Energy–consistent estimations of entrainment for fully-developed wind farms

The fluid flow within a fully-developed wind farm is a complex, three–dimensional field, which involves interactions between the individual turbine wakes and the boundary-layer shear. These interactions result in a transfer of turbulent momentum and kinetic energy from the ambient flow to the wind farm. In large wind farms, such transfers take place mainly in the vertical direction, resulting in an increase of the overall boundary layer roughness. Estimates of either the vertical fluxes or the effective roughness can be used to compute the mechanical energy extracted from the wind farm.

In the present work we propose the use of a horizontally–averaged entrainment coefficient (measured along the turbulent interface) as an equivalent expression of the effective roughness. For our calculations, we consider a fully developed, neutrally stable Atmospheric Boundary Layer (ABL) which is forced in a double periodic domain by a mean pressure gradient. The calculations are undertaken using the high-order finite-difference solver WInc3D/Incompact3D and compared to previous numerical studies and analytical models. Estimations for the horizontally–averaged entrainment coefficient are made by examining different wind farm layouts (aligned vs collocated) and their respective turbines spacing.