Wind Turbine Blade-Element Momentum

Raising the turbine...

Presets

Live readouts

Graphs

Cp vs tip-speed ratio for three pitch settings (Betz limit in green)

Steady power curve and pitch schedule

Spanwise induction factors a and a-prime (x10)

Spanwise angle of attack, lift coefficient and tip-loss factor

Wind speed and rotor speed vs time

Electrical power and pitch vs time

Spanwise power distribution and chord

Cp map: tip-speed ratio (x, 2 to 14) vs blade pitch (y, -4 to +20 deg); bright = high Cp, ring = operating point

Numerical checks

running...

How to use

Choose a preset, then change wind, rotor radius, blade count, design tip-speed ratio and the control strategy. Press Gust for a gust. The stream tube widens as the rotor slows the wind. The lower right panel shows the velocity triangle at the blade station you pick (r/R slider): see the angle of attack, lift and drag. The graphs show Cp against tip-speed ratio, the power curve, spanwise induction and angle of attack, and the time response. Space pauses, R resets.

Drag to orbit, wheel to zoom, camera buttons for presets.

The science

Momentum theory and the Betz limit. An ideal actuator disk that slows the wind from V to V(1-a) at the rotor and V(1-2a) far downstream extracts

P = 2 ρ A V³ a (1-a)², C_p = 4a(1-a)², C_T = 4a(1-a)

which peaks at a = 1/3 with C_p = 16/27 = 0.593, the Betz limit: no rotor can capture more than 59.3 % of the kinetic power of the wind passing through its area. The graph at the bottom left of the 2D view shows the corresponding velocity and pressure along the axis. The wake stream tube expands to radius R √((1-a)/(1-2a)).

Blade element momentum (BEM). Each annulus of the rotor is treated as an airfoil section whose local flow combines the axial speed (1-a)V and the tangential speed (1+a')Ω r. With inflow angle φ = atan[(1-a)V / ((1+a')Ω r)] and angle of attack α = φ - (twist + pitch), the section forces give normal and tangential coefficients C_n = C_l cosφ + C_d sinφ, C_t = C_l sinφ - C_d cosφ. Equating blade forces with momentum change gives, with solidity σ = B c / (2 π r) and Prandtl tip/hub loss factor F,

a = 1 / (4F sin²φ / (σ C_n) + 1), a' = 1 / (4F sinφ cosφ / (σ C_t) - 1)

solved by iteration for every station (with the Buhl correction when a > 0.4). Thrust, torque and power follow by integrating the section forces over the span: P = Ω Q, C_p = P / (½ ρ π R² V³). The blade chord and twist come from the Betz-Glauert optimum for the chosen design tip-speed ratio and angle of attack. The check below compares the numerical Glauert optimum with the textbook value, and the BEM result with the optimum for the same lambda.

Operation. Below rated wind speed a generator torque Q = K Ω² with K = ½ ρ π R⁵ C_p,max / λ³ keeps the rotor at the best tip-speed ratio (maximum power point tracking). Above rated wind a PI controller pitches the blades towards feather to hold rated rotor speed and power. The rotor obeys J dΩ/dt = Q_aero - Q_gen. Wind is the mean plus an Ornstein-Uhlenbeck turbulence of the chosen intensity (integral time scale 12 s) and an optional gust. Yaw misalignment is modelled as V cos(yaw) normal to the rotor (power falls roughly as cos³). Annual energy uses a Weibull wind distribution with shape 2 (Rayleigh) and the steady power curve.

Simplifications. Section lift and drag are an illustrative thin-airfoil model with flat-plate stall, no Reynolds-number or 3D stall-delay effects; no tower shadow, wind shear, blade flexibility, dynamic inflow or wake skew beyond cos(yaw); generator efficiency fixed at 94 %; drive-train inertia scales with R⁵ (illustrative). Results show trends and orders of magnitude, not a certified power curve.

3D display notes. The blade shape is generated live from the optimum chord and twist of the BEM design (so changing radius, blade count, design tip-speed ratio or chord scale changes the blades); pitch rotates each blade about its span axis. The wake tube and tip vortices use actuator-disk wake expansion with the thrust coefficient from the BEM solution and ignore wake skew. Tower, nacelle, terrain, trees and baked textures were made in Blender.