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SOME CONTRIBUTIONS TO THE STUDY OF SCOUR IN LONGCONTRACTIONS
(EQUIVALENT, SECTION, SEDIMENTATION).
Item Type text; Dissertation-Reproduction (electronic)
Authors ALAWI, ADNAN JASSIM.
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Alawi, Adnan Jassim
SOME CONTRIBUTIONS TO THE STUDY OF SCOUR IN LONG
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SOME CONTRIBUTIONS TO THE STUDY OF SCOUR
IN LONG CONTRACTIONS
by
Adnan Jassim Alawi
A Dissertation Submitted to the Faculty of the
DEPARn1ENT OF CIVIL ENGINEERING AND ENGINEERING ~1ECHANICS
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY HITH A MAJOR IN CIVIL ENGINEERING
In the Graduate College
THE UNIVERSITY OF ARIZONA
198 5
-
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that
we have read
the dissertation prepared by ADNAN J. ALAWI
entitled SOME CONTRIBUTIONS TO THE STUDY OF SCOUR IN LONG
CONTRACTIONS
and recommend that it be accepted as fulfilling the dissertation
requirement
for the Degree of Doctor of Philosophy
--~~~~~~~~~~------------------------------
Dat
IfW 1, fCJJj
Date
Final approval and acceptance of this dissertation is contingent
upon the candidate's submission of the final copy of the
dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared
under my direction and recommend that it be accepted as fulfilling
the dissertation
Dissertation Director
-
STATENENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of
requirements for an advanced degree at The University of Arizona
and is deposited in the University Library to be made available to
borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without
special permission, provided that accurate acknowledgment of source
is made. Requests for permission for extended quotation from or
reproduction of this manuscript in whole or in part may be granted
by the head of the major department or the Dean of the Graduate
College when in his or her judgment the proposed use of the
material is in the interests of scholarship. In all other
instances, however, permission must be obtained from the
author.
SIGNED:_-1..O-~, JI+'-, ....::;.da~~A.14'L.
-
ACKNOHLEDGMENTS
I am greatly indebted to Dr. Emmett M. Laursen and Professor
Margaret S. Peterson, not only for their assistance and guidance
during
this research, but for the knowledge and friendship which they
ha ve
shared with me during the entire period of my study at the
University
of Arizona.
The writer also expresses his appreciation to the other
members
who served on the doctoral committee, Dr. Simon Ince, Dr.
Edward
Nowotzki, and Dr. Paul King.
To the Ministry of Higher Education in Iraq, my
appreciati.on
for providing me the opportunity and support for obtaining an
advanced
degree at the University of Arizona.
To my family, I give special thanks for their encouragement
and
moral support.
iii
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TABLE OF CONTENTS
Page
LIST OF
TABLES"""",,""""""""""""""""""""""""""""""""""""""""""""""" vi
LIST OF
ILLUSTRATIONS............................................. i x
ABSTRA cr " " " " " " " " " " " " " " " " " " " " " " " " " " "
" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " xi
CHAPTER
1.
2.
INTRODUCTION ............•.................................
1
The Long Contraction.................................. 2 The
Equivalent Rectangular Section................. ... 5
PREVIOUS STUDIES OF SCOUR IN A LONG CONTRACTION .......... .
8
Subcritical Flow in a Long Contraction................ 8
Equations for Scour Depth in a Long Contraction ....... 10
Straub" " " " " " " " " " " " " " " " " " " " " " " " " " " " "
" " " " " " " " " " " " " " " 10
Griffith.""""""""""""""""""""""""""""""""""""""" .. " 11
Laursen" " "" " "" " " " " " " " " " " " " " " " " " " " " " " "
" " " " " " " " " " " " " 13 Silverston and Laursen
...............•............ 16 Komura. """" """" """"""""""""
"""""""" " """ """" "" "" "" " 18 Gill. " " " " " " " " " " " " " "
" " " " " " " " " " " " " " " " " " " " " " " " " " " " " " "
24
Supercritical Flow in a Long Contraction .............. 26
3. DERIVATION OF ADDITIONAL EQUATIONS FOR SCOUR IN A LONG
CONTRACTION: ANALYSIS AND DISCUSSION OF RESULTS .......... 29
Equations Investigated ......•......................... 29 Basis
of Analysis ..................................... 40 Scour in a
Long Contraction: Equations 3.1
Through 3.8."""",,""""""""""""""""""""""""""""""""""" 41
Derivation of Depth and Slope Ratios Using
the Manning Formula ............................. 42 Derivation
of Depth and Slope Ratios Using
the Chezy Formula............................... 44 Analysis
Using Equations 3.1 Through 3.4 .......... 45 Analysis Using
Equations 3.5 Through 3.8 .......... 47 Analysis of Einstein's
Bed-Load Equation .......... 51
Discussion" " " " " " " " " " " " " " " " " " " " " " " " " " "
" " " " " " " " " " " " " " " " " 55 Scour in a Long Contraction:
Equations 3.9
Through 3 .13. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .. 58
iv
-
v
TABLE OF CONTENTS -- Continued
Page
Effect of Width Ratio on Scour Depth in a Contraction. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Effect of Approach Depth on Scour Depth in a Contraction. . . .
. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .
70
Effect of Slope in Approach Channel on Scour Depth in a
Contraction ......•...... ~ ............ 75
4. THE EQUIVALENT RECTANGULAR SECTION ANALYSIS AND DISCUSSION OF
RESULTS..................................... 85
5. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 94
APPENDIX A: COMPUTER PROGRAMS FOR SELECTED SEDIMENT-TRANSPORT
EQUATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 97
List of Symbols....................................... 99
APPENDIX B: STUDIES TO DEFFINE AN EQUIVALENT RECTANGULAR SECTION
.............................................. 117
List of Symbols ....................................... 119
APPENDIX C: BASIS AND LIMITATIONS OF SEDIMENT TRANSPORT
EQUATIONS AND MEASURED SCOUR IN LONG CONTRACTIONS .... 137
Sediment Transport Equations.......................... 138
Measured Scour Data in Long Contractions .............. 145
REFERENCES. • • . . . • • • . . • • • . . • • . . • . . . • • •
• • • . . . . . . • . . . . . . • . • . • . • . . . • •. 151
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LIST OF TABLES
Table Page
3.1 Algebraic manipulation; depth and slope ratios using the
Manning equation.......................................... 49
3.2 Algebraic manipulation; depth and slope ratios using the
Chezy equation............................................ 50
3.3 Mathematical computations; effect of shear ratio on depth
and slope in a contraction using Einstein's (1950) sediment
transport equation ...••••..••.•••.•..••..• 54
3.4 Computer analysis; effect of width ratio on depth and slope
in a contraction using Laursen's (1958a) sediment transport
equation .............••.....•.•..••••........... 61
3.5 Computer analysis; effect of width ratio on depth and slope
in a contraction using ~lliddock's (1976) sediment transport
equation .•......•....•...•.•................•... 62
3.6 Computer analysis; effect of width ratio on depth and slope
in a contraction using Yang's (1973) sediment transport equation
.••••.•.•..•••.•••........•..•.....•.... 63
3.7 Computer analysis; effect of width ratio on depth and slope
in a contraction using Toffaleti's (1968, 1969) sediment transport
equation •...•••••.••..•.•.•.•..•..•.... 64
3.8 Computer analysis; effect of width ratio on depth and slope
in a contraction using Colby's (1964a, b) sediment transport
equation........................................ 66
3.9 Computer analysis; effect of approach depth on depth and
slope in a contraction using Laursen's (1958a) sediment transport
equation .•••••.••...••..••.•...•....•.• 71
3.10 Computer analysis; effect of approach depth on depth and
slope in a contraction using Maddock's (1976) sediment transport
equation .••....••........•.••..•....... 72
3.11 Computer analysis; effect of approach depth on depth and
slope in a contraction using Yang's (1973) sediment transport
equation........................................ 73
vi
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vii
LIST OF TABLES -- Continued
Table Page
3.12 Computer analysis; effect of approach depth on depth and
slope in a contraction using Toffaleti's (1968, 1969) sediment
transport equation ...••..•..••..•.•..•..•..••.... 74
3.13 Computer analysis; effect of varying slope in approach
reach on depth and slope in a contraction using Laursen's (1958a)
sediment transport equation ...••.......•••.•....•. 77
3.14 Computer analysis; effect of approach depth on depth and
slope in a contraction using Maddock's (1976) sediment transport
equation ...•.•.•.....•.......•......... 78
3.15 Computer analysis; effect of approach depth on depth and
slope in a contraction using Yang's (1973) sediment transport
equation........................................ 79
3.16 Computer analysis; effect of varying slope in approach
reach on depth and slope in a contraction using Toffaleti's (1968,
1969) sediment transport equation ...... 80
3.17 Computer analysis; effect of varying slope in approach
reach on depth and slope in a contraction using Colby's (1964a, b)
sediment transport equation ••.•.••..•.••....••. 82
4.1 Summary of cross-section characteristics ...••.............
88
B.l Computer analysis; comparison of characteristics of original
triangular section approximated by two rectangular subsections,
T-2, with rectangular sections (first approximation, run number 1)
•••..•......•. 128
B.2 Computer analysis; comparison of characteristics of original
triangular section approximated by eight rectangular subsections,
T-8, with rectangular sections (first approximation, run number 1)
••............ 129
B.3 Characteristics of sections with slope increased 20 times
(first approximation, run number 2) .............. 131
B.4 Characteristics of sections with slope increased 20 times
(second approximation, run number 2) ............. 132
B.5 Characteristics of sections with sediment size increased 10
times, and the slope increased 20 times (first approximation, run
number 3) .............. 134
-
Table
B.6
LIST OF TABLES -- Continued
Characteristics of sections with sediment size increased 10
times, and the slope increased
viii
Page
20 times (second approximation, run number 3) •••..••....•.
