Page 118 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Wave Energy Chapter | 5 109

wave mechanic textbooks. Here, we present the main assumptions, parameters,
equations, and wave properties associated with linear wave theory.
The main assumptions of linear wave theory can be summarized as [1]:
● The fluid is incompressible and inviscid.
● The flow is irrotational. 1
● The bed is horizontal with a constant depth; the bed is impermeable.
● The wave amplitude is small.
● A regular wave with a constant wave period is considered.
● Coriolis effects are neglected.
● Waves are long-crested: the hydrodynamic solution is provided for a 2D
vertical plane.
● The pressure at the water surface is constant.
Referring back to Chapter 2, we mentioned that the Euler equations represent
an incompressible and inviscid flow as follows:
∇· u = 0 (5.1)
du ∂u 1
= + u ·∇u =− ∇p − gk ˆ (5.2)
dt ∂t ρ
The Euler equations are the governing equations of an ideal fluid. A particular
case of interest is potential flow, in which the flow vorticity is also zero. The
other term, which is commonly used in potential flow theory, is irrotational flow
(i.e. zero vorticity). Irrotational flow is simply a flow in which the particles do
not rotate. It can be shown that the rate of rotation of particles in a flow field is
directly proportional to the curl (i.e. ∇×)ofvelocity
1
ω = ∇× u (5.3)
2
where ω is called vorticity (i.e. rotation vector); hence, irrotational flows have
zero vorticity. Referring to Eq. (5.2), if we take the curl of this equation, the
RHS will become zero, and this leads to
d(∇× u) d ω
= = 0 (5.4)
dt dt
In other words, the flow vorticity is constant for ideal flow. Therefore, by
assuming that the flow is inviscid and incompressible, we cannot necessarily
conclude that the flow is irrotational. Nevertheless, irrotational flow is a special
case in this context, where vorticity is not only constant, but it is also zero (i.e.
ω =∇ × u = 0). In hydrodynamics and wave mechanics, it is much easier
to deal with incompressible, inviscid and irrotational flows, because a scalar

1. The flow in an inviscid and incompressible fluid is not necessarily irrotational, as will be
discussed.