Movable-Bed Laboratory Experiments ^ `^^`^^^^ Comparing

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Movable-Bed Laboratory Experiments ^ `^^`^^^^ Comparing
' " ^ ^ ^ ^ WA"reRLOO»»KUNDia t^ABOHATOHIUM
Bibliotheek Laboraloiium
Vooisl
Poslbua 152 • 8300 AD EMMELOORD
MR 81-4
Movable-Bed Laboratory Experiments ^ '^^'^^^^
Comparing Radiation Stress and
Energy Flux Factor as Predictors
of Longshore Transport Rate
by
Philip V i t a l e
M I S C E L L A N E O U S REPORT N O . 81-4
APRIL 1981
Approved
for p u b l i c
distribution
releases-
unlimited.
U.S. ARMY, CORPS O F ENGINEERS
COASTAL ENGINEERING
RESEARCH CENTER
Kingman
Building
Fort B e l v o i r , V a . 2 2 0 6 0
MOVABLE-BED LABORATORY EXPERIMENTS COMPARING
RADIATION STRESS AND ENERGY FLUX FACTOR AS PREDICTORS
OF LONGSHORE TRANSPORT RATE
Philip
1.
Three-dl.enslonal
radiation
stress
INTRODUCTION
movable-bed
and e n e r g y
Y^tale
laboratory
flux
factor
tests
^'."„^f^f/i!
f "S„y
as P ^ ^ i " » "
S^Ïtal Engï-
::rrinrararorc;nte?.s ' T ^ ' k : : : ' ^ ^ .
r
r a ^ r , = / r : : - ; L e : L - t ^ i p o ^ i r r ; ^ ^ ^ ^ f.:
data,
and
performs
the data
r - i l a b i '
Center
analyses.
f^ol™
(CEIAC).
II,
Many
P^»"^^";^^"^^"
tslnrrïng
-tstÏl
l l Z l l .
of
and
test
.alysls
EMPIRICAL RELATIONS
two e x p r e s a r e r e l a t e d e - p i r i c a l l y to the
One, r a d i a t i o n s t r e s s , i s b a s e d on momens i o n s r e p r e s e n t i n g wave c o n d i t i o n s
Tum" f l u x ; the o t h e r on energy f l u x .
An' i m p o r t a n t c o n c e p t which i s a l s o used
i n the d a t a a n a l y s e s i s t h e s u r f s i m i l a r i t y p a r a m e t e r .
The
1.
longshore
transport
data
Momentum F l u x .
beS-^r^^. ' ^ r ^ ^ ^ ^
ro:g:e^\-vr<it.orxr~ r oore.t-r„r;oLnr.T. to t.
. res—obirr
g e n e r a t i o n of l o n g s h o r e
The
currents.
c o o r d i n a t e system
used
i s shown i n F i g u r e 1.
shorlfine^rtheTaxil
i s normal
the
a t the S t i l l w a t e r
T
z-axis
^
called
'
:
originates
:
^
^
the r a d i a t i o n
^
'
V
stress
Figure
1.
^
to the s h o r e l i n e
and
The y - a x i s
positive
i s a l o n g the
^^o^eward
an
^l^J\^''^^^^^^JZ%~x.ox.e
tTT-^Z^^
t h a t J com^only
level
tensor.
C o o r d i n a t e system f o r momentum
flux derivation.
11
A c c o r d i n g t o s m a l l - a m p l i t u d e wave t h e o r y , t h e components o f t h e water
p a r t i c l e v e l o c i t y i n t h e x- and y - d i r e c t i o n s f o r a wave t r a v e l i n g a t an
a n g l e , a, t o t h e s h o r e l i n e ( F i g . 1) a r e , r e s p e c t i v e l y .
velocity,
t h e wave group
E = - 4 ^
H gT cosh [ k ( z + d ) ]
„
f
^ ^ , . — ~ cose cosa
2 L
cosh kd
(1)
H gT cosh [ k ( z + d ) ]
„ .
V = -T- f
'• ^ , . — — cos6 s i n a
2 L
cosh k d
(2)
u =
where C i s t h e wave phase
the wave energy d e n s i t y
velocity,
and
E
(5)
where ^^-^g i s t h e root-mean-square (rms) wave h e i g h t .
The term i n parentheses i n e q u a t i o n ( 4 ) i s t h e f l u x o f wave energy per alongshore d i s t a n c e ,
F , assuming s t r a i g h t and p a r a l l e l b a t h y m e t r i c c o n t o u r s .
When zero wave
energy d i s s i p a t i o n i s assumed,
where
The
H
=
wave h e i g h t
g
=
a c c e l e r a t i o n of g r a v i t y
T
=
wave p e r i o d
L
=
d
=
water
k
=
wave number
6
=
wave phase.
F = EC cosa = c o n s t a n t
X
g
wavelength
I n t h i s r e p o r t , d i s s i p a t i o n i s assumed t o be zero up t o t h e breaker zone;
t h e r e f o r e , F^^ i s c o n s t a n t from deep water t o the breaker zone.