138
C.l Data for Hundred Foot River •......•.....•.•...............
146
C.2 Straub's experimental results ..••....••.....•....•........
149
C.3 Komura' s experimental resul ts. . . • . • . . . . . • . . •
• . • . • • . • . . . • .. 150
-
LIST OF ILLUSTRATIONS
Figure Page
1.1 Erodible bed profiles (subcritical flow) .•...•••.•..•.•...
4
1.2 Scour and fill with change in discharge ....••..••..•••..•.
6
2.1 Subcritical rigid contraction .•.....•......•....•........•
9
2.2 Contraction profile ...••.••...•..•.........•.....•........
9
2.3 Straub's solution for the long contraction ...•..........•.
12
2.4 Clear-water scour in a long contraction (after Laursen,
1980) ..••.•......•....•..•.....•....•........•... 15
2.5 Relationship between velocity and depth of scour in a long
contraction.......................................... 19
2.6 Gill's relationship between shear ratio and depth for B1 B =
2.................................................... 25
2 2.7 Scour in a long contraction (supercritical flow)
...•...... 27
3.1 Relation between median diameter of sediment and (d) as
found between different experiments (after Maddock, 1976)
..................................................... 34
3.2 Factors in Toffaleti equations............................
38
3.3 Colby's relationship for sand discharge in terms of mean
velocity for six median sizes of bed sands, two depths of flow, and
water temperature of 60°F (after Colby,
1964a)............................................. 39
3.4 Einstein's 1jJ~:. - 4).::- curve (after Ei.nstein, 1950)
........... 52
3.5 Comparison of computed deplh and slope ralios as a [unct ion
of widlh raLlo obtai.ned (Laursen, Haddock, Yang, and Toffa1eti
equations) and experimental and field
data................................................ h7
ix
-
x
LIST OF ILLUSTRATIONS -- Continued
Figure Page
3.6 Comparison between depth and slope ratios as a function of
width ratio obtained from the Laursen and Colby equations. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 68
3.7 Comparison between depth and slope ratios as a function of
particle (boundary) shear over critical tractive force obtained
from Laursen, Maddock, Yang, and Toffaleti equations
.•.•....•..••..•••.•..•....••.•..•..... 76
3.8 Comparison of depth and slope ratios as a function of the
ratio of particle (boundary) shear to critical tractive force
obtained from the Laursen, Maddock, Yang and Toffaleti equations
.•••.•.•..••••...•.•.•••..••.. 83
3.9 Comparison of depth and slope ratios as a function of the
ratio of particle (boundary) shear to critical tractive force
obtained from the Laursen and Colby equations; sediment size = 0.2
mm......................... 84
4.1 Triangular cross-section approximated by rectangular
sections T-2, T-2\v, and T-2E of same slope................ 86
4.2 Triangular cross-section approximated by rectangular
sections T-8, T-8W, and T-8E of same slope ........•.....•. 90
C.l Cross-sections of Hundred Foot River......................
146
C.2 Typical scour survey data .•••.•.•••.•.••••.•.•.....•••....
148
C.3 Sections in flume .••..•••....•.•..••..••••...•..••......•.
149
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ABSTRACT
The first objective of this investigation was to derive and
compare scour depth equations in a long contraction using the
most
widely used sediment transport equations and a variety of
other
equations. The second objective was to determine a procedure to
find
an equivalent rectangular section which would convey the same
water
discharge and sediment load at same slope as an irregular,
natural
channel in order to simplify numerical computations of scour
depth and
to allow appropriate application of long contraction scour
theory.
Some of the transport equations were manipulated
algebraically
to develop equations for scour depth and slope in a long
contraction;
others were manipulated using computer programs written
especially for
each equation, thus deriving scour depth equations.
A computer program was written to compare characteristics of
a
non-rectangular section with rectangular sections of different
widths
in order to derive a procedure to find an equivalent
rectangular
transport section (a triangular section was used in this
investigation)
but the procedure is equally valid for any irregular, natural
section.
This investigation indicated that depth in the contraction
is
greater than in the wider approach channel. How much greater
depended
on which sediment-transport equation was used. Nost of the
derived
scour equations, based on the different sediment transport
equations,
xi
-
Y2 predicted that the Yl ratio decreases as slope, velocity,
xii ,
LO c, and lC or
I~p increase; a few do not. Most of the analysis predicted S2
< Sl' but a few do not.
Field and experimental data provided extra evidence that the
depth in the contracted section is greater than in the approach
reach
and how much greater. The evidence that the slope is flatter is
not
sufficient to be completely convincing.
The equivalent rectangular transport section which can carry
the same water and sediment discharge at the same slope as the
natural
section has a depth which is a large fraction of the deepest
part of
the original section, and the width is considerably narrower
than the
top width of the origj.nal section. Resul ts of the investigaUon
also
indicated that the slope, velocity. sediment concentrati.on,
and
sediment size have little effect on the geometry of the
equivalent
rectangular section.
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CHAPTER 1
INTRODUCTION
Potential scour and the fluctuation in scour depth with
changing streamflow over time in a specific river reach are
basic
considerations in design, construction, and maintenance of
all
engineering works constructed in, across, and along banks of
alluvial
streams. Banks in successive reaches of an alluvial river are
composed
of different materials possessing different resistance to
erosion. In
reaches where bank materials are relatively resistant, the river
cross
section is narrower and deeper than in reaches where banks cave
readily
and the cross section is wide and shallow~
The relatively stable, narrower, deeper reaches can be
considered analogous to long contractions in a long wider and
more
shallow stream, and depth of flow in the contracted reaches can
be
computed as proposed initially by Straub (1934). Practical
examples of
engineering problems that can be analyzed by applying Straub's
method
for computing scour in a long contraction to achieve a more
reliable
and economical design include depths in river reaches that
are
stabilized and narrowed by revetments and dikes, in reaches
downstream
of railroad and high\vay crossings that are narrowed by
encroaching
abutments and piers, in reaches that are temporarjly narrowed
by
cofferdams, and in other similar situations. In the non-uniform
f 10\'/
1
-
2
of the transition at the beginning of a narrow reach the scour
is not
uniform; and, therefore, the scour locally can be greater than
the
long-contraction scour.
Natural river cross sections are highly variable and very
irregular, and analysis of sediment transport and scour can
be
facilitated if the natural cross sections ~an be approximated by
easily
determined rectangular cross sections carrying the same water
and
sediment discharges as the natural sections at the same slopes
as the
natural sections.
Various investigators have proposed numerous equations for
sediment transport over the years. The primary objective of
this
dissertation was to derive and compare scour depth equations in
a long
contraction using their equations for sediment transport, as
reported
in the literature. The secondary objective was to develop a
procedure
for determining an "equivalent" rectangular sediment-transport
section
to simplify and expedite sediment computations in engineering
practice.
The Long Contraction
The long contraction, long in the direction of flow, is of
interest in the study of river behavior and the design of
structures in
the river environment for both practical and theoretical
reasons. The
practical reason is that there are natural and man-made long
contractions and expansions in rivers and, therefore, the stream
bed
changes (scours and fills) with changes in flow. The
theoretical
reason is that the long contraction is the one case of scour in
which
the geometry is simple enough to a 110\-1 adequate mathemati
cal
-
3
description of the flow and sediment-transport characteristics,
thus
permitting analytical solutions describing the essential
characteris-
tics of the scour phenomenon.
In describing scour in a long contraction mathematically,
flow
in the transition sections from wide to narrow and back to wide
have
been ignored in this investigation because the non-uniform flow
in
those reaches is not pertinent to conditions in the long
contraction.
Assuming there is an equilibrium state of sediment transport,
the flow
and sediment transport can be described adequately in the two
uniform
flow reaches, the wide and the narrow.
If the widths are known, some characteristics of river
behavior
in the case of a long contraction can be predicted for a given
flow and
sediment discharge. If a narrow reach in an alluvial stream
is
preceded and followed by wide reaches (narrow because the
bank
materials are stronger and can withstand higher shear), the flow
in the
narrow reach will be deeper, the velocity will be slightly
greater, and
the slope will be less than in the wide reaches, as shown in
Figure 1.l.
As Laursen (1985) noted, this is the way things must be to
have
both wide and narrow reaches transport the total load. The
bank
material in the narrow reaches must, of course, be able to
withstand
higher values of boundary shear.
If the boundary shear is increased by increasing the flow
(but
not to the limit of causing bank erosion), the bed in the narrow
reach
will erode, and the eroded material will deposit in the wi.der
reach
-
;1.
b.
-
Energy grade line
.L
l'iRuro
1. \. Erodible beu profi leS (subcritical flo,,). --(a) Plan
dC'\~; and (11) Longitudinal section.
4
-
5
downstream. The wide reach upstream will also aggrade with
materials
removed from the next narrow reach upstream. The erosion and
deposition will begin at the heads of the narrow and wide
reaches,
respectively, and whether or not all reaches reach the new
equilibrium
condition depends on how long the higher flow lasts (Figure
1.2).
