Document 1948575

LECTURE NOTES FOR
TOPICS IN COASTAL
Compiled
ENGINEERING
and Edited
by
The Staff of ~oastal Engineering
for
lectures
given by
Prof. Dr. Ir. E.W. Bijker
Delft University
of Technology
Delft, "I'heNetherlands
August,
1972.
- iS. -
Chapter VI: Local Coastal Accretion
79
Introduction
79
Equation of Motion
80
Continuity Equation
81
Solution of Pelnard-Considère
83
Sandtransport Around End of Breakwater
86
Pertinent Philosop.hy
93
Non-Parallel Accretion
94
References
96
Numerical Example
97
Chapter VII: Beaches with Groins
References
Chapter VIII: Wave Forces on Piles
100
108
109
Drag Forces
109
Dynamic Forces
110
Lift Forces
112
Chapter IX: Offshore Constructions
114
Chapter X:
117
Offshore Mooring Structures
Chapter XI: Submarine Pipelines
119
List of Symbols
127
Bibliography
135
- iS. -
Chapter VI: Local Coastal Accretion
79
Introduction
79
Equation of Motion
80
Continuity Equation
81
Solution of Pelnard-Considère
83
Sandtransport Around End of Breakwater
86
Pertinent Philosop.hy
93
Non-Parallel Accretion
94
References
96
Numerical Example
97
Chapter VII: Beaches with Groins
References
Chapter VIII: Wave Forces on Piles
100
108
109
Drag Forces
109
Dynamic Forces
110
Lift Forces
112
Chapter IX: Offshore Constructions
114
Chapter X:
117
Offshore Mooring Structures
Chapter XI: Submarine Pipelines
119
List of Symbols
127
Bibliography
135
- ii~ -
Introduction
This set of lecture notes is intended to supplement
of Prof. Bijker covering Topics in Coastal Engineering.
information
in these notes will be amplified
These notes are written
in American rather than English.
list of references
literature
included at the back of this volume.
additionally,
In some cases,
in the lectures.
reader will see some words spelled differently,
A complete
the lectures
for example.
has been compiled
Some references
list of symbols.
and
are listed,
in the chapter where they are important.
also been made to compile a complete
The
An attempt has
- 12 -
depth of the channel should be greater if the channel bed is so hard
that damage has to be feared.
e. In case the bed is rocky so that serious damage has to be feared,
th is last derived encounter probability must be made extremely low,
10-8 to 10-10 for instance;if the bed is muddy, on the other hand,
the encounter probability can be rat her high, and only the steering
capabilities
count.
For very soft beds consisting of soft mud or sling mud even
negative under-keel clearances can be accepted.
In general, it is reasonable to assume that in channels in
shallow water with an average wave height, the desired water depth for
large ships is 10 to 20 percent in excess of the draft of the ship.
Most recent information:
Wicker, C.F.
Economie Channels and Manoevring Areas for Ships
Proc. A.S.C.E., Vol. 97, No. W.W.3 Aug. 1971
p.p. 443 - 453
Waugh,Jr. R.G. Water DepthsRequired
for Ship Navigation
Proc. A.S.C.E. vol. 97 No. W.W.3 Aug. 1971
p.p. 455 - 473.
Eden, E.W.
Vessel Controllability
in Restricted
Waters.
Proc. A.S.C.E. Vol. 97 No. W.W.3 Aug. 1971
p.p. 475 - 490.
- 2 -
a. The waterlevel to be taken into account is determined by the
frequency of entering of ships of a certain size. When the biggest
ships; say 250,000 OWT, enter the harbor only once every few days,
it is acceptable that ships of this size enter only at high tide.
A ferryboat, on the other hand, must be able to enter the harbor at
all time~ even during low water spring tide. A good estimate of the
required depth can be made only after all factors are considered,
including economic losses due to time ships have to wait before they
can enter.
b. The draft
of a moored shi.pis not only determined by its own
characteristics and its cargo, but also by the physical aspects of
the water such as its density,:salinitYJ and temperature. A moving
ship, moreover, is subjected to squat. Squat is the increase of the
draft with respect to S.W.L. due to the speed of the ship. This
phenomena can be explained easily with thè help of Bernouilli's
principle.
/ / ut!
/1/(/(//(/(////(//(////
c
'>
(//////
>
-
-
77/77777/777777777777777777777777777
The velocity of the water beside the ship increases due to the ship's
motion. According to Bernouillhthe
water level goes down as its
velocity increases. Squat is most pronounced, of course, in a relatively
narrow channel, but occurs also in a infinitely wide channel. Sometimes the draft increase is more at the bow that at the stern (this is
the case mostly with carriers having a large block coefficient) and
sometimes the reverse is true. Two other factors determining the
amount of squat are the depth of the fairway and the speed of the
ship. Velocities ranging from four to fifteen knots cause a squat
in the order of magnitude of 0.1 to 1.5 m.
c. Waves cause the following ship movements: The Dutch translations
are given in parentheses.
- 3 -
___
~ h__
t _
heaving (dompen)
pitching (stampen)
rolling (rollen)
swaying (verzetten)
surging (schrikken)
yawing (gieren)
wave crest
direction of wave propagation
- 34 -
In shallow water, 2 k h/sirih2k h ~ 1 yielding:
(h < L/25)
(32b)
Sxx has units of force per unit length of wave crest.
Transverse Radiation Stress Component
It now becomes necessary to examine the flow of momentum in the
Y Z plane. The Y axis is in the plane of the still water surface directèd
parallel to the wave crests. In contrast to the previous section, a unit.
thickness in the X direction will be assumed. This new radiation stress
component will be denoted by SYY' lts derivation closely follows that
for Sxx'
We consider the total flux of
crests through a plane Y
=
Y momentum
parallel to the wave
constant. lts mathematical definition
corresponds to (9) and is:
o
Ï-h
=
(p
+
p
v2)
dZ -
f
Po dZ
(33)
-h
Just as with SXX ' equations corresponding to (11) through (14) are:
SYY
S(l) + S(2) + S(3)
yy
yy
yy
=
s(l)
yy
=
1
p
v
2
dZ
(34)
(35)
-h
0
S(2)
yy
=
J
-h
(p - p ) dZ
(36)
p dZ
(37)
0
n
S(3)
yy
=
J
0
Here, the analysis becomes somewhat simpler, since for long-crested
waves, v
=
0 by identity.
Therefore, from (35) :
- 5 -
A continuously
decreasing depth due to shoaling should be taken
into account seperately.
The composition of the actual movement
is indicated in the
following figure:
design ship
at rest
desi
datum
water level
water level
beside ship
Actual
1- ---G
I
Squat
=
vertical mot ion
of deepest point
ship.
Z
h
I
I + A + C
C
bed surface
The total depth is
=
h
=
D
=
r
Z
max =
h
Z
G + I + A + C.
r + max +
total depth of channel
D
draft of the design vessel
max.squat of the design vessel at the speed allowed in the
channel
G
=
deviation of the water level from the predicted value
A
=
allowance for the bed fluctuations around the mean bed level
of the channel
-6 -
C
=
under-keel clearance that should be available to ensure a
convenient steering and propulsion of the ship
I
=
allowance for vertical motions of the ship due to wave action.
The following remarks ,can be made about the various components
of this equation.
D
r
: Although normally the summer draft in sea water can be
'taken, sometimes,when there is such a very important outflow of
fresh water that even the sea area is influenced, the draft in
fresh water should be taken.
G: This may be either gust oscillations or wind set up (wind
from sea) and wind set down (wind from land).
A: These are the fluctuations as they occur in the bed of the
channel. This is illustrated below.
Present design
bottom elevation -'-- charted bottom
elevation
---------------------
-
---------
x
configuration
Normal
Gauss
Distribution
C: In order to guarantee good steering and propulsions
capabilities of the ship this under-keel clearance should have a
certain minimum value. In the case of a stochastic process such as
this one, this value may be less during a certain percentage of the
time. The determination of this percentage is partly guess work and
can also be determined from tests with self-propelled and free
sailing and controled roodels.Either remote control or a helmsman
in the model ship may be used.
I: This movement, which is normally only of interest in the
case of wind from sea or with a swell irrespective of the wind
direction, should be considered together with A. The actual motion
is determinèd from the response curve of the ship to the wave motion.
- 65 -
role of the ripple factor, then the coefficient value, 5, will be
retained.
c.
Over most of its usable range, the computed sediment transport
increases with increasing grain diameter, D. This is especially true
when working with extremely small transports with a diameter of the
order of 0.2 mmo The vaiidity of this formula (with the given constants) seems doubtful when working with large grain diameters.
5. Verifications of the Formulas
Two example applications of the formulas are discussed here. The
first concernS a model beach shown in plan in figure 6. Figure 7 shows
beach profiles number 3 and 8 before and after testing. The distribution of wave height. water depth, and long shore current is shown for
various profiles in figures 8 and 9. The computed sediment transportation for sections 3 and 10 is compared in figure 10 to the measured sand transport at the downstream end of the model. The computations seem to agree reasonably with the measured values.
- 8 -
R(T)
=
response factor of the given point
at period T
f
HI~
n
--:t>
n(t)
Wave motion
R
T
1
Vertical ship motion
T
n (t) is the excursion of the water surface from the still water
level as a function of t.
Since it is assumed that the local (or instantaneous) wave
ordinate n (t) is distributed normally, the variance of this
excursion is
00
ff(n) . dt which is also equal to the total energy of the
o
wave spectrum.
The probability of nt between Hand
H + öH can be written
according to the normal distribution as
K
q
(K
<
n (t) <
K
+
ÖK)
=~
I-
+
ÖK
2
e
-T /2E
dT
K
In this expression the total energy equals p g E.
Similar relationships hold for the motion of the ship.
- 9 -
The total energy of the spectrum of the vertical motion, it'
of the ship is, when this total energy is again pgm, given by:
00
m
= f
f, dT
1.
0
and i is also normally distributed according to:
t
q
it
K <
<
K
+ t::. K) =
K
Here the mean value is zero, and the standard deviation
=
cr
=
~.s
The relationship between wave motion and ship motion can be
indicated by the following scheme,
Wave Motion
Ship Motion
~
rJVVVV'
f1(T)
~~I~R_e_sp_~_n~(;_:~F_a_c_t_o_r~I~--------------~3r-
1
E
/~
distribution
distribution
H
/b utl.on
°
~d'
Ibut
i
dl.strl.
l.strl.
utl.on
o
i(t)
s
For the actual computation of the depth a distinction should
be made between the situation with and without waves,
Case with no waves.
In this case the keel of the design vessel moves in a horizontal plane and only the irregularities of the channel bed have to
- 105 -
Al
Al
2x1.4xlO
2x2.8xlO
2x4.0xl0
-2
Fr
-2
0.78
n.56
6.66xlO
0.4
--
11:28
-2
1.58xl0 '
11.4
2.37xl0
h.s'
7.69xl0
r2
-2
2
Im/sec'
Y
-2
y
0.4
-2
rg
Al Y
Im/yr'.
m/sec
2.l0xlO
0.50xl0
0.75xl0
6
6
6
2 F 2
r g
2
2.66xlO
5.67xl0
8.74xl0
m/sec.yr
-2
-2
-2
8.26xlO
1.79xlO
2.76xl0
6
6
6
r2
2x4.0xl0
-2
0.5
-2
2.42xl06
29.4 xl0
-2
9.26xl0
6
table I
If the CERC Formula is used in these computations with regular
-2
waves then Al has the value 2.8 x 10 . Further, if y and Fr are
taken from solitary wave theory (0.78 and 1.25, respectively) then
t~e combination of constants Al
Y
2
'r"'
F
r
vg has the value 6.66 x 10
-2
m2/sec.
We have ignored the parameter q
in this discussion so faro
y
Indeed, no adequate theoretical relation has been yet determined
which would allow its evaluation. Swart has found values for q
y
from laboratory studies. However, these were restricted in applicability
by the fact that the wave properties were kept constant in all tests.
Another complication results when we observe that qy is also dependent
upon hl' and hence, upon our choice of separation between beach and
inshore,(see figure 3).
- 11 -
The probability
E
of hitting the bed is in this case:
=
1 - [1
- 1.2xl0 -8
which is, indeed, extremely
J
20
=
2.3 10-7,
low.
In the case of combination
of wav.es and bed undulations,
number of motions of the ship will normally
factor, since the bed irregularities
N is, in th is case, the number
be the only decisive
are assumed to be rat her long.
of oscillations
performed
during the journey of the vessel in the approach
to wave motion,
the
by the ship
channel. Analogous
the average period of the ship Qscillations
is:
~
['"f i dT
T
0
=
I® f.
i
dT
0
For a value of T of 10 sec, a length of the channel of 4000 mand
=
speed of the ship of 12 kn
N
=
4000/6
=
10
a
6 mis,
67
From the above discussion
about the case for waves the total ac-
cepted value for A is:
A
=
3.09 ok + 0.40
For 200,000 DWT ships and H
Ok
=
4.5 m, cr
movement,
is 3.09 x 0.426 + 0.40
The probability
The encounter
E
=
= 1 -
This percentage
m
~ 0.4
I 0,0225 + 0.16' = I 0.182' = 0.426
The total value of A (ship
clearance)
s
of exceedance
probability
G-
=
3 x 10
-sj
bed undulation,
1.718
=
and under-keel
4.03 Ok'
in th is case is 3 x 10
-5
.
of hitting the bed is in this case
67
=
2 x 10
-3
.
seems rat her high; from this point of view, the
- 12 -
depth of the channel should be greater if the channel bed is so hard
that damage has to be feared.
e. In case the bed is rocky so that serious damage has to be feared,
th is last derived encounter probability must be made extremely low,
10-8 to 10-10 for instance;if the bed is muddy, on the other hand,
the encounter probability can be rat her high, and only the steering
capabilities
count.
For very soft beds consisting of soft mud or sling mud even
negative under-keel clearances can be accepted.
In general, it is reasonable to assume that in channels in
shallow water with an average wave height, the desired water depth for
large ships is 10 to 20 percent in excess of the draft of the ship.
Most recent information:
Wicker, C.F.
Economie Channels and Manoevring Areas for Ships
Proc. A.S.C.E., Vol. 97, No. W.W.3 Aug. 1971
p.p. 443 - 453
Waugh,Jr. R.G. Water DepthsRequired
for Ship Navigation
Proc. A.S.C.E. vol. 97 No. W.W.3 Aug. 1971
p.p. 455 - 473.
Eden, E.W.
Vessel Controllability
in Restricted
Waters.
Proc. A.S.C.E. Vol. 97 No. W.W.3 Aug. 1971
p.p. 475 - 490.
- 13
The width of the channel can be determined in comparable way,
by considering the motion of the ship in the horizontal plane also
as a stochastic variable. By this method the chance that a certain
excursion from the idea_lcourse line will occur , and therefore, the
chance that a ship hits the bank of the channel or another ship can
be determined. In this way, we determine the width of the channel
in which the risk of collision is brought down to an acceptable
value.
There are a few principle differences with the case of the
vertical motion of the ship.
i. The probability distribution of the deviations of the theoretical (ideal) course line cannot be determined as easily since the
period of the movement of the ship around the theoretical course line
is normally long compared with the sailing time of the ship. The
deviations from this course line are, therefore, not stochastic, independent variables.
ii. As soon as two ships are so near to each other that a chance
on collision exists, the two movements are no longer independent.
Tests are necessary for investigation of this problem. Three possibilities are available:
a. From observations in the prototype the probability distribution
of the deviation from the theoretical course line d:.an be determined.
With this distribution the probability of a certain deviation from
the theoretical course line can. be determined.
With this distribution the probability of a certain deviation can be
computed. In this solution the mutual influence of the ships is not
yet taken into
consideration. It is, unfortunately, difficult to
obtain sufficient data under extreme circumstances.
b. With the aid of a model study it is more easily possible to obtain
sufficient data under extreme circumstances.
This model can be an hydraulic one with real ships on a scale of
about 1 : 25 to 1 : 50. In this case the mate and pilot will be normally in the model ship or the model ship will be steered with an
llt -
automatic pilot. The great advantage of this method is the very true
way in which bank influences and the mutual influence of the ships on
each other can be determined.
~. Another possibility is the use of a steering simulator. In such a
simulator the movement of the ship is made visible by solving the
dynamic equations of this motion using a computer. All required circumstances of wind, waves and current can be introduced into the equations. The pilot has all normal navigational aids at his disposal
so that a very realistic reproduction of the movement is obtained. The
great advantage of this method is the much greater number of trials
that can be made at acceptable costs. The reproduction of the mutual
influence of the ships and the influence of the banks of the channel
may still cause difficulties.
s:.Q)
Directional stability
critical
2.0
.--f
.--f
'.-1
+-'
en
:~~.
";'~
~
'.-1
ti::m
s:.
'"Ó
1.5
+-'
unacceptable
coïrib~nat~ons
Q)
'"Ó
s:.
Q)
___.._Lateral forces and
moments critical
..
:.
I
Acceptabie
combinations
~,:":;,,,: Bottom clearance
_
critical
+-'
m
6 ••
,
' ••
!.:..: : 'o! I.
;3:
1.0
2
3
4
0",
5
channel bottom width
ship beam
••
1·
.
6
- 15 -
The widthof
hydraulic
conditions.
importance
number
the approach
channel is also determined
largely by the
Of courses a channel with a cross current of
must be wider than a channel in still water. Alsos the
of ships that are expected
to sail at the same moment in the
channel will determine the width. It is difficult
to give fixed data.
As a general rule one can state that the path width required
by a ship
is about 1.8 times the beam of the vessel. Between two vessels meeting
each other a distance of about one beam
should be kept between
two paths. In a channel with banks almost up to the waterlines
distance
of 1.5 beams should be kept between
State of Knowledge
and Related Phenomena;
Department
of Factors Effecting
for Navigation;
of the Army: Corps of Engineers.
same width as the approach
itself should, in principles
channel
in a long approach
inevitable,
channel a meeting
of the breakwater
damage to ships and possibly
than the grounding
of two ships may be
a width slightly
to navigation
of a vessel in the approach channel.
blocking
itself. On
ends will involve greater
give greater hinderance
the ship from completely
have the
just in front of the entrance.
whereas this may be avoided in the entrance
the other hand, touching
prevent
Evaluation
Tidal Hydraulics
Chapter X, Design of Channels
The width of the harbor entrance
However,
a
the side of the bottom
of the channel and the path of the ship (see: C.F. Wicker:
of Present
In order to
the channel of the entrance
greater than the length of the ship may be used.