Since t h e
r a t i o o f s i n a t o C i s c o n s t a n t due t o S n e l l ' s law, e q u a t i o n ( 4 ) , which
r e p r e s e n t s the alongshore wave momentum e n t e r i n g t h e s u r f zone, i s c o n s t a n t
seaward o f t h e breaker zone.
depth
E q u a t i o n ( 4 ) can be r e v i s e d f o r a p p l i c a t i o n o f monochromatic waves, as i n
t h i s r e p o r t . For such wave c o n d i t i o n s , t h e average wave h e i g h t , H, measured
d u r i n g t h e t e s t s (and discussed l a t e r i n S e c t i o n IV) i s e q u a l t o ^j^g- By
r e w r i t i n g equation ( 4 ) ,
l a s t two terms a r e d e f i n e d as
-1^
..—2
S
= f£SH^ C cosa)
cosa] ^ _
xy
\ 8
g
/ C
and
= kx
where
t
i s t i m e , and
(6)
ü) the wave a n g u l a r
(7)
S
i s now d e f i n e d f o r use w i t h l a b o r a t o r y monochromatic wave d a t a .
Note
t h a t e q u a t i o n ( 4 ) i s v a l i d f o r any wave c o n d i t i o n ; e q u a t i o n ( 7 ) i s v a l i d o n l y
f o r c o n d i t i o n s where H equals 11^.^^^^.
frequency
2ir
2.
The y-momentum ( a l o n g s h o r e momentum) per u n i t volume i s pv where p i s
the water mass d e n s i t y .
The f l u x o f t h i s momentum i n t h e x - d i r e c t i o n
(onshore) per u n i t alongshore d i s t a n c e and u n i t water depth i s pvu. I n t e g r a t i n g over t h e water column and a v e r a g i n g over time produce t h e mean a l o n g shore momentum f l u x i n t h e x - d i r e c t i o n per u n i t alongshore d i s t a n c e
S
=
pvu dz
(3)
where the overbar denotes the mean w i t h r e s p e c t t o time and n t h e water
surface e l e v a t i o n .
S u b s t i t u t i n g e q u a t i o n s ( 1 ) and ( 2 ) i n t o ( 3 ) and d r o p p i n g
terms o f h i g h e r than second o r d e r produce
Energy F l u x .
I n l i t e r a t u r e , the longshore t r a n s p o r t r a t e has been e m p i r i c a l l y r e l a t e d
most f r e q u e n t l y t o a term found by m u l t i p l y i n g b o t h s i d e s of e q u a t i o n ( 4 ) by
the wave phase v e l o c i t y , C, t o y i e l d
Pj, = ("ËC^ cosa) s i n a
(8)
U n l i k e S , Pj^ i s n o t c o n s t a n t seaward o f t h e breaker l i n e ; t h e r e f o r e ,
s p e c i f y i n g where Pj^ i s being c a l c u l a t e d i s necessary.
This r e p o r t , f o l l o w i n g c o n v e n t i o n , determines
a t t h e breaker l i n e ,
P„, = (ÏC cosa), s i n a ,
£b
g
b
b
(9)
r e p r e s e n t i n g t h e v a l u e of
a t t h e p o i n t c l o s e s t t o where t h e longshore
t r a n s p o r t i s o c c u r r i n g . The s u b s c r i p t b denotes breaker v a l u e s .
The term
12
13
i n parentheses i n e q u a t i o n (9) has been shown t o be c o n s t a n t (see eq. 6)
seaward of the breaker l i n e ; t h e r e f o r e , s u b s c r i p t b may be r e p l a c e d by i
which r e p r e s e n t s any p o i n t seaward of the breake£ l i n e .
Making t h i s change,
u s i n g e q u a t i o n ( 5 ) , and l e t t i n g
H^^^
equal H f o r monochromatic waves,
e q u a t i o n ( 9 ) becomes
4.
Empirical Relations.
The e x p r e s s i o n s d e r i v e d i n the p r e c e d i n g paragraphs
following empirical relations
h
lb
V
8
="ji
^Is
sina.
^''^
(10)
cosa siaaj^
d e r i v e d i n G a l v i n and Schweppe ( 1 9 8 0 ) .
The r e l a t i o n s h i p
has been shown i n Longuet-Higglns (1952) t o be
- Vib
and
The Shore P r o t e c t i o n Manual (SPM) (U.S. Army, Corps of Engineers, Coastal
E n g i n e e r i n g Research Center, 1977) p r o v i d e s a term s i m i l a r t o P^,^ except
t h a t the wave h e i g h t used i s the s i g n i f i c a n t h e i g h t , H^. The term, c a l l e d
the l o n g s h o r e energy f l u x f a c t o r , i s d e f i n e d as
Is
a r e used t o s e t up t h e
(11)
between
(12)
assuming a R a y l e i g h d i s t r i b u t i o n of wave h e i g h t s as w e l l as a number of o t h e r
conditions.
Therefore,
where
and
this report.
are
coefficients
t o be
determined
from t h e t e s t
data i n
E q u a t i o n (15) i s based on the concept t h a t t h e work done i n moving
sand alongshore i s p r o p o r t i o n a l t o the energy which approaches the beach.
u n i t s a r e c o n s i s t e n t and
Kp i s d l m e n s i o n l e s s .
the
The
E q u a t i o n (16) i s based on the concept t h a t the sand t r a n s p o r t e d a l o n g s h o r e
depends on the alongshore f o r c e e x e r t e d by t h e wave m o t i o n on t h e bed i n s i d e
the s u r f zone. By the e q u a t i o n of m o t i o n , t h i s f o r c e I s r e l a t e d t o the change
of momentum i n s i d e the s u r f zone. The a l o n g s h o r e momentum,
s u r f zone t h r o u g h the breaker l i n e but cannot e x i t t h r o u g h the s h o r e l i n e
boundary.
