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

Vortex ring — where it appears

A closed loop of concentrated vorticity that propels itself through the fluid by its own induced velocity. It carries the fluid inside it bodily rather than entraining it, its speed depends logarithmically on the size of its core, and Hill's spherical vortex is the limiting member of the family in which the core fills the whole ring.

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 Hill's spherical vortex. A sphere of rotating fluid travelling steadily through fluid at rest, drawn in the frame that moves with it. Outside the sphere the flow is the ordinary potential flow past a sphere; inside, the vorticity is proportional to the distance from the axis and the fluid recirculates. The two solutions match in value and in slope across the surface, and there is no body anywhere — the boundary is a streamline and nothing else.

The one rotational solution anybody can write down

A sphere of spinning fluid travelling steadily through fluid at rest, with no body anywhere in it — the boundary is a streamline and nothing else. It is exact, it is two lines long, and the reason it is the famous one turns out to be the reason it is the only one a real fluid can settle into.

inviscid · Euler rotational At a fixed impulse every orbit is a curve of constant energy. Axial separation of two coaxial rings against the radius of the ring that started smaller, for six pairs with the same total impulse, all starting in one plane. Pairs starting nearer equal (inner curves) trace closed loops: the separation swings from one sign to the other as the rings take turns, and the radius swings with it. Pairs starting further apart run off to one side and never return. The pale curves are the separatrix, the energy of two equal free rings — it passes through the plane of the start at a ratio of 0.340 and its arms reach out to infinite separation.

Two rings leapfrog only if they start alike

Two coaxial smoke rings passing through each other in turn is the most famous thing vortices do, and it is not what two rings do in general. Set them off from one plane with different radii and below a definite ratio the smaller one draws ahead and never comes back. Which of the two happens is decided before either ring moves, by whether two separate rings could hold the pair's energy.

circulation · Vortex ring The number of passes grows in proportion to Reynolds number. Passes the two rings make before their cores are close enough to merge, against the vortex Reynolds number, marched with the merging threshold at 0.24 of the core separation and at 0.30, and the closed form [(0.24 dₘᵢₙ)² − σ₀²] Re ÷ 4Tₚₐₛₛ, with the closest approach dₘᵢₙ and the time between passes Tₚₐₛₛ read from the inviscid pair. Both thresholds give a count in proportion to Re: one or two passes at a few thousand, ten or thirty at thirty thousand.

Leapfrogging rings end by merging, not by parting

Two smoke rings that leapfrog in an ideal fluid do it for ever, because their energy binds them. A real fluid drains the energy, and the obvious guess is that the rings drift out of the bound state and part. They do not: diffusion drains the pair's energy and the energy of every possible pair of free rings together, and the margin that binds them never goes negative. What viscosity does instead is fatten the cores until, as one ring threads the other, the two are close enough to merge. That takes a number of passes in proportion to the Reynolds number — one to three at the few thousand of a laboratory smoke ring.

circulation · Vortex ring

Named alongside it

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

Kinetic energyVortex dynamicsCore modelHamiltonianImpulseInduced velocityModel limitAngular momentumThe Biot–Savart lawBoundary conditionCirculationExact solution

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