Page 40 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Chapter 2

Review of Hydrodynamic

Theory

The dynamics of tides, waves, and winds can be understood/simulated by
hydrodynamic-thermodynamic theory. In general, the conservation of mass,
momentum, energy, and entropy can be formulated mathematically to de-
scribe the motion of fluids (water/air). Unfortunately, these equations, which
are formulated by integral or partial differential equations, cannot be solved
analytically (i.e. using mathematics), unless they are significantly simplified.
Therefore, numerical models that are discussed in later chapters are employed
to provide us with approximate numerical solutions to these equations.
In this chapter, we discuss the basic equations that are used to describe
the dynamics of fluids. In addition, the simplified forms of these equations,
including two-dimensional (2D) flow that is popular for tidal modelling, and
one-dimensional (1D) equations that are used in the actuator disk theory of
wind/tidal turbines, are discussed. This chapter provides a very brief overview
of these equations, and more details can be found in other texts (e.g. see [1–5]).
As index/indicial notation is a popular and efficient method to present these
equations, we first briefly explain the index notation.

2.1 VECTOR AND INDEX NOTATION
A three-dimensional (3D) vector, such as velocity, can be represented using
several notations. Some notations are more efficient when writing out equations;
and we will use various notations in this book depending on the complexity of
the subject. Here, these notations are explained. Starting from a 3D velocity
vector, which has three components in the x, y, and z directions, we can show
the velocity (u) in vector notation as follows,

u = (u, v, w) = u ˆ i + v ˆ j + wk ˆ (2.1)
where u, v, and w are the components of velocity in the x, y, and z directions,
respectively, and ˆ i, ˆ j, and k are the unit vectors in the x, y, and z directions,
ˆ

Fundamentals of Ocean Renewable Energy. https://doi.org/10.1016/B978-0-12-810448-4.00002-1
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