Equation For Numerical Modeling Of Wave Transformation In Shallow Water

CHAPTER

3

EQUATIONS FOR NUMERICAL MODELING OF WAVE TRANSFORMATION IN SHALLOW WATER Masahiko Isobe Department of Civil Engineering University of Tokyo Bunkyo-ku, Tokyo, Japan

CONTENTS INTRODUCTION, 102 BASIC EQUATIONS AND BOUNDARY CONDITIONS, 103 Basic Equations and Boundary Conditions for Waves on a Fixed Bed, 103 Basic Equations and Boundary Conditions for Waves on a Permeable Bed, 107 MILD-SLOPE EQUATION, 109 Derivation of Mild-Slope Equation, 109 Alternative Forms of Mild-Slope Equation, 112 Physical Interpretation of Mild-Slope Equation, 115 Mild-Slope Equation on a Slowly Varying Current, 116 Mild-Slope Equation with Energy Dissipation, 117 Mild-Slope Equation on a Permeable Bed, 118 TIME-DEPENDENT MILD-SLOPE EQUATIONS, 121 Derivation of Time-Dependent Mild-Slope Equations, 121 Alternative Forms of Time-Dependent Mild-Slope Equations, 122 Time-Dependent Mild-Slope Equations for Random Waves, 123 PARABOLIC EQUATION, 125 Derivation of Parabolic Equation, 126 Alternative Forms of Parabolic Equation, 127 Parabolic Equation for Large Wave Angle, 128 Parabolic Equation in Non-Cartesian Coordinates, 129

101

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Weakly-Nonlinear Parabolic Equation, 131 Extension of Parabolic Equation Model, 133 BOUSSINESQ EQUATIONS, 134 Derivation of Boussinesq Equations, 134 Modified Boussinesq Equations, 135 Boussinesq Equations with Breaking Dissipation, 136 Boussinesq Equations for Waves on a Permeable Bed, 137 NONLINEAR SHALLOW-WATER EQUATIONS, 138 Derivation of Nonlinear Shallow-Water Equations, 138 Nonlinear Shallow-Water Equations on a Permeable Bed, 139 NONLINEAR MILD-SLOPE EQUATIONS, 141 Derivation of Nonlinear Mild-Slope Equations, 142 Relationship with Other Wave Equations, 144 VALIDITY RANGES OF WAVE EQUATIONS, 150 SUMMARY, 153 NOTATION, 155 REFERENCES, 157

Introduction Waves transform in shallow water due to shoaling, refraction, diffraction, reflection, transmission, bottom friction, breaking, etc. Predicting wave transformation is indispensable in coastal and ocean engineering practices because wave action causes various important phenomena such as forces on structures and sediment transport. Analytical solutions, numerical models, and physical models can be used for the prediction. Among them, numerical modeling has achieved remarkable progress owing to development of wave theory and computer technology. A comprehensive review has been given in [56] for numerical models based on the mild-slope equation and its parabolic approximation, and the Boussinesq equations. This chapter presents various numerical model equations and their extended forms for predicting wave transformation in shallow water. The mild-slope equation (MSE) includes the combined effect of refraction and diffraction of linear waves. Predictive models based on this equation have been used for a wide variety of engineering problems. Time-dependent forms of the MSE have been developed for improving numerical efficiency and treatment of boundary conditions and for extension to random waves. Parabolic approximations of the MSE have been developed to increase its computational efficiency remarkably.

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103

The Boussinesq equations are model equations for weakly nonlinear waves in shallow water. These equations have been solved both by direct numerical calculation and Fourier transformation. The equations have been modified to extend their applicable range to deeper water. The nonlinear shallow-water equations are fully nonlinear wave equations in very shallow water in which hydrostatic pressure distribution is assumed. Because these equations do not require any empirical formula for energy dissipation due to wave breaking, they have specifically been used to predict the wave transformation in surf and swash zones. Fully nonlinear and fully dispersive wave equations have been recently derived. These equations can reproduce even strongly nonlinear transformation. In addition, all the above model equations can be derived as special cases of the most generalized equations described in this chapter. The following sections of this chapter derive simple versions of the basic model equations to clarify the concepts behind various theories. Then, the chapter presents extended versions especially for the analyses of wave-current interaction, wave dissipation, and waves on a permeable bed. The last section describes the applicable ranges of the model equations.

Basic Equations and Boundary Conditions This section summarizes three-dimensional basic equations and boundary conditions for wave motions in a water body and permeable layer. The wave equations derived in the subsequent sections have a common characteristic in the sense that the basic equations are integrated in the vertical direction to yield horizontally two-dimensional (2D) equations. This reduction of the dimension simplifies the theory and makes it much easier to calculate wave transformation numerically.

Basic Equations and Boundary Conditions for Waves on a Fixed Bed Basic Equations in Terms of Particle Velocity. In describing water waves, the viscosity and compressibility of water can usually be neglected. Then, the basic equations are the continuity equation for an incompressible fluid and the Euler equations of motion [68]. To integrate the equations in the vertical direction, symbols are defined to distinguish the vertical direction from the horizontal directions: x 3 = (x, z) = (x, y, z)

(1)

u3 = (u, w) = (u, v, w)

(2)

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V 3 -- V,

where

-

(3)

0x'Oy'/)z

x, y = horizontal coordinates z = vertical coordinate u, v, w = components of water particle velocity in the x, y, and z directions, respectively

Vectors with subscript 3 denote three-dimensional quantities, whereas those without subscript are for horizontally 2D quantities. With the above notations, the continuity equation, and the momentum equations in the horizontal and vertical directions (in 3D) are written as follows: ~W

Vu + ~z = 0

~u

~t /)w ~t

(4)

~u + (uV) u + w . . . .

1

~z

+ (uV) w + w

/)w ~

=

p

V p

1 0p

- g

- - ~

p /)z

(5)

(6)

where p = pressure p = density of water g = gravitational acceleration. B a s i c E q u a t i o n s i n T e r m s o f V e l o c i t y P o t e n t i a l . Because the motion of an inviscid fluid starting from rest remains irrotational, the wave motion can be regarded as irrotational and thus the velocity potential ~3 exists:

U 3 = V3(~3

(7)

The continuity equation (4) is rewritten as Equation 8 and the momentum equations (5 and 6) can be integrated to yield the Bernoulli equation (9): ~2~3 - ' 0 V2~3 = V2@3 + ~Z 2

(8)

~9~3 1 )2 P + (V3~)3 + gz + -- = 0 /)t -2 p

(9)

The Laplace equation (8) and Bernoulli equation (9) are simultaneous partial differential equations in terms of ~3 and p, and equivalent to the continuity and momentum equations (4 to 6). Because the pressure p usually does not appear in

Equations for Numerical Modeling of Wave Transformation

105

the boundary conditions, the Laplace equation is first solved to obtain ~3 for a given set of boundary conditions, and then the Bernoulli equation is used to determine p. This means that any velocity field expressed by a velocity potential can be generated by a certain pressure distribution and without any shear stress. B o u n d a r y Conditions. The boundary conditions for water waves on a fixed bed consist of the dynamic and kinematic free surface boundary conditions (10 and 11), and the kinematic bottom boundary condition (12):

2

]

1 2 +W ) + g ~ = 0 p = - p [~(~3 --~---+-~(u

w=~

+ (uV)~j

w+(uV)h=0

(z=~)

(z = ~)

(10)

(11)

(12)

(z=-h)

where ~ = water surface elevation h = still water depth The dynamic boundary condition implies the constant pressure on the surface. The kinematic boundary conditions require that any water particle on a boundary remains on that boundary. One may obtain the equations for the latter condition by taking the total derivatives (D/Dt = ~//)t + (uV) + (w~/~z)) of z = and z + h - 0.

Non-dimensionalization. It is often important to know the order of magnitude of each term in the basic equations and boundary conditions. For this purpose, the dimensional quantities are non-dimensionalized as follows: X = LlX*,

z = hlZ*,

t=(L1/g~~)t*

- -

w

p = pghlEP*

~ = hieS*,

h = hlh*,

[i = h l / L 1

=

(13)

where L1, h 1 = representative length scales in the horizontal and vertical directions e, ~5- orders of the non-dimensional wave amplitude and relative water depth, respectively

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Then, Equations 4 to 6, 8, 10 to 12 are rewritten as ~)w* V'u* + ~ =0 /)z*

/)t*

+e

(14)

[

( u * V * ) u * + w*

/)z*J

= V*

P

+ e (u* V*) w* + w* /)w* /)z*

~t*

(15)

*

(16)

= - 1 -/)z-----~

1 t92~;

(17)

V2(~ 4" ~2 ~)Z,2 = 0 E ()(~ + - - ( U .2 + ~2W'2) + Z* + p * = O

(z* = e~*)

(18)

(z*=e~*)

(19)

w, = ~;* + E [(u, V'K*]

(z, = e;,)

(a0)

w* + (u* V*)h* = 0

(z* = - h*)

(21)

/)t*

2

[

P*= - Ot~; ~ - F + ~ (eu * 2 + 5 2 w * 2 ) +

;,

1

=0

~t*

E n e r g y C o n s e r v a t i o n E q u a t i o n . Because the mechanical energy is conserved for a flow of an inviscid fluid, the time rate of change of total energy in a certain volume V fixed in space is equal to the sum of the net energy inflow into it and the work done by pressure through its surface S:

0"-t"

(V3~)3

]

+ pgz dV = -

0

-~ (V3~)3

]

-/- pgz + p (u3) n dS

(22)

where the subscript n denotes the component of a vector in the direction normal to the surface. Because V is fixed in space and z is independent of t, the left side becomes

L.S. = p

~-

)

~3(~3 (V3(~3)dV

(23)

By rewriting the left side using the Bernoulli equation (9) and then converting the surface integral to a volume integral using the Gauss theorem, the right side becomes

Equations for Numerical Modeling of Wave Transformation

107

R.S. = D~s"-'~" ~r (V3(~3)nas

= PfvV3 (~'3 k ~9t V3,3) dV

(24)

Then the following energy conservation equation is obtained: ;v c3~3 V2~3dV = 0

(25)

-57

To derive this equation, the momentum equations have been used through use of the Bernoulli equation, but not the continuity equation. This implies that, even if the continuity equation may not be satisfied, the total mechanical energy is conserved only by satisfying Equation 25. This is the case for the MSE.

