Bruno Denet
email: bruno.denet@irphe.univ-mrs.fr

Guy Joulin
email: guy.joulin@lcd.ensma.fr

WRINKLED FLAMES AND GEOMETRICAL STRETCH

Keywords: number(s): 47.20.-k, 47.32.-y, 82.33.Vx

published or not. The documents may come

I. INTRODUCTION

Being of chemical and diffusive origins flame fronts propagating in premixed gases are markedly subsonic. Thus, because the fresh ( u ρ ) and the burnt ( bu ρ ρ < ) gas densities differ, flame deformations almost instantly modify the piecewise-incompressible surrounding flow, and conversely. This nonlocal hydrodynamic flame/flow feedback brings about the Darrieus [START_REF] Darrieus | work presented at La Technique Moderne[END_REF]-Landau [START_REF] Landau | [END_REF] (DL) wrinkling instability, and greatly complicates the free-boundary (hence nonlinear) dynamical flame-front problems, especially from a theoretical view point.

In the limit of small Attwood numbers 0< A () / ()

ub ub
ρ ρρ ρ = -+ <<1 the DL instability is weak, though. Sivashinsky's seminal work [3] showed how flat-on-average flames propagating into a quiescent pre-mixture then evolve according to a nonlocal nonlinear partial differential equation (PDE); subsequent works [4] essentially improved A -dependent coefficients therein.

Michelson's numerical early study [5] of the PDE showed that unforced flame fronts soon acquire the shape of parabola-like arches joined by sharper crests pointing towards the burnt gas;

at late times a single steady arch (or half-one) with a maximum wavelength compatible with periodic (or Neumann) boundary conditions generically survives in (not-too-wide) channels.

Further numerical investigations [6] of this (Michelson-) Sivashinsky (MS) PDE, exploiting the pole-decomposition method [7], evidenced a much richer manifold of steady solutions whose number increases with available lateral size noticeably faster than linearly. Most of those happen to be unstable and cannot be evidenced by more conventional time-marching procedures.

It would be nice to understand whether a simple mechanism underlies such a proliferation of steady solutions, which has so far not been done. The present analyses suggest that a geometrical stretch ( S ) induced at large enough scale by a front curvature of the proper sign, can noticeably contribute: it indeed generates new couplings with the mechanisms already at work in the MS equation, and generates novel solutions that are steady in a suitable frame. Though unstable, these may play a nontrivial role, like the long-lived weakly unstable states of flat-on-average flames forced by weak noise [6]; or as building blocks of the travelling bursts that are randomly 'emitted' near troughs of wide flames [8]. Hence a side question: is it a spontaneous emission?

A related problem of great practical importance is to understand how velocity modulations or fluctuations in the incoming flow of fresh gas, e.g. time-dependent or even turbulent ones, affect the front dynamics. The problem is unfortunately much too tough for a frontal theoretical attack.

Flames could conceivably be passively deformed at high intensities of forcing, but even the statistics of this is not yet fully understood theoretically [9]. Moreover, the inner cut-off length of front wrinkles in turbulent flows is experimentally known [10] to coincide with the neutral wavelength at which DL instability and curvature affects balance, strongly indicating that the smallest detectable flame wrinkles are not passive.

This hint of forcing-instability interaction at the cut-off length motivates another question: how do incoming velocity modulations affect instability-driven patterns at or about such scales?

The point is conceptually different from that of implantations of incipient wrinkles by the small-scale components of a forcing; these are indeed often present, purposely or because most numerical integration methods are noisy. Despite early attempts about passive forced propagations [START_REF] Frisch | Chaos and Statistical Methods[END_REF][12] the related mechanism of complex singularity implantation is not understood; unfortunately, it ultimately has a strong impact on flame dynamics [13]. Polesprinkling [14] mimics implantations and somehow bypasses the problem but, although tried for expanding flames [15], this trick is still hampered by its computational cost. To study space-or angle-periodic forced fronts, pseudo-spectral integration remains more flexible and faster [16].

The present work has a more modest scope: it focuses on already implanted patterns and inquires about their long term viability in the presence of a geometrical stretch caused by an underlying curved front, itself possibly influenced by large scale incoming velocity modulations.

The paper is organised as follows. An extended MS equation accounting for geometrical stretch is introduced in Section II. Pole-decomposed steady solutions (in a frame) are presented in Sec.

III, and specialized to a simple centred crest then to ones with larger amplitudes. The predictions are compared to numerical results in Sec. IV. Multi-crested 'steady' wrinkles are identified and analyzed in Sec.V, and their self-similar unsteady counterparts discussed in Sec.VI. Section VII takes up varying s S . Section VIII tentatively relates the cut-off scale of wrinkling to stretch, and Sec. IX gathers final remarks and open problems. Appendices A, B detail technical points.

II. NONLINEAR EQUATION FOR STRETCHED ISOLATED WRINKLES

The starting point adopted here to take up the aforementioned topics is a non-dimensional forced version of the (Michelson-) Sivashinsky [3]

(1 ²) 1 ² / 2 ... ss +-=+ , tan( ) s α = ∼ x ϕ A ,.
The stabilising effect of ² / 2

x ϕ can saturate the DL instability [5], soon leading to parabola-like arches when ( , ) 0 utx = .

