6 Z -Z
p v
(2D)
l+Z -D . 7 5 6 Z
v
p
i s computed solving equations
= GF
p
• • •
(148)
·
( 1 4 1 ) and
•
• •
(149)
F Toffaleti suggests a check on the concentration at the
level y * = 2 D with the intention of , qUite arbitrarily ,
avoiding unreali stically high values.
3
3
value i s 100 Ib/ft or 1 . 6 tons/m ,
( )
2D
= C
Lp d
i.e.
-0 . 7 5 6 Z
P
The suggested maximum
, 100 Ib/ft
3
•
.
•
( 150)
If condition ( 15 0 ) i s not satisfied Toffaleti sugge sts
making
=
3
3
100 ( lb/ft )
1 . 6 ( tons/m )
=
(C )
(C )
p y *= 2 D
P y * =2D
•
•
•
(151)
The total sediment transport rate (bed material ) i n a
s i z e fraction , p , i s given by
and the total sediment transport rate (bed mate ri a l ) for the
whole bed material
g st· = l: P g
1 stp
39
•
•
•
( 15 3 )
For computational purposes equations
( 1 30 ) and
(131)
were brought to the fol lowing analytical forms:PAM -
Denoting
[(
5
10 9: v
32 . 2
If 0 . 1 3
< PAM l( 0 . 50
A
If 0 . 50
< PAM
A
If 0 . 6 7
< PAM " 0 . 7 2 5 ; A
l
l(
0 . 67
If 0 . 7 2 5 < PAM " 1 . 2 5
If 1. 2 5
< PAM l( 10
Denoting
FAC
=
-PAM
FAC " 0 . 2 5
If
If 0 . 2 5
< FAC l( 0 . 3 5
If 0 . 3 5
<
FAC
.;
2 . 00
;
A
;
A
l
l
=
O . 6142
4 3 PAM
1 8 5 PAM
;
. . . (154)
4 . 2O
49
2 79
2 4 PAM •
=
2
D
gd
;
'J
10 PAM -1 . 4 8 7
=
l
10 V*
=
=
l
v*
'
lj
J
65
( ft)
k
=
1
k
=
5 . 3 7 FAC
k
=
0 . 5 0 FAC
. 10
• . • (155)
5
. . •
(156)
1 . 248
-1. 1
.
.
•
( 15 7 )
If the product A k < 1 6 , i t must be taken arbitrarily
l
as A k = 1 6 .
v * ' is computed as in the Einstein Theory .
l
The computations proceed as follows:1.
The shear velocity with respect to the grain , v* ' ,
i s computed a s described herein for the Einstein
methods using the relationships of Vanoni and Brooks
(Ref 2 3 ) .
2.
and k are determined
l
( 1 5 5 ) and ( 1 5 7 ) or the
The correction factors A
us ing the sets of equations
original plots of Toffal eti ,
( Ref 2 2 ) .
3.
The exponent Z i s computed using equation ( 1 2 9 ) .
,v
4.
The exponent Z
equations
(133) ,
i s evaluated for each fraction from
p
( 1 3 4 ) and ( 1 3 5 ) , the fall velocity
being derived using equation A4
( see Appendix 1 )
The nucleus load GF for each grain size fraction
p
i s evaluated using equation ( 1 3 9 ) .
5.
40
6.
The value of the concentration C
i s determined
Lp
.
from equation ( 14 9 ) and condition ( 1 50) .
7.
The bed load discharge and the suspension load
within each zone and for each grain s i z e fraction are
computed using equations
8.
( 1 4 1 ) to ( 1 5 1 ) .
The total sediment transport rate
(bed material )
is then determined from equations ( 1 5 2 ) and ( 1 5 3 ) .
TOTAL LOAD FORMULA OF P ACKERS AND W R WHITE
(1972)
The general function o f P Ackers and W R White (Refs
24 and 2 5 )
is one of the most recent formu l ae for the
evaluation of the total sediment transport rate .
It i s
based o n phys ical considerations and o n dimensional analysis .
The various coef fic ients were derived using a wide range of
The general function i s : -
flume dat a .
G
gr
c
=
Xd
sD
where
( sediment transport)
G
gr
F
gr
(mob i l ity )
[r J
=
=
m
F
_
C)
*
l
n
-;;;:�;:;:;::
IgD (s-ll
[--..!.V�l;-;O'"d'j(
l32 log
less particle si z e .
gr
=
D
(g (S_1)
2
\)
l
41
•
•
•
•
•
•
.
