The Sivashinsky equation for corrugated flames in the large-wrinkle limit

The Siv ashinsky equation for corrugated flames in the large-wrinkle limit Guy Joulin 1 , ∗ and Bruno Denet 2 , † 1 L ab or a toir e de Combustion et de D´ etonique, UPR 9028 du C NRS, ENSMA, 1 rue Cl´ ement A der, B.P. 40109, 86961 F utur osc op e Ce dex, Poitiers, F r anc e. 2 Institut de R e cher che sur les Ph´ enom ` enes Hors d’Equilibr e, UMR 6594 du CNRS, T e chnop ole d e C hˆ ate au Gomb ert, 49 rue Jolio t-Curie, 1338 4 Marseil le Ce dex 13, F r anc e. (Dated: Octo b er 2, 20 18) Abstract Siv a shinsky’s (1977 ) nonlin ear in teg ro-differen tial equation for the shap e o f corrugated 1- dimensional flames is ultimately reducible to a 2 N -b o dy problem, in v olving the 2 N complex p oles of th e flame slop e. Th ual, F risc h & Henon (1985) d eriv ed singular linear in tegral equ ations for the p ole densit y in the limit of large steady wr inkles ( N ≫ 1), which th ey solve d exactly f or mono- coalesce d p erio dic fronts of highest amplitude of wrinkling and approxima tely otherwise. Here we solv e those analytically for isolated crests, n ext for mono coalesced then bicoalesced p erio dic flame patterns, whatev er the (large-) amplitudes in v olv ed. W e compare the analytically pred icted p ole densities and fl ame shap es to numerical results deduced from the p ole-decomp osition approac h. Go o d agreement is obtained, eve n for m o derately large N s. The r esults are extended to giv e h in ts as to the dynamics of supp lemen tary p oles. Op en problems are ev ok ed. P ACS n um ber s : 47.20.K y , 47.54.-r, 47.70.Fw, 82.40.C k ∗ Corresp o nding author; Electro nic address: joulin@lcd.ensma.fr † Electronic addres s : br uno.denet@irphe.univ-mr s .fr 1 I. INTR ODUCTION Being able to describ e the nonlinear dev elopmen t of the L a ndau-Darrieus [1, 2] (LD) instabilit y of premixed-flame fron ts is a cen tral topic in com bustion theory . As early as 1977 Siv ashinsky [3] sho w ed, in the limit A ≪ 1 of small Att w oo d n um bers based up on t he fresh gas ( ρ u ) or burn t ga s ( ρ b < ρ u ) densities, 0 < A ≡ ( ρ u − ρ b ) / ( ρ u + ρ b ) < 1 , that t he shap e φ ( x, t ) of a flat-on-av erage, sp ontaneously ev olving wrinkled flame is gov erned b y φ t + 1 2 φ 2 x = ν φ xx + I ( φ ) (1) in suitable units. In (1) the subscripts denote partial deriv at iv es with r esp ect to time, t , a nd co o rdinate, x , normal to the mean direction of propagation, and the “ viscosit y” ν > 0 represen ts a recipro cal Pe clet num b er based up on the actual flame thic kness a nd the wrinkle w a v elength. The linear integral op erator I ( · ) is defined by I ( e ik x ) = | k | e ik x (whence I ( φ ) is the Hilb ert transform, ˆ H ( − φ x ), o f − φ x ) a nd stems from the LD instabilit y . The gr owth/deca y rat e of infinitesimal harmonics is | k | − ν k 2 , whic h iden tifies 1 /ν and ν as neutral w a v en um b er and minimum growth time, resp ective ly . The nonlinearity is geometrical, accoun ting as it do es for the cosine, (1 + s 2 ) − 1 / 2 ≃ 1 − s 2 / 2 + . . . , of the small angle (arctan( s ) ≃ s + . . . ) that the lo cal normal to the flame f r o n t mak es to the mean direction of propagation, where s ∼ φ x × A is the unscaled fro n t slop e. O riginally derive d in [3] as a leading order result f or A → 0 + , equation (1) happ ens to gov ern the shap e of steadily propagating f ron ts ev en when tw o more terms of the A -expansion are retained [4, 5]; its structure t hen remains v alid pra ctically up t o A = 3 / 4, i.e. , ρ u = 7 ρ b [4]. Numerics [6 ] rev eals that “steady” solutions of (1), corresp onding t o φ ( x, t ) = − V t + φ ( x ) are often ultimately reac hed. When (1) is in tegrated with p erio dic b oundary conditions for “not-to o-small” v alues of ν , ν > 1 / 25 sa y , the “steady” pattern has a single crest p er x - wise interv al of 2 π length, where φ xx is large and negativ e; without loss of generalit y o ne ma y a ssume t hat o ne is lo cated a t x = 0, in whic h case φ x = 0 when x is an integer m ultiple of π (i.e., x = 0 ( mo d π )) and φ xx ( ± π ) ≃ 1 /π . If Neumann conditions at x = 0 and x = π are used instead, still with a mo derately small ν , the fina l pattern obtained from n umerical (pseudo-sp ectral) in tegrations of (1) may also ha v e an extra crest lo cated a t x = π [7], with φ xx ( π ) large a nd negat ive. By the v ery wa y they are obtained as final state of an unsteady pro cess the 2-crested patterns hav e a finite basin of at traction, con trary to the case of p erio dic b oundary conditions [7] where the only stable patterns hav e a single 2 crest p er cell; ye t suc h “ ha lf-c hannel” solutions happ en to coincide with the restriction to 0 ≤ x ≤ π of prop erly shifted 2 π -p erio dic ones, for these are symme tric ab o ut x = 0 a nd x = π . If ν is to o small the widest patterns get v ery sensitiv e to noise, ev en when caused b y num erical rounding-off. In [8] the estimate µ ≥ O ( e − 1 / 2 ν κ ) ≡ µ c ( κ ) w as obtained for the noise in tensit y µ needed to trigger the app earance of extra-cells on top of the main ones with p erio dic b oundary conditions; the n um b er κ in the ab ov e exp onent is φ xx ( ± π ) ≃ 1 /π ; since the most rapidly g ro wing noise-induced disturbances ( with initial w a v en um b ers | k | ≃ 1 /ν [8]) of a nearly parab olic trough undergo an O ( e 1 / 2 ν κ ) amplification, they ultimately get visible as sub wrinkles of O (1) final amplitude if µ ≥ µ c ( κ ). Having a larger φ xx > 0 at their tro ughs (see Sec. 7), 2-crested pa t terns ar e presumably less sensitiv e to noise than the single-crest ones asso ciated with the same w a v elength, b ecause µ c ( κ ) increases dramatically with κ when ν is small. The numerical w ork of Ref. [9] also sho w ed that sums φ ( x 1 , x 2 , t ) = φ 1 ( x 1 , t ) + φ 2 ( x 2 , t ) of orthog onal, 2-crested one-dimensional patterns play a cen tral role in the s tudy of (1 ) g eneralized to 2-dimensional flames ( x → ( x 1 , x 2 ) , φ 2 x → | ∇ φ | 2 , φ xx → ∆ φ , I ( · ) ≡ m ultiplication b y ( k · k ) 1 / 2 in the 2 -D F ourier space k = ( k 1 , k 2 )) and to rectangular domains in the Cartesian ( x 1 , x 2 ) plane. Without noise suc h sums are exact stable solutions; with random additiv e forcing they recurren tly app ear as long - liv ed transien ts when Neumann conditions a r e adopted. F urther analyses on t he stabilit y o f solutions of Eq. (1) and their resp onses thus seem w arran ted, and getting the “steady” patterns that corresp ond to wide, hence large, cells ( or small ν s) is a pr erequisite. The presen t contribution is in tended to do this. It is organised a s f o llo ws. Se ction 2 in tro duces the p ole-decomp osition metho d, the discrete equations for the p ole lo cations, and the t w o in tegral equations that appro ximate them for large fron t wrinkles. The latter equations are next solv ed analytically fo r isolated crests (Sec. 3 ) then one-crested p erio dic patterns (Section 4), and the prediction compared to num erical results fro m the p ole- decomp o sition appro a c h. Sections 5 and 6 compute the flame sp eed from the density , and tak e up the dynamics of a few extra-p oles, resp ectiv ely . Section 7 generalizes the ab ov e in tegral equations to a pair of coupled o nes corresp onding to 2-crest p erio dic flames (and “half-channel” ones), then solv es them analytically; comparisons with nume rics are again pr esen ted. W e end up with concluding remarks and op en problems (Sec. 8). 