• TABLE OF CONTENTS
HIDE
 Front Cover
 Title Page
 Acknowledgement
 Table of Contents
 List of Figures
 Nomeclature
 Abstract
 Introduction
 Theoretical background and...
 Experimental program
 Analysis of results and discus...
 Conclusions
 Appendices
 Appendix I: Derivation of bed shear...
 Appendix II: Sample calculation...
 Appendix III: Sample calculation...
 References
 Biographical sketch






Group Title: UFL/COEL (University of Florida. Coastal and Oceanographic Engineering Laboratory) ; 69/021
Title: Modeling of the sediment transport in the vicinity of inlet and coastal region
CITATION PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00076156/00001
 Material Information
Title: Modeling of the sediment transport in the vicinity of inlet and coastal region
Series Title: UFLCOEL
Alternate Title: Sediment transport in the vicinity of inlet and coastal region
Physical Description: xii, 92 leaves. : illus. ; 28 cm.
Language: English
Creator: Lin, Pang-Mou, 1942-
University of Florida -- Coastal and Oceanographic Engineering Laboratory
Publication Date: 1969
 Subjects
Subject: Marine sediments   ( lcsh )
Sedimentation and deposition   ( lcsh )
Simulation methods   ( lcsh )
Coastal and Oceanographic Engineering thesis M.S
Coastal and Oceanographic Engineering -- Dissertations, Academic -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (M.S. in Engin.)--University of Florida.
Bibliography: Bibliography: leaves 90-91.
General Note: Manuscript copy.
General Note: Vita.
Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
 Record Information
Bibliographic ID: UF00076156
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 18271924

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Acknowledgement
        Acknowledgement
    Table of Contents
        Table of Contents 1
        Table of Contents 2
        Table of Contents 3
    List of Figures
        List of Figures 1
        List of Figures 2
    Nomeclature
        Unnumbered ( 9 )
        Unnumbered ( 10 )
        Unnumbered ( 11 )
        Unnumbered ( 12 )
    Abstract
        Abstract
    Introduction
        Page 1
        Page 2
    Theoretical background and analysis
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
    Experimental program
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
    Analysis of results and discussion
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Conclusions
        Page 73
        Page 74
        Page 75
    Appendices
        Page 76
    Appendix I: Derivation of bed shear stresses due to the combined action of waves and currents
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
    Appendix II: Sample calculation for depth scale
        Page 84
        Page 85
        Page 86
        Page 87
    Appendix III: Sample calculation of volume rate of bedload transport
        Page 88
        Page 89
    References
        Page 90
        Page 91
    Biographical sketch
        Page 92
Full Text




UFL/COEL-69/021


MODELING OF THE SEDIMENT TRANSPORT IN THE
VICINITY OF INLET AND COASTAL REGION




by



Pang-Mou Lin






Thesis


1969















MODELING OF THE SEDIMENT TRANSPORT IN THE

VICINITY OF INLET AND COASTAL REGION












By
PANG-MOU LIN


A THESIS PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE IN ENGINEERING














UNIVERSITY OF FLORIDA
1969















ACKNOWLEDGMENTS


The author is indebted to Dr. 0. H. Shemdin, without whose guidance

and encouragement, this thesis would not have been possible; the author

also wishes to express his gratitude to Dr. R. G. Dean for his review

of the manuscript and to Dr. B. A. Christensen for his review of the

analytical results and for his valuable suggestions.

The author also wishes to thank Mrs. Mara Lea Hetherington and

wife, Shiow-Shwu,:for the typing of the manuscript.

The research in this thesis is part of a larger study on Jupiter Inlet,

Florida, under the supervision of Mr. T. Y. Chiu. Financial support was

provided by the Jupiter Inlet Commission and the Department of Coastal and

Oceanographic Engineering.


L
















TABLE OF CONTENTS


ACKNOWLEDGMENTS

LIST OF TABLES

LIST OF FIGURES

NOTATIONS

ABSTRACT

CHAPTER

I INTRODUCTION

A. Statement of the Problem

B. Aim of the Investigation

II THEORETICAL BACKGROUND AND ANALYSIS

A. Similarity Laws for Open Channel Flows

B. Similarity Laws for Wave Motion

C. Similarity Laws foi Combined Motion Due to
Waves and Currents

1. Existing analysis of velocities due to
wave motion

2. Velocity field in the vicinity of an inlet

3. Evaluation of shear stresses

D. Summary

III EXPERIMENTAL PROGRAM

A. Test in the Flume

B. Test in the Model Basin


Page

ii

v

vi

viii

xii











Page


IV ANALYSIS OF RESULTS AND DISCUSSION

A. Flume Test Results

B. Model Basin Test Results

V CONCLUSIONS


APPENDIX I


- DERIVATION OF BED SHEAR STRESSES DUE TO
THE COMBINED ACTION OF WAVES AND CURRENTS


APPENDIX II SAMPLE CALCULATION FOR DEPTH SCALE

APPENDIX III SAMPLE CALCULATION OF VOLUME RATE OF
BEDLOAD TRANSPORT

APPENDIX IV SAMPLE CALCULATION OF EINSTEIN'S BEDLOAD
FUNCTION

APPENDIX V SAMPLE CALCULATIONS FOR LONGSHORE CURRENT
AND BOTTOM SHEAR VELOCITIES SHOWN IN TABLE 7


REFERENCES


BIOGRAPHICAL SKETCH
















2 Characteristics of bed materials


3 Results of sediment transport which were
tested in the flume

4 Test conditions in the model basin

5 Numerical results for the volume of bedload
transport per unit time per unit width


LIST OF TABLES


Tables

1 Computed values of x/b and y/b for various
values of p/Q and T/Q two-dimensional
orifice (After French [19])
I Kesuits or caicu.acea near ve.ocJ-LJes Jur
the site of weir trap in the model

8 Numerical results of time scales which were

6 Results of longshore sediment transport which
were tested in the model basin














caiculatea zrom equation Dou)


Page


I















LIST OF FIGURES


Figure Page

1 Generalized surface contours for stream expanding
past abrupt expansion (after Rouse, Bhoota, and
Hsu [14]) 15

2 General representation of outgoing flow 15

3 General representation of currents and waves 17

4 Channel entrance corresponding to a two-dimensional
orifice 19

5 Variation of velocity along streamline for flow
through two-dimensional orifice 19

6 Flow pattern through two-dimensional orifice 22

7 Schematic diagram of the flume 29

8 Velocity measurement in the flume 31

9 Velocity distributions over a plane bed when
sand is moving intermittently 33

10 Velocity distributions over a plane bed when
walnut shell is moving intermittently 34

11 Velocity distributions over a plane bed when
coal is moving intermittently 35

12 V vs. V* for plane and dune beds 36

13 Pan trap in the flume 37

14 q' p' curves for the three different bed
materials tested in the flume 40

15 Schematic diagram of model basin 41

16 Longshore bedload trap in the model basin 42

17 Bottom sand configuration before and after
test with combined wave and ebb flow 44


I










Figure


18 Bottom sand configuration before and after test
with wave only 45

19 Bottom sand configuration before and after test
with combined wave and flood flow 46

20 Bottom walnut shell configuration before and
after test with wave only 47

21 Bottom walnut shell configuration before and
after test with combined wave and ebb flow 48

22 Bottom walnut shell configuration before and
after test with combined wave and flood flow 49

23 Bottom coal configuration before and after test
with wave only 50

24 Bottom coal configuration before and after test
with combined wave and ebb flow 51

25 Bottom coal configuration before and after test
with combined wave and flood flow 52

26 The tested results in the model basin as com-
pared with the estimated (' curves 60

27 Typical plan view showing profile sections 64

28 Bottom profiles of channel and ocean after test
with combined wave and ebb flow 65

29 Bottom profiles of channel and ocean after test
with waves only 66

30 Bottom profiles of channel and ocean after test
with combined wave and flood flow 67

31 Sediment motion function for beach 68

32 Incipient motion function for sand
(d = 0.163 mm, Cota = 20) 69

33 Incipient motion function for walnut shell
(de = 1.6 mm, Cota = 20) 70

34 Incipient motion function for coal
(d = 0.78 mm, Cota = 20) 71


Page
















NOTATIONS


A,, B, Constants determined experimentally

BI Channel width

b Half width of the channel

CB Special "sediment parameter" and is considered
to depend entirely on sediment characteristics

C Chizy's coefficient
z

C Static volume concentration of bed material
o

C2 Empirical constant

c Celerity

c' Half width of the jet at infinity

d Water depth

db Water depth at breaker

d Grain size of bedload which is 50 per cent
e finer by weight

dl Water depth of the approach flow
d
d'= Dimensionless depth scale
L
F Force

F Dimensionless parameter
o

FI Froude number

f Darcy-Weisbach's coefficient

g Gravity acceleration

H, H Wave height

Hb Wave height at breaker


viii


I










k Wave number

k Roughness height

kl, k2 Empirical constants

L Wave length

n Empirical constant

nb Ratio of group velocity with respect to wave
celerity at breaker

P Probability distribution function

p Porosity of bed material

p Pressure

Pi Coefficient of horizontal orbital velocity
above the laminar sublayer
p
Po
p' =pV2 Dimensionless pressure scale

Q Rate of water discharge

q Velocity in potential flow field

qB' qs Volume discharge of sediment transport per
unit time per unit width

q Weight discharge of sediment transport per
unit width

qx Velocity component of q in x-direction

q Velocity component of q in y-direction

q* Shear velocity in potential flow

R Reynolds number

Se Slope of energy gradient line

Sf Specific gravity of water

S Specific gravity of bedload

s Distance measured along a particular streamline
from its intersection with the zero potential
line

