Numerical study of turbulent flame velocity

Combustion and Flame 151 (2007) 452–471 www.elsevier.com/locate/combustflame

Numerical study of turbulent flame velocity V’yacheslav Akkerman a,b , Vitaly Bychkov a,∗ , Lars-Erik Eriksson c a Department of Physics, Umeå University, S-901 87 Umeå, Sweden b Nuclear Safety Institute (IBRAE) of Russian Academy of Sciences, B. Tulskaya 52, 115191 Moscow, Russia c Department of Applied Mechanics, Chalmers University of Technology, S-412 96 Göteborg, Sweden

Received 15 August 2006; received in revised form 2 April 2007; accepted 10 July 2007 Available online 31 August 2007

Abstract A premixed flame propagating through a combination of vortices in a tube/channel is studied using direct numerical simulations of the complete set of combustion equations including thermal conduction, diffusion, viscosity, and chemical kinetics. Two cases are considered, a single-mode vortex array and a multimode combination of vortices obeying the Kolmogorov spectrum. It is shown that the velocity of flame propagation depends strongly on the vortex intensity and size. The dependence on the vortex intensity is almost linear in agreement with the general belief. The dependence on the vortex size may be imitated by a power law ∝ D 2/3 . This result is different from theoretical predictions, which creates a challenge for the theory. In the case of the Kolmogorov spectrum of vortices, the velocity of flame propagation is noticeably smaller than for a single-mode vortex array. The flame velocity depends weakly on the thermal expansion of burning matter within the domain of realistically large expansion factors. Comparison to the experimental data indicates that small-scale turbulence is not the only effect that influences the flame velocity in the experimental flows. Large-scale processes, such as the Darrieus–Landau instability and flame–wall interaction, contribute considerably to the velocity of flame propagation. Still, on small scales, the Darrieus–Landau instability becomes important only for a sufficiently low vortex intensity. © 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Premixed flames; Turbulent burning; Direct numerical simulations

1. Introduction Velocity of turbulent flame propagation is probably the most important problem of combustion science [1,2]. Still, this problem is far from a final answer in spite of intensive research, experimental, theoretical, and numerical. For a long time, starting with the papers by Damköhler and Shelkin, the research was dominated by an idea that the turbulent flame ve* Corresponding author. Fax: +46 90 786 6673.

E-mail address: [email protected] (V. Bychkov).

locity Uw may be described by a universal formula such as Uw /Uf = f (Urms /Uf ),

(1)

where Uf is the planar flame speed and Urms is the root-mean-square velocity of a turbulent flow in one direction [1]. An assumption such as (1) was very convenient for the theory, since it allowed the simplified model of constant gas density [3–10]. Flame dynamics does not affect a turbulent flow of constant density, which may be chosen in advance. However, the situation is much more interesting and complicated in reality. Because of strong thermal expansion,

0010-2180/$ – see front matter © 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2007.07.002

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

flame interacts with turbulence in a complicated way, which is not quite understood up to now. A famous review of experimental works by Abdel-Gayed et al. [11] clearly demonstrated considerable scattering of Uw /Uf measured by different groups for the same Urms /Uf , which refuted the possibility of a simple equation like Eq. (1). To save the situation, AbdelGayed et al. [11] modified Eq. (1) by including the Reynolds and/or Karlovitz number in the formula. Still, recent experiments [12–15] put into question the very possibility of a universal formula for turbulent flame velocity. Lee and Lee [13] demonstrated that the turbulent flame velocity in tubes depends strongly on the nonslip boundary conditions. Savarianandam and Lawn [15] have shown the importance of the Darrieus–Landau (DL) instability at low turbulent intensity. Filatyev et al. [14] formulated openly that experimental results on turbulent flame velocity are intrinsically different for different experimental setups, and even for different flow conditions within one setup. The problem may be illustrated in the following way. The experiments [13] employed a tube with rectangular cross-section 9 × 3.5 cm, while the integral turbulent length scale was only λT = 0.5 cm. In that case, all effects of turbulent burning (within the conventional meaning of this concept) are limited to small scales λ < λT = 0.5 cm. However, the flow on large scales λ > λT also contributes to flame wrinkling and to the flame propagation velocity. Wrinkling on large scales happens due to the flame–wall interaction, the DL instability, and other effects. The experimental studies [13] evaluated the influence of the large-scale flow by an empirical factor of about 2–4. Theoretical papers [16–19] described the large-scale effects in scope of the renormalization method and found close values. The situation is similar for other experiments [11,12,14,15]. Since the large-scale flow is specific to any experiment, then, indeed, no universal formula such as Eq. (1), even modified, can describe the diverse experimental data. Still, there is a common belief that combustion effects on small and large scales may be separated, with a certain universality expected for small-scale turbulent burning. In that case, limiting the study to small scales only, one may come to a formula like Eq. (1) for the local average turbulent flame velocity. As a next step in the analysis, one gets rid of small scales by increasing properly viscosity, thermal conductivity, diffusion, and flame thickness, and by replacing the laminar flame velocity by the turbulent one. Finally, one may study the large-scale effects for a flame with modified properties. Colin et al. [20] called such an approach “the method of thickened flames.” In order to justify the approach, researchers usually refer to the renormalization method. Though never proved rigorously within the combustion theory, this method

