Effect of the Darrieus-Landau instability on turbulent flame velocity. Abstract

Transcription

1 1 Eect o the Darrieus-Landau instability on turbulent lame velocity Maxim Zaytsev 1, and Vitaliy Bychkov 1 1 Institute o Physics, Umeå University, S Umeå, Sweden Moscow Institute o Physics and Technology, , Dolgoprudny, Russia Abstract Propagation o turbulent premixed lames inluenced by the intrinsic hydrodynamic lame instability (the Darrieus-Landau instability) is considered in a two-dimensional case using the model nonlinear equation proposed recently [1]. The nonlinear equation takes into account both inluence o external turbulence and intrinsic properties o a lame ront, such as small but inite lame thickness and realistically large density variations across the lame ront. Dependence o the lame velocity on the turbulent length scale, on the turbulent intensity and on the density variations is investigated in the case o weak non-linearity and weak external turbulence. It is shown that the Darrieus-Landau instability inluences the lamelet velocity considerably. The obtained results are in agreement with experimental data on turbulent burning o moderate values o the Reynolds number. Communication address: V. Bychkov, Inst. o Physics, Umeå University S , Umeå, Sweden tel. (46 90) ax: (46 90) vitaliy.bychkov@physics.umu.se

2 I. Introduction Turbulent lame velocity U w is one o the basic characteristics o premixed turbulent combustion, it is a key parameter or phenomenological models o the combustion process and a promising tool or multi-dimensional computations o turbulent burning in real industrial devices [, 3]. Turbulent combustion may proceed in several distinctive regimes with quite dierent properties []. Among them the socalled lamelet regime is the most typical or practical systems like gas turbines o power plants or car engines. In the case o lamelet burning a chemical reaction occurs at ast time-scales and short length-scales relative to the turbulent ones. A lamelet propagates as a relatively thin ront with the inner structure similar to the laminar lame o some thickness L, but turbulent low distorts the lame strongly on large length scales in comparison with L. Larger surace area o a corrugated lame ront leads to larger consumption rate o the uel mixture, larger total heat release and larger velocity o lame propagation U w in comparison with the laminar lame velocity U. As a result, turbulence increases power available rom a turbine combustor or internal combustion engine. The regime o turbulent lamelets has been studied theoretically or a long time and many interesting results have been obtained [4-9]. However, most o these results were restricted to the artiicial limit o zero uel expansion, when the density o the burnt matter ρ b is the same as in the uel mixture ρ with their ratio Θ = ρ ρ b =1. In this limit lame propagates passively in the turbulent low without aecting the low. Yet, most o the laboratory lames involve considerable thermal expansion Θ = 5 10, or which lame interacts with the turbulent low quite strongly. By this reason the artiicial limit o zero thermal expansion Θ =1 cannot provide quantitative description o turbulent lames. Besides, thermal expansion leads to qualitatively new

3 3 eects o lame-low interaction such as the Darrieus-Landau (DL) instability. In the case o laminar lames the DL instability bends an initially planar lame ront increasing the velocity o lame propagation [10-1]. Similar eects are expected or turbulent lamelets at least in the limit o weak and moderate turbulent intensity. The role o the DL instability or turbulent lamelets has been widely discussed, but in ew papers one could ind real attempts to solve the problem because o involved mathematical and computational diiculties [8, 13, 14]. However, even these papers considered only the limit o ultimately weak instability at small thermal expansion Θ 1<< 1 (or Θ < in [14]). I one takes the artiicial limit o zero thermal expansion Θ =1 like [4-7,9], then the DL instability disappears. Even in the case o laminar lames the nonlinear theory o the DL instability has been restricted or a long time to the limit o small thermal expansion Θ 1<< 1 [15,16]. Quantitative nonlinear theoretical description o the DL instability or realistically large expansion actors Θ has been developed not so long ago [11,17,18]. O course, the instability eect on turbulent lames could be studied by direct numerical simulations o the complete set o combustion equations, but simulations perormed so ar considered lame dynamics on small length scales below 10L [19,0], or which the DL instability is thermally suppressed. Recently a nonlinear equation has been proposed, which takes into account both inluence o external turbulence and the DL instability with realistically large density variations across the lame ront [1]. Preliminary evaluations o the turbulent lame velocity on the basis o the obtained equation [1,1] agree quite well with experimental results [,3]. Still, the particular solutions to the nonlinear equation [,3] did not include direct inluence o the DL instability. Though such solution may be used to describe lame dynamics or a rather low integral turbulent length

4 4 scale o about ( 30 50)L like in experiments [3], a more common experimental situation involves large length scales exceeding 100L considerably [], or which the DL instability aects lame velocity signiicantly. In the present paper we solve the nonlinear equation [1] in order to investigate the eect o the DL instability on dynamics o turbulent lames. II. Model equation We solve the ollowing model equation describing dynamics o a lame ront z = F(x,t) in a weakly turbulent low u( x,t) ˆ 1 Θ + 1 ( ˆ Φ F ) ( 1 ˆ F Θ + C L Φ + + CL Φ ) + 1 U w / U + ( F ) + Θ U t U t ( ) 3 1 ˆ λ c Θ 1 1 ( ) ( ˆ Θ ) 1 ˆ ˆ Φ uz F ΦF F Φ π Φ + =. (1) Θ U t U The equation (1) has been proposed in [1] in the section below we explain the origin and the main components o the equation. Besides, we consider some particular solutions to Eq. (1) obtained beore. Equation (1) is written in the reerence rame o the average position o a statistically stationary turbulent lame ront, with the average velocity U w o lame propagation being slightly dierent rom the laminar lame velocity U. The main advantage o Eq. (1) in comparison with earlier models o turbulent lames [4-6] is that it takes into account realistically large density variations across the lame ront described by the actor Θ = ρ / ρ b, which is the density ratio o the uel mixture ρ and the burnt matter ρ b. Parameters λ c, C 1, C are related to internal thermal-chemical properties o the lame ront: These parameters have been ound in the linear theory o the DL instability o a lame o inite thickness L

