C H A R A C T E R I S T I C S O F S O L I T A R Y W A V E B R E A K I N G

I N D U C E D BY B R E A K W A T E R S

By Stephan T. Grilli, 1 Member, ASCE, Miguel A. Losada, 2 and Francisco Martin 3

ABSTRACT: Laboratory experiments are presented for the breaking of solitary waves over breakwaters. A variety of behaviors is observed, depending on both breakwater and incident wave height: for emerged breakwaters, waves may collapse over the crown, or break backward during rundown; and for submerged break- waters, waves may break forward or backward, downstream of the breakwater. The limit of overtopping and wave transmission and reflection coefficients are experimentally determined. It is seen that transmission is large over submerged breakwaters (55-90%), and may also reach 20-40% over emerged breakwaters. Computations using a fully nonlinear potential model agree well with experimental results for the submerged breakwaters, particularly for the smaller waves (Hid < 0.4). For emerged breakwaters, computations correctly predict the limit of over- topping, and the backward collapsing during rundown.

INTRODUCTION

Emerged breakwaters are designed to offer protect ion on their seaward face (armor layer), by inducing runup, breaking, and part ial reflection of incident waves. Ext reme waves, however, may over top the structure and break on its upper part (crown or crest) or on its landward face. Hence , both of these must have proper re inforcements (e.g. crown wall). When overtopping occurs, a t ransmit ted wave may reform and still cause damage shoreward. Submerged breakwaters are designed to offer protect ion by inducing breaking and part ial reflect ion-transmission of large waves. For both breakwater types, assuming no structural damage, the percentage of wave transmission can be used as a measure of the degree of protec t ion offered by a breakwater against a given wave climate.

In the present study, labora tory exper iments and fully nonl inear com- putations are carried out and compared for the t ransformation, breaking, overtopping, and transmission of solitary waves over submerged and emerged breakwaters. Solitary waves are bel ieved to represent a good model for tsunamis (Goring 1978) and also for ex t reme design waves because of their large runup, impulse, and impact force on structures.

The propagat ion and runup of soli tary waves over shelves and slopes has been the object of numerous studies. A m o n g these, we will ment ion the works by Goring (1978) and Pedersen and Gjevik (1983) using Boussinesq equations, and by Synolakis (1987) and Kobayashi et al. (1987, 1989) using NSW equations. Based on the lat ter model , Kobayashi and Wur jan to (1989, 1990) calculated monochromat ic wave over topping, and wave transmission, over emerged and submerged t rapezoidal breakwaters , respectively, and found reasonable agreement with labora tory exper iments and empirical for-

1Assoc. Prof., Dept. of Oc. Engrg., Univ. of Rhode Island, Kingston, RI 02881. "-Prof., Univ. of Cantabria, Santander 39005, Spain. 3Grad. Student, Univ. of Cantabria, Santander 39005, Spain. Note. Discussion open until July 1, 1994. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on February 25, 1991. This paper is part of the Journal of Waterway, Port, Coastal, and Ocean En- gineering, Vol. 120, No. 1, January/February, 1994. �9 ISSN 0733-950X/94/ 0001-0074/$1.00 + $.15 per page. Paper No. 1428.

74

mulae used in design. Boussinesq and NSW theories, however, are only of first-order in the nonlinearity. For large waves and/or abrupt discontinuities in the bottom--such as submerged obstacles--interactions may become much stronger and require a higher-order theory for being accurately de- scribed. When there is induced wave breaking, clearly, a full nonlinear theory is required.

Losada et al. (1989) conducted experiments for solitary wave propagation over a step in the bottom. They observed four types of interactions when the wave height increases: (1) Fission; (2) fission and spilling breaking of the first soliton; (3) transition; and (4) plunging breaking over the step. For case (1) (small waves), the number of solitons and their amplitude was well predicted by the Korteweg de Vries (KdV) equation, except for the leading soliton. This is a case with weak interactions. For the other cases, strong interactions and breaking of the first soliton always occurred, and the KdV theory failed to predict experimental results. These cases were recently successfully addressed by Grilli et al. (1992) using a fully nonlinear model.

Efficient methods have been developed over the past 15 years for the numerical modeling of fully nonlinear waves, based on two-dimensional potential flow equations. The standard approach combines an Eulerian- Lagrangian formulation, to a boundary integral equation method (BIE). Initial space periodic models (Longuet-Higgins and Cokelet 1976; Vinje and Brevig 1981; Dold and Peregrine 1986) have now been extended to address problems with both arbitrary incident waves and boundary geometry (Grilli et al. 1989, 1990b). In particular, problems of wave shoaling over complex bottom geometry, and wave runup and interaction with structures have been solved with the latter model (Svendsen and Grilli 1990; Grilli and Svendsen 1991a, b). This model will be used in the present numerical applications.

Solitary wave interaction with a submerged semicircular cylinder has also been studied by Cooker et al. (1990), using an extension of Dold and Peregrine's (1986) model. Results showed a variety of behaviors, depending on wave height and cylinder size. A limited number of experiments was made, which confirmed the computations. In the present paper, we will extend this study to a more realistic class of emerged and submerged break- waters with trapezoidal shape, and we will characterize wave behavior over both emerged and submerged breakwaters, as a function of breakwater, and incident wave height.

In the following, dashes denote classical dimensionless variables of long wave theory, length is divided by d, time by V~-g, and celerity by V'-~.

DESCRIPTION OF EXPERIMENTAL SETUP

Experiments were conducted in the 70-m long, 2-m wide, and 2-m high wave flume of the University of Cantabria (Santander). Solitary waves were generated using a piston wavemaker, following the procedure introduced by Goring (1978) (Fig. 1). The flume width was divided into two subsections, the smallest one (in which a plywood breakwater was built), being 0.9-m wide. The distance from the wavemaker paddle to the breakwater was approximately 45 m. The breakwater was trapezoidal, with landward and seaward 1:2 slopes (cotg [3 = 2), a height hi = 0.4 m, and a width at the crest b = hi (Fig. 1). Fifteen values of the water depth d were selected, between 1.143 m and 0.195 m, which gave eight submerged breakwaters, with 0.35 < h'l - 1, and seven emerged breakwaters, with I < hl < 2.05, where hl = hl/d.