If the flow then decreases to the original value, the narrow
reach will deposit and the wide reach will erode, again starting
at the
head ends of the reaches and progressing downstream. If the low
flow
lasts long enough, equilibrium is established, with the beds
returning
to the original state. If the high and low flows do not last
long
enough to establish equilibrium conditions, but keep changing
back and
forth between high and low flow, different types of holes and
humps
(sand waves) can be moved through the river system,
gradually
attenuating, but persisting beyond ordinary expectations.
The procedure followed in this investigation was to
manipulate
several of the most widely used sediment-transport equations and
a
variety of other equations not as widely used with the Manning
or Chezy
flow equations, some by algebraic computations and others by
computer
programs written especially for this purpose, to investigate
the
effects of the most important parameters expected to influence
scour in
a long contraction.
The Equivalent Rectangular Section
The second part of this investigation was to determine a
procedure for finding an equivalent rectangular section which
would
convey the same water discharge and sediment load at the same
slope as
-
7
an irregular, natural channel to simplify numerical computations
of
scour depth.
The method of attack to obtain a solution was to write a
computer program to compare characteristics of a natural
non-rectangular section and a rectangular section. The effects
of the
number of subsections of the natural channel, flow velocity,
sediment
concentration, and sediment size on geometry of the
equivalent
rectangular section were tested.
Because of the variety and irregularity of natural channel
sections, a general procedure--not a mathematical equation--was
devel-
oped to derive an equivalent rectangular sediment-transport
section.
-
CHAPTER 2
PREVIOUS STUDIES OF SCOUR IN A LONG CONTRACTION
Ri vers tend to approach equilibrium conditions, and
sediment
supply and transport capacity must be in balance or scour or
deposition
\'Iill occur for either subcritical or supercritical flow
conditions.
However, supercritical flow behaves differently in some
situations, and
the differences can be important with respect to sediment
transport.
Specific values of velocity, depth, slope, and shear are
required to transport a given sediment load of a particular
sediment
size.
Previous work by earlier investigators on scour depth in a
long
contraction is summarized in this chapter, and supplemental data
are
given in Appendix C. There have been several studies of the
transport
phenomena in contractions for conditions of subcritical flow;
the case
of supercritical flow has not been addressed well.
Subcritical Flow in a Long Contraction
If the bed and banks are rigid and flO\.,r is subcritical,
the
water surface in the contraction will drop, and the velocity
will
increase as shown in Figure 2.1. If the sediment movement is
proportional to the velocity and the bed is movable, the
sediment
discharge capacity in the contraction will be increased. As a
result
8
-
Cl.
b.
---.-------------------------.--------------I " I '2
Figure 2.1. Suhcritical rigid contraction. (u) Plun view; and
(b) Profile.
HL ------.=-~'-----
d s
Figure 2.2. Profile, suhcritical movahle bed.
9
-
10
of imbalance between the sediment supply to the contraction and
that
which is leaving it, the bed (which is now considered erodible)
in the
contraction will start to drop, the velocity and the slope to
decrease,
and the sediment movement to decrease until the flow again
reaches an
equilibrium condition when the sediment supply to the
contraction
equals the sediment ].eaving it, as shown in Figure 1.1.
To evaluate the depth of scour for this condition, assuming
the
slopes are not very much different, the energy equation: 2
v2
+ --2g
can be used to define the scour depth as shown in Figure 2.2.
Depth of
scour is a function, of kinetic energy, head lost, and the two
depths of
flO','. If velocity heads and losses are small, the depth of
scour can
be defined as the difference between the depth in the
contraction and
the depth in the wide reach, ds = Y2 - Yl'
Equations for Scour Depth in a Long Contraction
Straub
The first analytical solution for the long contraction was
by
Straub (1934, 1940). Using the Manning equation to describe the
flow
and the Duboys formula to describe sediment ~ov2ment, and
assuming the
section is wide enough so that R = y, Straub1s equation is as
follows:
l l lc)B1 1/~ 3/7 _£ + {(~) + 4 (1 -Y2 (~)3/7 t II II ~B) } - l
(2.1) Y1 B2
2(1 + ~) II
-
11
where
1c = critical shear stress, Ib/ft2
11 = boundary shear stress in the approach section, Ib/ft2
Bl , Yl = width and the depth in the approach section, ft
B2, Y2 = width and the depth in the contracted section, ft
For general sediment movement, when 1c « l l ,Equation 2.1
simplifies to:
_Y1 = Bl 9/14 (_._) B Y2 2
(2.2)
Equation 2.2 implies no velocity effect, sediment size
effect,
or boundary shear effect on depth of scour. Figure 2.3 shows
Equation 1
2.1 graphically for....£. = 0, 0.5, 0.7, 0.9, 0.99. Equation 2.2
is a 11
good approximation if 1 c is less than one-half of 11 .
The Nanning equ~tion can also be manipulated with Straub's
sediment transport equ~tion to give the ratio of slope in
the
contracted section to slope in the approach section:
(2.3)
Griffith
Equation 2.2 agrees very well with Griffith's relationship
(1939), obtained from field measurements for three
cross-sections on
the Hundred Foot river (Appendix C).
(2.4)
-
12
Y/Yl TC/T 1
1.8 0.99 0.90 0.70 0.50
1.6 0.00
l.4
1.:2
1.0
1.0 1.2 1.4 1 . () 1.8 2.0
Figure 2.3. Straub's solution for the long contraction.
-
13
Laursen
For the case of general sediment movement with velocity well
above critical and considering sedi~ent size, Laursen (1958b)
developed
the following equation to predict depth of flow at a contracted
section
using the Manning equation to describe the flow and the Laursen
total
sediment-transport equation:
(2.5)
and his equation for the slope ratio is:
S2. B2.z.C a+3) TI2 .z.C 3+a ) -S = (-B )2 3a-1 (-)6 7-a
1 1 n l (2.6)
His exponent "a" represents the velocity effect on -, transport
with
different types of sediment movement,as defined below, because
it
depends on the ratio of shear velocity to fall velocity.
When
ITTP o w < 1/2 a = 1/4 (movement is bed load);
= 1 a = 1 (movement is bed load and some
suspended load);
> 2 a = 9/4 (movement is predominantly by
suspension)
ho/p = sheRr velocity in the approach channel
To = total boundary shear stress, Ib/ft2
w = the fall velocity of the bed material, ft/sec
-
14
n l , n2 = Manning roughness coefficient, where subscripts 1
and 2 represent conditions in the approach and
contracted sections, respectively
Laursen (1963) later proposed a formula for clear water scour.
l
Usi ng the Manni ng equati on and approximating the particle
(boundary)
shear as
where dsO is median grain size diameter (50 percent of the
material is
finer) .
Assuming T C = 4d sO for cohesionless bed material and R =
y,
Laursen found:
for cohesion less bed material and R = y, Laursen found:
~= (~)3/7 B1 6/7 (2.7) (-) Y1 Tc B2
Equation 2.7 is shown graphically in Figure 2.4.
Laursen (1980) states that the key to analysis of the
clear-water scour case is: "gi ven a wid th and depth of
approach, a
discharge, a sediment size, and a width of contraction, the
contraction
will scour until the depth is such that the particle (boundary)
shear
is equal to the critical tractive force."
For the clear-water scour case, the velocity of fImv and the
sediment size are very important. However, rather than separate
them
1. Clear water scour, as defined by Chaber Land Engeldillger
(1956) and Laursen (l9s8b),is \~here bed material is removed from a
local stream section and is not replenished by the approach
flow.
-
4.0
I
T· 0
Tc
1 . 0
::i.0
\' . ') v . 1
2.0
0.2
(l . 1
1.11 ~~~~~~~~~~--~~~--------~~~--~------~
l.0 2.0 ::i. 0
Figure 2.4. C1ca)'-\,'nter scollr in a loll~l contraction (after
Lnursen, 1980).
.1 . (1
IS
-
16
size (or fall velocity) effect, Laursen (1980) found it better
to think
of it as a single effect of the boundary-shear/critical
tractive-force
ratio because any increase in the velocity can be compensated
for by an
increase in sediment size, so that the ratio does not
change.
Silverston and Laursen
Silverston and Laursen (1976) extended Laursen's (1958b)
relationship for scour in the long contraction with general
sediment
movement, and they found that the exponents differed if the
channel was
not wide, and if the rate of sediment transport was not high.
They
showed that the hydraulic radius and critical tractive force
simplifying assumptions were unnecessary and could be handled
by
introducing the coefficients k and K, setting:
and
R = Ky
, '[
o - 1
'[ c
, 'T
o = k"T
c
Using the particle (boundary) shear '[ , which is less than the
o
total boundary shear, Laursen's modified equation for the depth
ratio
is:
Y2
Y1
and his equation for the slope ratio is:
(2.8)
(2.9)
-
17
Here:
and
k1 ~ = results from the shear approximation
2
~ = results from the hydraulic radius approximation.
Solutions from Equation 2.8 are shown graphically in Laursen
(1980) for
the conditions of bed· load transport, some suspended material,
and
mostly suspended material movement.
The effect of velocity on scour depth in a long contraction
as
a single parameter or mixed with other parameters was
investigated by
Alawi (1981), who explains four different ways in which velocity
can
affect scour:
1. For clear-water scour conditions in the approach reach
V 2 1
-T-c
- -1-2-0---'-17; -:; d--:2,...;r..3 Y1 50
is an important parameter.