5 5 [4,5]. The operator ˆΦ implies multiplication by absolute value o the wave number component along the lame surace in Fourier space, which may be presented in the case o a D low as 1 Φ ˆ F = k Fk exp ( ikx) dk π () Model equation (1) has been proposed on the basis o three rigorous theories: the linear theory o the DL instability by Pelce and Clavin [4], the nonlinear theory o curved lames resulting rom the instability by Bychkov [11] and the linear theory o lame response to weak turbulence by Searby and Clavin [6]. Below we explain briely the main results o the theories as well as components o the model equation (1). It has been shown [4] that development o small perturbations at an initially planar lame ront may be described by the equation F ( 1 ) ( ) Θ + 1 ˆ ˆ ˆ F Θ 1 λ ˆ ˆ Θ t t π c 1+ C L Φ + 1+ C L Φ U Φ 1 Φ U Φ F = 0 (3) Equation (3) has been obtained taking into account small but inite lame thickness L. The numerical actors C 1, C and the cut-o wavelength λ c depend on internal lame structure [4,5]. Particularly, in the case o a lame with Lewis number equal unity and a constant coeicient o thermal conduction the respective ormulas become λ c = πl 1+ Θ Θ + 1 Θ 1 ( ) ln Θ, C 1 = 0, C = ΘlnΘ Θ 1. (4) In the case o an ininitely thin ront Eq. (3) goes over to the Darrieus-Landau dispersion relation since λ c L [4]. According to Eq. (3) small perturbations o a planar lame grow exponentially in time i the perturbation wavelength exeeds the cut-o wavelength λ c. Equation (3) is the irst component o the nonlinear model (1).

6 6 I a lame ront propagates in a D channel o width R with ideally slip and adiabatic walls, then the DL instability is thermally suppressed in narrow tubes with R < R c = λ c /. In wider tubes R > R c perturbations grow exponentially until nonlinear eects come to play. In tubes o a moderate width R < ( 4 5)R c nonlinear stabilization takes place leading to a smooth curved stationary lame shape [11,1,6]. The curved lame shape and velocity may be described by the nonlinear equation derived in [11]: ( Θ ) Θ ( ) ˆ Θ λc 1 / 1 ˆ U ˆ w U + F + F ΦF Φ Φ F = 0 16Θ π, (5) ( ) ( ) where U w is propagation velocity o a curved (wrinkled) lame ront, which is larger than U. The nonlinear terms in Eq.(5) take into account both cusp ormation at the lame ront and vorticity production behind a curved lame. Equation (5) has been derived or arbitrary expansion coeicients Θ (even large ones) assuming small but inite lame thickness similar to the linear theory [4]. The thermal-chemical properties o the burning mixture are taken into account in Eq. (5) by the cut-o wavelength λ c, the analytical ormula or which coincides with the expression obtained in the linear theory. The combination o equations (3) and (5) determines the eects o lame-low interaction in the equation (1) in absence o external turbulence. Inluence o turbulence is taken into account according to the linear theory [6] or a lame ront in a weak external low Θ + 1 F ˆ F Θ 1 ˆ ˆ + U 0 Φ U Φ F U Φ + uz =, (6) Θ t t t where u z is the z-component o turbulent velocity at the surace z = 0. Assuming weak nonlinear eects, weak turbulence and a thin lame ront we can combine (3), (5), (6) into one model equation (1), as it has been proposed in [1].

7 7 In the present paper we consider a D lame propagating in a tube o width R with adiabatic boundary condition at the walls F x = 0 at x = 0, R, (7) so that the lame shape may be presented as F = F n cos πnx R (8) with the operator ˆΦ Equation (1) or a D low becomes ˆ π n π nx Φ F = Fn cos R R. (9) ˆ 1 Θ + 1 ( ˆ Φ F ) ( 1 ˆ F Θ F + C L Φ + + CL Φ ) + 1 U w / U + + Θ U t U t x ( ) 3 ˆ 1 F Rc Θ 1 1 ( ˆ Θ ) 1 ˆ ˆ Φ uz Φ F F x Φ π Φ + =. (10) Θ U t U Incompressible D turbulence in the laboratory reerence rame may be described by the representation [8] u z = u x = ( ) U i cos k i z + ϕ i ( ) U i sin k i z + ϕ i cos( k i x), (11) ( ) (1) sin k i x where we have taken the continuity condition into account. In Eqs. (11), (1) ki = πi R are the wave numbers o turbulent harmonics and ϕ i stand or random phases. The amplitudes U i o turbulent harmonics are determined by the Kolmogorov spectrum U i k i 5 / 6 with the rms-turbulent velocity in one direction given by the ormula U rms = U i / 4. (13)