3 I I. POLE-DE COMPOSITIO N(S) In 1985 Th ual, F risc h and Henon [10] (herein referred to a s “TFH”) disco v ered (see also [11]) that (1) p ossesses solutions φ ( x, t ) represen ting 2 π -p erio dic flame patt erns with slop es φ x in the fo r m φ x ( x, t ) = − ν N X α = − N cot  x − z α 2  , (2) in which the complex-v alued p o les of φ x ( x, t ), z α ( t ), are in v olv ed in conjug ate pairs ( z − α = z ∗ α , α 6 = 0) for φ x ( x, t ) to b e real when x is. F or this p ole-decomp osed expression to solve (1), the z ′ α s ( α = − N , . . . , − 1 , 1 , . . . , N ) m ust ev olv e according to the 2 N -b o dy problem dz α dt = ν N X β = − N β 6 = α cot  z β − z α 2  − i sign( I m ( z α )) , (3) where I m ( · ) denotes the imaginary parts of ( · ) a nd the signu m function (with sign(0) = 0) accoun ts for the LD instabilit y . Once (3 ) is solve d for the p ole lo cations, φ ( x, t ) is a v a ilable from (2) and the wrinkling-induced excess propaga tion speed V = −h φ t i > 0 f ollo ws from (1): V = 1 2  φ 2 x  , (4) where h·i stands for an av erage a long the x -co ordinat e; th us, V simply measures the wrinkling-induced fractional increas e in flame arclength, since h (1+ s 2 ) 1 / 2 − 1 i = h s 2 / 2 i + . . . ∼ A 2 × V . Beside p erio dic φ ( x, t )s, (1) also allow s [10] for isolated non p erio dic wrinkles that ha v e an infinite w a v elength, V = 0, cot( z ) replaced b y 1 / z , and dz α dt = ν N X β = − N β 6 = α 2 z β − z α − i s ign( I m ( z α )) . (5) In the lat ter situation, the precise v alue o f ν > 0 do es not matter since it could b e scaled out, and the in teger N ≥ 1 is ar bitrary . As for (2 ) (3), the maxim um allow ed v alue N opt ( ν ) of N in steady configurations increases with 1 /ν > 1 [10]. As show n b y TFH, steady flames obtained fro m (3) or (5) cor r esp o nd to p oles that “coalesce” (or align) along parallels to the imaginary axis, as a result o f the pairwise p ole in teractions that are attra ctiv e a long the real x -axis and repulsiv e in the normal direction. In the case of an isolated crest lo cated at x = 0, the po les ultimately in v olv e d in steady solutio n ar e of the form iB α , − N ≤ α ≤ N , 4 α 6 = 0, with real B α s satisfying coupled discrete equations deduced from ( 5): ν N X β = − N β 6 = α 2 B α − B β = sign( B α ) . (6) The autho rs of Ref. [10] a lso evidenced that the lar g er the n um b er N of p ole-pairs in suc h “ v ertical” steady alignmen ts, the smo other the in v olv ed p oles are distributed along the B co ordinat e, with B α +1 − B α w ell smaller than B N . This suggested TFH to replace the discrete sum in (6) (or its analogue deduced from (3)) by an inte gral ov er the con tin uous v aria ble B , with such a contin uous measure that P ( B ) dB is the n um b er of p oles lo cated b et w een B and B + dB ; a constructiv e definition of P ( B ) is sp ecified in (20) . In this con tin uous appro ximation the steady ve rsions o f (2) (3) are amenable to singular F redholm in tegral equations, sp ecifically: − Z 2 ν P ( B ′ ) B − B ′ dB ′ = sign( B ) (7) in the non- p erio dic situations (an isolated wrinkle at x = 0), a nd − Z ν P ( B ′ ) coth  B − B ′ 2  dB ′ = sign( B ) (8) for the mono coalesced 2 π - p erio dic cases (one single crest p er cell, at x = 0 ( mo d 2 π )). In (7) (8) B denotes the p ole imaginary co or dina t e, and the Cauc h y principal parts − R · dB ′ stem from the conditio n β 6 = α on the sums featured in (3) (5). Consisten t with their in terpretation as p ole densities, t he P ( B )s show ing up in ( 7 ) (8) b oth are non-negativ e ev en functions of t heir argumen t (fo r φ x to b e real when x is) a nd are no r ma lized b y Z P ( B ′ ) dB ′ = 2 N . (9) In (7)-(9) the in tegrals extend o v er the ranges (to b e determined as part of the solutio ns) where P ( B ) 6 = 0. The next sections will solve (7) (8 ) (9) analytically , star t ing with the simpler equation (7 ). I I I. ISOLA TED CREST Because isolated crests hav e φ x → 0 a t | x | → ∞ , we firstly anticipate the existence of some finite B max > 0 suc h that P ( | B | > B max ) ≡ 0 in (7). W e next recall the iden tit y − Z π / 2 − π / 2 cos((2 M + 1)Φ ′ ) cos Φ ′ sin Φ − s in Φ ′ d Φ ′ = π sin ( (2 M + 1)Φ) (10) 5 that can b e deduced, through the c hange of v ariable Φ → Φ + π / 2, f r o m a similar one app earing in the Prandtl theory of lifting lines [1 2, 13]. Iden tit y (10) allows one to solv e suc h singular in tegral equations as Wigner’s [14] (for the densit y , 2 ν P say , of eigen v alues of large real random matrices in the Gaussian Orthogo nal Ensem ble), written here as − Z 2 ν P ( B ′ ) B − B ′ dB ′ = B ; (11) its solution is the celebrated semi-circle la w 2 π ν P ( B ) = max( B max cos Φ , 0), [14], provided that one sets B = B max sin Φ , − π 2 ≤ Φ ≤ π 2 , (12) in (11). In terestingly , the same c hange of indep enden t v ariable in (7) pro duces − Z π / 2 − π / 2 2 ν P (Φ ′ ) cos Φ ′ sin Φ − sin Φ ′ d Φ ′ = sign(Φ) , (13) since sign( B ) = sign(Φ) for | Φ | < π . Ov er the same ra ng e (and hence o v er the narr ow er supp ort of P , | Φ | ≤ π / 2), the righ t-hand side o f (13) may b e expanded a s the F ourier series sign(Φ) = 4 π ∞ X M =0 1 2 M + 1 sin((2 M + 1)Φ) , (14) consisten t with our con v en tion that sign( 0 ) = 0. F rom (1 0) the solution to (13) can thus b e written as a F ourier series of cosines that all v anish at Φ = ± π / 2: 2 ν P (Φ) = 4 π 2 ∞ X M =0 1 2 M + 1 cos((2 M + 1)Φ) (15) = 1 π 2 log  1 + cos Φ 1 − cos Φ  (16) = 1 π 2 log 1 + p 1 − B 2 /B 2 max 1 − p 1 − B 2 /B 2 max ! , (17) and P ≡ 0 for | B | > B max ; to get (17) from (16), (12 ) w as explicitly employ ed. The cum ulativ e p ole distribution R ( B ) = R B 0 P ( B ′ ) dB ′ reads, after in tegration b y parts, as 2 ν R ( B ) = B max π 2  sin Φ log 1 + cos Φ 1 − cos Φ + 2Φ  , (18) whereb y the renormalization condition R ( B max ) = R (Φ = π / 2) = N fixes B max to b e giv en b y B max = 2 π N ν. (19) 6 TFH [10] fitted the cum ulativ e distribution they obtained from a n umerical resolu- tion of (5) for steady arrang ements of aligned p oles, b y the expression π 2 ν R = R B 0 log(1 . 2 8 N ν π 2 / | B ′ | ) dB ′ when | B | ≤ B max [10]. Equations (17) (1 9) sho w that 1.28 estimated fr o m their numerical p ole distribution at | B | ≪ B max actually was a nume rical appro ximation of 4 /π = 1 . 273 . . . Figures 1 and 2 compare the analytical findings (18) (19) to our o wn resolutions of ( 5 ), with N = 10 , and 100, resp ectiv ely . The TFH fit is also displa y ed for illustrat ion. The p ole densit y P is defined fo r α ≥ 1 by P (( B α + B α − 1 ) / 2) ≡ ( B α − B α − 1 ) − 1 , (20) in terms of the p ole lo cations (with B 0 = 0 by conv ention); it is sho wn in Fig. 3 fo r N = 10 0 , and compared with the con tin uous a ppro ximation (17) and the TFH fit. Once the cum ulativ e distribution is determined by (18) (19) in the con tin uous limit, a ppro ximations ˜ B α to the discrete p ole lo cations can b e retriev ed up on solving [10] R ( ˜ B α ) = α − 1 / 2 , α = 1 , . . . , N (21) n umerically ( e. g., b y the Newton-Raphson metho d, with the “exact” B α s as initial guess!). The resulting crest shap e ˜ φ ( x ) = − 2 ν N X α =1 log  1 + x 2 ˜ B 2 α  (22) is compared to the exact one (nu merical) in Fig . 