Wind Turbine Aerodynamics 3D

How much of the wind can a rotor take?

A rotor of radius R sweeps the area A = πR2. The kinetic power flowing through it is Pwind = ½ ρ A U3, so doubling the wind gives eight times the power and doubling the radius four times. The power coefficient Cp = P / Pwind measures how much the machine extracts. In one-dimensional momentum theory the rotor slows the wind to U(1 − a) at the disc and U(1 − 2a) far downstream, where a is the axial induction factor. Thrust and power are CT = 4a(1 − a) and Cp = 4a(1 − a)2; the maximum of the second is at a = 1/3 and equals 16/27 = 0.593, the Betz limit. The stream-tube picture in the 2D view widens exactly as continuity demands, A(x) u(x) = const.

Blade element momentum (BEM) theory

Real blades also spin the wake. Cut each blade into annular elements at radius r. The air meets an element with the relative velocity W, made of the axial part U(1 − a) and the tangential part Ωr(1 + a′), where a′ is the tangential induction. The inflow angle is φ = atan[ U(1 − a) / (Ωr(1 + a′)) ] and the angle of attack is α = φ − θ, where θ is the section pitch (built-in twist plus the collective pitch you set). The airfoil gives Cl(α) and Cd(α), which are projected on the axial and tangential directions:

Cn = Cl cosφ + Cd sinφ, Ct = Cl sinφ − Cd cosφ, dT = ½ρW2 B c Cn dr, dQ = ½ρW2 B c Ct r dr.

The same annulus must also satisfy momentum theory, dT = 4πrρU2a(1 − a)F dr and dQ = 4πr3ρU(1 − a)a′ΩF dr. Equating the two descriptions gives a and a′, found by iteration at every element (24 elements from 15 % of the radius to the tip). F is Prandtl’s tip and hub loss factor, which accounts for the finite number of blades B. For heavily loaded elements (CT > 0.96F) the Buhl empirical correction replaces the momentum relation. Finally P = Ω ∫ dQ and Cp = P / (½ρπR2U3).

Tip-speed ratio and stall

The tip-speed ratio is λ = ΩR / U. For a fixed blade, λ sets every inflow angle. At low λ the angle of attack exceeds the stall angle (14° in this model): lift collapses, drag soars and Cp drops (use the preset “Too slow”). At very high λ the angle of attack goes to zero, there is no lift to speak of and Cp falls again (“Too fast”). The optimum, near λ = 6.6 for this blade, is where the whole blade works near its design angle of 10° (third graph). The blade shape is the optimal one for 3 blades: c ≈ 8πr(1 − cosφd) / (B Cl,d) with φd = (2/3) atan(1/λdx) and the twist φd − αd (limited near the root).

Control: from cut-in to cut-out

Below rated wind speed the controller adjusts the rotor speed so that λ stays at the optimum (maximum power point tracking). Above rated power (2.5 MW at R = 40 m, scaled with R²) the blades are pitched to reduce the angle of attack, which keeps the power constant while cutting Cp and the loads; above 25 m/s the machine parks. Because P ∝ U3 the power curve rises steeply until rated and then flattens. Yaw error reduces the normal wind to U cosγ (a first approximation; measured losses are close to cos2).

Three in-page numerical checks

1. A scan of 4a(1 − a)2 finds the maximum 16/27 at a = 1/3, and the full BEM rotor with the drag and tip losses switched off at the design point must fall just below it (about 0.57 because of the finite hub and the cut root). 2. In every converged element with a < 0.38 the blade-element thrust equals the momentum-theory thrust, and the blade-element torque equals the angular-momentum torque (relative error below 10^-4). 3. The live operating point never exceeds the Betz limit.

Honest simplifications

The airfoil polar is an analytic model (linear lift with slope 5.9 per radian, a smooth transition to flat-plate behaviour after 14°, a parabolic drag polar), not wind-tunnel data; there is no dynamic stall, no 3D stall delay, no tower shadow, no wind shear or yaw dynamics, and no blade elasticity. The rotor speed follows the controller set-point through a first-order lag instead of integrating a torque balance with generator torque. The 3D page uses the same solver; its landscape and the animated flow are illustrative.