8 8 The number o turbulent harmonics in (11), (1) is a ree parameter o our model. In the case o a reely propagating lame ront the x-dependence o turbulent velocity also includes random phases. However, since we are interested in lame propagation in a channel o inite width with slip walls, then the boundary condition o the walls requires zero phases. In general, the turbulent velocity ield (11), (1) should also involve temporal pulsations. The inluence o temporal pulsations has been investigated recently in [7], where it has been shown that temporal pulsations do not lead to any qualitatively new eect in turbulent lame propagation. By this reason in the present paper we will use the Taylor hypothesis o stationary turbulence, or which pulsations caused by lame propagation are much stronger than real temporal pulsations o the low velocity in (11), (1). The Taylor hypothesis has been used in many papers on turbulent lame dynamics [4, 6-8, 8]. Since the model equation (1) is written in the reerence rame o a statistically stationary turbulent lame, then we have to go over to the same reerence rame in the ormulas or the turbulent velocity ield (11), (1). In the case o weak turbulence we have the longitudinal turbulent velocity ield at the ront position z = U t u z = ( ) U i cos U k i t +ϕ i cos( k i x). (14) The equation (1) has been solved beore in two particular cases: A) no turbulence U rms = 0, when dynamics o the lame ront is aected by the DL instability only; B) no direct inluence o the DL instability, when only external turbulence controlls lame propagation. Below we consider briely solution to Eq.(1) in both cases.

9 9 A) Inluence o the DL instability only I the turbulent intensity is zero U rms = 0, then the model equation (1) describes linear development o the DL instability at an initially planar lame ront and subsequent propagation o a wrinkled lame resulting rom the instability. As it has been pointed above, the instability developes and a curved lame shape becomes possible in suiciently wide tubes R /R c >1 with the critical tube width R c determined by thermal-chemical lame parameters: or example, R c = λ c / in the case o a D low in a channel with ideally slip and adiabatical walls. I the channel width is not too large R /R c < 4 5, then the DL instability results in a smooth curved lame shape propagating with velocity [11] U w 1= Θ( Θ 1) Θ 3 + Θ + 3Θ 1 M R c R 1 M R c R, (15) where M = Int[ R R c +1 ]. Figure 1 presents dependence o the wrinkled lame velocity on the tube width R /R c or the expansion actors Θ = 5, 7, 9. As one can see, the wrinkled lame velocity exceeds the planar lame velocity because o the larger surace o uel consumption. The dierence between the wrinkled and planar lame velocities becomes larger or larger thermal expansion Θ, since larger Θ (larger density dierence across the lame) leads to stronger DL instability. The analytical ormula (15) agrees quite well with results o direct numerical simulations o lame dynamics in tubes [1, 9]. As the tube width increases R / the R c wrinkled lame velocity tends to a limiting value U w 1= 1 Θ( Θ 1) Θ 3 + Θ + 3Θ 1, (16) which determines a maximal possible velocity increase or a stationary wrinkled lame. However, the maximal velocity may be achieved even or tubes o a moderate

10 10 width, or example, or R = R c. The maximal velocity increase (16) depends only on the expansion actor Θ. In the present paper we are interested, mostly, in lame propagation in channels o moderate width R/R c < 4 5, or which the DL instability results in stationary wrinkled lames. In much wider tubes the curved stationary shape becomes unstable with respect to the secondary DL instability o a small scale [17]. At that point it is interesting that the model equation (1) describes also the stability limits o the secondary DL instability, which are in good agreement with the results o direct numerical simulations [1]. B) Inluence o external turbulence only The model equation (1) has also a particular solution related to the external turbulence only with no direct inluence o the DL instability. Such solution has been ound in [1] or a three-dimensional (3D) turbulent low. We now obtain a D version o that solution taking into account the condition o weak turbulence. In the case o a turbulent low (14) we can rewrite the turbulent terms o Eq. (1) as ollows ˆ 1 Φ uz Ui 1+ = cos( U kit + φi ) sin ( U kit + φi ) cos kix = U t U U = U i cos U U k i t + φ i + π cosk 4 i x (17) and look or a lame ront position F = F( x,t) in a similar orm F = F i ( t)cos( k i x). (18) Substituting (18) into the model equation (1) with the accuracy o linear terms we come to a system o ordinary dierential equations or F i

11 11 Θ +1 Θ ( 1+ C L k 1 i) 1 k i U d F i dt + ( 1+ C L k i ) 1 U df i dt Θ 1 ( 1 R c k i /π)k i F i = U i cos U k i t + φ i + π U 4 (19) with the ollowing solution where F i ( t)= U i cos U k i t + φ i + π D i k i U 4 + γ i, (0) D i = Θ 1 ( 1 R c k i /π) Θ +1 Θ 1+ C L k ( 1 i ) ( ) + 1+ C L k i 1/ (1) and the phase shit γ i is determined by the expressions cosγ i = 1 Θ 1 ( 1 R c k i /π) Θ + 1 D i Θ 1+ C ( 1L k i ), () sinγ i = 1 D i ( 1+ C L k i ). (3) The average turbulent lame velocity is related to nonlinear terms o Eq.(1) ( Θ 1) 3 Θ Uw / U 1 = ( F ) + F ΦF 16Θ ( ) ( ˆ ), (4) where... denotes time and space averaging. Substituting representation (18) with amplitudes (0) into (4) we ind the turbulent lame velocity in the case o no direct inluence o the DL instability U w 1= Θ 4U U i. (5) D i In the artiicial limit o zero thermal expansion Θ = 1 and R the obtained ormula (5) agrees with the well-known Clavin-Williams ormula [4] written or the D turbulence model (11), (1): L