4 and to that issued from the contin uous appro ximation. The latter profile has φ x = − Z B max − B max 2 ν P ( B ) dB x − iB (23) = − 1 π sign( x ) log p x 2 /B 2 max + 1 + 1 p x 2 /B 2 max + 1 − 1 ! , (24) the second expression resulting from substitution of (17) in (23), then a lucky lo ok at p. 591 of Ref.[15]. As suggested b y the form of (2 3), and confirmed b y (24), φ x ( x ) is most simply deduced from P ( ± ix ) through con tour in tegration in the complex B - plane. A further in tegration b y parts of (24) yields the con tin uous-approx imation prediction for φ ( x ) (up to an additive constan t): φ ( x ) = − 1 π sign( x ) B max  sinh ξ log cosh ξ + 1 cosh ξ − 1 + 2 ξ  , (25) 7 where x = B max sinh ξ (compare to (12)). The integration constant was selected in F ig . 4 to achiev e go o d agreemen t with the exact φ ( x ) for | x | → ∞ . Tw o final remarks: (i) ν disapp eared a s a fa cto r in (24) as it should, b ecause ν can b e scaled out; (ii) φ ( x ) is of the form ν N F ( x/ν N ), and this scale-in v a r ia nce sho ws that the con tin uous appro ximation actually amoun ts t o describing φ ( x ) at large distances compared to the actual radius of curv ature (1 / R B max ˜ B 1 4 ν P ( B ) d B /B 2 = o ( ν )) of the flame tip, when N ≫ 1 ( that is, for large wrinkles). IV. MONO-COALESCED, PERIODIC C REST The follo wing simple remark will allo w us to solv e (8), i.e., in the case where all the p oles of φ x are aligned along the imaginary x -axis ( mo d 2 π ). Because P ( B ′ ) still is an ev en function of B ′ , only the eve n parts (at fixed B ) of coth(( B − B ′ ) / 2) will actually contribute to the inte gral o v er B ′ . Equation (8) may th us b e re-written as − Z B max − B max ν P ( B ′ )(1 − tanh 2 ( B ′ / 2)) tanh( B / 2) − tanh( B ′ / 2) dB ′ = sign( B ) , (26) up on use of the know n form ula for the tanh( · ) of a difference, and neglect of a term prop or- tional to − R P ( B ′ ) tanh( B ′ / 2) dB ′ = 0 . W e no w set tanh( B 2 ) = tanh( B max 2 ) sin Φ , − π 2 ≤ Φ ≤ π 2 , (27) con v erting (26) into − Z π / 2 − π / 2 2 ν P ( B ′ ) cos Φ ′ sin Φ − sin Φ ′ d Φ ′ = sign( Φ) , (28) whic h is nothing but (13). Therefore the sought a fter p ole-densit y is still given by (16), the only difference with the previous non p erio dic case b eing that B , B max , a nd Φ are now related b y (27) instead of (12 ). The new cum ulativ e densit y R ( B ) = R B 0 P ( B ′ ) dB ′ is giv en, after an integration b y parts, b y π 2 ν R ( B ) = 1 2 log 1 + A sin Φ 1 − A sin Φ log 1 + cos Φ 1 − cos Φ + Z Φ 0 log 1 + A sin Φ ′ 1 − A sin Φ ′ d Φ ′ sin Φ ′ , (29) 8 A ≡ tanh( B max / 2), whereb y the norma lizatio n (9) requires N ν π 2 = Z π / 2 0 log 1 + A sin Φ 1 − A sin Φ d Φ sin Φ . (30) As the ab ov e integral turns out to b e π ar csin A (p. 591 of [15 ]) the range of P ( B ), still giv en by R ( B max ) = N , now satisfies tanh( B max / 2) = sin ( π N ν ) (31) instead of ( 19). The latter and (31) coincide for π N ν ≪ 1, as do the a sso ciated p ole dens ities. The maxim um B max allo w ed b y (31), B max = + ∞ , has 2 N ν = 1 and cos Φ ≡ 1 / cosh( B / 2), whence P ( B ) resumes the form P ( B ) = 1 π 2 ν log  coth | B | 4  (32) obtained by TFH via F ourier tra nsformations. The fig ure 5 compares our predictions (29) and (3 1) with ve ry accurate solutions of (3) for N = 10 0 and 2 N ν = 1 . V ery g o o d agree- men t is obtained even if N is only mo derately la rge, a nd carries o v er to the p ole densities themselv e s. Again, appro ximate solutions ˜ B α can b e retriev ed from the analogue of (21), and an approximate flame front shap e from ˜ φ ( x ) = − 2 ν N X α =1 log(1 − cos x sec h ˜ B α ) + const . (33) Figure 6 sho ws of a comparison b et w een (33) , the exact flame shap e obtained from the exact (y et obtained n umerically) B α s satisfying (3), and the curv e deduced from the contin uous appro ximation, for which the flame slop e φ x ( x ) reads φ x = − ν Z cot  x − iB 2  P ( B ) dB , (34) again a real function b ecause P ( − B ) = P ( B ). With P ( B ) giv en by (16) (27) (31) the ab o v e in tegral can b e reduced t o one av ailable in p. 591 of [15 ] a nd yields (fo r − π ≤ x ≤ π ): φ x ( x ) = − 1 π sign( ξ ) log cosh ξ + 1 cosh ξ − 1 , tan x 2 ≡ A sinh ξ (35) thereb y confirming that φ x ( x ) is accessible from P ( B ) by analytical contin ua t ion to ± ix . In particular, t he TFH solution, eq. (32), has π φ x = − 2 sign( x ) log | cot x/ 4 | and φ xx ( ± π ) = 1 /π ; mo r e generally , φ xx ( ± π ) = A/π . A further in tegration by parts yields − i π φ ( x ) = sign( x ) log 1 + iA sinh ξ 1 − iA sinh ξ log cosh ξ + 1 cosh ξ − 1 +2 sign( x ) Z ξ 0 log 1 + iA sinh ξ ′ 1 − iA sinh ξ ′ dξ ′ sinh ξ ′ , (36) 9 whic h cannot b e ev aluated in simple closed form, but ma y b e compared to (29); of course φ ( x ) is real when x is, since the complex log( · ) in (3 6 ) a lso reads 2 i arctan( A sinh ξ ) = ix . Note that φ ( x ) ha s the form F ( x ; N ν ), in the presen t units where the pattern is 2 π - p erio dic. Adopting Λ 6 = 2 π as wa v elength w ould giv e 2 π φ = Λ F (2 π x/ Λ; 2 π N ν / Λ) with the same F . Accordingly , if ν N/ Λ is ke pt fixed, φ xx ( ± Λ / 2) scales lik e 1 / Λ as it should for ν → 0, whereb y halving the wa v elength renders the patterns less sensitiv e to noise (see the Intr o duction ). V. FLAME SPEED F R OM CONTI NUOUS POLE-DE NSITY Plugging (3 4) in to ( 4) allow s the wrinkling-induced increase in flame sp eed V to b e written as 2 V = ν 2 Z Z P ( B ) P ( B ′ ) h cot x − iB 2 cot x − i B ′ 2 i dB dB ′ . (37) Although the one-v ariable integrals inv olved when squaring (34) are o rdinary o nes, they ma y be written as principal parts. W e next inv o k e the t rigonometric identit y cot a cot b = − 1 + cot( a − b )(cot b − cot a ) and the av erage h cot x − iB 2 i = i sign ( B ) (38) to transform (3 7) in to 2 V ν 2 = − Z Z P ( B ) P ( B ′ ) dB dB ′ +2 Z − Z sign( B ) P ( B ) P ( B ′ ) coth  B − B ′ 2  dB ′ dB (39) The first double inte gral (= ( R P ( B ) dB ) 2 ) in (39) follows from the normalization (9), and is ( 2 N ) 2 . The second one is obtained from (8) after m ultiplication of b oth sides by P ( B )sign( B ) dB and subseq uen t in tegration ov er B : by ( 9 ), it is 2 N /ν . Th us the simple form ula V = 2 N ν (1 − N ν ) (40) ensues; no t ice that it was o btained without havin g to solv e (8). Actually (39) can b e sho wn from (3) to hold whatev er N and ν [16 ], again without solving the p ole-equations themse lv es. In view o f t he accuracy of (40) o ne ma y inquire whether t he solutions of (7) (8 ) satisfy the “invisc id” Siv ashins ky equation, i.e. (1) with ν = 0, in the steady cases. T o show they 10 do, for x 6 = 0 at least, one ma y set P = P ν and N = N ν to remo v e ν f r om (7 ) (9), then pro cess the Landau-Darrieus term o f (1) as follows in the case of an isolated crest: 2 iI ( φ ) = Z 4 P ( B ) sign( B ) x − iB dB = Z 2 P ( B ) x − iB dB − Z 2 P ( B ′ ) B − B ′ dB ′ + ( B ↔ B ′ ) = Z Z 4 i P ( B ) P ( B ′ ) ( x − iB )( x − iB ′ ) dB dB ′ = iφ 2 x , ( 4 1) where the notation ( B ′ ↔ B ) represen ts a second cop y of the