75

Incident /

~ Wave Paddle

H

d seaward 7

~. / 1 : 2 Breakwater

x v

landward

FIG, 1. Definition Sketch for Experimental Setup

Generation of Incident Waves Grilli and Svendsen (1990a, 1991b) showed that solitary waves generated

in a nonlinear potential model, according to Goring's (1978) first-order solution, changed their shape while propagating over constant depth. Pres- ent experiments and computations, essentially, show the same behavior: incident wave heights reduce slightly over 80 water depths or so of propa- gation, while small tails of oscillations are shed behind the leading wave crests.

Side wall and bottom friction effects were studied by Losada et al. (1989) for the propagation of solitary waves down the flume. They found that a small decrease in wave height always occurs, which is almost negligible for smooth-enough flume surfaces.

To eliminate both of the frictionless and friction effects of amplitude reduction from the experiments, Losada et al. (1989) measured incident waves sufficiently close (not quite close enough, however, to significantly interact with the reflected wave) to the step in the bottom they were study- ing, and made sure, by trial and error, the required incident amplitude was obtained at this location. A similar procedure is followed in the present experiments. Incident wave heights were measured at 7.5 hi in front of the breakwater axis, within the range: 0.06 -< H ' <- 0.59. Continuously measured incident wave elevation agreed to within 0.5% with predictions of the nu- merical model at the same location [see Grilli et al., (1992) for detail].

Sources of error in similar experiments made in this flume following an identical wave-generation procedure were also discussed by Cooker et al. (1990).

Measurements of Surface Elevation Surface elevations were measured using capacitance wave gages every

0.0351 s and for 10 s during each experiment, and a video recording was also made near the breakwater. Video images were digitized, and charac- teristics of wave breaking in the vicinity of the breakwater were analzyed. Due to the limited number of gages, identical experiments were repeated several times and gages, fixed on a movable carriage, were moved to dif- ferent locations until the full experimental region was covered. A high degree of repeatability of experiments was found, with differences in surface elevation of less than 0.5% between repetitions.

EXPERIMENTAL RESULTS Water depth d is fixed for each experiment (hence, h ~), and incident

wave height H ' is increased by small steps in successive experiments to

76

determine boundaries between different regions of wave behavior in the parameter space (h~, H' ) .

Characteristics of Wave Breaking Different wave evolution and breaking characteristics are observed, de-

pending on h'l: Case h'l < 1: Submerged breakwaters. Three regions can be distinguished

depending on H ' [Fig. 2(a)]:

1. No breaking with, mainly, wave transmission and reflection (TR), and crest exchange between incident and transmitted waves over the breakwater

2. Backward breaking of the wave tail onto the landward slope (BB), in the direction opposite to propagation of the transmitted wave, after the main crest has passed over the breakwater (in this case, a roller also prop- agates seaward as a bore, generally followed by a tail of oscillations)

3. Forward breaking (FB) of the transmitted wave, well beyond the breakwater crown. For a given h~, the FB may be spilling or plunging, depending on (increasing) H '

H ' (a)

0 . 5 ................... ~ ....................... i ....................... ~ ........... '~ .................. ~.B .................. L .............................................

' i ! ! ! i 0.4

0.3 -

0.2

0.1

0 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9

h t

1

H' (b)

0 . 6 i I

0 . 5 , i FC ~ ~ ~_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,

O4o O iiiiiiiiii o . 1 ............................ i .......................... ! ...................... i ............................. i .............................. i .............................

1.2 1.4 1.6 1.8 2 2 .2 1

FIG. 2. Measured Characteristics of Solitary Waves as Function of Incident Wave Height H ' : (a) Submerged Breakwaters ( h l -< 1); and (b) Emerged Breakwaters (h~, > 1)

77

In Fig. 2(a), (---o--) is the limit between wave transmission and reflection (TR) and backward breaking (BB), and (---c3----) is the limit between TR or BB, and forward breaking (FB).

In the FB region, and for h~ > 0.55, both types of breaking successively occur: FB of the main crest and BB of the smaller wave tail. On the boundary between BB and FB, both breaking types occur simultaneously. For h~ -< 0.55, only TR or FB occur, the breakwater crown being too deep for inducing BB of the (smaller) wave tail. Locations of breaking (BB and FB) have been measured for h~ = 0.8. These results are discussed later in the text, when compared to computations.

Most of the observations just reported qualitatively correspond to results obtained by Cooker et al. (1990).

The dashed curve in Fig. 2(a) separates breaking (FB), and nonbreaking solitary waves, over a vertical step hi in the bottom, as measured by Losada et al. (1989). In the present case, the step corresponds to a submerged breakwater of crest width b - ~, and one may thus conjecture that increasing b from h~ to ~ makes breaking occur for smaller waves and the BB region disappear, while the boundary BB-FB collapses onto the dashed line. Any other value of b, in between hi and ~, should thus lead to a BB-FB boundary in between those limits, and to a proportionally reduced BB region.

Case h~ > 1: Emerged breakwaters. Three regions can be distinguished depending on H ' [Fig. 2(b)]:

1. No overtopping (NO), with simple runup on the seaward slope 2. Overtopping (O), and sliding of the wave over the crown 3. Overtopping and forward collapsing (FC) of the wave over the crown

In Fig. 2(b), ( ~ ) is the limit between no-overtopping (NO) and ov- ertopping (O), and (----9--) is the limit between NO and forward collapsing (VC).

The rundown flow on the seaward slope may generate turbulence. When H' -> 0.2, breaking occurs during rundown by backward flow towards the slope (backward collapsing, BC, in the direction opposite to the retie, ted wave propagation).

