Numerical modelling of wave-induced currents in the presence of coastal structures

Coastal Engineering, 12 (1988) 63-81 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

63

N u m e r i c a l M o d e l l i n g of W a v e - I n d u c e d Currents in the P r e s e n c e of Coastal S t r u c t u r e s PIERRE GAILLARD*

SOGREAH, B.P. 172X, 38042 Grenoble C~dex, France (Received March 18, 1987; accepted for publication September 3, 1987)

ABSTRACT Gaillard, P., 1988. Numerical modelling of wave-induced currents in the presence of coastal structures. Coastal Eng., 12: 63-81. The hydrodynamic aspects of the general methodology of calculation of nearshore processes by means of numerical models are described. The paper focuses on the method implemented for calculating combined wave refraction-diffraction and reflection due to coastal structures and the associated radiation stresses. Results of numerical modelling are compared with experimental data obtained by Gourlay in the case of a shore-connected breakwater with periodic waves. A good agreement is found between both methods of investigation as concerns the spatial distribution of wave height and mean water level (wave-induced set-up). A good similarity of the wave-induced eddies in the lee of the structure is observed. A less satisfactory agreement is obtained between the velocity distributions in several profiles normal to the shore, although the overall order of magnitude is the same. A critical review of several wave-breaking criteria and sensitivity tests of the numerical model lead to advocate the use of the CERC criterion, particularly with steep beach slopes.

INTRODUCTION

The study of sand transport and of the resulting sea-bed evolution by means of detailed numerical models of the nearshore zone includes three main steps, which involve respectively the modelling of: wave propagation, including wave refraction, diffraction and breaking, nearshore currents and related water-level variations generated by waves, winds and tides, sand transport mechanisms and sea-bed modifications. The present paper describes the numerical methods developed by SOGREAH to cover the first two steps of this methodology, with particular emphasis on the study of the effect of coastal structures on wave propagation and nearshore currents. -

-

-

* Present address: CEFRHYG, 6 Rue de Lorraine, 3810 Echirolles, France. 0378-3839/88/$03.50

© 1988 Elsevier Science Publishers B.V.

64 METHODOLOGYOF CALCULATIONOF NEARSHOREPROCESSES

INITIAL SATHYMETRY

OFFSHORE WAVE CONDITIONS

1

1

ZI AVE PROPAGATION MODELSINVOLVING: - WAVEREFRACTION AND SHOALING - WAVEDIFFRACTION - WAVE-CURRENT INTERACTION

I WAVE FIELD PARAMETERS(WAVE HEIGHT, DIRECTION) AND DRIVING FORCES (RADIATION STRESSES)

1

2-DIM, CURRENTMODELINVOLVING: -

LOCALFLOWACCELERATION ADVECTION BED FRICTION HORIZONTAL DIFFUSION CORIOLIS EFFECT

OTHER CURRENT I G ENERATING MECHANISMS ~ I WINDS ~ TIDES

1

MEAN WATER LEVEL (SET-UP/SET-DOWN) AND CURRENT FIELD

I

1

~E._~EVOLUTO I ___~.N~ODEIN_._VOLVI ~.L NG:

I

SEDIMENT TRANSPORT LAWS I: ~ED-LON), SUSPENSION, EFFECT OF GRAVITY)~ '

1

SEDI MElT TRANSPORT RATES AND MODIFICATION OF NEARSHORE TOPOGRAPHY

Fig. 1. Methodology of calculation of nearshore processes.

Results of a comparative study between numerical and physical modelling on a situation involving wave refraction and diffraction phenomena will be presented. The scope of the paper is restricted to periodic waves for which experimental data is available in the above context. The general methodology of calculation of nearshore processes is sketched in Fig. 1. It involves the use of different numerical models for calculating wave propagation, nearshore currents and bed evolution with an interfacing of the computer codes for transferring data from one model to another. The spatial distribution of wave field parameters and of the wave-induced current driving forces, i.e. radiation stresses, resulting from the implementation of the wave propagation models are entered into a two-dimensional depthaveraged current model, which can include other generating mechanisms such as winds and tides.