Basic Equations and Boundary Conditions for Waves on a Permeable Bed For waves on a permeable bed as depicted in Figure 1, the motion is to be analyzed both in the water column and in the permeable layer. For the water column, the basic equation and the free surface boundary conditions are the same as for the waves on a fixed bed. The basic equations and the boundary conditions on the interface and bottom are described in the following section [55].

Basic Equations in Terms of Particle Velocity. First, by denoting the three components of the seepage velocity as

Up3 "-(Up, Wp)"-(Up, Vp, Wp)

(26)

h

u

Figure 1. Definition sketch for waves on a permeable bed.

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the continuity equation is expressed as follows: ~)Wp

(27)

Yap + - ~ Z = 0

When an obstacle is fixed in an accelerating fluid, it exerts a force to the fluid. This can be expressed by introducing the virtual mass coefficient C m. Then, by adding the mass of water in the permeable layer, the apparent total mass C r per unit volume is obtained as: (28)

C r = Ep + (1 - Ep)C M where Ep -- porosity By using C r, the momentum equations can be derived as

VOUp ~Up] Cr L/)t + (UpV)Up + Wp ()Z J [~Wp

Cr L ~)t

1 Epl.) --p Vpp - -~p Up

~Wpl = + (upV)wp + Wp ~Z

J -g

1 ~pp p ~z

E2pCf [Up3[Up

Ep_~ Kp Wp

E2pCf ~p

(29)

lu.I Wp (30)

where the last two terms on the fight side of these equations represent the linear and nonlinear resistance forces, respectively and K = intrinsic permeability cPf= turbulent friction coefficient = kinematic viscosity The values of K_ and Cf are given in [87]. Investigation of relative magnitude of each term has s~own that the nonlinear resistance force cannot be neglected in usual situations, so that equivalent linear resistance should be substituted even for small amplitude waves [22].

Basic Equations in Terms of Velocity Potential. For a small amplitude wave motion in a permeable layer, nonlinear terms can be neglected and the seepage velocity can be expressed by a velocity potential:

Up3 = V3(~p3

(31)

and the continuity and momentum equations are rewritten as

V2~p3 = V2(~p3 + ~2(~p3 ~z 2 --0

(32)

Equations for Numerical Modeling of Wave Transformation

~)p3 Pp Cr Ot + gz + ~ + fpG~p3 --0

109

(33)

P

where fp = coefficient for the linearized resistance force

Boundary Conditions. The boundary conditions at the interface are the continuity of the pressure and the flow rate: p = pp

(z = - h)

(34)

w + (uV) h = Ep [Wp w (UpV) h]

(z = - h)

(35)

An impermeable bed is assumed at the bottom of the permeable layer. This yields the boundary condition:

Wp + (UpV)h t = 0

(z = - h t )

(36)

where h t = h + hp = total water depth.

Mild-Slope Equation Two assumptions that enable a simple mathematical formulation of wave transformation on a sloping bottom are small amplitude and mild-slope assumptions. These lead to the mild-slope equation (MSE), which was first derived by Berkhoff [4] and then expressed in various ways (for example, [3, 33, 61, 86]). The MSE includes effects of both refraction and diffraction, and thus has widely been used in engineering practices. In the following, the MSE is derived from the energy conservation equation, which may give a clearer physical meaning. The procedure is somewhat similar to [3].

D e r i v a t i o n of M i l d - S l o p e E q u a t i o n The basic equation and boundary conditions for small amplitude waves on a fixed sloping bottom are obtained from Equations 8, 10 to 12 as V2~3 + 32~3 = 0 ~9z2 = __1 ~3~3 g Ot

(37)

(z = 0)

(38)

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1 ~21~3 g Ot2

(Z = O)

(39)

+ (V~3) (Vh) = 0

(z = - h)

(40)

3~___L3= OZ

3z

where Equations 10 and 11 are linearized and combined to yield Equations 38 and 39. Upon the assumption of a mildly sloping bottom, the second term in Equation 40 can be neglected in the zero-order of the slope (i.e., for a horizontal bottom). The velocity potential can be expressed as follows: ~3 = Z ( z ) , ( x , t)

(41)

where Z (z) o~ cosh k(h + z)

(42)

Equation 39 requires the following dispersion relation: 0.2 = gk tanh kh

(43)

and Equation 38 gives the water surface elevation in terms of the velocity potential at the still water level. This solution agrees with the small amplitude wave theory on a horizontal bottom. The vertical distribution function Z satisfies 32Z ~gz2

- k2Z = 0

()Z

0 .2 =~ Z 3z g

3Z 3z

=0

(44)

(z = 0)

(45)

(z = - h )

(46)

In the first order of the bottom slope (i.e., in the equations with terms proportional to the bottom slope), only the vertically integrated energy conservation equation is considered instead of satisfying the continuity and momentum equations at each elevation. The energy conservation equation is obtained from Equation 25 for a water column that has a projection area of dx x dy and a height from the bottom to the surface:

/

V21~3 q- ~Z 2

(47)

Equations for Numerical Modeling of Wave Transformation

111

where the velocity potential in the form of Equation 41 is considered and the upper limit of the integral is changed from the water surface to the still water level upon the small amplitude assumption. The vertical distribution function Z for a horizontal bed is regarded to give a good approximation even to that for a sloping bottom, but its derivatives may include a substantial error because of its sensitivity to the distribution function. Therefore, in the following derivation, Equation 41 is used for integration in the vertical direction but not in differentiated forms. By using the following identities:

ZV2~3 -- V(Z2V{~)+ Z(V2Z)O

(48)

Z2V~-----ZV(~3 Z(VZ)~

(49)

-

-

the first term in Equation 47 is rewritten as

0 V2~3dz _ V =V

2V@dz-- (Vh) (Z2Vt~)l_h +

(V2Z) @dz

2V,dz - (Vh) (ZV,3 - Z (VZ),)[_ h +

(vEZ),dz

(50)

The second term is reexpressed by using integration by parts twice and substituting Equations 39, 40, and 44 to 46 as qZ 02(~3 dz

0Z2

=

= f._q O2Z (~3dz + ( ~Z 3 h~ ()Z

~ ~ k 2Z2t~dzh

Z~ 02r162 g Ot2 g

OZ (~3/1~ -- ( ~Z) 3 ()Z t)Z

~Z I~3/! ()Z -h

+Z(Vt~ 3)(Vh)l_h

(51)

Then, substitution of Equations 50 and 51 into Equation 47 yields

V

E(Oz) i (SOz) 2dz Vt~ +k 2

2dz t~-

= - (~:?V2Zdz) t~ - (Vh) (VZ) Zl_ht~

Z 02 ~2~ g

g

/)t 2 (52)

where the terms on the fight side are of the second order in the bottom slope and therefore will be neglected hereafter. These terms have recently been included in a modified version of the modified slope equation MSE [8]. If the proportionality constant in Equation 42 is taken to give

112

z =

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cosh k(h + z) cosh kh

(53)

Equation 52 becomes V(CCgVr + (k2CCg - 0.2)~ _ ~

=0

(54)

which is the time-dependent form of the MSE [86]. Then, Equation 38 gives the water surface elevation:

1 /9r

. . . .

(55)

g ~t On assuming a sinusoidal oscillation: ~(x, t)= ~(x)e -i~

(56)

(x, t) = ~ (x)e -i~t

(57)

the MSE can be obtained from Equation 54: V(CCgVt~) + k 2 CCg~) = 0

(58)

From Equation 55, the complex amplitude of the water surface elevation is expressed as ~= i__~~) g

(59)

This relationship is independent of the water depth for a given wave frequency. In spite of the assumption that results in the vertical distribution of the velocity potential as given by Equation 41, the MSE can be applied up to a ~ slope [6], which covers most practical situations (Figure 2). Recently, a finite series expression of the velocity potential has been adopted to improve the accuracy of the MSE [67].

Alternative Forms of Mild-Slope Equation Alternative forms of the MSE can be derived by taking different proportionality constant in Equation 41. A useful example can be obtained by taking

Equations for Numerical M o d e l i n g of Wave Transformation

4-

113

§

O.2 -

4.

0.1

i q.

g tj

.~

O.O&

-

L

II.02

0.01

I

0.1

I

I

i

0.2

0./.

I

1

2

/.

,

2

1

0~

~2

0.1 t~loJ

Ws

Figure 2. Reflection coefficient as a function of b o t t o m inclination. Lengths are nor malized by d e e p w a t e r wave number. Curve: refraction-diffraction model. Crosses: threedimensional model [5].

(I)3 = Z' (z)(~' (x, t) Z

1

Z'-- ~fCCg "- 5/CCg

(60) cosh k(h + z) cosh kh

(61)

Then, the time-dependent form of the MSE and the expression of the water surface elevation become

V2t~'+(k 2

1

;=

0'2 )(~ ' Cfg ~'

g~/CCg bt

1 ~)2t~' CCg t)t 2 - 0

(62) (63)

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and the MSE and the complex amplitude of the water surface elevation become V2@' + k2@'= 0

(64)

(65)

gafCi2g where r (x, t) = t~' (x)e -izt

(66)

The Helmholtz equation (64) was obtained in [78] by transforming the dependent variable in Equation 58. An interesting feature appears in the expression of the energy flux in terms of ~'. The energy flux due to waves is obtained from the work done by the dynamic pressure to the leading (second) order of the wave amplitude: F=

dUdZ

(67)

where Pd = the dynamic pressure induced by waves Using the linearized Bernoulli equation and the definition of the velocity potential,

[

Pd = Re - p /)t .]

pt3

u = Re [V~3] = Z' Re

z' Re [i~' e -iot ]

[V~' e -i~ ]

(68)

(69)

Substitution of Equations 68 and 69 into Equation 67 yields F = po Re [i~' e -iot ] Re [V@' e -izt ]

(70)

which shows that the proportionality constant in the expression of energy flux is pt~ and thus independent of the water depth. This implies that if ~' and its normal derivative are continuous at the boundary, the energy flux is also continuous. The boundary condition used in [20] can also be derived from this result. Because the MSE is an elliptic equation, its numerical solution requires fairly long computational time, and thus some efficient numerical solution techniques have been investigated [51-54, 63, 75, 76, 96]. A technique that solves the wave ray and amplitude separately as in the case of refraction problem has also been developed [ 16, 98].