453

has been widely used already [2]. The renormalization method allowed even the analytical theory of turbulent burning, though under a much more restrictive assumption of a wide turbulent spectrum [4,5,16–19]. The wide spectrum implies that the integral turbulent length λT is much larger than the inner cutoff of flame wrinkles λc determined by the DL cutoff [16–19,21– 23]. The cutoff was calculated as λc = (0.2–0.5) cm for typical propane and methane–air mixtures in [17], which is comparable to the integral turbulent length scale λT = (0.5–1) cm in experiments [12–14]. As a result, the assumption of a wide spectrum is too restrictive to analyze the experiments [11–14] quantitatively, and one has to employ numerical methods to study the turbulent flame velocity. The numerical method of thickened flames was illustrated in [20,24] by investigating combustion in the complex geometry of gas turbines. The large-scale simulations of [24] were quite impressive. In contrast, investigations of the turbulent flame velocity on small scales were rather fragmentary so far. Systematic study of the turbulent flame velocity is still missing, though this is the basic step in the method of thickened flames. To obtain the turbulent flame velocity, Colin et al. [20] studied a collision of an initially planar flame front with a vortex pair (or with a single vortex, taking into account symmetry). The same flow geometry was also used in a number of papers [25– 27]; see also the reviews [2,28,29]. Still, papers [25– 27] considered rather small vortex sizes, well below the cutoff λc , which characterizes an effective flame thickness. In that case the flow is strongly influenced by thermal conduction and, therefore, vortices of extremely high intensity were used to obtain noticeable flame wrinkling (up to 50Uf in [27]). Combustion experiments typically involve much smaller turbulent rms-velocity, comparable to the planar flame velocity Urms /Uf = 0.5–5, with the turbulent length scale slightly above the cutoff, λT /λc = 1–3. Simulation parameters of [20] covered the domain of interest. Still, the geometry of a planar flame colliding with a vortex pair used in [20] is somewhat artificial, since in that case flame–vortex interaction is quite short. Flame propagation in a vortex array is much more representative, because it allows a statistically stationary regime of burning. The geometry of a vortex array was extremely popular within the model of constant gas density [6–10], but, to the best of our knowledge, it was never employed in direct numerical simulations of turbulent burning with realistic gas parameters. In the present paper we investigate systematically the turbulent flame velocity in a two-dimensional vortex array in direct numerical simulations. The problem was solved using the complete set of hydrodynamic and combustion equations including thermal conduction, diffusion, viscosity, and chemi-

454

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

cal kinetics. The flow imitated statistically stationary turbulent burning for different values of vortex intensity and size controlled by the channel width. Two cases of a single-mode vortex array and a multimode Kolmogorov turbulent spectrum were considered. We have shown that the scaled turbulent flame velocity Uw /Uf depends strongly on the scaled rms-velocity of the vortex flow Urms /Uf and on the size of the turbulent vortices. The dependence on the vortex rmsvelocity is almost linear for sufficiently high vortex intensity, which agrees well with the general belief and the theoretical predictions. Besides, flame speed increases with the vortex size approximately as D 2/3 . Such a tendency has not been predicted by any theoretical paper before; it creates a challenge for the theory. We also demonstrated that studies of flame interaction with a single vortex (or a vortex pair) lead to quantitatively different results than studies of statistically stationary burning in a vortex array. Comparison to experimental data demonstrated that small-scale turbulence is not the only effect that influences the velocity of flame propagation in the experimental flows. Other processes, such as the DL instability and flame interaction with viscous walls, also contribute considerably to the flame velocity. The present paper is organized as follows: in Section 2 we describe the basic equations used in direct numerical simulations; the idea of how to simulate a long-living array of turbulent vortexes is explained in Section 3; in Section 4 we describe the modeling results of turbulent burning; we conclude the paper with a brief summary.

