Dilatation — the series · Fluid Flow & Aerodynamics

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.

part 1 · kinematics

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.

part 2 · kinematics

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.

part 3 · kinematics

A wrinkled flame settles into arcs meeting at a cusp. The steady front of a flame in a periodic domain 5, 10 and 20 neutral wavelengths wide, from the exact pole solution of the Michelson–Sivashinsky equation, with the burnt gas below and the flame advancing upwards; each is drawn across one period, scaled to the same width. Every one is a single smooth arc bulging into the fresh gas, meeting its neighbour in a sharp cusp pointing back into the burnt gas, and in these units the three arcs nearly coincide: only the cusp sharpens as the domain widens. In physical units the arc's depth grows in proportion to the domain's width, so the three flames are the same shape at three sizes.

A wrinkled flame has one cusp and a speed limit

The linear theory of a flame says every long wrinkle grows and none is favoured. The weakly nonlinear theory — the Michelson–Sivashinsky equation — says where the growth goes: small wrinkles merge, the front settles into smooth arcs bulging into the fresh gas and meeting in sharp cusps, and in a domain of any width it ends with a single arc and a single cusp. That front is an exact solution made of poles in the complex plane, and its speed is a closed form that rises in steps as the domain admits more poles and then stops: beyond about five neutral wavelengths a wider flame is no faster, because it is the same shape at a larger size.

part 4 · kinematics

A quiet flame has one cusp; a noisy one keeps making more. Two flame fronts twenty neutral wavelengths wide, burning upwards, drawn apart for clarity: above, the quiet front, the exact pole solution with one cusp per period and the most poles it can hold; below, the same flame with a noise of a millionth kicking its growing wavelengths. The noise keeps seeding small wrinkles on the smooth arcs; each grows as it is swept along the arc into the cusp, so the noisy front carries a train of sub-cusps the quiet one never has, and advances at 0.862 against the quiet 0.5.

A flame's speed limit holds only in silence

A quiet wrinkled flame settles into a single cusp and a speed it never exceeds, however wide it grows. Add noise and the limit is gone. A disturbance of one part in a million million already speeds the front up; one in a million makes it a third faster; and a noisy flame, unlike a quiet one, keeps getting faster as it gets wider, because every added width is room for more wrinkles to be born on its arcs and swept into its cusp. The quiet flame's speed is a property of a silence no burner has.

part 5 · kinematics

Out to a hundred neutral wavelengths the noisy flame is still speeding up. The front's speed in excess of a flat flame's against its width, on logarithmic axes, at noise strengths of a thousandth and a millionth, with the quiet pole front's and a power fitted from 20 to 100. At a thousandth the speed climbs from 1.05 at 12.5 to 2.71 at 100; at a millionth from 0.668 to 1.71. Both follow the 0.42 and 0.43 power of the width to within 4 per cent, and neither bends over; the quiet front stays at a half.

A noisy flame is still speeding up a hundred wavelengths wide

A quiet wrinkled flame gains a fixed, bounded amount of speed over a flat one, however wide it is. A noisy flame keeps getting faster as it widens, and the open question was whether that ever stops. Out to a hundred neutral wavelengths it does not. The speed grows as about the 0.43 power of the width at two noise strengths a thousandfold apart. The noise sets only the prefactor, and the front's slope grows with the speed until it leaves the range the equation was derived for — which is where the growth stops being the equation's to decide.

part 6 · kinematics