in tegral that precedes it, with B and B ′ in terc hanged. The lines ab o v e successiv ely use (7), a ckno w ledge tha t ( B , B ′ ) are dumm y v ariables of in tegration that ma y b e in terc hanged, then employ (23) squared. Hence (25) satisfies (1) when ν = 0 and N is prescribed, if x 6 = 0. Thanks to (39) , a similar analysis applies to (8 ) , pro vided x 6 = 0 (mo d 2 π ). Beside providing one with an exact P ( B ), eq. (16) sho ws that (8 ) a dmits a contin uum of solutions, for there exists no t hing in (9) to tell o ne that N ough t to b e an inte ger; this will b e commen ted later (see Sec. 8). One finally sp ecializes (8) to B = B max to show that N is constrained b y 0 ≤ 2 N ν ≤ 1, since coth( B max − B ) ≥ 1 (see also (31)). VI. D YNAMICS OF SUPPLEMENT AR Y POLE-P AIRS In 200 0 , V a yn blatt & Matalon [17] addressed the li n e a r stability of p ole-decomp osed mono coalesced “steady” solutions − V t + φ ( x ) to (1). Up on writing φ ( x, t ) + V t − φ ( x ) ∼ exp( ω t ) ψ ω ( x ) ≪ 1 then ana lytically solving the linearised dynamics to get ω and ψ ω ( x ), the authors of [17] iden tified t w o types of linear mo des. The mo des of t ype I describ e ho w the 2 N p oles of φ x ( x ) ev olv e when displaced b y infin i tesim al amounts from equilibrium; all those a r e stable ( ω < 0), but one that has ω = 0 (see b elow ). The mo des of type I I w ere in terpreted [17, 1 8] a s r esulting fr om x -p erio dic arrays of p oles at ± i ∞ that ma y sp ontaneously approa c h the real axis if N is to o small fo r the selected ν < 1. The o v erall conclusion w as thus : when endo w ed with 2 π -p erio dic b oundary conditions, all the mono coa lesced solutions are linearly unstable, except a single one that has N = N opt ( ν ) ≡ ⌊ (1 + 1 /ν ) / 2 ⌋ ≃ 1 / 2 ν ( ⌊·⌋ ≡ integer part) and is neutrally stable ( ω = 0) against shifts a long the x -axis, the corresp onding a n ti- symmetric eigenmo de b eing ψ 0 ( x ) = φ x ( x ). F or N < N opt , mo des of t yp e I I can manifest themselv e s, t w o particularly dangerous ones corresp onding to incipien t secondary wrinkles 11 cen tred on t he main crests ( x = 0, mo d 2 π ) or tro ug hs ( x = π , mo d 2 π ). When Neumann conditions a re emplo y ed instead, the aforemen tioned shifts are not al- lo w ed any longer b ecause ψ 0 x 6 = 0 at x = 0 and x = π . Numerical in tegrations [7] o f (1) and (3) evidence that there ma y then exist stable bi-coalesced patterns comprising an extra crest lo cated at x = π . Ev en though the steady 2 π -p erio dic patterns also satisfy (1) with Neumann conditions when prop erly shifted to ha v e φ x (0) = 0 = φ x ( π ) no stabilit y analysis similar to [17] is y et a v aila ble in this case; ye t instabilities then necess arily require N < N opt ( ν ). Here w e a ddress a restricted asp ect of the problem, namely: w e study ho w the previously deter- mined mono coalesced “steady” solutions (25) (36) in teract with extra pairs of p oles. Since the free dynamics (3) conserv es the total num ber of p ole pairs at its t = 0 v alue, it make s sense to consider φ ( x, 0)s that inv olve them in a lar g er n um ber ( N + n ) than the N = O (1 /ν ) ones retained in a steady profile φ ( x ). Eac h of the n supplemen tary pairs at x m ( t ) ± iy m ( t ) con tributes a p erturbat io n φ m ( x, t ) = h φ m i − 4 ν P j ≥ 1 exp( − j | y m | ) cos( j ( x − x m )) /j to the flame shap es (this follows fro m (2) via a term-b y-term F ourier expansion [10]) and, as sho wn in [18 ], sup erp osing φ m s can repro duce virtually an y disturbance φ ( x, 0) − φ ( x ). In the presen t for mulation the only difference b etw een Neumann and 2 π -p erio dic b o undar y condi- tions deals with the initial phases x m (0): whereas the former require the x m s to b e compatible with the x ↔ − x and π − x ↔ π + x symmetries, the latter do not. Con trary to the more conv en tio nal normal-mo de metho d (to which it is equiv alen t if | y m (0) | ≫ 1 [18]), the p ole approach can follow the disturbances when significan t nonlinear effects set in. . . if one is able to solv e the 2 N + 2 n coupled equations for the p o le tra jectories in the complex plane. The next remark somewhat simplifies the task. In the limits N ≫ 1, ν → 0 + and ν N = O (1) that led to (8), accoun ting for n = O ( 1 ) extra p ole pairs – as is assume d here – exerts only a small O ( ν ) p erturbation on the 2 N p oles already aligned. Accordingly the distribution P ( B ) of p oles along the main alignmen ts at x = 0 ( mo d 2 π ) ma y b e k ept unc hanged, and giv en by (16) (27) (31), when computing the motion o f 2 n supplemen tary ones. In the illustrativ e examples that follow only t w o extra po les ( n = 1) lo cated at ± i y ( t ) ( mo d 2 π ), y > 0, then at π ± iy ( t ) are considered, to b egin with. 12 A. Extra-p oles at x ≃ 0 ( mo d 2 π ) When the tw o supplemen tary p oles are lo cated at ± iy ( t ) ( mo d 2 π ), their altitude y ( t ) is determined fr o m (3) – within O ( ν , 1 / N ) fractional errors in the limits N ≫ 1, ν → 0 + and ν N = O ( 1 ) – b y the OD E dy dt = − Z ν P ( B ′ ) coth  y − B ′ 2  dB ′ − 1 , (42) = 2 π arcsin(sin( π N ν ) coth ( y / 2)) − 1 , | y | ≥ B max , (43) where P ( B ) is the sam e as give n b y (16 ) (27) and (31), to leading order, and leads to the closed fo rm (43) o n integration [15]; for | y | ≤ B max , dy /d t = 0 b y (8 ) . T herefore, whenev e r 0 < 1 − 2 ν N = O (1 ) a nd ν → 0 + , an y initial y (0) > B max will ultimately lead to y (+ ∞ ) = B + max , thereb y adding one new incomer to t he already presen t con tin uum. Put in w ords: if 2 ν N < 1 initia lly , the main pattern is unstable to disturbances with p oles at ± i y ( t ), and the latt er pro cess tends t o mak e 2 N ν approa c h 1 from b elo w. P erio dic b oundary conditions w ould allow the supplemen tary pair to b e initially off the x -axis, say at x (0) ± iy (0) with 0 < x (0) < π (mo d 2 π ). The “ho r izontal” attra ction by the main p ole condensation at x = 0 ( O ( ν ), actually) [10] will make x ( t ) decrease, while y ( t ) still do es if 2 N ν < 1. Ultima t ely , the extra p ole pair will join the main p ole alignmen t (in finite time), and the previous conclusion is qualitativ ely unc hanged: the pro cess mak es N increase by o ne. When Neumann conditions are adopted, how eve r, at least two pairs ± x ( t ) ± iy ( t ) are needed if x ( t ) 6 = 0, to meet the requiremen t of symmetry a b out x = 0, and t w o p ossibilities are encoun tered as to their fate. In the first instance, corresp onding to not- to o-small x (0)s a nd mo derate v alues of y (0), the pro cess is qualita tiv ely the same as ab o v e, except that 2 pairs simu ltaneously join the main condensation a t | y | < B max , thereb y making N increase b y 2. If x (0) is small and y (0) w ell ab o v e B max , the horizon tal mutual attraction b et w een the pair mem b ers ma y mak e them hit the x = 0 axis at suc h a finite time t c that y ( t c ) > B max ; this is b est sho wn from (3) sp ecialized to x ( t ) ≪ 1, whereb y dx/dt ≃ − ν /x then x 2 ( t ) + 2 ν ( t − t c ) ≃ 0 for t . t c . The double p ole thus formed at iy ( t c ) then instan tly splits into t w o simple ones lying on the x = 0 axis at y ( t ) − y ( t c ) ∼ ± ( t − t c ) 1 / 2 , leading to a subsequen t dynamics that ultima t ely ends like a t the b eginning of this subsec tion if 2 N ν < 1. 