A wrinkled flame has one cusp and a speed limit

A flat flame is unstable at every size computes what happens to a small wrinkle on a premixed flame. Gas expanding through the flame bends the flow round the wrinkle so that the fresh gas speeds up where the flame bulges back and slows where it bulges forward, and every wrinkle grows, shorter ones faster, at a rate proportional to their wavenumber — the same unbounded growth that a vortex sheet shows at every wavelength at once, and for the same reason: the model treats the flame as a sheet of zero thickness. Only the flame’s response to its own curvature — the Markstein effect, a slightly slower burning where the flame is convex towards the fresh gas — cuts off the shortest. Everything longer than a few millimetres, for a methane flame, grows. The linear theory has no favourite wrinkle and no end state; it says only that a flat flame is not what anyone will see.

That is an unsatisfying place to stop, because a linear theory that predicts growth at every scale has told the reader nothing about the state the growth arrives at. A wrinkle whose amplitude grows exponentially must stop somewhere, and where it stops decides what anyone measuring a flame would see: its shape, its cell size, and above all its speed, which is what a burner designer and an explosion investigator both want. The linear theory cannot reach any of these, because every one of them is set by the term it threw away.

It closes on the equation that says what will be seen: the Michelson–Sivashinsky equation, the weakly nonlinear version of the same calculation, which is known to have exact solutions made of poles in the complex plane and to describe fronts of smooth arcs meeting in cusps. This essay solves it, marches it, and asks how fast such a front burns and how that depends on the flame’s size.

The equation #

For a front whose position is $\varphi(x, t)$ — its displacement towards the fresh gas — Sivashinsky derived in 1977, for a flame whose gas expands only a little,

$$\varphi_t + \tfrac12 \varphi_x^2 = \nu\,\varphi_{xx} + I(\varphi).$$

Each term is one of the linear essay’s ingredients. $I$ is the Landau operator: it multiplies each Fourier mode of the front by its wavenumber, and on its own it makes every wrinkle grow at a rate proportional to $|k|$ — the Darrieus–Landau instability. The term $\nu\varphi_{xx}$ is the curvature cut-off, which damps a mode at $\nu k^2$, so a wrinkle grows at $|k| - \nu k^2$ and those shorter than $2\pi\nu$ decay. The new term is $\tfrac12\varphi_x^2$, and it is geometry, not chemistry: a front propagating normal to itself at a fixed speed advances more slowly in the direction it is being tracked, by an amount proportional to the square of its slope. It is the same nonlinearity by which every compression becomes a shock — the equation without its Landau term is Burgers’ equation — and it is what stops the wrinkles growing.

The equation is written here on a domain of one period, $2\pi$, so that $\nu$ is the only parameter left: $1/\nu$ is the number of neutral wavelengths that fit across the domain, the flame’s width measured in the shortest wrinkle that can grow. For a methane flame with a neutral wavelength of about three millimetres, a burner six centimetres across is twenty wavelengths wide.

From noise, one cusp #

The first figure marches the equation from a flat front carrying random wrinkles a thousandth of a unit high, in a domain 12.5 neutral wavelengths wide, by a semi-implicit pseudospectral scheme. By time 3 half a dozen small wrinkles have grown out of the noise. By time 4 they have become cusps pointing back into the burnt gas, separated by arcs, and the cusps are moving: a cusp between two unequal arcs drifts towards the smaller one, and when two cusps meet they merge into one. By time 8 a single cusp is left, deeper than it will finally be; by time 300 it has relaxed onto a single steady arc and a single cusp.

That is the end state for every width tried, and it is an exact solution. The dashed curve over the last front is the pole solution with the most poles the domain allows, found without any marching; moved sideways to put its cusp on the marched one — the equation does not care where along the front a cusp sits — it lies on top of it.

A front made of poles #

In 1985 Thual, Frisch and Hénon found that the equation has solutions of the form

$$\varphi = -2\nu\sum_{\alpha} \ln \sin\frac{x - z_\alpha}{2},$$

a sum over complex numbers $z_\alpha$ in conjugate pairs — poles of the front’s slope, off the real axis. The equation becomes equations of motion for the poles: each is pushed by every other, as $\cot$ of half the complex distance between them, and drawn towards the real axis at a constant speed by the Landau term. The front is the shadow on the real axis of a gas of particles moving in the complex plane.

A steady front has all its poles on one vertical line, at heights $\pm B_j$, where the repulsion from the others exactly balances the pull towards the axis:

$$\nu\Big[\coth B_j + \sum_{k \ne j}\Big(\coth\frac{B_j - B_k}{2} + \coth\frac{B_j + B_k}{2}\Big)\Big] = 1.$$

These were solved by Newton’s method for up to forty pairs. The front they produce was then put back into the equation on a grid of 4,096 points, with the Landau operator applied by FFT: for a steady front, $-\varphi_t$ must come out the same at every point, and it does to four parts in $10^8$ at three different pole counts.

