Page 221 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

210 Fundamentals of Ocean Renewable Energy

8.3.1 Turbulence Closure
For large Reynolds numbers (>2000), the flow velocity experiences fluctuations
around the mean velocity. Common numerical models cannot simulate these
fluctuations, because they occur over very small timescales (e.g. fractions
of seconds). Alternatively, they replace the velocity by mean and fluctuating
components as follows

u = u +´u (8.29)
where u is the time averaged and ´u is the fluctuating part. Tidal models simulate
the temporal average rather than the actual velocity (u) to avoid this issue. The
argument is that these small and rapid fluctuations in velocity are not important,
compared with the mean velocity that is of relevance to tidal energy generation.
The time average of the fluctuating velocity is zero. Therefore, if the Navier-
Stokes equations were linear, we could just replace velocities by time averaged
velocities. To make this point clearer, consider the continuity equation, which is
a linear equation. Taking a moving time average leads to

∂u ∂v ∂w
+ + dt = 0 (8.30)
T ∂x ∂y ∂z
∂(u +´u) ∂(v +´v) ∂(w +´w)

+ + dt = 0 (8.31)
∂x ∂y ∂z
T
Also, for the x component of velocity (and similarly for the other components),
we have

∂u ∂ T udt ∂ T ´ udt ∂ T udt

dt = + = + 0 (8.32)
T ∂x ∂x ∂x ∂x
Because the average of the fluctuating velocity is zero, the time averaged
continuity equation simply becomes
∂u ∂v ∂w
+ + = 0 (8.33)
∂x ∂y ∂z
Unfortunately, this is not the case for the nonlinear parts of the momentum
equation. For instance, if we just consider the convective acceleration term in
the momentum equation, (see Chapter 2) we have
∂u ∂(u +´u)

u dt = (u +´u) dt (8.34)
T ∂x T ∂x
∂u ∂ ´u

= u dt + ´ u dt + 0 + 0 (8.35)
T ∂x T ∂x
∂u 1 ∂ ´u
2
= u dt + dt (8.36)
T ∂x T 2 ∂x
2
Because the time average of ´u is not zero (in contrast to ´u), additional terms
appear in the momentum equation. For instance, the convective acceleration
term after time averaging becomes