65 The nearshore current model itself provides the spatial distributions of the mean water level and of the overall current field. These distributions have an influence on wave propagation and should normally be accounted for in wave refraction and diffraction calculations. This feedback mechanism requires an iterative procedure of calculation, as separate models are used for waves and currents. The wave, water level and current fields are entered into the bed evolution model for estimating sediment transport rates and modifications of the nearshore topography. These modifications have also an influence on wave propagation and on the current field. This additional feedback mechanism requires a second iterative procedure of calculation for predicting the long term evolution of the nearshore topography. In the present paper we restrict ourselves to the presentation of the hydrodynamic aspects of numerical modelling. As concerns wave propagation, a distinction is made between areas where wave refraction, shoaling and breaking are the only phenomena involved and areas in the vicinity of coastal structures, where wave diffraction and reflection due to these structures are combined with the former. In actual applications of the method we may accordingly have to consider two kinds of wave models: (a) One model covering the whole nearshore area under investigation for calculating wave refraction without taking into account the impact of coastal structures. (b) Another model, of more restricted extent, covering the area where coastal structures may have an influence on wave propagation. This model is generally nested within the former one and its boundary conditions are obtained from the results provided by the refraction model. WAVEREFRACTIONMODELLING The modelling of wave refraction and shoaling is based on the method described by Bonneton and Gaillard (1985). Its main characteristics are the following: (a) Wave propagation is calculated using a wave front and wave ray tracing technique adapted to an arbitrarily shaped sea-bed topography. The numerical procedure has been chosen in order to incorporate either linear or nonlinear theories in the calculation of refraction patterns. For the prediction of waveinduced currents, the theory used for expressing the radiation stresses has to be consistent with the one used for calculating other wave properties and wave propagation, otherwise parasitic currents would result. At the present stage of development these applications are restricted to linear theory. (b) Outside the surf zone, the wave-height variation is derived from the energy flux conservation law, i.e. in terms of variations of wave group velocity

66

and of wave ray spacing. The corresponding mean water level variation is obtained from the formula of Longuet-Higgins and Stewart (1962). (c) Within the surf zone, the wave energy balance equation includes a sink term, which accounts for energy dissipation due to wave breaking and turbulence. The dissipation rate is modelled according to Stive's formulation for periodic waves (1984). The average horizontal momentum equation in the direction normal to the sea-bed contours is coupled to the above equation, so as to obtain a set of two differential equations for calculating simultaneously the wave height and the mean water level variations. (d) The components of the radiation stress tensor are calculated with the formulas of Longuet-Higgins and Stewart (1962), which are based on the assumption of a locally homogeneous wave field. MODELLING COMBINED REFRACTION-DIFFRACTION

In areas surrounding coastal structures, where combined effects of wave refraction, diffraction and reflection can occur, the calculation of the local wave field parameters is based on the boundary integral element method described by Barailler and Gaillard {1967). The representation of the sea-bed topography in the numerical model is the same as for refraction calculations, thus ensuring a good continuity between both types of models. A complex function ((xl,x2) of the horizontal spatial coordinates, related to the instantaneous water surface elevation ~ (x,y,t) and to the wave frequency o), is introduced:

~(xl,x2,t) --Re

{~(Xl,X2)e -ie°t }

(1)

The area investigated is divided into basins of convex shape, i.e. such that any wave ray propagating according to Snell's law from one boundary point to another is located entirely within the basin considered or along its boundary. In a given basin with boundary F , this function is expressed in integral form as:

- l!/0 ~n)

;" =~

62 (PM) ds

(2)

M

where P is the inner point considered, M a boundary point, and O~/Onis the normal derivative of ( associated with waves entering into the basin through a harbour entrance or waves reflected from a coastal structure located in M. Two kinds of entrances are distinguished: The first kind consists of "main" entrances, where the incident wave characteristics and the related normal derivatives are known. -

67

F'

MK I

.

-~

U~JK

6"p

~

" X

PM

Fig. 2. Definition sketch of combined refraction-diffraction calculations.

- The second kind consists of "secondary" entrances, which are boundaries connecting neighbouring basins, where the wave characteristics are not known beforehand. For the application of eqn. (2), the boundary is divided into segments of short length with respect to the wave length, and the normal derivative of ( has to be determined at the centre of each segment. The method requires in a first step the calculation of the unknown derivatives along reflecting structures and coastal features and along secondary entrances. This is achieved by solving a set of equations of the form:

where Ci is a constant representing the reflection coefficient in the case of a coastal structure or the transmission coefficient in the case of a seconda~ entrance. When a basin has a constant water depth h, and if the wave-induced setdown can be neglected, the ~ n c t i o n G~ and its normal derivative are obtained from eqns. (4) and (5) in terms of Hankel ~nctions of the first kind H~ ~ (kr) andH~ ~) (kr):

Ge(PM) =2 H~~) (kr)

(4)

OG~ On (MiMK) =2kH~ ~) (kr~x) cos ~ u

(5)

~ = ( ~,M~M~ ) where k is the wave number, related to w and h by:

w~ =gk tanh ( kh ) g is the acceleration due to ~avity, ny is the normal to the b o u n d a ~ at M~ (cf. Fig. 2 ), ri~ is the ~stance between M1 and MK.