12 1 U w 1= U rms. (6) U Both the velocity amplitudes U i and the actors D i in Eq. (5) depend on the wavenumbers k i, which, in turn, are determined by the tube width k i = iπ / R. As a result, equation (5) speciies dependence o the turbulent lame velocity on the tube width R. In order to understand the obtained dependence better we consider irst a simpliied case o a turbulent low (14) modelled by a single harmonic u z =U 1 cos( U k 1 t)cos( k 1 x). (7) Such a simpliied assumption about a turbulent low is oten used in direct numerical simulations [9, 7, 8]. In the case o a single turbulent harmonic the expression or turbulent lame velocity becomes 1 L w Θ 1 c Θ+ 1 L rms 1= Θ 1+ πc πc 1 Θ U R U U R R R U. (8) The dependence (8) is peresented in Fig. or the case o unit Lewis number Le =1 and a constant coeicient o thermal conduction, or which the cut-o wavelength λ c = R c and the numerical actors C 1, C are determined by the ormula (4). The turbulent lame velocity is shown or dierent values o the expansion coeicient Θ = 5, 7, 9 and or dierent turbulent intensity U rms = 0., 0.5, 1. As one can see, the dependence o turbulent lame velocity on the tube width diers considerably rom the previous case o a lame aected by the DL instability only. Now the critical tube width R = R c is not a point o sharp biurcation any more, but instead one has a smooth resonance at R R c [6]. In narrow tubes R < R c beore the resonance a curved shape o a lame ront is also possible and the lame velocity exceeds the planar lame velocity U w >U. Still, the narrower the tube, the stronger stabilizing eects o thermal conduction and inite lame thickness. According to Eq. (8) at

13 13 R 0 the lame velocity is equal to the planar lame velocity. For wider tubes R > R c ater the resonance the turbulent lame velocity decreases too, though or very wide tubes R >> R c it tends to a inite limit U w / U 1= 4Θ 3 ( ) 4Θ + Θ Θ 1 U rms U. (9) A curious point is that the limiting values or the turbulent lame velocity Eq. (9) decrease with increase o the expansion coeicient Θ. One can see the same tendency in Fig. a or Θ = 5, 7, 9 and a ixed turbulent intensity U rms = 0.5. The decrease o turbulent lame velocity in the case o weak turbulence with no direct inluence o the DL instability has been obtained already in [1]. Such tendency is opposite to the previous case o a wrinkled lame shape caused by the DL instability only, or which lame velocity increases with Θ. Besides, this tendency is also opposite to the situation o strong turbulence [1]: strongly turbulent lames propagate aster or larger expansion actors Θ. There is no physical explanation yet, why turbulent lame velocity depends on thermal expansion Θ in dierent ways or the cases o weak and strong turbulence. Finally, the dependence o turbulent lame velocity U w on the turbulent intensity U rms in the case o weak turbulence is the same as in the Clavin-Williams ormula: U w 1 U rms /U. The situation o a large number o turbulent harmonics in Eq. (14) is more realistic than a single turbulent mode. Particularly, Figure 3 presents dependence o turbulent lame velocity on the tube width or 150 turbulent harmonics. As in Fig. we take unit Lewis number Le =1 and a constant coeicient o thermal conduction. The main dierence between the plots in Figs. and 3 is that the resonance is practically missing or multi-scaled turbulence o Fig. 3. For example, or the turbulent intensity U rms = 0.5 and the expansion actor Θ = 5 the turbulent lame velocity depends on

14 14 the tube width in a quite monotonic way increasing rom U in narrow tubes R << R c to the limiting value Eq. (9) in wide tubes R >> R c. For larger expansion actors and larger turbulent intensity the slight resonance may be still seen. The reduced eect o resonance at R R c or multi-scaled turbulence may be easily understood, since now a considerable part o turbulent energy is spread between harmonics o a smaller scale, which have resonance at dierent points. Apart rom the resonance eect, other tendencies o the turbulent lame velocity remain qualitatively the same or the multi-scaled turbulence and or a single turbulent harmonic. III. Flame under simultaneous action o turbulence and the DL instability. A) The method o solution The main purpose o the present paper is to understand lame dynamics aected both by the DL instability and by external turbulence, that is, to ind a more general solution to Eq. (1). One possible way would be to perorm direct numerical simulations o Eq. (1). However, direct numerical simulations are typically characterised by a relatively low accuracy than a solution to an eigenvalue problem, and simulation results are more diicult or analysis. By this reason, we will solve the equation (1) as an eigenvalue problem in a semi-analytical way taking into account the conditions o weak turbulence U rms <<1 and weak nonlinear eects ( F / x) << 1 used in the derivation o Eqs. (3), (5), (6). In that sense we would like to stress that in tubes o moderate width with no external turbulence the DL instability results in a stationary curved lame ront (see Sec. IIA). On the contrary, i lame is aected by external turbulence only with no direct inluence o the DL instability, then the solution to Eq. (1) is strongly oscillating, see Eq. (0), Sec. IIB. Then it

