2
Theoretical Background
Governing Equations
BOUSS-2D is based on Boussinesq-type equations derived by Nwogu (1993,
1996). The equations are depth-integrated equations for the conservation of mass
and momentum for nonlinear waves propagating in shallow and intermediate
water depths. They can be considered to be a perturbation from the shallow-water
equations, which are often used to simulate tidal flows in coastal regions. For
short-period waves, the horizontal velocities are no longer uniform over depth
and the pressure is nonhydrostatic. The vertical profile of the flow field is
obtained by expanding the velocity potential, Φ, as a Taylor series about an
arbitrary elevation, zα, in the water column. For waves with length, L, much
longer than the water depth, h, the series is truncated at second order resulting in
a quadratic variation of the velocity potential over depth:
φ α + 2 ( zα - z ) [∇ φ α ⋅ ∇ h ]
Φ ( x, z, t)
=
(1)
2
( zα + h ) - ( z + h ) 2 ∇ 2 φ α + O ( 4 )
2
+
2
dispersion. The horizontal and vertical velocities are obtained from the velocity
potential as:
u( x, z, t) = ∇Φ = uα + ( zα - z)[∇(uα ⋅ ∇h) + (∇ ⋅ uα )∇h]
1
( zα + h ) - ( z + h ) ∇(∇ ⋅ uα )
2
2
+
(2)
2
∂Φ
= - [uα ⋅ ∇h + ( z + h)∇ ⋅ uα ]
w( x, z, t) =
(3)
∂z
where uα = ∇Φ|zα is the horizontal velocity at z = zα. Given a vertical profile for
the flow field, the continuity and Euler (momentum) equations can be integrated
over depth, reducing the three-dimensional problem to two dimensions. For
weakly nonlinear waves with height, H, much smaller than the water depth, h, the
vertically integrated equations are written in terms of the water-surface elevation
η(x,t) and velocity uα(x,t) as (Nwogu 1993):
4
Chapter 2 Theoretical Background
Previous Page
