Nonlinear refraction-diffraction of surface waves in intermediate and shallow water

CoastalEngineering, 12 (1988) 191-211 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

191

N o n l i n e a r R e f r a c t i o n - D i f f r a c t i o n of S u r f a c e Waves in I n t e r m e d i a t e and S h a l l o w Water OLE BIRGER RYGG

Department of Mechanics, University of Oslo, Oslo (Norway) (Received June 30, 1987; revised and accepted December 31, 1987)

ABSTRACT Rygg, O.B., 1988. Nonliner refraction-diffraction of surface waves in intermediate and shallow water. CoastalEng., 12: 191-211. A numerical scheme for solving the nonlinear Boussinesq equations is introduced. The numerical model is used to investigate nonlinear refraction-diffraction of surface gravity waves over a semicircular shoal. Results are compared with experimental data (Whalin, 1971) and previous reported numerical results by Liu and Tsay (1984) and Liu, Yoon and Kirby (1985). The present calculations reproduce the earlier results for shallow water waves, but are superior in intermediate water depth.

1. INTRODUCTION

In recent years considerable effort has been made in studying combined refraction-diffraction problems of water waves. The linear elliptic mild-slope equation introduced by Berkhoff (1972) formed the basis of later work on short-wave diffraction-refraction (Bettes and Zienkiewicz, 1977; Booij, 1981; Berkhoff et al., 1982; Tsay and Liu, 1982; Houston, 1981 ). To reduce the computation time and difficulty involved in the solution of the elliptic equation, Radder (1979) used the parabolic approximation method to the mild-slope equation. This method is based on the assumption of weak refraction and reflection. The parabolic method has been verified, comparing with laboratory data, by Radder (1979), Tsay and Liu (1982) and Berkhoff et al. (1982). Liu and Tsay (1983) have also developed an iterative numerical scheme to include weak reflection. Recently, the formulation has been extended to the case of second-order Stokes waves by Yue and Mei (1980), Kirby and Dalrymple (1983) and Liu and Tsay (1984). Liu, Yoon and Kirby (1985) used the parabolic approximation method on the dispersive long wave equation, i.e. the Boussinesq equation, to study combined refraction-diffraction of weakly nonlinear shallow-water waves. A nu0378-3839/88/$03.50

© 1988 Elsevier Science Publishers B.V.

192

merical method for the full Boussinesq equations was developed by Abbott et al. (1978), and further extended and tested by Abbott et al. (1984) and Madsen and Warren (1984). In this paper, a newly developed line by line iterative method for the nonlinear Boussinesq equations (Pedersen and Rygg, 1987) is tested on a combined refraction-diffraction problem. Results from computations are compared to the laboratory experiments performed by Whalin (1971) to investigate wave focusing behind a topographical lens. The model results are also compared to the numerical experiments performed by Liu and Tsay (1984), and Liu, Yoon and Kirby (1985). Basic equations are presented in Section 2. In Section 3 the finite-difference scheme and the computational technique are presented. Computational domain and boundary conditions for Whalin's (1971) experiment are given in Section 4. Comparison and discussion of numerical and experimental results are presented in Section 5. 2. BASIC EQUATIONS

A cartesian coordinate system with horizontal axes O-x and O-y and vertical axis O-z is introduced. A typical wave-length he, and a typical depth he, are used as horizontal and vertical length scales respectively. By introducing he/ g ~ as time scale, the non-linear non-dimensional Boussinesq equation may be written in the form, Peregrine (1972): 0~ -at

V. [ ( h + ~ ) v ]

(2.1)

{½dhgV- h - ~

--+v.

0ot

- ~h VV.-~ +O(Ae2,eA

(2.2)

where ~ is the surface displacement, h is the equilibrium depth, A is a nondimensional amplitude (~max~ A ), v = ui + vj is the mean horizontal velocity, .0 .0 V= z-z-+J-z-is the horizontal gradient operator, and e - / - - ~ / is treated as a (Tx

--\he)

(Ty

small parameter. The horizontal velocity v is irrotational only to the order ~o, if the depth is non-constant;

vx v=O(

Oh

Oh

)Ty

(2.3)

Rewriting the nonlinear and dispersion terms of eqn. (2.2) by using eqn. (2.3) leads to:

193 02

0+½~y(U2+V 2)=

0r+,

.[-0 2

. 0/0r.\-]

