Detached Breakwaters

DETACHED BREAKWATERS

T

OR

U

S

A

WARAGI

Pr

o

f

essor,

D

e

partm

e

nt of C

iv

il Engine

e

ring

O

sa

ka Un

iv

er

s

ity

Yam

ada-oka, 2-

1, Su

i

ta

-city

, Osaka 565,

J

A

PAN

I. FUNCIlON OFDETACHED DREAKW ATERIN CONTROLLING WAVES

AND WAVE-INDUCED CURRENTS 12-1

1.1 Hlstorical background and present situation ofshore proteetion works in Japan 12-1 1.2 Function ofdetached breakwaters in the control ofwaves 12·3 1.3 Punction of detached breakwaters in thecontrolofJongsherecurrents 12-4 1.4 Functionofsubmergod breakwaters with widecrownwidth in thecontrol of

waves 12-5

1.5 Function ofsubrnerged breakwater in the controlof nearshore currents 12-9 2. FUNCIlON OF DETACHED BREAKWATER IN TIlE CONTROL

OF SEDIMENT MOVEMENT 12-9

2.1 Mechanism of formation of salient behind detached breakwater 12-9 2.2 Function of detached breakwater in the trapping of sediment... 12-11 2.3 Numerical simulation Ior trapping of sediment by breakwaters 12-12 2.4 Topographical change on the shore-side of submerged breakwater 12-13

3. STABlLlTY OFDETACHED DREAKWATERS 12-13

3.1 Effectof incident wave irregularity and grouping on the stability of rubble mound

breakwater 12-14

3.2 Stable weight of rubble stones forsubmerged breakwaters 12-17

4. CONCLUSIONS 12-18

SYMBOLS : 12-18

REFERENCES 12-21

1. FUNCTION OF DETACHED

BREAKWATER

IN CONTROLLING

WAVES

AND WA VE-INDUCED

CURRENTS

1.1.

llistorical

background

and present

situation

of shore

proteetion

works

in Japan

Japan is surroundcd by sea with a total 34360 kilometers of shoreline. 47%

of the shoreline,

or 15991km,

requires proteetion works to avert potential disasters.

Since World War 11Japan has dcpendcd heavily upon hydraulic power for its electricity.

A

large number of hydroelcctric stations and dams were constructed, and the dams trapped a

huge amount of sediment,

resulting in significant erosion of many coasts.

The Japanese Coastal Law was enacted in 1953 to regulate two kinds of engineering works

that provide protection against coastal disasters.

One group mitigates storm surges and the

other bcach erosion. Needless

to say, there are also engineering works for

coastal disasters

caused by,wave overtopping, tsunamis, blockage of river mouths and so on.

Various kinds of

352

TORU SAW ARAGI

coastal structures such as sca dikes, seawalls, lockgates, detached breakwaters, groins,

artificial recfs, and wave absorbing block mounds have been constructed to provide proteetion

against these disasters (sce Table 1).

Table-I Totallength of shoreline with proteetion works (1988)

Structure

Sea walls

Sea dikes

Groins

Detached

Wave

absorb-breakwaters

ing blocks

Total lcnzth

2870 km

5960 km

367 km

630 km

1100 km

Figure 1 iIIustratesthc ratio of increase ofthe region where these kinds of coastal structures

were constructcd, using 1962 as a base year (Toyoshima, 1986).

20 10 16 14 12 10 9

'"

0

...

'"

L ₇ s: 6

...

_J: 0 L ₅ Cl 4 3 2

-

I T I I

.I

-0- offshore detached breakwat~ _ ...• sea di kes

_._.-

jetties

'/

-

.

-

._.-.- sea wells Jf -

/

-

.I'

I

/

7

·

-

!

-- T

-- p - ₁₇ ...

~~::::.:::_.;

---

_{_}

....;;..::·-L;.·-~

.-.-~7/

.

-

.

-

._.

=

bf

~

_7-/

.

-

""

19 2 1965 1970 1 75 1980 1985 Year

Fig.1 Rate of construction for various coastal structures (Toyoshima, 1986)

Detachcd breakwaters are increasing at a remarkable pace since they effectively rednee and

absorb incident wave energy.

Howcvcr, detached breakwaters. as weil as the wave absorbing block mounds in front of

seawalls,detract from the coastallandscape and prevent the effective utilization of many coastal

regions.

Recently,with the increasing concern for the preservationof coastal environments and easier

access to the shorcline, and with demands for pro-water front, new forrns of coastal proteetion

.

works have been devised in Japan:

1) gentIe slopc sca dikcs with permeabiesurfaces,

2) submerged breakwater with wide crown widths or artificial reef,

3) beach nourishment,

4) hcad-land defense works.

In this paper, thc hydraulic functions and stability of dctached breakwaters and submerged breakwaters with wide crown widths are discussed.

1.2. Function

of detached

breakwalers

in the control

of waves

Two wave forms are found behind detached breakwaters. waves transmitted the breakwater and waves diffracted from thc two ends of the breakwater. Wave height behind the breakwater is often estimated from the energetic mean of these two kinds of waves. In Japan, the wave hcight behind the group of detached breakwaters (H) is usually estimated by the following equation (National Association of Sea Coast, 1987):

H

All

I'

Hi =

\J

1+/'

Kt2 +

I+f

Kg2 (1)

whcre I is the lcngth ofthe detached breakwater, fis the length of the opening in the row of detachcd breakwaters, Kt is the transmission coefficient through the breakwater. and Kg is the diffraction coefficient at openings in the breakwaters (see Fig.2).

Ty

Oetached

Breakwater

Sea

Oike

Shoreline

Fig.2 Definition sketch of a plane arrangement of detached breakwaters

Various studies have been conductèd to determine the value of Kt, the transmission coefficient. Thc following is the empirical expression for

Kt

derived by Numata (1975) under

conditions where wave overtopping takes place: .