The second figure shows the poles of three steady fronts. A domain five neutral wavelengths wide holds two pairs; ten holds five; twenty holds ten: one pair for every two wavelengths, the rule Thual, Frisch and Hénon found, $2N - 1 < 1/\nu$. The poles spread up the imaginary axis, far apart at the top and crowded towards the bottom, and the lowest one sets the cusp: the closer a pole comes to the real axis, the sharper the front’s corner there. The lowest pole’s height falls from 0.185 of a period at five wavelengths to 0.081 at ten, 0.036 at twenty, 0.016 at forty and 0.0074 at eighty — each doubling of the width cuts it by a little more than half, so it shrinks roughly with the curvature cut-off. The cusp is as sharp as the Markstein effect lets it be. It is never a true corner: its radius of curvature is of order the pole’s height, and a real flame’s cusp is rounded further by the flame’s own thickness, which the equation — like the jump conditions across a discontinuity that has a thickness — does not contain.

The same arc at every size #

The third figure draws the three steady fronts across one period each, and the surprise is how alike they are. In the equation’s own units — lengths scaled to the domain, the front’s height scaled with it — the three arcs very nearly coincide; only the cusp sharpens. In physical units the front’s height scales with the domain’s width, so the three flames are geometrically similar: the same arc at three sizes, the wider one proportionally deeper, with the same slopes at corresponding points.

The same slopes are the key to the speed. The front advances, on average, by the amount its slope costs it in the $\tfrac12\varphi_x^2$ term — a tilted piece of flame covers more ground normal to itself than along the direction of mean propagation, so a wrinkled front burns more fuel per unit width and advances faster. A front whose slopes do not change with its size advances at a speed that does not change with its size.

The speed limit #

The fourth figure is the speed. For a steady front with $N$ pole pairs, the speed in excess of a flat flame’s comes out as

$$U = 2N\nu\,(1 - N\nu)$$

— 0.32 for one pair at $\nu = 0.2$, 0.3648 for three at $\nu = 0.08$, 0.42 for six at $\nu = 0.05$, exactly, from the pole front put into the equation. For a fixed number of pairs it is a parabola in $N\nu$, and it has a maximum of one half at $N\nu = \tfrac12$. The front with the most pairs a domain can hold has $N\nu$ just below a half whatever the domain, because the counting rule $2N - 1 < 1/\nu$ puts $N$ at about half the number of wavelengths. The formula can be rewritten as $U = \tfrac12 - 2(N\nu - \tfrac12)^2$, and the most-poles front has $N\nu$ within $\nu/2$ of a half, so its shortfall from the limit is at most $\nu^2/2$ — 0.02 at five wavelengths, 0.005 at ten, under two hundredths of a per cent of the limit at eighty. So its speed is one half, or very close to it, for every domain wider than about five neutral wavelengths: in the figure the envelope of the steps touches one half and never crosses it.

This is the answer to the question the linear theory left: a wider flame is not faster. The flame’s width enters the speed only through a whole number, the count of poles, and the counting rule and the speed formula conspire to cancel it. In the small-expansion limit in which the equation is derived, the speed limit is $\gamma^2/8$ of the laminar burning speed, with $\gamma$ the fractional density drop across the flame: the expansion sets the limit, the chemistry sets the unit, and the width drops out.

It is worth saying what the result is not. It is not a statement that wrinkling does nothing: a wide flame burns about $\gamma^2/8$ faster than a flat one, which for a modest expansion is a few per cent and is the whole of the effect the instability has on the speed. Nor is it a statement that the shape does not change with width: the cusp sharpens without limit as the flame widens. It is the narrower claim that the geometry of a steady front is similar at every width, and that a similar geometry burns at one speed — the same reasoning by which a jet whose profile is exactly similar at every distance has a spreading rate that no single measurement of its profile can distinguish from another.

The flame that settles there #

The fifth figure follows the marched front’s speed. While the wrinkles are small nothing happens — the speed excess is quadratic in the slope, and the slope is tiny. As the wrinkles grow and merge the speed climbs through the limit, and at time 8 — the moment of the third front in the first figure, when a single cusp has just formed and is far deeper than its final shape — it reaches 0.94, nearly twice the steady value. The excess is the deep cusp relaxing: its steep flanks burn fast, and as they flatten the speed falls back. It takes some fifty time units to settle. Over the last quarter of the run it averages 0.49986, against 0.4992 for the pole front with six pairs, the most a domain 12.5 wavelengths wide can hold: a difference of 0.13 per cent, from a march that knew nothing about poles.

