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

Perturbation — where it appears

A small departure from a known solution, expanded in the size of the departure so that each order can be solved in turn. It gives results a direct attack cannot, and it says nothing about how far the expansion can be pushed.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

A falling drop is a bun, not a tear. The shape of a drop pressed on by the air it is falling through, computed by matching the Legendre component of a sphere's own potential-flow pressure distribution against the change in curvature it produces. The result is oblate — flattened along the direction of travel — because the pressure is high at the poles and low round the equator. The teardrop of every illustration has the wrong symmetry entirely: a drop in free fall has no up, and one at terminal speed is being pressed on from in front.

The drop that is not a tear

A falling raindrop is flattened along the direction it is going, by the pressure of the air passing it rather than by its own weight, and the group that decides is the Weber number. The teardrop of every illustration has the wrong symmetry entirely — there is no up in the problem it is drawn for.

regimes · Drop shape A sinusoid, distorting on its way to a shock. A finite-amplitude sound wave at four fractions of the distance to shock formation, computed by inverting the implicit simple-wave solution. Each point of the waveform travels at its own speed, so the compressions catch up with the rarefactions ahead of them and the profile leans forward. At σ = 1 the front is vertical. The linear theory says the first panel is the answer at every distance, for ever.

Every compression becomes a shock in the end

Linear acoustics has no time scale in it, which is the sign that something has been thrown away. A 120-decibel tone shocks after three hundred metres and a jet engine after twenty; the distance goes exactly as the reciprocal of the amplitude, and nothing is exempt.

compressible · Characteristics A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

A swimmer that cannot go backwards

Taylor's waving sheet is the simplest self-propelled object in a viscous fluid, and its arithmetic contains a result that reads like a mistake — the work it does to travel a metre does not depend on how big its waves are. Doubling the amplitude quadruples both the speed and the power, and changes the bill for the journey by nothing at all.

viscous · Swimming Three exponents for one dimensionless group. The local slope of each error, measured over one decade at a time. The duct's is exactly 1, the long wave's is exactly 2, and the slender body's drifts from 1.900 to 1.733 across the range and never reaches either. The same geometric ratio, in three problems that look alike, and the third one has no exponent at all.

One group, three exponents

Lubrication theory, shallow water and slender-body theory are taught in three places and are one expansion in one group — a ratio of two lengths, with no speed, no viscosity and no fluid in it at all. The error is supposed to be second order. In three problems that look alike it is first order, second order, and an exponent that does not exist.

regimes · Slenderness The exact solution and its three approximations, at ε = 0.02. The outer solution is excellent everywhere except in a layer of width ε at the left, where it is wrong by a whole unit. The inner solution is excellent inside that layer and wrong everywhere else. The composite is their sum less the part they agree about, and it is within order ε of the exact solution across the whole interval — which is the entire content of matched asymptotics, drawn.

One formula for both ends

Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.

regimes · Crossover Three modes carry heat up to a ceiling of three; the rolls they were cut from keep going. The Nusselt number — heat carried across the layer over what conduction alone would carry — against r, the Rayleigh number over its critical value, for Lorenz's three-mode truncation, 1 + 2(r − 1)/r, and for steady rolls at the same wavenumber computed with 6 and with 42 temperature modes. At r = 2, r = 5 and r = 30 Lorenz gives 2.000, 2.600 and 2.933; the 42-mode rolls give 2.143, 3.323 and 5.970. Lorenz's value can never exceed three whatever the Rayleigh number; the rolls' keeps rising. The ceiling is not in the convection. It is in the three modes, which have only one way to thin the thermal layers, and half of it is used by the time the layer is twice past onset.

The truncation that cannot carry three times the heat

Lorenz's three modes give a convecting layer's heat flux in one line — one plus twice (r − 1) over r — and it can never reach three times what conduction carries. The same rolls computed with forty-two modes agree with that line exactly at onset, carry 7.1 per cent more heat at twice the critical Rayleigh number, and twice as much at thirty times it. The ceiling is not in the convection; it is in having one sine to draw the temperature with.

turbulence · Convection

Named alongside it

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

Model limitTruncationAsymptoticsBoundary layerConvergenceDimensionless numberScalingThresholdAdded massAeroacousticsBoundary conditionCharacteristics

All concepts