(6)

68

When a basin has a nonuniform water depth, the above functions are determined from refraction calculations within the basin. Considering the boundary point M as a source of wave radiation, wave rays are calculated in all directions, taking into account refraction effects due to the sea-bed configuration. The following expressions are applied in this case:

G2(PM) =4i I(PM) ei~(PM)

(7)

1 FC~M_~.~.I

I(PM'=2~] - - [_ C~pla, I _] o(PM) = ~

k(r)dr - a ~

e

4

aG~ (MjMK) = - 4 I(MyMK) Q(MyMK) ~. . . .,.~. U. an

Q(MyMK)= ~

~=

"~

[ ,( oo

~. i g ( M , M ~ )

(8)

Cg Or / AMj

-~

(n~,6)

in which: is the distance from M along the wave ray connecting M and P, are the wave group velocities at points M and P, is the wave length at point M, is the tangent to the wave ray at M~, is the spacing in the vicinity of P of two wave rays, originating from M with the angle 0M. Once the normal derivatives of ~-are determined on all boundaries, ~ and its spatial derivatives are calculated within the basins at the nodes of a rectangular network covering the area investigated. The spatial derivatives of ~ are parameters necessary for evaluating wave radiation stresses in situations where the wave height distribution has large variations over distances of the order of the wave length. These derivatives are given by: F

CgM,Cgp LM tr {Tp

0

Ox2-4i

~n M ~

(PM) ds

(9)

with

OG2 Oxy (PM)--2kH~ 1) (kr) cos ~PM j = l , 2 ~

~//PM =

~

( Pxj, PM)

(10)

69 or: OG2

Ox~ (PM) -- -4I(PM) Q(PM)

cos V/~Me i~(vM)

(11)

-~

~,M = (Pxj, tp) depending whether the water depth is constant or not in the basin. The components of the radiation stress tensor are calculated by the formulas given by Gaillard (1982) : -

1

C~/a ~-a ~

S11=pgh' +~ Pg ~ ~~

s~=¢~; ~+~eeV

/ 2

+a~ )

a2

)

(~

1 Ce a~a~ S~=~pg~ k~eos (O~-O~) ~f-a~ +a~-a~k ~ tanh ~ (kh) 4k tanh (kh)

~=a eio; 0O~_a~eio~, ~-

~=1,~

(la) (~4~

where: p is the water mass density, c is the local wave phase velocity, C~ is the local wave group velocity. The radiation stresses are derived from:

~1 ~

--1 ( OSll +OS12 ~ k OXl Ox2 ff - ~ ( o s ~ + os~ ~

r~=~ k Ox~ ox~ ]

(~

It has been shown in the above reference that these formulas are equivalent to those of Mei (197~) and that ~ can be expressed as:

~-

a~k + A(a~) (2kh) 8k tanh (kh)

(16)

2 sinh

The last term of this formula cannot be neglected in areas where wave reflection and diffraction due to coastal structures are observed. Because of this, the formulation of radiation stresses in terms of the local wave height and direction given by L o n ~ e t - H i g g i n s and Stewart is not valid in this case.

70 Bettess and Bettess (1982) have expressed the radiation stress tensor components in terms of the complex velocity potential and its spatial derivatives for similar applications. When the relationship between this potential and the complex water surface elevation is taken into account, these formulas reduce to eqn. (12), with ~ derived from eqn. (16). The behaviour of waves in the surf zone cannot be modelled in the same way as for refraction calculations, since in the combined refraction-diffractionmodel there is no explicit formulation of an energy balance equation. It is actually assumed that wave heights are limited by the local wave-breaking criterion. In order to account for bar-shaped beach profiles,, a further condition to be imposed is that the maximum wave height should be a monotonically decreasing function of distance along the direction of wave propagation. The wave-breaking criterion adopted in these calculations is the same as in the case of wave refraction. The choice of this criterion is discussed below. WAVEBREAKINGCRITERIA The wave-breaking criteria published in the literature can be classified into two main categories: The first category expresses the condition of initiation of breaking as exclusively dependent on local wave parameters and bathymetric features, thus assuming that phenomena involved in wave propagation between the offshore zone and the breakpoint need not be known, provided that the local kinematics are those of a progressive wave, and wave incidence is nearly normal to the shore. The criteria in this category generally give the wave height at the breakpoint in terms of wave period T, beach slope rn and local water depth h, relationship which can be expressed in nondimensional form as:

Hb h

- F (h/Lo, m)

with

Lo =gT~/2~

(17)

In this category are found the following criteria:

Battjes and Janssen (1978) kHb =Ta tanh ( ~ kh) 71 -~0,88; 72 =0.83

US-CERC h

(1977)

- B [l ÷Ah/Lo] -1

(18)

71 A=6.963 [ 1 - e -19"~] B = 1.56 [ l + e -19"5"~]-1

(19)

Goda (1975) h HbLo-0.17 {1 - e x p [ - 1.5~r ~oo (1 + 15 rn4/3) ] }

(20)

In the second category, the criteria give the wave height at the breakpoint in terms of beach slope and offshore wave steepness HolLo. In this category are found criteria of the form: -c

\Lo] Such a relationship is valid only in the case of wave shoaling with normal incidence on a straight beach, and cannot be applied in cases where refraction or diffraction effects are involved. However, when another relationship giving the water depth at initiation of breaking as a function of the offshore wave steepness is provided, the latter can be eliminated. This relationship is generally given in the form:

H-b=D mE ~Io ~ -F h \Lo]

(22)

The combination of both equations leads to:

Hb=cbmo,(h)~ h

(23)

Singamsetti and Wind (1980) have established the following wave -breaking criteria based on their own experimental data, referenced here as DHL1 and DHL2, respectively: A=0.575

B--0.031

C=0.254

associated with: 0.568 E-0.107 F-- 0.237 for 0.025 ~
D=

72

(a) P E N T [ DU SEA - ~[0

h

FONO ~>m.O,02 SLOPE

)

0.5

I

~,

~

I

~ I I III

I

I

I

iiiii

(b)

P[.,E ~,, F O N D } S E A - DEO

h

I

I

lO -~

10-2

10 -3

h/Lo

m .0.05

SLO~

---

=~--=~-~-_~__=j~__~-___~ •

~.-=_. ~ . ~ - _

.

0.5 I

I

10-3

I

I

', ~ ~ ~ ~

I

I

~0-z

I

I

I

~ ', ~ ~

~0 -~

F r o m t h e above, we derive t h e following c o n s t a n t s in eqn. ( 2 3 ) : DHL1 DHL2 Iversen

Cb=0.570 Cb = 0.872 Cb = 0.758

c~=0.089 c~= 0.137 c~= 0.110

fl----0.759 fl= 0.852 fl----0.847

I

h/L o

73

(c)

~

h

;~*-',;';:~,":,}-.°.,,,

2

-~---~-- -=~-~_-7.~~__~.~ I ~=~-~.

1-

0.5

I

:

~ I ~ ~ I

I

t

'

{

I

! ! !!

10-2

10 -3

I

I

h/L o

10 -1

PENTE DU FOND} m ,0,~0

(d)

SEA- BEO SLOI~

h

0.5

}

10-3

~

I

~

:

~ : ~ :

lo -2

;

;

~

:

:

; ,~ .' :

lO-1

;

h/io

Fig. 3. Breaker index Hb/h vs relative water depth h/Lo for several beach slopes m accordingto different wavebreaking criteria: {a) m=0.02; (b) m=0.05; m=0.10; m--0.20.

The breaker index Hb/h, given by the preceding wave-breaking criteria, is plotted versus the relative water depth h/Lo for several beach slopes in Fig. 3. This comparison leads to the following conclusions: {a) For beach slopes ranging between 0.02 and 0.10, and relative water depths ranging between roughly 0.02 and 0.10, all criteria give similar results. The

74 scatter is lowest for approximately m = 0.05 and increases with beach slope. As data used for validating those criteria are generally restricted within the ranges just mentioned, the agreement of the different criteria is not surprising. (b) The criteria involving constant power laws of the relative water depth lead to excessively high breaker indexes in very shallow water, i.e. below h/Lo = 0.02. Since they are also power laws of beach slope, they cannot be applied to constant depth areas, and to beach slopes lower than 0.02 approximately. (c) The criteria of Battjes and Janssen, CERC, and Goda give similar results for very mild slopes in shallow water. The first criterion however gives too low breaker indexes for steeper slopes. (d) The criteria of CERC and Goda give results in fair agreement for shallow water and beach slopes lower than or equal to 0.10. For steeper slopes Goda's breaker index takes unrealistically high values, whereas the CERC criterion keeps a finite value. The latter criterion thus appears to give the most reasonable estimate of the breaker index over a wide range of conditions. NEARSHORE CURRENT MODELLING The modelling of mean water level variations and nearshore currents is based on the method described by Benqu~ et al. (1982a, b), which was originally designed for calculating tidal currents, including in shallow zones uncovered at low tide. It has been extended to include when necessary in the same model wave- and wind-induced currents as well as tidal currents. The modelling is essentially two-dimensional and gives the spatial distribution of the depth-averaged mean horizontal velocities, u,v and its variation with time if the driving forces are time-dependent. This schematization precludes the calculation of secondary currents such as the undertow. The method is based on a finite-difference formulation of the basic equations in cartesian or curvilinear coordinates. These equations are:

o (gou~ OU 0 -~+~x~ (u~U) +gh ~-~x~ oxj \

0 V+ 0 Otl o ~ ~x~ ( us V) +gh Ox~ Ox~ k

Ox~] ~

rb~-r~, p p

FV=O

~FU=O

0~, OU, OV ,, -~-~"r~-~xl± ~-~x~= u where:

x,,x~_,t U, V F

are the space and time variables, are the unit-width discharges in the xl,x2 directions, is the Coriolis acceleration parameter,

(24)

(25)

75

......... _ , . ,

~

...... _. ..... ~......... ~ ~

~

\

. -~

~

/

//

\

', \

'~ '~ I ~

/

.I I,,__,,,

I~ / (/ / / / / / / / / / ~ / / / / / / / / ~

~

c |? I ~ !~

~x.~~~\ \ / ~ ~ \

/

/

\o \

\

'~ I

\

~

\ I \

~, '~

~

'

~ \

t' '

'~ ~ - ~ ~ ,

\

I

I

I

~

I

°...-. .

~c~r

WAVES

~

- ....

Fig. 4. Lay-out of physical model, with a shore-connected breaker, investigated by Gourlay (1974).

K is the coefficient of horizontal diffusion, ~ is the mean water level for a time interval of one wave period, ~b~,~b2 are the bed shear stresses, r~,r,2 are the wave- and wind-induced stresses. The numerical solution of these equations is based on a fractional step algorithm, in which advection is calculated by the method of characteristics, horizontal diffusion by an implicit finite-difference scheme, and wave type momentum transfer by an iterative alternating direction implicit scheme. Applications of the method to the cal-cul-ation of tidal currents and winddriven currents are described by Benqu$ et al. (1982a, b), Harem and U sseglioPolatera (1985).

COMPARISON OF NUMERICALCALCULATIONSWITH EXPERIMENTALDATA

A comparison of results of numerical modelling by the above methods and formerly published experimental data has been carried out. Tests related to wave refraction on an infinitely long beach, with a constant slope, have been reported by Bonneton and Gaillard (1985). The present paper considers a more complex situation in which combined effects of wave refraction and diffraction are observed due to the presence of a shore-connected breakwater. The lay-out of the physical model investigated

76

Ca)

..---- ~

j

~

f

-

~

~-~.,~< i

/

./

/ /

t

/

./~

/ !

~

/

(

--

%~

~

~

.

.

/~/~

~

~

. . . . . .

~

~

~

X

~f

(k .

-" ,'f i /1_

c',,

,l[ ,,']~

[

171

L/

I /ill ~

~(-, '-,-; I. , -

iIIi ,

~

/

/ /

~ /

/

"'" ~

~

. ")"" ~ ~

,~X/~

#

,~z

I/i f / l , ~.

(/

d [

/

ill ~1

~J/

.

,

~

k

~

~

.

"x

~

__ ~

. . . .

~

J i f

~-

s

)

~

_~

-

l t ~ b / l l } i / / / / / / / / / i / / ) / J

~-

_

~ , i

~--17-

~

~

a

~-

~/~

7 k--")

J

~

"

~

I

'~1

~ ~----

.

.....

~•

I

~-"~--""

-_

_ _ _ ~

~

.

__

~ _

.

.

.

- ~

-

.~----* . . . . ._-_.___--~-

X-.. ~'r--t<.---::~...... .,-_:-z~,-_ ..... \ ,," ~'"t r--.~'d---~_~ :-~--

,~ tk.

,~I',,?_~-~-:--~--_~---,'H_L.,Wi /a / I % -

<,\I~

¢1 .ii~

k ,';')4 I,A ~Ix ," ~ ,~. ~ I)IJ~,,,~\,L ~ ~_ /~r~ \~'~

I ~/I

,,

,- ,.

-~

--

~

~

~o. ~

I~

~O ~

--

"%... ~¢.._._,

':-"

~

zz=:~==-. ----

~

'-~

~

..

----~

i IA

~'~

~

.'<~ -'--- ...k.

~/

.