T h e r e f o r e , the change i n a l o n g s h o r e momentum i s S^^y and e q u a t i o n
(16) r e s u l t s . Kg has dimensions of l e n g t h over t i m e .
5.
Surf S i m i l a r i t y Parameter.
Kamphuis and Readshaw (1978) showed t h a t
the s u r f s i m i l a r i t y parameter.
Kp
and
a r e dependent upon
tan B
(17)
1/2
Is
2
lb
(13)
Since
^Jlb and
'Is a r e e s s e n t i a l l y the same terms, t h i s r e p o r t uses the
t e r m i n o l o g y and r e f e r s t o P.v as the l o n g s h o r e energy f l u x f a c t o r .
Longshore T r a n s p o r t
SPM
deepwater (d/L > 1/2) wavelength.
r e f l e c t s v a r i a t i o n s i n beach shape,
breaker t y p e , and r a t e of energy d i s s i p a t i o n . Using the r e s u l t s of l a b o r a t o r y
t e s t s , t h e f o l l o w i n g r e l a t i o n s h i p s were found by Kamphuis and Readshaw
Rate.
The l o n g s h o r e t r a n s p o r t r a t e , Q,
g i v e n i n t h e SPM i n u n i t s of volume per
u n i t t i m e , i s a l s o commonly shown as
I j ^ w i t h u n i t s of immersed weight per
u n i t t i m e . The r e l a t i o n s h i p between the two i s
For v a l u e s
pendent of
of ?t
5]^.
Kp = o.y^b
for
0.4
< 5t, < 1-4
(18)
Kg = O.OSCi,
for
0.4
< 5^ < 1-25
(19)
higher
than
the upper
limits,
and
K^
become i n d e -
(14)
The s u r f s i m i l a r i t y parameter i s e v a l u a t e d i n t h i s r e p o r t t o determine i t s
e f f e c t on the l o n g s h o r e t r a n s p o r t r a t e .
where p^ i s the mass d e n s i t y o f sand and a'
t h e r a t i o o f sand volume t o
t o t a l volume o f a sand d e p o s i t , which takes i n t o account the sand p o r o s i t y .
For d i s c u s s i o n s o f e q u a t i o n ( 1 4 ) , see Komar and Inman (1970) and G a l v i n
(1979).
Since the l a b o r a t o r y t e s t s d e s c r i b e d here measured
1» d i r e c t l y .
t h i s term i s used i n most o f t h e data a n a l y s i s .
111. EXPERIMENTAL SETUP
This s e c t i o n discusses the setup i n the SPTB ( F i g s . 2 and 3) and d e s c r i b e s
t h e wave g e n e r a t o r s , wave gages, and cameras and t h e i r p o s i t i o n s .
Also d i s
cussed are the sand-moving system, t h e method f o r measuring t h e l o n g s h o r e
c u r r e n t v e l o c i t y , and t h e s i z e d i s t r i b u t i o n of t h e sand used i n t h e e x p e r i
ment. The d e s i g n of the setup was based i n l a r g e p a r t on F a i r c h i l d ( 1 9 7 0 ) .
(Pg - p) ga'
14
Q
15
V»
DATA
The data c o l l e c t e d d u r i n g the e x p e r i m e n t s a r e p r o v i d e d i n Appendixes A t o
D.
Appendix A c o n t a i n s the h o u r l y and d a i l y d a t a f o r each t e s t .
Appendix B
l i s t s the beach s u r v e y d a t a , which a r e p l o t t e d i n Appendix C, taken a f t e r e a c h
test.
Appendix D p r o v i d e s 3 5 - T n i l l i m e t e r photos of the beach t a k e n d u r i n g a
t e s t w i t h the waves s t o p p e d .
Hourly
and
Table 4
Appendix A.
Run-time i s
n i n g of the
been run a t
would be the
D a l l y Data i n Appendix
A.
i s an example of how the d a i l y and h o u r l y d a t a a r e t a b u l a t e d i n
Column 1 l i s t s the run-time over w h i c h the d a t a were c o l l e c t e d .
d e f i n e d as the c u m u l a t i v e time of wave o p e r a t i o n from the b e g i n test.
A run-time of 05 10 means t h a t up to t h a t p o i n t , waves had
the beach f o r a c u m u l a t i v e t o t a l of 5 hours and 10 m i n u t e s .
This
c a s e even i f the f i r s t wave had been r u n 2 days b e f o r e .
Column 2 l i s t s the l e n g t h of time ( i n m i n u t e s ) waves were stopped to t a k e
o v e r h e a d photos of the beach.
The l e t t e r s CFD or TC i n d i c a t e t h a t the t e s t i n g
was completed f o r the day or the t e s t was c o m p l e t e d .
Between any two e n t r i e s
i n column 2, the waves were run c o n t i n u o u s l y .
For example, from the b e g i n n i n g
of the t e s t a t run-time 00 00 to run-time 01 00 ( s e e T a b l e 4 ) , the waves were
c o n t i n u o u s l y run.
At t h a t p o i n t the waves were s t o p p e d f o r 5 m i n u t e s to take
overhead photos of the beach.
The waves were then r e s t a r t e d and run c o n t i n u o u s l y u n t i l run-time 02 00.
Columns 3 and 4 l i s t the water t e m p e r a t u r e and the w a t e r depth,
tively.