15 15 would be reasonable to look or solution to (1) in the orm o a combination o a stationary term G( x) and a strongly oscillating term H( x,t): F = G( x)+ H( x,t). (30) Substituting (30) into Eq (1) and taking time-average we come to the ollowing equation: ( Θ 1) 3 Θ G G 1 U / ( ˆ w U + + Φ G) + x 16Θ x ( Θ ) 3 1 H 1 ( H ) Θ H ˆ Θ Rc 1 ˆ ˆ + Φ Φ Φ G = 0 x 16 x, (31) Θ π t where... t denotes time averaging. The obtained equation is similar to the stationary equation (5) describing the nonlinear stage o the DL instability with some correction terms produced by turbulence. Eliminating (31) out o (1) we come to the timedependent equation or turbulent part o the solution: ˆ 1 Θ + 1 ( ˆ Φ H ) ( 1 ˆ H G H + C L Φ + + CL Φ ) + Θ + Θ U t U t x x ( Θ 1) ( Θ 1) 3 3 G H ˆ ˆ H ( ˆ H ΦGΦ H + H ) ( ˆ Φ Φ H ) + 8Θ x x 16Θ x x t t ˆ 1 H Θ H Θ 1 Rc 1 ˆ ˆ Φ uz H x x Φ π Φ + = U t t. (3) U In the limit o weak turbulence and weak nonlinear eects all nonlinear terms in Eq. (3) may be neglected, since they give only small corrections to H, and we come to the linear equation: ˆ 1 Θ + 1 ( ˆ Φ H ) ( 1 ˆ H + C L Φ + + CL Φ ) + Θ U t U t

16 16 ˆ 1 Θ 1 Rc 1 ˆ ˆ Φ uz H 1 0 Φ π Φ + =. (33) U t U Then the oscillating term H( x,t) is speciied by external turbulence u z only independent o the stationary term G( x) (see Eq. (33)). On the other hand, averaged terms with H( x,t) play the role o external orce in the stationary equation (31). Solution to Eq. (33) coincides with the solution to Eq. (1) presented in Sec. IIB. Taking N T turbulent harmonics in the representation (14) we look or the oscillating term H( x,t) in the orm ( ) = H i H x,t N T ( t)cos πix R i=1 (34) and ind H i ( t)= U i cos( U k i t + φ i + π /4 + γ i ), (35) D i k i U where D i and γ i are determined by Eqs. (1) - (3). Formulas (34), (35) speciy the external orce ( Θ 1) 3 ( ˆ H ) Θ H H + Φ x 16Θ x t t (36) in Eq. (31). orm: We solve Eq. (31) numerically with the boundary conditions (7) taking G( x) in the ( ) G x N R πix = Gi cos. (37) i= 1 π i R We would like to stress that the number o harmonics N T in Eq. (34) and N in Eq. (37) are two dierent values. The ormer, N T, shows the number o turbulent modes in the representation Eq. (14) and plays the role o a ree parameter o the model.

17 17 Particularly, Eq. (8) has been obtained under the assumption o a single turbulent mode, N T =1. On the contrary, the value N in Eq. (37) shows the number o Fourier harmonics in the numerical solution to Eq. (31), which is determined by the accuracy requirements. In order to simpliy the numerical solution it is useul to perorm the ollowing auxiliary calculations: N N G 1 mx 1 lx Am cos π Bl cos π x = m= 0 R + l= N R, (38) ( ) N N N ˆ πix 1 π mx 1 πlx Φ G = Gi cos = Am cos + Bl cos i= 1 R m= 0 R l= N R (39) N ( ˆ G) G π mx Φ = Am cos, (40) x m= 0 R m 1 where A 0 = A 1 = 0, A m = G i G m i or m, B 0 = G i, B l = G i G m + i or i=1 i=1 i=1 1 l N 1 and B N = 0. Ater substituting (34), (37) in (31) and averaging in time N N l we obtain with help o (38)-(40) the ollowing system o algebraic equations or G i and U w 1: U w 1= Θ 4 B U N T i=1 U i / D i, (41) ( Θ 1) turb ( A + A ) = 0 ΘB 1 i Θ G (1 / ) i irc R Θ 1 4 i i or i 1, (4) Θ Θ where we have introduced the designation A turb i = D i / U i /, i imod = 0 and N T i; 0, i imod 0 or N T < i. (43)