T Time











t' =t
T

u

Ub
I U
U =
V

u

V

VL

V
m

V
x

V,

V
m,
V
x*
v

v'
V

w = p + iJP

? w
w' =
V

L

y' =L
L
t Z

a

C1

Ca2


B = (--)

Yf

Ys

K


Dimensionless time scale

Orbital velocity in x-direction

Orbital velocity at bottom

Dimensionless velocity scale in x-component

Orbital velocity near the oscillatory boundary

General term for fluid velocity

Longshore velocity

Velocity in the approach flow

Velocity in the region of channel expansion

Shear velocity

Shear velocity in the approach flow

Shear velocity in the region of channel expansion

Orbital velocity in y-direction

Dimensionless velocity scale in y-component

Complex potential

Dimensionless velocity scale in z-component

Dimensionless length scale in x-direction

Dimensionless length scale in y-direction

Dimensionless length scale in z-direction

Beach slope angle

Internal friction angle

Coefficient of orbital velocity measured from
original smooth plate

Scale parameter for characteristic length

Specific weight of water

Specific weight of bed material

Von Karman's universal constant














































TW
si-1 Q
= e, 0 = sinh e
C K
z
g


Normalized standard deviation of hydrodynamic
lift force

Dimensionless coefficient of net value inte-
grated from one wave period

Tractive force due to the fluid flow

Critical tractive force

Angle that the velocity vector in potential
flow makes with the positive x-ais

Angle of wave incidence at breaker

Parameter of dimensionless shear stress at
critical stage

Parameter of dimensionless shear stress
Scale defined by value in the model
value in the prototype

Dynamic viscosity

Kinematic viscosity

General term for mass density

Mass density of fluid

Mass density of bedload

Wave frequency

Potential flow parameters

Dimensionless coefficient for determination of
bottom velocity

Velocity potential

Dimensionless bedload transport intensity

Stream function

Dimensionless shear intensity
















Abstract of Thesis Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Engineering

MODELING OF THE SEDIMENT TRANSPORT IN
THE VICINITY OF INLET AND COASTAL REGION

By

Pang-Mou Lin

August, 1969

Chairman: 0. H. Shemdin

Major Department: Civil Engineering


Sediment transport in the vicinity of inlets and.coastal regions

depends on the combined bottom shear stresses due to both currents and

waves. The modeling of the movement of the bedload is controlled by

the Froude law, bottom shear stress, wave steepness, and friction factor.

Assuming Einstein's theory of bedload function can be applied to this

study, an analysis was performed after conducting experiments in the

flume and model basin. The results of bedload transport along the

beach were in reasonable agreement with the estimated theoretical values.

However, the sedimentological time scales for the three bed materials,

sand, walnut shell, and coal, were not in satisfactory agreement.















CHAPTER I


INTRODUCTION


A. Statement of the Problem

Problems dealing with sediment transport of inlets and coastal

regions are very complex and difficult, Very often, analytical solu-

tions fall short because of insufficient knowledge of the phenomena,

or because of complex geometry. In such cases, a model study with a

movable bed is desirable to arrive at a solution, Although it is

laborious and time-consuming to calibrate models to simulate adequately

processes in the prototype, a movable-bed model is a valuable guide to

the engineer in the design of coastal structures and navigational channels.

Often, it is desirable to use lighter movable-bed material to reduce

the operation time of the model. In this research, the similarity of

movable beds is studied by using sand, walnut shell, and coal.

B. Aim of the Investigation

A better understanding of the mechanics of sediment transport in the

vicinity of inlets and coastal regions is needed for proper design of

coastal inlets and coastal structures. The movement of sand along the

coast and in inlets is induced by both currents longshoree currents, flood

and ebb flows) and wave action.

Other investigators have arrived at a relationship between currents

and sediment transport [1] and, to a lesser extent, between waves and










sediment transport [2, 3]. The combined action of waves and currents in

sediment transport is not completely understood yet.

To arrive at a better understanding of this problem, a series of

tests was conducted in which different materials were used to verify the

similarity laws. The test program consisted of (1) a basic investigation

of grain movements of different materials in a one-dimensional flume, and

(2) a model study of waves and currents in an inlet and a coastal region

under conditions similar to the ones found in nature. Separate tests

were conducted for inlet currents (ebb and flood), waves approaching the

coastline at an angle, and a combination of waves and currents. The ac-

quired measurements consisted of currents in the inlet, wave heights, and

longshore sediment transports. After each run, the bottom configuration

was contoured and recorded photographically.














CHAPTER II


THEORETICAL BACKGROUND AND ANALYSIS


The motion of discrete sedimentary particles within a turbulent

fluid flow remains one of the most difficult problems.in sediment trans-

port. The parameters for a movable-bed model are numerous, and simili-

tude laws are not well established as compared to a fixed-bed model.

For example, the roughness coefficient is extremely variable due to the

formation of several types of movable-bed configurations, such as ripples,

dunes, sand bars, and anti-dunes. These bed configurations form as a

result of the interaction between the movable bed and the flow over it.

Empirical relationships among the average velocity, water depth, and

bed slope can be used in conjunction with sediment discharge analysis in

open channels. The relationships rely on empirical constants (Darcy-

Weisbach's f, Ch6zy's C or Manning's n) which, in turn, depend on flow

conditions., The latter effect must be taken into account when simulating

sediment transport.in a model.

The "sediment transporting capacity" of a channel was described by

Einstein [1] as a useful technique in describing sediment transporting

movable-bed channels. When the rate of sediment discharge is less than

the capacity of the channel to move it, erosion begins and propagates from

layer to layer, If the rate of sediment discharge is.larger than the ca-

pacity of the channel to move it, the surplus sediment settles down and

begins to cover the channel bottom. A similar description for inlets and

coastal regions is not valid because of the influence of waves, The










"sediment transporting capacity" of inlets and coastal regions depends

on the combined effects of waves and currents.

In order to simulate the fluid-sediment interaction processes in

inlets and coastal regions, an analysis of similarity laws will be dis-

cussed. The analysis will be conveniently subdivided into three cate-

gories: (1) channel flow only, corresponding to predominantly tidal

flow in inlets, (2) wave motion in the absence of currents, and (3) com-

bined motion due to waves and currents. The densimetric flow and salinity

intrusion will be ignored.

A. Similarity Laws for Open Channel Flows

The conditions for dynamic similarity of any flow system can be ob-

tained by writing the Navier-Stokes equation of motion in dimensionless

form. The dimensionless x-equation [4] is referenced to velocity V and

length L and can be expressed as


u' u' u' u'
S+ u' + vt + w au


= + ( ( + + ) (1)
V ax' 9x' VL x'2 y'2 3z'2

The quantities with superscript are dimensionless and are defined in the

following manner:

u'= x = = = t
u u x t t t
T L/V'

v' =u y, = p, = _P_
V L p pV2'

w d = p = const,
V L L
p = consto

g = const.

Since all of the quantities on the right-hand side of Equation (1) are








5

dimensionless, each of the two coefficient groups must be dimensionless

also. The square root of the inverse of the first dimensionless group is

called the Froude number and is expressed as

SV (inertia force ~
F1 gravity force

The Froude number is an important parameter whenever gravity is a factor

which influences fluid motion. The second group is the inverse of the

Reynolds number which can be represented as

1 viscous force/mass pV/pL2 =
R inertia force/mass V2/L pVL

The Reynolds number is important whenever viscous forces influence fluid

motion.

The dimensionless equation of motion may then be written as

qu+ u-- + v' -u + wt' 3


1 ad' 3p' 1 a2u' a2u' a2u'
= -+- (- + +-) >(2)
F12 i x' ax' x'2 + y' R z'2

The same dimensionless groups appear in the y- and z-components of the

equation of motion.

When the gravity force is more important than the viscous force,

the terms multiplied by the Reynolds number are ignored. Froude simi-

larity results under such conditions. Assuming that the fluid and the

acceleration due to gravity in the model and prototype are the same,

the similarity relationships for an undistorted model are expressed as

follows:

Time Scale: XT = d

Length Scale: XL = Xd

Velocity Scale: XV = X


I










5/2
Discharge Scale: X = 5/2

Force Scale: F = Xd3

A serious objection to modeling, according to the Froude law, is

that it does not take into account the relationship between bedload move-

ment and flow characteristics because the Froude law neglects the inter-

granular frictions during the bedload movement. A general discussion of

bedload transport is presented in the following paragraphs.

A study of the movement of bed particles shows that bedload transport

starts when the rate of flow has obtained a certain velocity. DuBoys [5]

introduced the idea of "critical tractive force" for the initiation of

sediment movement and the following formula for the movement of bedloads.


qB = Cg T(T T) when T > T (3)

q = 0 when T < Tc
where

qB = the discharge of sediment in motion and
is defined as volume of grains trans-
ported per unit time per unit width of
bed

CB = a special "sediment parameter" and is
considered to depend entirely upon sedi-
ment characteristics

T = tractive force due to the fluid flow

T = critical tractive force
c
Shields [6] constituted further contributions to the problem by labora-

tory studies. The following relationship was found.

d e

Yf(Ss Sf) de (4)
Equations (3) and (4) state that the tractive shear stress T and the

critical tractive stress T for starting the bed transport are functions









Vd
of the grain size Reynolds number (--). For practical purposes, the

function 41 may be assumed to be a constant which depends on the specific

weight ys of the sediment. The critical tractive force varies directly

with the first power of the diameter of sediment particles. Bagnold [7]

found that the bed shear stress T, required to maintain the motion of

bedload, is

T = (p pf) gde Cotanal (5)

where C is the static volume concentration (1 p), and p is the porosity.
o
The average value of C shown in his paper, was estimated to be in the

range 0.6 m 0.7. The friction coefficient tanal depends on the character-

istics of shear stress and grain size. Bagnold combined these two factors

in a dimensionless shear stress parameter e' where

8' = C tanai (6)

From his experiment, the critical bedload stage ec is likely to be widely

associated with the value of 6' at which bed features disappear or, at any

rate, cease to create appreciable drag. The experimental evidence [8]

shows the correspondence to be moderate. From Equations (5) and (6), it

follows that the similarity law for the bedload movement, due to shear

intensity, can be written as


S= (p- Pf) de e (7)


The bed shear stress can be expressed in terms of the shear velocity

V, and defined as

T = pv 2 (8)

The shear velocity V* can be related to the average velocity V by use of

Chezy's equation and Equation (8).


