Page 49 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

40 Fundamentals of Ocean Renewable Energy

Therefore,

∂(−d) ∂(−d)
dz
w(−d) =
= u(−d) + v(−d) (2.50)
dt
∂x ∂y
z=−d
By substituting Eqs (2.43), (2.44), (2.48), (2.50) into Eq. (2.42), several
terms cancel out; using the depth-averaged variables (Eq. 2.38), after rearrang-
ing the terms, the continuity equation becomes
∂η ∂(¯uh) ∂(¯vh)
+ + = 0 (2.51)
∂t ∂x ∂y
The previous equation has three unknowns: depth-averaged velocities in the x
and y directions, and the total water depth (h(x, y, t)). Additional differential
equations are therefore necessary, which will be formed by integrating the
momentum equations over the water depth.
A main assumption for SWE is that the vertical acceleration is small and
can, therefore, be neglected. This means that the main components of velocities
are horizontal. Using this assumption, the momentum equation in the vertical
(z-direction) can be considerably simplified. We will show that by neglecting
the terms that correspond to the vertical acceleration, the momentum equation
in the z-direction reduces to the hydrostatic pressure law.
Starting from Eq. (2.37) for i = 3, these assumptions imply,
dw 2
≈ 0, μ∇ w ≈ 0 (2.52)
dt
Therefore, the momentum equation in the z-direction becomes (B 3 =−g)
1 ∂p
− g − = 0 (2.53)
ρ ∂z
For hydrodynamic simulations, the pressure at the water surface is equal to
the atmospheric pressure (p a ). Integrating Eq. (2.53) over depth results in
η

p a − p(z) + ρgdz = 0 (2.54)
z
which is the hydrostatic equation law
p(z) = p a + ρg(η − z) (2.55)

If the atmospheric pressure is considered constant, then the pressure terms in
the momentum equation (at a constant z in the 3D case) can be replaced by
∂p ∂η
= ρg (2.56)
∂x i ∂x i
where i is 1 or 2, corresponding to the x or y direction. In some cases, such as
storm surge modelling, changes in atmospheric pressure are important, as they
can vary in the x or y direction. For these cases, we can write
∂p ∂p a ∂η
= + ρg (2.57)
∂x i ∂x i ∂x i