2. Basic equations and the details of modeling We performed direct numerical simulations of the two-dimensional (2D) hydrodynamic combustion equations including transport processes and chemical kinetics, ∂ ∂ ρ+ (ρui ) = 0, ∂t ∂xi ∂ ∂ (ρui ) + (ρui uj + δi,j P ) − γi,j = 0, ∂t ∂xj    1 ∂ 1 ∂ ρε + ρui uj + ρui h + ρui uj uj ∂t 2 ∂xi 2  + qi − uj γi,j = 0,   ∂ ∂Y ∂ (ρY ) + ρui Y − κ Le = −Ω, ∂t ∂xi ∂xi

(2) (3)

(4) (5)

where Y is the mass fraction of the fuel mixture, ε = QY +CV T is the internal energy, h = QY +CP T is the enthalpy, Q is the energy release in the reaction, and CV and CP are the heat capacities at constant

volume and pressure, respectively. We considered a single irreversible Arrhenius reaction of the first order, with the activation energy Ea and the constant of time dimension τR , Ω=

ρY exp(−Ea /Rp T ). τR

(6)

The diffusion vector qi and the stress tensor γi,j are given by   Q ∂Y ∂T qi = −κ (7) , + Le ∂xi CP ∂xi   ∂uj 2 ∂uk ∂ui + − δi,j , γi,j = μ (8) ∂xj ∂xi 3 ∂xk where κ is the thermal conduction coefficient, μ = ρν = Pr κ/CP is the dynamic viscosity, and Le and Pr are the Lewis and Prandtl numbers, respectively. In the present work thermal conduction was chosen as κ = 3.41 × 10−2 J/(m s K). To avoid the Zeldovich (thermal-diffusion) instability we used the unit Lewis number Le = 1. The choice of an appropriate Prandtl number will be discussed in the next section. We chose a relatively small activation energy, Ea = 7Rp Tb , which allows smoothing the reaction zone over few computational cells (Tb is the temperature of the burnt gas). The burning matter was a perfect gas of constant molecular weight m = 2.9 × 10−2 kg/mol with the equation of state P = ρRp T /m,

(9)

and with CV = 5Rp /2m, CP = 7Rp /2m, where Rp ≈ 8.31 J/(mol K) is the perfect gas constant. We considered a flame propagating in the z-direction in a 2D tube of width D open at both ends, with adiabatic boundary conditions and with slip at the walls, u · nˆ = 0,

nˆ · ∇T = 0,

(10)

where nˆ is a normal vector at the wall. We used the initial pressure and temperature of the fuel mixture Pf = 105 Pa and Tf = 300 K, respectively. We chose the planar flame velocity Uf = 3.47 m/s, which provided sufficiently slow flame propagation in comparison with the sound speed; the Mach number was about Ma = 10−2 . For methane and propane burning the Mach number is an order of magnitude smaller. Still, a small Mach number increases considerably the computational time, since the time step is proportional to the Mach number. At the same time, the physical results of the present simulations do not depend on sound waves and the Mach number, which allows using the increased value for the flame velocity. A strongly subsonic flow is almost isobaric, with the thermal expansion coupled to the energy release in the burning process as Θ = ρf /ρb = Tb /Tf = 1 + Q/CP Tf .

(11)

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

We considered Θ = 5, 8 typical for methane and propane burning. The flame thickness in our calculations was defined conventionally as Lf ≡ κ/CP ρf Uf .

(12)

