Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Blade-element — where it appears

A strip of a rotating blade, treated as a two-dimensional section at the velocity and incidence its own rotation and the induced flow give it. Requiring that account of the force to agree with a momentum balance on the same annulus is what turns an actuator disc into a rotor with a finite number of blades.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

One calculation is about the animal; the other is about how fast it happens to be going. The mean lift coefficient required of each animal's wings, computed two ways, on a log scale. Treating the wings as fixed and flying them at the animal's forward speed gives answers spanning a factor of 29 — from 0.88 to 26 — because the number is governed by a speed that has nothing to do with how the animal makes its lift. Doing the flapping arithmetic gives answers spanning a factor of 1.90. And in a hover the fixed-wing calculation has no answer at all: there is no dynamic pressure, and no coefficient however large will do. That is the version of the famous claim that is actually true, and it is a statement about the calculation rather than about the bee.

The bee that cannot fly

The claim has a traceable origin and the calculation behind it was a real calculation done with the wrong velocity. Doing it with the right one gives a number an aerofoil might plausibly produce — and still leaves a gap, and the gap is what took another sixty years to close.

misconceptions · Bumblebee Where four insects have to turn round. Wagner's function — the fraction of its eventual circulation a wing has built after a stated distance of travel — with the half-stroke of each of four insects marked on it. Every one of them reverses while the curve is still climbing, so no insect wing ever reaches the circulation a steady calculation assigns it.

A calculation with no memory in it

The bee calculation is famous for using the wrong velocity. Done with the right one it still falls short, and the reason is structural: a quasi-steady sum is a statement about a wing that has always been going, and an insect's wing travels between two and five chord lengths before it turns round.

misconceptions · Bumblebee Same disc, same solidity, different number of blades, different answer. Thrust and power coefficients for six blade counts at a fixed solidity. They span a factor of 1.101 in thrust, and every one of those rotors is the same actuator disc: same area, same blade area, same tip-speed ratio. The disc theory cannot distinguish them because the blade count is not one of its variables.

A disc that knows no blades

Momentum theory replaces a rotor with a surface across which the pressure jumps, and gets the Betz limit, the induced velocity and the whole energy argument out of it. It has no chord, no section and no number of blades — and at one fixed solidity, two blades and twenty give thrust coefficients ten per cent apart.

circulation · Blade-element The loading over the disc in forward flight, at μ = 0.4. Section lift per unit span over the rotor disc, with the flight direction upwards, the advancing side to the right and the reversed-flow region outlined. The loading is not axisymmetric and cannot be: at this advance ratio the advancing blade meets 1.40 times the tip speed and the retreating one 0.60. Every quantity a hover calculation reports as a function of radius is here a function of two variables.

The side that cannot keep up

A hovering rotor is axisymmetric, so one radial distribution of circulation describes the whole disc. Move it forward and the advancing blade meets one and a half times the tip speed while the retreating one meets a half — and lift goes as the square of that. The rotor does not roll over, because the pitch is made a function of azimuth.

circulation · Rotor A blade passing the vortex it shed a passage ago. The lift the blade section feels as it passes a tip vortex at five per cent of the radius. The pulse is a doublet rather than a bump — upwash on one side and downwash on the other — so the blade is pushed one way and then the other in the width of a few chords.

A blade that flies through what it shed

A rotor blade meets the tip vortex the blade in front of it left, a fifth of a second earlier, at a few per cent of the radius. What it feels is a doublet — five and a half degrees of upwash and then five and a half of downwash within a few chords — and the peak goes as one over the miss distance.

circulation · Rotor The induced velocity, after a step in thrust. A rotor's induced velocity following a thirty per cent increase in thrust applied at fifty milliseconds. It does not jump: the air the disc has to accelerate has an apparent mass, and the response is a first-order climb to the new momentum-theory value.

The inflow that takes time to arrive

Momentum theory gives a rotor's induced velocity from its thrust, instantly. It does not arrive instantly: the air the disc has to accelerate has a mass, and the response is a first-order climb with a time constant of 33 milliseconds — a twentieth of the time the wake itself takes to convect a radius.

circulation · Blade-element

Named alongside it

The objects these essays reach for when they reach for this one.

Induced velocityCirculationActuator discLift coefficientMemory kernelModel limitModel validityRegimeFlapping flightLeading edge vortexMomentum theoremQuasi-steady

All concepts