The forcing term u in Eq. (2.1) represents the y -wise component of a shear-flow type of modulation in the fresh-gas flow; ( , ) 0 utx ≠ may also result from an inhomogeneous and/or fluctuating fresh gas composition that makes n u change [START_REF] Barabasi | Fractal concepts in surface growth[END_REF]. Without its DL contribution ) and is henceforth omitted. Yet it must be kept in mind that the 'steady' patterns encountered later will only be so in a specific frame drifting at the uniform lateral speed U . A locally uniform x -wise gas velocity ( , 0) vt and the associated ( , 0) x vt φ could have been accounted for in Eq.( 2.3) then lumped in U and 'eliminated', but the S -dependent term cannot. The curvature S of the base pattern Φ is here termed "stretch intensity".

Incidentally, a

x Sxφ term appears when studying self-similar rational solutions to the Burgers equation [20]; / x XE = is then an abscissa measured in a t -dependent unit of length () E t .

III. CENTRED STEADY PATTERNS

Attention is from now on focused on finite-amplitude solutions to Eq.(2.3), namely: localized patterns φ that are peaked near the origin, and have ² was shown in [START_REF] Joulin | [END_REF] to also hold when , a useful check of accuracy for numerical resolutions.

B. Elementary centred steady crest

The case of one pole pair (

C. Large centred steady crests

In the stretch-free case, the uppermost pole altitude increases with the number N of pole pairs, and the typical difference

max max 2( ' )' sgn( ) , ' B B PB d B B SB BB ν + - ∫= - - (3.4)
where the principal-part integral complies with the constraint mn ≠ in Eq.(3.2). Although (3.4) formally is the difference between its 0 S = version and a Wigner equation (no sgn( ) B in the right-hand side, [START_REF] Mehta | Random Matrices[END_REF]) one may not subtract partial solutions: (3.

1 / 4 ( 1) / , 0 1, Sh h N h SN ν ν ν -≈ + - < < < (4.2)
to be compared with (3.3) gives the relationship between S and the upper barycentre bar B shown in Fig. 2

V. A VARIETY OF STEADY LOCALIZED PATTERNS

The previous centred isolated crests are unstable to lateral shifts. Writing the poles locations as

() () () jj zt D t Zt =+ leaves Eq.(4.1) invariant if ( ) (0) exp( ) Dt D S t =
, whereby a steady crest pattern can be dragged as a whole by a geometry-induced tangential velocity x Sx Φ≈ , away from the base flame trough 0 x = . Besides, [7] identified the reason why a population of nearby poles tends to align along parallels to the imaginary z -axis (see also [20]) and to build up a crest. This results from a 'horizontal' attraction encoded in the pair-wise interaction terms of Eqs.(4.1), combined with the 'vertical' repulsion featured in Eq. ( (5.1). The more populated crest feels the higher effective influence of stretch and will the first to loose remote poles if 0 SN ν ± > gets too high.

(1+) (2) (3) (1,1) (1,1,1) (2,1) (2+) (3+) (1+,1+) (1,1+,1) (1+,1,1+) (1+,1+,1+)
The second situation amenable to some analysis corresponds to a centred crest composed of 0 N vertically aligned pole pairs, symmetrically flanked by NN + lateral crests that have ) ) 0 ), and 0.06 S = .

Mk XS N ν = H , with 2 o r 2 1 MN N =+ , since 2 () N ξ H ∼ (1 / 2 ) (² ) N ξ - L and 21 () N ξ + H ∼ (1 / 2 ) (² ) N ξ ξ + L ([
Thus, the antagonistic actions of horizontal attraction between nearby crests and geometryinduced positive stretch near any flame trough (e.g., that of the base solution Φ we started from)

generates new equilibrium positions: other poles may sit there if the effective stretch intensity they feel is compatible with a vertical equilibrium, making secondary troughs appear on the flame profile…and so forth. This gives hints on how the stretch/nonlinearity competition is sufficient to contribute a complicated web of steady solutions to the MS equation [6]. As shown below, still more exotic configurations exist, for nearly the same reason.

B. Remote poles

In the above equilibriums between 'almost real' pole pairs in the presence of a weak enough stretch effect, Eq.(5.2), the DL instability mechanism little influence the crest-spacing directly, merely ensuring that nearly real poles remain so. Similar configurations could conceivably exist when the members of N poles pairs are markedly off the real axis, provided these lie at nearly the same altitudes ih ± ; an elementary configuration of this type (a single detached pair) was encountered in Sec. IV. The height 0 h > has to significantly exceed max x for this to be viable:

the repulsive influence that each of the N poles k x ih ≈ ± inside one single row feels from the complex conjugates (now at a distance of 2h ) must indeed be nearly uniform, and weak enough not to destroy the possibility of a "DL vs. stretch" balance with Comparisons of the above near-horizontal remote pole arrangements with references [6] [30] suggest that a similar stretch-based mechanism underlies the 'interpolating solutions' of the MS equation evidenced there, where approximately horizontal arrangements of poles are found to lie at a distance above and between the vertically-aligned ones pertaining to the main crests; the stretch is then due to the main crest curvature xx Φ and, in fine, results from the attraction by the array of vertically-aligned poles belonging to the space-periodic base front slope x Φ (instead of the pole-at-infinity of Sz in the present situation).

