2 η
η
ηt + ∇ ⋅ ∫
∆x ∫-h
u dz =
uo dz
(54)
-h
1
( zα + h )2 - h2 ∇ (∇ ⋅ uαt - 2uoα, t / ∆x ) = 0 (55)
(uα ⋅ ∇ )
uαt + g∇η +
uα +
2
Some previous investigators (e.g., Larsen and Dancy 1983) introduced a point
source for fluid flow into the continuity equation. It is, however, important to
apply a correction to the momentum equations to account for higher-order spatial
derivatives across the generation line.
Damping regions
Waves propagating out of the computational domain are absorbed in damp-
ing regions placed around the perimeter of the computational domain. Damping
layers can also be used to model the partial reflection from harbor structures
inside the computational area. Artificial dissipation of wave energy in damping
layers is achieved through the introduction of a term proportional to the surface
elevation into the right-hand side of the mass equation:
=
- ( x) η
Fdamping _ η
(56)
and a term proportional horizontal velocity into the right-hand side of the
momentum equation:
=
-( x) uα
Fdamping _ u
(57)
where (x) is the damping strength with units of s-1. Introduction of the damping
terms lead to the nonconservation of mass and momentum in the damping
regions. However, extensive tests with different combinations of the damping
terms showed that the inclusion of damping terms in both the mass and momen-
tum equation was more effective than just a damping term in the momentum
equation.
Numerical simulations to evaluate the performance of the damping layer
showed that waves could be effectively damped out in a layer half a wavelength
wide by employing a quadratic variation of (x) with a peak value of 30/T, where
T is the wave period. The damping strength has thus been nondimensionalized by
30/T, i.e.,
T
nd ( x) =
( x)
(58)
30
where nd(x) is a nondimensional damping coefficient that is allowed to vary
from 0 to 1. Figure 5 shows the variation of the reflection coefficient with damp-
ing coefficient for different relative widths (w/L) of the damping layer, where w
is the width of the damping layer and L is the wavelength. It should be pointed
23
Chapter 3 Numerical Solution
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