I








8


V= ()= (gRSe) V (9)
z

where C is the Ch6zy friction factor, R is the hydraulic radius, and
Z
S is the energy gradient. Rearranging Equation (9), it can be seen

that


V -z (10)
V
S*= g


Hence, the similarity law of the friction factor may be expressed as



= X (11)
V z

The movement of bed material under the influence of a shear flow was

treated by several authors. These studies resulted in a number of formu-

lae for bedload transport. Some of the well-known formulae are those of

Meyer-Peter and Mueller [9], Einstein [1], and Kalinske [10]. It appears

that most formulae may be written in terms of a relationship between two-

dimensionless parameters X and Y, defined as


X = -- q s (12)
[g(Ss l)de

where qs is sediment transport volume per unit time per unit width, and


Y = T (13)
(p Pf) g-d .e'

where 6'is the same as defined in Equation (6).

The basic relationship, governing the motion of sediment, was derived

by Einstein [1] and Brown [11] on the basis of experimental data. An

empirical relationship between shear stress and sediment transport rate










was deduced as

qs
s = 40( )
F [g(S 1) d 3] =s -1) e
o s e

where (14)

F 2 36v2 36v2
F= --o + 362-[
o + gd 3(Ss 1)] gd s(S 1)

and v is the kinematic viscosity of water.

The sediment transport rate qs in Equation (14) can be put into

a more convenient form in terms of weight (in water) per unit time per

unit width of sediment transport q It can be shown that


qw = Pwg(Ss 1) (1 p) qs (15)

If water is used in the model, the modeling relationship for sediment

transport q can be written as

h 2
L = s-) p) X t (16)
qw (Ss-1) (1-p) At

B. Similarity Laws for Wave Motion

The velocity potential, for a small amplitude progressive wave

traveling in the positive x-direction in an inviscid fluid, is found

to be


-H cosh ky
H 2 cosh kd cos(kx at) (17)
2a cosh kd

The horizontal component of orbital velocity is obtained from the first

derivative of p with respect to x

H cosh ky (
u = inh k sin(kx at) (18)
2 sinh kd











where


u = horizontal orbital velocity at an elevation
y above the bed

d = water depth
2i
k = wave number defined by --, where L is wave
length
21
a = wave frequency defined by 2-, where T is
wave period

H = wave height from crest to trough

T = time

From Equation (18), the similarity relationship for the orbital motion

becomes


X = A A cosh ky
u a H snh
sinh kd


(19)


The values of cosh ky and sinh kd are unity in an undistorted model

where X = Xd = L. This means that the values of ky and kd are equal for
y d L
both prototype and model at corresponding locations. Then,


Xu = X X H = (20)
u a H XT

The scale for the wave period can be obtained from the relationship be-

tween wave length and wave celerity c


L = c T


(21)


For linear wave theory, the general expression for wave celerity is given

by


c2 = tanh kd
k


The value of tanh kd is also equal in both prototype and model, so that
tanh kd


(22)





A 2 = 1 =
c k L

or

c = L (23)

From Equation (21), the similarity law for the wave period becomes


L -
xT x = XL (24)
c

and from Equation (20), it follows that

AH H
Au .= (25)
x T AL


The similarity laws, for the wave height and for the length

dimensions of the model, may still be chosen arbitrarily if simi-

larity of the wave steepness is not required. Otherwise, the following

relationship would have to be satisfied


AM = AL (26)

Under this condition, the breaking of the wave in the model will be

initiated at a depth corresponding to that in the prototype. The

breaking phenomenon may differ in the model, however, if the surface

viscosity is important. In summary, -the waves are to be reproduced

geometrically to scale at all locations in the model if the following

scale laws are satisfied.


AT = hL


AH = L
X1 x L 5/-*


A = AL
u L


(LZ/)









An analysis of sediment transport by wave motion was given by

Kalkanis [2] in terms of two-dimensionless parameters

P Pf d g
4' = s e (28)
Pf

qw 1 (29)
y- g(S S ) d3] (29)
Ys 5s f e

where u was defined as the orbital velocity near the oscillatory

boundary.


u = u [sinot ki(By)n sin(at a2lnk2Oy)] (30)

H 1 C;
o

where u = 1 and 8 = ( )o The values of the exponent n
o 2 sinh kd 2v
and of the constants k1, k2, and a2 could be obtained from experimental

data. Equations (28) and (29) are identical to Einstein's bedload

function. The remarkable characteristic of both equations is that the

bedload rate is indirectly related to the flow intensity through the

probability distribution curve.

Bp' A,'
1 no -t2 dt (31)
P =l- e 1 + A(1
S1
SB,1' ---
o

where n is the normalized standard deviation of the hydrodynamic lift

force. Kalkanis [2] obtained values of A*, B,, and from the experi-
o
mental data. These are shown as follows:

A, = 30.0

B* = C40


1- = 1.5 (32)
no










The equivalent Einstein universal constants correspond to A* = 43.5,

B, = 0.143, and = 2.0.
0o
C. Similarity Laws for Combined Motion Due to Waves and Currents

1. Existing analysis of velocities due to wave motion

Before discussing bedload transport, it is necessary to analyze

the bed shear stress due to the combined effect of waves and currents.

The most interesting area, from the point of view of this study, is in

the vicinity of an inlet where the net current is produced by tides and

longshore currents. The latter is produced by waves approaching the

beach. From Equation (18), a frictionless orbital velocity at a fixed

position along the bottom is deduced

H 1
ub = sinot (33)
b ~2 sinh kd

Bijker [12] introduced the orbital velocity above the laminar sublayer

plub, where pl is a coefficient which was found to be 0.39.

Eagleson [13] used the small-amplitude wave theory and the beach

geometry to produce an.expression for longshore currents.

2 3 b2 nb sina sineb sin2eb
V2 = -1 [ ] f (34)
L 8 db f
b

where

Hb wave height at breaker


nb = ratio of group velocity to wave velocity
at breaker

db = water depth at breaker

a beach slope angle

eb = angle of wave incidence at breaker

f Darcy-Weisbach's friction coefficient










The coefficient f is calculated by using the Karman-Prandtl resistance

equation for steady and uniform flow in conduits with rough walls. It

was found that

d b
f = [2 logo (b-) + 1.74]-2 (35)
e

where k is bottom roughness height.

2. Velocity field in the vicinity of an inlet

In the vicinity of an inlet, the local current at a position

depends on the expansion of flow as it leaves the inlet due to an ebb

tide, or on the contraction of flow as it enters the inlet due to a flood

tide.

Rouse et al. [14] developed a dimensionless diagram, shown in

Figure 1, to describe the most satisfactory boundary for an efficient

channel expansion under supercritical flow conditions. The channel flow,

under subcritical conditions however, is expected to behave differently.

Tollmien [15] was the first to investigate theoretically the turbulent

diffusion of a two-dimensional jet. Tollmien's analysis was based on the

momentum transport theory of turbulence. Later investigators (e.g.,

Albertson et al. [16] and Baines [17]) proceeded with analysis based on

the general momentum and energy considerations and the use of the error

function to represent the velocity distribution in an expanding section.

An analysis that approximately resembles the expansion process of

an ebb flow downstream of an inlet is the expansion of a circular jet into

a fluid of equal density that is initially at a state of rest. The flow

in an inlet is considered in analogy with one-half of the circular jet

flow field (i.e., the half beneath a horizontal plane through the axis
























B,F,
Fig. 1. Generalized surface contours for stream expanding
past abrupt expansion (after Rouse, Bhoota,
and Hsu (14) )


CHANNEL


OCEAN


Vm- di

1


d V-


Fig. 2. General representation of outgoing flow


T


Lx










of the pipe). The following analytical expression for the x-component

of velocity V was proposed by Baines [17] for pipe Reynolds number R

in the range 3 x 103 < R < 2 x 105.


x 1 B1
= B exp [- 81 Y (36)
V 2C2 x 2C22 (36)
m

m
where Vm is the mean velocity of the jet, shown in Figure 2; y is the

radial distance in an axialsymmetric flow; C2 is a dimensionless

constant to be determined experimentally (according to Corrsin and

Uberoi [18],C2 = 0.07 to 0.08 for no density difference flow); and B1

is the diameter of the circular jet and is considered here to be the

width of the channel. The coordinate system of the above equation is

shown in Figure 3. The lateral velocity V can be obtained by intro-
y
during a stream function


V x - f(- V ) dy
y 9x 3x j x

which yields

V B1 2
V= m y-- exp [- ] (37)
y 2C2 x2 2C22

The velocity components, V and V are used to approximate the channel
x y
velocity during ebb flows.

