a numerical model for nonlinear wave transformation ...

Coastal Engineering in Japan, Vol. 35, No. 1, 1992

A NUMERICAL MODEL FOR NONLINEAR WAVE TRANSFORMATION IN NEARSHORE ZONE BY MULTI-STEP FINITE CHARACTERISTIC METHOD Xiping Yu 1 Masahiko Isobe 2 A kim Watanabe 2

ABSTRACT Nonlinear equations governing wave motion in the nearshore zone are newly derived. The equations reduce under shallow water conditions to those for nonlinear long waves, and under small amplitude conditions to those for linear waves in an arbitrary depth. A numerical procedure for solving the equations is presented on the basis of the operator-splitting method and the finite characteristic method. Sample computations are carried out on the wave transformation due to a bottom topography and on that due to a detached breakwater. The computational results are compared with linear solutions and experimental data. I.

INTRODUCTION

Waves are regarded as linear only when the parameter gHIC 2 (H is the wave height, C the wave celerity and g the gravitational acceleration) is small. This condition corresponds to that the wave steepness HI L (L is the wavelength) is small for deep water waves and that the ratio HI h of the wave height to water depth is small for long wave~. Such a condition, however, is rarely satisfied in most of practical nearshore wave problems. We usually deal with a wave field including breakers where the wave height grows to as large as the local water depth. Then the neglect of nonlinearity in wave computation is obviously inappropriate and leads to solutions in which some significant phenomena are not simulated. A theoretical basis for the analysis of finite-amplitude nearshore wave motion is the well-known nonlinear long wave theory (See, e.g., Stoker, 1957; Mei, 1989). Application of the long wave theory, however, is severely limited because of two reasons: (1) the theory is valid only in very shallow waters and (2) it can describe only nondispersive waves. Since incident wave conditions are usually specified in a region where waves are free from the transformation due to bottom topography, a theory valid in various water depth is essentially required. Wave dispersivity is important even when the nonlinearity is strong because the nonlinearity steepens the front of a wave profile while the dispersivity plays an opposite role. Adequate description of 1 D. Eng., Tech. Res. Inst., Toa Corporation, Anzen 1-3, Tsurumi-ku, Yokohama 23 0, Japan; at present Dept. of Mech. Eng., Univ. of Hong Kong, Hong Kong. 2 D. Eng., Professor, Dept. of Civil Eng., Univ. of Tokyo.

36 the wave transformation thus depends on a proper assessment of both the nonlinearity and dispersivity. The long wave theory has been improved by Peregrine (1967). However, the theory, usually referred to as the Boussinesq theory, is still applicable to moderately long waves (Mei, 1989). For nearshore waves propagating over a wide range of water depth, any satisfactory nonlinear theory has not been developed. Currently, the analysis of a nearshore wave field relies mostly on a combination of linear theories and empirical formulas for processing nonlinear phenomena such as breaking. Successful examples of such treatment are found in Watanabe and Maruyama (1986), Isobe (1987), Watanabe and Dibajnia (1988) and Yu et al. (1992b). It is obvious that a nonlinear wave theory of practical use must reduce to the long wave theory for shallow waters and to the linear theory for small amplitude waves. The present study is devoted to the development and application of such a theory. We first derive the governing equations by integrating the basic equations of fluid motion in the vertical direction, and then describe a procedure for the numerical solution based on the operator-splitting method and the finite characteristic method. Applications to wave transformation due to a bottom topography and a detached breakwater are presented. Comparisons of the results with linear solutions and experimental data are also shown.

II.

THEORETICAL DEVELOPMENT

2.1 Derivation of Governing Equations Wave motion in an invicid and incompressible fluid is basically governed by the following continuity and Euler equations:

(1)

(2) ow+ o(wuj) 8t OXj

+ o(w 2 ) + .!!.._ oz

oz

(Pd) = p

0

(3)

where the subscripts i and j (i, j = 1, 2), to which the summation convention is applied, stand for the two horizontal directions; ui and w are, respectively, the horizontal and vertical velocity components, Pd the dynamic pressure, p the fluid density, Xi and z the horizontal and vertical coordinates and t the time. The boundary conditions on the free surface and the solid bottom are expressed respectively by

ory ory -+u--w=O ot J OXj oh u·-+w=O J

ox·J

(z=ry) (z=-h)

(4) (5)

where TJ is the water surface elevation and h the still water depth. The integration of Eq. (1) with respect to the vertical coordinate z from the bottom to the free surface results in

37

ary

at

+ aU1 = 0

(6)

ax1

where uj = J~h Uj dz represents the flow rate in the xrdirection. Similar integration of Eq. (2) gives

aui 8t

+ _!!___ OXj

17) uiuJdz + _!!___ 17) OXi

-h

-h

Pd dz - Pd p p

I .!!!!_ 7)