These measurements were t a k e n i n the morning b e f o r e the
s t a r t e d and i n the a f t e r n o o n a f t e r the t e s t i n g s t o p p e d .
Column 5 l i s t s the immersed weight of sand moved d u r i n g
p r e v i o u s e n t r y i n the column.
A value i s always l i s t e d with
s i n c e i t was o n l y a t the end of the day t h a t the b a l a n c e of
d u r i n g the time the waves were r u n n i n g c o u l d be p i c k e d up
Table 4, the v a l u e of 4,227 immersed pounds of sand i s the
t r a n s p o r t e d from run-hour 04 00 to 08 00.
T h i s column i s
l i s t i n g of sand t r a n s p o r t e d .
respectesting
t e s t i n g from t h e
a CFD or TC e n t r y
sand not weighed
and weighed.
In
q u a n t i t y of sand
not a c u m u l a t i v e
Columns 6, 7, 8, and 9 l i s t the wave h e i g h t s measured by gages 1, 2, 3,
and 4A or 4B, r e s p e c t i v e l y .
S e c t i o n I I I d i s c u s s e s the l o c a t i o n s of t h e s e
gages, w h i c h a r e shown i n F i g u r e 7.
Column 10 l i s t s the b r e a k e r a n g l e s measured from the P o l a r o i d 4- by 5 - i n c h photos of the b r e a k i n g waves ( s e e F i g .
16),
Column 11 l i s t s the l o n g s h o r e c u r r e n t v e l o c i t y measured by dye i n j e c t i o n s , as d i s c u s s e d i n S e c t i o n I I I .
Column 12 l i s t s the b r e a k e r t y p e , u s i n g
the f o l l o w i n g code:
s g , s u r g i n g ; p, p l u n g i n g ; c, c o l l a p s i n g ; and sp, s p i l l ing.
A double e n t r y i n d i c a t e s both t y p e s of b r e a k e r s were e v i d e n t w i t h the
f i r s t type p r e d o m i n a n t .
2.
Summary Data
Table.
For a c o m p a r i s o n of t e s t c o n d i t i o n s , T a b l e 3 p r o v i d e s the a v e r a g e v a l u e s
of water
temperature,
wave h e i g h t , wave b r e a k e r a n g l e , l o n g s h o r e
current
v e l o c i t y , and a v e r a g e l o n g s h o r e t r a n s p o r t r a t e i n immersed pounds per second
for each t e s t .
A l s o i n c l u d e d a r e the wave p e r i o d and g e n e r a t o r a n g l e .
33
3.
TphlP. 4.
Ex3n.T,1e o f h o u r l y ^nd d a i l y d a t a t a y ^ L n A E E g a d i x A , ,
•t3T IJ
sCLt
10 D i C - U S
PinlOQ 1.00 StCO-DS
-*11B
-«Tin
Survey
Data.
A f t e r each t e s t , t h e SPTB was d r a i n e d and t h e beach was surveyed.
The
d i s t a n c e and e l e v a t i o n p a i r s a r e l i s t e d i n Appendix B and p l o t t e d i n Appendix
C. The e l e v a t i o n datum i s t h e S t i l l w a t e r l e v e l (SWL), w h i c h corresponded t o a
0. 710-meter water d e p t h .
4.
Overhead
Photos.
Every hour d u r i n g t e s t i n g , t h e waves were stopped t o take an overhead 35m i l l i m e t e r photo of t h e beach (see F i g . 1 5 ) . The photos show t h e w a t e r l i n e ,
t h e l o n g s h o r e b a r , and t h e swash zone.
They a r e u s e f u l f o r a q u a l i t a t i v e
d e s c r i p t i o n o f how t h e beach responded t o t h e waves.
Appendix D c o n t a i n s a
s e r i e s o f photos f o r r u n - t i m e s 01 00, 08 00, 16 00, and 24 00.
1 10
VI.
11.T
U.2
i<r
10
11 Q
U
2
11 la
13 0
1Ï 10
IJ 0
1Ï •
19
3
DATA ANALYSIS
T h i s s e c t i o n i n c l u d e s t h e data a n a l y s i s t o determine t h e r e l a t i o n s between
Ij^ and S^y and I j ^ and Pg^^. The e m p i r i c a l c o e f f i c i e n t s found from these
r e l a t i o n s a r e t h e n , i n t u r n , r e l a t e d t o t h e s u r f s i m i l a r i t y parameter,
5,
which i s adapted t o t h e data c o l l e c t e d .
Also i n c l u d e d i s an e x p l a n a t i o n o f
the c a l c u l a t i o n s o f S^„, P^j,, 5, and Ijj^, a l o n g w i t h p l o t s o f t h e v a r i o u s r e l a tionships.
The wave h e i g h t used i n the c a l c u l a t i o n s i s t h a t measured a t t h e
toe o f t h e beach (average o f gages 1 and 2 wave h e i g h t s ) .
The breaker wave
h e i g h t , which would have been a b e t t e r v a l u e , was n o t used f o r t h e f o l l o w i n g
reasons.
The wave h e i g h t a t t h e t o e of t h e beach was measured f o r a l l 15
t e s t s ; t h e breaker h e i g h t was n o t .
A l s o , o n l y one gage was used t o measure
b r e a k e r h e i g h t , w h i l e two were used a t t h e beach t o e . The s i g n i f i c a n t d i f f e r ence i n h e i g h t between waves measured a t t h e two beach t o e gages (see App. A)
i n d i c a t e s t h a t some wave h e i g h t v a r i a b i l i t y e x i s t e d a l o n g t h e wave c r e s t .