The velocity distribution associated with flood flows can be

approximated in analogy with a flow from a large body of water into a

relatively narrow channel entrance. A convenient way to estimate flow

behavior in a contraction is to assume that it is approximately two-

dimensional so that use can be made of the potential flow theory. The

graphical method (see French [19]) becomes feasible, then, in estimating










x



jVm






Vy V


CHANNEL


qy
Li


J -- uoSin 4tCose

:-u.Sin at Sine


Fig. 3 General representation of currents and waves


V-


VL










the velocity magnitude and direction. The velocity potential ( and

the stream function p are related to the x-and y-velocity components

by the following relationships



Dx y


i_ = = V (38)
ay ax

A solution of the flow field was given by Greenhill [20]. A

definition sketch is shown in Figure 4. Use was made of complex analysis

where the flow in the z x + iy plane was mapped into the t plane by the

transformation


-= log 1 (39)


where c'is as shown in Figure 4. The velocity distribution in plane

was given by


dz n
S- V d e (40)

where V is the velocity along the streamline at infinity, W is the com-

plex velocity potential (( + iu), and 0 is a parameter which relates the

flow in terms of discharge Q.
w
sinh 0 = e Q (41)

The velocity field may also be described in cylindrical coordinates


V iO
= -- e (42)
q

where q is the velocity at a point (x, y). The velocity field may be

described conveniently in terms of Q, (, p, and V. The governing relation-

ship takes the form























FREE STREAMLINE


Fig.4 Channel entrance corresponding to a two-dimensional orifice


1.6 _



.2




, o ----


00




-2 -1 0 1 2 3 4


Fig. 5 Variation of velocity along streamline for flow through
two-dimensional orifice


-9-
vM


1,0


0.8


0.6
q
V
0.4


0.2


0


_


S--


1








20
2__(_ 2_(T 2Tr4
cosh2 log = {(e + 1) [(e + 1)2 4e sin2 ()] (43)
q Q

The velocity,at any point,may be determined in terms of the free stream-

line velocity V. In Table 1, this was done for various values of and
Q
It is, however, more convenient in many practical applications to
Q *
express the velocity field in terms of the mean velocity in the channel.

The latter is related to V by the equation

2 + n
V = V (44)
n m

Using Equations (43) and (44), values of and q- were plotted
V V
m
against in Figure 5, where s is the distance measured along a parti-

cular streamline from its intersection with the zero potential line.

The value of s for any given value of 4 and I was obtained from Figure 6.

3. Evaluation of shear stresses

Figure 3 defines the velocity notation used to calculate shear

stresses. The velocity components, due to flow expansion (ebb flow),

are given by V and V respectively. The velocity components, due to
x y
flow contraction (flood flow), are given by qx and q respectively. The

horizontal component of particle velocity, at the bottom, in the direction

of wave propagation is given by u sinat. The average velocity in the

inlet is given by V .

Prandtl gave the following expression for the turbulent shear

stress at any point in a fluid moving past solid wall


r= p2 (aV(v) )2 (45)
ay
where








21


Table 1
Computed values of x/b and y/b for various values of p/Q and ir/Q
two-dimensional orifice
(After French (19).)


n o
Q Q
0 -3
-2
-1
-112
-1/4
0
1/4
1/2
3/4
1
1-1/2
2
1/16 0

1/4
1/2
3/4
1
1-1/2
2
1/8 -3
-2
-1
-112
-1/4
0
1/4
1/2
3/4


x y
b b
1.03 0
0.605 0
0.11,3 0
-0.1.95 0
-0.380 0
-0.598 0
-0.855 0
-1.17 0
-1.56 0
-2.04 0
-3,42 0
-5.74 0
-0.574 -0.185
-0.835-0,221
-1.15 -0.272
-1,52 -0.338
-2.01 -0.426
-3,38 -0.685
-5.64 -1.13
1.03 -0.157
0.611 -0.176
0.129 -0.218
-0,172 -0.266
-0.34 -0.318
-0.54 -0.363
-0.78 -0.433
-1.07 -0.531
-1.44 -0.622


Y1
Q
1/8




3/16










1/4


















5/16


x y_
b b
-1.88 -0.831
-3.18 -1.35
-5.28 -2.20
-0.48 -0.53
-0.72 -0.64
-0.97 -0.76
-1.29 -0.955
-1.70 -1.21
-2.86 -1.96
-4.76 -3.21
1.04 -0.319
0.626 -0.345
0.159 -0.419

0.092 -0.508


n0o
Q
1
1-112
2
0
1/4
1/2
314
1
1-1/2
2
-3
-2
-1

-1/2
-1/4
0
1/4
1/2
3/4
1
1-1/2
2
0
1/4
1/2
3/4
1
1-1/2


-0.577
-0.675
-0.805
-0.986
-1.22
-1.54
-2.51
-4.07
-0.805
-0.955
-1.16
-1.44
-1.81
-2.94


2 -3.18 -4.80


-0.241
-0.401
-0.595
-0.621
-1.09
-1,44
-2.44
-4.04
-0.31
-0.46
0.65
-0.86
-1.13
-1.92


i at x y
Q Q b b
3/8 -3 1.06 -0.469
-2 0.67 -0.509
S -1 0.22 -0.599
-1/2 -0.01 -0.704
-1/4 -0.10 -0.785
0 -0,20 -0.896
1/4 -0.32 -1.06
1/2 -0.46 -1.28
1 0O.78 -2.01
1-1/2 -1.31 -3.26
2 -2.19 -5.32
7/16 0 -0.10 -0.969
1/4 -0.16 -1.14
1/2 -0.22 -1.38
3/4 -0.31 -1.70
1 -0.40 -2:15
1-1/2 .-0.672 -3.46
2 -1.12 -5.65
1/2 -3 1.05 -0.627
-2 0.665 -0.661
-1 0.261 -0.755
-1/2 0.102 -0.849
1/4 0.041 -0,915
0 0 0
1/4 0 -1.16
1/2 0 -1.40
3/4 0 -1.74
1 0 -2.18
1-1/2 0 -3.52
2 0 -5.79











































Fig. 6 Flow pattern through two-dimensional orifice










A = mixing length

V(y) = velocity at height y above the bed

p = fluid density

y = distance from the bed

For small values of y in a rough bed, A is given by

P = KY

where K is Von Karman's universal constant.

For a fully rough turbulent boundary layer, the velocity distri-

bution above the bed is given by [21]

V*
V(y) = log (46)
K k
e
where ke is the distance above the bed and is referred to as roughness

height. The velocity gradient becomes


Y) (47)
Dy Ky

Evaluating Equation (47) at y = yo, we obtain


aV(yo) V,
S= (48)
;y KYo

where V(y ) is defined as bottom velocity at the value of y (1 + e) k,
o 0 2 e
where e = 2.7183. If the flow in the layer k < y < e k is laminar [21],
e e
Equation (48) becomes


DV(Yo) V(Yo)
= (49)
y Y

From Equations (48) and (49), it follows that

V,
V(yo) = (50)










The above relationships are used in conjunction with Equation (10) to

obtain the bed shear stress under various conditions. The resultant

bed shear stresses during flood or ebb flow can be expressed as follows

(see Appendix I):

Ebb flow inside channel


PlU PlU
T(t) = pV 2 1 l-cosO sinat (1 cosO sinat)
m m

Ebb flow parallel to the shoreline


(51)


V V plu
T(t) = pV 2 L + sine sinct
L* V V
L L


V V PUo
(V-V + --- sine sinct)
L L


(52)


Flood flow inside channel


q plU qx
T(t) = pq,2 Ix+ cos sinot (- +


Flood flow parallel to the shoreline

V q pu sit
T(t) = pV 2 L + -- sine sinot
= VL VL VL
L L


PlU
S-- cose sinat)
4


(53)


V q plu
( + -- sine sinat)
VL V
L L
(54)


The combined bed shear stress can be expressed in terms of the

shear velocity V, and a dimensionless coefficient nx

T = pV2 x

where


T
1
x -i


(t) dt
pV,2


Substituting Equations (15) and (55) into Equation (14), the relation-

ship governing the combined motion of sediment due to wave and currents

is produced


(55)













q V 6 n 3
= 40 x (56)
w (1 P) F 03/2
w o [g(Ss 1) de]

The shear velocity V. can be related to the average velocity V by

using Equation (10). Equation (56) becomes


qw V 6 x 3/2
p (1 p) F C (S ) d57)
w o z s e

A modeling relationship can now be obtained by using Equation (57)


6 x
AV (58)
(l-p) *F C 6 3/2 A3/2 3
o z (Ss-) d

The net rate of erosion of a bed exhibits itself by the change in

the elevation of the bed. The equation of conservation of sediment

mass transport states that the change in the bed elevation, due to

erosion, must be proportional to the rate of change of q with distance.

Quantitatively, this is expressed as


S1 d q = 0 (59)
dt wg(Ss 1) (1 p) dx

where y is the bed elevation, t is time, x is the distance along the

direction of flow, and p is porosity of the sediment.

The time scale of sediment transport is obtained from the defi-

nition of q which is given by Equation (16). An expression for the

time scale is obtained by eliminating the weight rate of sediment dis-

charge scale in Equations (58) and (59).











S3/2 5/2
C z d (s-1) L
z e (Ss-l)
A =-- (60)
t AV6 A 3 AF
V nx Fo
x o

The above relationship satisfies the Froude law, laws of sediment

transport, and continuity of flow and sediment transport.


D. Summary

From the above analysis, it is possible to summarize the movable-

bed parameters by a functional relationship of the form

fl[de(Ss 1) 8e, d, V, g, V,, H, L] 0 (61)

The choice of parameters implies that the particle shape and its

size distribution, fluid characteristics, wave characteristics, and

movement of bedload are significant. Using the Buckingham r-theorem,

Equation (61) may be replaced by a functional relationship among four-

dimensionless parameters

V V* V (62)
2gd g( I)] ec V -* 0 (62)
( Isa 1) de c L
Assuming A 1, the shear velocity scale may be expressed as
g

A X4X Xi (63)
V (S -1) d e <
(S-l c

where 0 is a dimensionless coefficient determined from critical shear

stress for bedloads under incipient motion. The velocity scale is

given by Equations (11) and (63) as shown below.


