Tip-Speed Ratio

Tip-speed ratio, $\lambda = \omega R / v$, compares how fast the blade tips travel with how fast the wind arrives. It is dimensionless, it is the parameter $C_p$ is really a function of, and it explains at a glance why a twenty-bladed farm windmill turns lazily while a three-bladed turbine's tips move at highway speed.

The reason is timing. Each blade passes through air that the blade ahead of it has already disturbed, and it works best on air that has recovered. More blades therefore want a slower rotor: the classic multi-bladed water pumper peaks near $\lambda = 1$, a three-bladed electricity turbine near 6–8, and a two-bladed machine near 8–10. Run a modern rotor too slowly and the air slips between the blades before they can act on it; run it too fast and each blade flies through the wake of the last. $C_p$ falls away on both sides of the peak.

This is why turbines are variable speed. A rotor held at a fixed rpm sits at its optimum $\lambda$ at exactly one wind speed and drifts off it everywhere else; a variable-speed machine tracks the wind, holds $\lambda$ constant below rated power, and keeps $C_p$ at its peak across the whole working range. The gain is several percent of annual energy — worth a great deal, given that power goes as the cube of wind speed and every point of $C_p$ is multiplying a very large number.

Two hard limits sit on top of the aerodynamics. Drag rises with the square of tip speed, and aerodynamic noise rises roughly with its fifth power, which is why onshore machines are capped near 80 m/s at the tip and offshore ones are permitted to run faster. Since $\omega = \lambda v / R$, a fixed tip-speed cap on a growing rotor forces the rpm down — which is exactly why the biggest turbines look so unhurried, and why they are in fact turning their tips at close to 300 km/h.