Induced Velocity in Hover (Momentum Theory)

Take a hovering rotor away and put a disc in its place. The disc has no thickness, no blades and no drag; all it does is add pressure to the air passing through it. Draw a control volume around the streamtube: air arrives from rest well above, accelerates through the disc, and leaves in a contracted wake below. Conserve mass, momentum and energy across that volume and out falls $v_i = \sqrt{T/2\rho A}$, with the wake far below moving at $2 v_i$ through half the disc area. That is the Rankine–Froude actuator disc, published for marine propellers in 1865 and refined in 1889, and carried into the helicopter case by Glauert in the 1920s and 30s.

Notice what is not in the derivation. There is no blade, no chord, no number of blades, no twist, no aerofoil section, no profile drag, no tip, no swirl in the wake. Momentum theory does not know the rotor exists. This is a strength as much as a weakness — it means the result applies to any device whatever that pushes air down through a circle of that area, which is why the identical control volume gives Betz his 16/27 limit for a wind turbine running the other way. A turbine takes energy out of the disc and a rotor puts it in, and they are two halves of one argument. It is also why the answer is a floor and not a prediction, a point the hover-power page makes at length.

Three honest qualifications on the number itself. It is a uniform average across the disc, and no real rotor produces uniform inflow — the real inflow is small at the root and largest near the tip, and the excess out there is precisely where the induced power the theory misses gets spent. It ignores tip loss: the pressure difference must go to zero at the free end of the blade, so useful thrust is produced over roughly $BR$ with the tip-loss factor $B \approx 0.97$, meaning about six percent of the disc area does nothing. This page names tip loss and deliberately does not model it, because the moment you start correcting momentum theory piecemeal you are better off measuring a figure of merit instead. And the wake is not a smooth column of air but a helix of blade-tip vortices you can often see on a humid day.

THIS IS A HOVER FIGURE, AND CARRYING IT INTO FORWARD FLIGHT IS THE THIRD OF THIS SHARD'S CLASSIC MISTAKES. Once the aircraft is moving, the disc is fed by the free stream as well as by its own suction, and the mass flow through it climbs enormously. The induced velocity is then governed by $v_i = T / (2\rho A \sqrt{V^2 + v_i^2})$, an implicit relation that rearranges to a quartic and has no closed form; it is solved by iteration, and at any decent cruise speed it falls away roughly as $T/(2\rho A V)$. This is why induced power dominates the hover and becomes almost negligible in cruise, and why a helicopter can hover at a weight it could not lift vertically at altitude but can fly away with once it has translational lift. There is no forward-flight page in this shard because there is no formula to put on one.

And a word about autorotation, which is the reason for a page that does not exist here. With the engine gone, the airflow reverses and drives the rotor, and a helicopter becomes a glider at a lift-to-drag ratio of roughly four — the glide distance page in the aerodynamics shard handles that perfectly well. What this site will not do is compute an autorotative descent RATE. The familiar rule of thumb, $V_d \approx 1.8 v_h$, is an empirical constant fitted across the vortex-ring state, and in that region momentum theory is not merely inaccurate, it is invalid: the control volume assumption fails outright, because the flow recirculates through the disc instead of passing once through it. Air comes back up around the outside of the rotor and down through it again, thrust collapses unpredictably, and adding collective makes it worse. That is the region between powered hover and established autorotation, it is the one that kills people, and printing a confident number where the theory does not apply is the wrong kind of help. Train for it, do not calculate it.