Kt

=

0.123 log (43.12 um~~~c~ (2)

whcrc B is the breakwater width at still water level,

hs

is the crown height from still water level, lJc is the height of wave crest from still water level. The expression Umn represents maximum water particIe velocit at the wave crest and is expressed as:

:rtHi . (Hi)O.5 (h+lJC)3

cos"

k(ho+lJc)

Umax

= T

1+mi ho ho sinh kho

rnj

=

-0.644/og{ 1.562(hoIL)} : 0.07

<

holL

<

0.4

=

1.50 : halL

<

0.07

= 0.25 : 0.4

<

halL

where ho is water depth whcre the breakwater is constructed, Lis the wave length at depth h and k=2:rt/L. Numata also gave the relation between lJdho and Hi/ho as follows:

lJc _ Hl 0 415 (Hi)2.18 (4)

ho - 2ho +. ho

Although the exact valuc for Kg has to bc calculated from the diffraction pattern, values of between Ot7 and 0.9 are usually substituted in actual calculations of Eq.(I).

(3)

354 TORU SAWARAG!

In Eq.(I) the effect of interaction between waves and structures is not taken into account.

Recently, a boundary integral methou has been dcvclopcd to solvc a wave field around

structures such as detached breakwaters in which thc effect of wave-structure interaction is

fully considcrcd. For cxarnplc, Spring (1975) has solved the wave field resulting from a

regularly spaeed infinite row of vertical cylindcrs, and Pullin (1984) has developed a numerical

procedure for solving waves around a group of vertical structures. In these procedures, wave

fields

around

structures

are

caIculated by solving velocity potential as a boundary value

problem.

The

velocity potenrial

around structurcs <!lis expressed as the sum ofthe

velocity

potentialof

incident waves <!liand a scattcred wave potential <!ls,<!l=<!li+<!ls.The unknown potential <!Js is

usually evaluated by a finite clement method or a boundary clement method. Compared with

the former method, the lalter rcquires lcss computer capacity and CPU time because variables in the lalter metbod are one order lower in dimcnsion than thosc in the former. However, the detail of the numerical procedures is not referred here,

1.3. Function

of detached

breakwaters

in the control of longshore

currents

With thc norm al wave incidence, a pair of circulation cells of wave-induced current is

forrncd behind thc breakwaters. Since analytical solutions to mean water surfaces (wave set-up and set-down) and longshore currcnts were set forth IJy Bowen (1969) and Longuet-Higgins

(1970), much study on the tirnc-averagcd propertics of water-partiele motion has been

conducted.

Thc

basic equations

for

these f1uid motions are derived by temporally and vertically

averaging a continuity equation of mass flux and a N-S equation, and are expressed as follows:

all aU(h+ll) aV(h+ll) _

0

(5)

at+

ax

+

ay

-au

+

uau

at

ax

(~:x

+ ~)]}

sv

uav

+

vi}_Y= _ all __

1

_

[(asy

..!

+

~y

+

1:) _ (aRyx

+

a~yyy)]

at

+

ax

ay

gax

(h+ll)

iJx

iJy

Y

ax

u

wherc U and Vare the depth and time averaged veloeitics of wave-induced current in the x- and

y-direction, rcspcctivclyj r; a n d 1:yare the time averaged boltom shear stress es;

Rij,{(i,j)=(x,y)} is thc dcpth and time averaged Reynolds' stress tensor; II is the mean water

level and Sij,{(i,j)=(x,y)} is the radiation stress tensor introduced by Longuet-Higgins (1970);

and h is the dcpth in still water. .

Using Iinear wave theory, Sij is expressed as follows:

Sxx = ~ H2

[2~g

(cos20

+

1) -1]

(6)

Sxy = Syx =

t~

H2 si1l20

Syy

t-~

H2

[2~g

(s;1I2

0 +

1) - 1]

where p is the density of water, C and

Cg

are the celerity of wave propagation and group

velocity. respcctivcly, and 0 is thc angle of wave incidence.

The time averaged bottom shcar strcsscs 1:x ~nd"ty are usually evaluated from the

approxirnatc cxprcssion using friction factor, the water particIe velocity caused by waves at the bottom, anti thc velocity of wavc-induced curren!.

(7)

Thc Rcynolds' stress term is generally evaluated using a gradient-diffusion type expression with the eddy viscosity

(E)

and is rcfcrred to as "lateral mixing term". Sorne heuristic models for cddy viscosity in an uniform longshore cutrent on a long straight be ach have been proposed. They are summarized in Table-2.

Tablc-Z Mode! for eddy viscosity in uniform longshore current on a long straight beach Lateral mixing term Lateral mixing Remarks

roefficient

Bowcn (1969) d2V _E:constant

Rxy = Edx2 Longuet-Higgins

d ( dV) E = NxVgh 0< N < 0.016

(1970) Ryxph = dx PEhdx x: offshore distance

from the shoreline

d ( dV) H2

sr

Tbornton (1970) Ryx = dx Edx E=-_{8Jt2 h} cos2(J q: wave direction

d ( dV) E-James (1974) Ryxph = dx phe dx

~hIS :h:shb

S: bottom slope hhb21(hS):h>hb

d ( dV) 4 ab: excursion length

Jonsson et al. (1974) Ryxph = dx PhE-dx E =T~2 cos2(J

of water particIe d ( dV)

C

y2)1/3 4/3

{..fi,h

M: constant Balt jes (1975) Ryxph = dx phe dx E=M

16

S x g

_y

_{= H/(h+l1)}

r'v

E =

AF

l/3

yS'

xfi,h

A: constant

Kim et al. (1986) Ryx = Edx2 : h:s;hb

F=5.3-3.3Ç-O.07

IS

y=HfLI'JS

o :

h>1b

S'=(

-O.413Ç+O.98)S SOi!:1/60

The author ct al. conducted a series of experiments in a wave basin to investigate the function of breakwater length land distance from the initial

shoreline

Xerr in controlling longshore currents in cases where angle of wave incidence is 15ft. The results show that: I)When a breakwater with a length of less than two times the incident wave length is constructed within the breaker zone(XorrlXb=O.57, where Xb is the width of the breaker zone), thc velocity of longshore current decrcased to about 1/2 of that on a natural beach.

2)When the breakwater is constructed in alocation where XorrlXb>O.86, although shore-side waves deercase significantly, longshore currents on a natural beach are not affected by the breakwater just bchind the break water. However, a small weak circulation forms in the downdrift side of the breakwater.