18 18 The system (41), (4) has been solved numerically. However, instead o solving (4) directly we have introduced and solved the ollowing system o ordinary dierential equations involving virtual time ξ ( Θ 1) dgi Rc ΘBi 1 Θ = Gi 1 i + + Ai + Ai dξ R Θ 1 Θ 1 4Θ turb ( ) or i 1 (44) with arbitrary initial values but a ixed value o R c / R. We have ound numerically that solution to the system (44) tends to a unique set o stationary values G i with a suiciently small step o virtual time ξ independent o the chosen initial values o G i. The obtained set o stationary values gives the solution to Eq. (4) and the turbulent lame velocity (41). The spectral method o numerical solution to boundaryvalue problem (31)-(7) described above, as it is well-known act, provides the best accuracy among other methods. For example, even taking N = 30 Fourier harmonics in Eq. (37) we got 5% accuracy o calculations or the turbulent velocity U w 1. In most o the calculations we used a much larger value N =150 providing the accuracy ar better than 1% when the number o harmonics N is doubled, and the Fourier expansion Eq. (37) converged well. As an illustration Fig. 4 shows spectrum o the numerically obtained solution o Eq. (4) or R /R c = 5, U rms = 0.5, Θ = 7, N =150 and N T =150. As one can see, in that case the amplitude o the irst harmonic exceeds the amplitude o the 100th harmonics by 5 orders o magnitude. The chosen accuracy o numerical calculations was quite suicient taking into account that assumptions used in basic equation (1) (that is, in the derivation o the original theories (3), (5), (6)) hold with much worth accuracy o about 30% and less. As a test o the numerical method we have solved irst the stationary nonlinear equation (5), and the numerical result coinsided with the exact analytical solution Eq. (15). In order to imitate a multi-scaled turbulence we took the same number o

19 19 turbulent harmonics in Eq. (34), N T =150. Besides, we have also investigated the case o a single turbulent mode N T =1, which may be useul in the uture or comparison with direct numerical simulations. B) Results o numerical solution The main dimensionless parameters that determine dynamics o curved lames in ideal tubes are the scaled tube width R /R c, the turbulent intensity U rms and the expansion actor o the burning mixture Θ. To be particular, in all calculations we took unite Lewis number Le =1 and a constant coeicient o thermal conduction. First o all we have investigated dependence o the turbulent lame velocity U w 1 on the scaled tube width or dierent values o Θ and U rms. We have been interested mostly in tubes o moderate width R / 5, realistic expansion coeicients Θ = 5 10 and weak external turbulence U rms <1, or which the basic equation (1) may be valid. The results o numerical solution are presented in Figs. 5-7 or N T =150 turbulent harmonics. Particularly, Fig. 5 shows scaled turbulent lame velocity U w 1 versus the scaled tube width R /R c or some ixed expansion actor (Θ = 5, 7, 9 or Figs. 5(a), 5(b), 5(c) respectively) and dierent turbulent intensity. As one can see, the dependence remains qualitatively the same, as we had in the case o no inluence R c o the DL instability, Sec. IIB, Fig. 3(b). In all plots turbulent lame velocity increases rom the planar lame velocity U in narrow tubes R /R c << 1 to some limiting value in wide tubes R /R c >> 1. The increase may be quite monotonic, as we have, or example, or Θ = 5 and U rms = 0.5. For smaller turbulent intensity U rms = 0. one can easily see the trace o biurcation humps typical or the case

20 0 o the DL instability without turbulence, see Sec. IIA, Fig. 1. For larger turbulent intensity U rms =1 and larger expansion actors Θ = 7, 9 the biurcation humps vanish, but instead one can observe the resonance at R R c described in Sec. IIb, Fig. 3. Judging by the qualitative look o the plots in Fig. 5, one could conclude that the inluence o the DL instability becomes small already or U rms = 0.5. However, this conclusion is wrong as one can see comparing Fig. 5(b) and Fig. 3(b) quantitatively. Indeed, choosing the problem parameters R /R c = 5, U rms = 0.5 and Θ = 7 as an example we ind increase o the turbulent lame velocity U w taking into account the DL instabillity and U w without the instability. Thus, though the DL instability does not bring anything qualitatively new into the plot, it leads to considerable quantitative changes increasing the scaled lame velocity almost 3 times! As a matter o act, or such strong velocity increase the basic equation (1) does not hold any more, still it may show general eatures o lame ront dynamics under simultaneous action o external turbulence and the DL instability. To make the dierence between the cases with and without the DL instability more distinctive we present the characteristic velocity increase U w 1 at R = 5R c versus U rms in Fig. 6 or both cases. As one can see, the DL instability increases the turbulent lame velocity considerably or all turbulent intensities in the domain U rms <1. Besides, when turbulent intensity goes to zero U rms 0, the instability still provides a non-zero increase o the lame velocity U w 1= Obviously, this does not happen or a particular solution without the instability. A curious point o Fig. 6 is that the obtained instability inluence is much stronger than just ormal adding o the instability solution o Sec. IIA to the purely turbulent solution o Sec. IIB.

21 1 Figure 7 presents plots similar to Fig. 5, but now we ix turbulent intensity and study the scaled velocity o lame propagation U w 1 or various expansion actors Θ. At that point one has to remember, that U w 1 depends on Θ in a dierent way or the cases o a lame aected by the DL instability only or by weak external turbulence only. For the ormer case it increases with the expansion actor Θ (see Sec. IIA), while or the latter case it decreases (see Sec. IIB). One can observe both tendencies in Fig. 7 or the limiting values o U w 1 achived or relatively wide tubes, e.g. R /R c = 5. In Fig. 7(a) the turbulent intensity is rather small U rms = 0. and the characteristic lame velocity at R /R c = 5 increases with Θ similar to the nonlinear stage o the DL instability. In Figs. 7(b)-(d) the turbulent intensity becomes larger U rms = and the lame velocity at R /R c = 5 decreases with the expansion actor Θ as it happens or a weak turbulence with no direct inluence o the DL instability. We have also investigated the case o a single turbulent harmonic N T =1 popular in numerical simulations [9,7]. Figure 8 presents velocity o lame propagation or that case versus the scaled tube width or Θ = 5, 7, 9 and U / = 0., 0.5, 1. The rms U main dierence between the cases o a single turbulent harmonic and a multi-scaled turbulence is a much more pronounced resonance at R R c. Figure 9 shows how velocity o lame propagation depends on the number o turbulent harmonics N T : the more harmonics we take, the smoother resonance we get at R R c. IV. Discussion The results on turbulent lame dynamics obtained in the present paper may be interpreted in two ways. First, they may be used to understand lame dynamics in

