Page 54 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Review of Hydrodynamic Theory Chapter | 2 45

∂z
shown in the figure, and can be estimated as cos θ = . The force terms reduce
∂s
to
∂z ∂(pA)

F =−ρgA ds − ds (2.83)
∂s ∂s
The flux of momentum through the control volume can be written as

∂(ρQu)
ρuQ =−ρQu| + ρQu| s+δs =−ρQu + ρQu + ds (2.84)
s
∂s
We can further simplify the previous equation as follows
∂(ρQu) ∂u ∂u
ρuQ = = ρQ = ρAu (2.85)
∂s ∂s ∂s
Because the summation of forces (Eq. 2.83) should be equal to the net
momentum flux, therefore,
∂z ∂(pA) ∂u
− ρgA ds − ds = ρAu (2.86)
∂s ∂s ∂s
which results in
∂ ∂p d
p u 2
ρA g∂z + + u∂u = 0 ⇒ gz + + = 0 (2.87)
∂s ρ ds ρ 2
which finally leads to Bernoulli’s equation, as follows,
p u 2
gz + + = cons. (2.88)
ρ 2
p
This equation indicates that the work done by the pressure force will be
ρ
balanced by the potential gz and kinetic energy u 2 . The units of the terms in
2
Eq. (2.88) is the energy per unit mass of a fluid.
REFERENCES
[1] F.M. White, Fluid Mechanics (2003), 7th ed., McGraw-Hill, NY, USA, 2011.
[2] R.G. Dean, R.A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, World
Scientific Publishing, Singapore, 1991.
[3] N.S. Heaps, Linearized veritically-integrated equations for residual circulation in coastal seas,
Dt. Hydrogr. Z. 31 (1978) 147–169.
[4] J. Pedlosky, Geophysical Fluid Dynamics, second ed., Springer-Verlag, Berlin, 1992, 728 pp.
[5] J.H. Spurk, Fluid Mechanics, Springer-Verlag, Berlin, 1997, 513 pp.