(1994) derived similar expressions for the weakly nonlinear form of the

Boussinesq equations:

ω1ω2 (*k* h)2 cos ∆θ[1 - (α + 1/ 3)(*k* h)2 ]

2λ*k*1' k2' h3

(13)

ω [1 - α(*k* h) ][ω *k h*(*k*1h *k*2h cos ∆θ) + ω *k h*(*k*1h cos ∆θ *k*2h)]

2

'

'

+

12

21

2λ*k*1' k2' h3

where *Dq *= *q*1 *q*2, *k * = | *k*1 *k *2 |, *k *' = *k *[1 - (α + 1/ 3) (*kh*)2], and

(14)

λ = ω2 [1 - α(*k* h)2 ] - *gk* h[1 - (α + 1/ 3)(*k* h)2 ]

2

Figure 2 shows a comparison of the quadratic transfer function of the weakly

nonlinear Boussinesq model with that of second-order Stokes theory for unidirec-

tional waves where the wave group period is 10 times the average of the indi-

vidual wave periods, i.e., ω- = (ω1 + ω2)/20. The weakly nonlinear Boussinesq

at the deepwater depth limit by 65 percent and 45 percent respectively. Hence, it

cannot accurately simulate nonlinear effects in deep water. To reasonably simu-

late nonlinear effects, the weakly nonlinear model should be restricted to the

range 0 < *h/L *< 0.3.

The turbulent and highly rotational flow field under breaking waves is

extremely complex and difficult to model even with the Reynolds-averaged form

of the Navier-Stokes equations (e.g., Lin and Liu 1998; Bradford 2000). In

BOUSS-2D, we do not attempt to model details of the turbulent motion, but

rather, simulate the effect of breaking-induced turbulence on the flow field. We

have tried to develop a generic model that can be applied to regular or irregular

waves, unidirectional or multidirectional waves, and simple or complex bottom

topography without having to recalibrate the model each time. The key assump-

tions made in developing the model are (see Nwogu 1996):

velocity at the free surface, *u*η, exceeds the phase velocity, *C.*

vertical gradient of the horizontal velocity at the free surface, ∂*u*/∂*z*|z=η.

region with the horizontal velocity at the free surface.

8

Chapter 2 Theoretical Background

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