When a ship sails into a harbor and there is a current crossing
approach
the
channel,
sketch below.
the ship will follow a course as indicated
the
on the
- 16
1
I
I(
I
~
I
I
,
I
~
I
~
~
~
I
\,
\
Due to the fact that during the passage through the entrance the bow
will be in still water while the stern will be in the current, a moment
will be acting on the ship, forcing it to turn. Sufficient width must
be available inside the entrance.
Another general rule is that the approach line of the bigger ships
should be as straight as possible.
When a ship has passed the harbor entrance, it needs a certain distance
in which to stop. If there is some wave motion and current in front of
the harbor, the minimum velocity with which the ship can enter will be
in the order of magnitude of 3 to 6 KT. In a harbor, it is not possible
to give full astern, since the ship will then swing to starboard (when
the normal revolution direct ion of the propeller is clockwise when
looking forward).
It is, therefore, necessary that tugboats assist the ship in keeping
the proper course.
The following graph shows some stopping distances for tankers.
- 17 - .
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- 18 -
The usual procedure
is that the ship slows down, while the propeller
turns forward dead slow or is stopped. Reverse power is not allowable
since the ship becomes uncontrollable
in this case. When the
speed is brought down to 3 knots it is assumed that assistance
tug boats is obtained.
of
This will say that tug boats have connected to the
ship and have manoeuvered
themselves
into the position
to give as-
sistance. At this moment reverse power can be applied to the ship's
screws.
The valves above are normal stopping distances
keeping a straight
course. For example, when a 130,000 ton oil carrier'sailing
at four-
teen knots has to make an emergency
full
stop, giving continuous
astern, the stopping length is only 3 or 4 km (see the figure below.)
In this case, however, the direction
24 u
22 n
E-<
:J:
20n
~~
lH
0
180
tr>!
Q
..c: 120 ;+'
.-i-:
c
'H
,_
100
I ...~
'U
'--1
1.:1
.t:/
N
80
(/)
~
Q)
60
..!<:
~
40
0
IJ
I
/
riJ
a
o .2
/
,
.4 .6
./
/
~
VSi
'" b'
-
""erG.
" J(
1/ Q
I!J
E-<
20 1[,
/
;y
Q,)
'H
\
rIJ
C(
0
..,
/
"I
ei
140
is uncontrolled.
f-,D
~
(/)
-e 160
~
(/)
;j
;;,
(heading)
~
V
V
'j(~
t.>
/'
/'
•8 1.0 .1.2 1.4 1.6
La
.202 .2 2.4 2.6 2.8
Stopping distance in Nautical Miles
under extreme emergency conditions
(full power astern)
-
lS-
Some general ratios for ships are:
f or
· DWT
norma I shaps
BRT ~ 1.5
.
DWT
2
for very Iarge crude canr-i.er-s
BRT::=
for all ships TotalDWT
displacement ~ 1. 3 to 1.4
The ratio BRT varies from 1.7 for freighters to 1.3 for VLCC s.
NRT
Deadweight tonnage (DWT): the vesseï.!s
Ufting capacity or the number
of tons of 2240 lb. (= 1016 kg) that a vessel will lift when loaded
l.nsalt water to her summer freeboard marks.
Deadweight includes: crew, passengers, luggage, provisions, fresh
water, furniture, coal in bunkers,fûel'::oilin tanks and so on.
~oss
register tonnage (BRT) of a vessel consists of its total mea-
sured cubic contents expressed in units of 100 cu.ft. or 2.83 cU.m.
Net register tonnage (NRT) of a vessel is the carrying capacity
arrived at by measuring the cubic contents of the space intended for
revenue earning. One hundred cubic feet is the standard spaqe taken
as the accomodation for one ton of goods.
Disp1acement:
(1 ton
=
the number of tons of water displaced by a vessel afloat
2204 lb). The sum of light weight and dead weight is equal to
the displacement.
In accordance with Archimedes' principle, displacement and weight are
equivalent quantities for floating bodies. The displacement of a vessel i'S" the weight of water displaced at a given draft, and also the
weight of the vessel and its contents.
- 20 -
Chapter 11
Historical Development of Longshore Current Formulas
Amphibious landing during World War 11 were handicapped by currents
parallel to the coastline.
These longshore currents were generated by obliquely approaching breaking
waves. The war effort stimulated the start of investigations into this
phenomona. Later~ it was realized that this current is also of importance
to the transport of sand along the coast.
In this section, we attempt to examine some of the development
history of longshore current formulas. One of the more recent ones
will be developed in detail in later sections.
Both Galvin and Thornton have written good reviews of longshore
current formulas. Galvin uses a practical approach, while Thornton
takes a more theoretical point of view. The table on the following
page gives a summary of the most common formulas. No claim on completeness
is made. Most investigators used at least one of the following concepts
in their formula:
1. Emperical correlations~
2. Energy considerations,
3. Conservation of mass,
4. Conservation of momentum.
There are now discussed separately.
1. Emperical Correlations
The emperical formulas may be derived in either of two ways:
a. Develop a formula based, to some degree~ on methods 2, 3, or 4,
with undetermined coefficients. These coefficients are then evaluated
using available data. The Inman/Quinn~ Brebner/Kamphuis, and Galvin/
Eagleson formulas were derived in this way. These emperical formulas
except for that of Gal~in/Eagleson
are not dimensionally correct,
i.e. the coefficient are not dimensionless; this is not necessarily'
a practical disadvantage.
b. Linear statistical corre1ation of all available emperical data can
yield an equation; Harrison used this approach.
There are varying opinions about the elegance of these correlation
methods. By accepting these methods, one is accepting addition of
variables in place of multiplication which is more correct theoretically.
-21
--,
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- 22
Putting a wave period T
a wave height H
=
=
10 sec, a beach slope m
=
50, and
2 m into the Harrison equation yields a longshore
current velocity of 0.78 mis when the wave crests approach parallel
to the beach. This result is, of course, queer even though the input
data is realistic. Harrison himself noticed this problem and warned
that his formula may only be used when the angle of incidence of the
waves is between 20 and 150. Thus, in the above example, we are in
error by using his formula outside its desired range.
2. Energy Considerations
The authors of these formulas assumed that a percentage of the
incident wave energy is used to generate and maintain the longshore
current. Galvin comments that there is no theoretical justification
for the assumption that the ratio of current energy flux to incident
energy flux should be constant. He calculated this ratio from experimental data. He found that the ratio
s small, often less than 0.01
and usually less than p.l, and certainly not constant.
3. Conservation of Mass
The conservation of mass approach is based upon the idea that the
mass of a solitary wave swept over a bar is distributed in some way
to the longshore current. This title "conservation of mass" is a bit
misleading. It suggests that the other approach methods ignore
continuity. This is not true; all of the equations shown in the table
satisfy the continuity equation.
Bruun presents two methods: a rip-current theory and a continuity
theory based upon the Chézy Equation. His rip-current theory is based
upon the assumption that the entire mass flux of the solitary wave
is concentrated in the rip-current. These rips are spaced 300 m to
500 mapart
along the shore. Bruun's continuity theory, on the other
hand,is based upon an assumption of uniformly distributed return flows.
Here again, he assumes that the entire mass flux of a solitary wave
comes over a bar. This raises the water level uniformly between the bar
and shore over a longshore distance equal to the distance between
wave crests measured along the beach. Making a further very arbitrary
assumption that the other water surfaces remain undisturbed, yields
a value for longshore current when the Chézy Equation is applied.
- 23 -
The Inman/Bagnold approach differs in that they consider that only
the longshore component of the wave motion contributes to the current,
with the result that their predictions are much smaller than those of
Bruun.
Galvin/Eagleson emperically correlate the mass flux of the longshore
current with a fictious mass flux related to the wave. Since it turns
out that both fluxes are proportional to the square of the breaker height,
an equation results in which the longshore velocity is independent of
the breaker height.
The main obsection to all conservation of mass approaches is that
the equation of motion is completely ignored. One can say that only
a portion of the available information has been used. It is relatively
easy to postulate a flow system satisfying continuity which
cannot
really exist in practice.
4. Conservation of Momentum
Methods using conservation of momentum seem to be the most reliable.
Several are compared here. One of the more important is derived in
detail in the following two chapters. Both Bakker and Battjes have
published comparisons of various formulas using momentum conservation.
One will find in practice that the formulas of Thornton, LonguetHiggins, and of Bakker vary very little in result. The differences
among them are caused by a subjective preference for some second-order
terms, simplifications, or different bottom friction formulas.
Battjes shows that the numerical value of the friction factor, f, is
of the same order in all three formulas. Other formulas belonging
to this group, with the exception of Bowen's, are more or less out of
date. Bowen, however, does not specify his friction factor.
Putman, Munk and Traylorused solitary wave theory in their formula~.
They used a bottom friction term proportional to v2, but did not include
the effects of waves on the bottom friction. It soon became obvious that
their friction coefficient was much larger than the normal Darcy-\oleisbach
coefficient. Inman/Quinn proposed to take the friction factor as:
f
=
0.384
vlo5
•
Unfortunately, this did not solve the problem and violated the dimensions
as weIl.
- 24 -
Eagleson did better by taking the friction proportional to V2/sin ,.
This is better, but still a bit questionable •• is the angle of wave
incidence.
After Bowen applied radiation stress and Jonsson, Bijker, and
Kajiura explained bottom friction in separate papers, the formulas
by Thornton, Longuet-Higgins, and Bakker ware developed independent
of one another.
Radiation Stress is explained in the following chaper. After
that, it is applied to this problem of current prediction in chapter IV.
- 25 -
Chapter
111 Radiation
Radiation
stress is a term applied to a pressure
of the hydrostatic
This pressure
Stress
pressure
is revealed
force caused by the presence
only when second-order
to the square of the wave amplitude)
considered,
however,
remains
height differences.
derivation
The discussion
Higgins and Stewart
wave.
boundary
We shall use the results
the longshore
as a
results
from wave
of this radiation
stress
current caused by waves.
which follows is adapted
(1964) published
observed
or as a force difference
set-up~ This force difference
to determine
The wave profile
stress is most commonly
force acting on a wave reflecting
of waves.
terms (proportional
are considered.
a simple sinusoidal
The effect of radiation
causing a water-level
force in excess
trom that of Longuet-
in "Deep Sea Research",
Vol. 11
p. 529-562.
In all of the following
discussion,
these conventions
will be
observed:
a. The origin of coordinates
with the X axis directed
.,cl
wi11 be placed at the still water surface
positive
to the right,
in the direction
of
wave propagation. The positive Z axis extends upward trom the still
water surface.
b. The problem will be considered to be two-dimensional; a width of
1 meter perpendicular to the plane of the drawings will be assumed
(Y direction, initially)
c. The water density, p, is constant.
One should also remember that according to Newton's Law, a force
is equivalent to a rate of change of momentum. A pressure, or stress,
is equivalent to a flux, or flow of momentum. Tqis is also rate of
change of momentum per unit area.
Momentum Flux in Still Water
If we first consider a water body of uniform depth (h) having no
wave action, then the pressure at any point in the fluid is given by
p
o
= -
p g
Z
(1)
where g = gravitational acceleration and p is the density.
This is independent of x and is the flux of horizontal momentum
across a vertical plane (x = constant) per unit of vertical distance,
since a width of one unit is assumed.
-s,
26 -
The total flux across the section is obtained by integrating
o
o
=f
Po
(1)
Po dZ
f
=
-h
(2)
p g Z dZ
-h
This is, too, independent of x and, therefore, there may be no net
change in momentum as we progress trom one plane x
x
=
x
o
=
x
o
to another
+ dx.
This constant total flux (or force) P
is rea11y the hydrostatic force
o
which would be present on a rigid vertical wall extending to the depth
Z
=
-ho This hydrostatic
component, P , will latèr be subtracted from
o
the total computed flux with waves to reveal that component which
results trom the waves. This resu1tant will then yield the Radiation
Stress.
One will diseover, later, that this radiation stress does not
have the units of force per unit area. This is because a true stress
(pressure) will be integrated over a constant depth h, giving a force
per unit. length. Since the depth remains constant at each point wh ere
we shall examine the radiation
stress, then transformations
lègitimate
for true stresses can still be used.
Momentum flux in Waves
Consider a sinusoidal wave in water of finite constant depth h.
From short wave theory we find:
n
=
u
aw
= --,--:-~-:--~
sinh (k
w
=
(3 )
a cos (kx.- wt)
h)
aw
sinh (k h)
eosh k(Z + h) eos (kx - wt)
(4)
sinh k(Z + h) sin (kx - wt)
(5)
where:
wave profile
k
=
=
=
w
=
eireular frequeney
x
=
position along X axis
t
=
time
Z
=
distance (+ up) from water surfaee
n
a
amplitude of the surfaee wave
wave number
=
2n/wave length
=
2n/wave period
(6 )
- 27 -
h
u
w
=
=
=
water depth
instantaneous particle horizontal velocity
instantaneous particle vertical velocity
(see figure 1)
Z
=
n
u
w
t~
pu
2
-h
FIGURE 1 Definition Stretch
A rather general expression for the flux of horizontal momentum
across a unit area of the vertical plane in the fluid is:
(7)
The second term,
p
u 2 , represents a bodily transfer at horizontal
momentum as follows:
in a time dt the volume of water passing an element of the vertical surface is
u dZ 1 dt
its mass is: pul
dZ dt
the horizontal momentum then (mass • velècity) is
u . 1 . dZ • dt • u
Dividing by the area (dZ • 1) and time (dt) to get flux per unit
p
•
area yields the desired term.
We might note here that u may be positive or negative and may
have even a zero time average; while u2 is always positive and will
also have a positive time average. This fact will be of importance in
later discussions.
Fluid crossing this plane also, in general, will have a vertical
velocity component w.
A product p w could be used to represent the transfer of vertical
momentum across the plane x
=
constant. This term appears as a shear
stress and is commonly called a "Reynold Shear Stress" in theory of
turbulence. These Reynold ShearStre~will
not be considered further
here~however.
Returning to our more immediate problem~ we integrate (7) over
the depth to get the total flux of horizontal momentum across the plane
x
=
constant. Formally:
PI
J
=
+ p u2) dZ
(p
(8)
-h
Note here that the upper limit of integration is the actual water
surface, and not zero is often used in first-order theory.
Definition of Radiation Stress
The principal radiation stress component, denoted Sxx~ may now
be defined as the mean value of P1 with respect to time minus the
mean value of P with respect to time.
o
n
P
=
0
Sxx = P1
I
0
(p + p u2) dZ -
I
Po dZ
(9)
-h
-h
The bar is used to denote a time average. In the first integral,
we must be sure to take the time average afte!'-cómplèting-the
integration, since the limit n of the integration is, itself, a
function of time.
In the second integral, the bar may be omitted, since that
integral is constant, see equation (2).
(9)
now becomes:
o
=
J-h
(p
+
2
p u )
dZ
I
Po dZ
-h
(10)
There remains the problem of evaluating SXX.
We can now state a definition of radiation stress in more
sophisticated terminology as: The radiation stress is the contribution
of the waves to the time average of the vertically integrated
horizontal transfer of horizontal momentum.
;_29 -
Evaluation of Radiation Stress
As an aid to evaluation of equation (10), it is separated into
three parts as follows:
(2)
(3)
(1)
S
SXX = xx + Sxx + Sxx
(11)
0,) -S
xx =
(12)
J
u
p
2
dZ
-h
(2)
S
xx =
0
f-h (p - p )
0
(13)
dZ
(3)
S
=
xx
J
(14)
dZ
P
0
It may be verified that (11) through (14) are equivalent to (9) by
substitution. These terms will now be considered individually.
Equation (12) will be split again as:
(1)
0
Sxx =
f
-h
n
p
u
2
dZ +
I
p
u
2
dZ
(15)
0
the integrand (p u2) in both terms of (15) is already of second
order
proportional to a2; see equation (4). Since n is also a function
of a, then the second integral in (15) will yield only a term of third
order.
Since only first and second order terms are considered here, then (15)
may be approximated (to second order) by:
(1)
S xx
0
=f
P u2
dZ
(16 )
-h
Now that both limits of integration are constant, the time average may
be moved inside the integral
- 30 -
(1)
S
xx
o
=
f-h p
u2
dZ
(17)
This is a Reynolds Normal Stress integrated trom the bottom to the
still water level. It is, obviously, generally positive.
Since both limits of integration in equation (13) are constant,
we may again apply our previous technique. Equation (13) becomes:
(2)
S
xx
o
= f (p -
p ) dZ
(18)
o
-h
Po is excluded from the time average since it constant.
(2)
Sxx is caused by changes in the mean pressure p when compared
with the hydrostatic pressure p found in the absence of waves. p can
o
be evaluated by completing a second order analysis, but
p
may be more
easily evaluated indirectly from vertical momentum considerations as
follows:
Using arguments similar to those already used, the mean flux of
vertical momentum across a horizontal plane must be aqual to the weight
of water above that plane. The average water lev.elelevation is Z
= o.
Therefore:
p
+ p w2
= -
p g Z
=
Po
(19 )
or
= -
P - Po
p w
2
(20)
(20) substituted into (18)yields:
0
(2)
S
xx
=
f-
p
w
2
dZ
(21)
-h
this is, obviously, less than zero in general.
The third radiation stress term, equation (14), is the pressure
integrated from the still water level to the wave profile with this
integration averaged over time.