1.4. Functlon

of submerged

breakwaters

wlth wide crown width In the control

of waves

The powerfut effects of detached breakwater on wave transformation, especially the effect of diffraction, greatly affects the surrounding coast.

lt

has also been pointed out that the breakwaters reducc the exchange of sea water and detracts from the natural coastal view.

Recently, to.cope with these problems, sections of detached breakwaters are being replaced by submergod breakwaters. which are often referred 10as artificial reefs, in Japan. Figure 3 is a diagram of submergod breakwaters constructed on the Niigata coast facing the Japan Sea.

356 TORU SAW ARAGI

50.00 m HSL T.P.'0.49m core rubble stone 200-S00kgl!' T P -1 sOm

" S, ~. ), -,shee t

cross section

----

_{---. 5.}

----

...

_-,

---=-=:..-_--- - ..._

- __.

- - -

---'_

-_

-4 ..

--_----

_

--

----, 8 • " NO.96 t:__

o

o_.!_:

!.

0

!.,

0 lD:cr.:.JoC.=========:

::

::=======

HO.9~ NlG 0 00 0 0 0 0 0 • •

~.~

~.~

n

plane arrangement

c

=

=

=-:. -:'''':,':'~] '0 • •

n

NO.88

,r-1988.2.

o

100 200 m

Fig.3 Subrnergcd breakwaters (Niigata Coast)

Thc submcrged breakwater has two energy dissipation mechanisms that attenuate wave height. First, energy is dissipated when the wave breaks due to the abrupt change in water depth as it meets the suhmerged breakwater. Secondly, cnergy dispersion takes place on the surface and in thc pcrmcablc layer of the submerged breakwater.

Nowadays, a so-callcd mild slope equation is applied to predict wave transformation across submergod breakwaters (Mei,1978 and Yeang, 1982). However, the equation requires much CPU time.A procedure for prediering wave transformation over submerged breakwaters based on thc conservation of wave cnergy is introduced here.

The equation for energy conservation on a permeable layer in a stationary state is expressed by

a

a.

au (av au)

sv

ax {E(Cgcos6

+

U)}

+

ay {E(Cgsm8

+

V)}

+

Sxx

ax

+

Sxy ax

+

ay

+

Syy ay

=-01055

(8)

whcrc E is the energy density of incident waves and D10S5is the total energy dispersion rate.

D1o~~is estimated as the sum of the cnergy losses in the permcable layer Op and on the surface of the laycr Dr and energy loss causcd by wave brcaking Db.

The encrgy dispersion in the permeable layer Op is expressed by

1

T

Op

=

Tl

(WP}z:-h dt

(9)

where T is the wave period; wand p are thc vertical water particIe velocity and pressure at the surface of thc pcrmeable laycr, rcspcctivcly; and h is the dcpth on the pcrmeable layer.

The cnergy dispersion ratc pcr unit area on the permeabie surface caused by boundary shear Di is evaluatcd using thc following cxpression for boundary shcar stress ~ on the bottorn:

T/2

De

= ~

J

1:Uz=-h dt (10)

According to lincar wave theory, velocity potentiaion and in a permeable layer with pcrmeability of Kp is given as follows (Deguchi et al., 1988):

on the permeabie layer:

ct>

= ~

Re

[+OShkz - ~ Sinhkz} exp{i(kx- at)}]

(11)

in the permeable layer:

ct>d

=

g!!

Re[

{~coshk(h+z)(coshkh+

f

sinhkh)

zo

ttys

JI.

- ; sinhk(h+z) (sinhkh

+

!

coshkh } exp{

i(kx -

at)} ]

s

=

{(l-À)Cm

+

l}/À

(12)

where À, kp,d are the void ratio, permeability and the thickness of the permeable layer, rcspectively; h is thc dcpth on thc permeable layer;

a

is the angular frequency(=2Jt/T); Cm is

thc

added mass

coefflcicnt. F

=

-1;

and Re[ ] indicatcs

the

real part of the quantity in

the

brackets [ [,

The

expression y is

the

nondimensional permeability defined by the kinematic viscosity v, Kp and

a

in the form ofy =

kpa/V_

The expression ofk is the complex wave number (=(1+i(3), which satisfies the following dispersion relation on the permeable layer.

a

2 =

gk

(ys+i)sinh~hcosh~d

+

Ycosh~hsinh~d

(13)

(ys+i)coshkhcoshkd

+

ysinhkhsinhkd

The values of wand p in the permeable layer are expressed using

ct>d

as follows:

act>d

w

=Yai

(14)

Q_= -s

act>d _

:!__

ct>d

(15)

p

at

kp

From these relations, Eq.(9) is expressed as

DI = ~g

If213

(16)

where

13

is the imaginary part of the complex wave number k which indicates the attenuation rate of wave height on the perrneable layer. Figure 4 iIIustrates the relation between

13

and o2h/g when o2H/g=O.I, kpo/V=O.5 and d/(h+d)=O.5 (Deguchi ct al., 1988).

To evaluatc the value of Df, Jonsson's expression for the bottom shear stress (Jonsson, 1978) was utilized.

2

1:=

pfUz=

-

h / 2

Jonsson (1978) anti Riedel ct al. (1972) have provided empirical expressions for friction factor f. Wh en linear wave thcory is used to evaluate the horizontal water partiele velocity at a depth

358 TORU SAW ARAGI IO-l ,,211/g=0.1 A,,,/v=0.5 J /(h+J )=0.5 Cl --0.0 ---0.1 ---- 0.2 0.40 0.80 1.20 1.60 2.40 ,,21t/g 2.00

Fig.4 Relationbetween wave attenuation ratio ~ and

a

2hJg

of ze-h, Deis expressed as follows:

2

2 f(

H

)3

De

=

3

Jt

P

Tsillhkh

where k is the usual real wave nurnber at the depth of h.

On the other hand, various rates for cncrgy dispersion after wave brcaking have been

proposcd. In cstirnating thc value of Db, thc existing dispersion rate proposcd by Sawaragi et

al. (1984) was used, where

~

Db

=

0.18 FP

-

l/2(hHI)~

F _ { 5.3 - 3.~-

0.07/S : insi~e the breaker zone

-

0

: outside the breaker zone

where S is the bottorn slope

.

