Page 45 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

36 Fundamentals of Ocean Renewable Energy

Therefore, Eq. (2.24) for incompressible flow may be written as
∂u i ∂u i 1 ∂p 1 ∂τ ji
+ u j = B i − + (2.26)
∂t ∂x j ρ ∂x i ρ ∂x j
Eqs (2.19), (2.26) represent the continuity and the momentum equations for
incompressible flows. These equations can be expanded as follows
∂u ∂v ∂w
+ + = 0 (2.27)
∂x ∂y ∂z
∂u ∂u ∂u ∂u 1 ∂p 1
∂τ xx ∂τ yx ∂τ zx
+ u + v + w = B x − + + +
∂t ∂x ∂y ∂z ρ ∂x ρ ∂x ∂y ∂z
(2.28)
∂v ∂v ∂v ∂v 1 ∂p 1
∂τ xy ∂τ yy ∂τ zy
+ u + v + w = B y − + + +
∂t ∂x ∂y ∂z ρ ∂y ρ ∂x ∂y ∂z
(2.29)
∂w ∂w ∂w ∂w 1 ∂p 1
∂τ xz ∂τ yz ∂τ zz
+ u + v + w = B z − + + +
∂t ∂x ∂y ∂z ρ ∂z ρ ∂x ∂y ∂z
(2.30)
As the previous equations show, we have 4 equations but more unknowns
(i.e. velocity (u, v, w), pressure, and stresses) than equations. Therefore, we
need to either simplify the previous equations or include additional equations.
Let us look at a special case, where the shear stresses are ignored. This is
the case for inviscid flow, where we ignore viscosity and, therefore, the shear
stresses.

2.3.1 Euler Equations

If we assume that the flow is inviscid, then, τ ji = 0. Therefore, the hydrodynamic
equations reduce to

∂u j
= 0 (2.31)
∂x j
∂u i ∂u i 1 ∂p
+ u j =−δ i3 g − (2.32)
∂t ∂x j ρ ∂x i
or, in vector format,
∇· u = 0 (2.33)
du ∂u 1
= + u ·∇u =− ∇p − gk ˆ (2.34)
dt ∂t ρ
In the previous equations, it is assumed that the only body force is gravity,
which acts in the z direction; therefore, B 1 = B 2 = 0, and B 3 =−g. Euler