As if is not enough, for some given sets { , , } ); (ii) the existence of pole detachments from the main pile(s), which typically corresponds to the upper branches belonging to a given total pole number: e.g., a near-real (1,1,1) Hermite type of solution becomes a (1,1 + ,1) one as 0 Sν < decreases along the upper branch with same turning point, or (1,1,1,1) becomes (1 + ,1,1,1 + ) … .

To summarize Sec. VI: weak positive stretch allows for very numerous novel steady solutions.

VII. SELF-SIMILAR EVOLUTIONS

A. Nearly real poles

The steady patterns analysed in Sec.V are now shown to be special cases of analytically accessible evolutions. We firstly consider the situation of near-real pole arrangements analysed in the paragraphs below (5. eq LL < premature pair-wise coalescences occur instead of a single 7-pair crest being formed at once , see Fig. 6 ; a single crest will form soon after, however, by the alignment mechanism recalled at the beginning of Sec. V.

B. Remote poles

The above analysis can be adapted to the case of N remote pole pairs that are initially nearly aligned horizontally at (0) (0)

kk zx i h =±, (0) (1/ ) hS N ν = >> O
. To satisfy the imaginary part of (4.1), their current common altitude (i) Firstly, one must set 1 1 k NN == in (6.2)(6.5) , whereby the equilibrium scale factor now is

1/ 2 (2 / ) eq LN S ν ∞ ≡
. Yet like previously, steady nearly-horizontal remote pole arrangements are

unstable: | |~exp( ) LS t for 1 St >> if | (0) | eq LL ∞ > , whereas | (0) | eq LL ∞ <
leads to the collapse of all (simple-) poles into a single one (of order N ) at the finite

merger tt = , merger St ≡ ln(1 ( (0) / )²) eq LL ∞ --
. (ii) Another difference with near-real poles is that the dynamics (6.5) does not stop at merger tt = . ²( ) Lt, as is now defined by (6.6), may get negative and () Lt itself imaginary: the N poles then become vertically aligned, and will remain nearly so for merger tt > . k NN ≥ == . Solid (red) curves: analytical prediction, Eq.(6.3). Inset: like previously, with the same intentional "noise" on initial conditions as in Fig. 5.

The latter scenario of course assumes that the altitude ( ) ht has not shrunk to zero in the interim: whereas (0) 1/ ... eq hh S N ν >= -+ eventually leads to ~exp( ) hS t and poles that leak to i ±∞, any (0) eq hh < will drive the solution of (6.7) to ) a N -crest pattern will be seen to crop on top of the base flame before collapsing into a single one (or a -N crest pattern expanding laterally); on the contrary, if 0 merger tt > , only an isolated crest will be observed when acquiring a significant amplitude.

Thus, depending on the values of the initial altitude (0) h and scale factor (0) L , a variety of behaviours caused by nonlinear interactions may take place among initially remote pole arrangements, even though this can hardly be noticed from the real axis because the poles involved are too far from it when these 'off-stage' processes occurs. Moreover, slight differences in the initial conditions may result in completely different patterns as time elapses: the unstable equilibriums (e.g., remote poles at ke q e q Li h ξ ∞ ± , or near-real ones at 0 .( )

ke q k Li N ξν ± O
) play the part of 'shunting-' or 'saddle-' points for the system trajectories. Since the poles are indiscernible (identical residues) the existence of unstable equilibriums, whose number quickly increases as 1/ Sν →+∞, almost precludes one from tracing back the origin of sub-wrinkles of a weakly curved flame front from the sole observation of their shape, location and amplitude when they get visible. This likely contributes to the nearly random manner sub-wrinkles crop up on top of weakly curved flame troughs [8], like in a Galton box.

VIII. TIME-DEPENDENT STRETCH

A too intense constant stretch 0 S > , for example induced by a too strong steady ( ) ux in Eq.(2.1) , can moderate or inhibit the phenomenon of trough-splitting (i.e., crest formation) that the DL instability mechanism tends to induce. On the contrary, a constant compression, 0 S < , tends to pull the poles of x φ and to make them crowd near the origin 0 z = . What happens when S oscillates and possibly changes sign with time has so far not been investigated, even though this relates to the sub-wrinkles of flames subjected to a time-dependent, non-uniform (, ) utx in Eq.( 2 Before closing this section, a few remarks may be put forward.

The large-ω analysis can be easily adapted when the fluctuation '( ) St of stretch intensity contains R widely separated frequencies Whereas also allowing S 〈〉 to depend on slow time t is a harmless further generalization, it is not known whether the above elementary 'cascade renormalization' [31] can be extended to a continuous spectrum

IX. FINAL REMARKS, OPEN PROBLEMS

Combining analytical and numerical approaches based on the pole decomposition, this work revealed that inclusion of geometrical stretch markedly modifies the otherwise simple [7,[START_REF] Joulin | [END_REF] isolated solutions of the classical (Michelson-) Sivashinsky PDE for (weakly-) unstable flames.

