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

The Biot–Savart law — where it appears

The relation giving the velocity a filament of vorticity induces at a distance, borrowed unchanged from electromagnetism. It is how a wake's effect on a wing is computed, and it is why a curved vortex moves itself.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The speed depends on the core; the slope does not. A ring's self-induced speed against the logarithm of the cutoff used to compute it. The line is a quadrature of the reduced integral; the dots are an independent sum of cross products over a hundred and twenty thousand straight segments. Both diverge as the cutoff is thinned — a filament of zero thickness would move infinitely fast — so no answer here is a ring's speed until a core model is chosen. The slope, Γ/4πR, is the same whatever is chosen, and is what the check measures.

A ring moves because it is bent

A straight vortex filament induces exactly no velocity on itself — every element's direction is parallel to the line joining it to the point being evaluated, and the cross product is zero. Bend it into a ring and it drives itself forward, at a speed that diverges logarithmically as the core is thinned.

circulation · Vortex ring The downwash approaches twice the value at the wing — from above. The downwash behind an elliptically loaded wing of aspect ratio 8, as a multiple of the induced angle at the wing itself, against distance in spans. Every account of tail sizing quotes a factor of two here. Two is the value at infinity: the trailing legs of the horseshoe system contribute a factor (1 + x/√(x² + a²)) which is one at the lifting line and two far downstream. Close behind, the bound vortex dominates and the field is much larger, and the curve comes down to its limit. A tailplane sits two or three chords behind, which on this wing is 0.31 of a span — where the factor is 2.46, a quarter above the number in the formula.

The surface in the wake

A wing has no opinion about its own incidence, which is why it needs a second surface behind it. How much that surface is worth depends on how much of the wing's downwash it is sitting in, and the factor of two everybody quotes for that is the value at infinity — where no tailplane has ever been put.

circulation · Moment A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

The knot a flow cannot untie

Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.

kinematics · Helicity The local sweep of the isobars, across the span. The sweep of the half-load line at each spanwise station, for four geometric sweeps. Over the middle of the span it is the wing's own sweep, which is the simple theory being right. At the root it collapses — by twenty-one degrees at a geometric thirty-five — and at the tip it falls again. That root region is where the shock forms first on every swept wing ever built, and it is why they have waisted fuselages.

The sweep a root does not have

Simple sweep theory is one of the cleanest arguments in aerodynamics: an infinite yawed wing cannot know about the velocity along its own span, so only the normal component matters. A real wing has a root and two tips, and at the root of a thirty-five-degree wing the isobars are swept fourteen.

circulation · Sweep

Named alongside it

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

CirculationModel limitAerodynamic centreBeltrami flowConserved quantityCore modelCritical machDownwashHelicityHelmholtz lawsHorseshoe vortexImpulse

All concepts