Page 46 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

Review of Hydrodynamic Theory Chapter | 2 37

equations are the basis of linear wave theory and potential flow theory. Linear
wave theory will be presented briefly in Chapter 5 that discusses wave energy.

2.3.2 Viscous and Turbulent Flows

The shear stresses in Eq. (2.29) are associated with the fluid viscosity and
turbulence. For the case of laminar flow, where the shear stresses are caused
by the viscosity, Newton’s shear-stress relationship can help to evaluate the
stresses. Newton’s viscosity law in a simple form (horizontal flow) can be
written as
∂u
τ = μ (2.35)
∂z
where μ is dynamic viscosity and is the property of a fluid. In general, Newton’s
law expresses the relationship between the stress field and the deformations in a
fluid, and is called the constitutive relation. The simplest case of the constitutive
relation is the linear one (e.g. Newton’s viscosity law), in which the stress is a
linear function of the strain rate. Newton’s viscosity law, for incompressible 3D
flow, can be generalized as

∂u i ∂u j
τ ij = μ + (2.36)
∂x j ∂x i
Replacing the shear stresses by the previous equation, the momentum
equation (2.26) becomes
2
∂u i ∂u i 1 ∂p μ ∂ u i
+ u j = B i − + (2.37)
∂t ∂x j ρ ∂x i ρ ∂x k ∂x k
2
where ∂ u i , as mentioned before, is the Laplacian of u i and in the expanded
∂x k ∂x k
2 2 2
form is ∂ u i + ∂ u i + ∂ u i . Eqs 2.37 and 2.19 (continuity) are called the
∂x 2 ∂y 2 ∂z 2
incompressible Navier Stokes Equations.
For turbulent flows, the shear stresses would be dependent on the turbulency
in the flow field. One approach is replacing the dynamic viscosity with an ‘eddy
viscosity’ or ‘turbulent viscosity’. However, unlike dynamic viscosity, which is
constant and a property of fluid, the eddy viscosity is highly variable in a flow
field, depending on the level of turbulency. Some turbulence models introduce
additional equations to estimate the distribution of eddy viscosity in a domain.
More details are provided in Chapter 8.

2.3.3 Shallow Water Equations

The Navier-Stokes equations, which were discussed in the previous section, can-
not be directly solved by any mathematical method, unless many assumptions/
simplifications are made. Numerical methods have been developed to solve
the 3D form of the Navier-Stokes equations, for small-scale problems (e.g.