An entrainment model for fully-developed wind farms: effects of atmospheric stability and an ideal limit for wind farm performance

While a theoretical limit has long been established for the performance of a single turbine, no corresponding upper bound exists for the power output from a large wind farm, making it difficult to evaluate the available potential for further performance gains. Here we build a model describing the essential features of a large array of turbines with arbitrary design and layout, by considering a fully-developed wind farm whose upper edge is bounded by a self-similar boundary layer. The exchanges between the wind farm, the overlaying boundary layer, and the outer flow are parameterized by means of the classical entrainment hypothesis. We obtain a concise expression for the wind farm's power density (corresponding to power output per unit planform area), as a function of three coefficients, which represent the array thrust and the turbulent exchanges at each of the two interfaces. We assess the performance of our model by comparing the predicted power density to field data, laboratory measurements and large-eddy simulations for the fully-developed regions of wind farms, finding good agreement. We extend our model to include the effect of atmospheric stability on power output, by using an established parameterization relating entrainment coefficients to local Froude numbers. Our predictions for power variation with atmospheric stability are in broad agreement with field measurements. To the best of our knowledge, this constitutes the first quantitative comparison between an atmospheric-stability-dependent model and field data. Finally, we consider an ideal limit for array operation, whereby turbines are designed to maximize momentum exchange with the overlying boundary layer. This enables us to obtain an upper bound for the performance of large wind farms, which we determine to be an order of magnitude larger than the output of contemporary turbine arrays.