22 experiments with a relatively small integral turbulent length scale (comparable to the cut-o wavelength o the DL instability λ c ) and relatively low turbulence intensity. Experiments o this type have been presented in [3], where the ratio o the channel width to the lame thickness (the Peclet number) varied as R /L = Since the cut-o wavelenth is typically λ c = ( 0 50)L [4,5], then the experiment conditions in [3] are similar to the case o tubes o a moderate width 1 < R / R c < 5 considered in the present paper. Unortunately, one cannot compare results o the present paper to the experimental results directly, since the present results describe a D lame, while the experimental low is obviously a 3D one. Still, an indirect comparison is possible. For that purpose, one may remember that in the case o the DL instability the increase o lame velocity is twice larger or a 3D geometry in comparison with a D one [30]. The same relation holds or velocity increase o a weakly turbulent lame with no inluence o the DL instability [1]. Thus, it would be quite natural to expect that, when both eects work together, the velocity increase is also twice larger or a 3D low in comparison with a D low o the present paper. Such evaluation is shown in Fig. 10 or Θ = 7, R /R c = 5 by the dashed line A, which agrees rather well with the experimental results [3] presented by markers. Accurate investigation o the turbulent lame velocity in a 3D low requires time-consuming calculations and will be a subject o uture work. For comparison, Fig. 10 shows also turbulent velocity or a 3D particular solution to Eq. (1) without direct inluence o the DL instability (dashed line B). This plot goes noticeably below the experimental points, which demonstrates once more importance o the DL instability or turbulent lames. Another possible interpretation o the present results is to treat them as a sub-grid study o lame dynamics in a combustor o a large scale with some turbulent intensity,

23 3 which is not necessarily weak. In that case the tube width R considered in the present paper should be interpreted as the upper limit o the short-wavelength band o the turbulent spectrum, and U rms plays the role o a turbulent velocity on a length scale R. Designating the real integral length scale and the real turbulent velocity by L T and U T we have or the Kolmogorov spectrum U rms U T = ( R L T ) 1/ 3. (45) The limit o weak turbulence implies U rms <<1, and we took in our calculations U rms = Taking into account the evaluation or the cut-o wavelength [4,5] we ind that the tubes o moderate width R < 5R c considered in the present paper correspond approximately to R 100L, or R 100ν, where ν is kinematic viscosity with a characteristic value ν 0.15 cm /s. Then the integral velocity needed to produce the sub-grid turbulence with U rms = 0. 1 on the length scale R 100L may be evaluated as U T = U rms U ( ) 1/ 3 = U rms R /L T U L T U 100ν 1/ 3. (46) I we take typical laminar lame velocity U =100 cm/s and the length scale L T =100 cm comparable to the size o a gas turbine, then calculations o the present paper correspond to the integral turbulent intensity U T =1 10 and to the respective turbulent Reynolds number Re T = U T L T /ν = The obtained estimate shows that calculations o the present paper may be quite reasonable as a sub-grid model or a turbine combustor. Still in most o laboratory combustion experiments [3,,3] a smaller integral turbulent length scale (about 1 cm in [3]) and a smaller value o the Reynolds number Re T = are employed.

24 4 Acknowledgements This work has been supported by the Swedish Research Council (VR).

25 5 Reerences [1] V. Bychkov, Phys. Rev. Lett. 84, 61 (000). [] J.F. Griith and J.A. Barnard, Flame and Combustion (Blackie Academic and Proessional, London,1995). [3] A.N. Lipatnikov and J. Chomiak, Prog. Energy Combust. Sci 8, 1 (00). [4] P. Clavin and F.A. Williams, J. Fluid Mech. 90,589 (1979). [5] A. R. Kerstein, W. T. Ashurst, and F. A. Williams, Phys. Rev. A 37, 78 (1988). [6] V. Yakhot, Combust. Sci. Technol. 60, 191 (1988). [7] P. Ronney and V. Yakhot, Combust. Sci. Technol. 86, 31 (199). [8] B. Denet, Phys. Rev. E 55, 6911 (1997). [9] L. Kagan and G. Sivashinsky, Combust. Flame 10, (000). [10] Ya.B. Zeldovich, G. Barenblatt, V. Librovich, and G. Makhviladze, The Mathematical Theory o Combustion and Explosion (Consultants Bureau, New York,1985). [11] V.V. Bychkov, Phys. Fluids 10, 091 (1998). [1] O.Yu. Travnikov, V.V. Bychkov, M.A. Liberman, Phys. Rev. E 61, 468 (000). [13] P. Cambray, and G. Joulin, Combust. Sci. Thechnol. 97, 405 (1994). [14] B.T. Helenbrook and C.K. Law, Combust. Flame 117, 155 (1999). [15] G. Sivashinsky, Acta Astronaut. 4, 1117 (1977). [16] M. Frankel, Phys. Fluids A, 1979 (1990). [17] V. Bychkov, K. Kovalev, M. Liberman, Phys. Rev. E 60, 897 (1999). [18] V.V. Bychkov and M. A. Liberman, Phys. Reports 35, 115 (000). [19] T. Poinsot., S. Candel and A. Trouve, Prog. Energy Combust. Sci. 1, 531 (1996).