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115

Physical Interpretation of Mild-Slope Equation To give an idea of refraction and diffraction effects included in the MSE, the complex amplitude of the velocity potential is expressed in terms of the amplitude a and phase angle 0 [5, 16]: ~-ae

(71)

i0

On substituting this expression into the mild-slope equation (58), the following equations are obtained from the real and imaginary parts, respectively: ( V 0 ) 2 --

k2 +

V(CCgVa)

(72)

CCga V (a 2 C C g V 0 ) = 0

(73)

By invoking V0 = k, the term in the parentheses in Equation 73 becomes t~a2Cg, which is essentially the energy flux. Therefore, Equation 73 represents the energy conservation. Equation 72 is essentially the same as the eikonal equation to determine the wave direction in refraction problems, but it includes an extra term on the right side. If, for example, there is a local maximum of a in a constant water depth, the second term in Equation 72 takes a negative value, which makes the magnitude IV01 of the real wave number smaller than that determined from the dispersion relation and thus is equivalent to a larger water depth at the location. This results in increase of the distance between two wave rays and decrease of the energy density and thus wave amplitude. This phenomenon is interpreted as the dispersion of wave energy due to diffraction. For an alternative form of the MSE, substitution of the following to Equation 64: ~' = a' e i0'

(74)

results in ( V 0 ' ) 2 --

V2a ' k2 + ~

V (a'2V0 ') = 0

ap

(75)

(76)

This equation implies that the transport velocity of a p2 is equal to the wave number vector V0'. Thus, if only the wave number vector but not necessarily the group velocity is calculated accurately, the distribution of a' can accurately be predicted.

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Mild-Slope Equation on a Slowly Varying Current A mild-slope equation in the presence of a slowly varying spatial current has been derived [33]. The waves and current motions are separated as u=U+u

w,

w=W+w

(77)

w

where capital letters and suffix w denote the current- and wave-induced velocities, respectively, and Ww -" ~1~3 OZ

Ilw = Vl~3'

(78)

From the continuity equation and bottom boundary condition for current, W = - z (VU)

(79)

The assumptions on the magnitudes of the velocities are O (U) = 1 and O (Uw) = O (w w) = O (W) = e. Then, the basic equation and boundary conditions are written as V2~3 + ()2~3 = 0 ~Z 2 '

Ot

+ (UV)t~3 d- g~ = 0

0,3

O~

= -- + (uvK bz bt

0,3 ~z

(80)

(z = 0)

+ ~(vu)

+ (V@3) (Vh) = 0

(z = o)

(z = - h)

(81)

(82)

(83)

Equations 81 and 82 are rewritten as = _ 1 D~___ Z g Dt

/)z

=

where

Dt

(z = 0)

+ ~(vu)

(84)

(z = 0)

D ~ = m + (UV) Dt 3t

(85)

(86)

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117

With these definitions, one finds the following mild-slope equation for waves on a slowly varying current [33]:

V(CCgVr + (k2CCg - o 2)~ - (VU) De Dt

D2r - 0 Dt 2 -

(87)

where the intrinsic frequency o satisfies the dispersion relation (43) and obtained from the local angular frequency to as = to - kU

(88)

For a monochromatic and uni-directional wave field, the previous equation can be solved [49]. However, if a local wave field consists of more than one component due to more than one wave path from offshore region, the intrinsic frequency cannot be determined uniquely. This causes a problem in solving Equation 87. Various versions of Equation 87 for wave-current interaction have been reviewed and examined in [50].

Mild-Slope Equation with Energy Dissipation An energy dissipation term may be introduced into the MSE as follows [ 13]: V(CCgV$) + (k2CCg + io fD)$ = 0

(89)

By substituting Equation 71 into this equation, the following equation can be obtained from the imaginary part:

V(a2CCg V0) = - a2 fD

(90)

from which fD is understood as an energy dissipation coefficient. Various formulas of fD are given for a porous bottom, viscous mud bottom, laminar bottom boundary layer, densely packed surface film, and others in [13]. An empirical formula is often used for wave breaking [25]. The breaking point is first determined in [93]: Y =Yb

(91)

where y denotes the ratio between the water particle velocity u e at the still water level and wave celerity C: Uc ), = --~ (92) The value of y at the breaking point is given as Tb = 0.53 - 0.3 exp[-3~/h/L o ] + 5 tan 3/2 ~ exp [-45~/h/L o - 1)2 ]

(93)

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where tan 13 = bottom slope L ~ = deep-water wavelength Then, fD is given so that it yields constant wave height to water depth ratio 7s on the uniformly sloping beach and vanishes when the ratio becomes smaller than ~'r because of increase in water depth or others:

fD='~

(94)

y s 2 7~ tanl3

~/s = 0.4(0.57 + 0.53 tan 13)

(95)

~/r =0.135

(96)

Mild-Slope Equation on a Permeable Bed The MSE on a permeable bed was derived in a manner similar to the MSE on a fixed bed [79]. The basic equation in the permeable layer is the Laplace equation:

~2(~p3-- 0

(97)

V2@p3 + ~Z2 ....

The boundary conditions at the interface are Equations 34 and 35, which can be rewritten in terms of the velocity potential as

~ 3 = Cr ~---~

~3 Oz

~p3 + fpl3@p3 ~t "

+ (Vh) (V{~3) = Ep

(Z =

- h)

I~P3z 3 + (Vh) (V~p 3)]

(98)

(z = - h)

(99)

These interface boundary conditions do not include the continuity of the tangential velocity, which requires the introduction of boundary layer. However, the energy dissipation in the boundary layer is usually small compared to that in the permeable layer, so that it can be neglected [83]. The boundary condition (36) at the bottom of the permeable layer is also rewritten as

~r~z + (Vht) (V~)p3 ) = 0

(Z = -- h t)

(100)

By using the separation of variable technique, the analytical solution is first derived for a horizontal bottom and interface:

Equations for Numedcal Modeling of Wave Transformation

(~3 =

F (z)0(x, t)

119

(101)

~p3 = Fp (z) ~ (x, t)

(102)

where F(z)=

(103)

Ep

sinh (khp)exp [k(h + z)]- ~Sp cosh [k(h + z)] ep sinh (khp) exp (kh) -/Sp cosh (kh)

cosh [k(h t + z)] Fp(Z) = ep sinh (khp) exp (kh) - ~p cosh (kh)

(104)

~ip = ep sinh (khp) - (C r - ifp) cosh(khp)

(105)

0 .2 --

gk

I~p exp (kh) sinh (khp) - (Sp sinh (kh) ep exp (kh) s i n h ( k h p ) - ~p cosh (kh)

(106)

Next, by substituting Equations 101 and 102 into the following integral equation: f qhFV3t~3dz 2 + ~_-h s ht

(107)

the following MSE on a permeable bed is obtained: V (otV~) + k 2 tx~ = 0

(108)

where o~= tx1 + Ep(C r - ifp)O~2 ~1 = [32h { 2~~ [1 - exp(-2kh)] - 2~-~~[1 - exp (2kh)]- 2~2~3 }

1 [ sinh(2khp)] ~2 = - ~ 2 hp 1+ 2khp ~l = [ep exp(kh) sinh (khp) -/Sp cos (kh)] -1 ep exp(kh) sinh (khp) - (~Sp/ 2) exp kh [~3 =--(~p / 2) exp(-kh)

~ 2 -"

(109)

Figure 3 compares distribution of the root mean square (rms) water surface fluctuation calculated by using Equation 109 and measured data from a laboratory wave flume study. Instead of the wave height, the rms value, which represents the wave energy, is used for comparison because transmitted waves are not sinu-

120

Offshore Engineering

o

,

:

Dm

"~ "":...... ; "" "' 9.'..'7.' :.'" ": ; " ; . X-:"

'

5.8

,tl.l.,NrtB.'~.'%

9

L

'

3.0

o-~

Z.

E

X~

9

i

,..

9

CALCULAT

i

'9

"

1ON

EXI)I~Is IMENT

9

5

....

U

2.0

'

i

"

9

9

II i

-

4.

66

T

--

1.

8 2

pm el

D O

--

O.

60

,:m

Doo

"

87.

60

cm

B

--

s

O0

cm

@

1.0

0.8

X,~ .,A,

O.O 0

9

t

2

4

3

6

6

7

a

9

xo

x (m)

-

CALCULAT

9

A

E

0

EXPER

_

0

0

,

,,

1

2

-

t.

--

1.81

s

D s

--

8.

O0

cm

Doo

"-

O0

cm

B

--

OO

cm

B I

r

39. 238

X,h

...... s

47

I MENT

2

X,

!! i T

I ON

4

I

,

,

8

6

7

I. 8

9

10

X (m) Figure 3. Distribution of root mean square values of water surface fluctuation due to a submerged permeable breakwater [79].

Equations for Numerical Modeling of Wave Transformation

121

soidal due to nonlinear effect. A good agreement implies that both transmission and reflection coefficients are accurately calculated.

Time-Dependent Mild-Slope Equations Time-dependent forms of the MSE have been proposed for improvement of numerical calculation [10, 52, 63, 71, 92] or for application to random wave analysis [26]. The following presents time-dependent mild-slope equations for monochromatic waves and then introduces those for random waves.

Derivation of Time-Dependent Mild-Slope Equations For a sinusoidal oscillation expressed by Equation 56, the mild-slope equation (58) can alternatively be written in a hyperbolic form as

a2~ V(CCgVtD- n 0 - ~ = 0

where n -

cg 1( C

=-

1+ ~ 2 sinh 2kh

(110)

/

(111)

A hyperbolic equation can be split into two simultaneous first-order partial differential equations. By considering physical meanings, the following two quantities that correspond to the water surface elevation and flow rate per unit width are introduced:

1 ~)~ _ _

.

.

.

.

g at C2 Q=

g

V~

(112)

Then, by substituting the previous definitions into Equation 110, one can obtain an equation equivalent to the continuity equation. One can also obtain an equation similar to the momentum equation by cross differentiation of Equation(s) 112. These are written as ~9r 1 ~ +-V(nQ) = 0

/)t

n

aQ + c2vr = o at

(113)

122

OffshoreEngineering

These two equations constitute a set of simultaneous partial differential equations in terms of ~ and Q, and equivalent to the MSE. These are called timedependent mild-slope equations. The definitions of the two dependent variables are the same as Nishimura et al. [71], but Copeland [10] used a different definition for Q, which leads to a physical meaning different from the flow rate by the factor of n = Cg/C.