A h A A A A* A =A (64)
V V C (S -1) d C d
2 e c Z
From this equation, the depth scale can be determined. Since the undis-

torted model is nugCgertd from previous analysis, the time scale for










fluid and wave period can be expressed as


T = A
T d

The sedimentological time scale is shown.in Equation (60) from the

analysis of the equation of conservation of sediment mass transport.


(65)


I















CHAPTER III


EXPERIMENTAL PROGRAM


Experiments were conducted in a tilting recirculating flume and a

model basin. The experimental program is described in the following

sections.

A. Test in the Flume

A schematic diagram of the flume is shown in Figure 7. The flume is

60 feet long, 3 feet deep, and 2 feet wide and has a maximum slope of

2 per cent and a maximum discharge of 5.8 cfs. The tests in the flume

were conducted for three types of granular beds. These were made of sand,

ground walnut shell, and bituminous coal. Water was allowed to flow until

the flow was statistically steady and uniform in the test section. For

each bed material, a series of tests was carried out with various combi-

nations of mean depth and flow velocity. The characteristics of bed

materials are shown in Table 2.

Velocity measurements were obtained at different elevations in the

test section as shown in Figure 8. A propeller-type velocity meter,

manufactured by A. Ott Kempten (Germany), was used 'for this purpose. The

bed materials were arranged in the middle portion of .the flume at a thick-

ness of 2 inches along a 30 foot test section. The test bed is shown

schematically in Figure 7.

In order to obtain the velocity profile which produces the critical

shear stress, the water discharge was increased slowly in small increments












Explanation
a pumping motor
b valve
c venturi meter
d jacks
e pivot
f control gates


g tail box
h flume 3x2x60'
i flow exit from pipe
j recirculating pipe
k train
I velocity meter


f .h
fTi


I! I I l T M. -- 1
.q -v-


I- I -- f Y.


-C- a- Th -
o' em Tb -a


I~-- II


Fig. 7. Schematic diagram of the flume


II


I I


L...l L >J I


-


g I


R


SII f I f f I I I r II f f y CI f I I Il r r/ e l l Il "I If r






















TABLE 2

Characteristics of bed materials


Material Spec. Gravity D80 D50 D20 (mm)


Sand 2.67 0.23 0.16 0.14

Walnut 1.35 1.6 1.6 1.25

Coal 1.25 0.84 0.76 0.53





























































jI ri '



II

.' '
'--- Li


Fig. 8 Velocity measure

the flume


ement in







32

spaced by ten-minute intervals. When the bed material underwent inter-

mittent motion, the velocity profiles were measured at that stage. The

velocity profiles are shown in Figures 9, 10, and 11. The relationship

between the mean velocity V and the shear velocity V* for the three bed

materials is shown in Figure 12.

The bedload transport rates in the flume tests were measured by taking

samples from a pan trap which is shown in Figure 13. The quantity by

weight of the sediment was determined after drying. On the average, three

samples per run were taken. The mean sediment transport rates were con-

sidered to be the average of the three samples. The latter was assumed

to represent.the long-time average bedload transport prevailing in the

flume. The results of sediment transport tested in the flume are expressed

in Table 3 and Figure 14.

B. Test in the Model Basin

A schematic plan view of the model basin is shown in Figure 15. The

maximum still water depth in the constant depth portion was approximately

11 inches. The channel connecting the two basins was 10.75 feet long and

2.75 feet wide. The beach was arranged to have a 1:20 slope (i.e.,

tana = 0.05). The velocity profile in the channel was measured by the

propeller-type velocity meter at different depths. The mean velocity was

then taken by averaging the velocities over the full water depth. The

three types of sediment materials used in the flume were also used in the

model basin study. The water depth for each material was selected to

satisfy the similarity laws derived in Chapter II and summarized in

Equation (64) (see sample calculation in Appendix II). The water depth

was set by choosing an appropriate thickness for the bed material.
































































05 06 07 08 0.9
V (y)ft/sec
Fig. 9 Velocity distributions over a plane bed when
sand is moving intermittently


yin














S d = 10


d =6


Shear velocity
Mean velocity for 8 depth

Roughness height


101









Yin




10














-1
10


VI 1 =0.504
10


V,=0.040 ft/sec


ft/sec


ke=0.0013 ft

ke =0.00092 ft

ke =0.00083 ft


V ,, =0.521 ft/sec
8

V ,, =0.524 ft/sec
6


9c =0.0378
C


).3 0.4 0.5 0.6 0.7 0.8
V ( y)ft/sec

Fig.10 Velocity distributions over a plane bed when
walnut shell is moving intermittently


+

V.
V,

Ke


//
/
/
/


//
//
//


/1A
















S d =10

+ d=9"

V. Shear velocity
Vs., Mean velocity for 9" depth

ke Roughness height / /


* /


101










Y in





10


V,, =0.506. ft/sec.
10

V .. =0.538 ft/sec.
9


k =0.00047 ft.
e

k =0.00054 ft.
e


e =0.0436


i I I I I I I
0.3 0.4 0.5 0.6 0.7 0.8
V (y)ft/sec
Fig.11. Velocity distributions over a plane bed when
coal is moving intermittently


V,=0.0296 ft/sec.


10 1














c
)-



5


Cz41
S 0


V
ft/se

1.C


0.


0.




V
ft/se

1.C


0.!


1c


5-

5-


Cz=36.5


0.
.C






V
ft/sec

1.0-


0.5.


)


)2 .03 .04 .05 .06 .07

V4 ft/sec (Sand,


.08 .09 .10 .11
D0=0.163 m.m.)


CZ 55


Fig.12. V vs. V, for plane and dune beds


.02 .03 .04 .05 .06 .07 .08 .09 .10 .11

V.ft/sec (Walnut shell, De=1.6 m.m.)


.12 .13


.02 .03 .04 .05 .06 .07 .08 .09 .10 .11

V. ft/sec (Coal, De=0.78 m.m.)





.,


1


























Flow direction


BedloadN -Pan trap


12- 3 0 1



Fig.13 Pan trap in the flume













Table 3. Results of sediment transport which were tested in the flume


Test no. Water Mean Shear Bedload Dimensionless Dimensionless
depth velocity velocity transport shear intensity bedload
d V V, q x105 transport intensity
(in) (ft/sec) (ft/sec) (Ib/seclft) 0

I 10 1.075 0.099 7.627 2.927 309.14
A
Ig 10 1.075 0.099 7.870 2.927 318.99

IC 10 1.075 0.099 6.900 2.927 279.77
ID, 10 1.109 0.123 36.816 1.897 1491.05
ID 10 1.109 0.123 45.414 1.897 1839.27
1E 10 1.273 0.151 105.158 1.258 4262.05
IF 10 0.736 0.067 1.102 6.392 44.66
IG 10 1.010 0.107 37.918 2.506 1.23.68

IA 10 0.723 0.0660 8.906 13.526 160.477
IEB 10 0.672 0.0409 6.348 35.330 114.385
IC 10 0.729 0.0696 34.204 12.200 616.322
.D 10 0.861 0.0835 101.631 8.476 1831.289
IE 10 1.001 0.0870 420.304 7.808 7573.457
IF 9 0.659 0.0460 6.757 27.809 121.754
XiH 9 0.821 0.0545 61.904 19.897 1115.448
IG 9 0.742 0.0478 26.455 25.866 476.692

(Continued)













Table 3 (Continued)


7 *1 -


0.418
1.256
103.108
25.705
63.734
7.914
15.035
50.352


28.259
84.911
6970.575
1737.776
4308.711
535.023
1016.435
3404.027


sand
walnut shell
coal


"A

nrc



ME
XF
GII

mH


0.553
0.641
0.930
0.731
0.805
0.601
0.647
0.718


0.0383
0.0390
0.0617
0.0400
0.0592
0.0525
0.0572
0.0650


14.043
13.474
5.411
12.875
5.878
6.453
6.296
4.876


For
For
For































































Fig.14 curves for the three different bed materials
tested in the flume




















































Fig.15 Schematic diagram of model basin










































Fig.16 Longshore bedload trap in the model basin
















TABLE 4


Test conditions in the model basin


Conditions Sand Walnut Shell Coal


Depth in the channel 5.25 in. 5.62 in. 4.06 in.

Depth in the ocean 8.125 in. 8.57 in. 7.72 in.

Velocity in the channel

Flood 0.646 ft/sec. 0.665 ft/sec. 0.568 ft/sec.

Ebb 0.675 ft/sec. 0.724 ft/sec. 0.598 ft/sec.

Wave period 0.859 sec. 0.875 sec. 0.729 sec.

Wave height 0.093 ft. 0.102 ft. 0.073 ft.

Testing duration 180 min. 29 min, 85 min.


The test duration was determined experimentally as the duration
for accumulating equal volumes of bed material in the weir trap.




























