Wind Turbine Performance in Very Large Wind Farms: Energy and Momentum Budgets for a Streamtube

In this study, we analyze a wind turbine in a large wind farm with the same streamtube approach used to derive the Betz limit. Using large eddy simulations (LES) of a periodic array of actuator disks in a channel flow, we present mean momentum and mean kinetic energy budgets for a range of thrust coefficients. Our results compare favorably with the theory of Nishino (TORQUE, 2016) for predicting wind turbine power coefficient from only the wind farm layout, local thrust coefficient, and ground roughness height. The maximum power coefficient found is smaller than the Betz limit for an isolated turbine, and is maximized by a smaller value of the thrust coefficient as well. The optimal thrust coefficient is found to be the one which maximizes the work done by Reynolds stresses on the surface of the streamtube. Increasing the thrust coefficient promotes turbulent mixing and faster wake recovery, but also slows the mean flow, resulting in the observed tradeoff. We compare with LES of an isolated actuator disk interacting with homogeneous turbulent inflow, and find that turbulence and the presence of other turbines both contribute to deviations from the idealized Betz picture.