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

Velocity triangle — where it appears

The vector diagram relating the absolute flow velocity, the blade speed and the velocity relative to the blade in a turbomachine. The work a stage does follows from it through Euler's turbomachine equation, and the velocity a cascade's blades actually work in is the mean of the inlet and outlet ones.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Most power at exactly half the jet speed, found by search at 0.5000. The power a bucket takes from a jet, against how fast the bucket runs, for four deflection angles. Every curve is a parabola with roots at zero — where the force is greatest and the bucket is not moving — and at the jet speed, where the bucket is running away and there is no force at all. The peak is halfway between, at U = V/2, and it is there for every angle and every flow rate. A golden-section search that knows none of the algebra puts it at 0.500000.

Half the jet speed takes everything

A bucket standing still feels the largest force and does no work; a bucket running with the jet does no work either. Between them the power peaks at exactly half the jet speed, for every bucket shape and every flow rate — and at that speed a perfect bucket leaves the water motionless.

applied · Jet Deviation: how far the flow leaves from the blade angle, against solidity. The angle between the outlet flow and the blade, for a row of flat plates at 30° stagger meeting a flow at 45°. An open row barely turns the flow at all and the deviation is nearly the whole of the intended turning; a tight row guides it to within a thousandth of a degree of the blade angle. Nothing about the blade changed between the two ends of this curve.

A row is not a set of aerofoils

An isolated aerofoil's incidence is measured from the free stream. A compressor blade's cannot be, because an infinite row of identical blades above and below it has a circulation that is part of its own free stream — and the velocity the theorem uses is a vector mean that exists nowhere in the machine.

circulation · Cascade At 70 per cent speed the first stage runs at 0.64 of its flow coefficient and the last at 1.17. The flow coefficient of each stage of a compressor of 8 stages at 300 m/s mean blade speed, drawn for a flow coefficient of 0.5 and a work coefficient of 0.35, as a share of the value its blades were cut for, along the operating line a choked exit nozzle sets, at 110 per cent, 100 per cent, 90 per cent, 80 per cent, 70 per cent of design speed. At design speed every stage is at exactly one. Below it the front stages fall towards the stall limit (shaded below 0.82) and the rear ones rise towards the choke limit (shaded above 1.3): at 70 per cent the first stage is at 0.642 and the eighth at 1.170. Above design speed the pattern reverses, the front stages rising and the rear falling.

Matched at one speed and at no other

Every stage of a compressor passes the same mass flow, and the annulus behind each one is cut for the density the air will have reached there at design speed. Slow the shaft and the air is less dense than the metal expects, so the rear stages carry more volume than they were drawn for while the front ones starve — and below a definite speed no throttle setting keeps all of them working at once.

applied · Turbomachine

Named alongside it

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

Control volumeEfficiencyModel limitTurbomachineCascadeChokingCirculationConservationDensityDeviationFrame of referenceImpulse turbine

All concepts