Still, we stress that Lf is just a mathematical parameter of length dimension, while the characteristic width of the burning zone may be an order of magnitude larger. Another useful length scale is the cutoff wavelength of the DL instability λc . Perturbations of an initially planar flame in a uniform flow grow for λ > λc and decay in the opposite case [30]. The DL cutoff shows when thermal conduction becomes important for hydrodynamic flame phenomena and plays the role of an effective flame thickness even for turbulent flames [16–19]. The DL cutoff is proportional to the flame thickness Lf and depends on the thermal-chemical properties of the burning matter. In our simulations we found λc = 18.8Lf for Θ = 5 and λc = 17.2Lf for Θ = 8. In a tube with ideally adiabatic and slip walls (playing the role of symmetry axes), the DL instability develops for the tube width exceeding the critical value Dc = λc /2. We used a two-dimensional Eulerian code developed in Volvo Aero. The code is robust and it was utilized quite successfully in studies of laminar burning, hydrodynamic flame instabilities, flame acceleration, flame oscillations, and similar phenomena [31–35]. The numerical scheme of the code and the computational methods were described in details in our previous papers (see, for instance, [35]). In the present simulations we considered the tube widths 0.5 < D/Dc < 4. In almost all simulation runs we used a tube length 10 times larger than the tube width L = 10D, which corresponds to tube lengths up to 350Lf . The tube contained originally an array of 10 identical vortices. The planar flame front was created at the boundary between two vortices in the array, with two vortices initially in the burnt matter, and other eight vortices in the fuel mixture; see Fig. 1. Initially, the fuel mixture was at rest (in average), with the mean flow (Θ − 1)Uf in the burnt matter. Similarly to our previous papers [31,32,35], we used the Zeldovich–Frank–Kamenetsky solution for a planar flame front as an initial condition. As the flame starts propagating, it pushes a flow, which may drift some of the “unburned” vortices out of the tube. The tube length L = 10D was sufficient to observe the flame crossing at least 5–6 vortices in the array. A test simulation run for a longer tube L = 16D demonstrated the same flame behavior as in the case of L = 10D. We used a uniform square grid with the grid walls parallel to the coordinate axes and with the grid size Dc /40 ≈ 0.2Lf in both directions. Unlike in most of our previous simulations [31–36], the grid was uniform all over the tube since we have to resolve not

455

only the flame front, but also the vortices. We have demonstrated recently [35] that such a grid is quite sufficient to resolve the inner flame structure even in the case of noticeably curved flames. We imposed nonreflecting boundary conditions at the tube ends as described in [35]. Using such conditions we avoided reflections of weak shocks and sound waves from the tube ends, which otherwise might influence burning. The turbulent flame velocity was calculated as  1 Ω dx dz Uw = (13) ρf D with the reaction rate Ω defined by Eq. (6). We would like to stress that Uw may be quite different from the velocity of flame propagation in the laboratory reference frame; the value Uw characterizes the burning rate (amount of fuel mixture consumed per unit time) rather than any shift in space. One more parameter characterizing the flame velocity is relative increase in the flame surface area with respect to the tube cross section. In the standard model of an infinitely thin flame front, the velocity increase is equal to the increase in the flame surface area. In the 2D geometry the relative increase in the surface area is replaced by the ratio of the total flame length Dw versus the cross section of the tube, Dw /D.

3. Modeling the vortex array The first task of the simulations was to obtain a long-living array of turbulent vortices. Simulations in scope of the constant-density model [6–10] typically imitate turbulence by the Fourier decomposition, ux =

uz =

N  i=1 N 

Ui sin(ki z + ϕiz ) sin(ki x + ϕix ),

(14)

Ui cos(ki z + ϕiz ) cos(ki x + ϕix ),

(15)

i=1

where ki = iπ/D are the wavenumbers controlled by the tube width, Ui are the mode amplitudes, and ϕiz , ϕix are the random phases. The rms-velocity of the flow (14)–(15) is calculated as N  2 = u2  = u2  = 1 Ui2 . Urms x z 4

(16)

i=1

Within the model of constant gas density, the number of modes N and the amplitudes Ui may be chosen arbitrarily, for example, according to the Kolmogorov −5/6 . In scope of the model, the spectrum Ui ∝ ki turbulent flow is just an externally prescribed vectorfunction in the equations of thermal conduction and diffusion, and one does not face a problem of how

456

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

(a)

(b)

(c)

(d) Fig. 1. The initial flame shape and the flow with Urms /Uf = 1 in a tube of width D/Dc = 2. The colors in (a) show temperature from the cold gas (blue) to the burnt matter (red) for a single-mode vortex array. The white curves show the streamlines of the flow. The colors in (b) show vorticity in the flow: red for the positive counterclockwise direction; blue for the negative clockwise direction. (c, d) present similar values for the multimode flow with N = 5 harmonics and the amplitudes obeying the Kolmogorov law. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to keep vortices from decaying. Unlike that, in direct numerical simulations we cannot choose the flow (14)–(15), since it does not satisfy the basic equations (2)–(5). Taking into account viscosity in the Navier–Stokes equations, we obtain turbulence decaying either in time, or in space, or both [17]. At the same time, we would like to reproduce the velocities (14)–(15) as closely as possible and to observe flame propagation in the turbulent flow as long as possible. Considering one mode in the Fourier decomposition (14)–(15) and solving the Navier–Stokes equations analytically similarly to [17] within the first-order corrections in Lf /D, we find that the velocity and pressure field in the form ux = U sin(kz + ϕz ) sin(kx + ϕx ) × exp(−2νk 2 t),