,--3


-- -- -. -




-L


--- Before test


After test


Fig.17. Bottom sand configuration before and after test with combined wave and ebb flow


i


5',,,
















- -





4' n -- -'


,i II -


S-- Before test


After test


Fig.18. Bottom sand configuration before and after test with wave only



















' ----------- -. -
L- -q
*--s
4-'-=r-------- -'
/ '----- *5



;' -I
\I / -- S

ZL\?~


-I '


------ Before test


After test


Fig.19. Bottom sand configuration before and after test with combined wave and flood flow


-S






































Before test

After test





Fig.20. Bottom walnut shell configuration before and after test with wave only



































Before test


After test





Fig.21. Bottom walnut shell configuration before and after test with combined
wave and ebb flow


- -_-C-T_
























-- -=

3 -- -
/t
i-

-3---C--C-F -
J~ ~r ---, ~2~-5~--


-------- Before test


After test


Fig.22. Bottom walnut shell configuration before and after test with combined

wave and flood flow


.4



































r5-


------- Before test


After test


Fig.23. Bottom coal configuration before and after test with wave only


---


I
,'j
























--- nsJ
.--- --- ---


-3-





It ______
I S
i MSL


Before test


S- After test










Fig.24. Bottom coal configuration before and after test with combined

wave and ebb flow


i
















I -

-~ -' L
-I
--41


.- --- Before test

Vi -- After test


\\ I i.
Fig.25 Bottom coal configuration before and after test with combined
wave and flood flow


ff










Table 5 Numerical results for the volume of bedload transport per
unit time per unit width



Types of Bed Bedload transport Specific Porosity Bedload
test materials in dry weight gravity transport
sx0ld5 qs4x12 in volume
(Iblsecl ftgsem m3seccm)
______ ______ (lb/sec/ft) (glsec/cm) _________(cm3/sec/cm)


Wave









Wave
and
Ebb






Wave
and
Flood


sand


walnut
Shell


coal




sand

walnut
shell


coal


sand

walnut
shell

coal


437.23
542.33
476.19

104.06
114.42

248.44
347.44
373.68


192.35
119.05
141.64

63.82
58.42
155.88
244.08
253.52

782.63
763.73
274.52
229.94
285.72
252.84


8.505
8.071
7.087

1.548
1.706

3,701
5.171
5.564


2.861
1.772
2.106

0.951
0.866
2.322
3.635
3.753

11.646
11.364
4.087
3.425

8.503
7.520


2.67


1.35


1.25




2.67


1.35


1.25


2.67

1.35


1.25


0.607


0.496


0.524




0.607


0,496


0.524


0.607

0.496


0.524


__ __ __ L __ I. L __ __ j


0.0587
0.0729
0.0640

0.0216
0.0238

0.0588
0.0824
0.0888


0.0258
0.0160
0.0190

0.0132
0.0120
0.0372
0.0580
0.0596

0.1052
0.1026
0.0590
0.0476

0.1356
0.1202











Table 6 Results of longshore sediment transport which were
tested in the model basin


Types of Bed Bedload Calculated Dimensionless Dimensionless
test materials transport shear shear bedload
q x10 velocity intensity transport
intensity
(Ib/seclft) V, (ftlsec) 0'x165


Wave










Wave
and
Ebb







Wave
and
Flood


sand


walnut
shell


coal


sand


walnut
shell


coal




sand

walnut



coal

1


437.23
542.33
476.19

104.06
114.42

248.44
347.44
373.68


192.35
119.05
141.64
63.82
58.42

155.88
244.08
253.52


782.63
763.73

274.52
229.94

285.72
252.84


0.088


0,081


0.061




0.088


0.081


0.061



0.068


0.051


0,055


3,60


9.01


5.53




3.60


9.01


5.53




6.10


22.74


6.81


177
219
192

18 64
2060

1680
2328
2520


78
48
57
1152
1052

1052
1648
1704


320
310

4950
4140

3840
3412










The bed thicknesses were 2.37, 2,45, and 3,16 inches for sand, walnut

shell, and coal, respectively.

Waves were generated by a flap-type wave generator with variable

periods and wave heights. Wave measurements were obtained by using a

resistance-type wave gage. The waves were made to approach the shore at

an angle of ten degrees with respect to the normal of the shoreline. The

flood and ebb flows were controlled by weir boxes and gates. The water

surface elevations,upstream and downstream of the channel, were measured

by two point gages as shown in Figure 15. The bedload transport rates in

the model on the ocean side were measured by taking samples from a pan

trap located under the sand weir as shown in Figure 16.

The experimental conditions for the different sediments are outlined

in Table 4. The simulated conditions were based on the similarity laws

derived in Chapter II. It is emphasized that modeling relationships,

shown in Equation (64), are valid for an undistorted model with the same

beach slope and wave steepness for all bed materials. After the completion

of every test, the bottom configurations were contoured and recorded photo-

graphically (see Figures 17 through 25). According to Equations (I-lOa) and

(I-12a) in Appendix I, the results of bedload transport rates are given in

Tables 5 and 6. Sample calculations of bedload transport rates are given

in Appendix III. Sample calculations of Einstein's bedload function are

shown in Appendix IV.















CHAPTER IV


ANALYSIS OF RESULTS AND DISCUSSION


The present study has included both flume and model basin investi-

gations in an attempt to obtain quantitative results for sediment trans-

port in the coastal environment. The analysis of results and discussion

are presented below.


A. Flume Test Results

The results of the velocity profiles, as obtained in the flume and

shown in Figures 9, 10, and 11, were used to determine the dimensionless

shear stress coefficients e The value of e for sand in this study is
c c
approximately the same as that shown by Bagnold in Plot VI [8]. The

grain size used in Plot VI was 0.31 mm, and the Qc was found at the range

of values from 0.05 to 0.14. According to Bagnold [7], c is a function
c
of (1 p) and the internal friction angle of bed material. The magni-

tude of 9 does affect the intensity of the movement of bed material.
c
Considering the similarity law, e is one of the essential parameters to

simulate the bed shear stresses between two different bed materials.

In Figure 12, the linear relationship between the mean velocities

and the shear velocities is shown for the purpose of selecting the con-

stant friction coefficients; the results of the velocity measurements were

taken both in plane and dune beds. The reason for selecting the friction

coefficient as constant is that the test velocity was scaled up a little

higher than the critical velocity for bedload movement. The error










introduced cannot be very significant as compared to the approximation

made in Equation (49) where a linear velocity distribution in the boundary

layer was assumed.

Figure 14 is a graphical representation of the results shown in.

Table 3 which was obtained by using Einstein's bedload function theory

(see Appendix V). The data for coarse bed materials fit Einstein's bed-

load function very well. From the experimental results by Bishop, Simons,

and Richardson [22], it is now apparent that A* and B* are not universal

constants but are related to the median diameter of the bed material.

The results of the present study show that the values of A* increase with

increasing grain size, and the values of B* decrease with increasing grain

size and that A* and B* are independent of the types of bed material.

These characteristics of A, and B* are similar to the results shown by

Bishop et al. [22].


B. Model Basin Test Results

The accuracy of the results concerning the similarities among differ-

ent bed materials, as one would expect, depends entirely on the quality

of approximations made in the course of developing the various relation-

ships. Thus, it is necessary to analyze these approximations step by step

and make suggestions regarding their possible improvement.

The steps of calculation for all similarity scales were described

in Chapter II. However, it is to be noted that the depth scale was de-

termined by considering the similarity laws among the parameters which

include only the grain size, specific weight, packing, and movement of

the sediment grain as given by Equation (64). From the depth scale and

physical properties of sediment, the velocity scales are obtained. Two








58

time scales of special interest in the present study are wave period and

sedimentological time. The former is equal to the square root of the

length scale; the latter is determined from flow characteristics and

the conservation of mass transport of the bed as given by Equation (60).

It is assumed that the bedload transport, due to the combined

action of waves and currents, is a function of bottom shear stress.

Since the movement of bedload due to wave action is related to the

probability characteristics [2], the theory of Einstein's bedload

function can be assumed to be valid in this study provided the

dependence of the bottom shear stress on the oscillatory mean flows

is taken into account. Equation (51) through Equation (54) were derived

for this purpose. However, these derivations were based on the as-

sumption that the velocity profile, from the edge of the viscous sub-

layer to the granular surface, is linear. Somediscrepancy may occur

for low water velocity. The error introduced by this can be reduced

due to the effect of oscillating bottom water particle velocity, and

the error is usually insignificant [12, 21].

The basic analytical treatment of channel expansion of ebb flow

is assumed to be the same as that for a half circular jet expansion.

The theory of circular jet expansion was first described in detail by

Tollmien [15] and later investigators, for example, Albertson, Dai,

Jensen, and Rouse [16] and Baines [17]. It is reasonable to apply this

theory in the course of the analysis of current pattern. In this experi-

ment,there was no direct measurement of water velocity in the region of

channel expansion, but one would expect that the ebb flow conditions

contribute less to the longshore bedload transport. This may be one of

the reasons that inlets and channels always suffer from the sediment








59

problem. The results of the ebb flow study, as represented in Figure 26

and Table 6, show that the bedload transport rate along the shoreline has

the least amount of weight in comparison with the other two conditions.

This agreement appears quite satisfactory in similarity to natural

phenomenon.

Concerning the flood flow pattern, the basic analytical treatment

was presented by French [19] in his research of "tidal flow in entrances."

It is reasonable to apply potential flow phenomenon to the flood flow

condition. Although there may be some discrepancy in the analysis, the

error cannot be very significant for the small beach slope which was used

in this study. According to Equations (52) and (54), the longshore bed

shear stresses were calculated (see Appendix V) and shown in Table 7.

The bed shear stress, for flood flow condition, at the site of the sand

weir trap was found to be less than the other two conditions (i.e., waves

only and ebb flow with waves). This is reasonable because at the 'site of

the weir trap minimum flow exists under the condition of flood with waves

since the longshore current is balanced by the tidal current. The results

of longshore bedload transport for flood flow condition, as shown in

Figure 26, do not have close agreement with the estimated curve. However,

one reason may be due to the combined effect of potential flow and wave

diffraction which do affect the bottom particle's movement to leeward by

longshore current.

The velocity of longshore current, as shown in Table 7, was calcu-

lated according to Eagleson's formula shown in Equation (34). No direct

measurement of longshore current velocity was conducted in this study.

