A flat flame is unstable at every size

The air flows in against the heat finds that at constant pressure a joule is a volume. A flame is the extreme case: a hydrocarbon flame heats its gas sevenfold, and the burnt gas leaves the flame seven times faster than the fresh gas arrives, even though nothing in the flow approaches the speed of sound. That essay’s flame is flat and steady, and it ends by naming the consequence of the expansion that a flat flame hides: a curved flame deflects the gas coming towards it, and the deflection makes the curvature grow.

This essay computes that growth. The result, due to Darrieus and to Landau independently, is one of the cleanest instabilities in the subject — it needs no viscosity, no heat conduction and no chemistry beyond the fact that the gas expands — and it says that a flat flame is unstable to wrinkles of every size.

The gas bends round a wrinkle, and the bending feeds it #

A flame burns into the fresh gas at a fixed speed relative to that gas, the laminar burning speed $S_L$. Where it moves relative to the ground depends on how fast the gas it is burning into is moving. Now wrinkle it. Behind a forward bulge the burnt gas, expanding sevenfold, pushes back on the fresh gas in front, and the fresh gas ahead of the bulge is diverted sideways: the streamlines spread apart, and by conservation of mass the gas slows. The bulge is burning into slower gas, so it advances further relative to the rest of the flame. Behind a lagging part the streamlines crowd together, the gas speeds up, and the lagging part is carried further back.

Both effects make the wrinkle larger. Nothing in the argument depends on the wrinkle’s size, only on the expansion, and that is already the whole of the instability.

Four conditions across a thin flame #

To put a number on it, treat the flame as a surface of zero thickness across which the density drops by a factor $\sigma$ and nothing else happens. On the fresh side the flow of a small wrinkle $F e^{iky + \omega t}$ is irrotational, a potential decaying away from the flame. On the burnt side it is irrotational plus a vortical part carried away by the burnt gas — the flame makes vorticity, because the pressure and density gradients across a curved flame are not parallel, the baroclinic source that breaks Kelvin’s theorem. Across the flame four things must hold: the flame burns into the fresh gas at $S_L$; the mass flux through it is continuous; so is the normal momentum flux; and so is the velocity along it.

Those four conditions are four linear equations in the amplitudes of the three flows and the wrinkle, and they have a solution other than zero only when a four-by-four determinant vanishes. The growth rate is the root. Solved numerically, the root is Landau’s closed form,

$$\omega = S_L k\,\frac{\sigma}{\sigma + 1}\left(\sqrt{\sigma + 1 - \frac{1}{\sigma}} - 1\right),$$

at every density ratio from two to ten and every wavenumber tried, to three parts in $10^{16}$. The determinant was built from the jump conditions; the closed form was not used to build it.

The growth factor depends on the expansion alone. A density ratio of one — a “flame” that does not expand — gives none: at 1.001 it is $5 \times 10^{-4}$. At a density ratio of seven, typical of a hydrocarbon flame in air, it is 1.578, which means a wrinkle grows by a factor of $e$ in the time the flame advances one-tenth of the wrinkle’s wavelength. That is fast. A wrinkle a centimetre across on a flame burning at forty centimetres a second doubles in under two milliseconds.

The flame leaves vorticity behind it #

The determinant cannot be satisfied without the burnt side’s vortical flow, and that is a result in its own right. Four jump conditions need four unknowns; with only the two potential flows and the wrinkle there are three, and no growing wrinkle is possible. A flat flame produces no vorticity, but a wrinkled one must, and the determinant says how much.

Solved for its null vector, with the wrinkle’s amplitude as the unit, the vortical flow at a density ratio of seven has a streamwise velocity 3.7 times $S_L kF$ and a vorticity 3.5 times $S_Lk^2F$. At a density ratio of two the vorticity is 0.27 in the same units; at ten, 5.8. Its source is the one Kelvin’s theorem excludes and Bjerknes added: across a curved flame the pressure falls along the base flow while the density falls along the wrinkled front’s normal, the two gradients are no longer parallel, and the baroclinic torque between them spins up the burnt gas. The vorticity is carried away with the burnt gas at seven times the burning speed and, in this inviscid model, never decays.

The numbers are not small. A wrinkle a millimetre deep and a centimetre long on a flame burning at forty centimetres a second sheds burnt gas with a vorticity of about five hundred per second — comparable with the vorticity in the turbulence of a gentle stirred vessel. A wrinkled flame makes its own turbulence behind it, and a flame that has wrinkled spontaneously is no longer burning into quiet gas once its own products have mixed back round it.