(158)
(159)
l-n
m , C , A and n are , given in terms of D
D
•
n
V
v*
•
y /3
J
gr
( 1 60)
D
, the dimension­
•
•
.
(161)
For coarse sediments
(D
are as fol lows : n
=
A
=
m
=
>
gr
0. 0
0 . 170
1. 50
0.025
C =
For the transitional s i z e s
n
=
A
=
m
=
1 - 0.56 log
0.23 + 0.14
D
60)
(60
these four coefficients
)
1): ­
D
)
gr
•
•
•
•
•
•
(162)
gr
�
gr
9�66
gr
+
2. 86
log C =
1.34
log D
gr
-
( log D ) � gr
3. 53
(163)
, and the dimensionle s s grain
gr
s i z e , D . , express the square root o f the ratio shear forces/
gr
immersed weight and the cube root o f the ratio immersed
The particle mobility , F
weight/viscous forces respectively .
describes the
gr
inf luence o f viscous forces on the sediment transport
phenomenon .
Hence D
The sediment transport parameter , G
defined as : G
gr
=
gr
, is
Shear forces
::':;;:":::"= • E
:'::':::':'='=-::" TC::
"r'!:
t
d we i gh
mme r s e
where E i s an efficiency factor "based on the work done in
moving the material per unit time and the total
stream power " .
The coefficients
n
and A have phys ical meaning .
The
coefficien t n relates to the transition zone of parti c le
size;
n
=
1
for fine sediments
(D
=
gr·
i s with total shear , v* ;
1)
where correlation
0
of F . and G
for coarse
n =
gr
gr
sediments (D
where correlation o f F
and G
is with
gr
gr
gr
grain shea r , V and diD being the representative variabl e s .
) 60)
42
=
0 and equation ( 1 5 8 ) indicates
gr
Hence A is the critical mobility number .
At threshold conditions G
= A .
gr
Comparison with Shields original parameters yields the
2
relationship Yc = A .
that F
The general function of Ackers and White i s based on
flume data with sediment sizes in the range 0. 04 mm < D < 4 . 0
mm .
Correlation with lightweight sediments i s good.
limitation on Froude number of
further investigations .
�
A
0 . 8 i s suggested pending
For graded sediments the D
of the
35
bed material is suggested as the effective particle s i z e .
The equations are dimensionally homogeneous.
The calculation proceeds as follows : 1.
The value of D
2.
The values of A , C , m and n as sociated with the
gr
is determined u s ing equation ( 1 6 1 ) .
computed D
value are determined us ing either the s et
gr
o f equations ( 1 6 2 ) or the set ( 1 6 3 ) .
3.
The value o f the particle mobi l i ty number i s
evaluated u s ing equation ( 1 60 ) .
4.
The value of G
i s computed either us ing equation
gr
( 1 5 8 ) or the graphs provided by White in Ref 2 5 .
5.
The sediment concentration by weight i s determined
from equation ( 1 5 9 ) .
43
REFERENCES
1.
SHIELDS A
Anwendung der Aehnlichkeitsmechenik und der
Turbulenzforschung auf die Geschi ebebewegung .
Mittei lungen der Preussichen Versuchsanstalt fur
Was serbau und Schiffbau , He lft 2 6 , Berlin , 1 9 3 6 .
2.
KALINSKE A A
Movement o f sediment as bed load in
rivers .
Am. Geophys . Union Trans . , V. 2 8 , pp 6 1 5- 6 2 0 ,
1947 .
3.
LINDLEY E S
Regime Channe l s .
Proceedings , Punjab
Engineering Congres s , Vol. 1 7 , 1 9 1 9 .
4.
INGLIS CLAUDE C
Meanders , and their bearing o n river
training .
Maritime and Waterways Engineering Divis ion ,
Institution of Civi l Engineers , Paper No 7 , 1 9 4 7 .
5.
INGLIS CLAUDE C
Historical note on empirical equations
developed by Engineers in India for flow of water and
Proc . 2nd Congres s IAHR,
sand i n alluvial channe l .
Stockholm, June 1 9 4 8 .
6.
Formulae for bed load
MEYER-PETER E and MULLER R
Pro c . 2nd Congress IAHR, Stockholm, June
transport.
