Helical Wind Rotor

What is modelled

Each blade of a vertical-axis rotor sees a wind that is the sum of the free stream and its own motion, so its angle of attack swings through every value in one turn and its pull along the track is strongest at the front and the back. A straight blade therefore pushes in pulses. Twisting the blade into a helix spreads its pull over several azimuths and smooths the sum.

Equations

• Relative flow W = U (1 − a) √((λ' + cos θ)2 + sin2θ), α = atan2(sin θ, λ' + cos θ), with λ' = λ/(1 − a).
• Tangential force Ft = ½ ρ c h W2 (Cl sin α − Cd cos α), torque Q = R Σ Ft, averaged along the twisted span.
• Power coefficient Cp = ω Q̄ / (½ ρ (2 R H) U3), Betz 4a(1 − a)2 ≤ 16/27.
• Twist factor of a harmonic sinc(k Φ / 2), which vanishes for N blades at Φ = 2π/N.

Checks run in the page

• The mean torque is the same for any twist.
• The N-per-turn harmonic follows the sinc law and vanishes at one blade pitch of twist.
• The Betz maximum and the peak angle of attack sin(alpha) = 1/lambda.
• Power equals angular speed times mean torque.

Honest simplifications

One uniform induction factor stands in for the wake, the shadow of the front half on the rear half is ignored, the lift curve is a thin-airfoil line with a simple stall blend and the blade end losses and the arms are left out. It shows the trends, not a design.

• Press Straight 3 blades, then Helical 3 blades: the pulsing torque flattens and the mean stays put.
• Press Slow, stalled and watch the red blades and the collapsed Cp.
• Drag the twist slider and find the dips of the ripple plot at one blade pitch.