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

Leading edge suction — where it appears

The forward force carried by the square-root velocity singularity at a sharp leading edge, equal to half pi rho C squared. It is finite because the pressure diverges only as fast as the area vanishes, and whether the flow stays attached enough to collect it decides between a section with no pressure drag and one whose drag is quadratic in lift.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

A lift curve that keeps climbing to 49 degrees. The lift of a slender delta of aspect ratio 1, split into the potential term that an attached flow would give and the vortex term the separation adds, with a conventional wing's curve behind them. The delta's is nonlinear from the start and reaches 1.68 at 49 degrees, where a conventional wing stalled at fifteen. That is the whole design case for the shape: not that it is efficient — it is not — but that it still has lift at incidences where an ordinary wing has none, which is what a delta-winged aircraft needs on approach and in a turn.

Lift out of a failure

Separation is what ends an ordinary wing's lift curve. A slender delta with sharp leading edges separates on purpose, rolls the shed sheet into a pair of vortices above its upper surface, and takes most of its lift from the suction they induce — with a curve that climbs to forty-nine degrees.

circulation · Vortex lift Why a flat plate has no drag, drawn as a triangle. A flat plate at incidence with the three forces that must balance. Pressure can only act along the plate's normal, so the pressure force is the arrow perpendicular to the plate. Kutta–Joukowski says the resultant is perpendicular to the free stream. The difference between the two directions is the suction force at the leading edge, which acts forwards along the plate and is exactly L sin α. Without it the plate would have a drag of L sin α, and an inviscid fluid does not permit one.

A finite force from an infinite speed

Pressure on a flat plate can only act along the plate's normal. Kutta–Joukowski says the force is perpendicular to the free stream. Those two directions differ by the angle of attack, and the discrepancy is made up at a single point where the velocity is infinite and the area is zero.

circulation · Leading edge suction Four corners, and what the flow does in each. The local flow in corners of four different interior angles, drawn from the exact local solution ψ = r^(π/α) sin(πθ/α). The exponent of the speed is π/α − 1 and depends on nothing else: at a right angle the corner is stagnant, at a flat wall nothing happens, and at any angle greater than a straight line the speed has no bound at the corner. The last panel, at 360 degrees, is the flow round the edge of a plate.

Nothing turns a sharp corner

Near a corner the flow is fixed by the angle and by nothing else — not by the size of the corner, not by the flow far away, not by the fluid. The exponent is π/α − 1, and every sharp edge in aerodynamics is the one case where it comes out at minus one half.

inviscid · Wedge flow The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface.

The part of the flow inside the body

A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.

inviscid · Singularities The six bodies, drawn at the same scale. Two circles, two ellipses and two Joukowski sections, each at the incidence that gives it a circulation of exactly two. There is no family resemblance and no common parameter; what they share is one number, and the theorem needs nothing else.

One formula, and it does not ask what the shape is

Kutta–Joukowski gives the lift of any two-dimensional body from one number. Six bodies with nothing else in common are put at that number here and come out with identical lifts — and with pitching moments, load distributions and suction peaks that are not even close.

circulation · Lift An open rotor's slipstream shrinks; a duct holds it open. The radius of the slipstream behind a hovering rotor of radius R, against the distance behind the disc: for an open rotor, from the vortex-cylinder model, contracting towards R/√2 so that the wake ends with half the disc's area; and for a rotor in a straight duct, whose slipstream leaves at the full area, and in a duct that widens to 1.3 times it. The same thrust from a wider jet needs a slower one, and a slower jet wastes less energy.

A duct is worth the square root of two

An open rotor squeezes its slipstream to half its own area and pays for the fast jet that results. Put the same rotor in a straight duct and the slipstream leaves at the rotor's full area, the jet is slower, and hovering costs 29 per cent less power. The duct is not a passive guard: it carries half the thrust itself, on the suction round its inlet lip. In cruise almost all of the advantage disappears.

applied · Actuator disc Losing the leading-edge suction doubles a slender wing's drag due to lift and multiplies a long wing's by about half its aspect ratio. The drag due to lift with no leading-edge suction, Cₗ²/Cₗα, over the induced drag a full suction leaves, Cₗ²/(πAe), against aspect ratio, from a vortex lattice for rectangular, delta and elliptic flat wings; with the closed form 1 + √(1 + A²/4) that Helmbold's lift slope gives an elliptic wing. Rectangular: 2.15 at A = 1, 3.46 at 4, 5.33 at 8, 7.11 at 12; delta: 2.22 at 0.5 and 2.83 at 2. The slender limit is 2.

A wing that cannot hold its suction gains little from its span

The leading-edge suction is a small force concentrated at one edge, and on a finite wing it decides what the aspect ratio is worth. With it, the drag due to lift is the induced drag, which falls as the span grows. Without it, the drag due to lift is the lift tilted back by the incidence, and that falls only as fast as the lift slope rises — which stops at the section's 2π. A slender delta that loses its suction pays exactly twice its induced drag; a wing of aspect ratio eight pays more than five times it. Without the suction no span can lift a wing's best lift-to-drag ratio past a ceiling set by the section alone.

circulation · Leading edge suction

Named alongside it

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

Induced dragModel limitSeparationAspect ratioCirculationConformal mapd'Alembert's paradoxDrag polarKinetic energyKutta–Joukowski theoremPotential flowSuction peak

All concepts