Firstly, it was demonstrated that accounting for a uniform stretch intensity S is enough to generate novel types of isolated solutions. New steady (unstable-) solutions with arrays of horizontally aligned near-real or remote poles (or both) were evidenced; when 0 S = only one centred isolated solution, involving a single vertical pole alignment, existed whatever 1 N ≥ is [21]. The net result is a proliferation of equilibrium front shapes, especially at small s S , Fig. 9. law. By contrast, stretch effects acts on front singularities (hence on its shape) in an isotropic way. Its presence brings about the possibility of new " stretch vs. horizontal attraction" and/or " stretch vs. Darrieus-Landau instability" partial equilibriums, thereby noticeably contributing to the quick proliferation of (unstable-) steady solutions when 0 S > decreases. A similar mechanism operates in periodic solutions of the stretch-free MS equation: S is then effectively provided by the curvature of the main cell trough, and the many poles associated with its crests provide the means to pull extra wrinkles/poles away from the trough, in a nearly isotropic way.

The analogy is further substantiated by the fact that the new stretch-induced 'steady' states found above are unstable, like those evidenced in [6] [30]; their self-similar evolution, described analytically, will likely help handle those bursts travelling along wide front troughs.

It has also been shown that too intense stretch suppresses all isolated 'steady' wrinkles (Fig. 9), the larger/wider the easier: this is not caused by a local quenching of combustion processes inside the front structure [START_REF] Williams | Combustion Theory[END_REF], but results from an untenable balance between stretch, geometrical nonlinearity and nonlocal hydrodynamics (and curvature) . As the above analyses showed it, this effect is only quantitatively modified when the stretch intensity oscillates, crest suppression being just made somewhat easier. These findings have been used to suggest this mechanism as the reason why the wrinkle upper cut-off wave-number in turbulent flames experimentally coincides [10] with the neutral wave-number identified by linear stability analyses. Classical (orthogonal) polynomials were encountered when studying stretch-induced horizontal pole/crest equilibriums. Not unduly surprising, for these are not directly affected by the Darrieus-Landau mechanism, and hence are electrostatic-like. The needed polynomials obey differential local equations, yet the classical ones do not cover all 'horizontal' equilibriums at weak stretch; some available generalisations (e.g. Heine-Stieltjes polynomials [START_REF] Andrews | Special Functions[END_REF]) will hopefully do.

But concerning the 'vertical' equilibriums the situation is much less clear. The nonlocal DL instability mechanism indeed acts on the pole population in a way that is explicitly 'vertical', encoded as it is in an irremovable Hilbert transform. Further studies of the "stretched Sivashinsky polynomials", whose roots obey the steady pole equations, seem warranted: e.g., to theoretically elucidate the very nature of the miracle that allows the MS equation to have poledecomposed solutions (hidden symmetries or mere good fortune?), and because the Darrieus-Landau mechanism is an indispensable ingredient of stretch-induced wrinkle suppression.

Yet another important theoretical point is dangling. Implicit in the discussion about the influence of stretch on the wrinkle wave-number (Sec.VIII) was the assumption that the maximum admissible number of front-slope poles, corresponding to turning-point conditions, is the relevant one: since the pole dynamics conserves their number (when finite), how can noise supply 'enough of' them as the stretch intensity varies? This brings one back to a nonlinear

  Time-independent s S , as exist about the troughs (wide local minima) of steady base flame profiles Φ when unforced[21], are considered first ( t -dependent ones will be touched upon in Sec.VII). Localised solutions to the nonlinear (2.3), e.g. the steady ones (in a suitable frame) considered below, bring about a continuum of Fourier modes, which makes them difficult to handle numerically by spectral methods. The pole-decomposition technique recalled below bypasses the difficultyA. Pole equationsAs first shown in[7][22], the MS PDE (Eq.(2.3) with 0 US = = ) admits exact non-periodic solutions φ representing localized patterns that are 'steady' and have 2 . n B >0 for n = 1,2… N and nn B B -= -(φ is real when x is), are poles of the flame slope x φ continued in the complex z -plane, zxi B = + . These carry the same 'charge' (residue) 2ν -, fixed by the dominant balance 2 /2x

FIG. 1 (

 1 FIG. 1 (Color online) Numerical pole density ( ) num PB vs. pole altitude B (solid black), compared to the analytical prediction ( ) PB - , Eqs.(3.5) and (3.7) with max (, ) B BN S ν -=

FIG. 2 (

 2 FIG.2 (Color online) Barycentres of 100 N = pole pairs aligned along the imaginary axis, vs. stretch intensity S, for 1/ 199.5 ν = ( * 0.102... S = , Eq.(3.8)), without detached poles (lower curves) or with them (upper curves). Solid (black) and dash-dot (violet) lines: Newton-Raphson iterations from the steady Eq.(4.1); Dash (red) and dash-doubledot (green) lines: analytical predictions see text.