Alternative Forms of Time-Dependent Mild-Slope Equations Different forms of the MSE and different definitions of two dependent variables yield slightly different time-dependent mild-slope equations. By substituting the following definition of the vertical distribution function: ~)3 "- z u (Z) ~)u (X, t)

(114)

Z"

(115)

Z =~

= ~

1 cosh k(h + z) cosh kh

into Equation 52, the MSE becomes V(C2V(~ it) -- ~)t2, = 0

(116)

Then, by defining

1 /)~" ~= R=

g~-ff Ot C2

Vr

(117)

another set of time-dependent mild-slope equations is obtained:

VR =0 Ot OR + cEv(~/n~)~ =0 3t

~ +

~

v

qllp~

(118)

In [29, 90], because the definition of the water surface elevation is different by the factor of x/if, a correction factor becomes necessary after solving the equations. In Watanabe and Maruyama [92], the vertical distribution function is defined as (~3 -- z u ' (Z)~ u' (X, t)

(119)

Equations for Numerical Modeling of Wave Transformation

Z n

1 cosh k(h + z) n cosh kh

123

(120)

and then the MSE has the following form:

2 V(~") V-n

1 ~)2(~H, n ()t2

=0

(121)

By defining 1 ~" gn /)t C2 Q= V~" gn

(122)

the corresponding time-dependent mild-slope equations have the following form: --+VQ=0 Ot OQ c 2 --+ V(n~) = 0 Ot n

(123)

When breaking transformation is analyzed, an energy dissipation term is added to the second equation: o~Q --+ /)t

C2

n

V(n~) + foQ = 0

(124)

The numerical model based on the first equation of Equation(s) 123, and Equation 124 has been tested for wave transformation due to refraction, diffraction, and breaking to predict wave-induced nearshore current and sediment transport and resulting bottom topography change. Figure 4 shows the comparison between calculated and measured breaking lines and wave height distributions. T i m e - D e p e n d e n t Mild-Slope Equations for R a n d o m Waves A sinusoidal oscillation is assumed in the time-dependent mild-slope equations previously derived. Another time-dependent mild-slope equation is Equation 54, which has been applied to the propagation of wave groups [41 ]. Because the coefficients in the equation are evaluated at a certain frequency and their

124

Offshore

Engineering

4.0-=

Incident waves

3.0-! E

j./i v

9

{

g 2.0-

O m

Breaking point Computed ( ~o/C' 0

[ 0.35)

=

Measured (Om ~ y ~ 4m)

9 Measured (4m <.yr 8m)

1

0.t

off

5.0

z

4.0

"~

" ir

Y0.

w

!

9

0.0

9

9

,

1.o >-o0I ,'v 0.0

~ " r * ' ~ ,

1.0

3.01

M e a s u r e d (Ore ~ ; u ~ 4 m )

A

Measured (4m < tr~ 8m )

,

"

~ --

,

9i

I

2.0

4.0 "

,

,

l

"'

"'" ' ~ " t ;

3.0

9

4.0

4.0

..

Computed 0

' 3.0

,

v

I

,

y=4.

2.0

a

. . . . . . .9.

,

2.0

,

~:

4.0

e

1.0

0.0

".

,

"~

~

9

flu 5.0

~

,

.'

3.0 2.0 Distance onshore x (m)

1.o

0.0

y=2m _ _

..~...

......

"",

" "

"~ 1.0 0.0

0.4)

Figure

9

i

1.0

4. Location

,

L

~

.

~

2.0 3.0 Distance onshore x (m) of breaker

line and

cross-shore

,

,

4.0

distribution

of wave

height

[92].

changes due to the deviation of the frequency are correct to the first order, the model is valid only for a small range of frequency. Time-dependent mild-slope equations, which are applicable for wider wave spectra, are derived as follows [26]. By considering that random waves consist of an infinite number of component waves with different frequencies, the sinusoidal oscillation with a representative angular frequency ~ is removed from ~':

Equations for Numerical Modeling of Wave Transformation

t~' = ~' (X, t)e -iSt

125

(125)

Hence, for an arbitrary angular frequency o = i5 + (Ao), ~' is written as ~' (x, t) = ~(x)e -i(A~

(126)

and thus

/)r ~)t

- i(Ao)r

-

-

(A(~)

2

(127)

t)t 2 -

In an alternative form (64) of the MSE, k 2 is a function of the frequency and approximated by a Pad6 approximant: k2 = b 0 + b l ( A O ) + b 2(AO) 2

(128)

1 - a 1 (Ao)

where the coefficients are determined according to the spectrum range of the irregular waves. Substitution of this approximation into Equation 64 with use of Equations 127 yields

V2~)' - ial V2

+ bor + ib 1 - ~ - b 2 0t 2 = 0

(129)

All the coefficients in this equation are independent of the frequency. Therefore, it is used to calculate ~' composed of infinite number of waves with different frequencies, and thus enables direct calculation of random wave transformation. It is noted that, as seen from Equation 76, the transport velocity of a '2 in Equation 76 is the same as the wave number vector. Thus, as long as the Pad6 approximant to k 2 is accurate, the shoaling coefficient for a '2 and then the water surface fluctuation by Equation 63 can accurately be predicted. Figure 5 shows a comparison of water surface fluctuation of shoaling random waves.

Parabolic Equation Numerical calculation of the MSE requires a fairly long computational time due to the elliptic characteristic of the equation. Since Radder [78], various parabolic approximations have been proposed to save computer resources. This section presents alternative forms of the parabolic approximation.

126

Offshore Engineering

~, AAA (~n~)~ V V~V A . A. . . A ,A,.~ ~ v,A~1 2 ~ P,,

,

,,

An

-s

~,A

AAII vvv

I

t(s)

9111eas.

"I ",,. (cm)O[ ,5 O ' :j ' '''

A/

A -

i,/': ~'~;V" V' V~'10V V V:"~v2~5 V'~

]

t(s) Figure 5. Time history of water surface fluctuation of random waves in shoaling water; top figure: measured incident wave history, bottom figure: calculated and measured histories at 4m shoreward [26].

Derivation of Parabolic Equation Parabolic equation can be obtained by eliminating the second derivative in one (x) direction in the elliptic type MSE. This is based on the assumption that the change of amplitude in the direction (wave ray direction) is small compared to that in the direction perpendicular to it (wave front direction). An alternative form (64) of the MSE can be rewritten as

@2r @2r c3x-T + ~ - ~ + k2@'= 0

(130)

On considering waves that propagate approximately in the x-direction, the phase change in the direction can almost be removed by

(~'= ~' e iJkdxf

(131)

in which the change of ~ in the x-direction is assumed to be small. Then, in the expanded form of the first term on the left side of Equation 130:

O2C)2~,X-_ k( ~)2 ~)X2 I]/' + 2ik "~XO~/+' i 7oxOk - Xl/,-

kEy '

/ eil kdx

(132)

the first term is neglected in the parabolic approximation. Thus Equation 130 is approximated by a parabolic equation:

Equations for Numerical Modeling of Wave Transformation

2ik -~x + ~)y2 + i ~x

=0

127

(133)

which, for constant k, reduces to 2ik ~9~' +

t)X

~)y2

=0

(134)

This equation is the simplest form of parabolic equation.

Alternative Forms of Parabolic Equation In the previous derivation, it is not always convenient nor possible to use k. Thus, the modified wave number K, which is usually obtained from a constant depth or a uniformly sloping bottom, is used to reduce the spatial variation of the wave phase:

1~'= tIa'ei~Kdx

(135)

Then, on substituting the following relationship:

V, = qj,ei(~ Kdx- ~kdx)

(136)

into Equation 133, the following parabolic equation can be obtained:

t)tI~' t)2~IJ' [ ~)k

2ik /)--~-+ ~)y2 + i~x + 2 k ( k - K )

]

~g'=0

(137)

Another slightly different parabolic equation often appearing in literature is obtained by substituting Equation 135 into Equation 130 and neglecting /92W'/3x2 [91 ]: 2iK - ~ x + ~)y2 + i ~

+ (k 2

)

=0

(138)

A parabolic equation derived from the original mild-slope equation (58) is 2 i k ~~)~ -~

1

(9 ( CC ~ /

CCg Oy

g

+ [, i

CCg

~)(kCCg) + 2 k ( k - K ) 1 ~ = 0

~x

(139)

128

Offshore Engineering

where t~ = tlJe iIK dx

(140)

The difference in the basic parabolic equations (133, 137-139) previously described is also found in extended equations for weakly nonlinear waves or non-Cartesian coordinates.

Parabolic Equation for Large Wave Angle When the wave angle relative to the x-axis is large, the assumption to derive a parabolic equation should be modified. One choice is to consider the large angle of propagation [34]. For progressive waves expressed by t~' = ae

i(kxx+kyy)

(141)

the x-component k x of the wave number vector should satisfy

(142)

which is approximated by a Pad6 approximant:

a0+a kx -

(143) l+b 1

where a 0 = 0.998213736 a 1 = --0.854229482 b 1 = -0.383283081 These values assure the error to Equation 142 is within 0.2%. The corresponding parabolic equation is obtained as 2 i k / ) ~ ' + 2k 2 V' /)2~, ~x (a 0 - 1) + 2 (b 1 - a l) /) y2

O31lt' 2ibl k /)xOy 2

(144)

Figure 6 shows the validity of the previous equation for obliquely incident waves with an angle of 45 ~. Another method has been proposed for wide angle on the basis of Fourier transformation [12, 14, 89].

Equations for Numerical Modeling of Wave Transformation

//

~---~

ii!1// l/ / \\~\, \

\.\

._, [ ,

129

\\\\\ \1 //

/ ~/ I

Figure 6. Amplitude contours as calculated by large-angle parabolic equation model for incident wave angles of 0 ~ (solid line) and 45 ~ (dash line) [34].

Parabolic Equation in Non-Cartesian Coordinates Another way to consider the variation of wave direction is to use non-Cartesian coordinates, which almost trace the wave propagation directions.