2. For weak movement where cannot be ignored and scour actually
c
decreases with an increase in velocity, the parameter of
interest is
T 02 T 01 (- - l)/(-T- - 1)
Tc C
which combines velocity, depth, and sediment size in both
the
contracted and approach sections.
3. Depending on the mode of movement (whether movement is
entirely
as bed load, includes some suspended load, or is
predominantly
IT7P suspended load) the parameter of interest 0 is related
to
UJ
-
18
velocity because boundary shear and velocity are highly
correlated, as are fall velocity and sediment size.
4. The kinetic energy and loss terms in the full definition
of
scour depth are the only effects that can be put into a form
utilizing a Froude number.
Alawi's experiments (1981) showed three distinct conditions
of
scour in a long contraction. He found that the scour depth in
the
clear-water regime (with T' < T ) increases until the flow
reaches the o c
critical for movement. His experiments also showed that scour
depth in
the sediment transport regime (with T' > T ) decreased in the
upper o c
regime of subcritical flow and into the supercritical flow
regime, but
then increased with increase in velocity. Figure 2.5 shows
the
relationship between the velocity of the flow and depth of scour
in the
contraction.
Komura
Komura (1963, 1966) investigated the effect of the standard
deviation of size of sediment composing the bed and the
variation of
sediment size on depth of scour in the long contraction. He
found that
if the rate of sediment supply into the contraction area is
zero, the
limit of scour depth depends on the sediment gradation, and if
the rate
of sediment supply is not zero, the limit of scour depth depends
only
slightly on the sediment gradation. Laursen (1963) indicated
that when
the rate of sediment supply into the contracted area is zero,
the limit
of the depth of scour is when the boundary shear equals the erit
iea 1
-
r-..., 4-
h -a
0.20
O. IS
~ O.lO
4-a
..:: .... C;
O.IlS
S 1I her i tic a 1 F 1 01\ SlIpcrcriticul Flow
Clcur Water I~cgimc Sediment Trunsport Repime
().ooI-L1 -----t-------+-----1-------...;I-----
0.0 (l.S 1 . Il 1.5 2.0
Velocity Cft/sec)
Figure 2.S. 1~l'lationship hC'th'C'en velocity and dC'pth of
scour in a long contraction.
....... I.D
-
, T o
20
tractive force of the boundary material ~-c
~ 1), and that when the
rate of sediment supply is not zero, the depth of scour reaches
the
limit when the sediment supply to a section equals the sediment
leaving
that section; thus when the velocity is large or the sediment is
small , , T T
and -2,_ 1 ~ ~ velocity and sediment size do not matter. Tc
Tc
Komura's theoretical analysis of scour in a long contraction
was based on two concepts of scour which he termed (1) the
dynamic
equilibrium theory, and (2) the static equilibrium theory.
First, in investigating scour according to the dynamic
equilibrium theory, which is equivalent to scour by
sediment-
transporting flow as used by the previous investigators,
Komura
proposed:
· .... here
(2.10)
qs = the rate of sediment transport in volume of material
per
unit of time and unit width
P, a = constants s
U* = the friction velocity, ft/sec
D = mean diameter of bed material, mm 3 a = the density of
sediment particles, slug/ft
p = the density of water, slug/ft3
g = the acceleration of gravity, ft2/sec
As is true of most sediment-transport equations, this
relationship depends on the total boundary shear which is the
sum of
-
21
the "particle shear," the "dune shear," and any "other
losses."
Komura manipulated equation 2.10 with the continuity
equation
for sediment transport, qsl Bl = qs2B2' in which qsl and qs2 are
the
rate of sediment transit in the uncontracted and contracted
sections,
respectively.
For depth of scour, KOr.lura (1966) used the following
equation
presented by Laursen (1963):
(2.11)
in which WI = the Froude number in the uncontracted section,
and
where
HL = the head losses at the contracted section
If the difference in the velocity heads and loss through the
transition can be neglected, Laursen's equation simplifies
to:
(2.12)
Komura then obtained the equation:
(2.13)
To evaluate Equation 2.13, let the boundary shear T equal
the
particle shear T as defined by Laursen: o
-
, L = L
o = 30y1/3
Equation 2.13 then becomes:
From the continuity equation,
Therefore,
22
(2.14)
(2.15)
(2.16)
Substituting Equation 2.16 in Equation 2.14, the final equation
is:
(2.17)
Moreover, if L = yyS, the total shear, Equation 2.13 also
becomes:
which is the same as Equation 2.17. Comparing Equations 2.17 and
2.18
with Equation 2.13, it seems that Ll = L2 whether L is particle
shear
or total. This is unreasonable and a contradiction because qs2
> qsl
and therefore U*2 > U*l.
Second, in investigating scour according to the static
equilib-
rium theory, which is equivalent to scour by clear water, Komura
(1966)
used the Iwagaki formula to define critical tractive force:
-
L .-£= P
a [(2..) - 1] gD c p
23
(2.19)
in which a = a constant, and the other parameters are as defined
c
previously.
Lcl __ :l Assuming and substituting in Equation 2.13, Komura
Lc2 L2 obtained the following expression for equilibrium depth
of scour.
(2.20)
If a in Equation 2.19 is a constant, Equation 2.20 becomes:
c
Y2 B (...J..)6/7 (2.21) -YI B2
However, this is not the general case of clear water scour where
Ll <
Lel but the limit where L 1 = Note also that the
sediment-transport equation used by Komura, Equation 2.10, does
not
include a crj.tical term.
Gill
Gill (1981) investigated the validity of Straub's formula
for
predicting scour depth in a long contraction theoretically,
using the
Manning formula to describe the flow, and assuming the rate of
sediment
transport to be given by:
(2.22)
in which
qs = unit sediment discharge, lb/sec/ft
= constant
m = constant
-
24
s = specific gravity of the bed material s D = average size of
bed material, mm
Ll = bed shear stress, Ib/ft2
By manipulating it with the equation of continuity of
sediment
(2.23)
Gill (1981) indicated that the highest value of m in known
empirical formulas is 3 (as in the Einstein-Brown formula), and
the
10\,est value of m is 1.5 (as in the Meyer-Peter formula). If
the
average value of m (2.25) is used in the above equation and
the
relationship between shear ratio and depth ratio is drawn for a
typical B1
contraction ratio of B = 2, results \dll be as sho\{n in Figure
2.6. 2
If Equation 2.23 simplifies to:
Al though this expression is the same as Laursen's if L 1 =
(2.24)
, L ,
o the
method of derivation is not acceptable since in Equation 2.21 if
L < c
Ll' qs = 0 for any value of L 1 and the equation of transport
is
L1 meaningless. If 1, Equation 2.23 simplifies to:
LC
(2.25)
-
Y2 y . 1
5
4-
3
2
1 ~-
0' - -;;-------;~__;_~-7::---+---+---+---l-0.5 1.5 2.-5 3.5 4.0
2.0 1.0 3.0
Figure 2.(). R
Gill's relationship hetween sllear ratio and depth ratio for 81
2. 2
T/TC
N (J1
-
26
and if 1"1 » 1" c' Equation 2.23 simplifies to:
6 3 Y2 B (-- 7m)
- (-1..) 7 Yl B2
(2.26)
If m = 2.25, Equation 2.26 simplifies to:
(2.27)
Equation 2.27 is exactly the same equation as Straub (1934)
found.
Supercritical Flow in a Long Contraction
All of the previous discussion has been in reference to
subcritical flow. The solution to the long contraction at
equilibrium
does not recognize whether the flow is subcritical or
supercritical.
Manning's equation and a sediment-transport relationship are
presumably
adequate for both types of flow. However, the kinetic energy
terms and
losses in subcritical flow tend to be insignificant, but they
are large
in supercritical £10\". T .... 'o very different behaviors can
be speculated
upon for the supercritical condition.
If the bed and the banks are fixed, the water surface in the
contraction will rise, as shmIn in Figure 2.7. If this
condition
occurs with sediment-transporting flow, deposition will occur in
the
long contraction and the flow will be unstable.
The other possibility is for the contraction to scour suffi-
ciently to accommodate the velocity head and loss affects. In
this
-
( :t :
(b)
V 11'"
B..,
----------
+deposition
==-====-
Figure 2.7. Scour in a long contraction (supercritical flo\'.').
--(a) Fixed bed, plan view and longitudinal profile, respectively;
and (b) Erodible bed, longitudinal profile.
27
-
28
case the bed and the water surface would have to have adverse
slopes as
they leave the contraction for the next wide reach
downstream.
-
CHAPTER 3
DERIVATION OF ADDITIO:~AL EQUATIONS FOR SCOUR IN A LONG
CONTRACTION: fu~ALYSIS fu~D DISCUSSION OF RESULTS
Equations Investigated
In addition to the equations of various investigators for
scour
in a long contraction reviewed in Chapter 2, there are numerous
other
sediment transport equations in the literature that do not
directly
address the question of scour in contractions. This chapter
summarize~
new work in this current investigation in which selected
sediment-
transport equations were manipulated (either algebraically or
by
computer) to yield equations for the depth ratio and slope ratio
in a
long contraction. In view of the large number of
sediment-transport
equations in the literature, research was concentrated on the
most
widely used equations and a variety of other equations not
widely used,
as follo\vs. Additional information on the basis of derivation
of these
equations and any limitations on flow depth and sediment size
is
presented in Appendix C. (Note that the symbols and definitions
of the
originators have been adopted, with the result that there is
inconsis-
tencyamong equations.) Where symbols differ in definition from
those
in Chapter 2, they are redefined in this chapter for
particular
equations.