  Interestingly, the existence of a detached pole pair in equilibrium signals the appearance of a new type of crest structures that bifurcate from || B = ∞ at 0 S + = , in a sense generalizing the B + root of Sec.III (Eqs. (4.2) and (3.3) coincide if 1 N = ). More general arrangements involving several remote poles will be encountered in sections V.B and VI.B. Solving Eqs.(3.2) iteratively for different stretch intensities with a fairly large N (=100) and 1/ 2N ν ≈

  N ν = O and any S , the poles of a steady symmetric 2-crest pattern are not exactly aligned vertically but actually reside along two disjoint curves, () () zb xb i b =+ and () x bi b -+ , with 0( )( ) x bx b <=and max || bb < . Both curves share the pole density () p b = () p b per unit length along the b -axis, with max 0 = for normalization, and the real unknowns ( ) x b and ( ) p b obey a complex-valued generalization of (3.4) deduced from (3.2): we solve (5.1) analytically for ( ) zb : to two orders all the poles are found to remain aligned at 1

1 totN 1 totN

 11 conceivable "stretch vs.nonlinearity" nearly-real equilibriums rapidly grow with tot N . The function ( ) >> ,[START_REF] Andrews | The theory of partitions[END_REF]) of unordered integer partitions of tot N because unequal weights k N may be permutated; yet N grows less rapidly than that ( ]) of ordered partitions because x x ↔mirror images of admissible asymmetric patterns also are, which leads to double-counting: since most patterns are asymmetric, we conjecture that >> . At any rate, the above combinatorial reasoning suggests that weak positive stretch allows for a proliferation of steady solutions.

FIG. 4 (

 4 FIG.4 (Color online) A few curves giving bar B (barycentre of positive pole altitudes k B ) vs. S (stretch intensity) at fixed 1/10 ν = , for pole arrangements with 3 N ≤ . The label arguments encode the number of near-vertical pole alignments (at small Sν ) and the number of pole pairs in each; the superscript "+" indicates the presence of a pair of remote poles, | ( ) | z ℑ→ ∞ as 0 Sν + → : e.g., (1 + , 1, 1 + ) .

  pole pairs each. The (horizontal) equilibrium of the central crest at 0 0 X = is then guaranteed by symmetry. In the limit 1 01 SN ν < << the NN + lateral crest locations kk X from Stieltjes' classical analysis of electrostatic problems[START_REF] Ismail | [END_REF] (see Appendix B).

  either type of pole arrangement (all close to the real axis, or near the altitudes k hN ν± >>) can be obtained, depending on the initial seed chosen in Newton iterations ; each type may have its own turning point, at least when N is moderate .The two types can even coexist in some instances, which contributes to a further proliferation of solutions. As shown in Fig.4, six different patterns are already allowed for the same stretch intensity (e.g. 1 S = ) when 3 N = , not to mention those belonging to 1, 2 N = only. The situation does not simplify as 0 Sν > gets smaller, since larger s N are allowed for. It is not excluded that the arrangements obtained so far are only the first members of an endless family, with a backbone combining nearly vertical and nearly horizontal pole arrangements in a hierarchical manner at larger and larger scales as 01 Sν < << decreases. Two features complicate the matter (see Fig.9 for solutions with N up to 8): (i) the couplings between the x -and the B -wise interactions (e.g., see Fig.4 at 0.4 S ≈

  too small ) and the cumulated numerical errors are enough to break the self-similarity of the final collapse. Repeating the simulation with (0)s

FIG. 5 (

 5 FIG. 5 (Color online) Dotted (blue) lines: Evolutions of the scaled abscissas

FIG. 6 (

 6 FIG.6 (Color online) Dotted (blue) lines: Evolution of the scaled abscissas

  dynamics for the s n β thus has the same structure as the restriction of (4.1) to aligned poles nn zi B = , up to coefficients that only depend on ( / σ ω ), i.e. on the power spectrum of the integrated stretch fluctuations . In particular, the late time state of a two-pole ( 1 N = ) oscillating crest must satisfy 11 /0 KJ S ν ββ -+ 〈 〉 = instead of (3.3), and will be allowed only if S 〈 〉 is less than as in the non-oscillating, two-pole case, see (3words, an intense enough high-frequency oscillating component of the stretch intensity tends to flatten a two-pole crest ( 1 B β -〈〉 > along the lower branch 1β -of steady solutions to(7.

  influence of each frequency can be accounted for in its turn, starting from R ω . As a result, the J coefficient involved in the slowest dynamics (7products converge for R = ∞ whenever the power spectrum of the integrated stretch fluctuation satisfies the comparatively mild condition

FIG. 9 :

 9 FIG. 9: Compiled bar B (upper-pole barycentre) vs. 0 S ≥ (dimensionless stretch intensity) curves obtained in the present work for 1/10 ν = , up to 8 N = , for some of them. Curves with 3 N ≤ already appeared in Fig.4, e.g., the rightmost solution (3.3) ( 1 N = ).

.

  As first realized by Stieltjes[START_REF] Ismail | [END_REF][START_REF] Andrews | Special Functions[END_REF], the above sum over jk ≠ can also be written as lim { of the roots 1/ k η of () () / N N α η η L is deducible from the known coefficients[START_REF] Gradshteyn | Tables of Integrals, Series and Products[END_REF] of () (.)