Case h~ = 0(1):

1. For h'2 only slightly larger than 1: the incident wave flow reduces to a thin layer of water over the breakwater, whose front collapses when it passes over the crown and slides over the surface of the seaward body of water as a bore. The main crest then overtakes this bore well beyond the breakwater, and spills or plunges depending on the value of H ' .

2. For h~ only slightly smaller than 1: collapsing over the crown and forward breaking occur almost simultaneously, the former being dominant when H ' increases.

Wave Transmission and Reflection Transmitted H; and reflected H'r, wave heights were measured at 7.', 'zl

landward, and at 9.25hl seaward of the toe of the breakwater front slope, respectively. Measured transmission and reflection coefficients, Ct = HJH' and Cr = H'JH', are given in Figs. 3 and 4, as functions of h'l and H'. Because of continuous wave evolution over the breakwater, Ct and Cr only correspond to locations where they were measured. Results in Fig. 3 show:

78

H ~ submmted ~ emerged 0 . 6 ! , , ~ . , . . , , . . , ,, . . . . . . . . I . . . . i . . . . l / . . . ' ] ' ' ' j . . . . . . . . . . ~J ' . . . . I ' / " l " 'y.

o, . . . . . . . . . . . . . . . : : . . . . . . . . . . . . . . . . . . . o, ............... - ..... i i

i i i i i ! i ~176 ! o o ~ i .~-- ' 0 . 3 , ........ i ........................ ! ........... i ........ i ............. b ......... i ........ T ............ " " ; ; ~ ~ : o O i ! i : .~i i i i ..........

~ i ...~ ........ i ........... i ......... i ...... ', .......... i ............. i..-......~--.~..-<.-.:-i;. ......... i ............. i ............. i ........... 0 . 2 - ~ - .....

0 . 1 ..... -~ ......... i ........... ~ . . . . . . . ...... i . ~ ' ' i i i N O i i

i ~ ~ ' L . ' i " i i i i i i . i " . . i . . . . i . . . . i . . . . i . . . i . . . . i . . . . i . . i . . . .

0 l i i i ~ i ~ ' ~ i ~ " i h ' 0 . 6 0 . 8 1 . 2 1 . 4 1 . 6 1 . 8 2 I

FIG. 3. Measured Wave Transmission Coefficient C,, for Emerged (h~ > 1) and Submerged (h~ <- 1) Breakwaters, as Function of Relative Wave Height H '

H ' submerged ~ emerged o . 6 L . . . . ! . . . . ! . . . . ! . . . . ! . . . . ! . . . . LJ. . . . . . . ! . . . . . . . . . . . . ! . . . . . . . . . ! . . . . .

0.5 ............................................................. ~

~ ......... i ............. i ............ ! ............. I ........... i ............ t ............ i ........ : ~ ~ - i ............. ! ............. i ............. i ............ i ......... - i i i i i i i i i ! ! i i "

0.3

0 . 2 .......... i . . . . . . . . ~ ............ i ........... i ........... i ......... i ............. i.. '~ ' ~ ~ -~.~ ............ i~ ......... -

0 . 1 .o..,..!0 ........ + i i i i i ! i L { i i ............ .......... ........... .......... ............ ............. ............. ............. ............ ............. ............ ..........

0 i ' i 1 i i i i h '

0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2

FIG. 4. Measured Wave Reflection Coefficient Cr on Emerged (h~ > 1) and Sub- merged (h~) Breakwaters, as Function of Wave Height H'

1. For submerged breakwaters, transmission is large in all cases. Even for hi = d, transmission is still between 50% and 60%.

2. For emerged breakwaters, transmission vanishes, as expected, along the limit of overtopping (dashed line, reproduced from Fig. 2(b)]. For large overtopping waves, however, transmission may reach 20 to 40%.

In Fig. 3, (---) is the limit of overtopping from Fig. 2(b), and symbols • represent positions of computational results from Table 1 (corresponding experimental result is Ct = 0.93).

In Fig. 4, the symbol • represents positions of computational results from Table 1 (corresponding experimental result is C, -- 0.22).

79

TA

BL

E

1.

Nu

mer

ical

Res

ult

s fo

r S

ub

mer

ged

Bre

akw

ater

of

Hei

gh

t h

~'

= 0.

8

00

O

H' (1)

0.06

0.

10

0.20

0.

30

0.40

0.50

0.60

0.

70

Typ

e (2)

T

R

TR

B

B

BE

F

B

FB

F

B

FB

H "

a•

(3)

0.07

7 0.

126

0.24

4 0.

352

0.46

3

0.56

3

0.66

9 0.

765

H "

i.

(4)

0.04

6

0.07

7

0.16

1 0.

229

0.30

0

0.37

5 0.

458

0.51

6

H}~

(5

)

0.38

7 0.

477

0.47

5 0.

427

0.42

7

c,

(6)

0.95

0.

98

1.00

C x',

,b

x)b

(7)

(8)

(9)

0.28

--

--

0.29

--

--

0.

27

1.00

--

--

1.20

3.

59

--

--

2.40

--

--

1.

70

--

--

1.30

--

--

1.22

t; =j-

(1

0)

(11)

16.1

7 9.

28

14.7

2 8.

31

10.4

5 3.

74

8.81

3.

73

7.22

2.

98

5.94

2.

18

5.32

1.

66

5.20

1.

58

1 .................... [ ....................... i n .................. i ...................... ; ' ~ ................ f ....................... i ........................ i ........................ i

o . s .................. i ............... ' .............. . . i ........................................... [ ....................... J ....................... i ....................... i

- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . .

i o,

-1.5 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . L . . . . - 20 -15 -10 -5 0 5 10 15 20

F I G . 5 . D e f i n i t i o n S k e t c h f o r C o m p u t a t i o n s

D E S C R I P T I O N O F N U M E R I C A L M O D E L

The two-dimensinal (2D) model by Grilli et al. (1989, 1990b), based on fully nonlinear potential flow equations, is used in the computations. A brief description of this model is given in the following section.

