(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 ]
G (ω1 , ω2 ) =
2λk1' k2' h3
(13)
ω [1 - α(k h) ][ω k h(k1h k2h cos ∆θ) + ω k h(k1h cos ∆θ k2h)]
2
'
'
+
12
21
2λk1' k2' h3
where Dq = q1 q2, k = | k1 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.
Simulation of Wave Breaking
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):
a. The breaking process is assumed to be "spilling."
b. Turbulence is produced in the near-surface region when the horizontal
velocity at the free surface, uη, exceeds the phase velocity, C.
c. The rate of production of turbulent kinetic energy is proportional to the
vertical gradient of the horizontal velocity at the free surface, ∂u/∂z|z=η.
d. Breaking-induced turbulence is primarily convected in the near-surface
region with the horizontal velocity at the free surface.
8
Chapter 2 Theoretical Background
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