1948.
7.
EINSTEIN H A
The bed load for sediment transportation
in open channel flow.
US Dept Agriculture , Soil
Conservation , Ser . Tech. Bul l . 102 6 , 1 9 5 0 .
8.
KEULEGAN G H
Laws of turbulent flow in open channe l s .
National Bureau of Standard s , Journal of Researc h ,
Vol 2 1 , 1 9 3 8 , pp 701- 7 4 1 .
9.
EINSTEIN H A and CHIEN N
Transport of sediment mixtures
with large ranges of grain s i z e .
Missouri River
Division , Corps of Engineers Sediment Series No 2 ,
University of California , June 1 9 5 3 .
10.
BISHOP A A S IMONS D B and RICHARDSON E V
Total bed
material transport. ' Proc ASCE 9 1 HY2 , February 1 9 6 5 .
1 1.
ROUSE HUNTER (Editor) Engineering Hydrau lics.
XI I , John Wiley and Sons , New York , 1 9 5 0.
12.
BAGNOLD R A
The flow of cohesionless grain in fluids.
Phi losophical Trans . ROY . Soc. London , A, 2 4 9 , No 9 6 4
(1956)
•
44
Chapter
13 .
BAGNOLD R A
An approach to the sediment transport
problem from general physics .
Geol . Survey ,
Pro fessional Paper 4 2 2 - I , US Government Printing Office ,
Washington , 1 9 6 6 .
14 .
LAURSEN E
The total sediment load of stream . Journal
of the Hydraulics Divi sion, ASCE , Vol 5 4 , No HY1 ,
Proc . Paper 1 5 30 , February , 1 9 5 8 .
15 .
A formula for bed load transportation .
ROTTNER J
La Houille B l anche 4 , No 3 , 301-7 , 1 9 5 9 .
16.
YALIN M S
An expre s sion for bed load transportation .
Pro c . ASCE 8 9 , HY3 , 1 9 6 3 .
17 .
Mobi l e bed fluviology .
Dept of Technical
BLENCH T
Services , University of Alberta, Alberta , Canad a , 19 6 6 .
18 .
A monograph on sediment trans­
ENGELUND F and HANSEN E
Teknisk Vor l ag , Copenhagen ,
port in alluvial streams .
1967 .
19 .
ENGELUND F
Hydrau lics Resistance o f alluvial streams .
Journal of the Hydraul ics Divi sion , ASCE , Vol 9 2 ,
No HY2 , Proc Paper 4 7 3 9 , 1 9 6 6 .
20.
Hydraulics of sediment transport .
GRAF W H
Hill Book Company 1 9 7 1 .
21.
TOFFALETI F B
A procedure for computation o f the total
river sand d i scharge and detailed d i s tribution , bed to
Committee on Channel Stab i l i s ation , Corps of
surface .
Engineers , US Army . Technical Report No 5 , Vicksburg ,
Mi s s . , November 1 9 6 8 .
22.
TOFFALETI F B
Definitive computations o f sand
discharge in rivers .
Journal of Hydraulics Divi sion ,
ASCE , Vol 9 5 , No HYl , Pro c . Paper 6 3 5 7 , 1 9 6 9 .
23 .
Laboratory studies o f the
VANONI V A and BROOKS N H
roughness and suspended load of alluvial streams .
Report E- 6 8 , Sedimentation Lab , California Inst . Tech . ,
Pasaden a , Cal i forni a , December 1 9 5 7 .
24 .
ACKERS P
Sediment transport in channel s :
an
a lternative approach .
Hydraulics Research Station ,
Report No INT 102 , March 1 9 7 2 .
25 .
WHITE W R
Sediment transport in channe l s :
a general
function .
Hydraulics Research Station , Report No INT
104 , June 1 9 7 2 .
26.
RUBEY W W
Settling velocities of gravel , sand and
silt.
Amer . Jour . S c i . 2 5 , pp 3 2 5 - 3 3 8 , 1 9 3 3 .
45
McGraw­
27.
DUBOYS P
Le Rhone et les rivieres a lit affouil lable .
Annales des Ponts et Chausees , 5 Ser . , Vol 1 8 ,
pp 1 4 1- 1 9 5 ( 1 8 7 9 ) .
28 .
ROUS E HUNTER (Editor) Engineering Hydraulics .
