A noisy flame is still speeding up a hundred wavelengths wide

A flame’s speed limit holds only in silence took apart a result that had looked complete. A wrinkled flame has one cusp and a speed limit had solved the Michelson–Sivashinsky equation exactly. Its steady fronts are smooth arcs meeting in a single sharp cusp per period, built from poles in the complex plane, and they run faster than a flat flame by an amount that never exceeds a half, however wide the flame. Then noise was added — a random kick to every growing wavelength, a millionth or a thousandth in size — and the limit was gone. Small wrinkles were born on the arcs, swept along them into the cusp and swallowed, and a flame full of them ran faster. At a noise of a thousandth the speed climbed from 0.77 at a width of eight neutral wavelengths to 1.36 at twenty, about as the square root of the width, with no sign of stopping.

The question it left was whether it stops. There were two ways it might. The equation itself might saturate: once the wrinkles born on an arc grow to be as large as the arc can hold, more width might bring no more speed. Or the equation might stop being a description of a flame, because it is derived for a gently wrinkled front and a fast front is a steep one. The calculation the essay named was the obvious one: much wider flames, fifty and a hundred neutral wavelengths, at two noise levels. Here it is, and neither stop arrives first in the order expected.

Six widths and two noises #

The equation is the same, and so is the scheme that marches it. On a periodic domain, the front’s displacement $\phi(x,t)$ obeys

$$\phi_t + \tfrac12\phi_x^2 = \nu\phi_{xx} + I(\phi) + \eta,$$

where $I$ is the Darrieus–Landau operator that makes long wrinkles grow, $\nu$ the curvature term that cuts off short ones, and $\eta$ the noise, white in time and kicking every growing wavelength with the same strength. The domain’s width in neutral wavelengths is $1/\nu$. A flat flame is unstable at every size derived where those terms come from: the gas expanding as it burns, and the front’s own response to its curvature.

The widths are 12.5, 20, 33⅓, 50, 80 and 100 neutral wavelengths, and the noise strengths a thousandth and a millionth. Each run marches three hundred time units from a flat front and averages the speed over the last hundred and eighty, by which time every width has settled to a speed it flickers about. The grid grows with the width, from 256 points to 2,048, keeping about twenty-five points per neutral wavelength as the earlier calculation did. The widest runs take ten minutes each to march, so the speeds are stored, and the narrowest is re-marched each time the calculation is checked and must come back to the stored value to the last digit — the noise is seeded, so the march is exactly repeatable.

The growth does not bend over #

The first figure is the answer, on logarithmic axes so that a power law is a straight line and saturation a line bending over to flat. At a noise of a thousandth the speed rises from 1.05 at a width of 12.5 to 1.36 at 20, 1.72 at 33, 2.03 at 50, 2.45 at 80 and 2.71 at 100. At a millionth it rises from 0.668 to 0.860, 1.07, 1.34, 1.58 and 1.71. Both are straight. Fitted from twenty neutral wavelengths to a hundred, the first grows as the 0.42 power of the width and the second as the 0.43, and each fit passes within four per cent of every point. The quiet front, drawn for comparison, stays at a half.

So over a factor of five in width beyond where the earlier essay stopped, the growth neither saturates nor slows enough to see. The earlier essay’s square root was close. A half fits the narrowest pair, 12.5 to 20, at both noise levels — 0.54 — and the wider flames settle a little below it. A flame eight times as wide as one at 12.5 neutral wavelengths runs about two and a half times as fast in excess of a flat flame, at either noise.

What the straight line means physically is that the mechanism the earlier essay described has no natural size. A wrinkle is born on an arc, grows while it is carried along the arc towards a cusp, and is swallowed. A wider flame has longer arcs, so a wrinkle has longer to grow before it arrives, and arrives larger; and a wider flame also has room for more cusps, so the arcs do not simply lengthen without limit. The two effects together give a power, and the calculation says that power is a little under a half and does not change between twenty neutral wavelengths and a hundred.

The local power scatters but does not fall #

A fit over five widths can hide a slow bend, so the second figure takes the power apart: the slope of the logarithm of the speed against the logarithm of the width, between each pair of neighbouring widths. Saturation would show here as a fall towards zero. Between 12.5 and 20 both noises read 0.54. Beyond twenty the values scatter between 0.36 and 0.54, with no trend in either series.

