Two-scale momentum theory for very large wind farms

A new theoretical approach is proposed to predict a practical upper limit to the efficiency of a very large wind farm. The new theory suggests that the efficiency of ideal turbines in an ideal very large wind farm depends primarily on a non-dimensional parameter $\lambda/C_{f0}$, where $\lambda$ is the ratio of the rotor swept area to the land area (for each turbine) and $C_{f0}$ is a natural friction coefficient observed before constructing the farm. When $\lambda/C_{f0}$ approaches to zero, the new theory goes back to the classical actuator disc theory, yielding the well-known Betz limit. When $\lambda/C_{f0}$ increases to a large value, the maximum power coefficient of each turbine reduces whilst a normalised power density of the farm increases asymptotically to an upper limit. A CFD analysis of an infinitely large wind farm with 'aligned' and 'displaced' array configurations is also presented to validate a key assumption used in the new theory.