XII , John Wiley and Sons , New York 1 9 5 0 .
29 .
SHULITS SAMUEL The Shoklitsch bed load formu l a .
Engineering , London . p p 6 4 4 -6 4 6 , June 2 1 , 1 9 3 5 and
p 6 8 7 , June 2 8 , 1 9 3 5 .
30 .
MEYER-PETER E FARRE H and EINSTEIN A
Nevere
Versuchsresultate uber den Geschiebetrieb,
Schweizerische Bauzeitung, Vol . 103 , No 1 3 , pp 1 4 7 - 1 5 0
(March 3 1 , 1 9 3 4 ) .
31.
CASEY H J
Uber Geschl ebebewegung .
Doctoral
D i s sertation , Technische Hochschu le , Berlin, Germany
(1935) .
Translation No 3 5 �1 US Army Waterways
Experiments Station , Vicksburg , Miss .
32 .
STRAUB L G
Mi ssouri River Report , Appendix XV, US
Serial No 9 8 2 9 , House Document No 2 3 8 , 7 3 rd Congres s ,
2nd Ses sion ( 1 9 3 5 ) .
Chapter
33.
USWES Studies of river bed materials and their
"movement , with special reference to the lower Mi s s i s s ippi
River , Paper 1 7 ( 1 9 3 5 ) .
34.
HAYWOOD 0 G JR
Flume experiments on the transportation
by water of sands and l ight�weight material s .
D i s s ertation , D . S c . , Mas s . Inst. o f Tec h . ( 1 9 40 ) .
35.
SCHOKLITSCHE A
Handbuch des Wasserbaues t Springer­
Ver l ag , Vien"na, 2nd Ed . , Vol . I ( 1 9 50 ) .
36.
LEVI I I
Dynamics o f r iver beds .
1 9 4 8 ( in Russian ) .
37.
BOGARDI J L
European concepts o f s ediment transporta­
Journal of Hydraulics D iv i s ion ASCE No HYl
tion .
January 1 9 6 5 .
38.
ELZERMAN J J and FRIJLINK H C
Present state of the
inve stigation on bed load movement in Holland, IXth
Assembly of the International Union of Geodesy and
Geophys i c s , Brus sels , No 1 - 4 , pp 100- 1 1 6 ( 1 9 5 1 ) .
39.
COLBY B R and HEMBREE C H
Computations of total
sediment d i s charge , Niobrara River near Cody , Nebraska .
US Geological Survey , Water Supply Paper No 1 3 5 7 ,
Washington , DC , 1 9 5 5 .
46
Leningrad , Moscow,
40 .
EGIAZAROFF I V
Calculation o f non-uniform sediment
concentration s .
Journal o f the Hydraul ics Divi sion ,
ASC E , No HY 4 , July 1 9 6 5 .
41.
BOGARDI J L
Di scuss ion of the total sediment load of
streams by E M Laursen , ASCE , Vol 8 4 , No HY6 , Proc .
Paper 1 8 5 6 , 1 9 5 8 .
42.
D i s cussion o f the total
GARDE R J and ALBERTSON M L
sediment load of streams .
Proc . ASCE , Vol 8 4 , No HY6 .
43.
COLBY B R
D i scharge of sands and mean velocity
relationships in sand-bed streams .
US Geological
Survey , Pro f . Paper 4 6 2 -A, Washington , D . C . , 1 9 6 4 .
44.
WILSON K C
Bed-load transport at high shear stre s s .
Pro c . ASCE 9 2 , HY6 ( November 1 9 6 6 ) .
45 .
CHANG F M S IMONS D B and RICHARDSON E V
Total bed
material di scharge in al luvial channel .
Proc . 1 2th
Congres s , IAHR, Vol I t Sep tember 1 9 6 7 .
46 .
SHULITS S and HILL R D JR
Bed load formulas .
Pennsylvania State University .
December 1 9 6 8 .
47
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APPENDICES
APPENDIX 1
Evaluation of Ye ' w and v
Many of the theories inc luded in this report contain
the following three parameter s in dif ferent combinations : 1.
The critical dimens ionless parameter of Shields ,
o f which the movement of the sediment starts , Ye '
2.
The fall velocity , w .
3.
The kinematic viscosity , v .
Any o f these can be obtained from graphs d i splayed in the
exis ting l i terature .