IV. NUMERICAL vs. ANALYTICAL Two

  numerical approaches were employed to study Eqs.(3.2): (i) use Newton iterations or kin, that are delicate to initiate in case of multiple solutions yet give access to stable and unstable ones, and can benefit of analytically determined seeds; (ii) acknowledge that equations (3.2) are

	additive constant. The large steady crest thus has	(() ( 0 ) ) / x φ φ -	max B	=	max FxB (| | /	,	max S B	)	, and
	is ν -independent. The cumulative pole distribution	0 B R BP B d B () ( ' ) ' ≡∫	deduced from (3.5) is:
				max	=+ -	max	θ	+	θ	θ	,	(3.6)
	to be compared with the restriction to steady and pure imaginary poles n ( ) ii x πφ . The corresponding centre of mass of positive s B , bar B iB of more general equations [7][19] for the ≈
	max B PB Bd B N 0 (' )' ' / poles () ∫ () jj zt xt i Bt , has () j =+	max BB / bar of x π	= (1	-	/ 3) / (1 ww -	/ 4)	,	max wS B π ≡	, and can be notably
	less than the yet unknown max B ; the 'shape factor'	rms B	/	bar B also varies weakly with w .
	Finally, the normalisation of ( ) PB re-written as ( R θ π =	/ 2)	=	N	completes the resolution of
	Eqs.(3.4). It relates max B and N ν to the stretch intensity S by an analogue of (3.3):
						max NB ) / π νπ -+= , max (2 1 ( / 4) 0 SB	(3.7)
	the structure of which again parallels (3.2) and the 'sum rule' (expressed as (2	1) / NB ν bar --	1
	(/) ² rms bar BBS B bar += ). 0
	A single large crest again exists at fixed Nν whatever	0 S ≤ is, and has max 2 B πν ≤	N	. Just
	like with (3.3) two values B ± of max B	(hence two densities	( ) PB -	,	( ) PB +	) are obtained for
	0* SS ≤≤	, namely 2 πν	( , ) NB N S B ν -≤≤	*	and	( , ) B NS B ν + ≥	*	with:
										4) does not hold for	max || B B >	,
	where no pole lies, and max B	itself must be found as part of the complete solution, thanks to the * 1/ (2 ² ), * 4 SN B N π νπ ν ≡≡ . (3.8)
	overall normalisation max 0 B ∫ Similar to the 2-pole case, both branches merge at * 2 ( , 0) (' ) ' PB d B N = . BB N ν -=	when	SS =	*	. No real
	Equation (3.4) is solved by a Fourier method like in [21] (see Appendix A). In terms of an max B exists if * SS > , and the DL mechanism (middle term of (3.7)) again is needed for crest
	angle suppression. And larger wrinkles are easier to suppress: *~1/ /2 /2 π θπ -≤ ≤ + defined by max sin BB θ = , ( ) PB reads: SN ν	.
	max PB B 2 ² (| | π ν ≤= ) ln(cot ²( / 2)) θ Put differently, a given stretch intensity 01 Sν < << allows for a large centred steady crest iff max cos , S B π θ -(3.5)
	and 1* NN (| | PB B max ≥= . With ) 0 << ≤ 1/ 2 ² S π ν ≡ ( . ≡ integer-part). In case the number of pole pairs exceeds * max sin / B B θ = and 2 ( ) PB fixed by Eq.(3.5), a contour N ν integration in θ -plane (or p.591 of [24]) expresses max max 2( ' )' / ( ' ) B x B PB d B x i B φν + -=-∫ in terms of initially, at least * NN -of them will ultimately be expelled to | | B Nν >> as time elapses (see -below for the pole dynamics): the stretch intensity selects the width and amplitude of the
	max x B / surviving crest; both scale like the radius of curvature 1/ S of the base flame in any case : sinh ξ ≡ as sgn( ) x x πφ -= max ln(coth ²( / 2)) cosh SB ξ ξ -. The ensuing crest profile
	reads 2** 4 / sgn( ) ( ) / xx B max πφ 1 . 2 7 SB π =≈ for	sinh ln[coth ²( / 2) ] 2 ξ ξ ξ -1 N >> , 2 =-1.00 cc SB = for N = . SB max (s i n h c o s h ) / 2 ξ ξξ + + 1	, up to an

φ in unsteady situations, viz.:

  . A more complete determination of h accounts for the full pole =98.682 569 while the exact (numerical) value is 98.682 595. The 'centre of charge' estimate (4.2) gives 98.682 645 and is still a few percent accurate up to nearly

	distribution with density	() PB -	spread over [ max B -	,	+	max B	], instead of a mere global charge
	4( ν -sat at 1 ) N	0 B = . Once the integrals over	( ) PB d B -	are analytically evaluated (see pp.
	591 and 393 of [24]) the equation for h looks like (4.2), except for its last term that is replaced
	by	(2 / ) arcsin( π	max Bh S B max / ) -+ ² / ( ( ² h h B max -	1/ 2 ²) )	, with	max B	(( BN ν -=-	1 ) , ) S	. This
	resumes the form (4.2) when	1 SNν << and provides one with a useful test of the numerical
	method and convenient seeds for iterations: for	S =	0.01	,	ν =	0.1	and	4 N = , the more complete
	expression gives h

* SS = ;

this accuracy is to be used in Sec.VI .