Figure 5 shows a comparison between thc calculated wave heights and measurcd wave

hcights in a two-dimcnsional experiments on thc subrnerged breakwater shown in Fig.2

(Sawaragi ct al., 1989)

.

Figures (a) and (b) correspond to non

-

breaking and breaking

conditions on the breakwater, respectively.

(17)

(18)

o

water depth on the submerged breakwater:1m.

helght of breakwater:Jm. crown wldth:50m

1.0

-~rp~o;~

•• 0

o.

0.8 • x break Ing

'0,

;;:

0.6 scale 0.4 • 1/50 0 eXil·

~o~~.OO~

- theo. 1/10 • up. • •

- - - - th.e_...._o~'.nd!!HrlIrl.._{brea kwater}

0.2

3.0 4.0

x/L

(al breaklng condItIon

111=1.5m • T-6.4 sec 0.4 1.0 o~oog

_.geo~,

o~ ~ 0.8 non-brea~ng '''' ... x 0 ... 0.6 scale ~ 1/50 0 expo ~ ~ -theo. ~ 1/10 • expo • ---- theo. submergedbreakwater 0.2

o

-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 IIll [b] non-bruklng condition . HI-0.5111 • r-6.4 sec

F

i

g

.

5 Wave attenuation on thc subrncrged breakwater

1.5. Function

of submerged

breakwater

in the control of nearshore

currents

Wavc-induced currcnt

a

round thc submerged breakw

a

ter can be c

a

lculatcd using Eqs.(5)

and (6) in a similar mannor

a

s for currents on thc shore-side of thc det

a

chcd breakwaters

.

Ud a ct

a

l. (1988) conductcd c

x

perimcnts on wa

v

e-induced flow patterns around a

submergcd bre

a

kwater with leng

t

h

l,

opening wid

t

h

r

,

a

nd dist

a

nce fr

o

m

t

he shorcline X

o

rc.

Thc flow w

a

s c1assificd in

t

o four p

att

erns

,

as

s

hown in Fig

.

6.

Thc flow p

a

tterns occurred undcr the following cond

i

tions

:

Pattem I (Fig.ta)

:

I/Xoff

=

1 to 4 and Ilr < 4,

Pattern 11(Fig.ïb)

: llXoff> 4 and Ilr

<

4,

Pattcrn lII(Fig

.

(c»

:

llXorc- 1 and Ilr

<

4

,

Pattern IV (Fig.(d»

:

llXorc

=

1 to 3 and Ilr > 4.

Based on these rcsults, Ud

a

ct al. recommcnded that:

1) If thc region in the shorc

-

side of the submcrgcd breakwater is to be used as a swimming

area, or whcn uniform wave dec

a

y behind the breakwater is desired

,

the width of the

opening

r

must be less than 114.

2) When thcre is Iikcly to be a deposition of sediment on the shorc-sidc of the breakwater, the

valuc of I' should be gre

a

tcr than 114 and the Icngth of the bre

a

kwater / should be less than

4Xorr

.

(a) Pattern I

submerged

V

breakwater

I,,'"I~

1

CQ 00

(c) Pattern DI

submerged

V

breakwater

t

GJ

f[]t~]t

[JJt

~C~C

Fig.6 Patterns of wa

v

e-induced current around submerged breakwaters

2

.

FUNCTION

OF DETACHED BREAKWATER

IN TUE CONTROL OF

SEDIMENT

MOVEMENT

2.1.

Mecbanism

of formation

of salient

bebind

detacbed

breakwater

S

a

lients or a tornbolos forrncd on the shore-sidc of the detached breakwaters are brought

about by nonuniform long

s

hore sediment transport on the shore-side of the breakwater. The

topographic change caused by sediment transport in the Iittoral zone is expressed by the

following cquation:

ah

=

_l__ (~

+ ~)

(19)

at

1-

À

ax

ay

whcrc qx and qy are the local sediment transport rate in x(cross-shore) and y(longshore)

dircctions, rcspectively

;

and h is the water dcpth measured downward from still water level.

360

TORU SAW ARAGI

We invcstigatcd the rclation between the longshare sediment transport and topographic

change by integrating Eq.(19) in the region whcre the sediment transport takes place

.

Xo and

Xer were givcn the landward and the seaward limit of the significant sediment transport

,

respectively

.

The integration of Eq

.

(19) between X=Xo and Xcryields the following relation

because the longshore and cross-shore sediment transport at x=Xo and x=Xe- are zero:

a

x,

aXer

ax

o

1

(a

x,

)

at

x!r

hdx -

hx=x

cr

Tt

+

hx=x

o

Tt

= 1- À.

ay

x!r

qydx

(20)

Change in a sectional area below a reference level along the x-axis A and a tot al longshore

sediment transport rate Oy is defined as follows:

Xcr

A

=

10

hdx

(21)

Xcr

Qy

=

x[

qydx

(22)

When the characteristics of the incident waves are constant

,

the second and the third terms in

the left hand side of Eq.(20) approach zero. Under such conditions

,

Eq.(20) is written as

follows:

aA __

1_~

at - 1-

À.

ay

(23)

Furthermore,

when the change in sectional

area

t:.A

is expressed

as the product

of

representative dcpth of topographic change

ii

and shift in the shoreline

Ms,

Eg.(23) becomes

als

1

_

1

iJQy

at - 1-

À.

ii

ay

(24)

where Is is mcasured positive landwerd.

Eq.(24) impl ies that the longshorc gradient of total longshore sediment transport rate causes

thc change in shorcline contour. For example, thc shoreline retreats if iJOy

/

iJy

>

0 and the

shotcline advanccs if iJO

y

/iJy

<

O.

Figurc 7 schcmatically illustratcs wave pattems and changes in a shorcline. Oblique incident

waves are diffructcd by a dctachcd breakwater which breaks the uniformity

of longshore

sediment transport. As a rcsult, thc shorelinc is transformed according to the broken line in the

figure.

Qyr---40-.!.

loI.yes

/

Fig.7 Schematic illu

s

tr

a

tion of shorcline change on the shore-side of a detached breakwater.

