Page 50 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Review of Hydrodynamic Theory Chapter | 2 41

Similar to the continuity equation, we can integrate the momentum equations
in the x and y directions over depth. Integrating the momentum equation in the
x direction (Eq. 2.28) produces
∂u ∂u ∂u ∂u ∂η
η η
η
dz + u + v + w dz =− g dz + ···
−d ∂t −d ∂x ∂y ∂z −d ∂x
1 η
∂τ xx ∂τ yx ∂τ zx
+ + + dz (2.58)
ρ −d ∂x ∂y ∂z
Again, we can evaluate the integrals using Leibnitz’s law, which leads to the
cancellation of several terms. Therefore, for the local acceleration term, we may
write
∂u ∂ ∂η ∂(−d)
η η
dz = udz − u(η) + u(−d) (2.59)
−d ∂t ∂t −h ∂t ∂t
Referring to Eq. (2.25), the convective acceleration term in the momentum
equation can be written as
∂u ∂u ∂u ∂u 2 ∂uv ∂uw
u + v + w = + + (2.60)
∂x ∂y ∂z ∂x ∂y ∂z
Application of Leibnitz’s rule for the convective term leads to
η 2 η
∂u ∂ 2 2 ∂η 2 ∂(−d)
dz = u dz − u (η) + u (−d) (2.61)
−d ∂x ∂x −d ∂x ∂x
∂uv ∂ ∂η ∂(−d)
η η
dz = (uv)dz − u(η)v(η) + u(−d)v(−d) (2.62)
−d ∂y ∂y −d ∂y ∂y
∂uv
η
dz = u(η)w(η) − u(−d)w(−d) (2.63)
−d ∂x
If we add the above equations (2.61–2.63), and use Eqs 2.47 and 2.48,
several terms will be cancelled out. Integrating the hydrostatic pressure term
generates
η
∂η ∂η η
g dz = g dz
−d ∂x ∂x −d
∂η ∂η
= g (d + η) = g h (2.64)
∂x ∂x
Also, application of Leibnitz’s rule to the shear-stress terms results in
η η
∂τ xx ∂ ∂η ∂(−d)
dz = τ xx dz − τ xx (η) + τ xx (−d) (2.65)
−d ∂x ∂x −d ∂x ∂x
η η
∂τ yx ∂ ∂η ∂(−d)
dz = τ yx dz − τ yx (η) + τ yx (−d) (2.66)
−d ∂y ∂y −d ∂y ∂y
η η
∂τ zx ∂ ∂η ∂(−d)
dz = τ zx dz − τ zx (η) + τ zx (−d) (2.67)
−d ∂z ∂z −d ∂z ∂z