Parabolic Equation in Orthogonal Curvilinear Coordinates. Orthogonal coordinates that trace refraction patterns on a uniformly sloping bottom are used in [6 I, 91 ]. A more general expression is given in the following [24]. As shown in Figure 7, the coordinates ~* and rl* are defined so that rl* = constant and ~* = constant, respectively, coincide with the directions of wave rays and fronts of refracted waves due to a distribution of wave number K. Because the units of the coordinates are taken as phase in radians, the distances d~ and drl corresponding to d~* and dYl* are d~ = h~d~*,

dq = hndyl *

(145)

130

OffshoreEngineering

(wave front) ~

+

hnd~? ~

hcdE.*

+

(t~~ + d~',r/* + d r / ' ) (wave ray) Figure 7. Orthogonal curvilinear coordinates.

where the scale (conversion) factors h~ and h n are (146)

hg = hn = 1/K Then, substitution of the equation: r = ~iJei~Kh~d~*

(147)

into the MSE:

1 /) CCg h~h n /)~ * h~ ~)~*

t) CCg + k2CCgr = 0 h~h rl 0rl* hrl 0rl*

(148)

and neglecting the second derivative with respect to ~* yields 2 ik +

1 3~ h~ ~ * [

1

1

3 (

h~ 3W

~ - ~ ~ ~ CC ~ ~ CCgh~ h n ~ * ( g h n 3xl*

)

i 1 ~ ] (kCCghn) + 2k(k - K) ~ = 0 CCghn h~, ~9~*

(149)

This equation is modified for large angle and applied to wave refraction, diffraction, and breaking of random waves [25]. Figure 8 shows an example of application to a field in which the spatial distribution of the significant wave height and mean direction are compared between calculation and measurement.

Equations for Numerical Modeling of Wave Transformation

131

Ore< Hi/3=

7.

10

6 ..

90 100

500

im I

45

lO00m

Figure 8. Comparison of the spatial distribution of the significant wave height (length of arrows) and mean direction (direction of arrows) between calculation by parabolic model (dash lines) and measurement (solid lines) [27].

Parabolic Equation in Non-orthogonal Coordinates. Non-orthogonal coordinates were introduced in [36, 57]. As seen from an example below, some extra terms are introduced due to non-orthogonality of the coordinates ~ and 11.

2iK---~-+ 2iK

+~--~J"~-+

"~x

an2 +2ax

+ (k 2 - K 2 ) tIJ'

(150)

Figure 9 compares wave height distributions calculated by using orthogonal and non-orthogonal coordinates with measurement. W e a k l y - N o n l i n e a r Parabolic E q u a t i o n

A parabolic equation for diffraction of weakly nonlinear waves of Stokes type was derived for constant water depth [99]. For variable water depth, [38] and [58] derived the following nonlinear equation of Schr6dinger type: bY 2ik ~ + ~ 1 0 bx CCg Oy _k 2 C D~ =0 Cg

/

CCg

-I- ,

i

CCg

+ 2k (k - K)

]

Ox (151)

132

Offshore Engineering

o s .% . i

i-o, i

||

I I I I !

I I I I !

a ,

, I B, ,I

I

3 I

-

9

P

4 . 9.0

!

'\ - ] I. . . . . . . . i

I Ii

4.1.8

-~ B

!

eL_ o

;;

~'tl--'-9.0 i, O. 6 - - ' ~

1.5

I

i'

l

3.6 ~

I~o.=4

I "~"

~

.~

I! :~ I:

l

.-//r,

l

h!-,0.4 """

I

- . r

r . ~, . . - - / , ", ' ' r

1115

i

I

I

i

./'~'~.\.

[

i

[

I

1.o am

-r. 3:

0.5

0.0

I

-1.s

!

-1.o

I

-o.s

.... 1

o.o

I

!

o.s

~.o

! !

~.s

y(m)

Figure 9. Wave height distribution along section B-B'; circle: measured, solid and dashdot lines: non-orthogonal coordinates used in [36] and [57], respectively, dash line: orthogonal coordinates [24].

where D =

cosh 4kh + 8 - 2 tanh2 kh 8 sinh 4 kh

e2

(Y E~ = ~

'

k

g

IqJl

(15 2)

Figure 10 compares the results calculated by the linear and weakly nonlinear parabolic equations with laboratory data, which indicate the importance of wave nonlinearity.

Equations for Numerical Modeling of Wave Transformation .

~-o.o ~

I ~,

.0

.

.

.

.

133

.

i'",

s'%%

o

9

9

zlm)

,o i

TM'..d(Uc~,\

_ - ' ~

~

,'

9'

.'

,'

.o'

,, '

,,'

;

,:

1 ,;

L-_ 0

'

5

I0 ylm)

15

ZO

0

I

2

3

4

5 6 (X- 10.5)(..)

7

8

9

I0

Figure 10. Comparison among wave height distributions measured (circles) and calculated by linear equation (dash line) and by nonlinear theory (solid line) around a shoal [39].

Because Stokes wave theory, as mentioned later, is valid only when the Ursell parameter is smaller than 25, these nonlinear equations could give even worse results than linear equations in very shallow water. A modified nonlinear term is proposed in [40]. This agrees with the previous theory in deep water and gives correction of wave celerity in terms of relative wave height in shallow water. According to this modification, the coefficient D in Equation 151 is substituted by D': D ' = (1 + f~D)

fl = tanh5 kh,

tanh (kh + t'2e) - 1 tanh kh f2 = [kh/sinh kh] 4

(153)

(154)

where D = theoretical coefficient given by Equation 152 E x t e n s i o n of P a r a b o l i c E q u a t i o n M o d e l

A parabolic equation with energy dissipation term has been proposed and applied [13]. Wave reflection that is neglected in basic parabolic equations is considered in [34, 59]. Applications of various parabolic equations to field are found in [ 15, 85].

II

134

OffshoreEngineering

Boussinesq Equations Boussinesq equations are weakly nonlinear shallow water equations that include terms up to the orders of e 2 (relative wave height squared) and 152 (water depth to wavelength ratio squared). The Boussinesq equations for variable water depth were derived by Peregrine [77]. As the coefficients are independent of frequency, the equations can also be used for random wave transformation. Moreover, inclusion of the second-order terms in relative wave height allows to calculate wave-induced nearshore current simultaneously with the wave field.

Derivation of Boussinesq Equations Boussinesq equations can be derived by an iterative procedure. In reference to the non-dimensionalized basic equations and boundary conditions 14 to 21, zero-order terms are first considered to yield zero-order relations and then the Boussinesq equations are derived by considering the terms up to the order of e and 152.In the following derivation, dimensional equations (4 to 6 and 10 to 12) are used by considering the order of each term in the non-dimensional forms. As seen from Equation 16, zero-order terms in Equation 6 are the two terms on the right side, which can be integrated with the boundary condition 10 to yield the hydrostatic pressure distribution: p = pg(~ - z)

(155)

Then, from Equation 15, the horizontal components of the water particle velocity u are independent of z in the leading order, which enables the integration of Equation 14 with the boundary condition (12) to yield the linear distribution of the vertical component of the velocity: w = - z(Vu) - (Vhu)

(156)

In the next iteration, the first term on the left side of Equation 6 should be included because, as seen from Equation 16, it is of order of 152. Substitution of Equation 156 into Equation 6 and integration gives pressure distribution modified by the vertical acceleration term:

I

p = p g (~ - z) + - ~ ~- (Vu) + z

(hu)

1

(157)

In the horizontal momentum equation 5, the third term on the left side is of orders higher than e 2 because u is independent of z to the leading order. Then, by substituting Equation 157 into Equation 5 and taking the depth average, the following momentum equation is obtained:

Equations for Numerical Modeling of Wave Transformation

t)fi

h 2 ~9 h ~9 + (fi V) fi + gV~ . . . . V(V fi) + V [V(hfi)] ~t 6 0t -2 ~-

135

(158)

where fi is the horizontal components of the depth-average velocity. The depthintegrated continuity equation is derived by integrating Equation 4 from the bottom to the surface: "/..2 + V[(h + ~)fi] = 0

(159)

~t

The Boussinesq equations have been verified for wave transformation in shallow water [1, 17, 19, 65, 80]. A numerical calculation method based on the Fourier transformation has been developed [37, 60].

Modified Boussinesq Equations Because of the assumption, the previous Boussinesq equations can be applied only in shallow water. To maintain the error in the wave celerity within 5%, the water depth to deep-water wavelength ratio h ~ o must be smaller than 0.22. Modified Boussinesq equations have been developed by applying the method [94] to improve the accuracy of the wave celerity [64, 66]. Keeping the same order of theoretical accuracy, the equations have higher numerical accuracy in deep water. By using the continuity equation (159), the momentum equation (158) can be written as h 3 ~) V(Vu) ~}t [(h + ~)fi] + V{ (h + ~) [fi, fi]} + g (h + ~)V~ . . . . 6 Ot h 2 ~} + ~ - - V [V(hu)] 2 /)t

(160)

where the operation [, ] denotes the matrix generated from the two vectors as [a, b] = ab t (b t denotes the transposed vector of b). The previous equation is expressed in terms of the flow rate Q per unit width:

$

+

h 2 ~9 3 ~t

h~} + -[V, Q] (Vh)

h V (VQ) + o: (Vh) ~- (VQ) (161)

136

OffshoreEngineering

The equation of leading order terms in the previous equation:

~Q /)t

+ ghV~ = 0

(162)

is taken its divergence and then gradient and multiplied by h 2 to yield h 2 ~t) V(VQ) + gh 2 V(hV2~) + gh 2 ([V, V~]Vh) = 0

(163)

which has the same order of accuracy as Equation 161. Then, by adding B times the previous equation to the momentum equation (161), the following modified Boussinesq equation can be obtained: J-g~~

, +g(h+;)V~= h

B+

O

h2 -~0 V(VQ) + Bgh2V(hV2~) h b

+ Bgh 2([V, V;]Vh) + ~ Vh ~ (VQ) + ~ ~- [V, Q] (Vh)

(164)

Equation 164 with B = 1/15 can be used for h ~ o < 0.5 within a 5% error in the wave celerity [66]. Modified Boussinesq equations are derived by taking the dependent variable as the horizontal velocity at an arbitrary elevation [73]. The elevation za optimum for the wave celerity within the range 0 < h ~ o < 0.5 was found to be -0.39h, which closely agrees with -0.40h obtained from Equation 164 with B = 1/15. The equations have been applied to random wave interaction [74]. Parabolic models have been proposed by applying Fourier transform to the modified Boussinesq equations [9].