- 31 -
This integration leads, strictly, to difficulties when n is below
Z
=
asince, then, p is undefined in the range n < Z ~ a. This may
be most easily overcome, according to Longuet-Higgins, by "extending
the velocity field upward to the mean level" Z
=
a. Near the free
surface, p is very nearly equal to the hydrostatic pressure measured
from the instantaneous surface n. The pressure fluctuates in phase
with the surface elevation.
p
=
pg (n - Z)
(22)
Substitution in (14) yields:
(3 )
S
=
xx
I
pg (n - Z) dZ
(23)
o
Evaluating only the integral (neglecting the time average) and noting
that n is independent of Z (eqn (3)
I
pg (n - Z) dZ
=
):
[I I
n dZ -
pg
o
Z dZ }
J: ]
o
Taking the time average yields:
(3)
S
(24)
xx
This is generally greater than zero.
Since n
=
a cos (kx - wt) then
n
2
n
1
= 2a
2
;
(1
lr
f
o
2
cos x dx
= 1:.)
2
- 32 -
and
(3)
sxx
(25)
Interpretation of S
xx
(1)
Adding S
xx
xx yields, using (17) and (21:):
0
(1)
(2)
S
+ S
xx
(2)
and S
xx
=
0
u
p
J
2
dZ
-h
J
p
w2 dZ
-h
0
=
2
(u - w2) dZ
p
J
(26)
-h
From short small amplitude wave theory (equations 4 and 5)=
2"
u
>
2
w • Therefore, (26) is generally > O.
Before attempting further solution of (26) directly, we note
that for irrotational, incompressible flow:
a
a Z
-
=
2
2
(u - w )
=
2 (u aw
=
2
ax
a
ax
2 (u au _ w aw)
az
az
+ w au)
(u w)
ax
=
0
Therefore, (u2 - w2) is independent of Z, even though u and w
are both, themselves, functions of Z.
(26) becomes:
o
=
p (u2
- w2)
f
dZ
-h
(27)
-
33
-
Formal substitution of (4) and (5) in (27) gives
(2)
(1)
2 2
w
= ph _ a__;,__[ cosh2 k(Z+h )cos2Ckx-wt)-sinh2k(Z+h)sin2(kx-wt)]
xx + Sxx
sinh2 kh
S
Since on1y the trignometric functions depend upon time:
=
h a2 w2 [COSh2k(Z+h) cos2(kx-wt) _ sinh2k(Z+h) sin2(kx-wt)1
sinh2kh
p
'TT
Since
1
11
2
cos x dx =
f
1
2:
0
then this becomes:
=~
2
2
h a w
2 sinh2 kh
[cosh2 k(Z + h) - sinh2 k(Z + h)]
p
. h2 x
with cosh2 x - Sln
2
=~pha
w
2 sinh2 kh
=
1, this becomes:
2
for small amplitude waves, w 2
=
g k tanh k h,
(28)
yielding:
(1)
(2)
S
+ S
xx
xx
2
g a k h
sinh 2 k h
P
=
(29)
If we remember that the total energy density of the waves, E, may be
defined as:
E
= ~2
p
g a2 then we can determine
Sxx from (11), (25) and (29)
SXX
=
using (30):
2kh
1
E(sinh 2kh + 2)~0
In deep water, 2 k h/sinh 2 k h
(30)
(31)
+
0 yie1ding:
(h > L/2)
(32a)
- 34 -
In shallow water, 2 k h/sirih2k h ~ 1 yielding:
(h < L/25)
(32b)
Sxx has units of force per unit length of wave crest.
Transverse Radiation Stress Component
It now becomes necessary to examine the flow of momentum in the
Y Z plane. The Y axis is in the plane of the still water surface directèd
parallel to the wave crests. In contrast to the previous section, a unit.
thickness in the X direction will be assumed. This new radiation stress
component will be denoted by SYY' lts derivation closely follows that
for Sxx'
We consider the total flux of
crests through a plane Y
=
Y momentum
parallel to the wave
constant. lts mathematical definition
corresponds to (9) and is:
o
Ï-h
=
(p
+
p
v2)
dZ -
f
Po dZ
(33)
-h
Just as with SXX ' equations corresponding to (11) through (14) are:
SYY
S(l) + S(2) + S(3)
yy
yy
yy
=
s(l)
yy
=
1
p
v
2
dZ
(34)
(35)
-h
0
S(2)
yy
=
J
-h
(p - p ) dZ
(36)
p dZ
(37)
0
n
S(3)
yy
=
J
0
Here, the analysis becomes somewhat simpler, since for long-crested
waves, v
=
0 by identity.
Therefore, from (35) :
- 35 -
S(l)
yy
= 0
Also, comparing
(36) and (37) with (13) and (14):
0
S(2)
yy
=
S(3)
yy
1
= S(3)
xx = 4
S(2)
xx
=
f-h
-
p
w
p
g a
2
dZ
(38)
2
(39)
using (21) and (25) respectively.
Substitution of w from (5) in (38) yields:
=
S(2)
yy
-
p
a
2
w
o
2
f-h
2
sinh kh
2
cash k(Z + h) dZ
(40)
2
using the time average of cos , we get :
S(2)
yy
=
-
p
a
2
2
w
0
2
f-h cash
. 2k h
s1.nh
2
k (Z + h) dZ
the integral yields:
r
kh
.
2
cash k (Z + h) dZ
=
1
k
-h
cash
2
q dq
0
with q
=
f
1
= k(Z + h) .
[ sin~ 2 q
k
[
1
= k
Ik\
0
%
+
~h ]
sinh 2 kh
4
c]
(41) now becomes:
S(2)
yy
=
=
p
2 2
a w
{ sinh 2 kh + kh }
2 k
. 2
4
2
s1.nh kh
2
2
a2 w2 sinh 2 kh
p a w
h
p
8 k sinh2 kh
4 sinh2kh
(41)
- 36 -
or , using (28) :
=
-p g a
2
2
gak
h
2 sinh 2 kh
p
4
(42)
This does not look like much, but when we add S(3) to get Syy
(remembering
that S(l)
yy
=
0):
yy
= S(2) + S(3)
yy
yy
2
=
p g a
4
=
2
- P g a
= -
p g a
2
kh
sinh 2 kh
2
+ p g a
2
4
kh
sinh 2 kh
2
kh
sinh 2 kh
E
(43)
using (30).
In deep water, kh/sinh 2 kh ~ 0 yielding:
(h > L/2)
(44)
In shallow water, kh~sinh
Syy
= 21 E
(h
2 kh ~ 1/2 and:
L/25)
<
(45)
Syy has units of force per unit of length of wave orthogonal.
Shear Stresses
Finally,
for completeness,
transfer
of x - momentum
momentum
manifests
across the plane y
=
the possibility
constant.
of the
Since this
itself as a shear stress, the pressure at the point
does not contribute.
direction.
we must investigate
By definition,
This results
in
pressures
an equation
act only in a normal
sornewhat simpleI' than (9).
- 37 -
=
["
p u v dZ
(46)
-h
Since our waves are still long-crested, v - O.
Therefore, quite simply, (46) becomes:
(47)
Since the shear stress SXY is zero, then, from strength of
materials, we can conclude that SXX and Syy must be principal
stresses.
Transformations of Radiation Stresses
In the preceeding sections we have found the horizontal stresses
acting on vertical planes through our point oriented parallel and
perpendicular to the wave crests and extending from the water surface
to the bottom. These compoeents have been found to be principal
stresses.and may be expressed and transformed using the methods of
strength of materiais. Two common methods use tensors or the Mohr
Circle.
In tensor form, the total stress S may be expressed as:
h
S
=
[:xx
sJ
[ sinh
2 k 2 kh
=
+
1
2"
0
E
0
k h
sinh 2
J
(48)
The transformation via the Mohr's Circle will be illustrated
and used in the following chapter on determination of currents along
a coast.
- 38 -
Chapter IV
Determination of Currents along a Coast:
Computatdon of current velocities parallel to a coastline is
required in order to properly estimate the sand transportation along
a coast. Four force components, together, determine the magnitude of
the resultant velocity. These forces are:
1. Wave Forces resulting from the radiation stress
2. Tidal Forces
3. Friction Forces on the bottom, always acting to reduce the
current velocity
4. Turbulent Forces resulting from differences in velocity
between adjacent streamlines.
In the case of a fully developed long-shore current, these four
forces will determine a state of dynamic equilibrium with the current
velocity being constant.
The determination of each of the force components referred to
above will be explained separately in the following sections. The
axis and sign conventions used in the following wil1 be the same as
those used in the earlier derivation of radiation stress.
1. Wave Forces
The wave forces result from a shear component of the radiation
stress. The work of Longuet-Higgins presented earlier has been adapted
by Bowen (The Generation of Longshore Currents on a Plane Beach;
Journalof
Marine Research, V.27, 1969, p.206-216) for our application
(a sloping bottom, shallow water, and breaking waves).
A plan view of a coastal zone is shown in Figure 1.
- 39 -
x
coastline
~Q)
-'-~§'---~N
- -~~-- .
~::E
.
.- y_---- __......J
~."I------.--.---_."2--._--
Figure 1
Coastal plan view
vertical axes Zand
showing axis notation and angles. The positive
z are directed positive up from an origin at the
still water surface elevation. The depth contours are assumed to be
parallel to the coastline. The X Y Z axis system is the same as was
used in the Radiation Stress derivation. The y axis is parallel to the
coast with the x y plane in the still water surface.
Using Mohr's Circle, figure 2, the radiation stress components
acting in the x and y directions (perpendicular and parallel to the
coast) are:
S
xx
=
S
yy
=
S
=
xy
+
sin 2
4>
cos 2
4>
( 1)
cos 2
4>
(2)
- 40 -
=
Note S
xy
=
(Sxx - Syy) sin ~
S
yx
cos ~
(3)
in magnitude
~ is the angle between the wave crests and the shore.
Shear
Stress
S
Normal
stress
Figure 2
MOHR'S CIRCLE
- 41 -
Except in zones where the longshore current is not fully developed
the shear stress component S yields the only force of importance here.
xy
Remembering that we are now in shallow water, making the proper
substitutions from the radiation stress derivation in equation (3)
yields:
S = E
xy
=
_
1)
2
•
Sl.n
4> cos
4>
(4)
E sin 4>cos 4>
1
=8
p
where E
(1
2
2
g H , with H equal to the height of the waves at the
point in question (in the breaker zone).
As noted in the radiation stress derivation, S
has units of force per
xy
unit length (length of coastline, here). In order to be consistent with
the units of relationships developed later, this must be transformed
into a force per unit horizontal area. This can most easily be done by
considering the derivative of equation (4) with respect to x. In order
to carry this out, the following modifications are made to expose the
functions of x:
Since 4>is a function of depth, h, and hence, x, we substitute:
cos-4>=:: cos 4>b
r
sin
4>
=
with c
c
o ~
4> <
4>br«
1 rad.
(Sa)
sin
=
wave speed in shallow water
= rg-ï1
sin
sin '"
'f'br
(Sb)
Further, we assume a linear relationship between water depth and wave
height within the breaker zone.
H
=
2a
=
A h
(Sc)
- 42 -
8ubstituting (5) and the definition of E in (4) yields:
8
= -1
xy
8
p gA
2
h
2
~
v~~r
sin 4>brcos 4>br
(6)
having one variable, h •
Differentiating using the chain rule
F
r-
=
d
dx
F
r
=
5 p g A2 h3/2
1/2
16
hbr
(8 )
xx
sin 4>br cos 4>br tan ah
dh
where tan ah is the bottom slope dx
(7)
at depth h.
This has the desired units.
2. Tidal Forces
From the theory of long waves associated with tides, we find that
the tide force per unit of volume may be given by:
=-pgay-
a h'
(8)
where
V is a unit water volume
is the tide force acting on the unit volume
a h'
is the slope of the surface of the tide wave.
Cl Y
(measured on a profile parallel to the coast).
h' comes from:
h'
= Z cos
where
(~ t - K y)
~ and
K
(9)
are associated with the tide period and tide
wave length respectively.
- 43 -
Clh'
"
K Z
Cly =
sin (n t - K y) substituted in (8) yields:
,
FT
V =
- p
g K Z sin
(0
t - K y)
(10)
In order to compare this to the radiation stress force component,
the units of the equation must be modified to give force per unit of
horizontal area. This can most directly be accomplished by taking the
volume Y as a unit of area, Ab' times the local water depth h.
Equation (10) then becomes:
(11)
=
Multiplying through by h yields:
F '
-AbT =
FT
=-
P g h K Z sin (n t - K y)
(12)
where FT now has the desired units (force p.erunit area).
3. Friction Forces
Bijker has derived an expression for the bottom friction force
in a zone with waves. This derivation is valid provided that the breaker
angle ~br
=
T'
is less than about 200•
ub
T
C
[ 0.75
1.13
+ 0.45 (~ V )
]
where
T'
is the bottom shear stress (total)
T
is the shear stress caused by a current alone.
c
=
y2
p
g
C2
y is the stream velocity
C is the Chézy friction coefficient
(13)
- 44 -
~ =
K
0.45
C
K
rg
is the von Karmán Constant
~ =
0.0575 C
=
0.4, yielding
(in metric units)
ub is the water velocity component along the bottom caused by the waves.
The computation of ~
in the breaker zone is nearly impossible
theoretically. However, in order to determine a solution to our problem
it will be assumed that a sinusoidal wave still exists.
From short wave theory:
w H cosh (kh)
2 sinh (kh)
(14)
where H is the wave height.
=
with k
21T
L
w
=
21T
T
\
and the
approximations for shallow water, (14) becomes
21T
T H
= __;;----,.--
c H
2 • 2~ h
= 2 h
(15)
where c = L = l""g1i' for shallow water.
T
finally
û
-' H
b-2'
rt:
h
(16)
Substituting all of this into equation (13) yields i
[0.75
+ 0.45
(~
V
H
2
r;;"
{~
)
1.13
]
(17)
(Ej1)
91
[
(~
s/
81°1
d S
(
(Ll) pUE (L) mOJJ) ~u1AEaI auoz
Ja~EaJq a4+ u14+1M anJ+ ÁIIEJaUa~ s1 s141 °saoJoJ ~u1u1EmaJ Ja4+0
OM+
a4+ 0+ aA1+EIaJ IIEms s1 aOJoJ ap1+ a4+ 'saoEId ÁUEm uI
°Á+poIaA a4+ 0+
u01+1soddo u1 S+OE sÁEMIE 'ÁIsn01Aqo 'aoJoJ u01+o1JJ a4+ ~aA1+E~au JO
aA1+1sod Ja4+1a aq ÁEm aOJoJ ap1+ aq+ ~u01+oaJ1P Á aA1+1sod a4+ u1 +OE
sÁEMIE saoJoJ aAEM a41 °saoJoJ u01+o1JJ pUE 'saoJoJ IEp1+ 'saoJoJ aAEM
:Jap1suoo 0+ s+uauodmoo aOJoJ aaJ4+ ÁIuO sU1EmaJ aJa4+ 'aIq1~1I~au
SE JO pasods1P uaaq ÁpEaJTE aAEq saoJo] aouaInqJn+ a4+ aou1S
°umpq1unba
o1illEUÁP JO a+E+S E pa40EaJ a.t.Eq
sao.rojpaJap1suoo 84+ 'anIEA
mnm1xEm s+1 pa40EaJ SEq +uaJJno s14+ ua4M °passnos1P +snç saoJoJ
JnoJ a4+ JO aouanIJu1 a4+ mOJJ sdolaAap +uaJJno aJ04s~UOI a41
saoJoJ +uaJJno a4+ JO UOS1JEdm00
°pa+oaI~au ÁIa+aIdmoo ÁIIEnsn s1 +1 'aJoJaJa41 °aJa4
paJap1suoo saoJoJ aaJ4+ Ja4+0 a4+ 0+ paJEdmoo uaqM IIEms ÁJaA s1
aOJoJ +uaInqJn+ s14+ JO s+oaJJa aq+
'UMOU~
s
j
MOU s1 SE JEJ sV
Á
(81)
V
=
s
j
: SE uaA1~ aq UEO aOJoJ JEaqs +uaInqJn+
s141 °ÁJOa4+ JaÁEI ÁJEpunoq u1 pasn +E4+ 0+ IaI1EJEd ~u1UOSEaJ E s1 s141
0(S1XE
X
a4+ ~UOIE) +SEOO a4+ 0+ Jasol0 saAom auo SE sa1JEA +SEOO aq+ 0+
IaI1EJEd +uaJJno a4+ +E4+ +OEJ a4+ mOJJ +lnsaJ saoJoJ +uaInqJn+ a41 .
saoJoj +uaInqJn1 oh
°Á+100IaA JO u01+ounJ pa+Eo1Idmoo Ja4+EJ E s1 +1 'OSIV °au1IaJ04s a4+
mOJJ aouE+s1P 'aoua4 pUE '4+dap Ja+EM JO u01+ounJ E s1
-
Str -
I~
+E4+ a+oN
- 46 -
Since
we want
of distance
from
the
(19)
for V in terms
that
this
will
Bakker
coast,
in order
(17)
Bakker
begins
and hence
to more
with,
of the
dirficult,
simplifications
easily
(from
distribution
h, it would
of h. Examination
has made
17),
=
the velocity
be extremely
(egn.
TI
to find
right
as a function
be convenient
hand
side
to solve
indicates
at best.
to the
obtain
right
hand
a solution.
side
of (19),
In place
of
Bijker):
2
P g V
(20)
C2
(21)
as an equivalent
to equation
This is simplified
(13).
by assuming
that in the breaker
zone ~ ~
»
V;
this yields:
sin w t
= ab
where ~
Substituting
sin w t
:
0.45 K C
~ =
;g
and taking an average with
11"
1
11"
f
0
sin
e de =
2
11"
(22)
- 47 -
yields:
p
=
TI
g V
0.45
K
C
rg
C2
(2)(.45) P
2~
K
/g'V
~
=
(23)
C
1T
Substituting (16) in (23) :
=
TI
( 2 )( .45) P
C
K
r; V
;-ti
2
H
1T
rg =
.45 P
K
g V H
rb'
1T
(24)
C
Using (5c) this becomes :
=
TI
.45 P
K
1T
g V A
C
Iïl
(25)
or, in another form
= P g3/2
TI
ff
Ç, A
/h'v
(26)
C2
This is now much simpier to work with than equation (17).
