Analytical Solution for the Optimal Spacing of Wind Turbines

The lateral and longitudinal spacing between individual turbines in a wind turbine array must be large enough to minimize the wake effects caused by an upstream turbine on those that lie downstream from it. Here, the flow downstream of a single wind turbine is examined by modeling its far-field development as a turbulent axisymmetric wake which is well described in the turbulence literature. In particular, the velocity defect profile in the wake is approximated by a Gaussian function in the radial coordinate. Scaling laws are used to derive closed-form solutions for the wake diameter, wake velocity defect, and wake power recovery as functions of downstream distance from the rotor. Our results show that at a downstream distance of 10 rotor diameters, the wake centerline velocity will recover to 77% of the free-stream value. It is also seen that the power within the wake recovers quickly for small downstream distances, but beyond about 10 rotor diameters, the rate of power recovery slows down. Implications for the optimal spacing of wind turbines are discussed based on these findings.