2.2

.

Function

of det

a

ched

breakwater

in the trapping

of sediment

Indcsigning dctachcd breakwaters. thc [ocation, length and opening width must first be

dctcrmincd. Topogrnphical changes on the shore-side of the breakwaters rcsulting from

sediment trappcd by breakwaters closcly re[ated to these valucs. Numerical simulation procedures which will hementioncd later can bcof great help in the determination of these

values,

First ,the simplified relation between these values and the topographical changes are cxarnincd through field data. The Ministry of Construction of Japan studied the correlation between geometries of dctachcd breakwaters and the topography on the shore-side of these breakwaters through field surveys (National Association of Sca Coast, 1978). A definition sketch of the geomctry of dctachcd breakwaters is shown in Fig.2. The expression Xoff is thc distancc between the initialshoreline and thcbreakwater,

1

is the length of the breakwater,

I'

is the width of the opening, and Tx and Ty are the length and the width of the salient, respectively.

Figurc 8 shows the rclation between the geometry of the breakwater anti a representative profile of thc

c

onespond

i

n

g

bcach topography which is dcfined as the ratio between the area

of theshorc-sidc coast of thc breakwater and thearea ofsalient

As

:

_ Arca 0f saliant _ TxT

y

/2

As

-

Area of thc shore-side coast of the breakwater - Xoff I (25)

1.0 u-type 0.8

•

As 0.6

•

•

0.4

••

0.2

;~

0 0 0 0.5 1.0 1.5 0 hol hr C-t e

.

0

.

:

sandy coast A :gravel coast • :others 2 4. 6 8

o

3 4 5 6 til tlX

Fig.8 Topographic change on theshore-sidc of the dctachcd breakwater as a function of location anti length of dctachcd breakwater

In Fig.8,

ho

isthe water depth at the breakwater anti, hrand L are dcfincd using the averaged wave hcight and pcriod for thc fivc largest incident significant waves during thc year

[Hs]

s

and period [Tsjs as follows:

h

-

=

rH~]5 ; L

= ~

[Ts]s.

B- anti C-type coasts in thc figurc correspond to bar typc coasts with gcntlc slopes and planc coasts with stcep slopcs, rcspcctivcly. Thc values of

l

I

l'

for thc large part of break water in

Japan range between 0.3 to 0.5.

362

TORU SAWARAGI

2.3. Numerical

simulation

for trapping

of sediment

by breakwaters

Sediment movement in thc shore-side of the dctached breakwater depends on both waves and currents. Givcn thc wave and current fields in the shore-side of the breakwater, the rate of sediment transport there can be cstirnatcd using proper sediment transport formulas.

A number of forrnulas for prediering rate of sediment transport have been proposed by many

invcstigators based on various sediment transport modeis. Those models are generally

classified into two categories. Onc is the power model, originally proposed by Bagnold (1965)

(for

cxarnplc,

Komar, 1970; Walton et

al.

,

1979; Watanabe et al.,1982).

The ether

is

the

flux model in

which

the rate of sediment transport is expressed as tbc

product of sediment concentration

and

itsmigration speed (for example, Kana,1976; Tsuchiya

ct al., 1978;

Sawaragi

ct al., 1986). There is a third group of formulas based

on the

ra te of

sediment transport in a unidirectional flow (for

è

xarnple

,

Iwagaki et al., 1962; Bijker, 1968).

Here, the following forrnulas for bed load transport rate(qb) and suspended load transport

rate(qs) dcrivcd by thc authors(Sawaragi ct al., 1990) are used to examine the effects of

detacbed breakwaters on

longshore

sediment transport rate:

qb = 471tadso2('\j'm-'i'c)3/2(U

lUb)

(26)

qs =JCU dz

=

Co

(EST.

Iwc)

U : outside the breaker zone }

=

Co min

{Esz

Iw{,

D}U : inside the breaker zone (27)

where dse is mean grain sizc of bed material, '\j'c is the critical Shields' Number, U is the

velocity vector of mean eurrent, ub is maximum water particIe velocity at the bottom due to

waves,

Co

is the concentration of sediment at the refercncc level, Eszis thc diffusion coefficient

of susperuled sediment,

wcis the settling velocity of sediment,

D is the totallocal depth, and

min{

,

}

indieates the minimum value of the two quantitics in { , }. The values for'\j'm, Eszlwf

and

Co

are rclatcd 10the sediment and fluid propertics as follows:

"4'm= (f/2)IFbI2

I {(asla -

1)gd50} (28)

Eszlwc= O.021exp{0.5(~Fhf!2)1/2} (in cgs unit) (29)

whcrc IFblis the water particIe velocity due10waves and currents and is expressed as:

IFbl2= {ub2+ (2/1t)"b(UcosO

+

Vsillfl) + (U2 + V2) 14} (30)

Co

= 0.347

[O

.

688ubl

{1.13(aslo-l)gwfT1.77}] (31)

Figurc 9 compares the calculatcd and measured totallongshore sediment transport rat es Qyf

and Oycon thc shorc-sidc of thc dctachcd brcakwater(Sawaragi et al.,1990) . Thc value of Qyf

is calculatcd by inlegrating locallongshore sediment transport ratc qbYand qsy obtained from

Eqs.(26) and (27). The velocity of wavc-induccd currents and that of water particles at the

bottom are calculated by solving mild slope equations and the fundamental equations for

wave-induced CUTTen!.

The value of Qye is cstimatcd from topographical change Ó-h(x,y), measured over time

interval~I during the movable bed expcriments:

Qye(y+Ó-y)= Qyc(y) + Ó-A(y) lly (1-

À)

(32)

llt

~A(y) =

f

Ó-h(x,y) dx

where ~ y is the inlerval of thc measuring line.

-

--

Q,.

~ 2000 • - Q,t ~[j1000- ; : -;; 0

.

:

#r-...

·

---t

u 1.0 1:5_---~. er ~

o

0.5 2.0 2.5 (a) x,"/x.cO.86 t/l,-l.O Y/l, .~ 2000

l

(b) x,"Ix.-t.70 , • I

ÏJ

1000-

i

j

é

lIl,-l.O

>:

_{o O}

.

