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

Roll-up — where it appears

The process by which a shed vortex sheet curls into a concentrated vortex under its own induction. It is where a sheet's fine structure is lost, and a computation of it depends on a smoothing length that has to be chosen and does not go away.

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

Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

circulation · Tip vortex A vortex sheet rolling up, at five stages. One period of an initially flat vortex sheet with a small perturbation on it, drawn at equal intervals. The perturbation grows, the sheet steepens, and the ends wind into a spiral. Nothing is added to the sheet after the first instant.

A spiral is a legible record

When a vortex sheet rolls up, the fluid in it can never change places: two points on a sheet cannot pass one another. So the arms of the spiral are a map of the initial sheet, in order, and the picture is a record of the roll-up rather than a snapshot of it.

inviscid · Vortex sheet

Named alongside it

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

CirculationModel limitCentroidConserved quantityInduced dragInitial conditionMaterial lineMeasurementMemory kernelModel validityRegularisationRolling moment

All concepts