~- 6 - ~

.

~

> '

~

..~)/.,;

~" t,~-~ ~.-,

-~

.

-~

i . ~'j~ ~-~, ~' i /~,,~u ~ t < " : ; - - .

!

~

I ZI .~.--- ' !"..~ # 141 I I

~ ~

,

i~ ~

-

.

~

~

I

(b)

/

.

i,"

~/ k

f4

t

,

z

.

t~ ,0 ~ /

Villlililllillllllliilllillllll~

/

. 4 ~

~ / ~/

"~ /

",_)

.

~

~

,'J :~

~

~

g

~-~ " ~i x~k Ut ~ - k ~ . . . . . . ~ ~

2~ )

. i ~ ~

/ II

/~-~\

,~i

~

~

/il

I

7

. ' ~~

/

i

/

i

/

~ - -

~

.

/

~ -

., ~ -

~____

/

/

I,I /

I~ :ly

b ~

. ,"

,q

'.

"

~I .I

\\~i" *,.~--

S

/

~ --~

I

~.---,-~

~.

~ , -; ~ _ , ; ~ < - - ~ .

.

.

.

o

I

I

,,,

I

Fig. 5. Spatial distribution of wave-height measured by Gourlay and calculated by the present method: (a) BJ criterion; (b) CERC criterion calculated: ; measured: - - - .

by Gourlay (1974) is shown in Fig. 4. The structure is parallel to the general shoreline direction and connected to the shore by a circular beach centred on the breakwater tip with a beach slope

I

1 .... 1,5

1,5

1,5

•

~

=

,

~

Profite

-"

. ', ~ . 1

1

'

-

Profile

~1 ~

I

!

; 0,5

0,5

B.P

~

(o)

C

B

0,5

i

B,P

~

~

~ ~ ~ ~ :

~

:

:

"

:

....

"

i:::'

iiiiii~

70

~:::~ ~: a~

~ ;i

=~

E ._

~e

iiiii~=

~o

~ii ~:~ l o

O:~:

0

1.

0 "~:"

0 ':i~:"

'

-----

..:

.,

~ 1,5

I. 1,5

I ~,5

1,5

=

~

1

~ ~:

P~of~e

1 t

0,5

~

0,5

~

i

:

"

[L~: 0,5

:

"

0:~:

0

iiiiiii~

~o

ii~: ~

?.:i~.m~

E ._

; calculated - + - + - B J criterion

B.P

.

....... 0 ':::

