A flame's speed limit holds only in silence

A flat flame is unstable at every size: the gas expanding through a premixed flame bends the approaching flow so that any long wrinkle in the front grows, until a curvature effect cuts off the short ones. A wrinkled flame has one cusp and a speed limit followed the growth to its end with the Michelson–Sivashinsky equation, the weakly nonlinear model of that instability. The wrinkles merge; the front becomes smooth arcs bulging into the fresh gas and meeting in sharp cusps; and in a domain of any width it ends as a single arc and a single cusp — an exact solution built from poles in the complex plane. Its speed in excess of a flat flame’s is $2N\nu(1 - N\nu)$ for its $N$ pole pairs, which rises in steps as the domain admits more poles and never exceeds a half. A wider flame is the same arc at a larger size, and no faster.

That essay was careful to say what the limit was a limit of: the steady fronts of the equation, the speed of the quietest flame of a given width. It named noise as the first thing to break it — new poles fed in from far up the imaginary axis make small cusps on the arcs before being swept into the main one — and asked how the speed rises once they are, and how small the noise must be for the quiet limit to be what a burner sees. This essay answers both by marching the equation with noise.

The equation, with something to hear #

The front’s displacement $\varphi(x,t)$ on a periodic domain obeys $\varphi_t + \tfrac12\varphi_x^2 = \nu\varphi_{xx} + I(\varphi) + \eta$. The Landau operator $I$ multiplies each Fourier mode by its wavenumber and is the instability; the curvature term $\nu\varphi_{xx}$ damps short waves; the nonlinear term is the front advancing along its own normal; and $1/\nu$ is the domain’s width measured in neutral wavelengths, the wavelength that neither grows nor decays. The new term $\eta$ is a white noise in time, kicking every Fourier mode in the growing band — the wavelengths longer than the neutral one — independently, with a strength $a$ per mode. Wavelengths shorter than the neutral one are damped whatever kicks them, and forcing them only adds jitter.

The equation is marched pseudospectrally, as in the essay before: the linear part implicit in Fourier space, the nonlinear part explicit and dealiased, the step shortened wherever the front’s slope is steep enough to need it. The front’s advance is read off its mean, which the noise does not touch, averaged over the last three-fifths of each run.

Three checks come before any result. With the noise switched off after a single disturbance at the start, the march settles on the pole front’s speed to eight parts in a hundred thousand: the noisy equation, with its noise removed, reproduces the quiet limit it is about to break. With every mode stable — a domain narrower than the neutral wavelength — and a small noise, each mode is a damped linear oscillator kicked at random, an Ornstein–Uhlenbeck process whose variance is known exactly for the scheme; the measured variances match it to two per cent. And halving the largest step moves the noisy speed by 2.4 per cent, the first-order error of the noise scheme, measured on the mean of three seeds so that sampling scatter neither hides nor mimics it; it is a tenth of every effect reported below.

A noisy front keeps making cusps #

The first figure puts a noisy front beside the quiet one, both twenty neutral wavelengths wide. The quiet front is the exact pole solution: one long smooth arc, one sharp cusp, nothing else. The noisy front, kicked by a disturbance of a millionth in its growing wavelengths, has the same overall shape and a train of small sub-cusps along its arc.

Each sub-cusp is the life of one disturbance. A kick on the arc is a wrinkle, and on a flame every wrinkle longer than the neutral wavelength grows. But it grows on a surface that is itself moving: the arc’s slope carries anything on it sideways, towards the cusp, and a wrinkle is swept into the main cusp and absorbed within a finite time. The noise keeps supplying new ones faster than the cusp can swallow them, and the front carries a standing population of them. Every one adds a little extra surface to the front, and a front with more surface burns more fresh gas per second. The noisy front here runs at 0.86, the quiet one at 0.50.

Sensitive to a part in a million million #

The second figure is the answer to the essay before’s question: how small does the noise have to be? At a width of 12.5 neutral wavelengths the quiet front runs at 0.499. A noise of $10^{-12}$ per mode — a disturbance a million million times smaller than the front’s own size — already adds half a per cent. At $10^{-10}$ the front is four per cent faster; at $10^{-8}$, a seventh; at $10^{-6}$, a third; at $10^{-3}$, twice its quiet speed; at $10^{-2}$, three times.

