2 η

η

ηt + ∇ ⋅ ∫

∆*x *∫-*h*

(54)

-*h*

1

( zα + *h *)2 - *h*2 ∇ (∇ ⋅ *u*α*t *- 2*u*oα, *t */ ∆*x *) = 0 (55)

(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.

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

(56)

and a term proportional horizontal velocity into the right-hand side of the

momentum equation:

=

-( *x*) *u*α

(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

30/*T*, i.e.,

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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