1

2

2.r

+ ~ [ ~ u ) J - ~ e h V v+O(Ae 2) (2.5) Ou Ov where we have introduced the notations - ~ = ~ and ~ = O. 3. THE FINITE DIFFERENCE FORMULATION The numerical approximation to a quantity F at the gridpoint with coordinates ( o z z l x , f l z t y , T A t ) is denoted by F ~ , p . z l x , A y and At are the space and time increments respectively. The space discretization is done on a staggered Arakawa-C grid (Fig. 1). A finite difference notation is introduced where we define a difference operator ~x by ~xF ~,p = ~ x (F~+ ½,~-F~_ ½~)

(3.1)

and an average operator for the x-direction (F

--x

y

1

~'

y

).,p = ~ (F~+½,B +F~_½,p)

....

t ..............

0

--

I ............

O

I

--

I ............

0

I

(3.2)

4 ..........

--

0

r

I ............

--

0

1

I ......

--

0

I

e

: ~ - point

-

: u - point

I

I 0

--

0

--

0

I o I

-

o I

--

0

i -

o

Fig. 1. Grid discretization.

0

--

r -

I

--

o

I -

I

0

o i

I -

o t

i ,

c::,×

: v-

point

194

Similar difference and averaging operators are also introduced for the y-direction and the time. T h e equation of continuity eqn. (2.1), is solved numerically by a simple predictor-corrector approach. Predictor values for the surface elevation, t/-n, are obtained by:

[~t( ~_n_ on-l) = _g~{ (h+ (~-x)"-~)u~- ½} -~y{(h+ (~-~)"-')vn- ;}]~+½,.+ ½

(3.3)

and the corrected values, t/n, from

[~t~= -~x{ (h+~: x )u}-~y{ (h+~J)v} ]i+½,v+~ n-~

(3.4)

where we have used the definition ~/~.- ½= ½( t/- n + t/n- 1). T h e above procedure eqn. (3.4), is repeated until convergence is reached. T h e difference form of the m o m e n t u m equations reads: O = [ 5 + ~x (tl+ T ) - eDx ] ~ + ~

(3.5)

O = [5+~y (t/+ T) - ~Dy ]~%½,v

(3.6)

where T , D . and Dy are representations of the nonlinear and dispersion terms respectively and are defined as TiLt,v+½ = ½[{ (u-X)n-~}2+ {(v -y)n-½}2

+At{ ( u - X ) ~-~ ( 5 - x ) n + (v - y ) n - ~ (5 -Y)n} ]j+½,~+ ~

(3.7)

+ O ( zIt2,ZIX2,Zly 2 )

Dx =- [ - ~h{gx~

2 ( h u ) + 5y ( h - Y 6 y S ) } + ~,~h2t32,.x n +62y)5-½h~y(5-xSxh)]~,p+~

Dy

-

[ _ ~ 1h { g y 2( h v ) •+ g x ( h - X g x 5

(3.8)

~ 2 (dx2 )}+~h -t- gy2 ) v"

½ h ( ~ x ( 5 - Y ~ y h ) n b+ ~,,

(3.9)

T h e relation between the acceleration 5,5 and the velocities u,v are implem e n t e d by n , [ ¢~tU=5 ]?,p+ ½, [ (~tV=5 ]j+~, p

(3.10)

In the case of linear equations and uniform depth we obtain the stability criterion:

195

hoar 2 <~

1 - -14 Ax 2

1 Ay 2

~-~ eh~)

The implicity of eqns. (3.5) and (3.6) are essential for the stability of the scheme. The calculations for each time-step are split in a three-step procedure. At first we calculate qn by the predictor-corrector approach, eqns. (3.3) and (3.4). Secondly, the implicit equations, eqns. (3.5) and (3.6), are solved for u",b ~ by an iterative method. We have chosen a line by line iterative technique. Values obtained from integration of the linear non-dispersive shallow-water equations are used as initial values for the iteration. One iteration, resulting in the improved values uj'*,~+ ~ and b* + ~,p, consist of the following four steps: (1) Solve the sets of tridiagonal equations for the intermediate quantities " + l.~j,p + ½ defined by O=[5++rx(U+_iz)+5x(q,+T,[u,=u+)

- ~, h S x 2 ( h d + )

+ ~eh25~d + - ½eh(fy (h -Y(~ygZ) + leh25~ ( u ) - ½ehSy (b -XS~h)]j,p+ ½

(3.11)