T h e r e f o r e , t h e average of the measurements a t t h e two beach t o e gages i s
p r o b a b l y a more r e l i a b l e e s t i m a t e o f t h e e n t i r e wave passing t h e t o e than t h e
one gage measurement a t the breaker i s of t h e e n t i r e breaker wave. A compari s o n o f t h e data i n t h i s r e p o r t w i t h past s t u d i e s i s shown i n a Q v e r s u s
Pjjb gi^sph.
15 10
16 0
14 I
IB ID
IT 0
:T •
IT 10
1.
C a l c u l a t i o n o f S^„.
Equation (7)
S
= ( ^ C
xy
\ 8
g
was
22 0
Z2 Ï
22 10
21 0
ai
used t o c a l c u l a t e
S„„.
xy
Rearranging
cos^^^
J C
the equation,
3.7
^xy =TI»'^
'
2a
(21)
ID
ICFD = t e s t i n g completed
f o r day; TC = t e s t i n g
34
completed.
where n i s t h e r a t i o
Cg/C and a f u n c t i o n of t h e water depth and wave
period or l e n g t h . S
was c a l c u l a t e d a t t h e t o e o f t h e beach by u s i n g t h e
average of t h e wave h e i g h t s measured a t t h a t l o c a t i o n (see F i g . 7) , and by
u s i n g t h e g e n e r a t o r angle f o r a. This was c a l c u l a t e d f o r each s e t of wave
d a t a . Thus, f o r t h e s t a n d a r d 24-hour t e s t , 24 v a l u e s o f S^^ were c a l c u l a t e d
(see App. E ) . The average o f S
f o r each t e s t i s l i s t e d m Table 5.
35
Table 5.
Test
Total
run t i m e
I
2
3
4
5
6
7
8
9
10
12
13
14
15
(hr)
25
30
24
24
24
24
24
24
24
24
24
24
24
24
Test c y c l e c a l c u l a t i o n s .
(N/m) (J/m/s)
2.201
1.179
2.043
1.137
3.232
2.280
3.615
2.158
0.789
0.987
2.144
1.977
4.158
3.161
3.918
3.018
4.286
2.808
8.250 14.761
4.839
2.942
2.948
2.241
11.578 28.802
9.253 13.536
The same beach slope was used f o r a l l 15 t e s t s and was determined as shown i n
Figure 2 1 . A value o f C was c a l c u l a t e d f o r each t e s t u s i n g the average
H f o r the e n t i r e t e s t .
These values a r e l i s t e d i n Table 5.
S
h
P^b
^xy
(N/s)
0.6116
0.6889
0.8396
0.6188
0.7544
0.9966
0.7281
0.3446
0.5227
1.0605
1.6328
1.1941
3.2938
2.5502
(m/s)
0.5190
0.6058
0.3682
0.2868
0.7640
0.5042
0.2303
0.1142
0.1862
0. 1285
0.5550
0.5328
0.2845
0.2756
-0.4r
0.2779 0.6604
0.3373 0.6686
0.2598 0.3374
0.1712 0.4508
0.9557 0.8997
0.4648 0.6815
0.1751 0.4787
0.0880 0.4835
0.1220 0.3761
0.0718 0.3764
0.3374 0.6644
0.4051 0.9190
I 0.1144 0.6112
0.3934
1 0.1884
C a l c u l a t i o n o f P^;,*
Equation (10)
£b
C cos
Stotlon (m)
Figure 21.
„as used t o c a l c u l a t e
P^^"
" ™
^"
parentheses,
^xy™^
c a l c u l a t e d a t the toe o f the beach. However, t b e s i n e term used the t r e a k e r
Tn f as measured from t h e photos of t h e b r e a k i n g waves.
^^^^H^^^^^^
u s t d i n the c a l c u l a t i o n was the average o f the breaker « " S ^ " , / ^ l ^ ^ / " ^ J °
n i n u t e s b e f o r e and a f t e r the wave data were c o l l e c t e d (see Fxg. 14).^^P,,^ was
c a l c u l a t e d f o r each s e t of wave d a t a , 24 values o f P lb
f o r each t e s t i s
the s t a n d a r d 24-hour t e s t (see App. E ) . The average o f ^Ib
l i s t e d i n Table 5.
3.
C a l c u l a t i o n of
The
surf
similarity
parameter
of
Kamphuis
tan
B
and
Readshaw
(1978)
was
presented i n e q u a t i o n ( 1 7 ) as
1/2
For the data i n t h i s r e p o r t , a d i f f e r e n t s u r f ^ ^ " ^ ^ ^ " " ^ ^ J ^ ^ ^ ^ ^ ^ J ^ ^ ^ ^ t h t s
s i n c e H w i l l be s u b s t i t u t e d f o r H,, as d i s c u s s e d a t t h e b e g i n n i n g o f t h i s
s e c t i o n " T h e r e f o r e , the s u r f s i m i l a r i t y parameter i n t h e f o l l o w i n g a n a l y s i s i s
tan
'H/L
36
e
n1/2
(22)
4.
D e t e r m i n a t i o n of beach slope used t o c a l c u l a t e
the s u r f s i m i l a r i t y parameter.
Special Tests.