OXi

Pd I ah p -h OXi

=o

(7)

Equations (6) and (7) obviously constitute a closed set for solving 7] and Ui if Pd is related to them. Modifying the pressure solution of the small-amplitude wave theory, we obtain the following expression (Yu et al., 1992a): Pd cosh km (k + z) -=gry p cosh km (h + 7])

(8)

where km is the wave-number, which is determined from the angular frequency (J by the modified dispersion equation: (J 2 / g = km tanh km ( h + 7])

(9)

From Eq. (8) the following relations can immediately be obtained:

(10) (11)

Pdl =g'T] p

7)

gry cosh km ( h + 7])

Pdl p -h

(12)

where Cm = (J/km. Substitution of Eqs. (10) to (12) into Eq. (7) leads to

aui

a

---;) +~ ut ux1

17) UiUjdz + ~a (Cmry)- gry~ary 2

-h

ux,

ux,

COS

gry hk (h m

ah

+ 7] ) ux, !l

.

=

0

(13)

It may be convenient to introduce the following momentum correction factor into Eq.

(13): (14) where i and j to which the summation convention is not applied are indicated by the underlines. The value of the factor f3iJ depends obviously on the vertical structure of the horizontal velocity components ui. We assume that the vertical distribution of ui is expressed by the function coshkm(h + z), that is,

38

=

U;

where

U;b

U;b

cosh km ( h

+ z)

(15)

is the magnitude of u; on the bottoi?. Then it is verified that

(16) where km(h + TJ) a = -----'---.,.._..:...:---.,.. tanh km(h + TJ)

nm =

~ [1 + 2

2km(h+ry) ] sinh 2km (h + T])

(17) (18)

By taking into account the relation

oc~ = (~ _ ox;

a

!) (3

9 o(h+ry)

ox;

(19)

Eq. (13) is finally developed as

(20) where 2

1

KI = - - - -

a

2 a

(3

1

cosh 2

km(h + TJ)

1

1

(3

cosh km ( h + T])

K2 = - - - -

(21) (22)

Equation (20) has been obtained by assuming that the fluid is invicid. This assumption, however, is not appropriate for waves in shallow water, where the effect of bottom friction is usually significant. To take into account of the momentum loss due to friction, we modify Eq. (20) by adding a friction term as

where the friction coefficient fD will be determined by the following formula, which is derived to simulate the effect of a laminar bottom boundary layer (Dalrymple, 1984): fD = 2j'

v'1+f2

(24)

with

(25)

39

where v is the fluid viscosity. If the magnitude of fD is not unreasonably large in deep and intermediate water, the friction term will not cause any undesirable effects on the solutions of 77 and U;, whereas its role in stabilizing the numerical computation is vital. The continuity Eq. (6) and the equation of motion (23) constitute an initialboundary value problem for nearshore wave transformation. In addition, since the above equations include second-order nonlinear terms, the wave induced mean water surface variation and currents can be predicted. However, the detailed discussion will be made elsewhere because they are dependent on many factors such as the energy dissipation, friction factor and lateral mixing.

2.2 Consistency with Other Theories ( 1) Long wave theory For long waves, since (26) the following relations can be immediately obtained from Eqs. (9), (16), (21) and (22):

C! = g(h + 77)

(27)

/3=1

(28) (29)

Consequently, Eq. (20) becomes aU;+ a (uiuj) ) a 77 _ 0 - +g (h +77-at

axj

or, in terms of the mean velocity ii; aili at

h + 77

axj

(30)

= U;j (h + 77), _ ail; axj

877 ax;

-+uJ-+g-=0

( 31 )

if the continuity equation (6) is referred. Equation (31) is exactly the equation of motion for nonlinear long waves (See, e.g., Stoker, 1957; Mei, 1989). (2) Linear wave theory When linearized for waves of small amplitude, Eq. (20) reduces to au; !)

ut

877 , ah + c2 ~ + "'2g77~ uX; uX;

= 0

(

32

)

where C and "'~ represent, respectively, Cm and "' 2 under the small amplitude assumptions. Equation (32) is equivalent to the equation of motion derived by Watanabe and Maruyama (1986). The proof is simply an extension of the one-dimensional case given by Yu et al. (1992a).

40 If the gradient of the bottom elevation is much smaller than the wave steepness, Eq. (32) can further be simplified as

aui + 02 877 ·= 0 at axi

(33)

which has been presented by Tanimoto and Kobune (1975).

III.

NUMERICAL METHOD

3.1

The Operator-Splitting Method Application of the operator-splitting method to solving long wave equations may be found in He (1982), He and Lin (1986), Lin (1988) and Benque et al. (1982). The mathematical background of the method can be found in Marchuk (1975). In connection with the present study, we consider the following model equation for a brief illustration:

8B

Bt + Dl[B] + D 2 [B] = 0 where B is an unknown vector, D 1 [ ] and D 2 We rewrite Eq. (34) in a finite difference form:

[

]

(34)

are space-differential operators.