(2) Solve the equations for u*:

o = [u* + ry ( u * - u +) +Sx (~"+ T ~ lug=u+ ) - ½ehS~ (hu ÷ ) + l~h2~xU + - ½~hSy ( h - % ~ * )

'2 + ~1 n_ L 2 ~o~ (u*) - ½~hS~ (b-X~xh)]j,~+ ~

(3.12)

(3) and (4) Apply the same procedure to eqn. (3.6) after substituting u* for Step 1 results in sets of tridiagonal equations for the accelerations u + along lines in the x-direction. The solution along each line is obtained using the Gaussian elimination. In step 2 we obtain equivalently tridiagonal equation systems along lines in the y-direction. In steps 3 and 4, equation systems for the accelerations b + are solved. A suitable choice of the relaxation factors rx and ry may speed up the convergence towards a correct solution of the m o m e n t u m equations, eqns. (3.5) and (3.6). To complete the time-step procedure we calculate u n+~, v n+½ from eqn. (3.10). In all simulations presented in this paper two steps of corrections for the continuity equation, eqn. (3.4), and three iterations for the m o m e n t u m equations seemed sufficient. Extensive testing of the numerical model may be found in Pedersen and Rygg (1987).

196 4. T H E T O P O G R A P H I C A L L E N S P R O B L E M

Whalin (1971) conducted a series of wave-focusing experiments in a wavetank with slowly varying topography. The wave-tank had a length of 25.603 m and a width of 6.096 m. In the middle of the channel, (7.62 m-15.25 m ), eleven semicircular steps formed the topography, with the depth varying from 0.457 m to 0.152 m in the deep and shallow portion respectively (Fig. 2). Whalin gave a set of equations approximating the topography; ~0.4572, h(x,y)= ~0.4572+~(10.67-G-x), [0.1524,

(0
(lO.67-G<~x<~18.,29-G)

(18.29- G ~
where G(y) = [ y ( 6 . 0 9 6 - y ) ] !

(0~
(4.2)

The bottom topography is symmetric with respect to the centreline of the wavetank, y = 3.048 m. Owing to the symmetry of the problem the computational domain consists of only one half of the wave-tank. The grid system follows the computational Y (m) 6.096

ffffffff I

~ .~

3.048

I

0

I

I

i

2

4

6

L

I

i

.~

~

~ .~

1

.~

4

-

~

ff

1

1

1

1

.~

.~

.o

.~

I

I

8

I0

14

12

18

16 ~

I

20

i

22

I

24

Fig. 2. Bottom topography of wave-tank (Whalin, 1971 ). 3.o48r

L

H -

- - ~- -

o

~

L o

--

o

--

o

--

o

--

o

i

o

-

o

-

-

o

--

o

o

-

o

--

o

--

--

o

--

o

I

o

--

o

--

--

o

--

o

0

--

0

--

0

- -t - - - F - --

--

o

--

o

--

0

--

o

--

--

o

--

o

--

0

--

o

--

o

--

o

Fig. 3. Grid system in the wave-tank.

I

~

0

-

o

--

~

o

--

o

--

0

--

o

--

--

o

--

o

--

o

--

o

--

o

--

o

--

o

--

o

--

--

0

--

o

-

o

-

o

--

--

o

--

o

--

o

--

o

--

o

o

--

o

--

o

--

0

--

o

--

o

I

i

I

i

I

--

I

I

I

o

I

I

~

w ~'la l

o

I

I

I

!

o

I

l

I

I

--

I

l

I

o

I

i

i

I

0 i

i

I

I

0 I

i

I

I

--

I

I

I

0

- -t - - - , - - - , - - - , - - - , - - , - .

-I--

L

1

I

I

0 I

I

i

I

-

I

i

1

I

o

-

I

centreline

- t- - - - I -

I

I

~

I

-t - -

I

I

I

I

o

-

I

I o

- -

i

I

I o

-

L

I o

o

- -I-

[

I

D

I

I 25.603m

|

l

26

x (m)

197 domain as sketched in Fig. 3. The no-flux boundary condition is used along the centreline and the side-wall of the wave-tank:

v=O f o r y = 0 a n d y = 3 . 0 4 8 m

(4.3)

In the equation of motion, eqns. (3.5) and (3.6), we need fictive points (Ui,o,~i,o etc.) outside the computational domain. From the no-flux condition these points are implemented by: Ui,O :l'ti,1,

~i,O=ill,l, etc.