We can now return to the problem at hand -- _comparison of the
forces -- byequating
5 P g
16
h 1/2
br
(25) with (7),
sin ~br cos ~br tan ah
=
.45
P K g V A
1T
l1i' (27 )
C
Solving this for V as a function of h yields:
V
=
5 1T A
h
C
(16)(.45) K hbr1/2
(28)
sin ~br cos ~br tan ah
Svasek and Koele have found that for the Dutch coast A
=
0.4 to 0.5
for the significant wave and that A ~ 0.3 for the root-mean-square
wave.
- 48 -
•
Substituting for the constants
A
K
=
=
0.3
0.4
we get
v =
1.63
he'
~~2
S1n ~br cos ~br tan ah
(29)
which gives the velocity distribution as a function of distance from
the coast, provided that hand
are known.
tan ah' the beach profile parameters,
It should be pointed out that in the example just considered,
only the wave forces and the friction forces were considered. It is
entirely possible that under certain circumstances the tide force,
for example, might also be important. This could obviously be added
to the analysis, but shall not be done here.
Result of this development
We have determined the velocity profile along a horizontal line
extendipg out f~om the beach. This velocity profile will be used later
in conjunction with a sand transport formula in order to develop a
sand transport profile.
It has been found that when this is done for the Dutch coast, for
example, the sand transport at a distance of 200 m from the coast is
about three times as much as at a distance of 600 m. This is a result
of the combination of increased longshore current velocity and increased
wave forces on the bottom material which cause more severe stirring.
A sand transport formula will be developed in the following
chapter.
- 49 -
Chapter V
Longshore Sand Transportation caused by Waves and Current
1. Emperical Formu1as
derived from an energy balance
For some time now, sand transport along a coast has been related
in some way to the component of the wave energy along an axis parallel
to the coast. These methods have found wide application and are based
upon·sound physical reasoning.
In its most general form, such a relationship is:
=
S
(1)
A Ea
where
S is the total sand transported along the coast.
E
a
is the energy flux component parallel to the coast, measured
in the breaker zone.
A is a proportionality constant.
Unfortunately, in this purely emperical formula, A is not dimension-
1.
less but has dimensions [ L T2 M-1
The energy flux component, E , is given by:
a
Ea
=
E0 K2
r
sin ~b cos ~b
(2)
where:
~b is the angle between the breaking wave crests and the
beach line.
E
o
is the energy flux in deep water in the direction of wave
propagation.
K
r
is the refraction coefficient.
From short wave theory,
Eo
=
1
16
(3)
- 50 -
with
=
g =
H =
0
c =
0
p
the water density
acceleration of gravity
deep water wave height
wave speed in deep water
Back substitution of (3) in (2) and then in (1) yields
(4)
where all of the constants have been combined, and the constant, A,
evaluated using data such as that available from CERC. Conveniently, a
bit of dimensional analysis reveals that the coefficient 0.014 is now
dimensionless.
Formula (4) although reasonably trustworthy, does have a few
limitations. These are:
a. Only the total sand transport is computed. No information on the
sand transport profile along a line perpendicular to the coast can
be obtained. Especially on coasts subjected to spilling breakers, or
where more than one offshore bank is present, this limitation can be
serious.
b. This formula is independent of the type or size of bottom material.
This is, of course, not true on real beaches. This formula is still
valid, however, provided that it is used only for beach materials
similar to that for which it is derived, namely, uniform sand ranging
in diameter from 0.2 to 0.5 mmo
c. The slope of the beach does not enter the equation.
d. This formula computes transport caused by waves alone.
Influences of superimposed currents are not considered. This limitation
can be very important in river deltas, for example.
e. This formula may not be applied to shoals, dumping grounds, or
near dredged channels.
- 51 -
Svasek [~ has largely overcome the first limitation. He did this by
assuming that the sand transport in a given strip parallel to the
coast ~s proportional to the energy lost by the
waVQS
as they cross
this strip. This assumption seems plausible, but has not been proven
rigorously. Even so, this method is applied to give the sand transport
profile when a relationship between wave height and distance from
the coast is known in one fOr'mor another.
Otherwise, the limitations a through e have not been overcome.
Bijker has, therefore, begun again with a new, separate derivation
based upon the sediment transport formulas for rivers (i.e. currents
alone) modified to include wave effects. First, the transport formulas
for currents alone are discussed to lend completeness to the derivation.
2. Sediment Transport Formulas for Steady Currents
A general form for a river sediment transport formula is:
S
=
6.
f2 (}Jh
-
D
I)
(5)
J
where:
. mms
3/
S is the bottom material transport ln
6.
is the relative density of the bottom material in water
D
is the grain diameter in meters
I
is the slope of the energy level
).I
is the so called "ripple factor" giving the influence the bottom
form
f1, f2 are functions, as yet undetermined, and h is the water depth.
Often the term on the left of (5) is called the transport term; while
the right-hand term is called thè current term.
We might note, here, that S is a bottom material transpÇlrt.This
is intended to refer only to material transported in contact with the
bottom or very closely above it. Material carried primarily in suspension
is not included, therefore.
- 52 -
The ripple factor, J.l, is given as:
=
J.l
Here C
(
C
3/2
r
)
C
DgO
(6)
is the Chézy fri~tion coefficient for the actual river bottom.
r
is a ficticious friction factor that would exist on a plane bottom
consisting of grains having a diameter of DgO
(90% passing from a bottom material sample).
In physical terms, the coefficient J.l gives an idea of how much
energy remains to transport the bottom material af~èr the energy
abser-bedby the large scale bottom roughness (ripples) has been deducted,
We might also admit that this factor is also used to hide the fact
that we really do not understand all the contributing phenomona.
Experimental evidence indicates that the ripples are essential
to sediment transport. Also, it has been found that the ripple factor
may have a value greater than one.
A more indirect connection between ripples and transport comes
from:
T
=
2
v
P g-
(7)
C2
r'
where T is the bottom shear stress in N/m2, and v is the stream
velocity. Note that T increases as C
r
decreases or, trom (6), as J.l,
and thus, the bottom roughness, increase. By assuming, for now, that
the transport is proportional to
T,
our thought process is completed.
The factor h I can be written in an equivalent form:
h I
=
T
( 8)
P g
Here, with equation (8) we see that the influence of the current
can also be related to the shear stress,
T.
- 53 -
There remains a problem of defining the bottom. This seems to be
an absurd problem, but is a real one since we have already stated that
the bottom material transport takes place in a thin layer near the bottom
and we have also tacitly measured depth to some bottom, as yet, not
defined. Einstein (6J has assumed that the bottom material transport
over the bottom takes place in a flat layer having a thickness equal
to a few grain diameters. Examination of figure 1 shows that this moreor-Iess theoretical assumption has troubie in practice. When there are
FIGURE 1
Sand transport above ripples.
The current in this photo is caused by a long
wave.
ripples, the assumption of a flat bottom is certainly violated. It
is obvious from the photo that the current pickes up particles from the
ripples and transp9rts them some distance in.a layer of limited thickness.
This thickness is, however, much greater then that in Einsteids assumption.
- 54 -
Further, the photo shows that the bottom material transport takes
place in a layer that extends only slightly above the ripples.
Later, when waves are considered, the oscillations imposed
on the stream velocity will have the effect of increasing the layer
thickness (the material is more severely stirred up).
Lacking better and more complete experimental evidence, the
"virtual" bottom is more-:-or-Iess
arbritrarily placed at one half a
ripple height above the ripple valleys. AIso, the virtual bottom
roeghness, r, is fixed as one half the ripple height above this virtual
bottom. Thus, the top of the virtual roughness corresponds to the top
of the ripples.
In order to compute the suspended material transport, one must
know the suspension concentration profile over the depth. This profile
can be derived from the following equation describing the vertical
particle motion.
w
s
c +
€
(z) dc (z)
dz
=
0
(g)
Here,
w
s
=
fall velocity of the suspended material particles in
water
c
€
is the concentration
(z) is the mass diffusion coefficient which is a function of z.
z
is the distance measured vertically from the bottom.
Separation of variables and integration yields
z
In
(~)
c
= - w
a
where c
a
s
f
dz
€Tz)
(10)
is the concentration at some arbitrary height, a, above
the bottom.
For a plane bottom Einstein takes the thickness of the bottom
material transport layer (a few bottom material particle diameters) for
the value a.
c becomes, then, the concentration of bottom material
a
in this transport layer.
- 55 -
For rippled bottoms, it is assumed that the bottom material layer
has thickness r (equal to the virtual bottom roughness)
and that the concentration
equal to a
is the concentration of bottom material
a
in this layer thickness. Thus, this assumption
evenly distributed
c
is parallel to that of Einstein.
With a bit of boundary layer theory, one can determine that the average
velocity
in this bottom material transport layer may be given by:
r
1
v
= r
o-r
1
2
v(z) dz +-
Jer
v
er
33
er
33
(l1a)
33
= 6.35
V
( l1b)
H
where
c
a
e
=
vx
=~,
the base of natural logarithms, and
of ten called the shear velocity.
can now be computed
Sb
c =
6.35 v
a
H
(12)
r
The subscript b has been added to S to emphasize that it is
bottom transport. To solve (10) we must still know the function
[9] assumes
Ippen
v
E
(z)
=
2
H
E
(z).
the foilowing
(1
dv
dz
z
h
=
V
)(
K
(1-~
h
)
Z
here:
v
is the stream velocity at height z, and
K
is von Kármán
constant
=
0.4
(13)
- 56 -
Substitution of (13) in (10) yields
(
In
w
c (z)
C
w
=
=
a
s
K V
)
In
M
z
s
K V
J
a
M
h
z
(1 - -)
h
(14a)
z )
h
z (1
a
(1 - -)
h
(~)
{
dz
1
(2.)
h
(14b)
This can then be changed to:
c
z
c
a
where
=
z
M
r
z
M
z )
(2.)
h
(1
(~)
(1
-
h
a
-)
h
(15)
1
w
=--K Vs
X
After a bit more mathematics, Einstein comes to the following
resuIts:
S
s
= 11.6 vR ca
In (33 h )
r
+
(16)
here
Ss is the suspended sediment transport, and 11, 12 are defined
below in (18) and (19).
Substituting a
=
r
and (12) in (16)
(17)
- 57 -
where:
z
= 0.216
11
A x
z
(1 - A)
z
=
12
A K
0.216
z
1
- 1
K
J
A
f
.L:....z
y
}
K
dy
(18)
- 1
z
1n (y)
dy
(19)
(l - A) x
A
=
r
h
y
=
h
z
Equations (18) and (19) are given in graphical form in figures
2 and 3.
We might note that these formulas data from the pre-computer
era.
They are well suited to hand computation but can also be used
effectively on a computer. A program using these formulas has been
developed by the Delft Hydraulics Laboratory.
- 58 -
104
9
8
I'
7
6
5
4
"~ r'\
3
"
1
9
8
102
.....
I\..
,
~
~1"
Y
~
r-"I' ........ " .....
" 1'.....
""-.
'"
.....
i'ooi'oo
2
r-..~
~'" r--.
r-."",
~
~
~
r-... .....
~
........
'.....
5
4
ro-
- _ ...... -
(] 1.0
~
i""""o
....
3
1:n
2
,
-
I"
4
i"'"~
r0-t-
~
<,
'
..... ~
.....
ro-
I
dy
~
r-,
7
~.Ir:f
10' 1
8
lil
I"
~
~ 1'-"",
~
...... ......
-
z
(~)
~"
"" -,,
1
3
f
0.216 A •
z
'1 - A) •A
,
.....
,Q~
'"",
1 =
0
\..as a function of A for various values of z*
.....
6
5
4
10°
I
1
,
5
4
2
Fig. 2
~
-,
I
fI,
z - 1
-
.
<,
3
3
-
~
2
10
-
~
......
.....
........
-
r-~
~
....
"' " ""' ~,
.... "' ,"
_ .......
I\..
.....
....
~
...
..........
r-.....
-
I
7
6
5
4
3
"-
"- <,
......
1'""0....
" ,",
.... ....
10....
--.r.;-
.r.:T
......
........
"
.....
r--.r-,
I-..
~
..... I""""
!""""" I--.
"'"
2.b
1
~
i""' ....
""'iiiii
~
....""1'-
r-"""" ~""I'-
3.0
,
_ r-- ""
~
2
1Ó'
~
-1
1o
0&.
B
7
-.:
~
.5
4
3
2
2
3
I, 567891
2
10-1,
3
4 5 6 789'
10-3
2
3
I, 56789'
2
10-2
3
I, 5 6789'
10-'~
A
- 59 -
104
9
8
7
6
5
4
z - 1
3
Fig. 3
2
0.216 A
f
lfi
z
(l - A)
lfi
z
(l_:_z) lfi ln y dy
y
A
function of A for various values of z
'" as a
1
t
=
'\
3
10
I2
1
I
1
•
5
4
Iz'3
2
102
,
1
9
~"'"
6
5
4
3
2
101
~
...._ i"'---
-
-~....... ...
r- ~
8
~
- _...
I
5
.... .._
4
.....
~....
~~...
!'oI..",
......
""'-10... ~
~ ,
3
2
100
1
!
5
4
3
2
-1
10
1
9
8
7
6
5
4
3
2
-2
10
1
10-5
2
3
4 5 67 891
10-4
2
'3
2
4 5 6 7 e91
10
-3
3
4 5 6 7891
10-2
2
3
4 5 6 78 91
_1
10 __.
A
- 60 -
3.
Influence
of Waves
In formulas
suspended
for bottom
material
the shear velocity
investigating
on Sediment
Transport
material
transport,
(17),
transport,
the shear
and
stress,
in the formula
T,
for
appears via
, v. The influence of waves will now be included by
their effect on the stress T. To compute this influence
we must investigate the velocity gradient caused by the combined waves
and.current
in the bottom boundary zone. Figure 4 shows the wave and
current velocity profiles separately.
I
I
I
I
Vz
I
I
I
./
/
/
Velocity Profiles in the Boundary Layer
Figure 4
The solid line gives the velocity profile due to the constant current
The dashed line shows the velocity profile at some instant caused by
the waves. The value z' is the thickness of the laminar sublayer.
Using definitions
T
=
from boundary layer theory, we find:
(d v(z)t
dz
(20)
bottom
where 1 is the mixing length
Noting from fig.4that:
=
KZ
- 61 -
v
d v(z)
dz
x
= ZT
=
(21 )
bottom
where we have used v
x
=
dv
1'-dz
This can be reworked to "
(22)
In order to include the effects of the waves on the shear stress,
Bijker makes the following assumptions
:
a. There exists a turbulent boundary layer at the bottom in the
wave velocity profile.
b. This turbulent boundary layer is directly above a laminar
sublayer having the same thickness as that caused by the steady
current.
c. The orbital velocity of the water particles caused by the wave
at the height z' is assumed to be equal to pUb'
coefficient,
and ub
bottom. Experimental
=
where p is a
Uo Sin wt is the orbital velocity at the
data show that 0.45 is a reasonable
value
for p. Also, this value does not disagree with approximate
theoretical results.
The two velocities measured at a height z' above the bottom are
shown in a plan view in figure 5. This shows that, in general, the two
velocities,
caused by waves and current respectively,
the same direction.
need not have
- 62 -
Figure 5
Velocity Combination
at top of laminar sublayer.
From figure 5, using the law of cosines:
v
=
z'
/
vz'
2
+ p
2
2
ub + 2
(23)
r
where
v
z'
is the resultant combined velocity at some time t, and
r
~
is the angle between the wave crests and the constant stream
velocity.
Substituting vz'
nition for ub:
Let) =
pK
2
(v,
in place of vz'
2
+ p
Z
Since,
in equation 22 and including the defi-
r
2 2
Uo
sin2 wt + 2
p u
o
sin wt sin ~)
(24)
is now a function of time, we wish to obtain its average
value, Lr' over a wave period, T.
T
pK
C
2 2
T
(v 2, + p 22.2
u §ln
zoo
4
wt + 2 vz'
P u
..
)
Sln w Sln ~ dt
(25)
Using (22), the shear stress component due to the constant current
alone can be evaluated.
'c =
pK
2
(26)
Dividing (25) by (26) gives an indication of the increase in
shear stress caused by the waves.
- 63 -
T
T
T
r
C
2
P
(1 +
= T-
2
2
u
u
_ ____;o_ sin2 wt + 2 P ~
vz '
v ,2
z
4
Using (21), (7) and the definit'ion of v
v
z'
=
1
K
Define
~
=
=
(28)
C
K
r
C
K
r
(29a)
Ig
0.0575 C
(29b)
(in metric units)
r
using the known values for
K,
p, and g.
(28) and (29a) in (27) give:
T
2
u
u
T
0
2
0
2
r
sin wt ain ~) dt
(1 + ~2 2" sin wt + 2Ç;
=
v
T
T
v
c
- 4
_
("
=
(27)
we obtain:
v .;g'
.* =
v
p
*
sin wt sin ~) dt
1
1 + 2
u
(1;
0
)2
(30a)
(30b)
v
4. Sediment transport formulas for combined current and waves
The increased bottom shear stress derived in section 3 is now
substituted
into the formulas for bottom material transport and for
suspended material transport.
In the case of suspended transport this is done by modifying v
*
using equation 30 b.
(31)
This approach has been
found to give reasonable results when
compared to prototype measurements,
is not too fine.
«
provided that the bottom material
0.15 mm). For finer material extremely large
values of suspended material transport are computed.
It can be pos-
sible that the assumption that the bottom material transport takes
- 64 -
place in a layer of thickness r is no longer valid. Some measurements
have shown that under these conditions (fine material) this bottom
transport layer can have a thickness of 1.6 r
or even more if the orbital velocity amplitude of the waves is large
compared to the stream velocity.
The bottom material transport can be computed using most any
available formula in which the modified value of v , equation 31, is
M
incorporated. At the present time, all qf the available formulas are
largely emperical.