(35) where 61 [ ] and 62 [ ] are finite difference operators, fJ.t is the time increment, p, q (0 ::=; p, q ::=; 1) and n are the time step indices. By introducing an intermediate variable

B,

Eq. (35) is splitted into the following two equations: (36)

(37) As far as p is appropriately selected, Eq. (36) is equivalent to a finite difference equation for solving

8B

Bt+DI[B] =0 in the time step from nf1t to (n Eq. (37) is equivalent to solving

+ 1)11t with

Bn as the initial condition. Similarly,

8B

8t +Dz[B]

(38)

=

0

(39)

with the solution B of Eq. (36) as the initial condition. Therefore, as shown in Fig. 1, the increment of B in the time step from nf1t to (n + 1)f1t can be obtained by solving the coupled component finite-difference equations, Eqs. (36) and (37). In their physical meaning, Eqs. (6) and (23) express that the variation of water surface elevation 7] and the flow rate Ui with respect to the time t is a result of two

41

(n+1)6t

B

Fig. 1 Description of operator-splitting method.

processes: the advection and the propagation. Therefore, the equations are splitted into the equation sets corresponding to the following three steps: Step 1: advection (40)

(41)

Step 2: propagation in the x-direction

aTJ at

au ax

-+-=0

(42)

[( 1 - -o- 27)) C 2 + "'19TJ J -aTJ + "'297)ah CmU -au + at 9 m ax ax + fo h +-7] = 0

(43)

Step 3: propagation in the y-direction (44)

(45)

3.2 Scheme for Advection Step Equations (40) and (41) can be approximated by

au au au -at + (3uax + (3vay = o

(46)

av av av - + (3u- + (3vat ax ay

(47)

=

o

42

t (n+l)6t

p

(j+l)6y ~~--~--~---

y

Fig. 2 Description of scheme for advection step.

=

where u U/(h expressed by

+ ry)

V / (h

and v

dx dt

=

+ ry).

f3u,

By introducing a characteristic curve

dy dt

=

(3y

(48)

Eqs. (46) and (47) become du

dt

=

o,

dv dt

-=0

(49)

along the curve. The numerical procedure for solving u~j 1 and v~j 1 , where the subscripts i and j are the grid indices, is illustrated in Fig. 2. A finite characteristic line is issued from P to Q according to the following discretized form of Eq. (48):

(50) We can directly calculate and v~t from Eq. (49).

uQ

and

vQ

with the bilinear interpolation and obtain u~r

3.2 Scheme for Propagation Step Since the equations governing the propagation process in the x- and y-directions are the same in form, the numerical scheme is described only for the x-component equations. By introducing two families of characteristic curves defined by

dx -=±

(51)

dt

Eqs. (42) and (43) can be expressed in the following characteristic form:

dU

dry

- ±>.- +G=O dt dt

(52)

43

p

y

if::.x jf::.y

Fig. 3 Description of scheme for propagation step.

where

oh

G = K2g7Jr;ux

UCm

+ fn-h+ 1]

(53)

As shown in Fig. 3, finite characteristic lines are issued from P to Q and R using the following discretized form of Eq. (51): Xp- XQ =

A?,jL1t

Xp- XR =

-A?,jL1t

ua,

(54)

which determine the coordinates of Q and R. Then UR, TJQ, ry'fl_, AQ, >.R_, GQ and GR_ are calculated by the parabolic interpolation. On the other hand, Eq. (52) is discretized as

(55) (56) Solving Eqs. (55) and (56) simultaneously, we obtain

(ARTJR

+ AQTJQ)- (UR- Uq) + (GR- Gq) Llt AR+Aq

un+I

=

(>.qUR

2,)

IV.

+ ARUQ)- ARAQ (TJR -ryq)- (>.qGR + ARGq) Llt AR + Aq

(57) (58)

EXAMPLES OF APPLICATION

4.1 Wave Refraction Due to Bottom Topography A problem, studied by Nishimura et al. (1983) is illustrated in Fig. 4. The bottom topography is given in terms of the still water depth h by the following equation:

h

= 0.02 [-x- 1 + cos(Ky/0.8)]

(59)

44

x(m) -10.0

-9.0

-8.0

IF················ _/

.........._

·--

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

.~:···------+-~ ~ ---· .. . • ~•.·~·.· ~~:1: ~~~:: .·

.··

·------- ------- ------------ ----

1

I

50

Fig. 4 Definition sketch for wave refraction due to bottom topography.