13 B. Extra-p oles at x ≃ π ( mo d 2 π ) In case the supplemen tary p oles ar e lo cated at π ± iy ( t ) eq. (42) is replaced by dy dt = 2 π arcsin(sin( π N ν ) tanh( y / 2)) − 1 + ν coth y , (44) since coth( u + i π / 2) = tanh u . Ev en though ν ≪ 1 the last term in (44), stemming from the in teraction o f the extra-p ole with its complex conjugate, cannot b e simply discarded, for otherwise (44 ) would not b e uniformly v alid if y gets small. According to (44), any initial y (0) indeed ultimately leads to y (+ ∞ ) = ν + o ( ν ) and to a small ( O ( ν )) stable disturbance cen tred at x = π (mo d 2 π ) whenev er 2 N ν < 1 . Although the main pattern’s curv ature φ xx ( π ) > 0 is O (1 ), and the O ( ν ) con tribution to φ ( x, t ) of the extra p ole-pair is small, it is nev ertheless enough [7] to render t he flame shap e φ ( x, t ) conca v e down w ar d at x = π ; as sho wn in [7], this o ccurs as so on as the extra p oles en ter a thin strip ab out the real axis, | y | . √ 4 π ν . Incorp orating O ( N ) extra pair s will a lso do, but the pro cess of dynamical trough splitting is not within reach of such OD Es as (44) when n = O ( N ). The structure of 2-crested steady patterns with n = O ( N ) will b e studied in Sec.VI I . Lik e in VI A one migh t b egin generalizing the presen t discussion b y en visaging a single pair of extra p o les off the x = π axis, but this is a lr eady co v ered in the preceding par agraphs: if 0 < x (0) < π the pair ultimately joins the p o les at x = 0 ( mo d 2 π ). It is more reve aling to consider tw o suc h pairs a t π ± x ( t ) ± iy ( t ) with x ( t ) “ small enough”, in a w a y compatible with Neumann conditions, b ecause something new a pp ears. Comparatively large x (0)s will clearly lead to pairs that ultimately stic k at x = 0 (mod 2 π ), b ecause their mutual horizon tal attraction could not opp ose that of the main alignmen ts. The other extreme of very small x (0)s again leads to the for mation o f double p oles at some π ± iy ( t c ), then a subsequen t ev olution of the t w o pairs π ± iy 1 , 2 along the line x = π (mod 2 π ) until they settle at O ( ν ) distances to the real axis if 2 N ν < 1. The imp or tan t conclusion is that stable 2-crest patterns exist when Neumann conditions are used and 2 N ν < 1 . By contin uit y there exist separating tra jectories S ± , suc h tha t no ne of the ab ov e b e- ha viours is o bserv ed if the p ole pa irs initially sit on them. The lines S ± lead the tw o supple- men tary pairs to w ards an unstable equilibrium, a result o f a comp etition b etw een a ttraction b y the main p ole p opulation at x = 0 (mo d 2 π ), and the mutual attractions/repulsions among the pair members. F or ν ≪ 1, and N ν = O (1), using the steady vers ion of (3) 14 and the p ole-densit y giv en by (16) (27 ) (31), one can sho w that suc h equilibriums corre- sp ond to x (+ ∞ ) = ± (2 π ν / A ) 1 / 2 + · · · and y (+ ∞ ) = ± ν + · · · to leading order, aga in with A = tanh( B max / 2) = sin( π N ν ). This sho ws that there exist ev en more g eneral steady so- lutions than considered elsewhe re in the pap er and in the literat ure (except in [7] where a similar conjecture w as made on a n umerical basis). One could hav e included o t her pairs as w ell, some of which along the x = π ( mo d 2 π ) axis. Our la st remark is to again stress that the free dynamics (3) conserv es the total n um ber of p oles (if finite). By the same tok en, allow ing this n um b er to v ary with time is a means to study a for c e d v ersion of the Siv ashins ky equation: adding a pair of p oles x m ± iy m at t = t m amoun ts to a ccoun ting f o r a term φ m ( x ) δ ( t − t m ) in the righ t-hand side of ( 1), and combining man y φ m s with v arious phases (as to v ary t heir signs), amplitudes ( ≃ − 4 ν exp( − | y m | ) if | y m | ≫ 1) and times of implan tation ( t m ) could help one in v estigate the resp onse of fla mes to a ric h class of w eak random noises. W e understand that a similar prop osal w as dev eloped ab out the “kic k ed” Burgers equation [19], i.e. , (1) without the in tegral term in the o ne- dimensional case. VI I. BI-COALESCED PERIO DIC P A TTERNS W e now tak e up the structure of “steady” 2 π -p erio dic solutions of (1) that would ha v e N pairs of p oles i B α ( mo d 2 π ), α = ± 1 , ± 2 , . . . , ± N , and n = O ( N ) other pairs at π + i b γ (mo d 2 π ), γ = ± 1 , ± 2 , . . . , ± n . F or brevit y , w e will say that the p ole-a lig nmen ts reside “at” x = 0 or x = π , resp ective ly , lik e the t w o crests p er-cell they corresp ond to. Because coth( u + i π / 2) ≡ tanh( u ), the steady v ersions of (3) corresp onding to suc h bi- coalesced flame patterns read as ν N X β = − N β 6 = α coth  B α − B β 2  + ν n X δ = − n tanh  B α − b δ 2  = sign( B α ) , (45) ν n X δ = − n δ 6 = γ coth  b γ − b δ 2  15 + ν N X β = − N tanh  b γ − B β 2  = sign( b γ ) . (46) In the distinguished limits ν → 0 + , N ν = O ( 1 ), nν = O (1), the p oles at x = 0 and x = π get densely pac k ed (at the scale of the w a v elength), with densities P ( B ) and p ( b ), resp ectiv ely . Both P and p will b e nonnegative and, in general, compactly supp orted: P ( | B | ≥ B max ) = 0 = p ( | b | ≥ b max ). The ranges B max and b max are to b e found as part of the solutions to the contin uo us ve rsions of (45) a nd (46): − Z ν P ( B ′ ) coth  B − B ′ 2  dB ′ + Z ν p ( b ′ ) tanh  B − b ′ 2  db ′ = sign( B ) , (47) − Z ν p ( b ′ ) coth  b − b ′ 2  db ′ + Z ν P ( B ′ ) tanh  b − B ′ 2  dB ′ = sign( b ) , (4 8) that are v alid for | B | ≤ B max and | b | ≤ b max , resp ectiv ely . T o restore some symmetry w e set tanh( B / 2) = A sin Φ , A ≡ tanh( B max / 2) ≤ 1 , (49) tanh( b/ 2) = a sin ϕ, a ≡ tanh( b max / 2) ≤ 1 , (50) in (47) (48), t hen ac kno wled ge that b oth P ( · ) and p ( · ) are ev en functions, whic h allows one to suppres s some o dd parts of the in tegrands, view ed as functions of Φ ′ (or ϕ ′ ) a t fixed Φ (or ϕ ). Some cum b ersome a lgebra ultimately transforms (4 7) and (48) into − Z 2 ν P (Φ ′ ) cos Φ ′ sin Φ − sin Φ ′ d Φ ′ + Aa sin Φ × Z 2 ν p ( ϕ ′ ) cos ϕ ′ 1 − A 2 a 2 sin 2 ϕ ′ sin 2 Φ dϕ ′ = sign(Φ) (51) − Z 2 ν p ( ϕ ′ ) cos ϕ ′ sin ϕ − sin ϕ ′ dϕ ′ + Aa sin ϕ × Z 2 ν P (Φ ′ ) cos Φ ′ 1 − A 2 a 2 sin 2 Φ ′ sin 2 ϕ d Φ ′ = sign( ϕ ) (52) 16 where all the v a riables (Φ , Φ ′ ), ( ϕ, ϕ ′ ) are now t a k en in the common [ − π / 2 , π / 2] range. One ma y th us a dopt a common nota tion f or t hem, ( σ, σ ′ ) say , in b oth ( 5 1) and (52) a nd subtract the results to eliminate the sign( · ) functions in the right-hand sides. This pro duces an homogeneous equation for the difference P ( · ) − p ( · ), o f whic h one obv ious solution is P − p ≡ 0. Hence the imp ortan t result: if P = p is indeed a viable solution, then P ( B ) = J  σ = arcsin  tanh B / 2 A  , (53) p ( b ) = J  σ = arcsin  tanh b/ 2 a  , (54) where J ( σ ) is the same function fo r b oth. The eve n J ( · ) function itself is then found from (51) or (52) to satisfy − Z π / 2 − π / 2 2 ν J ( σ ′ ) cos σ ′ sin σ − sin σ ′ dσ ′ + Aa × Z π / 2 − π / 2 2 ν J ( σ ′ ) sin σ cos σ ′ 1 − A 2 a 2 sin 2 σ ′ sin 2 σ dσ ′ = sign( σ ) . (55) F urther c hanging the indep endent v ariable to θ , with sin θ = (1 + Aa ) sin