Page 48 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Review of Hydrodynamic Theory Chapter | 2 39

∂ β(x,y) β(x,y) ∂f ∂α ∂β
f(x, y, z)dz = dz − f(x, y, α(x, y)) + f(x, y, β(x, y))
∂x α(x,y) α(x,y) ∂x ∂x ∂x
(2.39)
or alternatively,
β(x,y) ∂f ∂ β(x,y) ∂α ∂β
dz = f(x, y, z)dz + f(x, y, α(x, y)) − f(x, y, β(x, y))
α(x,y) ∂x ∂x α(x,y) ∂x ∂x
(2.40)
where α and β are arbitrary functions.
Starting from the continuity equation (2.27), and integrating over the flow
depth results in,
η
∂u ∂v ∂w

+ + dz = 0 (2.41)
∂x ∂y ∂z
−d
where η is the water elevation above the datum (still water level), and d is the
water depth (bathymetry). Eq. (2.41) leads to
η ∂u η ∂v
dz + dz + w(η) − w(−d) = 0 (2.42)
−d ∂x −d ∂y
Referring to Leibnitz’s rule, the derivatives inside integrals can be evaluated as
follows
η η
∂u ∂ ∂η ∂(−d)
dz = udz − u(η) + u(−d) (2.43)
−d ∂x ∂x −d ∂x ∂x
η ∂v ∂ η ∂η ∂(−d)
dz = vdz − v(η) + v(−d) (2.44)
−d ∂y ∂y −d ∂y ∂y
The water free-surface is dependent on time and location (i.e. x, y), and can be
expressed as,
z s = η(x, y, t) (2.45)
The sea bed can similarly be expressed as (ignoring erosion and sedimentation)
z b =−d(x, y) (2.46)
Therefore, the vertical velocity at the free surface, and the bed, may be
expressed as

∂η ∂η ∂η
dz
w(η) =
= + u(η) + v(η) (2.47)
dt
z=η ∂t ∂x ∂y

∂d ∂d ∂d
dz
w(−d) =
=− − u(−d) − v(−d) (2.48)
dt
z=−d ∂t ∂x ∂y
In addition, if it is assumed that the sea bed does not change over time, then,

∂d
= 0 (2.49)

∂t
z=−b