where the unit of h, x and y is meter. Incident waves, 1.06 em in height and 1.2s in period, propagate in the positive x-direction. Since the offshore open boundary is located at a position where the nonlinearity is of insignificant, the water surface elevation consists of the incident and reflected wave components: ry = 'Tlo(x - Ct) + 'Tlr(x + Ct). Therefore, the boundary condition is

ary =cary

at

2a'T/o

ax+ 8t

(60)

where 'T/o corresponds to the incident waves. The lateral boundaries are solid walls and therefore the velocity component in the y-direction there vanishes. Along the moving shoreside boundary, the depth and the flow rate are zero. The movement of this boundary is determined by the following equations: dx

dy

dt =Us,

dt = Vs

_

(61)

where u8 and iJ 8 are fluid velocity components at the wave front. Shown in lower half of Fig. 5 is the computed wave height distribution. As a comparison, the linear solution obtained by the finite element method (Yu et al., 1992b) is presented in the upper half. Figures 6 and 7 show representative cross-shore and longshore wave height distributions of the solutions as well as the experimental data (Nishimura et al., 1983). The present results agree satisfactorily with the experimental data and the linear solution. This implies that the wave field including breakers can be well simulated by the present theory without introducing any empirical equation, which is usually necessary in linear theories (Isobe, 1987). This characteristic of the nonlinear wave theory may become remarkable when waves break under more complicated conditions, because the establishment of an universal empirical formula of breaker index will be very difficult.

45

unit: em

-9.0

-10.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

0

-1.0

-2.0

x(m)

Fig. 5 Wave height distribution due to a bottom topography.

3. 0 /""'-.

su

2. 0

'--"

~

1. 0 0 -6. 0

-4. 0

-5. 0

-2. 0

-3. 0

x(m) Fig. 6 Cross-shore distribution of wave height (y

= 0.7 m).

3. 0 r---;I:==::r:==::r::::;----,

s 2. 0 u ~

'--"

-Present ---Linear • Experimental

1.0 o~--~--~--~--~

-0. 8 -0. 4

0

0. 4

0. 8

y(m) Fig. 7 Longshore distribution of wave height (x

=

-5 m).

4.2 Wave Transformation due to a Detached Breakwater Figure 8 shows an example studied by Watanabe and Maruyama (1986). A detached breakwater of 2.67 m long is placed parallel to the bathymetric contour lines of a 1/50 plain beach. The incident waves are 2 em in height and 1.2s in period and propagate in the x-direction. We treat this problem by the present nonlinear model. The offshore and shoreside boundary conditions are imposed in the same way as in the previous section. Owing to the symmetry of the problem, computation is carried out for only a half domain, as shown in Fig. 8. Along the axis of the whole domain, namely the lower lateral boundary, the normal velocity component vanishes, as

46 x(m) -I 0. 0

-9. 0

-7. 0

-8. 0

-4.0

-5. 0

-6. 0

-3. 0

-2. 0

-1.0

r-----~--------~----~----._----~--------~----~----~4.0

3. 0

a

2. 0 ;;:;

J

- -

I_-

1.0

----~--

~~---~---~---~~

........ I

~

....

'

50

Fig. 8 Definition sketch for wave transformation around a detached breakwater.

unit: em

-10.0

-9.0

-8.0

-7.0

-6.0

-5.0

-4.0

-3.0

-2.0

-1.0

x(m)

Fig. 9 Wave height distribution around a breakwater.

0

47

6. 0

----s

u ....._.,

4. 0

t:q 2. 0 0 -5. 0

-4. 0

-3. 0

-1. 0

-2. 0

0

x(m) Fig. 10 Cross-shore distribution of wave height (y

= 0 m).

6. 0 ,..--.,.

su

'---"

t:q

-Present ---Linear • Experimental

4. 0 2. 0 0 -4. 0

-3. 0

-2. 0

-1. 0

0

y(m) Fig. 11 Longshore distribution of wave height (x

=

-3 m).

along the perfectly reflective surface of the breakwater. The upper lateral boundary is considered as non-reflective so that the water surface elevation at each time step in the numerical computation is determined by an explicitly discretized form of the continuity equation. The computed wave height as well as the linear solution of Yu et al. (1992b) is presented in Fig. 9. Wave height distribution in the cross-shore direction at y = 0 and that in the longshore direction at x = -3 m (behind the breakwater) are shown in Figs. 10 and 11. The characteristics of the nonlinear wave theory described in the previous section are again confirmed. It is also seen that the nonlinear method gives a smaller reflection than the linear theory. This implies that the nonlinear term reduces the fluctuation of wave height distribution, which gives the distribution smoother than the experimental result.

VI.

CONCLUSION

Basic equations governing the nonlinear wave motion in the nearshore zone were newly derived by integrating the continuity and the Euler equations of fluid motion in the vertical direction. The equations were proved to be consistent with both the nonlinear long wave theory and the linear wave theories. A numerical method for solving; the equations was developed on the basis of the splitting-operator method.

~----------------~-~~

-

~

"