Boussinesq Equations with Breaking Dissipation A model was proposed to incorporate the effect of wave breaking in the Boussinesq equation [84]. The model introduces excess momentum flux R b due to breaking waves: Rb =

~5r

[(C - fi), ( C - fi)]

(165)

1 -- ~ r / ( h + ~) where

~r = thickness of the surface roller determined in a heuristic geometrical

way C = the wave celerity vector Figure 11 shows the water surface elevation, cross-shore distribution of the wave height, and wave-induced current calculated by this model [82].

Equations for Numerical Modeling of Wave Transformation

137

0.1-

1

~,o.oa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "" "7 r"'~ ~r~ "~ 1 0.o,3=o.o2-

i

0.02

4

6

,

\

1 ]

8 10 12 14 16 18 20 22 24 26 28

0.1-

.~o.oa. . . . . . . .

0"02-.-

' [

0.02

4

6

8

10 12 14 16 18 20 22 24 26 28 Diitonce (m)

0.02 m,,m/s 15

16

17

18

19

20

21

22

23

24

25

Figure 11. The water surface elevation and wave-induced current calculated by Boussinesq equations incorporated with breaking model and comparison between calculated and measured cross-shore distribution of the wave height [82].

Other breaking models that are incorporated with the Boussinesq equations are proposed in [30, 31 ].

B o u s s i n e s q E q u a t i o n s for W a v e s on a P e r m e a b l e B e d Equations for weakly nonlinear shallow water waves on a permeable bed can also be derived upon assumptions similar to those in Boussinesq equations. By following the derivation in [ 11 ], the following Boussinesq equations on a permeable bed are obtained: - - + V[(h + ~)fi] + V [ E p h p U p ] - 0

/)t

3~ Ot

h 2 c)

+ (fir) fi + gV~ . . . .

6 Ot

h ~)

+ ~ -~-V[V(EphpUp )]

(166)

hb V (Vfi) + -~ ~ V[V(hfi)] (167)

138 Offshore Engineering [~p ] 1 cq Cr LOt + (UpV)gp + gV~ = ~ ~-~V[V(h2g)] + ~-~V[h(V(Ephpgp))] -- C r ~- + -~p

V(VUp) + -~- V(h - hp) (V~p)

hP2 V(Vhtgp)+(Vh)(Vhtup) - - ~ p uP

~p

(168)

Nonlinear Shallow-Water Equations When the wavelength is extremely long compared to the water depth, the basic equations and boundary conditions are significantly simplified and nonlinear shallow-water equations are obtained [88]. Thus, vertical acceleration can be neglected and hydrostatic pressure distribution results. As can be understood by using the characteristics method, the equations do not allow waves of permanent form on a horizontal bed, which is due to the neglect of vertical acceleration. Thus, the nonlinear shallow-water equations should be used when wave transformation due to other effects such as sloping bottom and energy dissipation are predominant [7]. One advantage of using the equations is that wave breaking can be modeled as a discontinuity of the solution and therefore the model does not need any empirical relationship or constant [18, 32]. This gives a primary reason why these equations are used to analyze breaking wave transformation including run-up [23, 81 ]. Derivation

of Nonlinear

Shallow-Water

Equations

Assuming that the water depth to wavelength ratio is extremely small, the momentum equation (16) in the vertical direction or, in dimensional form, (6) with the boundary condition (10) gives a hydrostatic pressure distribution: p = pg (~ - z)

(169)

Then the momentum equation (5) in the horizontal direction becomes

~u

+ ( u V ) u - - gV~

(170)

Ot

The continuity equation can be obtained by integrating Equation 4 from the bottom to the surface:

Equations for Numerical Modeling of Wave Transformation

~_2 + V[(h + ~)u] = 0 Ot

139

(171)

Adding a bottom friction term to the momentum equation, Kobayashi and his coworkers [43, 45-48, 95] studied extensively the wave transformation in the surf zones including run-up, set-up and reflection under various conditions of smooth and rough bottoms, gentle and steep slopes, and regular and irregular waves. Although discrepancy is found between calculation and measurement in wave shoaling and breaking inception, wave transformation in very shallow water has been accurately reproduced. Swash oscillation due to obliquely incident waves has also been analyzed by the nonlinear shallow-water equations [2, 42].

N o n l i n e a r S h a l l o w - W a t e r E q u a t i o n s on a P e r m e a b l e B e d Based on the long wave assumption, recently model equations have been derived for nonlinear shallow-water waves on a permeable bed [44]. In the following, the equations are presented for a three-dimensional case by using the present notations. The continuity equations for the water and permeable layers are obtained by integrating vertically Equations 4 and 27, respectively. t)~ ~-V[(h + ~) U] = - qb ~)t

(172)

V[Ephpup] = qb

(173)

where qb = flow rate per unit area from the water layer to the permeable layer The momentum equations for both layers can be obtained from Equations 5 and 29 upon the long wave assumption: /) 1 "tO[ ( h + ~)u] + V{(h + ~)[u, u]} = - g(h + ~)V~- =Lfb lulu-- qb Ub

EpV

t) [EphpUp ] + V {8php[Up , Up] } = - gephpV~ - Ephp

Ot

I

2Cf Ep luplup + qbU b

(174)

Up

V

(175)

where the second term on the right side of Equation 174 represents the bottom friction with the friction coefficient fb" The last terms on the right sides of Equa-

140

Offshore Engineering

tions 174 and 175 represent the momentum exchange between the two layers, and u b = u for qb > 0 and u b = up for qb < 0. Figure 12 shows a good agreement between calculated and measured waterline oscillations. The terms of momentum exchange consider the effect of boundary layer in some sense. However, it is also interpreted that the boundary of the two control volumes for the water and permeable layers changes depending upon the sign of qb: at the lower limit of the boundary layer during qb > 0 and at the upper limit during qb < 0.

2

Zr

'i

o -'1-

.........

Measured

"-2150

Computed

155

160

165

170

2 $i

o

1-

9

Zr

o-

Ii

i

i

e

'~1-

.........

Measured

-2150

155

Computed

160

165

170

21 Run P3 Zr -1 i "--2

|/

150

.........M e a s u r e d '"

i

155

-

Computed I

160

"'

I

165

"

170

Figure 12. Waterline oscillations due to random waves on a sloping beach [95].

Equations for Numerical Modeling of Wave Transformation

141

The continuity equations for the two layers are combined to yield the total mass conservation equation:

~ t + V[(h + ~) u] + V [EphpUp ] - 0

(176)

The momentum equations (174 and 175) are rewritten with use of the continuity equations (172 and 173) as

an --~ + ( u V ) u ~Up

+ gV~ = - 2(h + ~)

lulu- h qb-t- ~

EpV -k (UpV)Up + gV~ = - Ephp ~ U p ~t Kp

(ub -- U)

E2pCf qb ]uplup -b (u b - Up) ~p Ephp

(177)

(178)

As compared to Equation 166, the continuity equation is the same as in the case of the Boussinesq type equations. If C r is taken to be unity as often assumed, the differences in the momentum equations are that Equations 167 and 168 include the effect of vertical acceleration, whereas Equations 177 and 178 include the bottom friction and momentum exchange. However, if the two control volumes are taken constantly outside of the boundary layer, the last terms in the momentum equations should vanish because u b = u for Equation 177 and u b = Up for Equation 178 at all phases of wave motion.

Nonlinear Mild-Slope Equations Fully-nonlinear and fully-dispersive wave equations were first derived in [69, 70]. Other formulations are also found in [26, 72]. Equations are derived by expanding the dependent variable such as the velocity, velocity potential, or pressure into a series in terms of a set of vertical distribution functions, substituting the expression into the basic equations and then integrating them in the vertical direction. Because no assumption is made on the nonlinearity and dispersivity, the resultant equations can be used even for the analysis of strongly nonlinear wave transformation. Required accuracy can be achieved by increasing the number of terms in the series: usually by two or three terms. The following section introduces derivation by using Lagrangian. Then, it is demonstrated that the model equations including the MSE, Boussinesq equations, and nonlinear shallow-water equations are derived as special cases [26].

142

Offshore Engineering

Derivation of Nonlinear Mild-Slope Equations

A Lagrangian for water waves is written as [62] s

_ ~t + ,~]= ftt2ffA l_f~, ,[i~t~3

(Vr )2 + 1 (t)~z3)2 ~ + g z }, d z d g d t

(179)

1

For infinitesimal changes in ~3 and ~, the change in Lagrangian is expressed after integration by parts:

~L/~-'--fttl2IIAI_;hI V2r

02(I)3 0Z2 ) 80 dz dA dt

--I 1 f ~ l L c3t +

+ ~ + (V~) (V~)3)-

+ 2 ( ~ z 3) +g~ 8~z_;

8*

+ (Vh)(V~)3)+ z=~

0r

~

dA dt z=-h

;

t2

+ I~21 J~CI_~h'~-n (~ dz ds dt + I IA I_h [(~(I)]tl dz dA

(180)

To terminate the Lagrangian with respect to ~3 and ~, each term in Equation 180 must vanish, which yields all the basic equation and boundary conditions (8, and 10 to 12) for water waves. To derive two-dimensional model wave equations, the velocity potential (I)3in three dimensions is expressed as a series in terms of a set of vertical distribution functions Zct which should be given a priori:

N (~3(X, Z, t) = Z Za (z; h(x)) f(x(x, t) - Zafa

(181)

0t=l Substitution of this expression into Equation 179 and analytical integration in the vertical direction yields: s

~] = I]2 1 IAZ(f~, ~)dA dt

(182)

I

oqf~ r ar = ~g (r - h2 ) + ZI3 -all3 where Z f~,--~-, ~ + 1 AN (Vf~) (Vfl3) I D~cfvfl3(Vh) 2 + --I B#f~,f~) + C~f~ (Vf[~)(Vh) + ~2

(183)

Equations for Numerical Modeling of Wave Transformation

Aixl3 =

h

ZixZBdz'

ozoo.oz,o.