"

_....

-

-

_J

~!-e--:

_,

_{::-.c--. ~_.}

---o 0.5 1.0 l.5 2.0 2.5 (c) x Ix-0 86 lil • • _ y/l, t/l,-1.0 .~ 2000

1

-8/t-1.0 ;:;--_IJ 1000 _{- ••}

_·

_.

_-

_-

_-

_i

"

1

I __yA I._-::.-"'~~~ _ :;; 0 '=:;..:.:..:.=..J~~~ i----L ..---;--

-o

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (d)xlll/x,-1.70 ~ 2000E

l

•

.

• t/l.-1.0 1;_u1000-· _{_ .,.--}

_.j

_...~.

!~O

I I ,_

~

_{à 0}

--

_{___J---=.::J..}

."

-. --""'

-

f

_-

.

-,

_{- ~}

~~

o

0.5 1.0 1.5 2.0 Z.5 3.0 3.5 4.0 y/l, Y/l,

Fig.9 Distribution of longshore sediment transport rate on the shore-side of detacbed

breakwaters

2.4. Topographical

change on the shore-slde of submerged breakwater

The effects of su

b

m

e

r

g

od bre

a

kw

a

ters with wide crown width mentioned in the sections (1.4

and 1.5) on topographic

a

l ch

a

nge were examined using fie

l

d data from a location on the

Niigata Coast

.

The plane arrangement of thc breakwater is shown in Fig

.

3. Figure 10 shows

the

a

nnu

a

l change in the contour of the shorcline from 1986 (bcfore thc construct ion of thc

break-water) to

1

988 (

a

fter the complet

i

on of two breakwaters) (Jap

a

n Inst. of Construction

Eng

.

,

1989)

.

The two submergod breakw

a

ters were constructed in the gap bet ween the detached

bre

a

kwaters whcrc th

c

f

a

cing shoreline was subject to erosion. The shoreline facing the

submergod breakwater indicatc

s

a quitc different change comparcd with thc shorelinc facing the

dctachcd breakwaters

.

The lalter

a

dv

a

nccd seaward

,

resulting in a s

a

licnt. On the other hand,

the shorclinc facing thc

s

u

b

mergod breakw

a

ters did not significantly advance seaward showed

no marks of cro

s

ion

.

Thc sh

o

rclinc configur

a

tion is

a

lso smooth compared to that facing the

dctachcd bre

a

kwaters. This ind

i

cates that well-designed submerged breakwaters have a mild

but steady effect in maintaining shorelines.

3. STABILITY OF DETACHED BREAKWATERS

Det

a

ched breakwaters

a

re usu

a

lly constructed from rubble or wave absorbing blocks of

various kinds.The stabie weight of the rubble stone is usually determined by the so-called

Hudson

'

s formula

.

Howe

v

cr

,

as po

i

nted out b

y

m

a

ny researchers

,

the incident wave period is

not t

a

ken into account in thc fonnul

a

.

Co

nsequentl

y

,

somc modific

a

tions in Hudson

'

s formula

ha

v

e been proposed in whieh the significant wa

v

e heigh

t

and pe

r

iod are used to express the

364 TORU SAWARAGI

1986 before construction of submerged breakwaters

{ ISO 100 (11) 50

o

Fig.l0 Change in shorcline contour due to submerged breakwaters (Niigata Coast) (Japan Inst.

of Construct ion Eng.

,

1989)

char

a

ctcristic of irregul

a

r incident

w

aves

.

The effect of the duration of incident waves is also

taken into account in the modifications. These modifications together with the destruction

mechanism and rcliability of the rubblc mound structure will be d

i

scussed in lectures by Dr.

Magoon and Dr

.

J

.

Vander Meer in this short course.

The author et

a

l. also conductcd a series of experiments on thc destruction mechanism of

rubblc mound breakwaters b

y

irrcgular wa

v

es and found that the destruction rate of a rubble

mound breakw

a

ter dcpcnds largel

y

on the run of the incident waves. The run of irregular

incident wa

v

es is closcly related to the pcakcdness of the frequency spectrum of incident

waves. From these data, thc author ct al. deriv

e

d a design procedure to directly determine the

stabie weight of rubble stone from the frequency spectrum. The design procedure is outlined

bclow,

3.1. Effect

of incident

wave inegularity

and grouping

on the stabillty

of

rubble mound breakwater

Thc destructien ratc of a rubblc mound

Da'

is usuall

y

determined by the number of rubble

stones which are movcd from their former place per total number of stones in the reference

section

,

as fellows:

0

'

(

%

)

=

!1_!!!'1

_Q

~

QL~

~

Q!1~~

m

ov

~

om thci

ed

r

former p

~

siti

o

n

x

100

a

.t

otnl numbcr of ston

cs

111

the reference secnon

(33)

To cxprcss thcdogree of destructien more accurate Iy, thc author et al. (1985) proposed a ncw definition fordestructien ratcDa:

AJ

'

Da(%) =

-

A

-

x

100

(34)

o

whcrc An' is the

d

cstroycd

volume of thc cover laycr (revetment) and Ao is the destroyed

volume of the cover laycr whcn thedestructien rcachcs thc COiClaycr as shown in Fig.ll.

stones)

core layer

Fig.11 Definition sketch of damaged profile of rubble mound

The value of Da is roughly rclated to

Da'

by

Da'

=

0.2Da. (35)

Per Bruun (1979) pointed out that resonance on the slope of the breakwater strongly affects itsstabil ity. The author etal. (1983) found that resonance took place when the surf similarity parameter S is in the region of 2< S < 3.

Morcover, it is natura

I

to consider that

the

run of high waves affect the destruction of rubble mound. Therefore, in the design of rubble mound breakwaters. the probability of the occurrence ofboth high waves and the surf similarity parameter of zero-up or zero-down cross waves have to be considcred in determining the stabie weight of the rubble stone.