(4.4)

At the left boundary the velocity in the x-direction (u) corresponding to a sinusoidal wave input is specified. The fictive points needed for the velocity in the y-direction at the left boundary are implemented by the irrotational condition:

[~x~=Sya ]~j

(4.5)

The linear form of the equation of motion specifies the surface elevation beyond the left boundary. At the right hand boundary it is desirable to absorb all incoming waves. Following the derivation of higher order absorbing conditions for linear waves by Enquist and Majda (1977) we used the following second order condition: 02U

02U

C2 02U

Ot 2 "t-Co--~X- 2 -~y2=0

(4.6)

where c is the phase velocity of the wave. For the Arakawa-C grid the most successful finite difference implementation was:

[d2tu-X+cdtdxU

--t

C2¢~2

- ~ v y u-~=0]i~+½,~+~

(4.7)

This absorbing condition resulted in a small amount of reflection, less than 3% for even the most nonlinear waves in the experiments. Despite this the right hand boundary was extended to x = 37.5 m to be sure that the reflections would not influence the interesting area of computation. In all simulations presented in the next chapter, the grid increment was 0.125 m and 0.1524 m in the x- and y-direction respectively. For simulations with wave periods of 3 seconds this corresponds to 50 and 30 gridpoints per wavelength, in x-direction, in the deep and shallow portion of the channel respectively. A 2 second wave period gives equivalently 31 and 20 gridpoints per wave length. The discretization in time was At--0.047 s resulting in a Courant number of C = g x ~ o ~ < 0.8. In addition the model was run with half of these --/IX grid increments, but only negligible differences were observed in the first three

198

harmonics. A steady state was reached after simulation of 11 wave periods. A fast Fourier transform is used on the last wave period to evaluate the amplitude of the harmonics. 5. NUMERICAL E X P E R I M E N T S AND DISCUSSION

Whalin conducted three sets of experiments by generating waves with periods of 1, 2 and 3 seconds. Waves with different wave amplitudes were generated for each wave period. In Table 1 the experimental information is summarized. The Boussinesq equations are deduced under the assumption of shallow water depth but includes nonlinearity and dispersion to the leading order. If the ratio between the wave length and the waterdepth is sufficiently large the equations may apply both in the deep and shallow region in the experiment. The ratio are evaluated and given in Table 2 for different wave periods and for the shallow and deep areas of the wave-tank respectively. In Fig. 4 the general dispersion relation is compared to the dispersion relation for the Boussinesq equation and the Korteweg-de Vries equation. From Fig. 4 we may assume that the Boussinesq equation can be used for wave simulation of waves with wavelengths as short as four times the characteristic depth. This includes both the experiments with 2 and 3 second wave TABLE1 Experimental information at water depth ho = 0.4572 m (The wave number ko is defined according to the general dispersion relation: o)2 =gko tanh hobo) Wave period T (s)

koh(~

Wave amplitude ao

(cm) 1.0 2.0 3.0

1.922 0.735 0.468

0.97 0.75 0.68

1.95 1.06 0.98

TABLE 2 Wavelength compared to characteristic depth Wave period T(s)

ho(m)

~.o/ho

1.0 1.0 2.0 2.0 3.0 3.0

0.4572 0.1524 0.4572 0.1524 0.4572 0.1524

3.27 7.20 8.55 15.63 13.43 23.80

1.49 1.46

199 ¢

1.0

0.8-

0.6-

0.4-

0.2

5

10

Fig. 4. Dispersion relations: (a) general dispersion relation: 0) 2=gk tanh kh; (b) dispersion relation for the Boussinesq equation: 0) 2 - 1 +ghk2 ~k2h 2' (c) dispersion relation for the Korteweg-de

/

k2h2\ 2

Vries equation: 0)=ghk2~ 1----~--) .

periods. Waves with wavelengths varying from five to fifteen times the water depth are what we will call waves in intermediate water depth. With the assumption of slowly varying water depth, Liu and Tsay (1984) used a nonlinear SchrSdinger equation with variable coefficients to describe the forward-scattering wavefield in Whalins experiment. The method is based on the second order Stokes wave theory and is developed using the parabolic approximation method. Their numerical experiments were compared to Whalin's results for wave periods of 1 and 2 seconds. Figures 5, 6 and 7 compare numerical results for wave periods 2 s obtained by Liu and Tsay (1984), the present Boussinesq model and the experimental data by Whalin. In spite of the scattering in the experimental data we may conclude that the Boussinesq model gives a very good prediction of the first, second and third harmonic wave amplitude. For the first order harmonic wave, the model by Liu and Tsay overpredicts the amplitude in the focal zone. Our Boussinesq model follows more closely the experimental data, see Fig. 5a and Fig. 6a. Both models predict the maximum amplitude of the second order harmonics quite well, but contrary to Liu and Tsay's results, the Boussinesq model also reproduces the sharp maximum for the amplitude of the second harmonics. This deficiency may be due to the parabolic approximation which is used in deducing the nonliner SchrSdinger equation.