Governing Equations With the velocity potential defined as (b(x, t), the velocity is given by u

= Vr = (u, w), and continuity equation in the fluid domain fl(t) with boundary T(t) is a Laplace's equation for the potential (Fig. 5)

V2qb = 0 in l-l(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

Using the free space Green's function G(x, x/) = -(1/2-n)loglx - x/l, (1) transforms into a boundary integral equation (BIE)

fr [ O+ OG(x'x')]dF(x) (2) ~ ( x , ) + ( x , ) = (x) ~ ( x ) C ( x , x , ) - + ( x ) 0----U- . . . . . . . . .

in which x = (x, z) and xl = (xt, zt) = position vectors for points on the boundary; n = unit outward normal vector; and a(xt) = a geometric coef- ficient.

In the model, the BIE (2) is discretized using nodes on the boundary, and higher-order boundary elements (BEM) to interpolate between nodes (Brebbia 1978). Nonsingular integrals in (2) are calculated by standard Gauss quadrature rules. A kernel transformation is applied to weakly singular integrals, which are then integrated by a numerical quadrature, exact for the logarithmic singularity. An adaptive numerical integration is used for improving the accuracy of regular integrations near corners (A-D in Fig. 5), and in breaker jets, where elements on different parts of the boundary are close to each other.

Boundary Conditions On the free surface Ty(t), + satisfies the kinematic and dynamic boundary

conditions, respectively

D--f= ( ~ + u ' ~ ) r = u = ~ 7 + D t on Fr(t ) . . . . . . . . . . . . . . . . . . . . . (3)

81

Dqb 1 Pa Dt - 9 z + ~ Vqb" Vqb - - -

P on Fi(t ) . . . . . . . . . . . . . . . . . . . . . . (4)

with r = position vector of a free surface fluid particle; O = acceleration due to gravity; z = vertical coordinate (positive upwards and z = 0 at the undisturbed free surface); Pa = pressure at the surface; and P = fluid density.

In the model, time integration of the two free surface boundary conditions (3) and (4) is based on second-order Taylor expansions (Dold and Peregrine 1986), expressed in terms of a time step At, and of the Lagrangian time derivative [as defined in (3)]. The motion of free surface fluid par t ic les-- identical to nodes of the BE M discretization used in solving (2) - - i s cal- culated as a function of time (Grilli et al. 1989).

Two methods are used for generating solitary waves in the model:

�9 For H ' <- 0.2, solitary waves are generated by a numerical piston wavemaker, following Goring's (1978) solution (Fig. 5). This so- lution is quite accurate for small waves (Grilli and Svendsen 1990b, 1991b). The motion Y and normal velocity are specified over the paddle as

04, = xp; V + . n - On Up on Frl(t) . . . . . . . . . . . . . . (5)

where the overline denotes a specified value; and (xp, Up) = pre- scribed wavemaker motion and velocity, respectively.

�9 For H ' > 0.2, elevation ~q, and potential q~ for numerically exact solitary waves are specified on the free surface Fi(t0 ) at initial time to, following Tanaka 's (1986) method

~(x, to) = -q(x); $(x, to) = ~p(x) on Fr(t0 ) . . . . . . . . . . . (6)

Along the stationary bot tom Fb and other fixed boundaries Frl and/or Fr2, the no-flow condition states

O_+ = 0 on Fb and Fr~ and/or F r 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7) On

Numerical Accuracy Time step is automatically adjusted to ensure optimal accuracy and sta-

bility of computations (Grilli and Svendsen 1990b). Accuracy is checked by verifying conservation of wave volume and total energy. In all present cases, parameters have been selected for both of these to stay constant to within 0.05%, during most of the wave propagation. When breaking occurs, how- ever, errors in volume and energy increase, and computations were stopped when errors became larger than 0.5%.

NUMERICAL RESULTS FOR SUBMERGED BREAKWATER

Numerical Data A tank with depth d = 1, length 40d, and a submerged trapezoidal

breakwater at x ' = 0 (height h~ = 0.8d = b, and 1:2 side slopes) is used in the computations. Incident wave heights are: H ' = 0.06, 0.1 and 0.2 [generated with (5)] and H ' = 0.3, 0.4, 0.5, and 0.7 [generated with (6)].

82

The initial distance between free surface nodes is Ax = 0.25d for H ' -< 0.5, and 0.1875d for H ' > 0.5.

Free Surface Evolution and Breaking Types For H ' = 0.06 and 0.10, a crest exchange takes place over the breakwater

[Fig. 6(a) and 6(b)]: the incident wave shoals over the seaward face, up to a height H '~ , (curves a -c ) ; the incident crest then decreases and vanishes around x' - - 1.3, while a new crest emerges over the landward face (curves d-e) . Eventually, a transmitted wave, only slightly smaller than the incident wave propagates down the tank, while a reflected wave, about one-fourth the incident wave height, propagates backward in the tank (curves f -g) . This weakly nonlinear interaction is referred to as a transmission-reflection (TR). Higher frequency oscillations occur landward of the breakwater, and to a larger extent for larger incident waves [Fig. 6(b)]. This corresponds to the higher nonlinearity of the wave-breakwater interaction, when H ' in- creases. For both cases in Fig. 6(a) and 6(b), however, general features are almost identical.

In Fig. 6, times of curves (a -g ) are given in Table 2. Curves c correspond to maximum crest height H'ax. Vertical exaggerations are 84 and 42, re- spectively.