29
-
30
Depth and slope ratios developed by the previous
investigators
whose work is discussed in Chapter 2 and those derived in this
analysis
are all assembled for convenient reference in Tables 3.1 and
3.2.
1. Schok1itsch (1934)
(3.1)
in which
Qs = bed load, !bs/sec
q = unit flow discharge, cfs/ft
qo = discharge at which sediment transport begins, cfs/ft
= Q.Q638Q 54/ 3
D = average size of the bed material, ft
2. Schoklitsch (Shu1its, 1968)
Q = 156.2 S3/2 (q - q )B, Ibs/sec s 0 (3.2)
where
qo = discharge at which sediment transport begins, cfs/ft
= 1.088D3/ 2
57/ 6
3. Meyer-Peter (1934)
Q = (39.3 q2/3 S _ 9.94D)3/2B, Ibs/sec s (3.3)
-
4. Haywood (1940)
2/3 = r~q - 1.2D4/3}3/2B lbs/sec
Qs L 0.117D1/3 '
5. Meyer-Peter and Muller (1948)
in which
3/2 Qs = 9.23(Qk T - 4.84D) B, lbs/sec
Qk = 1 for a very wide and level bed channel
T = boundary shear, lbs/ft2
6. Waterways Experiment Station (1935)
1 T-Tc m Qs = -- { )., lbs/sec 3600n 62.4K J
in which
k, m = constants
n = Manning coefficient
7. Engelund-Hansen (Engelund, 1966; Engelund and Hansen,
1967)
L_d_S_O_ f To 13/ 2 Q = 0.05 Y V2~ ,- t 1 s s Vg (; _ 1) {Ys -
y)dSO
Ibs/sec
in which y = specific weight of sediment particle, Ib/ft 2 s
31
(3.4)
(3.5)
(3.6)
(3.7)
-
32
8. Ackers-White (1973)
o V2 1"D1 1 5 Qs
= 0.56 -y--- ( - A) • B, Ibs/sec hlp /DIGs - 1)
(3.8)
in which
F
A = value of F at nominal initial motion gr V*
= sediment mobility number gr IgO(Gs - 1)
V* = shear'velocity, ft/sec
G = mass density of sediment relative to that of water s
9. Laursen (1958a)
in which
-
" ; c = L P', -) 6 To loys (- _ l)f(-t->--- ) T W C
c = sediment concentration, percent by weight
p = fraction of bed material of diameter d
(3.9)
Igys The function f(-w-) is related to the ratio of the
shear
velocity and the fall velocity. The function values were
approximated
by the following equations for different modes of sediment
transport.
a.
b.
Igys f(/gys) When movement is bed load, -w- < 0.5, CD =
10.4(/:;5)0.25
1 d 0 5 Igys
When movement is bed load and some suspended oa, . ~--W--
2 0 f( /gys) ~ ., w
-
c. When movement is predominantly suspended load, /~s >
2,
Total sediment discharge was found from the equation
Q _ cQy 1bs/sec s - 100'
10. Maddock (1976)
c = 1/2
{ 3 VS keGs - l)g D eGs - l)gD 1/4,4/3
10 ¢d - ¢ 1/2 ( 2 ) J dy w
in ,,'hich
c = sediment concentration, parts per million by weight
33
(3.9a)
(3.10)
¢d = sediment-size function as shown in Figure 3.1. For
sedi-
ment size D = 0.15 mm (which was used in the computer
program) ¢d = 0.013.
k = temperature, of
Total sediment discharge was found from the equation
Q = cQy , Ibs/sec s 1, 000, 000 (3.10a)
11. Yang (1973)
wD V* log c = 5.435 - 0.286 log -- - 0.457 log -- +
\) W
wD V* VS VcrS (1.799 - 0.409 v - 0.314 log w) 10g(w - -w-)
(3.11)
-
(). 1 -
O.O}
"0 -e- O.(lOl
O.ODOI
O.ODOOl
O.OOOOI (l.OOI O.OI () . 1 1.0 10 lOll
~ledi;lIl Di ameter (mi 1 Ii Illl'tl'rs)
Fi!!ul'e :i. I. Relation bet\,"ecn median diameter of
sedillll'llt and ¢(d) as found from differellt l'xpl'riJl1ellts
(;Ifter ~lacldoc". I ~J7()) .
"34
-
in which
in which
c = sediment concentration in parts per million by weight
When the shear velocity Reynolds number is smaller than 70,
2.5 Vcr w = --(-v-*o-)--- + 0.66
log \J - 0.6
V{~ = shear velocity, ft/sec
When the shear velocity Reynolds number is greater than 70,
w 2.05
Total sediment discharge was found using Equation 3.10a.
12. Toffaleti (1968, 1969)
35
(3.12)
where
qb = bed load discharge, tons/day/ft, in the bed load zone of y
2d
relative thickness r = Jr
qsL' qsw and qsu are the suspended sediment discharges
tons/day/foot in the lower extending y 2d Y 1
zone from - = - to - ---' r r r -11.24'
in
in
the middle zone, extending from .l = _1_ to l.. = _1_. r 11.24 r
2.S and in the upper
d' f Y 1 zone,exten lng rom r = 2.5 to the surface,
respectively. Toffaleti's bed load transport equation is:
-
where
n = 0.1198 + 0.000048T v
c = 260.67 - 0.667T z
Toffa1eti's suspended bed load equations are:
qsL = M (11~24)1+nv-o.758Z _ (2d)1+nv-O.756z
O.244z l+n -z l+n -z r I"i r v r i\ v
(1l.24) \'(2.5) (1l.24~ = M
------------------------------~~-----
l+n -z v
O.244z O.5z l+n -1.5z l+n -1.5: ( r) ( r ) {V (~) v , 11. 24 2.5
r 2.5 J
= M --------------------------------------------~
36
(3.12a)
(3.12b)
(3.12c)
( 3 .l2e )
The value of ~1 is found by substituting the empirical value of
qsL
determined by the following equation into Equation 3.l2c and
solving
for M. That value of M is then used in Equations 3.12a, 3.l2b,
3.12d,
and 3.12e.
a.bOOP (3.12f)
-
37
'vhere
P = 1 for sediment of uniform grain size
A functiorr r,t (1 (,5",1/3
g: ,en in Figure 3.2a = a ---------lU\*
K function (105V) 1/3
:05Sd given in Figure 3.2b = a cd ------4 1 U\ *
TT = 1.10(0.051 - 0.00009T)
The equations used in the computer programs for A and K4 are
given in
Appendix C.
13. Colby (1964a, 1964b)
(3.13)
Hhen the flow temperature is = 1· , when the
concentration of fine sediment is negligible, K2 = 1; and when
average
sediment size, d50 lies between 0.2 mm and 0.3 mm, K3 = 100. For
these
conditions, Equation 3.13 shows that Qs = qsl B. Colby I s
relationship
for sand discharge for two flow depths as a function of velocity
is
shown in Figure 3-3. For a known flow depth and average
velocity, qsl
was found from Figure 3.3, and approximated by equations to be
used in
the computer program as given in Appendix C.
14. Einstein (1950)
Einstein I S equation for bed-load transport is based on the
relationship between the complex dimensionless parameters,
the
dimensionless measure of bed-load transport, ~ * which he termed
the
intensity of bed-load transport, and a flow intensity (shear
intensity)
-
3.0
2.0
1-: 1.0 ....... 0.8 r-. O.t) :;>1 -I<
tr.=~~ 0.4 -1-03
0..2
OJ
u.
c: 0.0 a .; OA ~ Q.3 f-, f-,
~----~~~~~~~~----~ 8 O~~~--L-~~~~--~ ::; ::; :::
('~ I.r; -:-
A
,.... ::; ~ ...:; co
b.
('I --'
('I
Figure 3.::? Factors in Toffaletti equations. -- (a) Factor !\
in Elllwtion 3.12f; and (b) Correction factor k4 in Eouation :;.12f
(after Toffa1eti, 19(9).
38
-
.j-J
c o
4-
'f. ::: c
.j-J
...... V,
c-
('j ..... U tf'. ......
C.
"0 ::: ('j
r.r:
IO
0.1
Depth I. 0 ft Depth lOft IOOO
I
/"/ ,'I, 1',1
I" ,,,,1 I,'l/ I I ~: I : ' ," /11, " 'II ,"
Mean Velocity (ft/sec)
lOll
III
II • I
AvailablC' data ExtrapolatC'd d:lt:1
Figure 3.3. Colby's relationship for santi disLiwrgl' ill terms
of 11le:1Il vC'locit)' for six median si:es of hed sands, t\\'O
depths of flO\~, and \~atcr tC'mperature of ()llor (after
Colh~',
1964a).
39
-
40
parameter, '¥ *. Solution of Einstein's equation involves a
graphical
procedure. It was, therefore, analyzed separately from the
other
sediment-transport equations in this investigation and is
discussed
in more detail later in this chapter.
Basis of Analysis
Because of the complexity of some of the sediment-transport
equations, computer programs and computations were used for
those
equations for which algebraic manipulations could not be used
to
establish the equations for scour in a long contraction.