It then follows that they are a bit untrust-
worthy when applied to extreme conditions.
Since the background work
for this derivation was done in connection with model investigations,
[71
the Frijlink Formula
was used. It was at that time the most trust-
worthy for th is use on modeis, implying very low sediment transports.
The Frijlink Formula, modified for our use is:
Sb
=
5 D
v
Ii'
C
r
exp
[
- O.27L:.D C2
r
2
uo)2
llv (1+ !U~
v
2
..
wh ere exp d enotes exponentlatlon
]
«(e
(32)
1) .
The following critical comments concerning equation 32 should be kept
in mind.
a. The term
5 D
v
rg
(33)
C
r
contains only the steady current velocity.
This is reasonable from the point of view that this current is
respnsible for the material transport; the waves simply act to stir
more material loose from the bottom. The ripple factor does not appear
in this term.
b.
The formula (32) does not degenerate to the original formula
given by Frijlink when u
o
is set equal to zero. Originally the ripple
factor did appear in the term before the exponential.
Perhaps the
value of the constant, 5, should be modified. This constant was determined by Frijlink based upon model and prototype data then available.
Since there is, until now, little agreement concerning the precise
- 65 -
role of the ripple factor, then the coefficient value, 5, will be
retained.
c.
Over most of its usable range, the computed sediment transport
increases with increasing grain diameter, D. This is especially true
when working with extremely small transports with a diameter of the
order of 0.2 mmo The vaiidity of this formula (with the given constants) seems doubtful when working with large grain diameters.
5. Verifications of the Formulas
Two example applications of the formulas are discussed here. The
first concernS a model beach shown in plan in figure 6. Figure 7 shows
beach profiles number 3 and 8 before and after testing. The distribution of wave height. water depth, and long shore current is shown for
various profiles in figures 8 and 9. The computed sediment transportation for sections 3 and 10 is compared in figure 10 to the measured sand transport at the downstream end of the model. The computations seem to agree reasonably with the measured values.
- 66 -
Other pertinent data concerning this test are:
1.55 seconds
Wave period
Average Grain Diameter (sand) Q.22 mm
0.3
mm
Estimated bottom roughness
1 cm.
Table I gives some results of the sediment transport computations.
TABLE I
Suspended and bottom sediment transports for sections 3 and 10
of beach model.
(Transports in 10
-3
3
m /hour)
Profile 3
Profile 10
,S
s
Sb
S
-.s
1
.008
.0027
.01l0
.214
.1034
.3172
2
.710
.4146
1.1250
5.494
5.3281
10.8224
3
7.802
8.5542
16.3564
9.330
1l.3323
20.6624
4
7.990
8.6924
16.6823
13.749
19.2661
33.0150
5
9.156
11.ll89
20.2745
18.684
30.8782
49.5623
6
11.796
15.5936
27.3895
9.472
13.5641
23.0357
7
12.179
16.2759
28.4545
13.671
33.2131
46.8839
8
11.177
22.7754
33.9527
5.272
17.0600
22.3325
9
7.338
22.4334
29.7718
1.745
5.4273
7.1719
10
3.788
10.2292.
14.0168
1.199
2.5170
3.7163
11
1.905
4.4168
6.3214
.752
1.1200
1.8718
12
.895
1.5568
2.4515
.351
.5067
.8572
74.744 122.0640
196.8075
79.932
140.3163
220.2486
Measuring
. ;Sb+S
:
S
Sb
Sb+Ss
Point
- 67 -
--~l~
Q. Q.
Q. :1
:1 '"
111
L..
"0
t)
C
....
~
~
111
C
....uo
t)
111
111
111
....o
U
c
0
z0
E
t)
>
H
....c
t)
el
C
L..
t)
.x
...el
....
.Cl
....
....
~
U
'M
:1
Q)
cr::
cu
....0
t.I
s:
111
0'1
C
0
H
p.,
Q)
r::
.c
o
cu
.c
o
cu
ÇQ
Q)
Q)
ÇQ
'0
0
:E:
'M
....:l
Q)
H
r--------,
L
--.;..--
.
_j
o
....
o
- 68 -
c
c
è
-
è
I
I
C)
N
co
CT)
(/)
(l)
r-i
OM
4-i
o
H
p..
..c
U
rel
(l)
p:)
4-i
o
~
o
(/)
OM
~
~
o
U
- 69 -
-- 1
-s
...'"
f--
",.c
.2"
&I
._ .c
Ol
.;;
f--
'"
s:
~
~~
Cl.
1.1
I
I
,__
~
...
1.1
~
/
I
I
•
I
\
\
(
~
X
/
/
V
/
,
,
\
/
\
\
\
r-.
<,
I
\
'\
"\\
/
(/)
(/)
zrdw
~
u
~ U'dw ~
0
(/)
w
ordw ....
--
/
-
ü
9
en
'--
w
>
S'dw
ClO
~
Z
9
e-,
>--9 'dw
I
w
o
Cl::
::I:
lil
~1. 'dw
,
,
I
1"\
V
'\\
\
--~:
N
L.
I
~-
--
M
u &I
o >
--".
o
z
<{
~S
U)
dw
r--""
dw
lil
I::I:
(!)
1.1\
V
/
w
::I:
W
~E' dw
~
-::;-Z' dw
~
z
dw
0
.....
::::>
m
a::
.....
~l'
M
lil
N
-z
(")
E
t"i
0
0
E
----
--
0
00
r-:
0
~
o
~
o
0
E
o
- 70 -
.--
lil
N
s:
...
s:
-.r:.
'ü
...
L..
--
~ "~
IJ
"'~
I
-
~I
Q..
IJ
"D
IJ-
0>
f--
.-
lil
._IJ,~IJ
I
I
\
I
,
I
..
I
\
\
\
('
\
I
I
~
\ -,
/
~
u
z
C/)
LLl
I-
ëi)S'dw
U
o
...J
>
LLl
0:
e-,
f--I.' dw
o
::r
C/)
(.!)
9'dw
CD
z
9
Cl
z
<{
U"I
I
,
I
\
f--'7'dw
J
\
'"
-OL' dw
f--S' dw
\
........
~ ~en
-9' dw
\
<,
dw
C/)
C/)
LLl
\
\
~~'
LLl
II
I
I
IU
C/)
I
\
I
o
.-
\
r-;
\
/
\
/
C/)
I-
::r
(.!)
...;r
I
E' dw
LLl
::r
LLl
-l'd w
('l'1
-l'd w
>
«
~
LL
o
z
N
o
;:::::::
:J
m
0:
.-
l-
C/)
Cl
-.
I
lil
E C"l
0
E
N
d
--:
0
0
0
.-
d
I
N
d
("1\
d
E
- 71 -
,
-
.c.
-
;:,-
cn
<'"
S!N
';~1i
"
~;:,Ut
~"'
.....c.
<"'u
'".c.'O
0'"'0-
CU
I
'-
.. '"
'0''0
cu
0.0."'
",,-0
-
'0
o
'u
I)
';:,
f-Ut
::J
0.
"'
-\ \- ~t
0
..... ,;c
~"" ~
\
( ....__ ............
.....-!C ~
..
-
...
~_:--
~
i...
' ......
____.
~
~
~
~
IJ
.....
0
en
L..
, Ut "'
ü,-..c
u
::::J
~ 0
,!!!
Ut.c.'O
Ut 0 .....
~
S'dw
_;_
/"dw
-
~.
\.
\.
~.
1"" ......
'
Cl)
..__ "'dw
",
.-
\
~
I.
IJ'\
I
0
/J
-
~ E
0
o
o
o
Ol
o
Cl)
o
r-.
0
C.D
o
LIl
o
.-
o
d
I
L
('dw
':.:::: ..I't·"", - Z'dw
~
N- l'dw
...
en
9'dw
..__ S'dw
..
ti
".c.
.c._
ordw
..__ 6'dw
-\- ~
o-!! -
'0
C7\
e-,
........~ .. "" ...
.~
0
zrclw
CID
13;),
s:
Col
,., JfI'"
V
V
"" "\.
r--..r-r-
'-
.' /
... ...-.../,1
/
~
~
!
.~ J
.?
0
1---__...._-
E
I)
I.
~II1
.
"'- ...
'0
I)
J
'-
men
M
I
ti
_g~
I---
J
~
S!!
Ut.c.
'--
E
;::
<'"
o
- 72 -
The second example involves the computation of sand transport
along the coast of Queensland Australia - figure 11. These computations were carried out by the Delft Hydraulics Laboratory.
Within the breaker zone, the longshore current was computed
using the formula developed by Eagleson
(51
Outside the breaker
zone, the velocities are taken from prototype measurements. A bottom
roughness of 0.17 m is chosen for the computations, based upon observations made at the site. The beach is composed of sand having an
average diameter of 0.225 mm with DgO
=
0.350 mmo
The influence of bottom roughness values was verified for the
profiles (alpha 1 to 31) just south of the Tweed River. Use of bottom
roughnesses of 0.1 mand
0.25 m in place of the value 0.17 m gave a
variation in sediment transport of + 21% and -17% respectively.
Figure 12 shows the final resulting computed values compared with
values derived from measurement of changes in the actual beach profile.
In order to properly locate the given line vertically, it is assumed
that the computed value at Tugun is correct.
- 73 -
8o
s
::;\
--"
"
"}
_.-
UH,.
'TOl!
,. _,
_,
_--
_
-
_,-'
'
,_
__
._60
-'-
.~"
,.
Figure 11
Coast
0f Queensland,
.
Australla
QUEENSLAND
;'
oSCAL.E
.2
M1LE$
- 74 -
i
I
•
I
I
I
.,
I1
IC
-,
I
)j:>aNMOJJDN.
-
.- ~
I
I
I
r
\te
I
"\
...
1 ,)\
I
PH IP!31Jng
I
B
o
o
,...
s
§
§
I
~
è
§§
Z
0
c
.2
4: z
4:
I-
'"
0
&..
Lil
0:: til
0 Z
n, W
...
0
.-"E
"
u
.00
)jJ u!qwnJJnJ
In
..
un6nJ.
Ja6uoQlU!Od
11D\fdiV
80
8..
o
-J
..
c
"ti
"
0
Ol
I
I
-e
Cl
<,
0
~>
•
4)
CD
/)(1
\
0
U
,(qqoN 'flJON
I
0
AqqoN 'flnos
~~
,!
4:
.2
~&..
U
U
I
I
(/)
N
I
~
l-
e
I
\
(
°laz
I
I
)(
o
N
I.
Jog .ij 6uoJaN
I
m
....J
"ti
til W
"
~
~
Q.
e
0
u
.
N
rl
(j)
...;::1
)(
I
)(
a
•
ba
·rl
~
z ::::>
4: CJ
a:: u,
I-
....J
4:
0::
0
II-
....J
0
- 7-5 -
6. Simplification of the longshore sediment transport formula.
Several simplifications concerning the determination of the
current velocity have already been introduced in the section
Deter-
mination of Currents Along A Coast.
Aresult
v
=
from that section, equation 29 there,is
h
1.63 h 1/2
Cr sin~b
cos~b
tanuh
(34)
b
after modification of the notation to comply with current usage.
In the transport formu1a, equation 32, a portion of the exponential factor is :
u
~)
v
2]
(35)
Using here the same assumption as in the development
v < < ~ u
(34), namely:
o
then the term (35) becomes nearly independent of v and, therefore,
Sb becomes nearly linearly dependent upon v - equation 32.
Often, in describing the sediment transport formula, term 33 is called
the transport term while the exponential term is referred to as the
stirring term.
7. Results of this development
In the scct i on on current determination along a coast, we determined the velocity profile as one moved
to the beach. It is often
derivation
along a line perpendicular
th is velocity which is put into the present
as the constant stream velocity. When this is done, we
can develop a profile of sand transport along a line extending out
from the beach.
- 76 -
It has been found that for the Dutch coast, for example, sand transport
200 m from the coast is about three times as much as at a distance of
600 m. This is a result of the combination of increased longshore
current velocity and increased.wave
sulting in greater stirring.
forces on the bottom material re-
- 77 -
Literature Pertinent to the Longshore Current and the development of
sediment transport formulas involving both waves and currents.
[ 1 ]
Bowen, A. J .
The Generation of longshore eurrents on a plane beach.
Journalof
[ 2 J
Marine Research. 27, pp. 206 - 215. 1969.
Bij ker, E.W •
Some eonsiderations
about seales for eoastal models with movable bed.
Delft Hydraulies Laboratory. Publieation No. 50. 1967.
[3
J
Bijker, E.W. en Svasek, J.N.
Two methods for determination
of morphologieal
changes indueed by
eoastal structures.
Proeeedings
[ 4]
22nd Int. Congr., Paris, 1969. S 11, Item 4.
Bijker, E.W.
Littoral drift eomputations on mutual wave and eurrent ineluenee.
Comm. on Hydr. Delft University of Teehnology.
Report No. 71-2,
[ 5]
1971.
Eagleson, P.
Theoretieal
study of longshore eurrents on a plane beaeh.
M.I.T., of Civ. Eng. Hydr. Lab., Report No. 82,1965
[ 6 JEinstein,
H.A.
The bed load funetion for sediment transportation
in open ehannel flow.
U.S. Dept. of Agr., Teeh.Bull. No. 1026. 1950.
I7
]
Frijlink, H.C.
Diseussion des formules de débit solide de Kalinske, Einstein et
Meyer-Peter et Mueller eompte tenue des mesures réeentes de transport
dans les revières Néerlandaises.
me
2
Journ. Hydr. Soc. Hydr. de Franee,
Grenoble 1952. pp. 98 - 103.
[ 8
J
Graf, W.H .
Hydraulies of Sediment Transport.
pp. 123 - 242.
Me.Graw - Hili Book Company.
- 78
[ 9]
-
Ippen, A.T.
Turbulence and diffusion in hydraulic engineering.
Proc. 12th Congr. of I.A.H.R. Fort Collins. Vol. 5 pp. 152 - 182. 1967
- 79 -
Chapter VI
Local coastal accretion
due to the construct ion of break-
waters etc.
Introduction
Wh en an impervious
longshore
breakwater
transport,the
is built on a coast that has a
local equilibrium
The area in front of the breakwater
capacity
condition
has a smaller sand transport
and this will cause sedimentation
the breakwater.
sand transport
be coastal erosion.
the original
in the area, so the re sult will
In this chapter we will determine
rate of the accretion.
side of
is restored but the only sand supply is
formed by the sand already present
basically
on the upstream
On the other side of the breakwater
capacity
will be disturbed.
Of course, the derivation
the shape and
for erosion is
the same.
+y
breakwater
ori inal coastline
accretion
so
-x
+x
fig. 1
Definition
S
o
~
sketch
=
sand transport
=
angle of wave incidence
=
angle between the new coastline and the breakwater
To determine
along the undisturbed
the theoretical
of motion, a continuity
accretion
coastline
curve we need an equation
equation and a number of boundary
conditions.
- 80 -
The
equation
of motion:
Previously
we have
1. The C.E.R.C.
formula:
introduced
two
sand
transport
formulas;
namely
S
=
1.4 x 10-2
c
sin 4>b cos 4>b
o
It should be noted that in this particular
to express
S as a function
case it would be better
of 4> instead as a function
of 4>b' as in this
case the 4>b will change continuously
as it is dependent
of the changing
"constant"
coastline.
4>remains
at a point where the direction
of the wavefront
on the direction
as it is determined
is not yet influenced
by the new coastline.
For small values of 4>we can say:
sin 4>
cos 4>
so
=
=
4>
1
4>
S
2. Bijker formula:
S
bedload
=
5 D
y_
exp
C
(- 0,27 __ ll_D_C_2
j.JV2 [1
=5DY
C
the ratio
considerably
Sbedload
va
}
+ ~ (~vo)
]
occurs inside the breaker
zone and in
is rather large. This simplifies
this formula
as A is approximately
V.
2__
rg'A-
Most of.-the-sand transport
this region
__
U
1 and makes
- 81 -
The suspended load is approximately directly proportional to
Valso. In other words, 5total :: V. A more exact derivation results
. 5tot :: v1.1• It has also b een shown that V ::~.
~
ln
Both equations give almost the same result"S
q
=
.
S0
o
.. 41 or S
0
= q~.
is called the coastal constant.
cl»
When the influence of refraction and diffraction is neglected
we see that the angle of wave incidence becomes smaller at the rate ~
as we move aloDg the zone of accretion towards the breakwater thus:
(
5 = q
x
At
-
.:!z)
dx
x = 0 , S is zero:
x
.:!z
dx
x=O
For
cl»
x = -
00
=
, .:!z
dx =
a
= cp
0
and
S
<p.
x=-co = S 0 = q
The equation of motion now becomes:
=5
_q.:!z
x
0
dx • In this equation the values of <p, q and .:!z
dx are
positive as a result of the accepted coordinate system.
S
The continuity equation:
Before we can write this equation we first have to know the
characteristics of the coast and the way the accretion will take place.
Looking at beach cross sections one finds, in general, the slope
of the bed profile is fairly uniform between the waterline and a depth
of -7 to -12 m. Below this depth the gradient is very low.
The profile can be schematized by two straight lines, as is done in
fig. 2 •
- 82 -
to 12 m
Figure 2
Beach Profile Schematization
Pelnard-Considère established his continuity equation by assuming
a parallel accretion (fig. 3). With this assumption the continuity
equation for a slice dx is:
(S
as x
+-ax
x
dx) dt - S
x
dt
= - ~at
h
dx
dt.
~
at dt
as
sx
s + ~ax
--t--.x
dx
_--
h
~
êx dx
.
dx
Plan
Figure 3
The continuity equation becomes:
as x
äX
+
h
=
Profile
0
However:
1. Sand transport by wind on that part of beach that becomes dry
during low tide is not considered.
2. In reality the accretionprofilè can he steeper than the original
equilibrium profile (see fig. 4) so h becomes a function of y.
- 83 -
+y
Figure
The
4
equation
Accretion
of the
profiles
theoretical
steeper
than
accretion
original
curve
beach
according
to Pelnard-
Considère.
The accretion
will
occur
according
to the
line
given
in fig.