26 6 [0] P. Renard, D. Thevenin, J. Rolon, and S. Candel, Prog. Energy Combust. Sci. 6, 5 (000). [1] V.V. Bychkov, M.A. Liberman, R. Reinmann, Combust. Sci. Tech. 168, 113 (00) [] R. Abdel-Gayed, D. Bradley, M. Lawes, Proc. R. Soc. Lond. A 414, 398 (1987). [3] R.C. Aldredge, V. Vaezi, P.D. Ronney, Combust. Flame 115, 395 (1998). [4] P. Pelce and P. Clavin, J. Fluid Mech. 14, 19 (198). [5] G. Searby and D. Rochwerger, J. Fluid Mech. 31, 59 (1991). [6] G. Searby and P. Clavin, Combust. Sci. Technol. 46, 167 (1986). [7] V. Bychkov and B. Denet, Combust. Theory Modell., submitted. [8] R.C. Aldredge, Combust. Flame 106, 9 (1996). [9] V.V. Bychkov, S. Golberg, M. Liberman, and L.E. Eriksson, Phys. Rev. E 54, 3713 (1996). [30] V. Bychkov and A. Kleev, Phys. Fluids 11, 1890 (1999).

27 7 Figures FIG. 1. Scaled velocity o a curved stationary lame U w 1 vs the scaled tube width R /R c or dierent uel expansion Θ = 5,7,9 according to Eq. (15).

28 8 FIG.. Scaled turbulent lame velocity U w 1 vs the scaled tube width R /R c given by the solution o Eq. (1), not including the DL instability, with one turbulent harmonic in representation (14): (a) or the ixed turbulent intensity U rms = 0.5 but or dierent uel expansion Θ = 5,7, 9 ; (b) or the ixed uel expansion Θ = 7 and

29 9 dierent turbulent intensities U / U : curves A, B, C correspond to values rms U rms / U = 0.,0.5, 1 respectively. FIG. 3. Scaled turbulent lame velocity U w 1 vs the scaled tube width R /R c given by the solution o Eq. (1), not including the DL instability, with 150 turbulent

30 30 harmonics in representation (14): (a) or the ixed turbulent intensity U rms = 0.5 but or dierent uel expansion actors Θ = 5,7, 9 ; (b) or the ixed uel expansion Θ = 7 and dierent turbulent intensities U rms / U : curves A, B, C, D correspond to values U / = 0.,0.5, 0.7, 1 respectively. rms U FIG. 4. Spectrum G i o the numerically obtained stationary solution G ( x) o Eq. (4) or R /R c = 5, U rms = 0.5, Θ = 7, N =150 and N T =150.

31 31

32 3 FIG. 5. Scaled turbulent lame velocity U w 1 versus the scaled tube width R /R c given by the solution o Eq. (1), including the DL instability, or some ixed uel expansion actor (Θ = 5, 7, 9 or Figs. 5(a), 5(b), 5(c) respectively) and dierent turbulent intensity U / U : curves A, B, C, D correspond to values rms U rms / U = 0.,0.5, 0.7, 1 respectively.

33 33 FIG. 6. Characteristic increase o the turbulent lame velocity U w 1 or R = 5R c versus turbulent intensity U rms. The solid and dashed lines show the solutions with and without the DL instability, respectively.

34 34

35 35 FIG. 7. Scaled turbulent lame velocity U w 1 versus the scaled tube width R /R c given by the solution o Eq. (1), including the DL instability, or some ixed turbulent intensity ( U / = 0.,0.5, 0.7, 1 or Figs. 7(a), 7(b), 7(c) 7(d) respectively) and rms U dierent expansion actors Θ. To distinguish plots we used dashed line in Fig. 7(a) or Θ = 7.

36 36

37 37 FIG. 8. Scaled turbulent lame velocity U w 1 versus the scaled tube width R /R c given by the solution o Eq. (1), including the DL instability, with one turbulent harmonic in representation (14) or some ixed turbulent intensity ( U / = 0.,0.5, 1 or Figs. 8(a), 8(b), 8(c) respectively) and dierent expansion rms U actors Θ.

38 38 FIG. 9. Scaled turbulent lame velocity U w 1 vs the scaled tube width R /R c given by the solution o Eq. (1), including the DL instability or dierent number o turbulent harmonics N (curves A, B, C, D correspond to values N = 1, 5, 30, 150 T T respectively) and the ixed uel expansion actor Θ = 7 with U / =0. 5 and N =150. rms U

39 39 FIG. 10. Scaled velocity o turbulent lame U w 1 vs turbulent intensity U rms / U or Θ = 7 and R = 5R c (solid line). The dashed lines A, B show respective evaluation o lame velocity or a 3D case with and without the DL instability, respectively. The markers show the experimental results (Aldredge et al, 1998).