How far ahead the fresh gas knows #

On the fresh side, the deflection that drives the instability is a potential flow decaying as $e^{kx}$ away from the flame. The gas a whole wavelength ahead has been deflected by $e^{-2\pi}$, two-tenths of a per cent, of what it is at the front. The fresh gas knows about a wrinkle only over a distance of about a sixth of the wrinkle’s wavelength, which is why the mechanism acts locally and the growth rate scales with the wavenumber alone.

At the flame the deflection is strong. Its velocity is $\Omega S_L kF$ — the growth factor times the burning speed times the wrinkle’s slope — which is the burning speed itself once the slope reaches $1/\Omega$, about 0.63 for a density ratio of seven. A wrinkle whose amplitude is a tenth of its wavelength already makes the gas ahead of its crest move as fast as the flame burns; that is where the linear theory must fail, and it is where real wrinkles stop growing and sharpen into cusps. The linear calculation therefore predicts not only that wrinkles grow but roughly where they stop: at depths of about a tenth of their length, which is what photographs of cellular flames show.

No favourite wrinkle #

The growth rate is proportional to the wavenumber, exactly — at ten times the wavenumber it is ten times the growth, to the precision the determinant is computed at. So a thin flame has no preferred size: every wrinkle grows, and the shorter it is, the faster.

That is the same pathology a vortex sheet has, for the same reason. The model has no length in it. A sheet of zero thickness and a flame of zero thickness both describe a discontinuity, and a discontinuity’s instability can only scale with the one length the disturbance brings, its own wavelength. For the vortex sheet the missing length is the shear layer’s thickness. For the flame it is the flame’s own structure, and it enters through a single number.

A real flame’s size comes from its response to curvature #

A real flame is a fraction of a millimetre thick, and its burning speed is not quite fixed: where it is convex towards the fresh gas, heat conducted forward spreads over a wider front of fresh gas and the flame burns more slowly, and where it is concave, faster. To first order the change is proportional to the curvature, $S = S_L(1 - \mathcal{L}\kappa)$, and the constant of proportionality $\mathcal{L}$ is the Markstein length — a few flame thicknesses, a few tenths of a millimetre for methane and air.

Put that into the first jump condition and the determinant acquires a term that rises as the square of the wavenumber and opposes the growth. The growth rate now rises as $k$ for long wrinkles, turns over, and falls to zero at a cut-off. In this model the cut-off is exact: $k\mathcal{L} = (\sigma - 1)/2\sigma$, which the determinant returns as 0.428571 at $\sigma = 7$ against the closed form’s $6/14$ to six figures. Wrinkles shorter than $4\pi\sigma/(\sigma - 1)$ Markstein lengths are smoothed out; longer ones grow, and the fastest has about half the cut-off’s wavenumber.

For a density ratio of seven the fastest wrinkle is 27.5 Markstein lengths across and the shortest that grows is 14.7. For a methane–air flame with a density ratio of 7.4 and a Markstein length of 0.2 millimetres, the fastest wrinkle is 5.4 millimetres across and it grows by a factor of $e$ every 2.9 milliseconds. The size comes entirely from the Markstein length; the expansion sets how fast.

The Markstein length can also be negative. Lean hydrogen flames, in which the light fuel diffuses faster than heat, burn faster where they are convex, and then nothing opposes the hydrodynamic instability at short wavelengths: the flame breaks up into cells a millimetre or two across whatever its size. That is a second instability, thermodiffusive rather than hydrodynamic, and it adds to this one rather than replacing it.

Why large flames are wrinkled and small ones are smooth #

An instability with a growth time of three milliseconds only matters if the flame lasts long enough for a wrinkle to grow. A convenient measure is the number of e-folds the fastest wrinkle gains in the time the flame takes to burn through a distance equal to its own size. That number is proportional to the size: for methane and air it is about one for a flame a millimetre across and ninety-five for one ten centimetres across.

The prediction is that small laboratory flames stay smooth and large ones wrinkle spontaneously, and that is what is seen. A spherical flame ignited in a large vessel of premixed gas starts smooth and, beyond a radius of a few centimetres, develops a pattern of cells over its whole surface. The cells add flame area, the extra area burns more gas per second, and the flame accelerates — which is one reason a large unconfined gas explosion burns faster than its laboratory burning speed predicts, and why that speed cannot simply be scaled up. A small Bunsen flame is also kept smooth by the stretching of its front by the flow, which the model here leaves out, but its smallness would keep it smooth anyway.