For H ' = 0.20 and 0.30, a crest exchange also takes place [Fig. 7(b) and 7(c), curves c, and d, e, respectively], similar to Fig. 6(b), curves d, e [plotted at the same scale in Fig. 7(a)]. Nonlinearities, however, are larger, and higher frequency oscillations in the back of the transmitted wave evolve into

T / #

0 . 0 8

0 . 0 6 -

0 . 0 4 -

0 . 0 2 -

0 -

- 0 . 0 2

(a)

. . . . . . . . . . i . . . . : ................................... ~ .................. i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i - f . . . . . . . . . . . . . . . . . . . . . . . . . . . . - l

................................... i _ . -

5 - 1 0 - 5 0 5 1 0 X '

f l ' (b)

0 . 1 5 . . . . . . . . . . . . . . . . 1 ~ i i /

0.1

0 . 0 5

0

- 0 . 0 5 x '

1 5 - 1 0 - 5 0 5 1 0

FIG. 6. Computed Results for Solitary Wave Transformation over Submerged Trapezoidal Breakwater at x' = O, with hl = 0.8, h ' = h'~, and Slopes 1:2: H ' = (a) 0.06 (TR); (b) 0.10 (TR)

83

TABLE 2. Time t' Computed for Curves a-g in Figs. 6-8 (from Instant Incident Wave Crests Pass x' = - 6 )

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Curves 6(a) 6(b) 7(a) 7(b) 7(c) 8(a) 8(b) 8(c) 8(d) 8(e)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (1 O) (11 ) a -0.75 -0.90 - - 2.44 2.29 - - - 3.18 2.86 2.56 2.72 b 1.42 1.25 - - 4.97 4.07 4.07 4.60 4.27 3.79 3.69 c 5.07 4.98 4.98 6.85 5 . 2 2 5.22 5.47 5.14 4.77 4.65 d 7.56 7.24 7.24 8.38 6.04. - - 6.21 5 . 7 9 5.30 5.20 e 9.54 8.99 8.99 10.45 6 . 7 8 6 . 7 8 7.06 . . . . f 12.99 11.59 11.59 - - 7.51 - - - - - - g 16.17 14.42 - - - - 8.26 8.26 . . . . . .

a small backward propagat ing wave train, whose leading crest eventual ly breaks towards the breakwater (curves d, e, and f, g, respectively). This terminates the computat ions and is referred to as backward breaking (BB). In Fig. 7, the small line around x ' = 0 marks the breakwater crown. Vert ical exaggeration is 3.8. Initial stages of a T R can also be seen on Fig. 7(b) and 7(c). For H ' -- 0.3 in Fig. 7(c), the BB occurs almost s imultaneously with a FB of the t ransmit ted wave (curve g). This is globally referred to as BF. Fig. 8(a) shows results of Fig. 7(c) in undis tor ted scale, and the initial evolution of the BB can be t te r be seen over the landward face (curve g).

For larger waves (H ' > 0.30), a crest exchange still occurs [Figs. 8 ( b - e ) , curves c], but the emerging crest rapidly evolves into a forward (plunging) breaker (FB, curves d, e). With increasing incident wave height, the break- ing location in Fig. 8 ( b - e ) gets closer to the b reakwate r crest, and the breaking wave height decreases. See also Table 1. Occurrence of a FB terminates computat ions before a BB, or a TR, can develop. This is also observed in experiments. In Fig. 8, t imes of curves ( a - g ) are given in Table 2. Results are given in undis tor ted scale.

Wave Envelopes and Detailed Results Wave envelopes for the cases in Figs. 6 - 8 , all exhibit a maximum H ' , x

in front of the breakwater , a round x ' = - 1.35 (Fig. 9). In Fig. 9, ( ) = positive envelope; ( . . . . ) = negative envelope; exper imental results for H ' = ([]) 0.10, (A) 0.20, (o) 0.40; ( - - - - - ) = average location of Htma• ( - - 1 . 3 5 ) , and H ' i n (~0.7) . The small line around x ' = 0 marks the breakwater crown. Vert ical exaggerat ion is 2.6.

Table 1 summarizes wave-breaking behaviors: smaller incident waves transmit and reflect (TR), whereas larger waves (H ' -> 0.20) b reak backward (BB), at x ~,b, or forward (FB), at X~b, with a height H~b at breaking. Breaking is assumed when the computed free surface has a vertical tangent; x~b, x~o denote locations of maximum surface elevations at breaking (breaking height). In Table 1, H 'ax and H'min are positive wave envelope ext rema in F " t . . . . . lg. 9; H~o Is the forward breaking height; Ct and Cr are transmission and reflection coefficients at x ' = 4 and - 9 . 4 , respectively; x~b and x~o are locations of backward and forward breaking, respectively (BB cannot be calculated for H ' > 0.3, since computat ions have to be s topped as soon as FB appears); and x} and t} are locat ion of the last computed crest and the time of propagat ion from the instant the crest passes x ' = - 6.

Fig. 10 shows a plan view of locations of H 'ax and H ' i . , and breaking

84

0.5 0 .4

0 .3 0.2

0.1 0

-0.1 - 0 . 2

(a)

................... t ...................................................................... i ....................... { T

. . . . . . . . . . . . . . . . . . . . T ....................................... ~ ............................ ~ ....................... r ....................................................................

~ i ~ ~ ' I , , . ~ . . . . ~ ................... i ................................ ~ .......... ! ...................

.................... ~ ....................... ~ ....................... i ............................................... { ..................................................................... I I I i I I I I I I J I i I I I I I I I I I I I ~ I I

4 - 3 - 2 -1 0 1 2 3 4 x I

r/

0 .5 0.4 0 .3 0.2 0.1

0 -0 .1

- 0 . 2

' (b)

~ ...................... ~ ~ ~ ' ' .... i ....................... ; .................... i ............. ~ ............

4 3 - 2 -1 0 2 3 4 x '

0.5 0.4 0.3

0.2 0.1

0 -0.1

- 0 . 2

(c)

' ' ' ' ' ' ' ' ' I ' ' ~ I ' ' ' ] ' I ' ' ' I ' ' ' :

i . ! i : ............ ; ~ - - i ........................ i .............. ~ ~ ~

..................... } ....................... i ........................ ! ....................... ~ ....................... ~ ....................... t ....................... i .....................