(17)

uz = U cos(kz + ϕz ) cos(kx + ϕx ) × exp(−2νk 2 t), ρ  P − Pf = U 2 cos(2kx + ϕx ) 4 − cos(2kz + ϕz ) exp(−4νk 2 t),

with k = π/D, and with the rms-velocity Urms =

U exp(−2νk 2 t). 2

(20)

The vortex intensity Eq. (20) decays in time, with the characteristic decay time τd =

1 D2 Re D = = , 2 2 2νk 2π Pr Uf Lf 2π 2 Uf

(21)

where Re = DUf /ν = D/ Pr Lf

(22)

is the Reynolds number. In order to observe statistically stationary burning in a vortex array we have to obtain the vortices “living” much longer than the characteristic time of flame propagation through a vortex τ = D/Uf . The condition τd  τ leads to

(18)

Re  2π 2 .

(19)

In the case of typical propane and methane flames with Pr = 0.5–0.7, propagating in an industrial burning chamber with D/Lf = 104 –106 , the condition

(23)

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

457

Fig. 2. The rms-velocity of the vortex flow Urms scaled by the initial value Urms,0 versus the scaled time Uf t/D for different Prandtl numbers: Pr = 10−3 , 0.1, 0.3, and 0.7. The solid lines present the analytical formula (20); the markers show the simulation results: empty for the single-mode flow and filled for the multimode one.

(23) is obviously fulfilled. However, it is not so easy to satisfy (23) in direct numerical simulations because of inevitable restrictions on the tube width. For example, in the present paper we were able to consider tubes as wide as D/Lf = 35. In that case one has to choose a really small Prandtl number in order to observe long-living vortices. Luckily, it was shown that turbulent flame speed depends only slightly on the Prandtl number [17]. In the present simulations we used Pr = 10−3 , which corresponds to μ = 3.4 × 10−8 N s/m2 , ν ≈ 2.9 × 10−8 m2 /s, with Re = 104 –105 . We verified the above analytical estimates by direct numerical simulations. Fig. 2 shows decay of vortex intensity in time for different values of the Prandtl number from the realistic one 0.7 to Pr = 10−3 . The solid lines correspond to the analytical solution (20); the empty markers show the results of numerical simulations obtained for a singlemode vortex array. The analytical results are in very good agreement with the numerical ones. Still, flame propagation influences the vortices and modifies the rms-velocity in comparison with the initial values. In most of the simulation runs we used a tube of length L = 10D with an array of 10 vortices of size D × D shown in Fig. 1. Although multimode turbulence with the Kolmogorov spectrum is closer to reality, the geometry of a vortex array (17)–(18) is quite typical in numerical and analytical studies of turbulent burning [6,8–10,17]. The configuration of

a vortex array is popular, since the first harmonic in the spectrum (14)–(15) is of primary importance for the turbulent flame velocity; see [17]. For comparison, in some runs we have also studied a multimode array (14)–(15) with the Kolmogorov spectrum. The respective initial vortices are shown in Figs. 1c, 1d. The filled markers in Fig. 2 show the time decay of Urms in the multimode flow calculated numerically for Pr = 10−3 , 0.1, 0.3, and 0.7. Even for realistic Prandtl numbers Pr = 0.3–0.7 the time dependence of vortex intensity in multimode flow resembles qualitatively that of a single-mode vortex array. When the Prandtl number is low, e.g., Pr = 10−3 , the filled markers practically coincide with the empty ones, which means that the multimode array may live as long as the single-mode flow. There is another small but important detail in the simulation of a vortex array. In order to obtain longliving vortices, we had to choose a zero z-component of the circulating velocity at the tube ends at the beginning of calculations uz,vortex (zentry , x) = uz,vortex (zexit , x) = 0.

