A simple stationary semi-analytical wake model

We present an idealized simple, but fast, semi-analytical algorithm for computation of stationary wind farm wind fields with a possible potential within a multi-fidelity strategy for wind farm topology optimization. Basically, the model considers wakes as linear perturbations on the ambient non-uniform mean wind field, although the modelling of the individual stationary wake flow fields includes non-linear terms.

The simulation of the individual wake contributions are based on an analytical solution of the thin shear layer approximation of the NS equations. The wake flow fields are assumed rotationally symmetric, and the rotor inflow fields are consistently assumed uniform. Expansion of stationary wake fields is believed to be significantly affected by meandering of wake deficits as e.g. described by the Dynamic Wake Meandering model.

In the present context, this effect is approximately accounted for by imposing suitable empirical downstream boundary conditions on the wake expansion that depend on the rotor thrust and the ambient turbulence conditions, respectively. For downstream distances beyond approximately 10 rotor diameters (at which distance the calibrated wake expansion boundary conditions are imposed), the present formulation of wake expansion is believed to underestimate wake expansion, because the analytical wake formulation dictates the wake expansion to behave as x1/3 with downstream distance, whereas wake expansion as primary controlled by wake meandering develops approximately linearly with the downstream distance.

The link from a non-uniform wind farm wind field, consisting of linear perturbations on the ambient non-uniform mean wind field, to a fictitious uniform wake generating inflow field is established using two different averaging approaches – a linear and a non-linear. With each of these approached, a parabolic system are described, which is initiated by first considering the most upwind located turbines and subsequently successively solved in the downstream direction.

Algorithms for the resulting wind farm flow fields are proposed, and it is shown that in the limit of very large downstream distances, the simulated field is shown to recover to the undisturbed ambient wind field upstream the wind farm.