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While a theoretical limit has long been established for the performance of asingle turbine, no corresponding upper bound exists for the power output from alarge wind farm, making it difficult to evaluate the available potential forfurther performance gains. Here we build a model describing the essentialfeatures of a large array of turbines with arbitrary design and layout, byconsidering a fully-developed wind farm whose upper edge is bounded by aself-similar boundary layer. The exchanges between the wind farm, theoverlaying boundary layer, and the outer flow are parameterized by means of theclassical entrainment hypothesis. We obtain a concise expression for the windfarm's power density (corresponding to power output per unit planform area), asa function of three coefficients, which represent the array thrust and theturbulent exchanges at each of the two interfaces. We assess the performance ofour model by comparing the predicted power density to field data, laboratorymeasurements and large-eddy simulations for the fully-developed regions of windfarms, finding good agreement. We extend our model to include the effect ofatmospheric stability on power output, by using an established parameterizationrelating entrainment coefficients to local Froude numbers. Our predictions forpower variation with atmospheric stability are in broad agreement with fieldmeasurements. To the best of our knowledge, this constitutes the firstquantitative comparison between an atmospheric-stability-dependent model andfield data. Finally, we consider an ideal limit for array operation, wherebyturbines are designed to maximize momentum exchange with the overlying boundarylayer. This enables us to obtain an upper bound for the performance of largewind farms, which we determine to be an order of magnitude larger than theoutput of contemporary turbine arrays.

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