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

Actuator disc — where it appears

A model of a rotor as a discontinuity in pressure across a permeable disc, with no blades in it anywhere. It gives the momentum theory of propellers and turbines, and the Betz limit follows from it in a page.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

The tube widens because the air slows. The streamtube through an actuator disc at an induction factor of 0.333. The three radii are not drawn to taste: each is fixed by requiring the same mass to pass every station, and the slowest station is therefore the widest. The tube widening in front of a wind turbine is why some of the wind goes round it rather than through it, and it is the whole reason a disc cannot take everything.

The most a disc can take

A wind turbine cannot extract more than sixteen twenty-sevenths of the energy passing through the circle its blades sweep. That is not a limit on turbines — it is a limit on anything at all, and it follows from three conservation laws and no engineering.

applied · Actuator disc Efficiency is decided before the engine is chosen. Froude's propulsive efficiency against the ratio of jet speed to flight speed. It is 2/(1+σ) and nothing else — no engine, no fuel, no combustion. A turbojet with a jet at three times flight speed cannot exceed 50% however good its core is, and a propeller moving a great deal of air slowly is above 90% before anybody has designed anything.

A big slow push

The same thrust can be had from a lot of air moved a little or a little air moved a lot, and the two are not equivalent. One number decides which, it contains no engine, and it is why every airliner built since 1970 has a fan far larger than the machine driving it.

applied · Actuator disc Betz's ceiling, and the rotor that cannot reach it. The power coefficient of Glauert's optimum rotor against tip-speed ratio, with Betz's 16/27 drawn as the ceiling it is. The gap is wake rotation: a rotor that extracts power applies a torque, a torque leaves the wake spinning, and that rotational energy never reaches the shaft. It falls as the rotor is geared up and is never zero — which is why large wind turbines turn so slowly and yet have such fast tips.

The wake that has to spin

A rotor that takes power out of the wind must apply a torque to it, and a torque applied to air is angular momentum left behind. The axial theory has nowhere to put that energy, so Betz's ceiling is unreachable at every finite tip-speed ratio — and the gap is computable.

applied · Actuator disc 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 Momentum theory answers every descent rate except those between hover and twice the hover inflow. The induced velocity at a rotor disc against its climb speed, both in units of the hover induced velocity √(T/2ρA), at fixed thrust. The climb branch (thick) solves v(V + v) = 1 and is a streamtube for every climb and for hover, where v = 1. Continued into descent (dashed) it still has a root, but the air it describes leaves the tube at both ends. The windmill-brake branch (thin) solves v(V + v) = −1 and is real only for descent faster than two hover inflows, where it meets v = 1 again. Between V = −2 and V = 0 (shaded) neither is a streamtube. The faint diagonal is v = −V, where the rotor would need no power: it crosses the band and touches neither valid branch.

Between hover and twice the hover inflow

A rotor's momentum balance has an answer for every climb and for every fast descent, and none for descending at anything between zero and twice its own hover inflow. There one root sends air out of both ends of its streamtube and the other root is not a real number, and at each edge of the band one end of the tube stops moving — which is where the vortex ring state lives, and where a wind turbine's thrust coefficient of one sits.

applied · Actuator disc 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 An open rotor's slipstream shrinks; a duct holds it open. The radius of the slipstream behind a hovering rotor of radius R, against the distance behind the disc: for an open rotor, from the vortex-cylinder model, contracting towards R/√2 so that the wake ends with half the disc's area; and for a rotor in a straight duct, whose slipstream leaves at the full area, and in a duct that widens to 1.3 times it. The same thrust from a wider jet needs a slower one, and a slower jet wastes less energy.

A duct is worth the square root of two

An open rotor squeezes its slipstream to half its own area and pays for the fast jet that results. Put the same rotor in a straight duct and the slipstream leaves at the rotor's full area, the jet is slower, and hovering costs 29 per cent less power. The duct is not a passive guard: it carries half the thrust itself, on the suction round its inlet lip. In cruise almost all of the advantage disappears.

applied · Actuator disc The suction a diffuser needs to beat Betz on its own exit. The best power per unit of exit area against the suction held behind the exit, as a pressure coefficient −c. The curve is (2/3√3)(1 + c)^(3/2) up to c = ½ and the dots are a direct search. It starts at 0.385, sixty-five per cent of Betz's limit, and crosses 16/27 at c = 1/3 exactly — the pressure on the back face of Betz's open disc. A diffuser with no suction at its exit is worse than a bare rotor as large as its exit; one that beats the limit on its exit area is holding more suction there than the open disc holds behind itself.

A duct beats Betz only on the area it chooses

Put a wind turbine inside a flaring duct and it can take more than 16/27 of the wind's power through its rotor, which is the claim. Measure the same power against the duct's exit, the area the device actually fills in the wind, and a plain diffuser takes less than a bare rotor of that size would. It crosses the limit only when the duct holds a suction behind its exit, and the suction it needs is exactly the pressure on the back face of Betz's own disc.

applied · Actuator disc

Named alongside it

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

Momentum theoremControl volumeModel limitThe Betz limitEfficiencyKinetic energyStreamtubeThrustBlade-elementInduced velocityTip vortexCirculation

All concepts