200 ]Harmonic . . . .

-0.020

I

. . . .

i , , , , I

. . . .

I

. . . .

I , , . , r

. . . .

I i

I . . . .

b I..

k i ~0.0t5

0,0t0 i

/ O • 005

o.ooo

....

0

i ....

s

i. is

i ....

~0

. . . . i. . . . . . | .

20

i

2s

....

i . . . .

~0 ALOng

~s

40

CmorneL

Im)

2 HalImmi, E o

. . . .

.020

I

. . . .

{

I

I

I

I

. . . .

I

. . . .

5, I

I

I

I

~

I

I

I

I

....

I

/

J

~o.ots

//

o

i°~o" " \

,ooo

,

0

5

~0

~5

20

25

~0 ALong

~s

40

::hammeL

[m ]

3.lJarmonh" ....

~ o ,o2o

I ....

I,,

,I

. . . .

I . . . . . . . .

2

. . . .

I

....

d

<~0,015

0 .o05

o

0

5

i0

~g

2o

25

30 ALong

~5 Cmo~n~L

~0 [m)

Fig. 5. Wave amplitudes along the centreline of the wave-tank for T = 2 . 0 s and a o = 0 . 0 0 7 5 m: ( o o o ) = e x p e r i m e n t a l data ( Whalin, 1971 ); ( - - - ) = numerical results (Liu and Tsay, 1984 ), ( ) = present numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

201 ~0.020

....

i

i ....

~,,,,I,,,,I

o_

1.Harmonic ....

I ....

p ....

r ....

i

/


i

~

/ t

0.0~0-

o

o

/

s

0,005

O.O00

~

....

0

i ....

5

i ....

i ....

~0

i ....

~5

20

i ....

25

i ....

i ....

50 55 ALOng C h o n n e L

~O [m]

2.Harmonic

z

o .020

' ' ' , l , r , , l , , J J l , , , , l , i , , l , , , i l i , , , J i , , ,

~ 0 . 0 t 5

0.Ot{3

//

°°~\

o

O .005

~

r

/

z

j

O .000

, ,

O

5

~O

~5

20

2S

50 5£ #0 ALong ChommeL (m)

3.Harmonic

To .020

....

J ....

i ....

i , , , , i

....

i ....

i,,,,i,,,,


o

O.000

~

~

~ 5

, I0

~5

~ 20

2£

~0 ~5 ALong ChonneL

, ~O [m}

Fig. 6. Wave amplitudes along the centreline of the wave-tank for T = 2 . 0 s and n o = 0 . 0 1 0 6 m: ( o o o ) = experimental data ( W h a l i n , 1971 ); ( - - - ) = numerical results ( L i u and Tsay, 1984 );

(

) = present numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

202 l,Harlnonic ~O.OgO

,

.

i ....

i ....

i ....

i ....

J ....

t ....

I ....

i

i .... gO

i ....

i

~0.025

0.020

0.045

\

\ r

....

I .... 5

i .... I0

i .... ~5

i ' ' 20

p .... 25 ALong

55

Cmonne[

40 (m)

2Harmonic

e

O.OgO P #

h)

~o.o2s

O.02O-

O .005

-

~

0.000 ~ 0

5

~0

i ~5

,,,i

~

, 2O

i .... 25

k~"-~

i .... 50

ALong

i .... g5

ChommeL

J ~0 [mJ

3.1tarmunic

F

~O.02S< 0.020-

L

0,005

F 0,00o

--

,I 5

0

~S

2O

25

5O

5S

ALong ChonneL

~0 [m/

Fig. 7. Wave amplitudes along the centreline of the w a v e - t a n k for T = 2 . 0 s and a o = 0 . 0 1 4 9 m: ( o o o ) = experimental data (Whalin, 1971 ); ( - - - ) = present numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

203 l.llarmonic

<

0.0~0

o

0.000 qO

t5

20 ALong

25

ChonneL

[m)

2.Harmonic

~0.015

,

,

,

,

I

,

,

,

,

I

,

,

,

,

E

,

,

,

,

. . . .