σ 1 + Aa sin 2 σ , (56) fortunately conv erts the seemingly hop eless (55) into a form equiv alen t to the already solve d Eq. (28), θ pla ying the part that the for mer Φ did there (most easily sho wn by starting from (28)). Accordingly , t he solution to (55) is av ailable in terms of the already found p ole- densit y p ertaining to the isolated crests, then the mono coalesced ones: f r o m (16) one indeed has 2 ν J ( σ ) = 1 π 2 log 1 + cos θ 1 − cos θ , (57) with θ defined in (56). As said earlier, eqs. (53) (54) , P ( B ) is immediately retriev ed up on setting sin σ = tanh( B / 2) coth( B max / 2) in (56) (57); same op era t io n to get p ( b ) from J ( σ ), up on setting sin σ = tanh( b/ 2) coth( b max / 2) in (57 ) (56). The first step to get B max and b max again is to compute the cum ulativ e p ole- densities. F or example R ( B ) = R B 0 P ( B ′ ) dB ′ is computed a s follows from (57) (5 6): 2 π 2 ν R ( B )= Z Φ 0 log  1 + cos θ ′ 1 − cos θ ′  2 A cos Φ ′ d Φ ′ 1 − A 2 sin 2 Φ ′ , (58) = log 1 + A sin Φ 1 − A sin Φ log 1 + cos θ 1 − cos θ + 2 Z θ (Φ) 0 dθ ′ sin θ ′ log 1 + A sin Φ ′ 1 − A sin Φ ′ , (59) 17 again with the understanding that Φ (o r Φ ′ ) is view ed as a function of θ ( o r θ ′ ) via (56), and con v ersely ; (58) is obtained from the definition of R ( B ) up on setting ta nh( B / 2) = A sin Φ, and (59) results fro m an integration by parts. The cum ulativ e densit y p ertaining t o p ( b ) is obtained in the same w a y from (57) (56), no w thanks to tanh( b/ 2) = a sin ϕ : the result is lik e (59), except for the substitutions A → a , Φ → ϕ , B → b , R ( B ) → r ( b ). The normalisations R ( B max ) = N and r ( b max ) = n th us imp o se the t w o conditions N ν π 2 = Z π / 2 0 dθ sin θ log 1 + A sin Φ 1 − A sin Φ , (60) nν π 2 = Z π / 2 0 dθ sin θ log 1 + a sin ϕ 1 − a sin ϕ , (61) that ma y be compared to the former equation (30), and reduce to it when Aa = 0. Although w e could not compute the ab ov e normalization in tegrals in closed forms, this can b e done n umerically without difficult y to get A and a as function of N ν and nν (or con v ersely). Note that N ≥ n is equiv alent to A ≥ a . N > n also implies that R ( · ) > r ( · ) when b oth are ev aluated at the same argumen t, Fig. 8. Before closing this section, it remains to compare the a b o v e predictions to direct n umerical resolutions of (45) (46), b y the Newton-Raphson metho d. This is done in F igs 7 and 8. Figure 7 will hop efully convince the reader that b oth P ( B ) and p ( b ) can b e expressed in terms of the single function J ( σ ) given by (57). No w that the p ole densities are av ailable, o ne ma y try to compute the cor r espo nding increase in flame sp eed, V , from (47) (48) without solving them (lik e in Sec. 5), to pro duce V = 2 ν ( N + n )(1 − ( N + n ) ν ); (62) this simple form ula reduce to (4 0) if n = 0, and could hav e b een deduced directly from the discrete p o le- equations, without solving them. The sum N + n pla ys t he part N did for mono coalesced patterns and, as is show n up on sp ecializing (47) to B = B max , has to satisfy 2( N + n ) ν ≤ 1. As men tioned earlier, the flame slop e φ x ( x ) p ertaining to the con tin uous approxim a- tion(s) can b e obtained directly fr o m the corresp onding p ole-densit y(ies) via an analytical con tin uation fr o m the r eal B (or b ) a xis to ± i x . Using the same pro cedure here giv es, for 18 0 ≤ x ≤ π : φ x = − 1 π sign( ¯ x − x ) log cosh ψ + 1 cosh ψ − 1 (63) sinh ψ = (1 + Aa ) t a n x/ 2 A − a ta n 2 x/ 2 (64) for bicoalesced flames, ¯ x b eing the p oint where sinh 2 ψ → ∞ in (64) and, therefore, φ x ( ¯ x ) = 0: ¯ x = 2 ar cta n s tanh B max / 2 tanh b max / 2 . (65) A t the flame fro nt tro ugh, κ = φ xx ( ¯ x ) = 2( A + a ) /π (1 + Aa ) > A/π : the corresp onding critical noise amplitude µ c ( κ ) needed to trigger the app earance of sub wrinkles marked ly exceeds that p ertaining to mono coalesced fronts (see Sec. 1). Tw o extra p ole-pairs initially placed at the p oin ts ± ¯ x ± iν (mo d 2 π ) w ould sta y there in unstable equilibrium. There exist separating tr a jectories S ± passing through them, whic h delineates the basins of attraction of the main p ole condensations at x = 0 or x = π . Only the tra jectories of initially remote extra-p oles that are close enough to S ± will en ter the O ( √ ν ) strip adjacent t o t he B = 0 axis where their direct influence on the main pattern b ecomes visible [7]. As seen from the real axis, the pro cess then manifests itself as extra sharp sub-wrinkles seemingly “ emitted” suddenly at x ≃ ± ¯ x (mo d 2 π ) b efo r e trav elling to one of the main cusps where they eve n- tually j o in a main condensation. The N/ n -dep enden t shap e of suc h separating t r a jectories th us controls the fate of “supplemen tary” p oles of whatev er origin, initia l conditions or f orc- ing; this will b e exploited elsewhere, though one can already confirm that stable 2- crest steady patterns with n = O ( N ) ∼ 1 / ν exist if Neumann conditions are emplo y ed. With 2 π - p erio dic conditions t hese are unstable ev en if 2 ν ( N + n ) = 1, as is seen b y considering initial conditions where the 2 n p o les are sligh tly shifted to the left of x = π ( mo d 2 π ): b oth crests will ultimately merge. Comparisons with accurate nume rical resolutions of the p ole equations (45) (46) are again go o d, Fig.9. F or ν = 1 / 199 . 5, N = 80, n = 20 they yield ¯ x = 2 . 0539 73 whereas our prediction (65), with A and a iteratively obtained from the normalization conditions (60) (61), giv es ¯ x = 2 . 053888 . Like R ( B ) a nd r ( b ), φ ( x ) cannot b e obt a ined in closed form, y et is readily accessible n umerically . Also, if A = a , elemen tary tr igonometry shows that the predicted flame slop e (63) resumes the result (35), up to a tw o - fold reduction in x - and B max -scales. 19 VI I I. CONCLUDING REMARKS & OPEN PR OBLEMS The ab ov e analyses may con v ey the feeling t ha t the p ole densities obtained so far hav e a family lik eness, whic h is true b ecause they w ere all deduced from the solution (16) p ertaining to isolated crests via adequate c hanges of indep enden t v ariable. Whereas (16) itself basically follo ws from standard F ourier analysis combined with a luc ky re- summation of the series thus obtained (15), it w ould b e in teresting to understand why the c hanges of v aria ble (2 7 ) then (56) w ork so well. Admittedly the in tegral equation (26) b ears some formal resem blance with (7), whic h guided us to prop ose the new v a riable (27); but that introduced in (56) lo oks more strange to us, a nd was actually discov ered b y trial-a nd-error a f ter the ’resolv en t’ in tegral equation (55) is obtained. Y et (56) unlik ely solely comes “out of the blue”. In effect, one may note that (56) is equiv alen t to tanh( β ) = tanh( β max ) sin θ if one defines tanh 2 ( β max ) ≡ tanh( B max / 2) tanh( b max / 2) and sets tanh( β / 2) = ta nh( β max / 2) sin σ , whic h clearly mirro rs what w as employ e d to map the mono coa lesced p erio dic case on to the isolated- crest problem. Hence (56) r ests on the celebrated comp osition la w for h yp erb olic tangen ts ( τ 1 , τ 2 ): τ 1 ∗ τ 2 = ( τ 1 + τ 2 ) / (1 + τ 1 τ 2 ). It w ould b e interes ting to kno w whether the asso ciated