However , for computational purposes
it is neces s ary to expre s s these sediment and fluid
properties in a convenient analytical way.
1.
The Shields curve
In the original paper of A A Shields
(Ref 1 ) the
initiation of motion i s given as a relationship between the
parameters
Y
=
v*
2
and
--,-."...=.
g (s-ll D
Ye
=
v
--
f (R* l
•
•
•
(Al)
Here a more convenient form has been used eliminating the
value v* as a variable quantity , from the term of the right­
hand side of equation (All .
pu tting
•
•
•
•
•
•
(A2 )
the relationship (Al ) can be rewritten as
Ye .
The Shields diagram,
=
f eD
gr
)
(A3 )
( Ref 1 ) , as expressed above in
equation (A3 ) was approximated by three curves with terminal
49
points at D
gr
If
D
2 , 1 6 and 7 0 .
2 . 16
gr .;
I f 2 . 1 6 .; D
gr
Y
!
e
10
=
-0 . 6 3 - log D
gr
D
70
<
D
" 70
gr
If
2.
=
Y
e
0 . 05 6
=
•
•
•
gr
-0 . 60
(M )
The fall velocity
There are various effects which compli cate the uniform
motion of a particle in a fluid , such as particle shap e ,
multi-particle inf luence , turbulence , etc .
There i s no
equation which describes th� uniform motion of a single
particle with comp lete prec i sion .
So , in order to compute
analytically the settling velocity one is compelled to
choose one of the several approximations to the problem .
The Rubey ( Ref 2 6 ) equation seems to be fairly reliable for
.-�
particles with the shape of natural sands and it was used
in this report to express the settling velocity .
w
It is
F ( s - l ) � g� D �
=
•
•
•
(A5 )
where F i s the dimension less fall velocity and is given
see equation (A2 ) .
below as a function of D
. gr '
F
3.
(1
=
3
+
)
lL � D 3,.
J
(�6
3
gr
)
�
·
•
.
•
•
( A6 )
The kinematic viscosity
I n order to relate the kinematic viscosity to the
temperature the we l l known expre s s ion of Poiseui lle
was used a s fol lows : v
=
2
1
+
0 . 03 3 6 8 T
c
+
0 . 000 2 2 1 T
c
2
•
(A 7 )
in which v is in m / s and the temperature , T , in degrees
c
centigrade .
50
APPEND IX 2
solution of integrals used in the Einstein methods
The bed load function and the total load function of
H A Einstein include the integrals expressed in equations
(33) ,
( 5 4 ) and (55) whose solution is somewhat cumbersome .
The original work o f Einstein ,
(Ref 7 ) , gave special graphs
to help in solving these integrals and readers interested
should refer to them.
Here a numerical integration i s
presented suitable for u s e i n computer programs .
(i)
'
is o f the form : -
The expression ( 3 3 )
b
whi c h , expanding e _
t
/
2
e
_t
2
dt
a
,
�n
Maclaurin ' s series and
integrating term by term, become s : b
/
e
_t
2
. dt
a
a
and so
/
b
e
_t 2
dt =
a
[
t
t
3
3
7
t
+
3!7
5
t
+
2:5
.
.
.
•
The series in expre s s ion (AB )
=
( n - l ) : ( 2n - l )
=
2
t ( 2n - l ) = 0
n- l
n ( 2n+ l )
( -1 )
( _l )
51
n
•
•
b
a
(AS )
is convergent for any
value of the variable t because in the limit (n
n 2n+l
(-1)
t
n
n
2
)
+ 1..!..
(
!
_..::::..:c---->.;;:.:.;:..;.
.;;; ---.
2n-l
n
l
(_l) - t
..J
n-l 2n -'1
(_1)
t
+
+
( n - l ) ! ( 2n- l )
+
00
)
( i i ) The integrals given in equat ions ( 5 4 ) and ( 5 5 )
are solved by Simpson ' s rule in which the curve
represented by the function who se integration i s sought
i s divided up into an even number of strips , m , o f
equal width and each portion approximated b y a parabo l a .
Thus the integral is evaluated a s fol lows : i=m
I =
\" b.