Equations 3.1
through 3.8 were investigated using algebraic computations,
and
Equations 3.9 through 3.13 were investigated using specially
developed
computer programs, Appendices A and C. To investigate the
Einstein
equation, the graphical relationship was approximated as a
series of
linear relationships.
The concept of continuity of both flow and sediment load,
which
was first used by Straub (1934), was used in all computations.
The
solutions, therefore, are for the limiting state. In a real
flood the
upstream end of the contraction will scour almost to the limit,
and the
next succeeding wide reach will fill beginning at the upstream
end. As
the long contraction is actively scouring, the length not yet
scoured
out will have a higher-than-normal velocity and, consequently,
a
higher-than-normal slope and water surface. After the entire
contraction is scoured out, the water surface will lower and
there will
be additional deepening of the streambed. With a continuously
changing
discharge, this active scour phase is exceedingly complex, and
the
-
41
limit may never be attained. The limit concept can, however,
serve for
design purposes if used prudently and with understanding.
Scour in a Long Contraction: Equations 3.1 Through 3.8
Algebraic computations were used to derive scour equations in
a
long contraction based on sediment-transport Equations 3.1
through 3.8.
The Manning and Chezy equations (Equations 3.14 and 3.15,
respec-
tively), were used to describe the flow
(3.14)
Q = CIRS A (3.15)
in which
A = cross-section area, ft.
C = Chezy coefficient
n = Manning coefficient
Two sets of assumptions were used for the Manning and Chezy
equations, as follows:
Assumption one conditions
Hydraulic radius is equal to depth of flow (R = y)
Roughness coefficients are the same in the wide approach
reach
and the contracted reach (nl = n2 or Cl = C2).
Assumption two conditions
Hydraulic radius is some fraction of the flow depth (R = Ky)
Roughness coefficients are different in the wide approach
reach
and the contracted reach (n l ~ n2 or Cl ~ C2 )
-
42
The derivation of depth and slope ratios for the various
sediment
transport equations, based on these assumptions and using the
Manning
and Chezy equations, is discussed below.
Computed ratios of depth and slope are summarized in Tables
3.1
and 3.2.
Derivation of Depth and Slope Ratios Using the Manning
Formula
For the first assumption (R = y and nl = n2), the Manning
equation in the approach channel and in the long contraction can
be
written as follows:
(3.13)
(3.14)
(3.15)
For the second assumption (R = Ky, nl f n2, and K is a
constant
defined below), the Manning equation is:
where
and
= _By
R = Ky B+2y
B 1 K=--=--
B+2y 1+2Y. B
(3.16)
-
43
(3.17)
(3.18)
Kl The effect of the ratio of -- on the slope and on the depth
is almost
K2
always small, as shown in the following example.
If we assume that:
Bl = 900 feet
B2 = 450 feet
Yl = 10.0 feet
According to Straub (1934), Y2 can be found from
K in the
or
10( 900)0.643 = l~ 6~ f ). '- ee t .:150
I{ Bv K - - --~,~-)"(B+2)') y
approach and in the
1 Kl = =
1 2xl0
+ 900
2v 1 + _.!_. B
contracted
0.978
1 K2 = ---=--- = 0.935 1 + 2x15.62
-Ts-o -r
1 0.978 K = (). 935 = 1.046
)
reach, respectively, is
-
44
or
(0.978)1.33 = 1 062 0.935 .
The effect is 3 percent on the depth ratio and 1.5 percent on
the slope
ratj o.
Derivation of Depth and Slope Ratios Using the Chezy Formula
For the first assumption (R = y and Cl = C2), the Chezy
equation in the approach channel and in the long contraction can
be
written as follows:
(3.19)
(3.20)
Y2 (3.21)
For the second assumption (R = Ky, Cl i: C2, and K is the
constant defined above), the Chezy equation is:
(3.22)
(3.23)
(3.24)
-
45
The apparent difference between using the Manning or the Chezy
equation
is less than it seems because n1/n 2 ~ C1/C2·
Analysis Using Equations 3.1 Through 3.4
The Schok1itsch equation (1934)
(3.1)
was investigated initially using both the Manning and Chezy
equations
for assumption one and assumption two conditions.
For assumption one and equilibriur:J flow condition, the
total
sediment discharge in the approach reach and in the contracted
section
are equal, and
25 (3.25)
For a high flow (flood), q » q , and the sediment size in the
t\W o
sections is the same (DI = D2), Equation 3.25 becomes:
(3.26)
From the continuity equation, the flow discharges in the
approach reach
and in the contracted section are the same, and
(3.27)
Substituting Equation 3.27 into 3.26, the slope ratio
becomes
-
46
From Equation 3.28 and the Manning flow equation, (Equation
3.15), the
depth ratio becomes:
v • 2 (3.29)
From Equation 3.28 and the Chezy flow equation, Equation
3.20,
the depth ratio, becomes:
(3.30)
Equations 3.29 and 3.30 sho\, that the £10\, depth in the
contracted
section is greater than in the approach reach, which is
compatible with
the literature (Straub, 1934; Griffith, 1939; and Laursen,
1958b),
Table 3.1. Equation 3.28, however, shows that the slope in
the
contracted section and in the approach reach is the same,
which
contradicts the literature, Table 3.1.
For assumption two conditions \,hen the discharge is not
much
greater than the critical discharge (q - qo = kq), where k is
a
coefficient equivalent to the shear coefficient, and following
the
above procedure, the slope ratio becomes:
0.11 )
Should S2 be less than, equal to, or greater than Sl? The
approximate
"theoretical" answer depends on the construction of the
approximate
sediment-transport equation used. However, because the
coefficient k2
will always be larger than kl (whether this is the k used wHh
the
Schoklitsch equation or some other k), the S2 will be less \vhen
the
-
47
flow is just a little greater than required for sediment
movement than
when it is much, much greater. Hhat few measurements there
are
indicate that S2 is less than Sl (Straub, 1934).
Substituting Equation 3.31 in Equations 3.18 for the Manning
equation and 3.24 for the Chezy equation, the depth ratios for
the
Schoklitsch equation (1934) become:
and
\" . ') (3.32)
(3.33)
E\·en thou!i· tlwy appear to be different, the sediment-
transport equations of Schoklitsch (1943), Meyer-Peter (1934),
and
Haywood (1940) give the same solutions for the depth and slope
ratios
for both sets of assumptions as Schoklitsch (1934), Tables 3.1
and 3.2.
Analysis Using Equations 3.5 Through 3.8
The Meyer-Peter and Muller (1948), Haterways Experiment
Station
(1935), Engelund-Hansen (1966-1967), and Ackers-\vhite (1973)
sediment
transport equations resulted in very similar solutions for depth
and
slope ratios for both of the sets of assumptions, as
follows.
For assumption one conditions, the depth and the slope
ratios
obtained by manipulating any of these sediment transport
equations with
the Manning and Chezy flow equations, respectively, are:
(3.34)
and
-
48
(3.35)
The slopf' ratios, respectively, are
(3.36)
and
(3.37)
Equation J.34 shows that the depth in the contracted section
is
deeper than in the approach reach, but slightly shallower than
with the
previous set of equations and similar assumptions, and Equation
3.36
predicts that the slope is slightly greater than indicated by
the
previous set of sediment-transport equations and the Manning
equat ion,
Equations 3.28 and 3.29. Indeed, the slope in the contraction
is
slightly steeper than in the wider approach. Equations 3.35 and
3.37,
using the Chezy equation, show that the depth and the slope are
the same
as for the previous set of equations and similar assumptions,
Equations
3.30 and 3.28.
For assumption two conditions, the depth and the slope ratio
prediction equations using the Manning and Chezy equations,
respectively,
are
and
\' . 2 (3.3H)
(3.39)
(3.40)
(1.41)
-
Table 3.l. Algebraic manipulation; depth and slope ratios using
Manning equation. (a) Assumption one conditions,
R = y, and for a high flood (t) L 0 » LC; Assumption two
conditions, n1 = n2 , R = Ky, and 1 0 = ki 0
Author and Date
(a) From the Literature
Straub/Duboy (1934)
Griffith's (field measurement) (1939)
Laursen (1958b)
Gill (1981)
Fror:1 this Investigation
Schoklitsch (1934) J Schoklitsch (1943) Heyer-Peter (1934)
Haywood (1940)
(b) From the liter;,: J:.£.
Depth Ratio
)'2 (B I )0.643 v-= ~ ·1 Y2 (Ill )0.637 YI = 112'
8· 69} Y2 B .64 • ( 1)0.59 Yl • ll2 ' ,
)':, B U.71,) •• vet ( 1)0.571 ·1 b,
... .., BI -= (_)0.(,1) ~ I B2
Y. B n -9 n, n O' k, 0 " ~I ° 01 Silverston and - = (.:.1).0'
(-=-)" '(...:.) "(-) •
Eqn. Slope Ratio Eqn.
2.2 ,3
2.4
~.5
2.26
3.29 J.28
3.34 c.36
49
the nl = and
1 C
Laursen (l97G) )'1 B2 n I k I k2 2.8
KOr:1ura (1966)
Gill (1981)
Schokli t sch (1934)
~Ieyer-Peter (1934)
Jlaywood (1940)
y.,
~Ieyer-Peter an~l Nuller (194J)
\Ia.terways Experiment y,
2.24
(~)o. 57 (n~ )0. g", k~ ,0.43'~1 0.57 Station o 93':J' ~. L.:
;'1 '" 1 ".::::'.:;1}
\t!:er !\'h it ( (1973)
Engeland/Han., r, (1966, 1'.1(7)
~ Exponent depends on mode of scdlment movement.
