Wind Energy

40 J.C.L. da Costa et al.
Sε = 1
2 ραCD
�
ε � �3 � � �
C4εβp
�U� − C5εβd
�U� ε , (7.2)
k
where CD and α are the tree drag coefficient and density foliage, and the
closure constants are defined according with the model being used.
Model βp βd C4ε C5ε
Svensson et al. [6] 1.0 0.0 1.95 –
Liu et al. [4] 1.0 5.1 1.5 0.4
k − ε − v 2 − f [3] 1.0 6.75 1.5 1.5
Note that when C4ε = C5ε, theSε formulation becomes
ε
Sε = C4ε
k Sk , (7.3)
which corresponds to the formulation in [6]. For simplicity, the equations
above were written in Cartesian coordinates, whereas the code is in general
coordinates, appropriate for computer simulation of complex terrain, cf. [1].
The ground surface was modelled by wall laws. The mean and turbulent
fields at the inflow boundary followed from horizontally homogeneous
atmospheric boundary layers. The velocity was tangential at the top and lateral
boundaries. At the outflow boundary the velocity was obtained by linear
extrapolation from the inner nodes, constrained by global mass conservation.
The pressure at the boundaries was obtained by linear extrapolation from the
inner nodes. At the lateral and top boundaries the turbulent quantities were
obtained by linear extrapolation.
7.3 Results
Computer simulations of the flow over a forested (CD =0.8, α =6.25 m −1 )
sinusoidal hill and along a flat clearing, downstream of a forest (CD =0.3 and
α between 6.0 and 57.5 m −1 ) region are compared with wind tunnel measurements
[4, 5].
The results are a sample of an ongoing appraisal of turbulence models of
flow over forested regions, which we are showing here with the main purpose
of illustrating the difficulties that one is currently faced with.
Flow over a modelled forested sinusoidal hill
The trees can account for an increase of both the turbulence and its dissipation
rate, yielding lower turbulence within the canopy height and higher
turbulence values above the tree top. This is the trend in Fig. 7.1, for both
the experimental data and the model results.
Downstream of the hill top, the k distribution along the vertical, and the
location of the maximum, is set by the mean vertical shear, due to the mixing