Bix~ -----

dz, :Za =

h fh

/)z ix

/)z dz, Cix~

=

h

143

- - ~ Zi~ dz,

dz

(184)

To terminate the Lagrangian with respect to fix and ~, the following Euler equations are required: =~

I [ ]

(185)

,3(3fa/3t ) + V 3(Vfix)

(186) On substituting the definition (182) of ~, the following nonlinear simultaneous partial differential equations can be obtained:

a~

Z~ --~ + V (Aixl3 V f13)- B~I~ f~ + V (C~f~V h) - C~I~(V f~) (V h) - D~f~(Vh) 2 = 0

087)

o3fl3 1 1 ,9Z~ ~9Z~ g~ + Z~ --~ + -~ Z~ Z~ (Vf~') (Vf[3) + 2 /)z /gz f'r f~ c)Z~ 1 ,gZ~ ~9Z~ + - ~ Z~ fr (Vfl3) (Vh)q 2 ~)h ~)h f~, fl3 (Vh)2 = 0

where Z;a : Zaiz=;

/)Z;a -- OZa[ '

~z

~z

088)

(189) z=;

Upon the mild-slope assumption, terms of second order in the bottom slope are neglected to yield the following nonlinear mild-slope equation:

a;

Z;a ~ + V(Aal3Vfl3) - Bal3 f13 + (CI3a - Cixl3)(Vfl3)(Vh)

~z~

+- ~ Z~f, (V~)(Vh)= 0

(190)

144

Offshore Engineering

g; +

Of13

+

1

Cvf

)r

1 ~Z~ r

+ 2 Oz

Oz f~' f~

az~

+ Oh Z~ f~,(V fB) (V h) = 0

(191)

The unknowns in these equations are ~ and fa (~ = 1 to N). Once the equations are solved numerically, the velocity potential is determined by Equation 181. Although the mild-slope assumption has not been used to derive Equations 187 and 188, simplification to Equations 190 and 191 may usually be consistent with the selection of the vertical distribution functions Za because they are usually taken from the theory for waves on a horizontal bed. Figure 13 compares calculated and measured water surface fluctuation at the shoreward edge of and behind a submerged breakwater. With only three terms, calculation gives a good agreement with measurement. Figure 14 compares calculated and measured water surface fluctuation and bottom velocity on a sloping bottom. In spite of strong nonlinearity of waves just prior to breaking, the agreement is good for both surface fluctuation and velocity.

Relationship with Other Wave Equations Mild Slope Equation. Linearized forms of the nonlinear mild-slope equations ( 190 and 191) are

i9~ o z~ .-~-+ V(A~I 3 7 f13)- B~ f13 + (CI3a - C~13) (Vfl3) (Vh) = 0

(192)

/)f~ = 0 g~ + Z~ ~ t

(193)

When only one component is taken in the series and the vertical distribution function is taken from the small amplitude wave theory: ~3 (X, Z, t ) = Z 1 fl (x, t)

(194)

cosh k(h + z) cosh kh

(195)

Z1=

Equations for Numerical Modeling of Wave Transformation

145

2.Ore O.Tml.5mO.7m2.1m waves_

P1 _-----

I

P5

T

I

15cm 50cm

,.~//~.~

9

ii~li

I_

-J

_Case 4 (T = 2.01 s, Ho = 5.0 era)

P3

-I

0

1

t/T

2

Figure 13. Water surface fluctuation at the shoreward edge of and behind a submerged breakwater. N indicates the number of terms taken in the series expansion of the velocity potential [26].

146

Offshore

Engineering

,,

..~,,

,,,

%

40.1cm L .L

.!.

_,

0.4m l.Om

U

9.03m

1 - 2 (fp = 0 . 5 0 H z , H i t/3 = 5.4 c m )

Case in

,

x = -40 /

cm

9

....... I

meas;

l

l

N

= 2;

I

....... N = 3 1

I

t

k~,

-

-1

nL._

101

i

0 l

0

1

i

z2o

....

.-~.,

t

10

,

,

!

J

5

5 0 z = 720 cm

u.b

," 9

z

5 =

-10/

.- . . . . . :'.-_ . . . .

..... . . . . . .

-"--_

i

i

15

T

,q

/

20

,

,

~

t

r

r

r

10

t(s)

15

10

.'.

~

15

,

~

/

20

,

0

5

Figure 14. Water surface fluctuation ~ and bottom velocity shoaling water [26].

Ub

2 0

of random waves on

E q u a t i o n s 192 a n d 193 b e c o m e

~+ at

V

af~

g~ + - ~ = 0

Vf 1 +-(k g

2 CCg

) fl

=0

(196)

(197)

Equations for Numerical Modeling of Wave Transformation

147

which yields a time-dependent form of the MSE:

a2f,

V (CCg V fl ) -k- (k 2 C C g -- (y2) fl - ~ = 0 Ot 2

(198)

If more than one component is taken for which the vertical distribution functions are cosh k a (h + z)

Za =

(Ya 2

(199)

cosh kah gka tanh kah

_

(200)

then Equations 192 and 193 become (201)

-0r- + g~t~ V 2 f13 - a ~of ~ = 0 Ot

g~ +

13=1

af~=o 3t

(202)

where 2

10"a - ~ A~ -

k~ - k~

ca2 na

2 2

(a ~ 13)

2 2

1 kaO'13 - k130'a B~ =

(a = ]3)

k~ - k~

o 2 a ( 1 - n a)

(203) (a = 13)

From Equations 201 and 202, ~ can be eliminated to yield o V 2 f~ - Ba~f o ~= 0 + Aa~

(204)

g 13=1 0t2 Then, by~ assuming progressive waves with the angular frequency 8 and wave number k" fa = aa ei(~-&)

(205)

148

OffshoreEngineering

Equation 204 becomes _ Bal3 o al3 =

f~2

13=1

AaBal3 o

(206)

13=1

To have a nontrivial solution, k 2 is determined as an eigenvalue for a given 0. It can easily be proved that 1< = ka for 8 = Ga, and therefore the dispersion relation is exactly satisfied at the frequencies Ga (o~ = 1 to N). This suggests that the dispersion relation is accurately satisfied even if the frequency is not equal to either of the selected frequencies. Therefore transformation of random waves with a wide spectrum can accurately be calculated by Equations 201 and 202.

Nonlinear Shallow-Water Equations. These equations are obtained by taking one component with Z 1= 1

(207)

Then, All = h + ~,

Bll = 0,

Cll = 0

(208)

and the nonlinear mild-slope equations (190 and 191) become "/_2 + V[(h + ~) V t'1] = 0 bt

(209)

c3f1 1 )2 g~ + .-~- + ~ (V fl = 0

(210)

By rewriting these equations in terms of u: u = V ~ 3 =Vf~

(211)

the nonlinear shallow-water equations are obtained:

~)t

+ V [(h + r~) u] = 0

(212)

~u ~ + ( u V ) u + gV~ = 0 ~t

(213)

Equations for Numerical Modeling of Wave Transformation

149

Boussinesq Equations. For two components with the following vertical distribution functions: (h + z) 2

1,

Z 1 =

Z 2 = ~

(214)

h2

the nonlinear mild-slope equations become

~9~

I

~+Ot V ( h + ~ ) V f 1+ 2(h + ~)~

h3

-

f2 ( V ~ )

(h-3~ + ~~)3 Vf 2]

(h + ~)2 (h - 2~) (V f2) (Vh) 3h 3 (215)

(Vh) = 0

(h+~)2 ~)~ [(h+~) 3 (h+~)5 ] 4(h+~) 3 h 2 8t + V 3h 2 Vfl + 5-h~ V f2 -- 3h 4 f2 -

(h + ~)2 (h - 2~) 2 (h 4- ~ ) 3 ~ 3h 3 (V fl ) (V h) h5 f2 (V~) (Vh) = 0

g~+~+

,{

h2

Ot +-2 Vfl + ' 5 ~hV f 2

) 2 ) h------T~Vf2 h 3 - 2 ( h (+ ~h) ~ +{ f2 ~ Vf~+ (Vh)=O

)2 f

1 2 (h + ~) +-2 h2

(216)

2

(217)

By invoking the orders of magnitude as

V h --, 0 (q~-), ~ - fl ~' 0 (E), f2 - 0 (e 2)

(218)

Equations 215 to 217 are simplified as 8t ~ V (h +~) V fl +-~ Vf2 + ~ (Vf2)(Vh) = 0

(219)

~+V ~t

(220)

Vfl -

~

f2-

-3

(Vfl)(Vh)=O

150

Offshore Engineering

t~fl /)f2 1 )2 g~ +---~- + - ~ + ~ (V f, = 0

(221)

Then, because (h + z ) 2 u=Vt~3 =Vfl + ~ V f 2 hE

-

2z(h + z) 3 f2 v h h

1 1 ~=vf, +gvf: +gf~Vh

(222)

(223)

Equations 219 to 221 are combined to yield the Boussinesq equation:

a~

(224)

+ V [(h + ~) fi] = 0

/)t h 2 ~)

~+(fiV)fi+gV~ /)t

h~

V (Vfi) + ~- ~ V [V (hfi)] . . . . 6 /)t

(225)

Validity Ranges of Wave Equations In the MSE, the wave steepness HA, (H = wave height, L = wavelength) is assumed to be small, but the relative water depth h/L is arbitrary. On the other hand, in the Boussinesq equations, the relative wave height H/h and relative water depth squared (h/L) 2 are assumed to be small quantities with the same order of magnitude, i.e., the Ursell parameter U r = H LE/h 3 is of order of unity. These two sets of assumptions, respectively, lead to the Stokes and cnoidal wave theories for waves that propagate on a horizontal bottom without deformation. There are only two independent non-dimensional parameters for waves of permanent form: H/L and h/L, or any combination of products of these two parameters. Perturbation expansion might be done with respect to two independent parameters instead of one parameter as in the Stokes and cnoidal wave theories. A regular perturbation solution can be found if the Ursell parameter U r = n L 2 ] h 3 = E 1 and relative water depth squared (h~) 2 - E2 are chosen as the two parameters, i.e., in the velocity potential expanded into double power series [28]: oo

oo

(226) m=l n=l

~)mncan be solved to any order of

e1

and

E 2.

Equations

for Numerical

Modeling

of Wave Transformation

151

0 (~2 - (h/L) 2) Stokes

waves f

A

)- - --Q- - --i )---~

)----1

)----0----4

)----~

)---..1

. . . . .

)----0---~

)----4

)----4)

.....

)----0---.

)----~

)_--.~)

.....