The

author et al. represented

this

probability using the conditional run length j(So*IHs) which express the run length ofso *

=

SISo

=

2 under the condition of H<!:

Hs,

where so is the

surf similarity parameter of the maximum wave and

Hs

is the significant wave

he

ight. Furthermore, wave energy directly relating to the stability of the rubble mound breakwater is

represented by the ave rage of the energy sum of each wave in each of the runs derived above over each run Icngth, defined by

00

N

~m

=

l

2:

pgH;2 /

2:

Nj

1=1 1=1

where Nj is thc number of the run whose length isj, and Hr is the wave hcight of i-th wave in

t

hc

run.

The

au thor ct al. obtained

the

following relation between

&um

and the mean run length j(So*IHs):

&um

= O.78j(S0*IHs} - 0.44

pgHs2

/

8

The

destruction rate Da of a uniform slope breakwater is related to

&um

by

[

&um

talla]

Da = 153.8 --2 - -30.1 Psgla

tamp

(36)

(37) (38) where Psis the density of the rubble stone, la is the representative diameter of the rubble stone,

'l'

is the friction angle of the rubble stones, and

tana

is the slope of the breakwater. 12-15

366 TORU SA WARAGI

Tbc mean run lengthj(l;(J·IH~) defined above is closcly related to thc pcakcdness of the frequency spectrum of incident waves Or by the following relation:

j~·IHs) = 30pl16 + 0.81 (39)

whcre

E(f)

is the frcqucncy spectrum of incident waves,

2

00

Op=

-

ffE(j)df

mOl) (40)

Assuming that thc wcight of thc rubblc stone W is expressed by W = Psg/a3,the stabIe weight of rubblc on the uniformly sloping mound is determined from Eqs.(37) to (39) as follows:

W [

pg(6.150p+20.0)

talla]3/2

3 .

= 1/3 --. Hs : for uniform slope

(Psg) (Da+30.1)

tam.'

When thc rubhlc mound breakwater of uniform slopc is damagcd, the slope of break water dcforms into a composite shapc as shown by a thick solid line in Fig.12.

(41)

Fig.12 Definition sketch of

rubble

mound breakwater of composite slope

If

the original slopc of the rubble mound breakwater is a composite shape as shown by the dotted line in Fig.12, the breakwater will bc more stable and the wcight of rubble needcd can bc reduced. The author et al. carried out experiments on thc stability of a rubble-mound break water of compositc

slopc whose

hypotheticat stope tafia' is 1/2.3 and obtained

the

following rclation for determining the stable weight for the rubble stone: W

[

pg(5.460p+17.73)

talla']3/2 Hs3"

.

I

= --: lor composite s ope

(p~g)1/3(Da+36.3) tall~'

Figure 13

illustratcs

the relation between the

valuc

of

Op

and

thc stable

wcight of the rubble stone W.

It

canhesccnthat thc stablo weight W increases in proportion to

9P

.

Comparing thc

results for

thc

cases with different

values

of

In,

the stabihty becomes higher with

incrcasing In

.

As

to thc influcncc

of

h2

on the

stability,

the experiments with

h2/ho

=

114

show better rcsults th,1I1thosc with h:zlho = 1/2. The reason for this is considered that the hydrodynamic fotces bocome lcss impulsivo and the resonance on the slope occurs less frcquently, with hzdccreasing, Tbc quantitative estimation of the influence of

In

and hz must he studied furthcr.

Figure 14 shows the rclation obtaincd between the stabIe weight of the rubble stone W and destruction ratc

Da

whcn Op=2.5 and H~=7m.

In the figure, the stabIe weight of the rubble stone calculated from Hudson's formula is also shown. The calculated weight from Eq.(41) with an allowable destruction rate (Ba) of 20% scerns to

cortespond

to that evaluated by Hudson's formula. The determination of allowable destruction rate is a practical problcm for future study.

(42)

5.0 - uniform slope ----composHe 5lope 2.0

o

o

1.0 2.0 3.0 4.0 Q, 5.0

F

ig.l} St

a

bie weight of rubble stone as a function of Op

20

-

i- -·-Ifudson formula

I~

-

-- uniform(tanCJ=I/lEIJ·(41) .3). t-- --- composite slope (tana·~1/2.3).Eq.(42) t- Q,~2.5 1- H.~ 7m

-f\

\;-

1""-~

~

I'--<, _'-

r--.

t----

_.-

t=-

_-

--100 80 60 40

o

o

20 40 60

0.(1)

80 100

Fig.14 Relation between stabie weight of rubble stone and destruction rate

Stabie weight for composite type breakwaters is approximately 1/2 that for uniform slope

breakwaters. gi

v

cn

t

hc samc rate of destruction

;

in other words, smaller materials can be uscd

in constructing rubble-mound breakwaters of composite slopc.

3.2. Stabie weight of rubble stones for submerged breakwaters

Thc Public Research Institute of the Ministry of Construction of Japan (Uda

,

1988) conducted

a series of cxpcrirncnts tn determine the stabie weight of ruhble stones for submergod

breakwaters beföre storting construction

o

f the submergod breakw

a

ters on the Niigata coast

shown ahovc in

Fig.J,

The

y

found that ruhble stoncs on the submerged bre

a

kwater were first

lifted up hy the lift force

a

nd Ihen moved from the surface of the breakwater. From thc

experiment

,

the follow

i

ng

Iormula

for determining stabie weight w

a

s proposed

,

bascd on the

368

TORU SAWARAGI

balance between thc weightof the stones and the lift force acting on them, in the cases where

Ho'Ih>0.3

:

(

f

u

)3

3

Ws

=

Sn

PsK

v

R

(Ps

/

p-l) cos a

in which

Sn

=

QKal2K

v

(St

a

bility Number)

Umax

(HO'

R)

fu

=

{ij=

8 exp -l

.

~

- 2.8Ho'

+

0.2

Kv

=

Vr/d

3 ,

Ka

=

Ar/d

2

wherc CLis thc lift force coefficient

,

V

r and

Ar

are the volume and sectional area of the rubble

stonc,

Ho'

is the equivalent dccp water wave height of incident waves,

ho

is the depth of the

sen

f100r

at the submergod breakwater and R is the depth at the crown of the breakwater

.

The

valucs of Sn and Kv depend on the material used for rubblc. The following values are given by

the Millistryof Construction of Japan

:

Sn

=

0.9, Kv

=

0

.

