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

Rotor — where it appears

A set of rotating blades that exchanges momentum with the air passing through them. Its inflow is not imposed but produced by the rotor itself and takes a measurable time to arrive, so a blade is always flying in the wake of what it did a moment ago.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The one term in ground effect that really is about carried air. The added mass of a plate approaching a plane, as a multiple of its free-air value. In free air a plate borrows ρπc²/4 per unit span — exactly 0.79 for a unit chord in unit density, which is the mass of the circle its chord spans. Near a wall the fluid in the gap must leave sideways through a narrowing passage, so it moves faster than it would in the open and carries more energy: the borrowed mass grows as the inverse of the gap and diverges as the gap closes. At half a chord it is 1.21 times, at a tenth 2.06, at a twentieth 3.12. This is a cushion in the ordinary sense — fluid that has to be got out of the way and resists being — and it is the term the steady argument has none of.

The cushion that is there after all

A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.

misconceptions · Ground cushion A blade root at 412.51 megapascals, and the chord is not in it. The centrifugal stress at a blade root against shaft speed, for one annulus of 0.36 m², in four materials. Integrating the blade's own weight outward gives σ = 2π ρ_b A N² with a taper relief — and the chord has cancelled, the blade count has cancelled, and every property of the gas has cancelled. What is left is an area times the square of a speed, capped by a material. The horizontal lines are each material's allowable stress, and where a curve crosses its own line is the fastest that annulus may be turned in that metal. At 12000 rpm this blade carries 412.51 MPa with a tip speed of 439.82 m/s.

The stress that picks the aerodynamics

The free term in a rotor's pressure rise wants radius and speed, and both are capped by something with no fluid in it. Integrate a blade's own weight outward and the root stress comes out as the annulus area times the square of the shaft speed — with the chord cancelled, the blade count cancelled, and every property of the gas cancelled.

applied · Turbomachine 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.

Added massBlade-elementCorrelationInduced velocityMemory kernelModel limitModel validityRegimeActuator discAnnulusCirculationControl

All concepts