. . . . . . . II II ,4) 11 II

" cnoidal ~ ,, w a v e s l!

)- ---0-

Figure

)----4)

_ _.~ ) - - - - ~

.....

I

I

i

I

I

1

2

3

4

5

15. Illustration

of the solution

in a double

0 (~1 ~HL2/h3) power series.

The final solution can be illustrated in Figure 15. If the summation with respect to n in Equation 226, i.e., summation in the vertical direction in the figure, is taken first, the resultant single power series becomes equivalent to the Stokes wave solution. The order of the solution agrees with the number of solution lines included and the terms corresponding to the points on the lines are included in the solution. On the contrary, if summation is taken first with respect to m, i.e., in the horizontal direction, the resultant single power series becomes equivalent to the cnoidal wave solution. Each theory reflects its own process in taking the double summation. Then, it is understood that, even though other various wave theories can be established by first taking summation in inclined directions, finite-order solutions contain only finite number of terms in the double power series and therefore inferior to both the Stokes and cnoidal wave theories. For example, the theory for O[H/L] = O[(h/L) 4] corresponds to taking the first summation in the direction of 45 ~ and the first-order solution includes only one point in the figure. Thus, the Stokes and cnoidal wave theories are the only two useful theories for waves of permanent type. As can also be understood from the previous discussion, the Stokes wave theory is valid in relatively deep water, whereas the cnoidal wave theory is valid in relatively shallow water. Figure 16 compares wave profiles calculated by a 5th-order Stokes wave theory, 3rd-order cnoidal wave theory, 5th-order stream function wave theory, and small amplitude wave theory in various water depths. In an intermediate water depth, all the three finite amplitude wave theories predict almost the same wave profile. However, in shallow water, the Stokes wave theory gives an unrealistic wave profile because the given wave parameters are out of the validity

152

OffshoreEngineering

S-5 SFM-5--~.

h/Lo =0.641 H/L, =0.128

SFM-S~ 5 - 5 ~

tt/ Lo =0.096 H / L o =0.038

-

- 0 ,

~0

-1 -0.5

I

I 0

I

-1 -0.5

0.5

i

X/L (a) h=100rn. T=10s. H=20m

I

0.5

X/L (b} h=lSm. T=lOs. H=6m

,.'",, -

1 0

/ ~

~,/ ',,~.-.~..,:,'

s

h~ Lo = 0.032 ,'H/Lo =0.013 ./,h

=0.4

:z: ~- 0

/

/

\c-3

~', /~"ss -1 -0.5

I

' I~o.St "~ ,

I

I

0 X/L (c) h=5m. T=10s,H=2m

0.5

Figure 16. Comparison of wave profiles calculated by various wave theories (S-1- small amplitude wave, S-5: 5th-order Stokes wave, C-3: 3rd-order cnoidal wave, SFM: 5thorder stream function wave theory).

range of the Stokes wave theory. For the cnoidal wave theory, although the profile agrees fairy well with other theories even in deep water, other quantifies such as the velocity on the bottom cannot be predicted reasonably. To assure the validity of the perturbation theories, the series solution should be convergent, which implies that higher order terms should be smaller than lower order terms. By taking the ratio between the second- and first-order terms for various quantities and using the approximations in deep and shallow water ( h ~ >> 1 and << 1), the following results can be obtained:

~H/L (h/L >> 1) t Ur (h/L << 1)

2nd order 1st order

Stokes

2nd order oc

1st order

cnoidal

f h/L [H/h

(h/L >>1) (h/L << 1)

(227)

(228)

Equations for Numerical Modeling of Wave Transformation

153

From these results, the validity range of the Stokes wave theory is limited by the wave steepness in deep water and by the Ursell parameter in shallow water, whereas that of cnoidal wave theory by the relative water depth in deep water and relative wave height in shallow water. This can be investigated further in a quantitative manner by calculating the errors generated by finite-order solutions in the two nonlinear surface boundary conditions. Figure 17 shows constant error lines for the dynamic surface boundary condition as well as breaking criteria obtained theoretically [97] and empirically [21]. Based on the error calculation, a diagram for the selection of a wave theory is proposed as Figure 18. The validity ranges of the wave equations described in this chapter can roughly be understood from the previous discussion. Especially, the weakly-nonlinear MSE of Stokes type can be used only for small Ursell parameters, and the original Boussinesq equation cannot be used for large relative water depth. Although modified versions of these equations may give reasonable results for particular wave properties such as the wave celerity, modification consistent for all properties may not be possible. For more strict calculations, fully nonlinear mild-slope equations might be most promising.

Summary This chapter deals with several types of equations for numerical modeling of wave transformation in shallow water. The mild-slope assumption, weakly nonlinear shallow-water wave assumption, and long-wave assumption lead to the MSE, Boussinesq equations, and nonlinear shallow-water equations, respectively.

0.1

0.01 0.001

0.01

0.1

1

h/Lo Figure 17. Constant (1% of gH) rms error lines for dynamic surface boundary condition (S: Stokes wave, C: cnoidal wave, SFM: stream function theory).

154

Offshore Engineering

tan#= 1/10.1/20, 1/30, 0-1150 God-, (1970)

~'"--v--,-TL..-.2L._~./.H/h=O.B3 Y. am,.ada and -'-"

. ~ ",-""?7"--.....:..~'r"~.-~ .~ t

l::inlotan~

i

i

H/h=0.4

C-3 0.1 2

0.01t

v

0.001

t,~

/

I,,,,,l

~ S-5

,

,

, ,,,1,,1

0.01

, ,,,,H

0.1

I

h/Lo Figure 18. A diagram for selection of a wave theory.

The fundamental form of the MSE is derived for small amplitude monochromatic waves, but it has been extended for wave transformation on a slowly varying current. The Boussinesq equations and nonlinear shallow-water equations do not necessarily assume oscillatory motions and thus can be used for predicting wave-current interaction even by their fundamental forms. The energy dissipation, especially due to wave breaking, has high priority in extending the model equations derived on the basis of the momentum equations for an inviscid fluid. Terms representing the effect of breaking have been introduced into the MSE and Boussinesq equations. The nonlinear shallow-water equations have the advantage that the wave breaking is mathematically treated as the discontinuity of the solution and the energy dissipation is automatically considered. Equations for waves on a permeable bed are also presented for the three models. They can also be used for predicting wave transformation over submerged breakwaters. To increase computational efficiency, parabolic approximations have been proposed to the MSE of an elliptic type. Varieties of parabolic equations were presented to clarify the difference among them. Time-dependent mild-slope equations have been proposed to improve numerical efficiency and treatment of boundary conditions, or to deal with random wave transformation. They were also described in this chapter. As described in the previous section, each set of model wave equations has its own validity range. Reliable prediction cannot be performed outside of the

Equations for Numerical Modeling of Wave Transformation

155

range. Nonlinear mild-slope equations are valid as long as the series expression for the dependent variable in terms of vertical distribution functions gives a good approximation. Strongly nonlinear and strongly dispersive wave transformation can be predicted accurately by the equations. Because the progress in the numerical modeling of wave transformation in shallow water has been remarkable, reliability of predictive models has greatly improved. Numerical model equations will also be used for developing techniques to control not only wave height but also wave period and direction, which cannot be predicted by linear models.

Notation a C Cg CM Cf Cr

amplitude wave celerity group velocity mass coefficient turbulent resistance coefficient in permeable layer apparent mass per unit volume of permeable layer F energy flux per unit width fa(x, t) coefficient for Z a in Equation 182 fo enery dissipation coefficient bottom friction coefficient linearized resistance coefficient in permeable layer g gravitational acceleration H wave height h still water depth h 1 representative vertical length scale hp depth of permeable layer h t = h + hp, total depth K, intrinsic permeability k wave number k wave number vector s Lagrangian defined by Equation 179 L wavelength Lo deepwater wavelength L1 representative horizontal length scale P pressure in water layer Pa dynamic pressure pressure in permeable layer flow rate per unit width flow rate per unit horizontal projection area from water layer to qb permeable layer

156

Offshore Engineering

Rb t U, V, W

u3 u u

ub Uc Up, Vp, Wp Up3

excess momentum flux due to wave breaking time components of water particle velocity in water layer in x, y, and z directions; u, v = horizontal components; w = vertical component = (u, v, w) = (u, v)

depth-average horizontal velocity in water layer water particle velocity transported between water and permeable layer water particle velocity at still water level in wave propagation direction components of seepage velocity in permeable layer in x, y, and z directions; Up, Vp = horizontal components, Wp = vertical component --= (Up, Vp, Wp)

= (Up,

Uw

W Ww

x, y, z X X3

= (x, y, z)

Z Z" Z"

vertical distribution function defined by Equation 53 vertical distribution function defined by Equation 61 vertical distribution function defined by Equation 115 vertical distribution function defined by Equation 120 a set of vertical distribution function = Uc/C, ratio of water particle velocity at still water level to wave celerity value of 7 at breaking point value of Y on uniform slope value of Y at recovery point = hl/L 1, relative water depth = a/h l, relative wave amplitude porosity water surface elevation kinematic viscosity general coordinates; either orthogonal or non-orthogonal coordinates non-dimensional general coordinates water density

Z PtP

Za 7

~s Vr 13

V

~,

Vp)

depth-average horizontal velocity in permeable layer horizontal two-components of velocity induced by currents horizontal two-components of velocity induced by waves vertical component of velocity induced by currents vertical component of velocity induced by waves Cartesian coordinates; x, y = horizontal coordinates, z = vertical coordinate = (x, y)

1"1~

P

Equations for Numerical Modeling of Wave Transformation

157

Ij

intrinsic angular frequency representative angular frequency = (~ - D, deviation from representative angular frequency % velocity potential in water layer r velocity potential at still water level; defined by Equations 41 and 53 amplitude of velocity potential 4) at still water level velocity potential defined by Equations 60 and 61 0' amplitude of (l)' defined by Equation 125 defined by Equation 140 defined by Equation 135 v defined by Equation 131 o.) apparent angular frequency; = (~ in absence of current G

V3

Ox Oy Oz =

D

Dt (vector)3 (vector) () ( j~ ( )0 ( )*

,~yy

0 = -- + (uv)

Ot

three-dimensional vector horizontally two-dimensional vector quantity in permeable layer quantity at water surface quantity at still water level nondimensional quantity References

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