group pro p erties giv e access to still more general solutions to the Siv ashinsky equation (1). That the scale-in v a rian t sign um function featured in (7) (8) is left unchanged b y the success iv e c hanges of v ariables also is a ke y prop erty that tr aces bac k to the presence of the Hilb ert tra nsform ˆ H ( − φ x ) in (1): in fine , it expresses that the complex velocity a b out the flame is a sectionally-analytic function in the complex x -plane, whic h is indeed a robust statemen t fo r it is little a ffected b y conformal changes of v ariables that would lea v e the real axis globally inv arian t. Normalizing P ( B ) to 2 N brings ab out the g rouping N ν a nd, as long as the integral equations (7) (8) of the con tin uous approxim ation are concerned, there is no reason wh y N should b e a n integer. Th us, (7) ( 8 ) effectiv ely admit a 1-parameter contin uum of solutions. The situation is – in a sense, analogous to the Saffman-T a ylor problem of viscous fingering and related ones (see [20] and the references therein): when surface effects (here curv a ture) are omitted, a con tin uum of steady patterns is found. The equation (1) for flames is p eculiar, ho w ev er, b ecause one kno ws fro m the v ery b eginning that only a discrete set of steady mono coalesced solutions exist, corresp onding to N s that are in tegers less than a w ell-defined ν -dep endent v alue, N opt ( ν ). The Siv ashins ky equation (1) thus offers the opp ortunity to see 20 ho w the WKB a ppro ac hes to finger-width selection dev elop ed [20] for the Saffman-T a ylor problem, or kin, can b e transp osed to the presen t system to obtain a quan tization condition on N ν ; for here “invis cid” solutions ar e no w av ailable a nd one knows the answ er in adv ance. This analysis lik ely is a key step to study flame resp onse to noise, but has not ye t b een completed. Because WKB approaches essen t ially lo ok for solutions of a linearised equation in the f o rm exp( i R x k ( x ′ ) dx ′ ), where k ( x ) = O (1 /ν ) dep ends on the “invisc id” solution, it is seen that obtaining the latt er to leading ( O (1)) order in ν is not enough. Hence the question: ho w to compute the leading ( O ( ν )?) correction to the flame profiles obtained ab ov e? Ob viously this would require to b etter understand the nature of the con tin uous appro ximation leading to the in tegral equations (7)-(9) o r (47) (48) for p ole densities. I n this contex t one may p erhaps adopt the – ra ther un usual – view p oin t that the exact p ole equations (5), once sp ecialized to z α = iB α and steady patt erns, constitute Gauss-lik e quadrature for m ulae to ev aluate (7) num erically . Ho w to define a “ b est” w a y of c ho osing the piv otal v alues, i . e . the B α s, naturally leads [21] to the notion of orthogo nal p o lynomials asso ciated with the Siv ashinsk y equation (1). In the case of Wigner’s equation (11) the Hermite p o lynomials are inv oked [14], but w e are not aw a re o f suc h mathematical analyses ab out (1) a nd (7) (8). Next, w e recall that 2-crested pa tterns studied in Sec.VI I also b elong to a contin uous family of solution profiles, now indexed by tw o indep endent parameters N ν, nν . Ev en if N + n is assumed to b e given by the optim um v alue N opt ( ν ) ≃ 1 / 2 ν , there still remains the question of how N /n is selecte d in n umerical resolutions of (1) with Neumann conditio ns at x = 0, π . The ratio N /n can undoubtedly b e c hosen b y t he initial flame shap e φ ( x, 0 ). In the case of forced pr o pagations, the noise intensit y ( µ ) might w ell play a lso a role, for one can imagine situations where exp( − π / 2 ν ) ≪ µ ≪ exp( − π / 4 ν ): the noise is then in tense enough to break mono coa lesced patterns (see Intr o duction ), y et to o w eak to noticeably a ffect the more curve d 2-crested patterns with N = n . T o tailor a global criterion as to compare the 2-crest patterns and their resp onse to noise, the follow ing remarks could b e of some use. Let us collectiv ely denote t he B α s and b γ s as B and b , resp ectiv ely . The unsteady v ersions of (4 5) (46) – the p ole equations f or bicoalesced patterns – may b e re-written a s d B dt ′ = − ∇ B U, d b dt ′ = − ∇ b U, (66) 21 in terms of U ( B , b ) = V ( B ) + v ( b ) + w ( B , b ), with V ( B ) = ν X α | B α | − 2 ν 2 X α, β <α log     sinh B α − B β 2     , (67) w ( B , b ) = − 2 ν 2 X γ , β log  cosh b γ − B β 2  , (68) and an expression similar to (67 ) for v ( b ) ; t ′ = t/ν is time scaled b y the shortest growth time of small-scale wrinkles (see Intr o duction ). Accordingly , when the right-hand sides of (66) are supplemen ted with statistically iden tical, indep enden t random ( e.g. , G aussian) additive forcings, the joint probabilit y densit y o f ( B , b ) will tend to a quan tit y ∼ exp( − U ( B , b ) /µ 2 ), where µ ≪ 1 c haracterizes the no ise intensitie s. Because U ∼ 1 in the small- ν limit (since P ( B ) a nd p ( b ) are O (1 /ν )) the ab ov e exp onential is strongly p eak ed ab out the steady solutions. One can thus think of emplo ying the N /n -dep enden t scalar U ( B , b ), ev aluat ed at steady state, as a n ob jectiv e means to discriminate the v arious 2 -crested patterns in the presence of forcing. The task of ev aluating U in the con tin uous appro ximation has not y et b een completed. Neither is the analysis required to handle situations where the p oles are sligh tly misaligned... y et still symmetric ab o ut x = 0 and x = π for compatibility with Neumann b o undary conditions. One m ust finally stress t ha t the presen t ana lyses did not exhaust all the p ossibilities of “steady” solutions of (1), ev en with 2 π as minimal p erio dicity . The interp olating s o lutions disco v ered in [7, 2 2] constitute another class, comprising ( p ossibly many-) extra p oles, nearly ev enly distributed [17, 18] along sinuous curv es at a distance from the real axis. In our opinion suc h unstable equilibriums are also w orth analyzing in detail f or ν → 0, as are those men tioned in Sec.VI and generalizations of (1) itself [23]. As an end to a numerical w ork on (1), with noise included in the r ig h t hand side [7], one of us concluded tha t “. . . it is lik ely that new a nalytical studies of the Siv a shinsky equation should b e p ossible: ev en if the equation is now almost 30 y ears old, many things remain to b e explained”. The w ords still hold t r ue. 22 Ac kno wledgmen ts One of us (G.J.) thanks H. El- R abii (P oitiers Univers it y) for discussions, helps in the Calculus... and the L A T E X v ersion of the pap er. [1] L. D. Land au, Acta Physicoc himica USSR 19 , 77 (1944). [2] G. Darrieus (1938), unpu blished work presente d at L a T e chnique M o derne , P aris. [3] G. I. Siv ashinsky , Acta Astron. 4 , 1177 (1977). [4] K. A. Kazak o v, Ph ys. Fluids 17 , 032107 (2005) . [5] G. I. Siv ashinsky and P . Cla vin, J. Ph ys. (F rance) 48 , 193 (1987). [6] D. N. Mic helson and G. I . Siv ashinsky , Acta Astron. 