Exponent dppvnds on m.
~. '1~
s, - . (~10.()9\~\C'.H(,,~\I.:.O, I.: ,r-." ~l il.. I,,: "..
r'l
:,,':' '
-
Table 3.2. Algebraic manipulation; depth and slope ratios using
the Chezy equatio~. -- (a) Assum?tion one conditions,. C] = C2' R.
~ y, and for a hJgh fJood, T >? T ; and (b) i\ssumptlon two
condltlons, CJ o c I C
2, R = ky, and T - 1 = kT . o c ()
Author and Date Depth Ratio
Fro~ This Investigation:
(a)
Schok1itsch (19340 Schok1itsch (1943) Meyer-Peter (1934) Haywood
(1940)
)' B 2 = (-1.)0.67 )'} 1>2
Muller (l':i .. ti) ~ ~!eyer-P('t pr 'In.-l } \" Waterways
Experiment .} Station (1935)
(b)
III r,J,-\-,
B2
Eqn.
3.30
j.JJ
Schok1itsch (1934)J yZ Schok1itsch (1943) ~ = Meyer-Peter (1934)
}}
B C k, K (~)0.67(~)0.67(~)0.22(~)0.34
B2 C2 k} K2
Haywood (1940)
Meyer-Peter ;Jnn J v Z ~!uller (194' '. - =
v Waterways Experiment .} Station (19:f'» -.J
B C k K (...!.)0.67 (...!.)(~)O. 5(+) 0.5 BZ C2 k} KZ
3.33
3.39
Slope Ratio
52
-
51
Analysis of Einstein's Bed-Load Equation
Einstein's bed-load equation (1950), which relates a
sediment
transport parameter, ¢i~' to a flow parameter, ¢*, is very
complicated
and difficult to apply to derive the depth and slope ratios for
a
contracted channel.
In this analysis the method used to derive the depth and
slope
ratios was to divide the Einstein (1950) ¢-l~ - q.r-l~ curve on
a log-log
plot into eight straight line segments, or power relationships
as shown
in Figure 3.4. The equation for each straight line segment was
then
manipulated with the Einstein bed load equation (1950), which
is:
in which
IB = fraction of bed load in a given grain size
Ib = fraction of bed material in a given grain size
qB = bed-load rate, lbs/sec/ft of width
G = specific gravity of the bed material s
g = acceleration due to gravity, ft/sec 2
D = grain size diameter, mm
(3.42)
S = ratio of specific gravity of the bed material to specific s
gravity of water
The equations were applied to two successive flow sections
and
were manipulated with the Manning flow equation. The assumptions
used
were that the sediment size was uniform, the channel was so wide
that
the hydraulic radius equaled the depth of flow, and the
Manning
-
=
:r. I •
:>-, -
:c 0
1
, . ;.-L-. " • "'" 1.1'. I, • "'"
.r. I • C
;..
;;.: .r,
=
::.
::. -::: -::. =
':
'.l.)l.JW
I!.ll!d ~\OLI
c cr. ~.
e: :..; '-
..... '.I. -
e: ~ i.:
:.... -
V
-.....
.. ~
" J. :: i.:
+
?~
-"'.
:r. '.I.
e: V ..... 'f. e:
...,..
,I",
V
:.r.
-
53
coefficients in the approach reach and in the contracted section
were
the same. The depth and slope ratio equations for the range
of
Einstein's flow parameter and sediment transport parameter
are
presented in Table 3.3. The particle shear as used by Laursen
(1958b)
in ratio to the critical tractive force corresponding to the
range of
the flow and sediment-transport parameters used by Einstein are
also
listed in Table 3.3 to give a sense of the relative magnitude of
forces
acting. Data in column 4 of the table indicate that the depth in
a
contracted section is greater than in the approach reach, which
agrees
with results when using the sediment-transport equations of
most
investigators. The depth ratios decrease with increasing shear
ratio
in the approach reach (as they also would \d th increasing
velocity,
sediment concentration, or total shear velocity in ratio to
fall
velocity) . Data in column 5 of the table indicate that the
slope in
the contracted section is flatter than in the approach reach for
lower , , LO LO
ratios of L' but steeper for high values of L' c c
In the medium range of Einstein's curve, the depth ratio is
close to that obtained from most other sediment-transport
equations. , L
Hhen --2. approaches unity, however, the depth ratio is larger,
as \dth LC
other equations, but critical trac ti ve force is unclear in
Einstein's
concept and equation. In practice, this condition will not be of
great
interest because it is not to be expected in floods.
At higher rates of movement, Einstein's equation predic ts
less scour and a steeper slope in the contraction. This may be
due to
ignoring the suspended sediment load. Einstein's suspended
load
-
Table 3.3. Mathematical computations; effect of shear ratio on
depth and slope in a contraction using Einstein's (1950) sediment
transport equation.
, Tal
'I' ~~ Tc
cP ~~ Depth Ratio Slope Ratio
, T
Y2 = (~)0.842 S B 0.6 < -21 < 0.67 22 < 'I'~~ < 26
0.0001 < CPu < 0.001 -2 = (-1.)0.807 Tc
.. ,;-Yl B2 Sl Bl ,
T Y2 = (~)0.804 S B 0.67 < -21 < 0.82 16.5 < 'I'v <
22 0.001 < ¢u < 0.01 -1. = (-2)0.679
Tc •• ... Y1 B2 Sl B1 ,
S B T Y2 = (~)O. 754 0.82 < ~ < 1.18 9.5 < '1'* <
16.5 0.01 < CPv < 0.1 -2 ~ (-1.)0.515
Tc .". Y1 B2 Sl B1 , S B T
Y2 = (~)O. 715 1.18
-
55
depends on the bed load, converting the bed load to a
reference
suspended load concentration and then integrating the product of
the
concentration and the velocity in the vertical. The variation to
be
expected between the approach reach and the long contraction
would
probably be mostly in the bed load (which has been considered)
and the
vertical distribution of the suspended sediment concentration.
Because
the shear velocity would not be greatly different in the
approach and
contraction, the ratio of total load to bed load should not be
greatly
different in the approach and contraction. This leaves the
question of
the validity of using Einstein's equation for high rates of
sediment
movement, as discussed by Laursen (1956).
Discussion
The several sediment-transport equations used in this part
of
the investigation result in almost the same predicted depth in
the
contraction even though it is known (Vanoni, 1975) that they
predict
quite different absolute sediment loads. Consideri.ng the
first
simplifying assumptions, so that only the width ratio determines
the
depth ratio, the exponent of the width ranges from 0.57 to
0.714.
Thus, if the depth of approach flow is 5 feet and the width
ratio is 2,
the .shallowest and deepest predicted depths in the contraction
would be
7.42 feet and 8.20 feet. This is not a large dif ference, about
10
percent, and a designer could use the larger value for one part
of the
design and the smaller for another part of the design so that
the most
conservative value would be chosen for each aspect of the
design.
-
56
The other ratios have a small effect on the predicted depth.
I The K values themselves \vill always be just slightly less
than unity,
and the ratio will be even closer to unity. Even though the
exponent
is noticeably different, it is always small, and the effect on
the
depth in the contraction will be very small, as discussed
earlier.
The resistance coefficient, whether the Manning n or Chezy
c,
ordinarily should not be too different in the approach channel
and the
contraction. For a real project, it should be possible to make
field
measurements during different flows in reaches with different
widths to
have some reasonably reliable estimates of n or C.
Unfortunately, it
is not generally possible to wait years to obtain measurements
for the
big floods. It should be noted that if the n values in the
approach
and contraction are equal the C values are not, and vice
versa.
The ratio of k values 2 can have a noticeable effect when
the
flow is only a little above that of the critical tractive force.
This
will generally be a low flow and may not be important for the
project
design. Indeed, the contraction may not fill to this limiting
state
because the approach flow does not supply enough sediment load.
The
deposi tion that does take place will be in the upstream end of
the
contraction and may have to be dredged out if maintaining depth
is
important. The problem is more in how low the flow might be and
how
shallow the approach depth. The effect of the k ratio will be
to
1. K R where R is hydraulic radius and y is depth of flow. \" I
'.,.
2. k = (-' ! L·
, T
1 )/~ T ~
'-
-
57
predict a slightly larger depth in the contraction--which is
good, of
course, except that the effect will probably not be sufficient
to solve
the problem if there is one.
Understanding what happens to the slope in the contraction is
a
more perplexing problem. However, the indications are that the
slope
in the contraction is not much different from that in the
approach. In
a relatively short contracted reach the slope difference, and
the
variation in slope difference forecast by different
sediment-transport
equations, would probably be unimportant over a mile or two. In
an
extensive contracted river reach, as for a navigation project,
even a
small difference can finally become important. All in all,
the
preponderance of evidence based on the formulas so far used is
that the
slope in the contraction is the same or less (but only slightly
less)
than the slope in the wide approach.
The exponent for the width-ratio effect on slope is small,
but
the effect can be of some significance. The exponents for the
other
ratios in the slope-ratio equations (hydraulic radius,
critical
tractive force, and other similar terms) are larger than in
the
depth-ratio equations, but still are probably not important
in
practice. The effect of the difference in the resistance
coefficient,
n or C, must be considered important. The best way of solving
this
dilemma is probably to obtain field measurements for reaches
of
different widths and for as great a range of flows as occurs
during the
time available for gathering field data. Data from other
"similar"
streams can also be used. Because the exponent for the
resistance
-
58
coefficient ratio is about unity, the n or C values need to
be
established better than ± 5 percent.