4 - 3 - 2 -1 0 1 2 3 4 x '

FIG. 7. Computed Results for Solitary Wave Transformation over Submerged Trapezoidal Breakwater: H ' = (a) 0.10 (TR); (b) 0.20 (BB); (c) 0.30 (BF)

locations, as a function of H ' , that illustrates previous observations. In Fig. 10, ( + ) = computed H~,• ( - -< ) - - ) = computed H'~,; ( - - • = computed BB (x~b for lower curve) or FB (x)b for upper curve); and (o) and (A) = experimental locations of BB and FB, respectively. For H~nax and H~nin, symbols denote individual results, and small lines show LMS trends. Bold lines denote the breakwater plan view. Fig. 11 shows that H'ax/H' and H'in/H' slightly decrease with H ' , with averages of about 1.20 and 0.65 respectively, and H)JH' decreases with H', from 1.30 to 0.60. In Fig. 11, for H~max/H ', (�9 = experiments, and ( _ _ ~ i ) = computations and LMS trend; for H'iJH', (A) = experiments, and ( - - q - - ) = computations and LMS trend; for H)JH', ( - - • = computations and LMS trend.

85

1"/' (a)

F o.s .................... ~b ................. i ....................... i ...................... i ....................... i ...................

o ................... !..,~ .............................

-O.S ~ i .............. ~ 1 ;

- 3 - 2 - 1 0 1 2 3

T/' (b) 1 0 . S ~ l . . . .

0

- 0 . 5

1 - 3 - 2 - 1 0 1 2 3

r / ' (c)

1 . . . . I . . . . i . . . . . . . . r . . . . I . . . .

0 . 5 a i ....... b ............. r ............... ! ................ d....i ...................

0 .......................................................................

3 - 2 - 1 0 1 2 3

:/ (d) 1 . . . . ! . . . . ! . . . . . . . . . . . . . . . .

O.5 i ~ . . ~ ,

- 3 - 2 - 1 0 1 2 3

T/' (e) 1

O. o

- 3 - 2 - 1 0 1 2 3

X J

X I

X '

FIG. 8. Computed Results for Solitary Wave Transformation over Submerged Trapezoidal Breakwater: / / ' = (a) 0.30 (BF), (b) 0.40 (FB), (c) 0.50 (FB), (d) 0.60 (FB), (e) 0.70 (FB). Times of Curves (a-g) are Given in Table 2. Results are Given in Undistorted Scale.

Comparison with Experiments Transmission and reflection coefficients, calculated for nonforward break-

ing waves (TR or BB), are given in Table 1, at the same locations as in the experiments. Experimental results in Fig. 3 and 4 (cross symbols) show that computations overestimate reflection and transmission by 6-7%, and 2 -

86

~'max

0.8 ~ - ~ . . ~ ~ I . . . . . . . . .

O. 6 - ~ .................... ~ ~-.---.---i----~ ............................... , .............................

0 . 4 - ~ ~ ~ ~ . - - . - . -

0 . 2 . . . . . . . . . . . . . . . . . . . . . . . L.I. - a

0 ............................ ~ .................................................... & ............... t._..~ ............................... ~ ............................ = = = = : : i : : : : ~ : -~ : = ~ - ~ - - - . : < " ~ t i i

-0.2 , , , , I , , , , , , , , , , ] , I , , , , I , , , ,

- 3 - 2 - 1 0 1 2 3 X'

FIG. 9. Envelope ~ ~a• of Surface Elevations from Figs. 6 -8 , for H ' = (a) 0.06; (b) 0.10; (c) 0.20; (d) 0.30; (e) 0.40; (f) 0.50; (g) 0.60; (h) 0.70

_

3 -

2 -

1 -

0 -

- 1 -

"2 -

- 3 -

0

FIG. 10.

X e

. . . . . I I I I I J -

. . . . . . . . . . . . . . . . . i~ ~i . . . . ~ ~ ................. i .... ]. i --t_~-

i i o o i ............... i o - - ~ : ......................... i .......................... ~ ............................ : : : :

. . . . i . . . . i . . . . i . . . . i . . . . ~ , , i . . . . 1 0.1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7

Locat ion x ' as Funct ion of H' for H~,,x, H'~n, BB, and FB

H,

1 . 2

1

0 . 8

0 . 6

i i E ' ' :i iiiiiiiiill ........................ i ...................................................... ~ .......................... ~ ..................................................... i .......................

L i i i i i T I [ I I I ] I I I I I ~ ~ I I I I I I I ~ I I

0 0 .1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 ) .7 H ,

FIG. 11. Resul ts as Funct ion o f H ' for H ' . x / H ' , H ' i . / H ' , and H ' ~ / H '

8 7

7%, respectively, likely due to the lack of energy dissipation at the break- water in the potential flow calculations.

Wave envelopes have been measured at a few locations between x ' = - 2 and 2, and forh} = 0.8 (symbols in Fig. 9). For H ' = 0.1 and 0.2, the agreement with computations is quite good (curves b and c). For H ' = 0.4, however, discrepancies are larger, with measurements systematically falling below numerical predictions (curve e). Direct observation of experiments for the largest waves showed that significant flow separation occurs at the breakwater crown, a phenomenon that leads to energy dissipation and am- plitude reduction. As these cannot be accounted for in potential flow cal- culations, the model overpredicts surface elevations over the breakwater .

Measured locations of breaking are compared to computat ions in Fig. 10. The agreement between both is quite good, especially for the BB, which is somewhat more local than the FB and, hence, can be bet ter identified in the experiments.

Measured maximum and minimum of positive wave envelopes are com- pared to numerical results in Fig. 11, for H ' = 0.1, 0.2, and 0.4. As expected from the preceding, the agreement is quite good for the first two waves, and poorer for the largest wave.