I

< 0.0~0

/~o /t

0

S

I0

o

° o °

o

o

2D

~S

2S

ALon0 ChonneC [m) 3.Harmonic ~0.0tS

,

,

,

,

[

,

,

k

,

I

,

,

,

,

J

,

k

,

,

I

,

,

,

,

<

0.010

O.OOS

0.000

1

'

,

. .

~

S

. . . .

i

40

. . . .

i

15

. . . .

i

. . . .

20 ALOng ChonneL

2S (m)

Fig. 8. Wave amplitudes along the centreline of the wave-tank for T = 3 . 0 s and ao=0.0068 m: ( o o o ) = experimental data (Whalin, 1971 ); (- - - ) = numerical results (Liu, Yoon and Kirby, 1985); ( ) =present numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

204 l.Harmonic

~ 0.020

. . . .

J <~0,0~5

5

~0

~

20 ALong

25

ChonneL

{m)

2.liarmonic

~0.020 e

~o

.o t5 •

oo o

0,00S-

27 %

O.O0O

. . . .

;

0

. . . .

F . . . .

i

~0

. . . .

f

15

. . . .

20

ALong

25

Cho~neL

[m]

3.Harmonic ~0.020

~0.0~5

0.000

J

L

J

L

[

,

,

L

i

I

,

i

,

,

,

J

,

,

,

I

,

L

,

,

-

~

. . . .

i 5

. . . .

-"=~, ~0

.

.

. . ~S

.

.

ALong

i ZO

. . . .

ChonneL

25 (m}

Fig. 9. Wave amplitudes along the centreline of the w a v e - t a n k for T - - 3 . 0 s and a o = 0 . 0 0 9 8 m; ( o o o ) = e x pe r i me nt a l data (Whalin, 1971 ); ( - - - ) = numerical results (Liu, Yoon and Kirby, 1985); ( ) = p r e s e n t numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

205 1 .Harmonic ~0.025

,

,

,

,

I

. . . .

I

. . . .

I

. . . .

I

.

.

.

.

~< 0 . 0 2 0 0.0

15 -

0.0~0-

O.OOO 0

~0

t5

2O 25 ALong Cho~n(~L (m)

2.Harmonic ~0.025

b) o

~ 0.020

o

<

o

0.01S

O.OlO

O.OOS

0.000 5

t0

t5

20 25 ALong ChonnoL ( e l

3,Harmonic

~0.025

d i0.020 <

0,0~5-

o°o oo

0.0~0-

0.00S -

~//

997~/

/

/°~ /

7 ALong ChonneL (m)

Fig. 10. Wave amplitudes along the centreline of the wave-tank for T = 3 . 0 s and a o = 0 . 0 1 4 6 m; ( o o o ) = experimental data ( W h a l i n , 1971 ); ( - - - ) = numerical results (Liu, Yoon and Kirby,

1985); ( harmonic.

) =present numerical results; (a) first harmonic; (b) second harmonic; (c) third

206 1 .Harmonic

J <

0.0 IO

t

xlj

\

g°°si 0.000

/

F

. . . .

~ 5

D

. . . .

i ~0

. . . .

i ~5

. . . .

i 20

ALong

. . . . 25

ChopneL

(m)

2.Harmonic ,~0.0

t5

. . . .

I

,

,

,

,

t

,

,

,

,

I

,

,

,

,

k

,

,

,

,

i,)

< 0.0~0

0.00S

0.000 S

t0

~S

2O

25

ALong-ChonneL

lm)

'

3.Harmonic ~O,O~S

,

,

,

,

I

,

,

,

,

k

,

,

,

,

I

,

,

,

,

,

,

,

,

,

J < 0.0~0

S

~0

~S

2O ALong

ChonneL

25 (m)

Fig. 11. Wave amplitudes along the centreline of the wave-tank for T = 3 . 0 s and a o = 0 . 0 0 6 8 m; ( - - - - ) = numerical results Boussinesq equation (Liu, Yoon and Kirby, 1985 ), ( - - - ) = numerical results modified K.-P. equation (Liu, Yoon and Kirby, 1985); ( ) = p r e s e n t numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

207 1.Harmonic

10.020

'

'

'

'


0.010-

0.000

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

ALong

i

. . . .

ChonneL

(m)

2.Harmonic

~0.020 e

o.