L "6
i=l
[
f
(t
2i-l
)
+ 4f
(t
2i
)
+ f
(t
2i+l
)]
•
•
•
•
•
•
•
•
•
(A9 )
where m = the number o f strips
b. = width of each strip
f ( t ) are : (AlO )
(Al l )
with the integration limits equal to A and 1 , see
equations
( 5 4 ) and ( 5 5 ) .
nature of functions
Because of the particular
(AIO) and (Al l ) it was found
convenient to achieve the integration in two part s :
(i)
Between t = A and t + 10
E
where E denotes the
entire part of log A .
( U ) F rom t = 10
E
up to t = 1 making the computation in
each logarithmic cycle in an independent form .
Thus the value of m , number of portions , is taken as a
constant equal to 10 in each interval
cycl e ) and the va�iable t
(or logarithmic
is computed at the extreme
i
and middle points of each portion.
Then the function f ( t . )
l.
is determined at the same
points and the integral evaluated using formula (A9 ) .
52
APPENDIX 3
Theories not evaluated
The fol lowing less widely used theories were not
evaluated in the present exercise .
The extra vo lume of work
was not considered worthwhile .
Bed load equation o f P DUBOIS
( 1 8 7 9 ) , Ref 2 7 .
Bed load equation o f A SCHOKLITSH ( 1 9 3 4 ) , Refs 2 8 - 2 9 .
Bed load equation o f F MEYER-PETER ( 1 9 3 4 ) , Ref 3 0 .
Bed load equation o f H J CASEY ( 1 9 3 5 ) , Ref 3 1 .
Bed load equation of L G STRAUB ( 1 9 3 5 ) , Ref 3 2 .
Bed load equation of WATERWAYS EXPERIMENT S TATION ( 1 9 3 5 ) ,
Ref 3 3 .
Bed load equation o f 0 G HAYWOOD ( 1 9 4 0 ) , Re f 3 4 .
Bed load equation o f A SCHOKLITSH (19 4 3 ) , Ref 3 5 .
Bed load equation o f I I LEVY ( 1 9 4 8 ) , Re fs 3 6 -3 7 .
Bed load equation o f J J ELZERMAN and H C FRIJLINK ( 1 9 5 1 )
Ref 3 8 .
(
Total load equation o f B R COLBY and C H HEMBREE (Modi fied
Einstein Method) ( 1 9 5 5 ) , Ref 3 9 .
Total load equation of I V EGIAZAROFF (19 5 7 ) , Ref 4 0 .
Total load equation o f J L BOGARDI ( 1 9 5 8 ) , Ref 4 1 .
Total load equation o f R J GARDE and M L ALBERTSON ( 1 9 5 8 ) ,
Ref 4 2 .
Total load equation o f B R COLBY ( 1 9 6 4 ) , Ref 4 3 .
Total load equation o f K C WILSON (19 6 6 ) , Ref 4 4 .
rotal load equation of F M CHANG , D B S IMONS and
E V RICHARDSON ( 1 9 6 7 ) , Ref 4 5 .
I t should be noted that computations using the Modified
Einstein method and the Colby method require the knowledge
53
of the concentration o f the suspended load .
above formulae are summar ised in Ref 4 6 .
54
Many o f the
TABLE
TABLE 1
Summary of theoretical methods
Graded
sediments
Date
Type
Bed or
total load
A Shields
1936
Deterministic
Bed load
No
A A Kalinske
1947
Deterministic
Bed load
Yes
C Ing l i s
1947
Empirical
Total load
No
E Meyer-Peter
and
R Muller
1948
Deterministic
Bed load
No
H A Einstein
1 9 50
Sto chastic
Bed load
Yes
H A Einstein
1 9 50
Stochastic
Total load
Yes
H A Einstein
and
C B Brown
1 9 50
Stochastic
Bed load
A A Bi shop ,
D B Simo.ns
and
E V Richardson
1965
Stochastic
Total load
Yes
R A Bagnold
1956
Deterministic
Bed load
No
R A Bagnold
1966
Deterministic
Total load
E M Laursen
1958
Deterministic
Total load
J Rottner
1959
Deterministic
Bed load
No
M S Yalin
1963
Determi nistic
Bed load
No
T Blench
1964
Empirical
Total load
No
F Engelund
and
E Hansen
1967
Deterministic
Total load
No
W H Graf
1968
Deterministic
Total load
No
F Tof faleti
1968
Deterministic
Total load
Yes
P Ackers
and
W R White
1972
Deterministic
Total load
No
Originator
No
. No
Yes
I
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j