Computed types of breaking (BB, BF, and FB), in Table 1 and in Fig. 12, are in full agreement with observations, except for a small discrepancy in the prediction of BF ( H ' = 0.34 would have been better). In Fig. 12, solid lines are experimental limits as in Fig. 2, and symbols are numerical results. Predicted breaking types are: ( x ) = TR; ( 0 ) = BB; (A) = BF,

H a

0 . 6 , , Q . . . . . . . . . . . . .

O. 5 .................. ~ ................. ~ ............................................... ..- ............................................... .- ............................................. i ] i :

0.4 . . . . . . . . . ~ ................. ~ ............................................... ~ ............................................. ~ . . . . . . . . . . . . . . . . . . . . . . ~!ii i~e ' i iii- O. 3 ......................... i ............................................. ~ .............................. -;--:e~---T ........................................... --

0 .2

0.1

0 h ' 0 . 5 1 1 . 5 2 2 . 5 i

FIG. 12. Breaking Types: Solid Lines are Experimental Limits (as in Fig. 2) and Symbols Are Numerical Results

TABLE 3. Computed Runup and Rundown for Solitary Waves H' on a 1:2 Slope

H' R~' ~' . t'~ R~ t~ (1) (2) (3) (4) (5) (6)

O. 10 0.260 0.225 7.38 - 0.041 12.78 0.20 0.573 0.536 7.96 BC 11.63 0.30 0.924 0.889 8.37 BC 11.79 0.40 1.304 1.274 8.77 BC 11.82

88

and ([3) = FB. Limits of over topping are [see next section]: ( . . . . ) = Synolakis (1987) NSW runup law h) = 1 + 3t'u (Table 3); and (o) = model results h'l = 1 + R', (Table 3).

NUMERICAL RESULTS FOR EMERGED BREAKWATER

Numerical Data A tank, with depth d = 1, length 20d, and a 1:2 slope at its rightward

extremity, representing the seaward face of an emerged breakwater is used in the computations. Initial distance between nodes is Ax -- 0.25d on the free surface and 0.50 on the bot tom. Solitary waves are generated as de- scribed previously,

Runup and Limit of Overtopping Computed maximum runup height R ' , and runup time t" are listed in

Table 3, and free surface elevations at maximum runup are given in Fig. 13(a). Results show that larger waves, although faster, runup for a longer time than smaller ones. Based on NSW equations, Synolakis (1987) derived a theoretical runup law for solitary waves: ~ " -- 2.831 (corgi3) 1/2 ( H ' ) 5/4. This expression is in good agreement with numerical results, particularly for the largest waves (Table 3): the relative difference between R', and ~ ' , varies from 15.5% to 2 .4%, as H ' increases. This confirms observations by Svendsen and Grilli (1990): on a small slope, the NSW solution gives a good prediction as far as runup is concerned. Svendsen and Grilli also

1.5

1 -

0.5

0

-0 .5

(a)

, , J , p i , , J P ' ' ' ' J ' I ' ' ' ] ~ ~ ' . , ~

i i i i c i

........ ............................. i i i i .............................. i ............................... ............ ......................... i .............................

- 3 -2 -1 0 1 2 X '

77' (b) 0.5

i J ~ , , , , q r ~ I , , I , I ,

0 i : b =

-0.5 . . . . x ' - 3 - 2 -1 0 1

FIG. 13. Computed Surface Elevation ~q' on 1:2 Slope, for Solitary Waves with H ' = 0.1, 0.2, 0.3 and 0.4 (Curves a-d, Respectively) at Time of: (a) Maximum Runup; (b) Maximum Rundown (Curve a) or Backward Collapsing (BC, Curves b-d)

89

found, however, that velocities and pressure are not so well predicted by NSW. In Table 3, t', is the time at maximum runup (from the instant the crest passes at x ' = - 6 ) ; t) and R) are the time of and the rundown value or backward collapsing event (BC), respectively (time is defined as for t ') .

The breakwater height, for which no overtopping occurs for a given wave, is h~(H') = 1 + R ' , (H ' ) (limit of overtopping). These numerical values and the values based on NSW equations, are compared to experi- ments in Fig. 12. Both theoretical predictions agree well with experimental results.

Rundown and Backward Collapsing Computations in Fig. 13(a) have been pursued further than maximum

runup to analyze the rundown on the slope. Results in Fig. 13(b) show:

1. For H ' = 0.1, rundown occurs with a negative surface elevation (Table 3) on the slope (curve a), and a reflected wave propagates back in the tank.

2. For H ' = 0.2 and above (curves c -d) , waves collapse backward towards the slope (BC). This has also been observed in the experiments.

Computations have been interrupted in each case, at time t} for which errors on wave mass became larger than 0.5%. Considering this limitation, only curve b clearly shows the initial stages of a BC.

CONCLUSIONS

The present study has identified breaking characteristics of solitary waves over submerged and emerged trapezoidal breakwaters. Wave transmission and reflection coefficients can be estimated from the results, as well as the limit of overtopping, breaking types, heights, and locations.

For submerged breakwaters, in particular, results show that wave trans- mission is always large, ranging between 55% and 90% for hi = d to 0.8d, and hence, little protection is offered against large waves, such as tsunamis. For emerged breakwaters, when there is overtopping, wave transmission may reach 20-40% for large waves. Hence, one may also question the protection offered by these structures against large waves.

Finally, increasing the width b at the crest of a submerged breakwater makes breaking occur for smaller waves. This, and the influence of the side slopes, however, have not yet been investigated in detail, and are left for further studies, along with detailed analysis of the velocities and pressure above the breakwaters.

ACKNOWLEDGMENTS

Miguel A. Losada was sponsored by the Direcci6n General de Puertos y Costas (M.O.P.U.), under the Programa de Clima Maritimo, and by the Comision Interministerial de Ciencia y Tecnolgia, contract no. PB- 87/0800. St6phan T. Grilli and Miguel A. Losada acknowledge support of the Nato Collaborative Research Grant Program, grant no. CRG-901019, and St6phan T. GriUi acknowledges support from the NSF grant no. BCS- 9111827.