5.
+y
Figure
5
successive
beach
lines
-x
Boundary
t
+x
values
=
=
0 :y
=
sx
t >
are:
0
-
0
S
0
for all values of x
at x
=
at x
= -
0
0
t
f
f
00
=
+ q,
=
0
S
S
y
dx
=
dt
=
y
0
=
x
0
0; Sx
=
S0
t
h
0
00
The equation of mot ion is
as
ax
x
=
S
q~ a2
. 2
ax
x
=
S
lY
0
q ax
(1)
- 84 -
The continuity equation is:
as x
--+
ax
(2)
Substituting (1) in (2)
or
q
iY ax2
h ~
at
a
iY2
~
-
-
at
ax
= 0
(3)
0
where:
S
o
a=.9.=
h
4>h
By using the boundary values we can solve equation (3).
The solution is :
y
=
4>
!4at
{
e
TI
-u2 -
urn e
}
where:
CIO
e =
u
=
2
yÇ'
2
f e
u
-u
du
-x
(x
14at'
<
0)
therefore:
CIO
o =
2
yÇ'
2
f
u
e
-u
du
u
CIO
=.1....
yÇ'
{
f
0
e
-u2
du -
f
0
2
e
-u
du }
(4)
- 85 -
2
=-
{
fi
u
rn
T-
I
e
2
du }
0
u
=
-u
1 -
I
2
-
R
e
-u
2
du
0
where:
u
1
-
R
J
e
-u
2
du
is the probability
integral which can be
found in tables.
0
In the following table the values of
2
e and {e -u -
ï
4YTI
e}
for
several values of x are given:
TABLE I
-u
2
u
e
0
1.000
1.000
1
4
0.724
0.620
-12
0.482
0.348
3
4
0.289
0.185
1
0.157
0.090
3
2
0.034
0.015
2
0.005
0.002
Equation (4)
because
e
- 4/ITe
is rather complicated, but it simplifies at the breakwater,
her-e x = 0;
u = 0 and y =
<I>
!4at'
(5)
TI
In fig. 6: OB
=
2
<I>
/
at'
TI
and OA'=
1. OB =
<I>
2
F7T
(<I>
is small)
- 86 -
Figure
6
-x
0
A
Area OA'B
= 1.2 A'O
S
Area OAB
't
0
=
h
x OB
=
=
2
.at
TT
2 Socp
cp =
7Tcph
2 S
t
=
t
0
TTh
Area4
area OA'B
= 2
area OAB
TT
Therefore:
Approximately
OA
=
2.7
OA'
2.7
Point A corresponds to u
=
-x
!4at'
TT
=
=
1.52
According to table I the corresponding y is only 1.5% of the value to
y at the breakwater.
Sandstransport
around the end of the breakwater
By using (5) we can determine the moment that the sand starts to
pass around the end of the breakwater
(see fig. 7).
Figure 7
Fr-om (5):
y
y
= ~
=
/4:t')
TT
(6)
4
L
From this moment,
tL,
on
the angle
S (fig.
1) which has been
constant and equal to cpup till now, will decrease. It is clear that
after the moment tL is reached formulas (4)
and (5) are no longer valid;
- 87 -
we have to derive another formula satisfying the same differential
equation (3,) which meets the following new boundary conditions:
= 0: y = L, thus S varies with time and
t > 0: x
=-00:
x
S
x
=
S .
0
The solution of this differential equation now becomes:
=
y
(7)
L 8
in which:
u
2
= 1 - Tn
8
f
e
-u
2
(see table I)
du
0
u =
-x
/4at'
ss =
ax
at x
=
0:
L_1_
(-
expo
3x
=
)2
/4at'
Irrat'
_dL
x
--
L
=
S
Irrat'
S becomes equal to cp when tcp
L2 1
- cp2a
(8)
n
In this derivation it has been assumed that the slope of the end of
the breakwater is the same as the beach slope. This is, of course,
not true, but is assumed as a simplification.
When we compare equation (8) with (6) we find that t~
<
tL·
This can be explained as follows:
So long as t
<
tcp
S > cp. This means that sand is being supplied
to the beach from a source at the outer end of the breakwater. This
source, obviously, does not really exist, but we need not worry about
this since we do not use this solution for t
<
shows a plan sketch.
tcp anyway. Figure 8
+y
L
-x
o
Figure 8
- 88 -
In fig.
9 the theoretical
accretion
curves
for the
equations
(4)
'TT
"4 ) and (7) (for t <I> =
are compared.
y
L
V
?
Equat on (
(
Equat on
x <I>
)
= ~cP
at t
=
) at t
2.6 2.4 2.2
2.0
V/
- -1.8
1.6
1.4
/~
/
1.2
1/
0.6
0.4
0.2
o
1\ tj)
1.0 0.8
0.8
~
.>;/V /
l+
Il.L1
L-J- _I..--- /
~
T
~
1.0
0.6
0.4 0.2
0
Figure 9 Coast Profiles according to equations (4) and (7)
The two curves show considerable
differences.
To bring these two
curves into better agreement we say that the areas enclosed by the
axis and each curve must be equal. By doing this, we introduce another
mistake in (7): for t
=
t<l>and x
=
0: S
>
<1>.We correct for this
mistake later with the help of a correction table. The area under the
curve (4) has already been given:
y
=
S t
0
Area4 =
h
'TT
4 ' thus
=
L for tL
S
=
.,
0
h
L2
'TT 'TTL
=
4
4<1>
-
L
<I> 2a
The area under the curve of (7 ) comes from:
+y -
-x
So
Figure 10
- 89 -
=~ f
Area7
(So - SS) dt
}
-+
SQ
>'
=
S
l)
(eq.
cp
(1 0
Area7
=
1
h
S
o
4>
f
B
dt
of motion)
L
wi th 13
Ix=o
=
/'TTat'
we get:
S
Area7
=
o
cph
=
2aL
f
/~at
dt
/t '
'TTa
If we denote the moment
at which Area4 is equal to Area7 by tN, then
we get:
=
Area71
t=t
2aL
S
1Ta
N
therefore:
/2
1TL2
yielding:
1Ta = 4 cp
2aL
tN =
=
2
1T L4 1T a
2
16 cp2 4 a L2
~L2
2
1T
16
1T
4
cp2a
with tL =
L2
cp2a
tN =
2
1T
ï6
tL
=
2!.
4
we get:
0.62 tL
- 90 -
We have now .established a relationship
between the time scales of
equations (4) and (7). This relationship
is shown, again, in figure 11.
The heavy line shows the range of validity of the two equations. A
human example of th is same time shift would occur when a traveiler
arrived in New York with his watch still set to Amsterdam time.
0
0
0.38tLI
Equation (4) valid
tL
(4)
time
II tN
(7 )
t~me
Equation (7) valid
Figure 11
The quantity of the sand passing around the end of the breakwater
can be determined with the help of the equation of motion:
sx =
For
x
=
S
(1-0
0: S
x
_q
dX
1
cp
=
Ss
=
8
2L)
dX
Ss
=
S
0
(1
- .ê_)
cp
- 91 -
With
L
a =
a =
S
Inat'
S
L
(l-
0
(9)
)
4/nat'
Because (9) is applied when t >tL (we are on the lower time axis of
fig. 11) we need to correct t. When t is taken as the time from the
very beginning of the accretion (upper axis), equation (9)
becomes:
at
with L
=
I ~tr
2~
The values of
Pelnard-Considère
we get:
Sa
computed from equation (10) are still incorrect.
gives a tab Ie showing the values as computed from
equat ion (10) along with corrected values. _Unfortunately, his explanation of the computation of the corrected values is not too clear.
Meanwhile, Bakker has recently verified the correctness of PelnardConsidèdere's
table by a different, more exact theoretical
computation.
The results of this work are shown in table 11 and in figure 12.
Sa
Table 11 Computed and corrected values of
~
S
t
tL
equation
(l0)
0
corrected
value
1
0.189
0
1. 25
0.315
0.298
1. 50
0.397
0.394
2.0
0.498
0.500
3.0
0.605
0.607
4.0
0.665
0.667
5.0
0.703
0.704
Table 111 gives some pertinent philosophy.
- 92 -
1.0
0.8
0.6
/
0.
~
so
1o.
--
-
~
I
2
----
~
I
0
o
1.0
2.0
3.0
4.0
S
Figure 12
Corrected values of ~
S
o
5.0
- 93 -
TABLE 111.
THE TEN COMMANDMENTS FOR COASTAL PROTECTION
1)
Thou shalt love thy shore and beach.
2)
Thou shalt protect it gainst the evils of erosion.
3)
Thou shalt protect it wisely, yea, verily and work
with nature.
4)
Thou shalt avoid that nature turns its full forte
gainst ye.
5)
Thou shalt plan carefully in thy own interest and in
the interest of thine neighbour.
6)
Thou shalt love thy neighbour's beach as thou love st
thy own beach.
7)
Thou shalt not steal thy neighbour's property, neither
shalt thou cause damage to his property by thy own
protection.
8)
Thou shalt do thy planning in cooperation with thy
neighbour and he shalt do it in cooperation with his
neighbour and thus forth and thus forth. So be it.
9)
10)
Thou shalt maintain what thou has built up.
Thou shalt show forgiveness for the sins of the past
and cover them with sand. So help thee God.
- 94 -
Non-parallel
accretion
If the gradient
of the profi~e
original
equilibrium
function
of y.
Figure
h
13
=
equation
profile,
Non-parallel
C Y
C
y ~
is higher
as is shown
than
in fig.
the gradient
13, then
of the
h is a
accretion
=
tan y = accretion gradient
=
equilibrium gradient
The continuity equation in this case is:
as
~+cY~=O
ax
An approximate
Y
=
(11)
at
solutionlf of (11) is:
1,59 M
3
,j,
(0,72M - x)2
(12)
'I'
(M - x)4
6 S t
where: M
o
=
)1/3
Ccp2
at x
= 0: y =, 1
_S
5 (
tcp
)1/3
0_
C
According to Pelnard Considère at x
y
=
cp ( 4at)~
n
When we compare
=
S
1,37 (
(13)
=
0:
tcp
o
h
see eqn , (5)
(13) and (5) it turns out that in the early stages
y increases faster when we use (13) .
• after many pages of mathematics
- 9S -
This'~s easy to understand when we compare fig. 13 with fig. 7. In
fig. 13 we need only small amounts of sediment to bring the shoreline
ahead; while in the other case long layers of sediment are necessary
to bring the shoreline ahead. After a number of years, however, ys'
the value of y from equation 5, overtakes Y13 and stays ahead until
the sand starts to pass around the breakwater and yS reaches its
maximum value. Now Y13 will surpass yS again, since its maximum value
is 'larger due to the higher gradient of the profile. Figure 14 shows
the gradients of the accretion
and the original equilibrium profiles
measured at Abidjan.
o
- S
- 10
- lS
- 20
o
- S
o
- S
Figure 14
Beach Profiles near Abidjan
Horizontal scale: 1:S00 Vertical scale 1:10,000
- 96 -
References
- Eagleson:
Theoretical study of longshore currents on a plane beach.
M.I.T. Dept. Civ. Eng. Hydr. Lab.
Report 82 (1965).
- Pelnard Considère:
Essai de theorie de l'evolution des formes de rivages en plages de
sables et de galets.
- W.T.J.N.P. Bakker:
The dynamics of a coast with a groyne system.
Ch. 31, Vol. I, Proc. 11th Coastal Engineering
Conference,
- E. van Hijum:
Kustaangroei
voor een dam bij niet-evenwijdige
aangroei.
T.H.D. 1972.
- Per Bruun:
The history and Philosophy of Coastal Protection •
•
London 1968.
- 97 -
Numerical example according to the method of Pelnard-Considère.
Given:
S
o
=
0.5xl0
~ at h
= -
6
3
m /year
10 m is 100
= 10 m
max
average beach gradient up to h
= -
length of breàkwater at S.W.L.
=
h
slope end of breakwater
10 m is 2%
1000 m'
1 : 2
See figure 15.
Questions
a. How long will it take before sand transport around the breakwater
will start?
b. Up to what distance from the breakwater will the effect of
accretion be noticed?
c. Assuming that a transport of 0.2 x 10
6
3
m /year around the
breakwater is admissable, determine the time, t, that this
transport is reached.
d. What can be done to prevent an increase sand transport after
time t determined in question c?
Solution
o
Figure 15
Profiles for
Sample Problem
-10
To compute tL we need the accretion curve of P~C (equation (4».
At the dam x
=
0 and the expression for tL becomes:
- 98 -
tL
=
2
.L
7f
,
4
4>2a
L
= 520 m
4>
=
a
=
S
(see figure 14)
7f
= -180
100
0
<Ph
=
therefore: tL
=
in which
K
O.~x106
7f
18
7f
= 18 rad.
10
x 10
=
9 x 105 2
m /year
7f
24 years.
b. Theoretically,
the accretion is zero at a point x
= -
00.
Practically speaking this point is at a distance 2.7 /4at'
7f
from the breakwater.
2.7 ~
7f
= 8100 m
c. Af ter time tL the accretion will occur according to (7) as a
result of changed boundary conditions. To circumvent a discontinuity between the two accretion curves, the accreted sand volumes
according to both curves are equated. This forces us to jump to
another time axis (fig. 11) and also to make a correction, because
the angle
<p
is not quite correct (table 11).
~
S
o
According to table 11: t
tL
=
1.5: t
=
_ 0.2 = 0.4
0.5
1.5 x 24
d. 1. Remove the excess sand mechanically;
=
36 years.
for example by agitation
dredging.
2. Lengthen the dam(temporary
solution)
3. Build a groin on the accreted beach. (see fig. 16)
Groin
Figure 16
breakwater
- 99 -
3
is also
between
a temporary
the groin
which
will
Sa/So
curve
transport
around
soon as a
= ~.
(a
= ~).
a storage
beyond
16b).
but
and the breakwater
become
(see figure
solution,
B will
area
it is better
will
than
be first
2. because
an area
later
on. Also,
the
be ~ess than
the slope
of the first
The erosion
the breakwater
will
occur
as long
because
a < ~.
slope
of erosion
there
will
of the
part.
be
This will stop as
Point A in the figure 16b marks the end of the erosion
At point B there will be transport around the groin:as well
as around the breakwater.
The solution for a beach having groins will
be discussed in more detail in the following chapter.
- 100 -
Chapter VII Sand Transport Along Beaches Protected by Groins
The theory presented in the previous chapters, derived from the
so-called "one-line theory" of Pelnard-Considère,
does not give ade-
quate results when thesand transport is partially interrupted. All
movements of sand perpendicular
to the beach line, so far, have been
neglected.
In order to include the effects of partial transport blockage,
Bakker has developed a so-called "two line theory". In this theory,
the shore is divided into two zones: The beach, and a deeper inshore.
Each of these zones is schematized by a single line; these lines may
behave nearly independently
of each other.
On and off shore transport of sand is allowed, thus, the assumption
that the beach maintains an equilibrium profile is no long er necessary.
beach
inshore
h
....
'"
"
Figure 1. Beach Profile showing the definition of variables.
The schematization
is shown by the dot-dash line.
The depth beyond the inshore is assumed to be so great that sandtransport no long er takes place. The boundary between inshore and beach
is usually chosen at a depth co incident with the depth at the toe of the
groins, the distances to theschematized
,
inshore line Y2 are defined from:
=
1
hl
f"
y(h) dh
o
( h1+h2
~
1
y(h) dh.
beach line, Yl' and to the
- 101 -
The transport of sand to or from the beach (perpendicular
to the beach
line) is assumed to be dependent upon the shore slope.
= 'ly"
S
y
da
I
Y
=
1
- Y2 - W
qy
~ h
= 'ly
(Y1 - Y2)
in which:
ex
W
is the shore slope
= 1h/(equilibrium
the distance
slope)
is
I
(Y2 - Y1) for which no on or off shore transport
takes place.
The dynamic equations are much the same as those of PelnardConsidère:
The Continuity
equations for beach and inshore respectively
=
+ S + hl lil
y
at
~1
ax
~2
S
ax
y
are:
0
= 0
+ h2 ~2
at
These last four equations lead to a set of two simultaneous
partial differential
equations with unknows Y1 and Y2. Analytical
solutions of these two resulting equations are sometimes possible.
For example, when q1/h1
=
q2/h2' these equations can be separated
into a "coastal equation" and
an "offshore transport equation".
The "coastal equation" is equivalent to that of Pelnard-Considère
with variabie
Y
h
= ~
hl
Y1 +
h
Y2
- 102 -
where h
The'bffshore
=
hl + h2, see figure 1.
equation" has y
=
Y1 - Y2 as a variable.
Another example yielding an analytical
solution occurs when Sl
= o.
This implies that there is no transport along the beach. The beach
serves only as a storage space for sand. This can happen whenever
groins are very closely spaced along a beach.
In other cases, numerical solution are necessary.
cussed in references
[1]
[3]
through
These are dis-
at the end of this chapter.
The "two line theory" presented above has the following advantages:
1. It allows the computation of the shape of the beach and inshore of a coast protected by groins.
This shape can be determined as a function of the parameters
~
ql' q2'
and the distance X between the groins. An important resulting para-
meter seems to be the distance
,
"2
for when X
(Sl
=
=
X
o
nearly all of the transport along the beach is stopped
0).
2. It allows prediction of the effect of a proposed groin system
on an existing shore. This prediction can include, also, the deposition
on the "up-drift" side of the first groin as weil as the scour at the
opposite end of the system. The effect of temporary changes or construction phases can also be investigated.
3. The effect of rip currents and the sand which they transport,
S . , from the beach to the inshore can be included with only minor
r:lp
modifications to the equations (see [6]).
The practical usefulness of this theory seems nearly unlimited.
However, we must first investigate the degree to which the assumptions
inherent
in this theory are satisfied in a practical problem.
Some of these assumptions are discussed further below.
The original "two-line theory" proposed by Bakker completely
ignored diffraction
effects resulting from obstacles
(groins).
