|
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.
|
RSS
HIDE