  One might indeed have assumed that the previously considered isolated crest centred at More generally, let M be the number of crests involved in a steady arrangement and k N the number of pole pairs belonging to the k -th crest. This kind of configuration will henceforth be referred to as a ( 1 ,..., ,... + ) will indicate that a pair of poles separates from the considered pole pile as 0

	expanded for	|| 2 bX <<	1	to produce a linear term	(	2 )(2 .2 ) / (2 ) ( 11 1 N X ν -= -zX zXS 1 ) / 2	that
	ultimately contributes the extra /2 S 0 0 xX == , with its 0 N pairs of poles 0,n . As net consequence the results for isolated crests, e.g. iB , still is present: a steady 3-crest pattern is thus
	those encoded in Eqs.(3.5) (3.7) for conceivable, with the lateral ones (again with 1 1 N >> , still hold for twin crests once N pole pairs each) staying at a distance 01 / SN ν 1 << < ±	is 1 X
	replaced by 3/ 2 S given by 10 ; this replacement is required whatever 1 N is. If 11 1 (2 .2 ) / (2 .2 ) / (2 ) SX N X N X ν ν =+ if 0,1 01 SN ν < << . The construction may be 1 0 SN ν < gets (1) O the
	vertical pole alignments deform, with | ( ) | x b decreasing faster at smaller | | b . pursued with 5 crests peaked at the abscissas 012 0, , X XX = ±±, or 4 crests at	12 , X X ±±, ... etc.
	Multi-crested steady solutions also are unstable against shifts (~exp( ) St ) as a whole. They
	then provide one with localized burst-like disturbances, akin to those invoked by Zel'dovich et kM NNN )-solution; and, whenever needed, the superscript "+" (e.g.,
	al. [26] but here of finite amplitudes, travelling along the nearly-parabolic base flame front k SN ν →+ .
	For	0 S > small enough that the typical crest spacing (anticipated to be ∼	1/ 2 () k NS ν	) noticeably
	exceeds max () k B	∼ k N ν , the crest abscissas k X must satisfy the conditions of equilibrium of M
	'charges' 4 k N ν ,	1,..., kM =	, subject to attractive horizontal 1/ X interactions and all sat on a
	common quadratic potential barrier	-	SX	²/2	:
								4	0.
								1
				SX	11 (2 .2 ) / (2 ) 1 N X ν ≡	. Their density ( ) p b still obeys (3.4) with Sx ( 1 S = , 1/10 ν = , 4 N = ). The 4 pole locations in
	upper x iB

1 {() , } p bN in lieu of {() , } PB N , but with S replaced by /2 SS + : as the () zb are not exactly real, the largescale influence they feel from the other distant crest is not quite uniform, yet it may be Taylor-2 /2 Sx ; Fig. 3 shows a sample travelling burst comprising 8-poles. These bursts admittedly are also unstable with respect to modifications of the crest mutual distances, yet the presence of several nearby crests may help some survive longer as a result of the 'horizontal' interactions: to wit, many-crested pattern may be steady if properly centred (in the frame evoked below (2.3)).

x FIG.3 (Color online) Solid (blue) line: Snapshot of a finite-amplitude, 4-pole-pair lateral "burst" (, ) tx φ superimposed to the base shape ²/2 + plane are shown as open circles above the x -axis. The scale is identical in all directions k N

  ], this result is compatible with the experimentally known near-equality between neutral and cut-off wave-numbers for the wrinkle spectrum of flames propagating in turbulent flows.