The scatter is the measurement’s own. Each speed is an average over a hundred and eighty time units of a quantity that flickers by two to seven per cent from one twenty-unit window to the next, so it carries an uncertainty of one or two per cent. Between widths as close as 80 and 100, a ratio of 1.25, that becomes an uncertainty of about a tenth in the local power. The two last points of the millionth series, 0.36, sit below the fit by about that much, and the thousandth series’ last point, 0.44, sits on it. A bend that set in beyond eighty neutral wavelengths and was slow enough to hide in that scatter cannot be ruled out by six widths. A saturation of the kind the earlier essay imagined — a ceiling reached once the wrinkles fill the arcs — would have been visible by now.

There is a parallel worth drawing in a quite different calculation. At higher orders the bottleneck hides intermittency found that a local slope read off a finite range can mislead by more than the effect it is meant to measure, and the range a real Reynolds number does not have measured how little room a laboratory flow gives a power law to show itself. A flame a hundred neutral wavelengths wide has one decade of width above the narrowest flame here, and the exponent it gives is a measurement over that decade, not a statement about infinite flames.

The noise sets the prefactor, not the power #

The third figure divides one noise level’s speed by the other’s at each width. A thousand times the noise makes the flame between 1.52 and 1.60 times as fast, at every width from 12.5 to 100, with no trend. The noise strength moves the whole line in the first figure up or down without tilting it.

That is a statement about how weakly the speed depends on the noise. Three decades of noise buy about half as much speed again, so the speed grows roughly as the logarithm of the noise, or as a very small power of it — the sixteenth or so. It confirms what the earlier essay found going the other way: that noise many orders of magnitude too small to see already breaks the quiet limit. Once the speed has escaped the limit, the strength of the noise that freed it hardly matters. What decides how fast a noisy flame runs is overwhelmingly its width.

Where the noise comes from in a real flame matters less than that it is there. Here it is an imposed random kick. In a burner it is the flow’s own unsteadiness, the mixture’s small inhomogeneities, sound — and, in a large enough flame, the front’s own sensitivity to the smallest differences in its starting shape, the same amplification of ignorance that the randomness that is not in the equations found at the root of turbulence’s statistics. A flame wide enough to wrinkle has no shortage of seeds. The calculation says it hardly matters how strong they are.

What the power would mean if it held #

A power a little under a half is easy to state and easy to over-read, so it is worth following it out. Doubling a flame’s width raises its excess speed by a factor of $2^{0.43}$, about 1.35. A flame a thousand neutral wavelengths wide, ten times the widest computed, would on the same line run at about 7.3 in the equation’s units — and its rms slope, by the identity below, would be $\sqrt{14.6}$, about 3.8. The power law extrapolates straight into the regime where the equation has nothing to say. So even if the line in the first figure continued for ever as a statement about the equation, its physical reach is set by the slope, not by the width.

There is a suggestive echo in experiment. Large spherical flames expanding freely in a premixed gas are observed to accelerate, their radius growing faster than linearly in time — about as $t^{3/2}$ in Gostintsev and his colleagues’ compilation of 1988, which is a speed growing as the cube root of the radius. The equation’s power here, 0.43, is of the same kind and not far from it. But an expanding flame is not a periodic one, its noise is its own, and its density jump is large, so the agreement is a reason to take the mechanism seriously rather than a measurement of the exponent.

What does grow without limit is the slope #

The fourth figure shows the quantity that decides whether any of this describes a flame. The front’s speed and its slope are not independent. The curvature term and the Darrieus–Landau term both average to nothing over a period, so the mean rate at which the front advances is exactly half its mean-square slope:

$$U = \tfrac12\langle\phi_x^2\rangle, \qquad \phi_{x,\mathrm{rms}} = \sqrt{2U}.$$

A faster flame is a steeper one, by an identity. The quiet front’s speed never passes a half, so its rms slope never passes one. The noisy front’s rms slope rises from 1.45 at a width of 12.5 to 2.33 at a hundred, and its steepest point — the flank of a cusp — from about four to nearly seven.