The extreme sensitivity has a clear source in the pole picture. The quiet front’s poles sit at definite heights above the real axis; a new pole, fed in by a disturbance, starts very high up — its height is the logarithm of the disturbance’s smallness — and falls towards the axis as it grows, on a time that depends on that logarithm. A disturbance a thousand times smaller arrives only a fixed, modest time later, and in the meantime it has still produced a cusp. The front hears noise on a logarithmic scale, which is why eleven decades of noise strength move the speed by a factor of three rather than by nothing or by everything. No real burner, whose flow is turbulent at the level of a per cent or so, is anywhere near the quiet end of the figure.

How one kick becomes a cusp #

The logarithm in the last section can be followed through a single disturbance. A kick of size $\varepsilon$ at a wavelength in the growing band grows exponentially, at the rate the instability gives that wavelength — up to $1/4\nu$, about three per unit of the equation’s time at this width. It does nothing to the front’s speed until it has grown to the size of the curvature scale, $\nu$, which takes a time proportional to $\ln(\nu/\varepsilon)$: a disturbance a thousand times smaller needs only about two more time units. Meanwhile the arc is carrying it sideways towards the main cusp at the arc’s own slope, and the arc’s length divided by that slope is the time it has.

A kick survives to become a sub-cusp if its logarithmic growth time is shorter than its sweeping time. Both are modest numbers, and the growth time depends on the kick’s size only through a logarithm, so kicks of wildly different sizes survive almost equally well. That is the whole of the flame’s sensitivity: not a weakness of the quiet front, which is stable to every disturbance it can absorb, but a race between two times in which the disturbance’s size barely matters. It is also why width helps: a longer arc gives every kick more time before the cusp swallows it.

A surface that amplifies whatever reaches it #

The arc of a wrinkled flame is an instance of something that turns up in quite different flows. A sloping ceiling carries a film that is unstable, but whose disturbances are swept downhill as they grow; such a film is a noise amplifier rather than an oscillator, and what it does depends on what it is fed. The flame’s arc is the same: every disturbance on it grows and every one is swept away, into the cusp. A quiet arc stays smooth because nothing is fed to it, and the pole front is the shape of that silence; a fed arc amplifies whatever reaches it into a stream of sub-cusps.

It is a different kind of unpredictability from the randomness that is not in the equations, where a deterministic system makes its own disorder, or from a flow that mixes with no randomness in it. The quiet Michelson–Sivashinsky front is not chaotic; it settles and stays settled. Its disorder in a real flame is borrowed from outside and amplified, and that is why its speed depends on how much disorder is borrowed.

A noisy flame has no speed limit #

The third figure is the other half of the result, and the part that matters most for large flames. The quiet front’s speed against width is a staircase that stops: a step each time the domain admits another pole pair, climbing to a half and staying there. The noisy front’s speed has no staircase and no ceiling. With noise of $10^{-3}$ it runs at 0.77 at eight neutral wavelengths, 0.92 at ten, 1.05 at 12.5, 1.21 at fifteen and 1.36 at twenty; with $10^{-6}$ it climbs from 0.53 to 0.86 over the same range. Over the factor of two and a half computed, the speed excess grows roughly as the square root of the width.

The reason is the same population of sub-cusps. A wider flame has longer arcs; a disturbance born on a longer arc has further to travel before the cusp absorbs it, and grows for longer on the way. The quiet front’s arc is the same shape at every size and so the same speed; the noisy front’s arc carries more, and larger, sub-cusps the longer it is. Large flames — the flame across a wide burner, the front crossing a fuel-air cloud — have been measured to accelerate as they grow, and the noisy equation is the simplest account that gives the growth without anything but the instability and a little disturbance to feed it.

A speed that flickers #

The fourth figure follows the speed in time rather than on average. Started from a flat front, each noisy flame takes about forty time units to wrinkle up and reach its mean, and then it does not settle: its speed flickers about the mean by a few per cent, the flickers growing with the noise. Each flicker is a burst of new cusps being born together on the arc and swallowed together by the main cusp. The quiet front’s speed, in contrast, is one steady number.

The flicker is a measurement of its own. The standard deviation of the windowed speed at noise $10^{-3}$ is seven per cent of the mean, at $10^{-6}$ about one per cent, and a flame’s speed measured over a short window scatters by that much for no reason but the population of sub-cusps passing through.