That agreement is the evidence that the pole front is not just a solution but the one a flame arrives at. The random start contained every wavenumber; the fronts with fewer poles, which are also exact steady solutions, are slower, and the march passed them by.

Why one cusp wins #

The merging of cusps has a clean description in the pole picture. The interaction between two poles is not the same in every direction: expanding the cotangent for two poles close together, it is attractive along the real axis and repulsive across it. Two poles side by side draw together; two stacked one above the other push apart. So two groups of poles on two vertical lines drift together, and the poles within a line spread up it. Once they are on the same line they stay: the single vertical line is the configuration the dynamics settles into, which is why a steady front has exactly one cusp per period. A front with two cusps is always unsteady — its smaller arc is eaten by its larger neighbour. The ending is the same as for two vortex rings that leapfrog: a configuration of several structures that looks as if it could persist, and a slow drift that ends with one.

The same process is visible on real flames in wide tubes and on the expanding surfaces of spherical flames: cusps travel along the front, merge, and leave cells whose size grows with time. On a spherical flame the front is growing, so the domain it offers keeps widening and new wrinkles keep appearing between the old ones; the steady single cusp of a fixed periodic domain is the simplest case of a process that on a real flame does not stop.

Where the speed limit fails #

The speed limit is a statement about the equation’s steady fronts, and two things break it on real flames. The first is noise. The pole front with many pairs is very sensitive: a small disturbance on the long arcs of a wide flame brings in new poles from far up the imaginary axis, and they create small sub-cusps on the arcs before being swept into the main one. A wide flame in a noisy flow never quite settles; it carries a population of small cusps on its arcs, and its average speed exceeds the steady front’s — the increase that measured large flames show, and that has been attributed to exactly this mechanism. The limit computed here is the speed of the quietest flame of a given width.

The second is the equation itself. It is derived for small expansion, and a real methane flame expands its gas about sevenfold — a flow whose density changes sevenfold across a sheet and which is, away from that sheet, incompressible in every sense that matters. At that expansion the flame’s own vorticity — which the air flowing in against the heat records — and the full hydrodynamics matter, and the cusps are sharper and the speed larger than the equation says. The structure survives: single-cusp cells, merging, a speed set by the expansion; the numbers do not.

What was checked #

The sixth figure is the ledger. The pole fronts were checked in the equation itself, not in the pole equations: the front was built from the solved poles, differentiated and passed through the Landau operator spectrally, and the resulting $-\varphi_t$ is constant to four parts in $10^8$. Their speeds match $2N\nu(1 - N\nu)$ to eleven figures at three pole counts. And the march from noise, which uses neither the poles nor the formula, arrives at the pole front’s speed to 0.13 per cent.

What the picture cannot show #

Small expansion. The equation is the first term of an expansion in the density jump, and the numbers are exact for it and approximate for real flames.

One dimension of wrinkling, in a periodic domain. The front is a curve, not a surface, and the domain repeats. Real flames in tubes meet walls, and three-dimensional wrinkles form cells rather than stripes.

No gravity and no turbulence. A flame propagating upwards is further destabilised by buoyancy, as a heavy layer over a light one is; one propagating downwards is stabilised. A turbulent flow wrinkles the front far more than its own instability does.

A single Markstein number. The cut-off here is a constant; real flames’ curvature response depends on stretch and on differential diffusion, and lean hydrogen flames are unstable for a different reason as well.

The convention the numbers depend on #

The front $\varphi$ is its displacement towards the fresh gas, and the figures draw it with the burnt gas below. The domain is one period, $2\pi$, and $1/\nu$ is its width in neutral wavelengths, those at which a wrinkle neither grows nor decays. Speeds are in the equation’s units, where the Landau operator’s coefficient is one; the pole heights are in the same units as the period.

Who found it, and when #

The instability is Darrieus’s and Landau’s, from 1938 and 1944, and the curvature cut-off Markstein’s, from 1951. Gregory Sivashinsky derived the nonlinear equation in 1977, and Daniel Michelson and Sivashinsky studied its numerical solutions the same year. Olivier Thual, Uriel Frisch and Michel Hénon found the pole decomposition in 1985, and the counting rule and the speed of the steady fronts followed from it; the role of noise in making large flames faster was developed through the 1990s.

Still open: the noise a flame can feel #

The speed limit holds for a quiet flame, and real flames are not quiet. The next calculation marches the same equation with a small random forcing added, at a stated amplitude, and asks how the average speed rises with the domain’s width once poles are being fed in from the noise faster than the cusp absorbs them — whether the speed then grows without limit as the width grows, as some measurements of large flames suggest, or saturates at a new value set by the noise, and how small the noise has to be before the quiet flame’s limit is what a real burner sees.