Scour in a Long Contraction: Equations 3.9 Through 3.13
Some of the more recent sediment-transport equations are of
a
form which cannot be easily manipulated algebraically, and they
W8re
accordingly investigated using specially developed computer
programs,
Appendix A. The equations of Maddock, Yang, Toffaleti, and Colby
were
investigated by calculating the flow characteristics in a
particular
contraction compatible with the flow and sediment load in a
specific
approach. Inasmuch as predictions of scour in a long
contraction
computed using the Laursen equation are within 10 percent of
values
predicted using all equations summarized in Tables 3.1 and 3.2
and all
predictions of slope are approximately equal, as discussed
earlier, the
Laursen equation was used as being typical of all equations and
as the
basis for comparison with results obtained using the equations
of
Maddock, Yang, Toffaleti, and Colby.
The principal results and flow characteristics determined
from
the computer analysis are listed in Tables 3.4 through 3.17, and
are
shown in Figures 3.5 through 3.9. The same approach conditions
were
used for all cases except Colby. Depths of 1 ft. and 10 ft. were
used
for Colby to avoid an additional approximation for his
curves.
The sediment load was specified by using the total sediment-
transport equations of Laursen (1958a), Maddock (1976), Yang
(1973),
Toffaleti (1968, 1969), and Colby (1964a, b), and the Manning
equation
-
59
was used to specify the flow discharge and flow conditions.
The
channel width of 500 ft. was assumed to be sufficient so that
the
hydraulic radius equals the depth of flow. The channel was taken
as
rectangular. The sediment size and the Manning roughness
coefficient
were kept the same in the approach reach and in the contracted
section.
Laursen's particle (boundary) shear (1963) and the critical
tractive
force were used to more fully describe flow conditions. The
equilibrium state was achieved when the sediment concentration
in the
contracted section was equal to the sediment concentration in
the
approach reach. After the flow and sediment discharge in the
approach
reach were computed, the technique was to assume a depth in
the
contraction, then (knowing the flow disL.'.arge), slope was
calculated.
Hith that slope and other flow characteristics, the
concentration or
sediment load was calculated. Hhen it was not the same as that
of the
approach flow, the depth was reassumed, and the process
repeated.
Effect of Hidth Ratio on Scour Depth in a Contraction
The effect of the width ratio on scour depth in the
contraction
was investigated first. Tables 3.4 through 3.8 list results of
the
programs run with channel-contraction ratios of 1.25, 1.50,
1.75, and
2.00. The tables also list the width ratio, depth in the
contracted
section, depth ratio, relationship between the depth and width,
slope
in the contracted section, slope ratio, relationship between the
slope
and width, velocity in the contracted section, velocity ra~io,
ratio of
the particle (boundary) shear in the approach reach to
critical
-
60
tractive force, and the particle (boundary) shear ratio. The
subscripts 1 and 2 represent the approach reach and the
contracted
section, respectively.
There is a similarity of data in the five tables in that
depth
in the contracted section was always greater than the approach
reach,
and the slope was less. Despite this general agreement, these
five
equations for sediment load did not all give almost the same
depths and
slopes in the contracted reaches. As can be seen in Tables 3.4
and
3.5, the Laursen and Maddock equations gave results which were
almost
exactly the same, differing only in the third significant
figure.
Table 3.6 shows that Yang's equation predicts depths a little
smaller
and slopes a lit tIe larger than Laursen or Maddock. Table 3. 7
shows
that Toffaleti's equation predicts depths significantly larger
and
slopes significantly smaller.
It is interesting, but not particularly important to this
investigation, that the sediment load was only slightly larger
for
Maddock than for Laursen, Yang was 1.5 times greater, and
Toffaleti was
2.51 times the Laursen predicted load. It is also noteworthy
that
according to Toffaleti's equation velocity in the contraction
is
slightly larger than in the approach, but the particle shear is
less.
These results are difficult to accept as reasonable.
Computations with the Laursen, Maddock, Yang, and Toffaleti
equations were performed with a sediment size of 0.15 mm and a
depth of
approach of five teet, a nominal depth of real interest. For
Colby's
equation, two depths were used, 1 foot and 10 feet, and a
sediment size
-
1'ab1(' 3.4. ComputC'r analysis; effect. of width rat io on
depth and slope in a contraction using Laursen's (J958a) sediment
transport equation. a
v2 , ,
~ Y2 ~ * 52 ~ ** :0. ~ ~4 B2 (ft) Yl ex SI B (ft/sec) VI TC
Toi
Y1 = 5.0 ft
1.25 5.84 1.168 0.694 0.000466 0.940 0.320 3.48 1.07 8.26
1.09
1.50 11.64 1.328 0.700 0.000437 0.910 0.332 3.67 1.13 8.26
1.16
1. 75 7.40 1.480 0.701 0.000414 0.863 0.335 3.84 1.18 8.26
1.23
2.0 8.12 1.624 0.700 0.000397 0.828 0.332 4.00 1.23 8.26
1.29
Y1 = 10.0 ft
1.25 11.67 1.167 0.692 0.000467 0.934 0.307 5.52 1.07 13.64
1.09
1.50 13.26 1.326 0.969 0.000439 0.878 0.320 5.83 1.13 13.64
1.17
1. 75 14.76 1.476 0.696 0.000418 0.836 0.319 6.11 1.18 13.64 1.
26
2.00 16.20 1.620 0.696 0.000401 1.298 0.320 6.36 1.23 13.64
1.30
a. Characteristics of approach section:
Items Y1 = 5 ft Yl = 10 ft
D, mm 0.15 0.2 B1, ft 500 500 51 0.0005 0.0005 n 0.03 0.03 Q,
cfs 8118 25774 V l' ft/sec 3.25 5.16 Q1' 1b/sec/ft 0.5 0.5
Y B * -.1. _ (.--!. ) ex Yl - B2 Q\ S B ~ "* ~ _ (-.1.)5 !:i 1
1)1
-
Table 3.5. Computer analysis; effect of width ratio on depth and
slope in a contraction using Maddock's (1976) sediment transport
equation. a
B1
B2
1.25
1.50
1. 75
2.00
Y2
eft)
5.82
6.64
7.40
8.13
Y2 Y1
1.164
1. 328
1.480
1. 626
a~~ S2
0.681 0.00046
0.700 0.00044
0.701 0.00041
0.701 0.00040
a. Characteristics of approach section:
Y2 -1'. Yl =
52 ~H:- S =
1
D = 0.15 lOrn B1 = 500 ft Y1 = 5.0 ft Sl = 0.0005
n = 0.03 Q = 8118 cfs
VI = 3.25 ft/sec q = 0.52 1b/sec/ft
s
B1 a (82)
(B2) B B1
, , 52
~ Lo1 L02
Lc -, Lo1
B** V2 (ft/sec)
V2 .
~
0.920 0.374 3.47 1.07 8.26 1.08
0.874 0.332 3.67 1.13 8.26 1.16
0.829 0.335 3.84 1.18 8.26 1. 23
0.791 0.338 4.00 1. 23 8.26 1. 28
(J\
N
-
Table 3.6. Computer analysis; effect of \dd th ratio on depth
and slope in a contraction using Yang's (1973) sediment transport
equation. a
Bl Y2 Y2 a-l!- S2 B2 (ft) Yl
1. 25 5.78 1.156 0.650 0.00048
1. 50 6.51) 1. 312 0.670 0.00046
1. 75 7.31 1.462 0.679 0.00043
2.00 8.02 1.604 0.682 0.00041
a. Characteristics of approach section:
D = 0.15 mm B1 = 500 ft Y1 = 5.0 ft Sl = 0.0005
n = 0.03 Q = 8118 cfs
VI = 3.25 ft/sec q = 1.26 1b/sec/ft
s
"Y2 Bl a ;C - = (_)
Yl B2 S2 B28
-lH!- _ = (_) Sl Bl
, , S2
8-lH :-V
2 V2 -
SI (ft! sec) VI
Lol L02
LC L ' 01
0.960 0.183 3.51 1.08 8.26 1.11
0.920 0.206 3.71 1.14 8.26 1.19
C.860 0.270 3.89 1.20 8.26 1. 26
0.820 0.286 4.05 1. 25 8.26 1.33
Q\ V-l
-
Table 3.7. Computer analysis; effect of \ddth ratio on depth and
slope in a contraction using Toffa1eti's (1968, 1969) sediment
transport equation. a
~ Y2 Y2
tl~< S2 B2 Y1 (ft)
1.25 6.17 1.234 0.942 0.00039
1.50 7.33 1.466 0.943 0.00031
1. 75 8.48 1.696 0.944 0.00026
2.00 9.61 1. 922 0.943 0.00023
a. Characteristics of approach section:
~_ Y2 = Yl S2
';H:-~=
D = 0.15 mm B1 = 500 ft Y1 = 5.0 ft Sl = 0.0005
n = 0.03 Q = 8118 cfs
VI = 3.25 ft/sec q = 1.26 1b/sec/ft
s
B (.:.l)tl
B2 B
(-=.2.) S Bl
, , S2
S-1:--1:- V2 V2 Tol T02
Sl VI ,
ft/sec T c Tol