The equation was derived on the assumption that those slopes are small, in a specific sense. Its $\phi$ is a scaled displacement, and a real front’s slope is of the order of the gas-expansion parameter times $\phi_x$: the fractional drop in density across the flame. Sivashinsky’s derivation is an expansion in that parameter, and it requires the product to be small. A real hydrocarbon flame expands its gas five to eight times — the volume change that the air flows in against the heat traced to the heat released at constant pressure — which puts the parameter, one minus the density ratio, near 0.8 to 0.9. At those values, a scaled rms slope of two means real slopes of order one or more, and the steepest flanks are far steeper than the expansion assumes. The weakly nonlinear equation is then outside the range it was built for.

So the growth does stop being the equation’s to decide, but not by saturating. It stops because the equation’s own picture — a gently wrinkled sheet — fails before anything in the equation’s dynamics has turned the speed over. For a weakly expanding mixture the equation holds further out, and its answer there is the power law of the first figure. For a real hydrocarbon flame the steepening reaches order-one slopes at modest widths, and what happens beyond is a question for the full equations of a front with large density jump, where the Darrieus–Landau mechanism keeps acting but the nonlinearity that limits it is no longer the simple $\tfrac12\phi_x^2$.

What a front a hundred wavelengths wide looks like #

The fifth figure draws one front, at twenty neutral wavelengths so that its detail is visible, beside the quiet pole front of the same width. The quiet front has one cusp per period and smooth arcs between. The noisy one has five cusps in the same period at the moment drawn, of unequal depth, and it spans 3.7 from crest to cusp against the quiet front’s 2.4. The cusps are not fixed: new ones form on the arcs and are carried into the largest, as the earlier essay watched happen. A wider front has more of them at once, and that population of cusps — not a single cusp sharpened further — is what the extra width buys.

The picture also says why the slope identity matters. A front with five cusps has five steep flanks where the quiet front has one, and the mean-square slope adds them up. That is the speed, by the identity. The speed is not a separate property that happens to grow alongside the wrinkling; it is the wrinkling, measured as the extra length of front per unit width, burning at the flat flame’s speed. The flow on either side of the front stays incompressible in the sense that matters however large the density jump across it, which is why so simple a geometric statement can carry the speed.

What was checked #

The sixth figure is the ledger. The narrowest run at a noise of a thousandth is re-marched whenever the site is checked and returns the stored speed to the last digit, because the noise is seeded. The identity between speed and slope is checked on the exact pole front of width twenty, whose slope is known in closed form and whose speed is computed separately from the equation: half its mean-square slope equals that speed to $10^{-13}$. One width, fifty neutral wavelengths, was rerun on twice the points: its speed came back 2.3 per cent higher, 2.079 against 2.033, which is about one and a half times the uncertainty either run carries on its own. The grids differ in points per neutral wavelength between widths, so a bias of that size could move a local power by about the scatter the second figure already shows; it moves the fitted power, which every width enters, by less than a hundredth. The quiet limit, the noise scheme’s variance and the step were checked in the earlier essay with the same scheme and are not repeated.

What the runs leave out #

One seed per point. Each speed is one noisy run, averaged in time. Its scatter is estimated from the run’s own windows, not from independent runs, and the local exponents carry it.

A periodic domain. A real flame of a given width has edges, or is curved into a cone or a sphere, and the cusps then drift towards the edges instead of circulating. A periodic front is the cleanest place to ask about width and the least like a burner.

Small expansion, throughout. Every number here is the equation’s, and the equation’s right to describe a flame ends where the fourth figure says it does.

From Michelson and Sivashinsky to wide noisy fronts #

Sivashinsky derived the equation in 1977 and Michelson and Sivashinsky integrated it numerically the same year. Thual, Frisch and Hénon found the pole solutions in 1985. That noise breaks the pole front’s speed limit, and that a noisy front’s speed grows with its width, was found in simulations by Joulin, Cambray and their collaborators around 1990, and has been studied since without an agreed law for the growth. The runs here say what the equation does out to a hundred neutral wavelengths, and where the equation stops being the flame.

Still open: the front with a real density jump #

The equation answers for weak expansion only, and its slopes leave that range at widths far smaller than a real burner’s. The next calculation keeps the noise and the width but replaces the equation with one that holds for a finite density jump — the Frankel or Sivashinsky–Clavin second-order equation, which adds the next term in the expansion parameter — and asks whether the extra nonlinearity it carries bends the speed’s growth over within the widths where the simple equation’s growth stays straight, and so whether a real flame’s speed has a ceiling the weakly nonlinear one does not.