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

Free-streamline — where it appears

A streamline that bounds a region of still fluid at constant pressure, so that the speed along it is constant and its shape is part of the answer rather than given. Free-streamline solutions produce a drag in an inviscid theory, at the price of one number — the pressure inside the wake — that the equations do not supply.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The jet is 61.1% of the hole. Flow out of a slot in a plane wall, solved by Kirchhoff's free-streamline method. The outer curve is not a wall and not a guess: it is the streamline on which the pressure is ambient, and where it goes is part of the solution. It leaves the edge of the slot travelling straight down the wall and turns through ninety degrees, settling to a jet whose width is π/(π+2) = 0.6110 of the opening. Every streamline drawn is a level set of the streamfunction the conformal map supplies.

The hole that halves the flow

A jet leaving a sharp-edged hole is narrower than the hole, and by an amount that is not measured but computed. One geometry gives exactly one half from momentum alone; another gives exactly π/(π+2) from a conformal map in which the shape of the free surface is part of the answer.

applied · Jet The Kirchhoff flow past a flat plate. A uniform stream meeting a flat plate held across it, with two streamlines leaving the edges and never returning. Between them is a wake of fluid at rest at a constant pressure. The equations solved are the same equations that give d'Alembert's paradox for a closed body, and this flow has a drag coefficient of 0.8798.

Drag in the theory that forbids it

d'Alembert's paradox is a theorem about flows that close behind the body. Stop requiring that, let two streamlines leave the edges and never come back, and the same equations — no viscosity, no vorticity — produce a drag coefficient of 0.8798.

inviscid · Free-streamline The hodograph plane, where the unknown boundary is the known one. The same flow drawn in the plane of its own velocity, ζ = (u − iv)/U. The plate, whose shape is known in the physical plane, becomes a segment of the imaginary axis; the axis of symmetry becomes a segment of the real one; and the free streamline — whose shape nobody knows — becomes an arc of the unit circle, because the speed on it is exactly the free stream. The unknown and the known have changed places, which is why the problem can be solved at all.

Where the unknown boundary is the known one

A free surface is the hardest kind of boundary — its shape is part of the answer, so the region the problem is posed in is not known until the problem is solved. Draw the same flow in the plane of its own velocity and the shape becomes an arc of a circle, known in advance and exactly.

inviscid · Free-streamline Three bluff bodies, whose Strouhal numbers agree once the wake's width is used. The measured Strouhal number of each body, and Roshko's universal number formed with the wake's width and the speed on the free streamline that bounds it. The raw numbers span a factor of 1.462; the collapsed ones span 1.0011, with a mean of 0.16281. The shedding was never body-dependent — the length in the number was.

The frequency a wake chooses

Bluff bodies shed at Strouhal numbers from 0.145 to 0.212, and the spread is not a fact about shedding — it is a fact about which length went into the number. Change the length to the wake's own width and three bodies agree to a tenth of a per cent. Then let the body move, and the number stops deciding anything at all.

regimes · Strouhal Stalled from its leading edge, a flat plate keeps a quarter of its attached lift slope. The normal force on a flat plate over the dynamic pressure on its chord against incidence: Kirchhoff's stalled flow, by quadrature through its hodograph, with Rayleigh's closed form 2π sin α/(4 + π sin α); and the attached flat plate's 2π sin α. At 5° the stalled plate carries 0.128 against the attached 0.548; at 20°, 0.423; at 90°, 0.8798, Kirchhoff's drag.

A stalled plate keeps a quarter of its lift slope and moves its centre aft

Kirchhoff's free streamlines gave a plate held across a stream a drag with no viscosity in it. Tilt the plate, let the flow leave both its edges, and the same construction is a wing that has stalled from its leading edge. Rayleigh solved it in 1876, and the hodograph confirms him to seven figures: near zero incidence the stalled plate lifts at a quarter of the attached plate's rate, its centre of pressure sits at five sixteenths of the chord instead of a quarter, and its force is normal to it, so its lift-to-drag ratio is the cotangent of its incidence. A real stalled wing lifts more, by a factor the wake's pressure decides.

inviscid · Free-streamline

Named alongside it

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

Boundary conditionConformal mapWakeHodographModel limitPotential flowSeparationVena contractaBase pressureBernoulli's equationCavitationCentre of pressure

All concepts