5

: natural stone

Sn

=

0

.

9, Kv

=

0

.

5 to 1.0

:

wave absorbing block

(43)

4. CONCLUSIONS

.

Thc functions of detached breakwaters and submerged breakwaters in controlling waves,

wavc-induccdcurrents and sediment movement were discussed. As has already been reported

by many rcscarchcrs

,

dctachcd breakwaters are very effective in controlling incident waves and

salient or tomboio formation due to deposition of sediment sometime result in sever erosion of

downstream coasts.

On thc other hand, submerged break waters have a relatively mild but steady effect in

retaining shorc-sidc sediment and have little effect on the surrounding coasts. Therefore,

submergod breakwaters are recently replacing det

a

ched breakwaters in Japan. However, there

are somc issues related to submergod brcakwatcrs: l)They may become fatal obstacles for

fishing boats or smalt plensureboats

;

2)The effect of a subrnerged breakwater on a coast with a

wide tidal range is not obvious bccause the hydr

a

ulic function of the submerged breakwater

depends on the water depth at thc crown.

In determining thc stabie weight of rubble stones for a submerged breakwater, it is important

to considcr thc effects of irregularity and grouping of incident waves

.

Naturally, the wave

period should also be taken into considcration, as was pointed out by Bruun and the author

.

From this point of view

,

the formula for determining the stabie weight of rubble for

submerged breakwaters as proposed by thc Ministry of Construction of Japan needs to be

further studied.

SYMBOLS

Change in sectional area below a reference lovel along x-axis

Sectional area of rubble stone

Areal ratio of salient and shorc-side coast of breakwater

Destroyed volume of rubble mound breakwater when the dcstruction reaches the core

layer

Destroyed volume of cover layer of rubble mound breakwater

Excurtion length of water partiele

Breakwater width at still water level

Celerity, Suspeneed sediment concentration

Group velocity

Lift force cocfficient

Cm

Co

Da

Addcd mass coefficient

Rcferencc

concentration

Destructionrateofrubble

mound

breakawter determinedfromthe volume of deformed

slope

Destruction rate of rubble mound breakwater determined from the number of rubble

stones moved from their original place

Energy

loss

due to breaking

waves

Energy loss caused by boundary shear

Total energy loss

Energy loss took place in pcrmeable layer

Thickness of permeable layer

Mean grain size

Energy density of incident waves

Water

partiele

velocity caused by waves and currents

Frequcncy

spectrum of incident waves

Wave energy directly relatingtothe stabilityofrubble mound breakwater

Friction factor

Gravity accerelation

Incident wave height

Significant wave height

Average of

fivc largcst

incident significant wave heights

Equivalent deep water wave height

Depth in still water

Representative depth of topographic change

Water depth at breaking point

Water depth at the foot of breakwater

Representative depth

Crown height from still water

level

Berm depth of

rubble

mound breakwater of composite slope

Wave number

Complex Wave number on permeable layer

Coeffisient

relating shape

of

rubble

stone

Diffraction coefficicnt

Permeability

Transmission coefficient

Coefficient relating

shape

of

rubble

stone

Wave length

Length of detachewd breakwater

Lcngth of opening in the row of detached breakwaters

Represcntativc

diameter of rubble stone

Berm width of rubblc

mound

breakwater of composite

slope

O-th moment of

frcquency

spectrum

Pressure

Peakedness of frequency spectrum

Total longshore sediment transport rate

Totallongshore sediment trasnport rate estimatcd from toporgaphic change

Totallongshore sediment transport rate calculated from flux model

Local bed load transport rate

Local suspended sediment tmasport rate

Local sediment transport rate

Depth at the crown of subrnerged breakwater

Depth and time averaged Reynolds' stress tensors

Bottom slope

Radiation stress tensors

Da'

Db

De

DIoss

~

d50 E

Fb

E(f) Esulil f

g

Hi

Hs

[Hs]s

Ho'

h

ii

hb

ho

hr

hs

h2

k

k

Ka

Kg

kp

Kt

Kv

L I

r

la

lil

mo

p

Op

Oy Oye Oyf

qb

qs

qi

R

Ri,j

S

su

12-19

370 TORU SA WARAGI Sn T

[Ts]S

T

x

il

ub umax V

Stability number for submerged breakwater Wave period

Avcrage of five largest significant wave periods Lcngth of salient

Width of salient

Time and vertically averaged velocity of wave induced current in x (offshore) direction Maximum water partiele velocity due to waves at bottom

Maximum water particIe velocity at thc wave crest

Time and vertically averaged velocity of wave-induced current in y (longshore) direction

Volume of rubble stone Weight of rubble stone

Water partiele velocity in vertical direction Settling velocity

Offshore distance Width of breaker zone

Distance between initial shoreline and break water Landward limit of significant sediment transport Seaward limit of significant sediment transport Longshore distance

Vertical distance Slope of breakwater

Hypothetical slope of rubblc mound breakwate of composite slope Imaginary part of complex wave number

Friction angle

Critical Shields' number Shiclds' number Eddy viscosity

Diffusion coefficient of suspended sediment Velocity potential

VcIosity potcntial in permeable layer Velocity potentialof incident waves Velocity potencialof scattered waves Void ratio

Displacement of mean water level

Hight of wave crest from the still water level Kinematic viscosity

Ratio of wave height and total water depth or Nondimensional permeability (=k~/V) Density of water

Density of rubble stone Angle of wave incidence Angular frequency

Time averaged bottorn shear stress Surf simularity parameter

V

r W w

Wc

x Xb Xoff

Xo

X

cr

y

z

a

a'

ESZ y

P

Ps

8

(J "ti 11-20

S

o

Surf similarity parameter of themaximum wave

So·

= SIS

o

REFERENCES

BAGNOUJ,R.A. (1965). The flow of cohesionless grains in fluids, Proc. Roy.Soc. Series

A.964, Vo1.249, pp.235-297.

BAlTJES,J.A. (1975) A

n

o

te

on modeling of turbulence in the surf zone, Proc. Symp. Modeling Technique, pp.1050-1061.

BliKER,E. W. (1968). Littoral drift as a function of waves and current, Proc.

l lst

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