-

O.O~O

~/

O .O05

/

O

5

t0

,"

t5

20 ALong

ChonneL

ALong

ChonneL

2S [m)

3Harmonic

-0.020 e

0.010"

S

I0

t5

20

25 Im)

Fig. 12. Wave amplitudes along the centreline of the wave-tank for T = 3.0 s and a o = 0 . 0 0 9 8 m: ( - - - - ) = numerical results Boussinesq equation (Liu, Yoon and Kirby, 1985 ); ( - - - ) = numerical results modified K.-P. equation (Liu, Yoon and Kirby, 1985 ); ( ) = present numerical

results; (a)first harmonic; (b)second harmonic; (c)third harmonic.

208 1.Harmonic 0.025

J

~

0,020

<

o.o~s

0.005

0.000 40

s

0

~5

2O ALong

25

ChommeL

[m)

2,Harmonic

0 . 0 t 5

-

~ r//J

,

O.OiO

0.005

0.000

. . . .

S

~0

~S ^Long

, 20

I

. . . .

ChonneL

25 (m)

&Harmonic ~o .025 E

~

,

,

,

,

I

,

,

,

,

I

. . . .

I

. . . .

0,020

<

,,'

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

,,

/,~ ......... 5

I0

5

20 *&Long

ChonnBL

(m)

Fig. 13. Wave amplitudes along the centreline of the wave-tank for T = 3 . 0 s and ao=0.0146 m: ( - - - ) = numerical results Boussinesq equation ( Liu, Yoon and Kirby, 1985 ); (- - - ) = numerical results modified K.-P. equation (Liu, Yoon and Kirby, 1985); ( ) =present numerical results; (a) first harmonic; (b) second harmonic; (c) third harmonic.

209 Boussinesq ~ 0 . o 2 o

. . . .

i

. . . .

i

. . . .

i

. . . .

i

. . . .

i

. . . .

~

. . . .

i

....

i

. . . .

~

. . . .

i

£

o.


0.0

~0 -

0.00S

-

0.000

~-~

0

, :/i

, , ~,

S

i°,-r

10

;7"

I ....

15

i , , ;','T

. . . .

2S

20

~S

~0

ALong C h o m n e L ~-0.0t5

.........

y°u?~7~

~,0 (m)

. . . . . . . . . .

b)

e

J

<

0.0 ~0

O.OOS

5

10

t5

20

ALon 9 ChomneL

25 (m)

Fig. 14. Two-dimensional topography: ( ) =first harmonic; ( - - - - ) = s e c o n d harmonic; (- - -) -- third harmonic; (a) T=2.0 s and ao=0.0075 m; (b) T=3.0 s and ao=0.0068 m.

For the most nonlinear waves (ao--0.0149), where Liu and Tsay do not give any numerical results, the Boussinesq model overpredicts the first harmonic wave amplitude. The frictional dissipation which is neglected in the model may be significant in this case. In 1985 Liu, Yoon and Kirby presented numerical results obtained using the parabolized Boussinesq equation and the Kadomtsev and Petviashvili equation. To obtain the parabolized Boussinesq equation they derived evolution equations for spectral-wave components in a slowly varying two-dimensional domain. In Figs. 8, 9 and 10 we have compared the present numerical method with the Boussinesq model of Liu et al. ( 1985 ), and with Whalin's experiments for

210

waves with three second periods. The present results seem to underpredict the second and third harmonic wave amplitude. In the constant depth area behind the shoal the result follows the experiments more closely than the results obtained by Liu et al. In this region we must expect the three-dimensional shoaling effects to be most dominant. In Figs. 11, 12 and 13 we compare our numerical results with the Boussinesq model and the modified K.-P. model of Liu et al. There is surprisingly good agreement between the K.-P. model and the present Boussinesq model as opposed to the Boussinesq model by Liu et al. Further investigation is needed to explain these differences. To recognize the effect of focusing we made some tests with the topography varying only along the channel from ho=0.4572 m to ho=0.1524 m. This was made by setting the function G in eqn. (4.1) constant equal to its value at the centerline of the channel, y-- 3.048 m. The wave modulation is then only due to the bottom variation along the direction of propagation. Figure 14 shows the first, second and third harmonic wave amplitude for simulation with, (a) T = 2 . 0 s, ao--0.0075 m and {b) T = 3 . 0 s, ao--0.0068 m. If we compare these results with Figs. 5 and 8 it is evident that the focusing is very weak for the sets of experiments with a wave period of three seconds. The rapid growth of the second and third harmonic waves in the focal zone is in this case mainly due to the variation of the topography along the wave-tank. In the experiments with a wave period of 2 seconds the three dimensional effects are on the contrary more dominant. The wavelength is shorter compared to the variation in the topography and the waves are therefore able to adjust sufficiently. This is most likely the reason why the present Boussinesq model, which treats the two horizontal directions of propagation as equally important, is superior in the case with wave periods of two seconds. 6. CONCLUDING REMARKS