The same growth in different fuels #

Because the growth rate is the growth factor times the burning speed times the wavenumber, the fuel enters twice, through its expansion and through how fast it burns. A lean methane flame expands about fivefold and has a growth factor of 1.17; a stoichiometric one, at 7.4, has 1.65. A stoichiometric hydrogen–air flame expands a little less, about 6.9, and has a factor of 1.56 — but it burns at about two metres a second against methane’s forty centimetres, so the same wrinkle grows on it five times faster. Hydrogen’s burning speed, not its expansion, is why its flames wrinkle so readily, and its small or negative Markstein length is why the wrinkles are so fine.

The growth also has an acoustic side. A wrinkling flame changes its own area, and so the rate at which it turns fresh gas into burnt gas, and at constant pressure that rate is a rate of change of volume. A flame whose area fluctuates is a fluctuating source of volume, which is the one kind of source that radiates sound efficiently — the monopole of acoustics. The roar of a large premixed flame is in part its cells forming, merging and forming again, heard.

Two instabilities with the same arithmetic #

The flame’s instability and the hanging layer’s have the same shape — an interface between a dense fluid and a light one, a destabilising mechanism that grows with the wavenumber, a surface effect that grows faster and cuts it off, and a fastest wavelength in between. They also meet: a flame propagating upward has its heavy fresh gas above its light burnt gas, and gravity adds a Rayleigh–Taylor growth to Landau’s at long wavelengths; propagating downward, gravity stabilises the longest wrinkles. Landau’s own analysis included gravity, which is why downward-propagating flames in tubes stay flatter than upward ones.

The difference is where the energy comes from. The hanging layer releases potential energy by falling. The flame releases nothing extra by wrinkling; it redistributes the momentum the expansion has already produced, and the deflection of the fresh gas is the whole mechanism. That is why its growth rate contains only the density ratio and the burning speed — and why a flow whose volume changes can be unstable in a way an incompressible one of the same geometry cannot.

What the picture cannot show #

A flame of zero thickness. The hydrodynamic model is valid for wrinkles much longer than the flame, and the Markstein term is its first correction. Near the cut-off the wrinkle is only a few tens of flame thicknesses long and the model is at the edge of its range.

Linear growth. Every rate is for an infinitesimal wrinkle. Real wrinkles saturate: the front develops sharp cusps pointing into the burnt gas, where two sides burning towards each other meet, and a cusped front is what a cellular flame actually looks like.

Incompressible on each side of the flame. The only density change is the jump itself; sound is absent. A flame in a tube or a combustor also couples to the tube’s acoustic modes, and that coupling — the flame’s heat release responding to a pressure wave and feeding it — is a separate and often stronger instability, the one that makes combustors hum.

No gravity, no stretch, no turbulence. Each would add to or subtract from the growth, and the essay’s claims are about the instability the expansion alone produces.

The convention the numbers depend on #

The density ratio $\sigma$ is the fresh gas’s density over the burnt gas’s. $S_L$ is the burning speed relative to the fresh gas. The Markstein length multiplies the curvature in the flame-speed law, with the sign chosen so a positive length slows a front convex towards the fresh gas. The methane numbers — a burning speed of 0.37 m/s, a density ratio of 7.4 and a Markstein length of 0.2 mm — are stated round values for a stoichiometric flame, and the Markstein length in particular is uncertain by a factor of two.

Who found it, and when #

Georges Darrieus described the instability in an unpublished lecture of 1938, and Lev Landau published the analysis independently in 1944. George Markstein added the curvature-dependent flame speed in 1951, which gave the instability a length. Michelson and Sivashinsky derived in 1977 the nonlinear equation that describes the cusped front a growing instability settles into, and Thual, Frisch and Hénon found its exact solutions in terms of poles in 1985.

Still open: the cusps a wrinkle becomes #

The linear growth says every long wrinkle grows; it does not say into what. The Michelson–Sivashinsky equation, the weakly nonlinear version of the calculation above, has exact solutions made of poles in the complex plane, and they describe a front made of smooth arcs meeting in sharp cusps, advancing faster than a flat flame by an amount that depends on the domain size. The calculation that follows solves that equation for a flame of growing width and asks how the cusped front’s extra speed scales with its size — the number that decides how much a large flame self-accelerates before turbulence takes over, and whether a gas that has not decided to react yet can be pushed across into detonation by the acceleration alone.