Diffraction has been included in later studies, however, by making
SOl' ~02' q1 and q2 functions of the position along the shor e , x,
- 103 -
instead of holding these parameters constant. One is referred to the
literature references
[SJ
[4J and
for more details on this matter. We
might point out, here, that this same technique could also be applied to
the "one-line theory".
The special effects of radiation stress in diffraction ~ones has
not been considered. In these zones, reversal of the longshore current,
and hence the
sand transport, can occur. This has not yet been included in the theory,
however.
The effects of rip currents on the incoming waves has been neglected.
This might be important since the waves provide the driving force for the
sand transport. However, experimental evidence indicates that this effect
of rip currents may safely be neglected for now.
There remains, still, a very practical problem. The parameters q1'
q2' qy, SOl' S02 and Srip must be expressed in terms of known or measurable
wave parameters. Figure 2 shows a schematized coast, in plan and profile
views. For this situation Bakker in
[SJ
has derived the following formulas
for the necessary parameters. They include Svasek's assumption concerning
the velocity profile of the longshore current (see chapter V). It has been
shown that this assumption does not cause significant deviations from
Bijker's results.
shoal at depth hbr
at depth hl
inshore
A
r-
x
A
offshore
contours
Plan
Figure 2
Profile A-A
Schematized Shore Plan and Profile.
- 104 -
SOl
= Al
y
S02
= Al
y
q1
= Al
y
q2
= Al
y
2
F
r
2
F
2
2
r
!_L'
~r
I
h3 sin
1
g , (h3
hbr
br
F
r
Ig h5/2
F
/
r
<Pbr
h3 ) sin
1
<Pbr
1
.h
g '(h3
br
br
h3 )
1
where:
<Pbr
is the angle of incidence of the waves at the outer limit of the
shoal.
1
n
A1
=
n
= ~c
A
o
c
o
y
evaluated, here, in deep water.
is the ratio of break er height to water depth at that
point.
is the wave Froude Number
in the breaker zone (F
r
=
1 according to linear theory and F
r
= 12
x 0.78
according to limiting solitary wave theory)
c
c
A
is the wave celerity
gr
is the wave group velocity
=
0.014 is the constant in the CERC formula (equation 4 chapter V).
Table I compares some values for the various parameters. This table
was computed starting with chosen values for Al' y, and Fr'
- 105 -
Al
Al
2x1.4xlO
2x2.8xlO
2x4.0xl0
-2
Fr
-2
0.78
n.56
6.66xlO
0.4
--
11:28
-2
1.58xl0 '
11.4
2.37xl0
h.s'
7.69xl0
r2
-2
2
Im/sec'
Y
-2
y
0.4
-2
rg
Al Y
Im/yr'.
m/sec
2.l0xlO
0.50xl0
0.75xl0
6
6
6
2 F 2
r g
2
2.66xlO
5.67xl0
8.74xl0
m/sec.yr
-2
-2
-2
8.26xlO
1.79xlO
2.76xl0
6
6
6
r2
2x4.0xl0
-2
0.5
-2
2.42xl06
29.4 xl0
-2
9.26xl0
6
table I
If the CERC Formula is used in these computations with regular
-2
waves then Al has the value 2.8 x 10 . Further, if y and Fr are
taken from solitary wave theory (0.78 and 1.25, respectively) then
t~e combination of constants Al
Y
2
'r"'
F
r
vg has the value 6.66 x 10
-2
m2/sec.
We have ignored the parameter q
in this discussion so faro
y
Indeed, no adequate theoretical relation has been yet determined
which would allow its evaluation. Swart has found values for q
y
from laboratory studies. However, these were restricted in applicability
by the fact that the wave properties were kept constant in all tests.
Another complication results when we observe that qy is also dependent
upon hl' and hence, upon our choice of separation between beach and
inshore,(see figure 3).
- 106 -
I
1.0
",,-
0.8
2
<ly
,
,,
0.6
,
max.
0.4
..,,
•
,
I
"
""
-o
_- -
+
\
+
t hr-ee
.
t~o diPlens onal model
,
\
•
\
,
I
,
•
•
I
o
<,
""
,
0.2
,
,_ .... '"
,-,
•
1.0
1.5
moe el
...
'-'-
0.5
kiimersina
- -.- - 2.5
2.0
-
1-• -
3.0
hl
hbr
Figure 3
Relation between q and depth of separation between inshore
and beach.
y
is another parameter that must be evaluated. This can be
S'.
rlp
estimated by using the theory of Bowen [7] to find the rip current
Bijker's method of computing the transported quantity of sand may then
be applied using this current.
We have seen above that the "two-line theery" has the practical
difficulty
that not all of its necessary coefficients
can be evaluated.
Thus, this theory must be considered to yield qualitative rather than
quantitative
results, at least, for the present. Further developments
are going on.
Certain elements of this "two-line theory" may, however, be worked
into the "one=-line theory". With this, qualitative results may still be
possible.
Several computer programs have been developed to carry out the
computation
involved in this work. These have been developed cooperatively
by the Delft University of Technology and the Ministry of Public Works
(Rijkswaterstaat).
Figure If shows some results obtained from the "two-line theory".
+
- 107 -
Wave direction
16 /:;
8 t:"
4
4t:.t
st:.t
16t:.t
16/:;t
8t:"t
4/:;t
groin
groin
Figure 4
Development
of beach and inshore lines
(two-line theory)
- 108 -
Related
literature:
1. Bakker,
W.T.:
One Aspect
of The Dynamics
of a Coast,
Partly
Protected
by a Row of Groynes.
2. Bakker,
Coa~t
W.T.:
with
The
Influence
a Harbour
W.T.:
The Dynamics
4. Bakker,
W.T.:
The
5. Bakker,
7. Bowen,
of a Coast
Influence
with
W.T.:
W.T.:
A.S.:
Klein
on the Dynamics
of Diffraction
a Groyne
Near
Shape.
The Dynamics
6. Bakker,
Transport
of a
Mole.
3. Bakker,
Coastal
of Offshore
Breteler,
of a Coast
with
De Dynamica
Rip-Currents.
van
E.H.J.;
a Groyne
Kusten.
Roos,
A:
System.
System.
a Harbour
Mole
on the
- 109 -
Chapter vnI
Wave Forces on Piles
Forces on piles have become a major topic of interest
engineering
since the construction
The circular cylindrical
insensitive
both longitudinal
and transverse,
of large drilling platforms
pile is most commonly used because
to the direction
components
in coastal
began.
it is
of waves or current. Wave forces have
(in the direction
of wave propagation)
or lift, forces. These lift forces act in a direction
parallel to the wave crests.
In the following equations,
it has been assumed that the ratio
of pile diameter to wave length is small
«
0.1) and that the wave
height is small relative to the water depth.
Longitudinal Forces
The longitudinal forces can be divided into:
a. Drag Forces, and
b. Dynamic Forces, sometimes called inertia forces.
a. Drag Forces
A fixed pile in a current experiences a force resulting from
friction and eddy formation. For a cylinder of unit length with
its axis perpendicular to the current~ this drag force per unit
length may be expressed as:
FD =
1
IJ
.2
V2 CD D
(1)
where:
IJ
is the density of the flowing fluid (water)
V
is the stream velocity
D
is the pile diamater
CD
is an experimental drag coëfficiënt
Equation (1) was originally derived for the case of a constant strcam
velocity V. For our application we must make the following modifications:
1. V is no longer constant. At a given position in waves,
V
=
wH
2
cosh k(z+h)
sinh (k h)
cos (wt)
(2)
These parameters are weIl known (see chapter 111 equations 4-6).
- 110 -
2. Since the drag force always acts in the direction of the
instillltaneousvelocity,
cosh
2
v2
in (1) should be replaced by V
k(z+h)
cos ( ert )
I
cos (wt)
I CD
lvi.
(3)
D
sinh2 (k h)
The coefficient
CD has various values, but its magnitude is usually
in the range 1.1 to 1. 5.
b. Dynamic Forces
Because the pile is placed in an unsteady flow, additional force
components arise resulting from the continuously
changing acceleration
of the fluid.
One of these dynamic force components can most easily be investigated by oscillating a pile of unit length in still water. The velocity
of the pile is made equal the corresponding wave partiele velocity,
(2). Figure 1 shows potential flow around such an oscillating
equation
pile at various times.The observer is assumed to be moving with the
pile.
wt
wt
=
=
0
TI
wt =
2
+ ..!
2
Figure 1 Flow patterns around a pile
It is obvious from figure 1 that fluid around the pile is being
continuously accelerated:. The force necessary to pro duce this
acceleration
can, theoretically,
00
be determined from
00
J
J
-00
-00
dx dy
(4)
where a unit thickness (pile length) is still assumed and v is the
water partiele stream velocity at point x, y at some time.
- 111 -
Evaluation of this integral is messy. Lamb suggested another procedure.
His procedure was to compute the energy in the flow pattern at an instant
and set this energy equal to the energy of a virtual mass of water moving
with the same instantaneous velocity, V, as the pile. The virtual mass
found in this way is equal to
m
.:_P_ 'fT_.:,.- D2
-
v
(5)
4
Thus, from Newton's Law:
F
I
av
= m vat
Using (2) we can evaluate
av
at
w2 H
= --2
(6)
av
at
cosh k{z+h)
sinh (k h)
sin (wt)
(7)
Yielding, for a unit pile length:
cosh k{z+h)
sin (wt)
sinh (k h)
(8)
A second longitudinal dynamic force component results from the
fact that in the real situation (fixed pile and waves) there is a
change in water surface elevation as we move across the pile.
This results in a "quasi-hydrostatic force" which can be shown to be
in phase with the inertia force. This can easily be combined with
equation (8) if we introduce a coefficient. This coefficient, experimentally determined, can also correct for the fact that our potential
flow assumption is very idealized. (8) becomes:
F
I
tr D2
C ,,-P--:---
= -
M
4
cosh k{z+h)
sinh (k h)
sin (wt)
(9)
The total lateral force per unit of pile length may be evaluated
by adding (9) and (3).
FX'
=
FI
+ FD
P
=
+
CD
D2
'fT
CM
4
.e..J2
2
cosh k{z+h)
sinh (k h)
sin (wt)
2
cosh k{z+h)
sinh2 (k h)
cos (wt)
I cos{wt) I
(lO)
- 112 -
This equation is sometimes called the Morrison Equation.
Fortunately, the two force terms, FI and Fn' are not in phase
with one another. Thus, this keeps the maximum force a bit lower
than it might otherwise be.
It has been found that the Morrison Equation often gives conservative predictions.
This may result from the assumption that the pile
remains rigidly in p Lace, This may be corrected by reducing the magnidu
tude of the inertia force by an amount mp dt' where mp and u are the
mass and velocity of the pile, respectively.
Equations
(10), when
corrected is:
Fx
w2 H - cosh k(z+h)
sin (wt)
2
sinh (k h)
=
+ C
+m
n
.e......Q
2
W
2 H2
4
2
cosh k(z+h) cos (wt) Icos(wt)1
sinh2 (k h)
du
p dt
( 11)
The total longitudinal
force acting on the pile may be determined
=
=
by integrating F over z with integration limits z
-h and z
o.
x
The moment acting on the pile can be obtained from a similar integration
including the moment arm.
Until now, we have assumed that the pile was small relative to
the wave length. Another procedure must be used if this assumption
cannot be met. Using potential flow theory and superposition,
a circle
of sources is placed along the perimeter of the pile. Each source
emits waves of proper amplitude and phase to guarantee that the resultant velocity component perpendicular
to the pile wall remains zero.
Integration of the resulting superimposed pressure fields around the
pile perimeter results in the desired force.
Lift Forces
Lift forces on a vertical pile act horizontally
line of the longitudinal
perpendicular
to the
forces. The frequency of these lift forces is
determined by the frequency of vortex shedding in the Kármán Trail
behind the pile. Thus, the frequency of this force has little if any
relation to the wave frequency. These forces act in a constant stream
as weil as in waves and are relatively easily found in nature.
- 113 -
The lift force frequency
is sometimes five times the wave frequency.
This can lead to unexpected
resonant vibrations of the structure
or its parts. These vibrations
can quickly result in fatique failure
of structural members or their connections.
A significant parameter
for these lift forces is the Keulegan-
Carpenter Number, Kc. This parameter
Kc
=
n H
D
is defined as:
cosh k(z+h)
sinh (k h)
and looks somewhat like the reciprocal
(12)
of the Strouhal Number.
The lift force becomes noticable when Kc is greater than about
3. This force then increases with Kc, reaching a value of around ~ the
longitudinal
force for Kc Z 15. These are the results of laboratory
studies more completely
bibliography).
reported in a paper by D.D. Bidde (see
- 114 -
Chapter IX
Offshore constructions
As a result of the ever increasing human activity on the bottom of
the sea there has been
rather·hectic recent development in the field
of off-shore constructions.
The designer must consider the following factors:
a. weight of the structure
b. wind, wave, and current forces
c. resonance of the structure
d. scour around the legs
e. necessary height above the S.W.L.
Each of these is discussed in more detail·below.
a. The weight on the structure ca~ run into the thousands of tons making
a good foundation necessary. The vertical elements of the supporting
structure are usually hollow through which piles can be driven into the
bottomthus providing horizontal as weIl as vertical support.
b. wave and current forces are treated in the section on forces on
piles. Wind forces have been treated in other courses. An analytical
solution of the mechanics of the structure's response to wave forces
has not yet been found, therefore the structural design of the member
connections is based upon past experience.
c. Resonance of the structure which can result in failure because of
metal fatigue can be avoided if the natural frequency of the structure
is considerably different from the wave frequency.
f
=
1
T
=
struct.
w
=/fm
f
=
k
m
=
=
w
27T
spring constant
mass of the platform + 0.23 mass of the piles
if the wave period is 6 sec.
Here, w is in radians/sec. and f is in Hertz.
d. Bottom scour around the legs can be avoided when filter layers are
used which can reduce the water velocity to such an extent that sand is
no longer transported. Sometimes the individual vertical elements can
be driven further into the ground. This is mainly done with semi-permanent structures where the larger capital investment in the form of
- 115 -
jacking installations and heavier legs is justified by a savings
resulting from eliminating filter constructions. In this case the
supporting structure cannot be built as a frame work.
e. The height above the S.W.L. has to be sufficient to prevent the waves
trom reaching the platform since this structure is not design to act as
a breakwater.
The original offshore structure was a fixed structure and is still
used in the oil industry as a exploitation platform. Several oil wells
are connected with the platform from which the oil is pumped ashore or
into tankers with the help of S.B.M. s. Sometimes the platform is built
to act as a mooring tower also. Recently more and more of these fixed
structures are being used to replace lightships and weather ships.
Semi-permanent offshore structures are used for oil exploitation
and in the construction industry. When the platform hasto be moved to
another location, the platform is lowered into the water after which
the legs are hoisted. The high center of gravity and the wind forces on
the legs make the system unseaworthy; in case of longer hauls, the legs
are often lifted out of their soekets and carried on the platform.
Since this is a difficult operation, the legs are often provided with
hinges.
In the construction industry the semi-permanent structure is often
self-propelled. The leg soekets are mounted eccentrically in turntables
and the legs are in turn jacked free of the bottom and turned in the
directions of movement and jacked into the ground again. When all legs
have been moved forward all turntables turn simultaneously to move the
platform (see figure 1). Other systems employ legs that move in slots
or pairs of legs on each turntable to insure continuous movement of the
platforms. The last system is used in dredging. The velocity of the
platform is rather low, the maximum at this moment is approximately
8 m/hr. The other systems are even slower and can be used for tunnel
and breakwater construction. Transport to other construction sites
still has to be done by towing.
- 116 -
3
2
1
Figure 1
6
5
4
Sequence of operations of a walking platform
As the oil industry feit the need for exploitation
in water deeper
than 100 - 120 m the serni-submersible platform was developed. The semisubmersible has to satisfy the following conditions:
1. easy towing
2. good anchoring possibilities
3. minimum movement when in operation
These conditions depend on shape, dimensions
and anchoring systems.
The last condition can be met when the legs are thin relative to the
floating bodies that provide the buoyancy for the system. The floating
bodies have to be deep enough so as not to experience too much wave force
resulting from orbital motion. If this is the case and the natural period
is two to three times as high as the wave period and the center of
gravity is low enough, the movement of the systern is usually small.
(see fig. 2)
Figure 2
semi-submersible
platform
- 117 -
Chapter X
Offshore Mooring Structures
Single Buoy Moorings.
The development of the S.B.M. is due largely to the increase in
size of oil tankers. Since the costs of building relatively
harbors that meet the depth requirements
small
of large tankers are prohibi-
tive, the idea rose to load and unload tankers off-shore.
Figure 1 shows a conventional
Single Buoy Mooring.
mooring cable
floating hose
Figure 1
More complex systems, developed to dampen horizontal
motion of a
tanker, turned out to be ineffective because the mass and water
displacement
of the mooring buoy is negligible
with respect to the
mass of the ship. The S.B.M. shown is anchored to the bed by a
series of cable or chains attached to fixed anchors. A pipeline on
the botton is connected with the S.B.M. by means of a flexible
pipeline, or hose. A floating hose runs from the buoy to the tanker.
The tanker is moored to the buoy by means of bow hawsers. This
system can withstand
considerabie wave-action.
possible up to a waveheight
Mooring is still
of 2.4 m and load transfer can continue
until the significant waveheight
reaches about 5 meters. Higher waves
can cause breaking of the flexible pipelines and endangers the lives
of the crew on the deck of the tanker.
This system has two disadvantages
1. Forces in the bow hawser can be higher than when the tanker is
moored to a fixed point. The tanker hardly reacts to the individual
.short waves(Length of the tanker is much greater than the wave length).
- 118 -
The S.B.M., however, will try to follow all of the water motions
resulting in high cable forces. We might say that the buoy is
moored to the ship.
2. The system needs two expens~ve pipelines. The hoses depreciate
rapidly and need frequent replacement.
A further development was the single point mooring tower. This tower
has a rotating platform about 20 m above S.W.L. supported by a
number of piles.
