a global sense over the entire record. The method also does not guarantee a sym-
metric directional distribution even when the target distribution is symmetric.
The velocity and flux boundary conditions along the wave generation lines
are obtained from the surface elevation time-histories using the linear transfer
function approach described previously in Equations 40 to 44.
Internal wave generation boundaries
In applications where there is significant wave reflection from bathymetric
features or structures within the computational domain, it is desirable to absorb
the waves that propagate back to the wave generation boundary to prevent a
buildup of wave energy inside the domain. This can be achieved by modifying
the boundary conditions along the wave generation boundary to simultaneously
generate and absorb reflected waves (e.g., Van Dongeren and Svendsen 1997). A
different approach proposed by Larsen and Dancy (1983) is to generate the
waves inside the computational domain and absorb reflected waves in a damping
layer placed behind the generation boundary. This approach has been adopted in
BOUSS-2D with the governing equations modified to allow for the generation of
waves inside the computational domain.
Consider the generation of waves along a horizontal line by a distribution of
sources that extend from the seabed (z = -h) to the free surface (z = η). The veloc-
ity potential associated with the fluid motion satisfies the Laplace equation
everywhere in the fluid except for generation line (x = xg) where there is a point
source of fluid mass. The governing equation for the fluid motion can thus be
written as:
∇2φ = q( y, z, t )δ( x - xg )
(51)
where q(y,z,t) is the volume flux density. Assuming that the water depth is
constant along the generation line, a modified form of the second-order Taylor
series expansion of the velocity potential about an arbitrary elevation z = zα in the
water column (Equation 1) can be written as:
1
( zα + h ) - ( z + h ) ∇2φα - qδ( x - xg )
2
2
φ = φα +
(52)
2
The horizontal fluid velocities are obtained from the velocity potential as:
1
( zα + h ) - ( z + h) ∇ ∇2φα - qδ( x - xg )
2
2
u = ∇φα +
(53)
2
On a rectangular grid with a finite grid spacing ∆x, the delta function can be
replaced with 2/∆x. To generate waves with a given velocity profile uo(y,z,t), the
mass and momentum equations along the grid generation line and adjacent
velocity points can be written as:
22
Chapter 3 Numerical Solution
Previous Page