Results based on numerical integration of the nonlinear Boussinesq equation are in good agreement with Whalin's (1971) experimental data for the topographical lens problem. The model is superior to earlier reported results for waves in intermediate water depth, i.e. wave length between 5 and 15 times the water depth. The present study indicates that the non-parabolized Boussinesq equation must be applied for investigating nonlinear refraction-diffraction where the focusing effect is dominant. In Whalin's experimental data set for shallow-water waves the focusing effect is weak. This is expected to be the reason why the present Boussinesq model is not superior to the parabolized Boussinesq method in this case. The numerical solution method demands no periodicity in the wave field and it is therefore also applicable to study transient wave propagation problems. If only the total wave amplitude is desirable and not each harmonic com-

211

ponent, the discretization may be set to 15-20 gridpoints per wave length. This is suffcient if the higher harmonic components do not dominate the problem.

REFERENCES Abbott, M.B., Petersen, H.M. and Skovgaard, 0., 1978. On the numerical modelling of short waves in shallow water. J. Hydraulic Res., 16: 173-203. Abbott, M.B., McCowan, A.D. and Warren, I.R., 1984. Accurcy of short-wave numerical models. J. Hydraulic Eng., 110: 1287-1301. Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction. Proc. 13th Int. Conf. Coastal Eng., Vancouver, BC, pp. 471-491. Berkhoff, J.C.W., Booij, N. and Radder, A.C., 1982. Verification of numerical wave propagation models for simple harmonic linear waves. Coastal Eng., 6: 255-279. Bettes, P. and Zienkiewicz, O.C., 1977. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Numerical Meth. Eng., 11: 1271-1290. Booij, N., 1981. Gravity waves on water with non-uniform depth and current. Dissertation, Delft University of Technology, Delft, The Netherlands. Enquist, B. and Majda, A., 1977. Absorbing boundary conditions for the numerical simulation of waves. Math. Comp., 31: 629-651. Houston, J.R., 1981. Combined refraction and diffraction of short waves using the finite element method. Appl. Ocean Res., 3: 163-170. Kirby, J.T. and Dalrymple, R.A., 1983. A parabolic equation for the combined refraction-diffracJ tion of Stokes waves by mildly varying topography. J. Fluid Mech., 136: 453-466. Liu, P.L.-F. and Tsay, T.-K., 1983. On weak reflection of water waves. J. Fluid Mech., 131: 5971. Liu, P.L.-F. and Tsay, T.-K., 1983. Refraction-diffraction model for weakly nonlinear water waves. J. Fluid Mech., 141: 265-274. Liu, P.L.-F., Yoon, S.B. and Kirby, J.T., 1985. Nonlinear refraction-diffraction of waves in shallow water. J. Fluid Mech., 153: 185-201. Madsen, P.A. and Warren, I.R., 1984. Performance of a numerical short-wave model. Coastal Eng., 8: 73-93. Pedersen, G. and Rygg, O.B., 1987. Numerical solution of the three dimensional Boussinesq equations for dispersive surface waves. Res. Rep. No. 1, University of Oslo, Inst. of Mathematics, Oslo, Norway. Peregrine, D.H., 1972. Equations for water waves and the approximation behind them. In: Meyer (Ed.), Waves on Beaches, Academic Press, New York, pp. 357-412. Radder, A.C., 1979. On the parabolic equation method for water-wave propagation. J. Fluid Mech., 95: 159-176. Tsay, T.-K. and Liu, P.L.-F., 1982. Numerical solution of water-wave refraction and diffraction problems in parabolic approximation. J. Geophys. Res., 87: 7932-7940. Whalin, R.W., 1971. The limit of applicability of linear wave refraction theory in a convergence zone. Res. Rep. H-71-3, U.S. Army Corps of Eng. Waterways Exp. Station, Vicksburg, MS. Yue, D.K.P. and Mei, C.C., 1980. Forward diffraction of Stokes waves by a thin wedge. J. Fluid Mech., 99: 33-52.