Wind Energy

56 I. Sládek et al.
10.1.1 Turbulence Model
A simple algebraic turbulence model is applied to the system (10.2) in order
to close the problem. The turbulent diffusion coefficient K is expressed by
K = ν + νT , νT = l 2
�
�∂u�2
� �2 ∂v
κ(z + z0)
+ , l =
∂z ∂z 1+κ (z+z0)
(10.3)
l∞
where νT , ν are the turbulent, laminar viscosities and l refers to the Blackadar’s
mixing length, κ =0.41 is the von Karman constant, z0 is the surface
roughness length and l∞ represents the mixing length for z →∞.
10.1.2 Boundary Conditions
The mathematical model (10.2) is equipped with the following stationary
boundary conditions:
– Inlet: u-component is prescribed according to the power law with some
exponent and v = w =0.
– Wall-ground: the wall-function (10.4) or the no-slip condition (u = v =
w = 0 at wall) are applied.
– Outlet, top face and side faces of computational domain: homogeneous
Neumann conditions for all quantities are imposed.
The grid should be fine enough close to the wall to resolve all the gradients
when the no-slip condition is applied. However, this increases the CPU-cost
of such simulation due to a stability time-step limit for the explicit scheme
that is actually used, see Sect. 10.1.3. On the other hand, the wall-function
approach is less CPU-time consuming since the grid can be coarser at wall.
The first inner grid cell is placed within the near-ground layer of a typical
thickness about 50 m. The wall-function then reads
� u
u2 + v2 + w2 = ∗
κ log
� �
z1 + z0
, (10.4)
where u ∗ denotes the friction velocity and z1 refers to the distance of the
center of the computational cell (where the wall-function is applied) from the
wall.
10.1.3 Numerical Method
The mathematical model (10.2) is solved in the computational domain on a
non-orthogonal structured grids made of hexahedral computational cells. It
is expected to obtain a converged solution to the steady-state for all the
unknowns and for the artificial time t →∞.
z0