Three t e s t s were performed under s p e c i a l c i r c u m s t a n c e s .
Test 2 was a
r e p e a t of t e s t 1; t e s t 8 was a repeat o f t e s t 7, except the sand feeder was
moved shoreward; and t e s t 11 was done w i t h a g e n e r a t o r angle of z e r o .
Tests 1 and 2 were both run w i t h a p e r i o d o f 2.35 seconds, a g e n e r a t o r
a n g l e of 10°, and a g e n e r a t o r e c c e n t r i c i t y o f 5.97 c e n t i m e t e r s . Test 1 r a n
f o r 25 h o u r s , t e s t 2 f o r 50 hours. A t w o f o l d comparison o f t h e two t e s t s was
o r i g i n a l l y planned.
The f i r s t 25 hours of t e s t 2 data was t o be compared t o
the t e s t 1 d a t a , and t h e n , both sets o f data were t o be compared t o the l a s t
25 hours o f t e s t 2.
U n f o r t u n a t e l y , due t o an e x p e r i m e n t a l e r r o r , only the
f i r s t 30 hours of the t e s t 2 longshore t r a n s p o r t data was c o l l e c t e d accurately.
T h e r e f o r e , t h e o n l y comparison made was t e s t 1 t o t h e f i r s t 30 hours
of t e s t 2.
Reference t o t e s t 2 i n the remainder of the r e p o r t r e f e r s t o t h e
f i r s t 30 hours o n l y . Appendix A c o n t a i n s a l l 50 hours o f t e s t 2 d a t a .
Table 6 compares the r e s u l t s of the two t e s t s .
The d i f f e r e n c e s l i s t e d
g i v e an i n d i c a t i o n o f t h e r e p e a t a b i l i t y of t h e data c o l l e c t i o n .
The longshore
t r a n s p o r t r a t e changed by 12.6 p e r c e n t , which i s a s i g n i f i c a n t v a r i a t i o n . This
i s an i n h e r e n t problem o f longshore t r a n s p o r t t e s t s , i n d i c a t i n g t h a t some
i m p o r t a n t unknown f a c t o r s are a t work.
37
Table 6.
Pet
Comparison o f t e s t s 1 and
local
run-time
(hr)
Test
Avg
Ti
%
(cm)
(degrees)
(N/s)
(N/m)
1
25
8.17
8
0.612
1.18
2
30
8.03
7
0.689
1.14
difference^
^Pct
2.
h
Avg
-1.7
difference
^Test
-
1 - Test
Test 1
2)
-3.4
+12.6
-12.5
(J/m/s)
2.20
2.04
-7.3
100
Tests 7 and 8 were both run w i t h a p e r i o d o f 1.90 seconds, a g e n e r a t o r
angle of 20°, and a g e n e r a t o r e c c e n t r i c i t y o f 5.97 c e n t i m e t e r s .
The o n l y
d i f f e r e n c e was t h a t the sand f e e d e r , which was l o c a t e d a t the SWL f o r a l l
o t h e r t e s t s , was moved shoreward 1.4 meters f o r t e s t 8.
The f e e d e r was moved
because the s h o r e l i n e a t the end o f t e s t 7 s i g n i f i c a n t l y angled
shoreward
toward Che d o w n d r i f t s i d e o f the beach. This can be seen i n t h e t e s t 7 photos
i n Appendix D.
The feeder was moved shoreward t o see i f a s t r a i g h t s h o r e l i n e
resulted.
I t d i d , as the photos i n Appendix D f o r t e s t 8 show. Another major
e f f e c t was the change i n I j ^ from 0.728 newton per second f o r t e s t 7 t o 0.345
newton per second f o r t e s t 8, a decrease of 53 p e r c e n t .
Test 8 i s excluded
from the remaining data a n a l y s e s .
Test 11 was run w i t h a p e r i o d of 2.35 seconds, a g e n e r a t o r angle of 0°,
and a g e n e r a t o r e c c e n t r i c i t y of 5.97 c e n t i m e t e r s .
The t e s t was meant as a
c o n t r o l to determine the amount of sand moved by the d i f f u s i o n caused by
b r e a k i n g waves.
This value o f
I^
f o r t e s t 11 was 0.089 newton per second.
A comparable q u a n t i t y of sand, 0.059 newton per second, a l s o moved u p d r i f t .
Test 11 i s a l s o excluded from the remaining data a n a l y s e s .
5.
D a l l y Cycle Graphs.
As discussed p r e v i o u s l y , longshore t r a n s p o r t c o u l d be measured o n l y on a
d a i l y c y c l e or t e s t c y c l e b a s i s . For the t y p i c a l 24-hour t e s t , s i x values o f
l o n g s h o r e t r a n s p o r t r a t e were c a l c u l a t e d .
Each r a t e covered a p e r i o d of 4
run-hours.
D u r i n g t h i s time p e r i o d , f o u r v a l u e s of S^^.^ and
Pj^jj were
c a l c u l a t e d , averaged, and r e l a t e d t o t h e c o r r e s p o n d i n g v a l u e of I j j .
These
values are l i s t e d i n Appendix F and p l o t t e d i n F i g u r e s 22 and 23.
Table 7
l i s t s the i m p o r t a n t s t a t i s t i c a l parameters.
Table 7.
Figure
Daily cycle s t a t i s t i c s .
No.