The ship moors to the platform which turns to allow the ship to
seek an optimum heading in the waves and currents. A ho se connects
I
from the platform to the ship.
This system has three advantages over the S.B.M.:
1. Only one flexible hose is needed.
2. There are smaller forces in the mooring lines.
3. An extra pumping station can be installed on the platform acting
as a booster.
The booster pumping station can increase the unloading speed of
the ship if the distance to shore is significant. This is important
because of the high cost of ship operation
(as high a 2000 Dutch
Guilders per hour).
On the other hand, the initial cost of such a mooring tower is
much more than a buoy mooring. Also, a buoy mooring may be more easily
moved to a new location, if desired.
Moorings are undergoing rapid development.
One is referred to the
literature for news of these developments and of specific applications.
Recommended literature:
E.H. Harlow
- Offshore Floating Terminals
Proc. A.S.C.E. WW3, vol. 97, aug. 71, pp. 531-548
Ch. Foulladosa - Collo Int. sur l'exploitation
Super-Petroliers
des Oceans/Amarrage
des
- Bordeaux, Mars 1971
Grubas, Marras - New Design Concepts and Advanced Technical Solutions
for Gil Marine Terminals
Coll.Int. sur l'Exploitation
des Gceans.
Lawrence Solna and David Cho - Resonant Response of Offshore Structures
Proc. A.S.C.E. W.W.1, Febr. 72.
- 119 -
Chapter XI
Submarine Pipelines
Off-shore submarine pipelines may be laid on the sea bottom or
buried in the bottom material. Pipelines laid on the sea bottom in shallow
water are subjected to wave and current forces which are much the same as
forces on vertical piles. These force components are:
a. Drag
b. Inertia
c , Lift
Because of the close proximity of the sea bottom, the lift force
component becomes somewhat more complex. The lift force resulting from
the vort ex trail remains, but the situation is complicated by the presence
of other effects.
Figure 1 shows the flow patterns around horizontal pipelines near a
fixed, flat bottom.
---------------------
,,'<IS, «\<
( ""<'<I
\ ","
....<:<: "" "
\,
'" < " "c; "" ,
Figure 1
Flow patterns around submerged pipelines.
In figure la the distance between the pipe and the bottom is
relatively large. Because the velocity above the pipe is higher than that
under the pipe (from the conventional velocity profile), the pressure
differences result in an upward lift force. When the clearance between the
pipe and bottom becomes smaller (figure lb), flow concentration under the
pipe increases veloeities in that region; a downward lift force results.
Finally, when the pipe rests on the bottom as in figure ie the lift force
again acts in the upward direction.
- 120 -
In reality,
nor fixed.
the
Whenever
increase
flow
of the bottom
this
time.
figure
Figure
Since
2
velocity
will
Result
does
however,
spreading
figure
imply,
to other
3.).
cross
that
force
1b will
equal
is neither
in figure
result
flat
1b is present,
in local
is exerted
erosion
on the pipe
come to look more
equilibrium,
to the pipe
at
like
the space
diameter.
scour.
decreases
cross-section
will
this reaches
be approximately
at a particular
not
lift
of figure
When
of bottom
the velocity
the pipe
A downward
some time.
on a bottom
such as is shown
under
The cross-section
the pipe
rests
a condition
material.
2 af ter
under
our pipeline
that
as the clearance
tends
the situation
to come to an equilibrium.
everything
sections
increases,
along
is stable.
the
length
The scour
of the pipe
This
continues
(see
- 121 -
a
Pipeline sa~ed
scour continues
b
Figure 3
Result of scour along pipe length
As the situation shown in 3b is approached, with the points of support
becoming more widely spaced, the pipe sags between the supports due to
its own weight. This reduces the clearance re-initiating
the erosion
cycle. This sag of the pipe can also lead to structural failure of the
pipe itself. In the breaker zone, this process of erosion and sag
results in the pipe finally burying itself. It has not yet been determined with certainty whether this also happens offshore.
All of this discussion leads us to a conclusion that it might be
better to bury the pipe initially. The pipe is then protected from
wave forces, fish nets, and anchors. Some anchors can dig as much as
3 meters into the bottom.
The pipeline is usually buried in a dredged trench, after which
it is covered as soon as possible to prevent scour from still taking
place. The pipe is exposed to scour-causing currents while laying in
the trench because the trench must be very wide with flat slopes. The
bottom width of a trench must usually be 5 to 10 times the pipe diameter.
This is necessitated,
largely, because it is so difficult to accurately
position a pipeline at sea. The side slopes of the trench must usually
be flatter that 10 percent. This is to prevent the natural currents
from back-filling the trench before the pipe is laid.
- 122 -
It is, as yet, difficu1t or even impossible to predict the rate of this
natural re-deposition
of material in a given situation. Such wide trenches
are obviously expensive to dredge because of the large quantity of
material to be removed.
Back-filling of such a trench presents another set of problems,
however. When loose sand is dumped above the pipeline, it can, by mixing
with the water, behave as a very dense liquid (p ~ 2). Since this
mixture is heavier that the pipeline
(p ~ 1.1), the pipe can float on
this mixture; thus, we can find our pipe on the sea bottom above a now
filled trench! Since coarse gravel is much less likely to exhibit this
dense liquid behavior, this is sometimes dumped in place of sand. This,
however, involves the danger that a large mass of gravel falling against
the pipe may crush the pipeline.
Another, entirely different possible means of burying a pipe
involves forcing it to sink into a region of artificially-made
quicksand.
Figure 4 shows such a scheme.
Pulling force
\.
J \'
pipeline in place
Figure 4 Sinking pipe in artificial quicksand.
Heavy U-shaped weights are placed over the pipe. Water jets are used to
create alocal
quicksand condition allowing the wèight to sink the pipe
to the desired depth. The entire system of jets and weights moves slowly
along the pipe. This method works best, of course, in sandy soils.
In clay or peat soils, water jets mayalso
However,·since
be used to sink a pipe.
the jets permanently alter the properties of the distllrbed
soil, there is a good chance that this soil will not regain its stability
after disturbance.
Erosion
usually results, leaving the pipe exposed.
All of these jetting methods can run into difficulty when large rocks or
boulders are present in the soil. Since this is often the case, water jet
techniques are not commonly used as yet.
- 123 -
Of ten , in water
a pipeline.
deeper
(The danger
is still
necessary
concrete
collars
Figure
These
5
ripples
long.
Little
work
weIl
bottoms
and remain
similar
to more
If this
common
be deeply
to a depth
of 20 meters
designer
can only
pray
bottoms.
to bury
Ie ss there.)
may
be done
If it
with
5).
They
about
pipelaying
them.
bottom
Some
Other
however,
on
problems
experts
experts
in the North
high
contend
claim
Sea.
and 200 meters
that
that
these
they
are
move
ripples.
is true,
buried
fail,
mud.
can be 5 to 20 meters
for pipeline
alternately
this
(see figure
as sling
special
claim
necessary
is much
in place,
on rocky
in position.
latter
problems
and anchors
anchors
such
on the bottom
is known
it is no longer
anchor
even
present
stabie
nasty
screw
of screw
soft unstable
These
from nets
or with
Megaripples
50 meters
to fix the pipe
Concept
methods
very
than
then megaripples
foundations.
and exposed.
is not practical
that problems
that
Sections
Initial
can present
of the pipe
burying
and perhaps
arise
will
some
will
of the pipe
impossible.
take
A
care of
themselves.
Laying
of Pipelines
Three
methods
available
a. Sinking
b. Pulling
c. Use of a lay barge.
to bring
a pipe
into place
under
water
are:
- 124 -
The sinking method involves lowering an entire pipeline into
position in one operation. Floating cranes can accomplish this work,
but the length of pipeline that can be placed is severely limited.
This method is well suited to river crossings, however. Care must be
taken during lowering of the pipe to prevent air pockets from forming
inside the pipe. Resulting bouyant forces can cause severe stresses in
the pipe walls.
The second method uses h~avy winches to pull a section of pipe
into position by dragging it from the shore. The leading end of the
pipe must be fitted with a sled to prevent its digging too deeply into
the bottom. A properly shaped sled can act as a plow, helping to
partially bury the pipe. (see figure 6).
I
\\","", " 1..6L._p_i_p_e
__ +r
Figure 6
_...
,<
<\
"'\
v v , \\'
Sled for leading end of pipeline
Very powerful winches are required to overcome the friction forces.
These forces can become extremely large if the pulling mot ion has been
stopped.
The distance over which a pipe can be pulled into place is limited
by this friction force. For distances greater that about 10 km a lay
barge will be required. Figure 7 shows a conventional
lay barge.
- 125 -
Pipe assembly
stations
barge
figure 7
Lay barge
When working at great depths, the bends in the pipeline can result
in high pipe wall stresses. Prestressing
by placing the entire suspended
pipe under tension can reduce this bending effect. The upper of the two
bends is sometimes eliminated by assembling the pipe segments along a
slope. Unfortunately,
heigh limitations
restrict the number of working
places available along the pipe, if a slope is used.
Specifications
for a particular
lay barge include:
barge: length: 105 m
beam:
draft:
24
m
7 m
stinger length:100 m
maximum laying depth: 200 m
pipe diameters handled: 20 cm to 80 cm.
5 welding stations and 1 X-Ray station are provided.
70 meters of concrete coated pipe can be laid per hour.
The reel-type laying barge represents
developments.
one of the latest technical
A long length of pipe, prefabricated
onto a barge-mounted
on shore, is wound
reel. At the desired location, the pipe is unrolled,
straightened with rollers, and placed very quickly in continuous lengths.
Concrete coatings, normally used to ballast the pipe, cannot be used now;
Thicker steel pipe walls accomplish this same effect.
- 126 -
Presently available reel systems can carry 31 km of 15 cm diameter
pipe or 7! km of 30 cm pipe. These systems can lay 450 meters of pipe
per hour. This is a practical limitation; anti-corrosion
field-installed
anodes must be
at about 350 me~er intervals. Coatings must also be
checked for damage caused by the straightening rollers and repaired if
necessary. If coatings are applied on the barge after the pipe is
straightened,
speeds of 3000 meters per hour can be possible. This
latter proposal is still plagued with technical difficulties,
however.
Another practical handicap for reel-type pipe laying is the pipe
size limitation - presently 30 cm. The radius of the reel core must be
at least 40 times the pipe diameter. This yields a reel core diameter
of 24 meters for 30 cm pipe. Larger pipes will require immense reels
and barges.
- 127 -
List of Symbols
The following
is a list of the most important symbols used in these
lecture notes.
Syrnbol
A
Definition
Dimensions
Measure of fluctuations
ln channel bottom
elevation
A
Parameter
L
H -br
hbr
A
Parameter r ln Einstein's
h
A
Coefficient
relating
Formula
sand transport
to
wave energy
A
0.014
Coefficient
Unit of horizontal
L2
area
a
Wave amplitude
a
Coefficient
C
Minimum ship keel clearance
C
Chézy friction
C
Relative beach slope
L
3.
L2 T
h
L
1
L2 T
coefficient
Drag coefficient
Chézy friction coefficient
for plane rough
bed
CM
Added mass coefficient
C
Chézy friction coefficient
r
c
Wave speed
c
Concentration
c
a
Concentration
-1
for river
at a distance,
a, above bottom
-1
- 128 -
Symbol
c
o
Definition
Wave speed in deep water
Dimensions
L T-1
D
Grain size
L
D
Pile diameter
L
D
r
Ship draft
L
E
Wave energy per unit area
M L2 T
E
Wave energy flux component
M L2 T
E
Wave energy flux in deep water
M L2 T
FD
Drag force per unit pile length
M T
Fr
rnertia force per unit pile length
M T
F
r
Radiation stress effect
F
Wave Froude number
F
Turbulent force
a
o
r
s
M L
FT '
Tide force
F
Total longitudinal force per unit length
x
F'
Total longitudinal force per unit length
f
Darcy-Weisbach friction factor
f
Frequency of vibration
f.(T)
Spectral density of ship motion
x
Hertz
1
f (T)
n
Spectral density of wave motion
G
Water level set-up
g
Acceleration due to gravity
H
Wave height
L
L
-3
-3
-2
-2
-1
Tide force per unit area
-3
-2
T
- 129 -
S)Tlbol
Definition
Dimensions
H
Wave height in deep water
L
h
Channel depth
L
h
Water depth
L
h
Water depth at toe of accretion
L
h
Depth at outer edge of inshore
L
h'
Height of tidal wave
L
Depth at outer edge of break er zone
L
Depth in beach zone
L
Height of inshore zone
L
I
Allowed vertical ship motion
L
I
Slope of energy line
o
Parameters ln Einstein's Equation
i
Vertical ship displacement
K
c
Keulegan-Carpenter
L
number
K
Refraction coefficient
k
Wave number
k
Equivalent spring constant
L
Wave length
L
L
Length of accretion to breakwater end
L
1
Mixing length
L
r
2
-1
L
'TT
L
6 S
M
Coefficient
[ __
t
0_
1
J~
C q,2
m
Energy of ship mot ion spectrum
m
Beach slope
L
- 130 -
Syrnbol
m
Definition
Dimensions
Total equivalent
m
p
m
v
nb
Mass of pile per unit length
Virtual
o
or added mass per unit length
Ratio of energy speed to wave speed in
breaker
n
M
mass
zone
Ratio of energy speed to wave speed in
deep water
P
o
Hydrostatic
Horizontal
force per unit leng~h
momentum
flux
=
p
Proportionally
D
.
HY drostatlc
pressure
q
Statistical
expectation
q
Coastal constant
=
Coastal constant
in y direction
q'
Coastal constant
in y direction
q"
y
Coastal constant in y direction
-0
y
Coastal
constant
S
0.45
M L-1 T-2
o
constant for beach
Coastal constant for inshore
R (T)
Response
r
Height of virtual bottom roughness
S
Bottom material
S
Total sand transport
Transport
factor
transport
per unit width
of bottom material
L
- 131 -
Definition
Symbol
S
Sediment transport along bottom
be d.Ioa d
-1
L3 T
Sand transport in rip current
L3 T
S
Transport of suspended material
L3 T
Total sediment transport
L3 T
Total sand transport at point x
L3 T
-1
S .
rlp
s
S
tot
S
x
sxx
(1)
SS,
xx'
-1
-1
M T'"'2
Radiation stress component
M T
(2)
(3)
Sxx
Components of SXX
-2
-2
M T
Radiation shear stress
M T
S
xy
Radiation shear stress component
M T
S
Sand transport perpendicular to beach
Syy
Principle radiation stress
M T
S
yy
Radiation stress component
M T
-2
SXy
(1)
SS,
yy'
(2)
yy
(3)
S
yy
Subcomponents of Syy
-1
S
Sand transport along undisturbed coastline
L3 T
S
Ol
Sand transport along undisturbed beach
L3 T
S02
Sand transport along undisturbed inshore
L3 T
S2
Sand transport along inshore
L3 T
T
Wave period
T
Wave period with respect to moving ship
T
Natural period of oscillation of structure
T
s t r uc t,
-2
-1
L3 T
e
-2
-2
M T
Sand transport past breakwater
0
-2
L3 T
Ss
T
-1
Principle radiation stress
xx
y
T
Dimensions
-1
-1
-1
-1
- 132 -
Symbol
Definition
Dimensions
t;
time
u
Horizontal
component
Horizontal
wave velocity
T
L T-l
of velocity
component
along bottom
-1
L T
amplitude
of ub
L T-1
u
Amplitude
of wave orbital velocity near bottom
L T
v
Velocity
v
Unit volume
v
Longshore
v
Stream velocity
L T
vs
Forward speed of ship
L T
v
Velocity
L T
v
Current velocity
L T
Current velocity at top of laminar sublayer
L T
Average velocity
L T
o
v
er
L T
-1
-1
L3
current velocity
L T
component
-1
-1
-1
-1
-1
-1
33
v
o-r
in boundary
layer
Current velocity at top of laminar sublayer
L T
Resultant
L T
velocity at top of laminar sublayer
-1
-1
-1
-1
Shear velocity
L T
VI
Modified
L T-1
W
Equilibrium
*'
shear velocity
distance between beach and inshore
lines
w
w
X
Vertical velocity
s
o
L
component
Fall velocity of particle
in water
Parameter for groin spacing
- 133 -
Symbol
Definition
Dimensions
x
horizontal coordinate
y
Parameter z
L
h
Distance to beach line
L
Distance to inshore line
L
Distance to inshore line
L
Computed value from eqn. 5, ch. 6
L
Computed value from eqn.15, ch. 6
L
z
Vertical coordinate
L
z
Ship squat
L
z
Vertical coordinate
L
z'
Laminar sublayer thickness
L
z
Amplitude of tidal wave
L
y'
2
Beach slope
Cl.
Wave direction relative to ship
Slope of beach at depth h
(3
Angle of beach line at breakwater
y
Ratio
Hbr
hbr
Psand - P
Relative density of sand in water ---------P
dz)
Mass diffusion coefficient
n
Wave profile
L
K
Wave nurnber of tide
L
K
Van Kármán constant
Wave length
=
0.4
L
-1
- 134 -
Syrnbo1
~
Dimensions
Definition
Ripple factor
Coefficient
P K C
;g
M L
-3
p
Density of water
°a
Standard
deviation
of channel roughness
Ok
Standard
deviation
of channel roughness
T
Bottom shear stress
M L
T'
Total bottom
M L
T
Bottom shear stress caused by current
M L
Resultant
-1 -2
M L
T
T
c
r
L
and waves
shear stress
average shear stress
~
Angle of wave incidence
~b
Angle of wave incidence
~
't'br
Ang1e of wave incidence at outer edge of breaker
in breaker
L
-1
-1
-1
zone
zone
~,
't'br
Ang1e of wave incidence at seaward
Circular
frequency
w
Circu1ar
frequency
w
Wave circular
w
Circular
side of shoal
T
of tide
frequency
T
2
T
TI
frequency of structure
T
2
T
TI
struct.
T
-1
-1
-1
-1
T
T
T
-2
-2
-2
- 135 -
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All references used in compiling these notes are listed below. All are
written in English unless otherwise noted.
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