	1 ω ω ≥ ), and to turbulent-like fluctuations of S ; the analogues of J and K could even be propagating along the centreline ( y -axis) of left/right symmetric two-dimensional channels with c > DD criterion, and the presence of
	tabulated numerically, if '( ) St St ( ) =-〈 〉 is available and contains no beat with S a variable width ( ) 0 y Λ> [32]. The scale factor is then () (. )/ L Et tu = Λ〈 Λ 〉 , ˆ~/ (1) ων = O . L ttu l ) ( a A (weight of nonlinearity in (8.1)) in c D further substantiates this.	A	²
	being a dimensioned time, whereby The unsteadiness caused by a time-dependent Σ will admittedly bring about numerical factors 0 S 〈 〉= and 1 E 〈 〉= ; the analogue of / JK again is less
	than1. One can show that the number N of pole pairs present 'in' the channel must be less than 2 (e.g., the grouping ²/ JK in (7.2)) in the above criterion; yet those will stay (1) O unless () L ut Σ
	(1 () y + Λ≡ 〈 Λ 〉 (i.e., JK = ), and no wrinkle is allowed durably in the wavy channel if / / ) / 2 JK νκ if a cellular pattern is to survive; this is more stringent than when JK κν ≤ and the frequencies ω it involves are well higher than ( ( ) ) Ln uk Ω OA . That, however, is unlikely 1.5 . for turbulent flames propagating through actual reactive gaseous pre-mixtures in conditions when 1.5 numerical solution 1 a thin front of ( ) l O thickness can be identified as such, for this requires () L ut Σ and ω to be
	0.5 well smaller than	theoretical prediction VIII. STRETCH vs. SPECTRAL CUTOFF u l [33], and ( ) / L (1) Ω= AO in practice: the estimate cutoff kk = O ( ) n	is then
	0.5 1 To put the findings obtained so far in a more physical perspective, and relate them to the second problem evoked in Sec. I, it is useful to restore dimensions in the 'stretched' MS equation (2.3) . This gives: 1 ˆˆˆ2 ˆˆ( [ ² ] ( [ / ( , ) ] , ). ) xx L x x n t L k u ax u I x φ φ φ φ φ ++ = Ω + Σ AA (8.1) where the over-hats of , , ... x t denote dimensioned variables, and the uniform drift velocity -1 -0.5 0 ψ expected to still hold true in such regimes, which it does [10][34]. As shown in Section III, the approximate result 2 max( ) 1 SB ≈ was obtained at turning points regardless of the number N of pole pairs involved, thereby suggesting that (1/ ) Σ O typical wrinkle size (amplitude and width) could be selected by the fluctuating stretch intensity () L ut Σ , when / 4 cutoff n ka Σ<<Σ ≡Ω . This militates in favour of a (mean-) power spectrum of wrinkling first pole vertical position ˆ()~() Ut Ut is again omitted : ˆ.²/2 x represents the parabolic background front, on top of -1.5 tied to that of stretch intensity at moderately small wave-number ratios / n kk: the estimate Φ≡Σ which sub-wrinkles of local amplitude ˆˆ(, ) tx φ grow. The dimensionless grouping (() )1 a ≈ + -6 -4 -2 0 2 4 6 ˆ1/ 2 B ≈Σ for the typical sub-wrinkle amplitude B prevailing at larger scales than1/ cutoff k might AA and the DL coefficient 1/ 2 ) ((1 ) / (1 ) 01 () ≈+ -≤ -Ω AA A only depend on the Attwood number ξ then form the basis of a scaling-law of the form 1 () ( / ) cutoff kfk k -Σ for the (mean-) power
	13 (see below Eq.(8.2) ). 0 < A () / () 1 14 ub ub ρ ρρ ρ =-+< , and are known from separate analyses [1]-[4]. 15 16 time FIG.8 (Color online) Solid (blue) curve: total flame front shape at turning point spectrum of wrinkling, with a nearly constant (or very small) (.) f at / Σ=Σ n kk<< cutoff 1 0 two-pole crest, ( ) ² / 8 2 ln(1 ² / 4) ψ ξξ ξ ≡ -+ , vs. ˆn xk ξ = . Dashed (black) line: osculating cosine 17 ( o r of the /1 n kk>> ).
	FIG.7 (Color online) Black solid line: oscillating solution 1 () B t -integration of equation (4.1) for 2 N = (, | | 1 , 2 kk zi B k = = ) for obtained from numerical 1/10 ν = , ( ) c SS ν == in (3.3)), see Figs 4 and 9. It corresponds to a total (i.e., base-flame + sub-This result cutoff n kk ≈ is not a mere dimensional consequence of having the Markstein length L St = 0.2 30sin(20 ) t + . Dashed (blue) line: analytical curve 1 exp( ( / ) cos ) T β σω -, 1,2 wrinkle) dimensioned flame profile φ φ ≡ Φ+ of the form: as reference in problems of flame dynamics, since n k L depends on the Attwood number: to wit, β being steady solutions of (7.1). (² ² ˆ() , ( ) 2 l n ( 1 ) ) ). (8 4 cn xk ak ξ ξ ϕψ ψ ξ Ω =≡ -+ A (8.2) () ñ k ≈Ω LA A for 1 << A . Next, the condition for suppression of all steady sub-wrinkle by A n geometrical stretch ( c SS > in the notation of Sec.III) can be re-written as 1/ c a >≡ DD , where
	1 ω ω intensity are fairly general. They actually extend to fronts that are 2( ) 1 Et π (/ 23 ) 0 . 9 ≥> > (replacing the series for ln J and ln K by integrals over κ / -periodic ( 0 κ > ) At any rate, the convexity of exponential functions guarantees that 1 E 〈 〉≥ , ²1 E The function (3 / 4 ln 4)(1 cos( / 12)) πξ --accurately osculates () ψ ξ at and between its min or D is a Damköhler number [33]. It is defined as the ratio of the rate-of-strain L u Σ based on flat-〈〉 ≥ for any (ln ) / '( ) dE d t S t = implies ²/ ²/ ² 1 JK E E =〈 〉 〈 〉≤ , whereby the trends revealed with harmonic variations of stretch caused by distortions at larger scale, visible wrinkles with wave-numbers || cutoff kk ≥= such that latter time-averages exist. The Cauchy-Schwartz inequality max, located at 0, 12 ξ =± , see Fig.8. This suggests that, as a result of the geometrical stretch flame speed L u and the background flame curvature Σ (tributary of the forcing function in (2.1)
	in x , in which case '( ) St d E d t (ln ) / =	. This pertains to x	x →-invariant flame fronts

The 'ultimate' steady sub-wrinkle (in the proper frame) that can survive as the stretch intensity L u Σ (now a reciprocal time) increases is the two-pole solution at its turning point (the same as at 1/ 4 nn kk π ≈ unlikely exist, provided the local wrinkle structure be considered quasisteady. Since the crest radii of curvature essentially scales like 1/ n k regardless of their amplitude [7), to the maximum DL growth rate ( ) Ln uk Ω A ; this suggests a dynamical origin. That a balance of nonlinearity, DL instability and stretch is a sine qua non of wrinkle suppression (see (3.3)) confirms the genuine flame-dynamical origin of the

² ( ) / sin ln[cot ²( / 2)] 2

problem already evoked in Sec.I: how does noise, even if weak when seen on the real axis, implant complex poles (incipient wrinkles)? Answering this question is one of the most challenging open theoretical issues about flame dynamics, not to mention its statistical aspects.