However, in the procedures of calculation, the breaking depth was obtained

by assuming that the wave height Hb breaks at the depth of 0.78 db. The



















Sand
\ A,=85
B. =0.78

























+1

+1
\



































SI I I


*\ \ I
Walnut shell

\ \
0 A.=852
B,=0.08


4-


I i I I I I I I



Legend
+ Sand
Wave -o- Coal
S Walnut shell


Wave
and
Ebb

Wave
and


Sand
Coal
Walnut shell


Sand
Coal
Walnut


Flood






, \







Coal

i A,=144
B. =0.22


shell-


I ,III


10 102

Fig. 26 The tested results in the model basin as compared
with the estimated 0'- U' curves


I
I -
I \I I I I II)













Table 7 Results of calculated shear velocities for the site of weir trap in the model


Bottom
Bed Measured Wave Water Assumed Friction Velocity wate Potential Calculated
materials wave period depth at absolute coeff. of particle velocity shear
height at breaking roughness longshore velocity along velocity
breaking current at breaking
breaking line

Hb(ft) T (sec) db(ft) k (ft) f VL(ftlsec) pu(ft/sec) q(ftlsec) V. (ftlsec)
b Db e L
0.088
(Wave)
0.088
Sand 0.085 0.859 0.108 0.0055 0.0515 0.208 0.158 0.050 (Wave
and
Ebb)
0.068
(Wave
and
Flood)
0.081'
Walnut (Wave)
shell 0.093 0.875 0.119 0.0060 0.0532 0.213 0.163 0.052 0.051
(Wave
and
Flood)
0.061"
(Wave)
Coal 0.068 0.729 0.087 0.0030 0.0476 0.193 0.142 0.041 0.055
(Wave
and
Flood)

Same as for the condition of Wave and Ebb









friction coefficient f, in Equation (34), is obtained by using the

Karman-Prandtl resistance equation for steady, uniform flows. The values

of absolute roughness height k in the present study were assumed ac-

cording to both Eagleson's report [23], and the velocity profiles measured

in the flume tests. The calculated longshore current, however, was

found by Eagleson to be in satisfactory agreement with observation in the

field. Some discrepancy may occur for the combined action of wave and

flood;.nevertheless, it is reasonable to apply his formula in this study.

The derivation of Eagleson's formula was based on the conservation of

momentum transported parallel to the shoreline. In the vicinity of an

inlet, a similar analysis of momentum normal to the shoreline may be made

and is seen to give a net flux of momentum into the inlet. The latter may

be interpreted in terms of net velocity into the inlet in analogy to the

longshore current in Eagleson's analysis. Consequently, the value of

bottom shear stress should be higher than the one calculated by Equation

(I-llb) as shown in Appendix I.

The coefficient pl of horizontal orbital velocity above the laminar

sublayer was determined experimentally in this study. The value of pl was

determined from the duration of a test to accumulate a predetermined volume

of sediment. From Figure 26, the shear stress was estimated, and using

Equation (I-9b), pi was evaluated, Using the three different bedload

materials, three values of p] were obtained for three test durations which

corresponded to the same volume transport rate. The final value of pl was

evaluated according to results that have the best fit both in sediment

transport rate and sedimentological time as compared with the estimated

curves and the test durations. The value of pl, in this study, was found

to be 0.24. Bijker [12] derived the value of pl on the basis of Prandtl's










shear stress shown in Equation (45), and pl was found to be a constant

value of 0.39. His experimental result of p], however, was 0.48. The

reason his experimental result of Pl was higher than the one found

in this study may be due to many sand traps introduced during the test.

Too many sand traps introduced during the test may change the bottom

roughness and increase the value of pi.

The bedload transport rate in the inlet was not measured directly.

However, the bottom configurations are shown from Figure 17 through

Figure 25 as recorded photographically. It is possible to analyze the

variation of bottom configurations by using these figures. Three bottom

section lines were indicated in Figure 27. According to Figure 27, the

section profiles were drawn for each condition and shown in Figure 28

through Figure 30. Hence, a comparison of similarities can be made among

these three different types of conditions. Generally speaking, the re-

sults of hydrographical variation are similar to each other for flood and

ebb flow with waves but exhibit a difference for walnut shell under the

action of waves only. Although a complete justification for this differ-

ence is not known, the existence of a small current may interact with waves

to produce a significant difference in shear stress,

Based on Newton's second law, Eagleson, Glenne and Dracup [23] de-

rived the equilibrium conditions for a stable beach. It is important

to consider this equilibrium condition if one tries to obtain reasonable

results. Figure 31 through Figure 34 were plotted for this purpose. For

a given grain size de, specific weight Ss, beach slope a, wave period T,

and water depth d, the equilibrium condition can only exist for a certain

value of wave heights If the wave height is higher than this value, the

beach slope changes. In this study, all the tests with coal, walnut shell,

















-3

-2

S-- < -- ----1 -


MSL


z-3 ---


Zero distance line


I I

2


Fig.27 Typical plan view showing profile sections















REFERENCE DISTANCE
0 1 2 3 4


Ocean
5 6 7 8 9 10 11 12


*2-

3"

54" .. -
15 ------
6"
rO


Section *1-1
-. -Coal
\ "- Walnut shell





Sand ,--.


Coal

y \


2
'3"-_-. /


- ---~ Sand --
5---------------
6" Wall
0
1"
Coal -7
2.. --- .---


'' --/_. Sand- .


.4 --
-5 -- -
6"


Section 0-0


nut shell- "-.


Section -1-1


Wal nut shell -- -
Walnut shell --7.


Fig. 28. Bottom profiles of channel and ocean after test with combined wave. and ebb flow


Channel
-3 -2


13 14


f
















Channel
7 6 5


REFERENCE DISTANCE
4 3 2 1 0 1 2 3


Ocean
4 5 6 7 8


Coal //



XWalnut shell
Sand / -
/ i
/


1 Section *1-1
2

.3" -
.4"

5"
6"


0

1 o' Section 0-0
-2
Coal """ "
.n- -W. Walnut shell -- -
Sand -- 4
5 --
U 6"
-5.


Walnut shell I-.--


a.
0


.. Section -1-1




4 --.


Fig. 29, Bottom profiles of channel and ocean after test with waves only


~-1

-..Coal


S- \- San


dj



















Channel
9 8 7


REFERENCE DISTANCE
6 5 4 3 2 1 0 1 2 3


Ocean
4 5 6 7


__ Coal-.._
Col --- ------.. .__
Walnut shell


0-r


C
0



.a_..C -o
-- --- -- -_ .. o _..
- -- --- W.II
---- IWalnut shell


Sand-


S -. Section 1-1
2"
2 \ -- -


4" ----

5"

-0


Section 0-0
2"


.4

6-- -"
6"


Coal 1
Coal _1 Section-1-1

Sand -3
------------_--- 3 -
Walnut shel I ,- -----

a.
u 5 .----
w 5
6
Fig. 30 Bottom profiles of channel and ocean after test with combined wave and flood flow



















9
-
7
6-

5

4








2 -


10-

9
9-
8
7

-


4-


1

Sinh2 kd Tnh kd )2


2.0-
Z.O--





1.0-

o.8-

.0.6-


5 6 7 8 9 I6'


d
Lo
motion function


f2() = ( kdTanh kd
2 L.


I


3 4


2 3 4 5 4


7 a 9 1o


2 3


Fig.31 Sediment


for beach








69






3/2 de g sS- sf 1.3+Sinok
Ho= T 258.7 11/2 Sf f2 )






10-

^^REAtO \NDEX




Ho (f-)



10-












o see
10 ..'
















8 9 10 11 12 13
d ( in)
Fig, 32 Incipient motion function for sand ( d -0.163 mm, Cot --20)
--'

^---



10 ^^^^^/-^'----' -







70






3/2 de g s- Sf 1.3+ Sind
H -T ( S )
258.7 p112 s 2 d






lo'
-~~ -. e

BREA<14

Ho(ft) .. .







Sse

1:0 Sec
.---






















I I I
SeC




















7 8 9 d n) 10 11 12


Fig. 33 Incipient motion function for walnut shell


( de=1.6 .mm CotoL=20)


















3/2 de- g ss- sf 1.3*Sind
258.7 )1/2 s 2 d)
f2- L-











BREA NDEX

If "


H,(ft)


















to


r --


- -


7 d (n) 10 11 12


Fig. 34 Incipient motion function for coal ( de=0.78 mm, Cot E= 20)


5Sec
0--







0.5c-- se-


Sec,--


I


-e
r








72

and sand had this consideration. The final results concerning the long-

shore sediment transport appeared satisfactory with the estimated curves.
















CHAPTER V


CONCLUSIONS


From the basic study of bedload movement and the test results, the

parameters involved in the modeling of sediment transport can be ex-

pressed as the function of Froude number, bottom shear stress, friction

factor, and wave steepness. The parameters, so selected, give better

correlation for various movement of bedload in the coastal environment.

The dimensionless bedload intensity ( was obtained according to the theory

of Einstein's bedload function 4' = f(p'). The dimensionless shear in-

tensity I, however, was determined from the combined shear stresses due

to the waves and currents. The results of longshore sediment transport

rate were favorably compared with the estimated values for each bed material

However, they were not in close agreement for the tests of walnut shell and

and sand under the combined action due to waves and ebb flow.

In this study, the values of the friction coefficient C for each bed
z
material were obtained with reasonable accuracy in the flume test. However,

if the friction coefficient C in the prototype cannot be well estimated,

the results obtained in the model test will be influenced. Since it is

always difficult to predict these values with sufficient accuracy, compu-

tation of the scale factor will have to be performed with different values

of C From these computations, the possible variation in the scale factors,
z
resulting from a wrong evaluation of the friction, can then be predicted.

For sediment transport research, most people are interested in knowing










how accurate the sediment time scales are by introducing a weir trap

in the model basin. Table 8 was calculated according to the time scale

as presented by Equation (60). The results show that the predicted

sedimentological time ran 31 per cent less than the test durations for

walnut shell and 36 per cent for coal. The scaled sedimentological time

for each bed material will affect the changes in bottom configuration.

