Physics
Betz limit
An actuator disc in the wind: brake the flow too little and you let it go, too much and the wake chokes the supply. The crest between them is 16/27 - and no blade can climb over it.
What happens
The whole module is one number and two closed forms: the induction factor a, Cp = 4a(1-a)² and Ct = 4a(1-a). The stream tube is continuity made visible - sqrt(1-a)R upstream, R at the disc, up to sqrt(2)R in the wake at the optimum - and the flow arrows are coloured by their own local speed. The rotor spin is kinematic garnish at a fixed tip-speed ratio, declared here; momentum theory itself dies at a = 1/2, where the wake would reverse, so the slider stops short of it.
Equations used in the module
Equations C_p = 4a (1 − a)², C_t = 4a (1 − a) max C_p = 16/27 = 0.593 at a = 1/3 (Betz) P = ½ ρ A U³ C_p, v_wake = U (1 − 2a)