90

APPENDIX I. REFERENCES

Brebbia, C. A. (1978). The boundary element method for engineers. John Wiley & Sons, Inc., New York, N.Y.

Cooker, M. J., Peregrine, D. H., Vidal, C., and Dold, J. W. (1990). "The interaction between a solitary wave and a submerged semicircular cylinder." J. Fluid Mech., 215, 1-22.

Dold, J. W., and Peregrine, D. H. (1986). "An efficient boundary integral method for steep unsteady water waves." Numerical methods .for Fluid Dynamics H, K. W. Morton and M. J. Baines, eds., Clarendon Press, Oxford, England, 671-679.

Goring, D. G. (1978). "Tsunamis--the propagation of long waves onto a shelf." Report No. KH-R-38, W. M. Keck Lab. of Hydr. and Water Resour., California Inst. of Technol., Pasadena, Calif.

Grilli, S. T., Skourup, J., and Svendsen, I. A. (1989). "An efficient boundary element method for nonlinear water waves." Engrg. Anal. with Boundary Elements, 6(2), 97-107.

Grilli, S. T., and Svendsen, I. A. (1990a). "Computation of nonlinear wave kine- matics during propagation and runup on a slope." Water wave kinematics, A. Torum and O. T. Gudmestad, eds., NATO ASI Series E: Applied Sciences, Vol. 178, Kliiwer Academic Publishers, Norwell, Mass.

Grilli, S. T., and Svendsen, I. A. (1990b). "Corner problems and global accuracy in the boundary element solution of nonlinear wave flows." Engrg. Anal. with Boundary Elements, 7(4), 178-195.

Grilli, S. T., and Svendsen, I. A. (1991a). "Wave interaction with steeply sloping structures." Proc. 22nd Intl. Conf. on Coastal Engrg., Vol. 2, ASCE, New York, N.Y., 1200-1213.

Grilli, S. T., and Svendsen, I. A. (1991b). "The propagation and runup of solitary waves on steep slopes." Research report no. 91-4, Ctr. for Appl. Coast. Res., Univ. of Delaware, Newark, Del.

Grilli, S., Losada, M. A., and Martin, F. (1992). "The breaking of solitary waves over a step: modeling and experiments." Proc., 4th Intl. Conf. on Hydraulic En- gineering Software, W. R. Blain and E. Cabrera, eds., Computational Mechanics Publications, Elsevier Applied Science, New York, N.Y., 575-586.

Kobayashi, N., DeSilva, G. S., and Watson, K. D. (1989). "Wave transformation and swash oscillation on gentle and steep slope." J. Geoph. Res., 94(C1), 951- 966.

Kobayashi, N., Otta, A. K., and Ryo, I. (1987). "Wave reflection and run-up on rough slopes." J. Wtwys., Ports, Coast. Oc. Engrg., 113(3), 282-299.

Kobayashi, N., and Wurjanto, A. (1989). "Wave overtopping on coastal structures." J. Wtwys., Ports, Coast. Oc. Engrg., 115(2), 235-251.

Kobayashi, N., and Wurjanto, A. (1990). "Wave transmission over submerged break- waters." J. Wtwys., Ports, Coast. Oc. Engrg., 115(5), 662-680.

Longuet-Higgins, M. S., and Cokelet, E. D. (1976). "The deformation of steep surface waves on water-I. A numerical method of computation." Proc. R. Soc. Lond. A350, 1-26.

Losada, M. A., Vidal, C., and Medina, R. (1989). "Experimental study of the evolution of a solitary wave at an abrupt junction." J. Geophysical Res., 94(C10), 14557-14566.

Pedersen, G., and Gjevik, B. (1983). "Run-up of solitary waves." J. Fluid Mech., 135,283-299.

Svendsen, I. A., and Grilli, S. (1990). "Nonlinear waves on steep slopes." J. Coast. Res., SI 7, 185-202.

Synolakis, C. E. (1987). "The runup of solitary waves." J. Fluid Mech., 185,523- 545.

Tanaka, M. (1986). "The stability of solitary waves." Phys. Fluids., 29(3), 650-655. Vinje, T., and Brevig, P. (1981). "Numerical simulation of breaking waves." Adv.

Water Resour., 4, 77-82.

91

APPENDIX II. NOTATIONS

T h e f o l l o w i n g s y m b o l s a r e u s e d in t h i s p a p e r :

A , B , C , D =

b =

cr= c , = d =

G = 9 =

H = h I = n = p =

R d =

R u =

r =

t = u =

x =

x ~-

xp , u p =

Z

OL ~ -

F = At =

A x = .q =

P = + = 1~ =

Subscr ip t a = b =

b b = d =

f = Fo=

l =

m a x =

min = p =

r =

r l , r2 = t =

bt =

S u p e r s c r i p t t z

fluid domain corners; breakwater width at crown; reflection coefficient; transmission coefficient; constant or reference water depth; Green's function; acceleration of gravity; solitary wave height; breakwater total height; outward normal vector on the boundary; pressure; maximum rundown value; maximum runup value; maximum NSW runup value on a slope [3; position vector of a free surface point; time; (u, w) = velocity vector on the boundary; horizontal coordinate; (x, z) = Cartesian coordinate; horizontal motion and velocity of piston wavemaker; vertical coordinate; geometric angular coefficient in boundary integrals; breakwater slope angle; fluid domain boundary; time step; distance between two discretization nodes; local wave amplitude; fluid density; velocity potential; and fluid domain.

atmospheric; bottom; backward breaking wave; rundown free surface; forward breaking wave; collocation point (or node) on the boundary; maximum value; minimum value; piston wavemaker; reflected wave; lateral boundary (1: leftward, 2: rightward); transmitted wave; and runup.

dimensionless variables; and imposed boundary condition.

92