What the noise stands for #

The noise in the equation is a stand-in, and it is worth saying what for. In a real burner the fresh gas arriving at the flame is not uniform: turbulence in the approach flow, acoustic waves from the burner and its surroundings, small variations of mixture and temperature, all perturb the front at every scale. The flame itself makes a further source, since the air flows in against the heat and the vorticity generated at the curved front feeds back on it. The white noise of this calculation is the simplest version of all of these, and its strength $a$ is not a number any of them hands over directly; what the figures show is that the answer depends on it only logarithmically at the quiet end and that any realistic level of disturbance is far from the quiet end.

Speed is surface #

It helps to be plain about why wrinkles make a flame faster, because it is the mechanism every number here measures. A premixed flame burns into the fresh gas at a fixed speed normal to itself, set by its chemistry; the front is a surface, and each piece of it advances along its own normal. That is a wave nothing in it travels with: the front moves, the gas does not move with it, and the rate at which fresh gas is consumed across a given width is the burning speed times the front’s actual area. A wrinkled front has more area than a flat one across the same width, and it consumes more gas per second, so its mean position advances faster. The speed excess in every figure is the extra length of the front over the domain’s width, expressed as a speed.

The quiet front’s limit is therefore a limit on how much extra length a single smooth arc can have, and the noisy front’s excess is the length its sub-cusps add. That is also why the gains here are not small in practice. A front half as long again burns half as fast again, and turbulent flames, which are noisy fronts at every scale, burn several times faster than laminar ones for the same reason.

What was checked #

The fifth figure is the ledger. The noise-free march lands on the pole front’s speed, 0.49924 against 0.49920. The variances of two damped modes kicked by the noise match the scheme’s Ornstein–Uhlenbeck values to two per cent and a tenth of a per cent — a check that the noise enters with the strength it is meant to and nothing else is added. Halving the step changes the speed by 2.4 per cent in means over three seeds, the noise scheme’s first-order error, a small fraction of the effects above. The quiet speeds against which every noisy one is compared are the essay before’s exact pole solutions.

What the noisy equation leaves out #

Real expansion. The equation is derived for small gas expansion; a real flame expands its gas several times, in a flow that is nonetheless incompressible in every sense that matters away from the front, its cusps are sharper and its speed larger, and whether its noise sensitivity is larger or smaller is not decided here.

A single front in one dimension. The front is a curve in a plane, periodic across its width. A real front is a surface, its cusps are lines or points, and its noise acts in two directions at once.

The noise’s shape. White in time and flat across the growing band is a choice. Noise concentrated at long wavelengths, as a turbulent approach flow’s is, feeds the arcs differently from noise at short ones.

Finite runs. Every speed is a time average over a finite run with a sampling error of a per cent or two, larger for the stronger noises whose speeds flicker more.

The convention the numbers depend on #

Speeds are the front’s mean advance in excess of a flat flame’s, in the equation’s units, in which the quiet limit is a half. Widths are $1/\nu$, the domain’s period in neutral wavelengths. The noise strength $a$ is the standard deviation per unit time of the kick to each Fourier mode’s amplitude in the growing band, $1 \le |k| \le 1/\nu$; the mean mode is not kicked.

Who found it, and when #

Michelson and Sivashinsky derived the equation in 1977; Thual, Frisch and Hénon found its pole solutions in 1985. The sensitivity of the pole fronts to noise, and the sub-cusps it breeds, was worked out by Joulin in the late 1980s and early 1990s, with Cambray and others; Karlin and Sivashinsky studied the noise-driven acceleration of wide flames, and Denet among others computed its dependence on width. The measurements of large flames accelerating as they grow, which this mechanism has been used to explain, go back to the explosion experiments of the 1980s.

Still open: whether the growth with width ever stops #

Over the widths computed here, a noisy flame’s speed grows about as the square root of its width with no sign of stopping. Two things could stop it. The equation might saturate at larger widths, once the sub-cusps are as large as the arc can hold; or its weakly nonlinear derivation might fail, since a fast flame’s front is steep and the equation assumes it is not. The next calculation marches much wider domains — fifty and a hundred neutral wavelengths — at two noise levels, and asks whether the speed’s growth with width follows a power, bends over to a new ceiling, or runs into the front slopes at which the equation stops describing a flame.