Ij^ v e r s u s
S^^^
Least
r2
squares l i n e s
Standard I Y-lntercepc
slope
Tlirough o r i g i n
slope
22
0.74
0.21
0.38
0.28
23
0.73
0.09
0^58
0.13
r e p r e s e n t s the f r a c t i o n
The square of t h e c o r r e l a t i o n c o e f f i c i e n t s .
1» about i t s mean which i s e x p l a i n e d by the a b s c i s s a
of the v a r i a t i o n of
and
P^^ are 0.74 and 0.73, r e s p e c t i v e l y .
These numbers
term. r2 f o r S
show t h a t
1^ c o r r e l a t e s
well
with
both
terms
to approximately
equal
degrees.
The l e a s t squares l i n e s l i s t e d i n Table 7 a r e i n F i g u r e s 22 and 23,
which a l s o i n c l u d e the l e a s t squares l i n e s c a l c u l a t e d w i t h the l i m i t a t i o n t h a t
the l i n e s pass t h r o u g h the o r i g i n .
The slopes of these l i n e s a r e 0.28 f o r
graph and 0.13 f o r t h e
1„ versus
P Jib graph.
the
xy
38
39
6.
Test Cycle Graphs.
The average longshore t r a n s p o r t r a t e f o r each t e s t was c a l c u l a t e d and
compared w i t h the t e s t average o f S^.^ and P^j^. These values are l i s t e d i n
Table 5 and p l o t t e d i n Figures 24 and 25. S t a t i s t i c a l v a l u e s a r e i n Table
8.
r 2 f o r 1^ versus S^^ and l£ versus P^^ are 0.72 and 0.74, respectively.
As w i t h t h e d a i l y c y c l e c a l c u l a t i o n s ,
I j ^ i s shown t o c o r r e l a t e w e l l
w i t h both terms t o a p p r o x i m a t e l y equal degrees.
Figures 24 and 25 i n c l u d e
b o t h the s t a n d a r d l e a s t squares l i n e and the l e a s t squares l i n e f o r c e d t h r o u g h
the o r i g i n .
The slopes o f the l a t t e r l i n e s are 0.26 f o r the
versus S^^^
graph and 0.13 f o r the I ^
versus P^j, graph.
Table 8.
Relation
LONGSHORE ENERGY FLUX FACTOR
F i g u r e 23.
Figure
No.
r2
Text c y c l e s t a t i s t i c s .
Least squares l i n e s
Y - i n t e r c e p t Through o r i g i n
slope
0.40
0.26
h
^«rsus S^y
24
0.72
Standard
slope
0.21
h
versus P^^
25
0.74
0.09
0.58
Kg v e r s u s ^
26
0.70
0.82
-0.07
Kp v e r s u s ^
27
0.56
0.89
-0.22
0.13
J^n/S
R e l a t i o n between l o n g s h o r e t r a n s p o r t r a t e ,
1^,
and longshore energy f l u x f a c t o r ,
Pj^^,, u s i n g
d a i l y c y c l e data ( t e s t s 8 and 11 e x c l u d e d ) .
14
iftDIATION STRESS
F i g u r e 24.
40
IVB
R e l a t i o n between l o n g s h o r e t r a n s p o r t r a t e ,
I^,
and r a d i a t i o n s t r e s s , S^^, u s i n g t e s t c y c l e
d a t a ( t e s t s 8 and U e x c l u d e d ) .
41
1.00
- 1 — i — \ — I — I — r
1—^—1—^—I—1—r
K s = 0.82
f-0.07
r2 = 0.70
?0.50
0.25
n
^ e
Ay
[ I I I I
@
I
I I
I I
5
I
1©
I I I I
I
I I I
15
I
I
I I I I
a®
LONGSHORE ENERGY FLUX FACTOR
F i g u r e 25-
7.
I
25
I
I I I I
3®
I
I I I I
F i g u r e 26.
35
J/B/S
R e l a t i o n between l o n g s h o r e t r a n s p o r t r a t e ,
Ij^,
and longshore energy f l u x f a c t o r , Pj^j^,
using
t e s t c y c l e data ( t e s t s 8 and 11 e x c l u d e d ) .
Surf S i m i l a r i t y R e l a t i o n .
1.00,
1
R e l a t i o n between K^ and t h e s u r f s i m i l a r i t y
parameter,
using t e s t cycle data
( t e s t s 8 and 11 e x c l u d e d ) .
43
1—
,
T
1
1
1
1
1
r-
0,75
F i g u r e s 26 and 27 were drawn t o t e s t t h e dependence o f
and Kp on 5.
Test numbers a r e i n d i c a t e d i n t h e f i g u r e s .
Table 8 l i s t s the s t a t i s t i c s .
The
K terms were c a l c u l a t e d u s i n g e q u a t i o n s ( 1 5 ) and ( 1 6 ) .
These graphs
show t h a t K i s f a r from being c o n s t a n t , as i s commonly assumed, and t h a t i t
i s s t r o n g l y r e l a t e d t o ^.
8.
I
Kp=0.89f-0.22
r2
= 0.5S
Comparison t o Past Data.
The u n i t s o f I j ^ and
were c o n v e r t e d t o those used i n t h e SPM and
p l o t t e d i n F i g u r e 28, which i s taken from F i g u r e 4-36 o f t h e SPM. The SPM
f i g u r e was m o d i f i e d by s h i f t i n g t h e x - a x i s t o c o n v e r t from
Pj^^ t o P^j,.
