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

Dilatation — where it appears

The rate at which a material element's volume grows, per unit volume, which is the divergence of the velocity. Continuity makes it minus the material rate of change of density, so an incompressible flow is one in which each parcel keeps its own density rather than one whose density is uniform.

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

A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

kinematics · Transport theorem A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Incompressible is not a property of the fluid

A flow whose density varies three to one across it, with a divergence of 10⁻¹⁰ everywhere. And a flow of air at Mach 0.1, whose divergence is three per cent of U/a and which every textbook calls incompressible. The word is about what the flow does to a parcel's volume, and about nothing else.

kinematics · Dilatation The gas flows in, against the heat. The radial velocity of the gas, in units of α/a, at the same four times — from the motion of the gas shells themselves — with the heat flux, reversed in sign and in the same units, drawn as dots. The two coincide: at constant pressure the gas moves at (γ − 1)/(γp) times the heat flux, against it. Heat leaves the spot outward and the gas comes in, everywhere.

The air flows in against the heat

A gas moving at a ten-thousandth of the speed of sound is as incompressible as anything gets, by the usual arithmetic. Heat one spot of it and let the spot cool: the gas changes its volume threefold, and it flows towards the hot spot while the heat flows out, at a speed fixed by the heat flux alone. At constant pressure, a joule is a volume.

kinematics · Dilatation The unburnt gas bends round a wrinkle, and the bending makes it grow. Streamlines of the unburnt gas flowing up towards a flame with a small sinusoidal wrinkle, one wavelength across, for a density ratio of seven, with the wrinkle exaggerated. The expansion behind the flame pushes back on the gas ahead of it where the flame bulges forward, so the streamlines spread there and the gas arrives slower, and they crowd together where the flame lags, so the gas arrives faster. A flame that burns into the gas at a fixed speed therefore advances further where it was already ahead.

A flat flame is unstable at every size

A flame expands the gas it burns, and the expansion pushes back on the fresh gas ahead. Where the flame bulges forward the fresh gas slows, so the bulge burns further forward; where it lags, the gas speeds up and it falls further back. Every wrinkle grows, the shorter ones faster, and what finally gives a real flame a size is how its burning speed responds to its own curvature.

kinematics · Dilatation

Named alongside it

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

Mass conservationModel limitIncompressibleMaterial derivativeCompressibilityDensityDivergenceMach numberConservationControl volumeDeformation gradientDispersion relation

All concepts