Page 51 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

42 Fundamentals of Ocean Renewable Energy

The first terms on the RHS can be replaced by depth-averaged quantities as
follows, for example

η
¯ τ yx h = τ yx dz (2.68)
−d
Also, because the depth-averaged variables are independent of z,in
∂(¯ zx h)
τ
Eq. (2.67), = 0.
∂z
By implementing the previous equations, replacing every term in Eq. (2.58),
and rearranging the terms, the depth-averaged momentum equation in the x
direction becomes
2
∂ ¯uh ∂ ¯u h ∂ ¯uvh ∂η
+ + =−g(d + η) + ···
∂t ∂x ∂y ∂x
τ
τ
1 ∂(¯ xx h) ∂(¯ yx h) 1 ∂η ∂η
+ + − τ xx + τ yx + ···
ρ ∂x ∂y ρ ∂x ∂y
water column z=η, surface
1 ∂(−d) ∂(−d)
+ τ xx + τ yx (2.69)
ρ ∂x ∂y z=−d,bottom
The shear-stress terms have been grouped into the bed shear stress, surface
stress, and water column stress. The bed shear stress can be estimated by several
empirical relationships. Using the Manning’s equation,
U|U| 2 U|U|
2
τ b = ρghn = ρgn (2.70)
h 4/3 h 1/3
√
2
2
in which U is the total depth-averaged velocity (i.e. U = ¯ u +¯v ) and n is
the Manning coefficient. Eq. (2.70) indicates that the shear stress always acts
2
in the opposite direction of the velocity (i.e. U|U| instead of U ). Therefore,
for the component of the bed shear stress in the x direction, we have
√
2
¯ u 2 ¯ u ¯u +¯v 2
τ bx = τ b √ = ρgn 1/3 (2.71)
2
¯ u +¯v 2 h
The wind shear stress can be computed using an empirical quadratic friction
law as follows
τ w = ρ a C wd W|W| (2.72)
where W is the wind velocity, usually specified at 10 m above the water surface,
ρ a is the air density, and C wd is the wind drag coefficient. Consequently, the
wind shear stress in the x direction can be evaluated as
W x 2 2
τ w , τ wx = ρ a C wd W x W + W (2.73)
τ wx = x y
2
W + W 2 y
x
where W x and W y are the components of the wind speed, W,inthe x and y

2
2
directions, respectively (i.e. W = W + W ). By replacing the bed and surface
x
y
shear-stress terms, the depth-averaged equations can finally be written as