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

Swirl — where it appears

The azimuthal component of a flow along a pipe or a vortex core, usually collapsed onto a single swirl number. Above a critical value the flow can carry a standing axisymmetric wave and information can travel upstream, and the critical value is a property of the radial distribution rather than of the number.

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

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 The two long-wave speeds, and the swirl at which one of them stops. For uniform axial velocity the wave speeds follow from the criticality condition by a Galilean boost: c = W(1 ± 2S/j), with j the first zero of J1. The upstream-running root crosses zero exactly at S = j/2 = 1.9159, and above that swirl no disturbance can travel upstream — which is what subcritical and supercritical mean here and in an open channel.

The swirl that holds a wave still

A swirling flow down a pipe carries waves, and above a certain swirl one of them stops moving. Below it, a disturbance downstream can send information upstream; above it, the flow has outrun its own waves. The words are open-channel flow's words, and they are the same words for the same reason.

inviscid · Swirl

Named alongside it

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

Model limitActuator discAngular momentumThe Betz limitBragg hawthorneCirculationConservationControl volumeCriticalityEigenvalueFroude numberHydraulic jump

All concepts