E q u a t i o n (13) shows t h e r e l a t i o n between ?^-^ and Pjjg. Test numbers f o r t h e
data p o i n t s o f t h i s r e p o r t a r e noted i n F i g u r e 28.
Two major o b s e r v a t i o n s a r e i m m e d i a t e l y a p p a r e n t .
The f i r s t i s t h a t t h e
l a b o r a t o r y data i n t h i s r e p o r t , as i n l a b o r a t o r y data from past r e p o r t s , have
considerable s c a t t e r .
Since t h e s u r f s i m i l a r i t y p a r a m e t e r , i, i n t h i s
r e p o r t v a r i e s by a s i g n i f i c a n t amount f o r t h e d i f f e r e n t t e s t s , as shown i n
F i g u r e s 26 and 27, some s c a t t e r i s expected.
The s u r f s i m i l a r i t y parameter,
of course, does n o t e x p l a i n a l l o f t h e s c a t t e r i n t h e l a b o r a t o r y d a t a .
There
are s t i l l some l a b o r a t o r y and s c a l e e f f e c t s which a r e n o t y e t u n d e r s t o o d .
42
F i g u r e 27.
Rel,
a t i o n between Kp and t h e s u r f s i m i l a r i t y
parameter,
using t e s t cycle data
( t e s t s 8 and U e x c l u d e d ) .
43
The second o b s e r v a t i o n i s t h a t most of the data f a l l beneath the SPM curve
c o n n o t i n g low values o f Kp. Since t h e SPM curve i s based on f i e l d d a t a ,
m o s t l y from Komar and Inman ( 1 9 7 0 ) , a p o s s i b l e e x p l a n a t i o n i s t h a t t h e f i e l d
data were c o l l e c t e d under c o n d i t i o n s of h i g h e r values o f 5 than those f o r
the
l a b o r a t o r y data.
Kamphuis and Readshaw (1978) suggest t h a t Komar and
Inman's data were indeed c o l l e c t e d under c o n d i t i o n s of h i g h i^. I t seems
reasonable t o assume t h a t t h e C values were a l s o h i g h .
VII.
SUMMARY AND CONCLUSIONS
An a n a l y s i s o f t h e r a d i a t i o n s t r e s s ,
S^^, and t h e energy f l u x f a c t o r ,
Pj^[j,
shows t h a t both p r e d i c t longshore t r a n s p o r t r a t e ,
I^, t o comparable
degrees.
A p p r o x i m a t e l y 70 percent of t h e v a r i a n c e o f 1^ about i t s mean i s
e x p l a i n e d by each term.
There appears t o be no major advantage i n choosing
one over t h e o t h e r t o p r e d i c t t h e longshore t r a n s p o r t r a t e .
However, S^y
has the advantage o f being c o n s t a n t seaward o f the breaker zone w h i l e Pj;,^^
i s not.
T h i s makes t h e c a l c u l a t i o n of S^^ more c o n v e n i e n t than
Pj^j^,
which must be determined a t t h e breaker l i n e . fti the o t h e r hand, Pg^^ has
the advantage of h a v i n g t h e same u n i t s as I j ^ ,
which means t h a t
Kp i s
dlmensionless.
The e m p i r i c a l c o e f f i c i e n t s , K^ and Kp, a r e f a r from c o n s t a n t a l t h o u g h
K
i s commonly assumed t o be so i n p r a c t i c e .
P a r t o f t h e v a r i a t i o n of t h e
c o e f f i c i e n t s can be r e l a t e d t o the v a r i a t i o n of t h e s u r f s i m i l a r i t y parameter,
5, as shown i n Figures 26 and 27. These f i g u r e s show t h a t K^ and Kp w i l l
i n c r e a s e w i t h 5. The c o n s i d e r a b l e s c a t t e r e v i d e n t i n F i g u r e 28 can be p a r t l y
e x p l a i n e d by t h e r e l a t i o n between t h e e m p i r i c a l c o e f f i c i e n t s and
The data
i n t h i s r e p o r t and past l a b o r a t o r y and f i e l d data a r e compared i n F i g u r e 28.
The l a b o r a t o r y data g e n e r a l l y p r e d i c t lower values o f Ij, f o r a g i v e n Pj^j^
compared t o t h e f i e l d d a t a .
P a r t of t h i s t r e n d can be e x p l a i n e d by t h e d i f f e r e n c e s i n t h e s u r f s i m i l a r i t y parameters, assuming t h e f i e l d data were
c o l l e c t e d under c o n d i t i o n s of h i g h £;.
A l s o , l a b o r a t o r y and s c a l e e f f e c t s
p r o b a b l y c o n t r i b u t e t o t h e lower l a b o r a t o r y t r a n s p o r t r a t e s .
The r e l a t i v e
importance of these f a c t o r s i s suggested as a s u b j e c t o f f u t u r e r e s e a r c h .
e Present Lab Dota
*
Past F i e l d
•
Past L a b Data
( Test No. I n c l u d e d )
Data
Longshore Energy Flus F a c t o r , P^^
(ft-lb/s/ft)
F i g u r e 28. Comparison o f data i n t h i s r e p o r t t o past r e p o r t s ,
u s i n g SPM Figure 4-36 ( t e s t s 8 and U e x c l u d e d ) .
44
45
LITERATURE CITED
°^
^° Movable-Bed E x p e r i m e n t s , " V o l .
V I I I , MR 77-7, Lahovatory
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46

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