.~iiiiiiii~

_;~'~: ~o~

- ~~~,,~

(b)

o

Profile C

~;,, ~

~

~

Profile B

!1 .~ 1

BP

P,ofi~ ~,

Fig. 6. Comparison of calculated and measured wave-induced mean water levels in profiles A to D: measured . . . . . (left) ; C E R C criterion (right).

~

t

1,5

Profile A

-~

78

, ,~_._--..,."---..---...--...'~,,..'~,..-,..\ \

,.,'-"-,,--~~.,?.

500

.,....~.D..~___/,~..~,.~.......~ _ ~.Z-,..~--,..~-,

,i~'iili;,~..

vT. ~ l / t ( ( i i l ~ ' - ~ '"_ L _ ~ *" . . . . . . .

400 i

-

300

~

"~

Z

~f/t

• ~'

¢1~]'i

,1

¢"

t , "

t

.

.

.

. .

. . . . . . .

~.

.

,

~ "

,

,

,

,

~ ,,

,

t

. . . . . . . . . . . .

. . . . . . . .

'

. . . .

. . . . . . . . . . . . .

. . . . .

', ". " - " - " "

.

~

~

' - i . . .~ . . . . . . .i . . . . . . . . . . . . .

ii

iii Illll

~

~

"

\

~

=::

i " '

i

~

"

I t

" ..

,

\

..

~.

~._,

7 )' I ,-__,

{i ' '

"~

'-"~

-~

,

~ ~ ',~

i

~

,

,

, ,

,

t

~

~

I

,,

,

I

~

~

~

~

"

~

#

~

~

I

I

~"

..

f

f.,._

, , , ' , , '

~..,"

. . . . . . .

,,-,~,.,,,.~,,-,~,,~-~--,-,,~,.,-~'.--~'

,r,

.... \

..

~

"" t" t i ~

~. ,"

l' ,' ,' ," .,* ,-",

- \ \ - ~t ..-...,

~

I',".,-..' 200

_ ~ . f i l l . . .

\

I

~

~

,..~

. . ,

--'--

,, l ! l f . . . .

, ,,,,',,,

~

,

,

~

-'t "~,

I

Fig. 7. Depth-averaged wave-induced current pattern calculated with the present method using the CERC breaking criterion.

of 0.10 in the whole model. A constant still water depth of 0.20 m is set offshore of the beach. Regular waves of 1.5 s period and 9.1 cm wave height with normal incidence to the shore are considered. Two wave-breaking criteria have been implemented in the numerical model for testing the sensitivity of the results to the choice of this criterion, namely those of Battjes and Janssen {referenced as BJ hereafter) and of the CERC shore protection manual. Figure 5a shows the spatial distributions of wave height measured by Goutlay and calculated by the present method with the BJ criterion. Figure 5b shows the same distributions, with the CERC criterion. The numerical results are the same in both cases outside the surf zone. The calculated and measured wave heights are in good agreement outside the surf zone, both in the sheltered area behind the breakwater and in the unsheltered area. With the BJ criterion, the surf zone extends farther offshore in the numerical model than it does in the experiment, and the wave-height gradient is too low in this zone. With the CERC criterion, the extent of the surf zone and the waveheight. gradient are in good agreement in the numerical and physical models. Figures 6a and 6b show the distributions of the measured and calculated mean water levels along profiles A to D shown in Fig. 4. The calculated levels are associated with the BJ and the CERC criteria respectively.

~

1-0

~

/

~-0

0.5

.'

/

/

~

~

~

0.5

~,~,I

\

=~

__

,,~

1,5

,~,'~ ~ ~

\

~

(a)

0.5

~

• r~

0

0

0

*"*

\

~ ) ~

~

,

\\

\\,

' ,

l 05

05

_=

~

20

o

:>

20 ~ ,~ ,~

40

~

-~o

-J.o

|

0-5

|

~x0 I

~

--~, ~r__l~, ~ 1.0

Profile 4

e

0

J

.-:-\

/~'~X",

~ *

~ ' ' - --' . ~ 11:0 . / ~ ' -~.0:5= 5B : =

:

/

/ ,~ ~~

Profile 3 ~

~

-

-~|40

t ........

2.0

Z.O

~'~2,0

_ _ 1

2-0

~-0

0.5

1,5

~ 1-0 (b)

0

-

. ~r_. ___ 0.5

~ 0

_

J 0

,:; .... 0,, 0.5

..%

\

lk

I

0.5

~ - - 0.~

05

\ ~ ~==

40

0

0

2o

0

2o

I

>

,~ -~

~

.=_

E

~2o

-•40 ,1o 0,5

\~%

[',

.,/-~,,,,,

..~---:~.._/ ~ 1.5~ _ 1.0 __- 0 .tB? 5-

Pro,,,e

J,5

1-0

/

/ / " ~

- ~*,-- -~ -_~.-~- ,~.=.~" "L_ ~ ~ - ~ ==

1-5

Profile 1

Fig, 8. Comparison of calculated and measured depth-averaged wave-induced currents in profiles 1 to 4: measured . . . . ; calculated - + - + - BJ criterion (left); CERC criterion (right).

2.0

~

1-5

-

~I

Pr°'"e2

1-5

l

...~.-~-

2.0

2.0

2-0

L

~_ _ ~ . ~ , ..... ~,~,.~ - ~

/,-. ;,,.--,,~,,~o

Profile 1

80 In the course of this comparison, it was found that there is an incorrect scaling of the horizontal distance from the SWL in Fig. 7 of Gourlay's paper. This can be checked by comparing the positions of the experimental breakpoint and plunge point in profiles C and 2, which have the same location, or in profiles D and 4, in Figs. 7 and 11 of the above reference. Taking into account the correct scaling of the measured set-up distribution, it is seen that the calculated set-up starts too far off-shore and that its slope is too low with the BJ criterion. With the CERC criterion, there is a good agreement between calculated and measured levels in all profiles, the calculated values being slightly overestimated. Figure 7 shows the wave-induced current pattern calculated with the CERC breaking criterion. As observed in the experiments, the numerical model shows a large induced eddy, turning counterclockwise and located partly in the sheltered and exposed areas. Other secondary eddies are induced near the sidewall and in the area where the beach and the breakwater intersect. Figures 8a and 8b show the distributions of the calculated and measured wave-induced currents along profiles 1 to 4 shown in Fig. 4, the calculated currents being associated with the BJ and the CERC criteria respectively. In each case, the bottom friction coefficient has been adjusted in order to obtain a correct order of magnitude of the maximum velocity. With both wave breaking criteria, the numerical model gives a velocity distribution which extends somewhat farther offshore than the measured distribution. The CERC criterion however leads to numerical results closer to the experimental data.