4 , 1207 (1977). [7] B. Denet, Phys. Rev. E 74 , 036303 (2006). [8] G. Joulin, J. Ph ys. (F rance) 50 , 1069 (1989). [9] B. Denet, Phys. Rev. E 75 , 046310 (2007). [10] O. Thual, U. F r isch, and M. H ´ enon, J . Phys. (F r ance) 46 (1985). [11] Y. Lee and H. Chen, Phys. Scr. 2 , 41 (1982). [12] L. Landau and E. Lifsc hitz, Fluid Me chanics (Pergamo n Press, Oxford, 1979). [13] G. K. Batc helor, An intr o duction to fluid dynamics (Cambridge Un iv ersit y Press, Cambridge, 1967) . [14] M. L . Meh ta, R andom Matric es (Academic Press, Boston, 1991), 2nd ed. [15] I. S. Grads hteyn and I . M. Ryzhik, T able of Inte gr als, Series and Pr o ducts (Elsevier-Academic Press, Amsterdam, 2007), 7th ed. [16] G. J oulin, Combust. S ci. T ec hn. 53 , 315 (1987). [17] D. V a yn blatt and M. Matalon, S IAM J. App l. Math. 60 , 703 (2000). [18] O. Ku p erv asser, Z. Olami, and I. Pr o caccia , Phys. Rev. E 59 , 2587 (1999). [19] J. Bec, U. F risc h, and K. K hanin, J. Fluid Mec h. 416 , 239 (200 0). [20] P . Pelc ´ e, Th´ eorie des formes de cr oissanc e (EDP Sciences - CNRS Edition, P aris, 2000). [21] G. E. An d rews, R. Ask ey , and R. Roy , Sp e cial F unctions (Cam bridge Univ ersit y P r ess, Cam- bridge, 2000 ). [22] L. F. Guidi and D. H. U. Marc hetti, Phys. Lett. A 308 , 162 (2003). 23 [23] G. J oulin, Zh. Eksp. T eor. Fiz. 100 , 428 (1991). 24 List of figur es Fig. 1: Numerical vs ana lytical cumu lativ e p ole densities , for an isolated crest with 1 /ν = 19 . 5, N = 10. If exact, the theoretical curve (dot-dashed line, eq. (18)) w ould pass through the middle of the risers of the n umerical staircase (solid line, eq. (5 ) ). The dashed line is the TFH fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Fig. 2: Same as in Fig . 1, for 1 /ν = 199 . 5, N = 100. Only the upp er hu ll (solid line) of the exact staircase is sho wn, for readabilit y . The dashed line is the TFH fit. . . . . 24 Fig. 3: Numerical ((20), solid line) vs analytical ((17), dot-dashed line) p ole densi- ties P ( B ) fo r an isolated crest with 1 / ν = 199 . 5, N = 100. The dashed line is the TFH fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Fig. 4: Shap es of an isolated crest with 1 /ν = 19 . 5, N = 1 0: con tin uous approxi- mation ((25 ), dot-da shed line), exact (solid line), and smo oth approximation from eq. (22) (dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Fig. 5: Numerical (solid line) v s analytical (eq . (29), dot-dashed line) cumu lativ e p ole densities for a mono coalesced p erio dic crest, for 1 /ν = 199 . 5, and N = 100 (= N opt ( ν )). Only the upp er hull o f the nu merical staircase is shown. . . . . . . . . . . . . . . . . . . . 27 Fig. 6: Shap es o f a mono coalesced p erio dic flame with 1 /ν = 19 . 5, N = 10 (= N opt ( ν )): con tin uous appro ximation ((3 6), dot-dashed curve ) vs exact result (solid line) and smo oth appro ximation ((33 ) , dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Fig. 7: Cum ulativ e p ole densities R ( B ) (upper curv es) and r ( b ) f o r a bicoalesced p erio dic patt ern with 1 /ν = 600 . 5, N = 200, n = 100: the solid and the dotted lines a re from eqs. (53) (54) and ( 5 7) (56); the dashed and the dot-dashed ones are the upp er h ulls of the exact staircases (see Fig. 1). As ( N + n ) = N opt ( ν ) , B 200 = ∞ . . . . . . . . . . . . . . . . . . . . 2 9 Fig. 8: Theoretical p ole densities P ( B ) (resp. p ( b )) plotted as dot-dashed or dashed 25 lines vs θ , eq. (57), with sin σ replaced b y (tanh B / 2) / A (resp. (tanh b/ 2) /a ). The solid and t he dotted lines ar e the n umerical p ole densities. All are for a bicoalesced p erio dic flame with N = 200, n = 100, 1 /ν = 600 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Fig. 9: Shap es of a bicoalesced flame with 1 /ν = 199 . 5 , N = 80 , n = 20: exact (solid line) vs con tin uous approximation (from integration of (63), dot - dashed). . . . . . . . . . 31 26 0 1 2 3 pole vertical position 0 5 10 pole cumulative distribution numerical cumulative distribution theoretical cumulative distribution (formula 18) TFH cumulative distribution FIG. 1: Numerical vs analytical cum ulativ e p ole densities, for an isolated crest with 1 /ν = 19 . 5, N = 10. If exact, the theoretical curv e (dot-dashed line, eq. (18)) would pass through the middle of the risers of the numerica l staircase (solid line, eq. (5 )). The d ashed line is the TFH fit. 27 0 1 2 3 pole vertical position 0 50 100 pole cumulative distribution numerical cumulative distribution theoretical cumulative distribution (formula 18) TFH cumulative distribution FIG. 2: Same as in Fig. 1, for 1 /ν = 199 . 5, N = 100. Only the upp er hull (solid line) of the exact staircase is sho wn, for readabilit y . Th e dashed line is th e TFH fi t. 28 0 0.5 1 1.5 2 2.5 3 pole vertical position 0 100 pole density numerical pole density theoretical pole density (formula 16) TFH pole density FIG. 3: Num erical ((20), solid line) vs analytic al ((17), dot-dashed line) p ole densities P ( B ) f or an isolated crest with 1 /ν = 199 . 5, N = 100. T h e dashed line is the TFH fit. 29 -1 0 1 x coordinate -2.5 -2 -1.5 -1 -0.5 0 flame shape exact solution continuous approximation (formula 25) smooth approximation (formula 22) FIG. 4: S hap es of an isolated crest with 1 /ν = 19 . 5, N = 10: con tin uous approximati on ((25), dot-dashed line), exact (solid line), and smo oth app ro ximation from eq. (22) (dotted). 30 0 5 10 pole vertical position 0 20 40 60 80 100 pole cumulative distribution numerical cumulative distribution theoretical cumulative distribution (formula 29) FIG. 5: Numerical (solid line) vs analytical (eq. (29 ), dot-dashed line) cum ulativ e p ole densities for a mono coalesced p erio dic cr est, for 1 /ν = 199 . 5, and N = 100 (= N opt ( ν )). Only the u p p er h ull of the numerical staircase is sho wn. 31 -2 0 2 x coordinate -1 -0.5 0 0.5 1 1.5 flame shape exact solution continuous approximation (formula 36) smooth approximation (formula 33) FIG. 6: Shap es of a mono coalesced p erio dic fl ame with 1 /ν = 19 . 5, N = 10 (= N opt ( ν )): con tin uous appro ximation ((36), dot-dashed curve) vs exact result (solid line) and sm o oth app ro ximation ((33), dotted). 32 0 2 4 6 8 10 pole vertical position 0 50 100 150 200 pole cumulative distribution numerical cumulative distribution at x=0 theoretical cumulative distribution at x=0 numerical cumulative distribution at x= π theoretical cumulative distribution at x= π FIG. 7: Cumulativ e p ole densities R ( B ) (upp er curv es) and r ( b ) f or a bicoalesced p erio dic pattern with 1 /ν = 600 . 5 , N = 200, n = 100: the solid and the dotted lines are fr om eqs. (53) (54) and (57) (56); the d ashed and the dot-dashed ones are the upp er hulls of the exact staircases (see Fig. 1). As ( N + n ) = N opt ( ν ) , B 200 = ∞ . 33 0 0.5 1 1.5 2 theta 0 100 200 300 400 500 pole density numerical pole density at x=0 theoretical pole density at x=0 numerical pole density at x= π theoretical pole density at x= π FIG. 8: Th eoretical p ole d en sities P ( B ) (resp. p ( b )) plotted as d ot-dashed or d ashed lines vs θ , eq. (57), with sin σ replaced b y (tanh B / 2) / A (resp. (tanh b/ 2) /a ). T he solid and the d otted lines are the numerical p ole densities. All are for a bicoalesced p erio dic flame with N = 200, n = 100, 1 /ν = 600 . 5 . 34 0 1 2 3 4 5 6 x coordinate 0 0.5 1 1.5 flame shape exact solution continuous approximation (formula 64) FIG. 9: Shap es of a bicoalesced fl ame with 1 /ν = 199 . 5 , N = 80, n = 20: exact (solid line) vs con tin uous app ro ximation (from in tegration of (63), d ot-dashed). 35