If the considered parameters were perfect in scale, the bottom configu-

ration should result in the same scaled topography during the test of the

scaled sedimentological time. The results of the bottom configuration in

this study are similar to each other except for the wave test on walnut

shell.

Although certain difficulties remain for predicting the sedi-

mentological time scale, owing to the inadequate knowledge of flow near

rough boundaries, Equation (60) is still useful in estimating test

durations in model studies.























< o







d o d o











sO OO
x r
>m O
r< 0 C
,- 0- -

















0 Od
S 0 0 0



--6

















S(V) Ln

,A n L n
Sr d LA
*dcw LA(


______________ _____________ .1 1


(1






N

To
<




















































APPENDICES


I















APPENDIX I


DERIVATION OF BED SHEAR STRESSES DUE TO
THE COMBINED ACTION OF WAVES AND CURRENTS


A definition sketch is given in Figure 3,. The resultant bottom

velocity V(y ), due to the combined motion of waves and currents, can be

expressed under different conditions as follows (note that V and V
x y
represent components of ebb veiccities and that qx and q represent

components of flood velocities):

Ebb flow inside channel

V
V(Yo) r plu) sinct cos (I-i)

where
C *K
-Z


Ebb flow parallel to the shoreline

V V
V(y ) L plu sinot sine (I-2)

Flood flow inside channel

q-
V(Yo) = + p.lu sinot cos6 (I-3)

Flood flow pa.sra!el to the shoreline

V q
V(y ) L + pu sinct sine (1-4)

Using Equations (I-i), (1-2), (1-3), and (1-4) and applying

Equation (50), the respective bed shear stresses become:











Ebb flow inside channel


T(t) = pK2y 2
o0


V
m
(--- PlU cose sinct


- cosO sinot (1 -
m


PlUo
1 --- cosO sinat
m


PlUo
--cosO sinot)
m


Plu
(1 "-0 cosO sinat)
m


(I-5)


Ebb flow parallel to the shoreline


V V
L +
, E


T(t) = pK2Y 2
0


PY
Yo


sinO sinot


PlU
+ E -P sine sinot
VL


V V pIU
(-- + sine sinot)
L VL


V2 V
p- \2 +
SVL, VL


-L sine sinot
VL


V V
V-L + v
VL


P Uo
L sine sinat)
(L
(1-6)


Flood flow inside channel


+ plu cosO sinot
o


T(t) = pK2y 2
o


= P(-) -a
gC q
1z

8


PlUo
+ --- cose sinat
7


(- + E U cosO sinat)
9 9


V 2
= p(m-)
z

g



=pVm 2


VL 2
= (-)
z
g










q
= pq* q-


Plu
+ -- cose sinot
q


plu"
+ E cos8 sinat)
q


Flood flow parallel to the


VL q
S + Pluo


sinO sinrt


I


T(t) = PK2Y 2
O


Pluo
+ -p- sine sinat
L


PlUo
+ C Pv-- sine sinot
L


V q
VL
L


VL q
(-LV L
L


P1Uo
+ --- sine sinat)
L


PlU
+ VL
VL


sine sinat)


(1-8)


Equation (1-5) through Equation (1-8) can be integrated for one wave

period by the following considerations.

Ebb flow inside channel


Pluo
If (1 p cose) > 0,
m


T



0


If (1 C


T(t) dt =


1 2
T m*


1 plo 2
[1 + ( cose) ]
2 V
m


(I-9a)


PlU
--- cose) < 0,
V
m


T

T T m


1 Pluo
2 {1 + (5- cose)
2 V
m


shoreline


(1-7)


VL 2
= p(-)
C
z

g


= pVL 2
*


I












Plio 2
(1 5 cosa sinat) dt}
m


1 -1 m
tl = sin- ( m -)
o Plu cosa


t2 = f tI


Simplifying Equation (I-9b), it follows that


SpV 2 {1
T m


1 PIU
+ ( -- cose)
M


- (27T 4t,)


+ Pluo
+ Q( 0 cosa)
V ,


1 p 2.
T mT


cosatI (w 2tl) (Q


1 Pluo 2
{(1 + 1 ( -ose) ]
m


PlU 2
-- cose) }
m


[1 (2n 4tl)]


8 Pluo
+ ( --
a V
m


cose) cosatl}


Ebb flow parallel to the shoreline


T(t) dt = pV
T L


V2 -V 2
2 [(VL Vy)2
VL


1 1 ino 2
+ E sine)
+ ( V--_]


(I-1Oa)


where

"+" sign for



"-" sign for



V V
If 0 < ( L ) < 1
E piu sine


V V
L Y i1
plu sine -

V V
L < 1
Plu sine -



V V
or 1 < ( L --- ) < 0
piu sine
o


t2

- 2


where


(I-9b)


T =
Tf
0


T(t) dt =


T



0


(I-9c)












1 p 2 VL
T L* V


[1 (2r 4tl)] + (5
al


1 PUo 2
+ ( -- sine) ]
L


PlUo
VL


sine cosotl}


(I-lOb)


where


1 -1 VL
tl = sin ( sin
o


V -V
"+" sign for 0 < < 1
6 plu sine

V V
"-" sign for 1 < L I <
P plu sine

V V
If = 0
SPluu sine

T

T = T (t) dt = 0

0


Flood flow inside channel


qx

If -- > 1
Plu
S--cose
q


T



0


r(t) dt = pq 2


q 2
q


1 (PUo 2
+ cos)


qx

If 0 < -- < 1
PlUo
S--cose
q


T

T=

0


T(t) dt =


(I-lOc)


(I-1a)












T(t) dt = pq,2 {[()2
T q


1 Puo 2
+ ( -- cose) J
q


[1 (27 4tl)] + -
a


Pluo
P(- cose) cosoti}
q


(I-11b)


where


1 -1
tl sin
a


q
( plu cos)
0


Flood flow parallel to the shoreline


T(t) dt = + pVV 2
TT L*


VL q) 2

L


1 PIUo 2
+ ( sine) ]
L
(I-12a)


"+" sign for



"-" sign for


VL q
If 0 < L 1
Splu sine


T



0


VL q- > 1
Splu sine -


VL -'Y < -1
SPlu sine -


VL 0
or 1 < < 0
Splu sine
-t~t dt 1 2 Lo


1 VL q) 2
T(t) dt = pV 2 V L V 2
T L VL


8
[1 (2,a 4tl)] + (
CY


1 PlUo 2
+ (E --- sine) ]
L


PlUo
V -- sine) costt}
L
(I-12b)


where


1 .-1 VL
ti = sin ( )
a E plu 0 sine
O


and


"+" sign for


V q
O L V
4 Plu sine
o


T


0


T

1
T = T

0


where








83

V q
"-" sign for 1 < L < 0
Splu sin9

VL q
If = 0
VL

T

T = T(t) dt = 0 (I-12c)

0















APPENDIX II


SAMPLE CALCULATION FOR DEPTH SCALE


The sample calculation for the depth scale of coal to sand, based

on the similarity law shown in Equation (64), is presented. The given

conditions are:


Given Conditions

Specific gravity Ss

Grain sizes d
e

Dimensionless coefficients
(obtained from Figures 9 and 11)

Chezy's coefficients Cz
(obtained from Figure 12)


Sand

2.67

0.163 mm

0.0914


36.5


Coal

1.25

0.78 mm

0.0436


55.0


Then, the water depth scale between these two materials is

Water depth for coal
d water depth for sand

S025 0.0436 0.78 55 2
1.67 0.0914 0.163 36.5


= 0.787
















APPENDIX III


SAMPLE CALCULATION OF VOLUME RATE
OF BEDLOAD TRANSPORT


The sample calculation shown in this appendix is representative

of the results shown in Table 5. The calculation is according to the

equation

Ys(1 p) = qs

where

f = volume rate of bedload transport

p = porosity of bed material

Ys = specific weight of bed material

qs = weight (dry) rate of bedload transport


Example: (For the test with wave, on only sandy bed)

Conditions Explanation

s = 2.67 x 980 g/cm3 Given data

p = 0.607 Given data
-2
q = 8.505 x 10 g/sec/cm Obtained from weir trap

V = 0.0587 cm3/sec/cm Calculated from above
equation


I















APPENDIX IV


SAMPLE CALCULATION OF EINSTEIN'S BEDLOAD FUNCTION


Einstein's bedload function [1] is indirectly related to the flow

intensity through the probability distribution curve


P = -
P=1


1



- B,1' L-
o o


A,'
-t2 dt = -
e 1+ A'


where the bedload transport intensity is


= -qw 1 [
Y g(S S ) e

and the flow intensity is

(Ss S ) g de
~ =2


(IV-1)


(IV-2)





(IV-3)


A*, B,, and n are universal constants with A* = 43.5, B* = 0.143, and

no = 0.5.


Example: (For the test with wave, on a sandy bed)

Conditions Explanation

d = 0.163 mm Grain size
e
S = 2.67 Specific gravity of sand
S
Sf = 1 Specific gravity of water

g = 32.2 ft/sec2 Gravitational acceleration

V = 0.088 ft/sec Bottom shear velocity obtained
from Table 7











S= 437.23 lb/sec/ft

' = 177 x 10-5


S' = 3.60


Obtained from weir trap

Calculated from Equation (IV-2)
in this Appendix

Calculated from Equation (IV-3)
in this Appendix


The point corresponding to the calculated p' and *' is shown in

Figure 14. The best fit line passing through similarly calculated

points in specified by A* = 35.7, B* = 0.40, no = 0.5. The difference

between these values and Einstein's values is noted.




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