CONCLUSION

A general methodology of calculation of nearshore processes based on the use of a combination of numerical models for the various phenomena involved has been presented. A modular structure of the codes implemented allows for further improvements in the schematization of the physical processes, the methods of solution or the range of applications. In view of the results of this investigation, the methods applied in the calculation of wave propagation, mean water level variations and wave-induced currents give on the whole satisfactory results in a complex situation where combined effects of wave refraction, diffraction and breaking occur in the vicinity of a coastal structure. The critical review of several wave-breaking criteria leads to advocate the use of the CERC criterion, particularly with steep beach slopes, as is the case in Gourlay's experimental set-up. The numerical tests performed with the BJ and CERC criteria show that the latter leads to the best agreement of the

81 n u m e r i c a l a n d p h y s i c a l m o d e l s as e x p e c t e d f r o m t h e a b o v e a n a l y s i s . T h i s a g r e e m e n t is v e r y good for t h e w a v e h e i g h t a n d s e t - u p d i s t r i b u t i o n s , b u t o n l y fair for t h e c u r r e n t d i s t r i b u t i o n . F u r t h e r r e s e a r c h is n e e d e d to i m p r o v e t h e n u m e r i c a l m o d e l l i n g o f n e a r s h o r e c u r r e n t s , w h i c h p r e s e n t l y a p p e a r to e x t e n d f a r t h e r outside t h e s u r f zone as c o m p a r e d to m e a s u r e d c u r r e n t s , a l t h o u g h t h e o r d e r of m a g n i t u d e of b o t h curr e n t fields is in good a g r e e m e n t . T h e m e t h o d o l o g y d e s c r i b e d h e r e for periodic w a v e s c a n be e x t e n d e d to irr e g u l a r w a v e s w i t h a n a p p r o p r i a t e m o d i f i c a t i o n of t h e n u m e r i c a l m o d e l l i n g o f wave breaking.

REFERENCES Barailler, L. and Gaillard, P., 1967. Evolution r~cente des modules mathdmatiques d'agitation due h la houle - Calcul de la diffraction en profondeur non uniforme. La Houille Blanche, 8: 861-869. Battjes, J.A. and Janssen, J.P., 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Coastal Engineering Conf. Hamburg, pp. 569-587. Benqud, J.P., Chenin-Mordojovitch, M.I., Hauguel, A. and Schwartz, S., 1982a. A new two-dimensional modelling system. Proc. 18th Coastal Engineering Conf. Cape Town, pp. 582-597. Benqud, J.P., Cunge, J.A., Feuillet, J., Hauguel, A. and Holly, F.M. Jr., 1982b. New method for tidal current computation. ASCE J. Waterways, Port, Coastal Ocean Div., 108 (WW3): 396-417. Bettess, P. and Bettess, J.A., 1982. A generalization of the radiation stress tensor. Appl. Math. Modelling, 6: 146-150. Bonneton, M. and Gaillard, P., 1985. Numerical calculation of wave-induced currents. Proc. 21st Congress IAHR, Melbourne, vol. 4, pp. 68-73. Gaillard, P.M., 1982. Kinematic and dynamic properties of diffracted waves in constant water depth. Engineering Applications of Computational Hydraulics, vol. 1. Pitman Advanced Pub. Program, London, pp. 131-183. Goda, Y., 1975. Irregular wave deformation in the surf zone. Coastal Eng. Jpn., 18: 13-26. Gourlay, M.R., 1974. Wave set-up and wave generated currents in the lee of a breakwater or headland. Proc. 14th Coastal Eng. Conf. Copenhagen, pp. 1976-1995. Harem, L. and Usseglio-Polatera, J.M., 1985. Two-dimensional modelling of wind-induced currents in coastal and harbour areas. Int. Conf. on Numerical and Hydraulic Modelling of Ports and Harbours, Birmingham. Longuet-Higgins, M.S. and Stewart, R.W., 1962. Radiation stress and mass transport in gravity waves with applications to surf beats. J. Fluid Mech., 13: 481-504. Mei, C.C., 1973. A note on the averaged momentum balance in two-dimensional water waves. J. Mar. Res., 31 {2 ) : 97-104. Singamsetti, S.R. and Wind, H.G., 1980. Breaking waves. Delft Hydraulics Laboratory, Rep. M.1371. Stive, M.J.F., 1984. Energy dissipation in waves breaking on gentle slopes. Coastal Eng., 8: 99-127. U.S. Army Waterways Experimental Station. Coastal Engineering Research Center. Shore Protection Manual, 1977.