(24)

An array placed “incorrectly” in the tube produced a noticeable drift flow, which washed the vortices away out of the tube much faster than predicted by Eq. (21). The condition (24) should hold even in the case of a Kolmogorov spectrum, which does not allow random phases in Eqs. (14), (15). As a result, the vortices in

458

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

(a)

(b) Fig. 3. Flame shape/position and the flow with Urms,0 /Uf = 1 in a tube of width D/Dc = 2 at the time instant Uf t/D = 1.28. The origin (z = 0) is related to the initial flame position shown in Fig. 1. The colors in (a) show the temperature from the cold gas (blue) to the burnt matter (red). The white curves show the streamlines of the flow. The colors in (b) show vorticity in the flow: red for the positive counterclockwise direction; blue for the negative clockwise direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Figs. 1c, 1d look well-correlated in spite of the multimode Kolmogorov spectrum. The restriction Eq. (24) does not concern the mean flow. We stress that, in general, gas velocity at the tube ends was not zero in the simulations; the tube was open at both ends with nonreflecting boundary conditions. A propagating flame-front-induced flow out of the tube in both directions. The flow drifted some vortices out, which, therefore, became “lost” for the study. To overcome this trouble and to make the flame cross at least 5–6 vortices, we had to take a tube sufficiently long.

4. Investigation of the turbulent flame velocity We studied flame propagation through an array of turbulent vortices in a tube with both ends open and with slip adiabatic boundary conditions at the walls. The main parameters of the problem are the tube width D (scaled by the critical width Dc = λc /2) and the rms-velocity of the vortex flow Urms (scaled by the planar flame speed Uf ). We performed simulations for a tube width in the domain of 0.5  D/Dc  4, with the rms-velocity from Urms /Uf = 0.05 up to Urms /Uf = 5.3. All simulation runs started with an initially planar flame front, which, however, became corrugated in a very short time due to the flame– vortex interaction and the DL instability. Figs. 1 and 3 present the flame shape and the characteristic streamlines and vorticity in a tube of width D/Dc = 2, with the initial rms-velocity Urms,0 /Uf = 1. Fig. 1 shows initial conditions of the simulation, while Fig. 3 presents a typical snapshot during the statistically stationary burning (Fig. 3 corresponds to the time instant Uf t/D = 1.28). Figs. 1a, 3a show temperature by color (from blue in the fresh gas to red in the burnt

matter), while the streamlines are shown by white curves. One can clearly see vortices in the fresh gas, while the streamlines behind the flame look much smoother. Such an effect happens because of thermal expansion of the burning matter. Indeed, initially the fuel mixture was at rest on the average uz x = 0, while the burnt matter was drifted as uz x = −(Θ − 1)Uf due to the gas expansion. As a result, vortices with Urms /Uf = 1 dominate in the flow ahead of the flame front, but they provide only slight sinusoidal deviations from the strong average flow behind the flame. Figs. 1b, 3b show vorticity in the same flow. The situation remains qualitatively the same during the whole process of burning, though vortices in the burnt matter may be washed away out of the tube rather fast because of the strong outflow. Propagating flame modifies the vortices, which makes vortex intensity ahead of the flame front different from the initial values. In each simulation run we investigated time variations of the rms-velocity ahead of the flame. Hereafter the value Urms denotes the instantaneous rms-velocity of the flow averaged over a square D × D just ahead of the flame front 1  2   2  (25) u˜ + u˜ z , 2 x where u˜ i = ui − ui . The region of averaging is illustrated in Fig. 4. In some of the simulation runs the instantaneous rms-velocity Urms differed noticeably from the initial value Urms,0 . The changes were especially considerable in the cases of weak turbulence. Time dependence of the scaled instantaneous rms-velocity Urms /Uf is shown in Fig. 5a for a tube of width D/Dc = 2 with the initial rms-velocity Urms,0 /Uf = 1. In that case Urms oscillates slightly around the initial value during almost the whole simulation run and it goes down as the flame approaches 2 = Urms

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

459

Fig. 4. The dashed line illustrates the region ahead of the flame for calculations of instantaneous Urms . The flow parameters are the same as in Fig. 3.

the tube end. The other solid line in Fig. 5a presents the relative increase in the flame velocity Uw /Uf − 1. We observe pulsations of the flame velocity as it travels through vortices; the central four pulsations in the sequence look quite regular and statistically stationary. On the contrary, the first pulse and the last one or two pulses may be noticeably different from the main sequence. In some of our simulation runs, the first/last pulse in the flame velocity was 2–3 times stronger in comparison with the statistically stationary sequence. Presumably, the first/last pulse should resemble flame interaction with a single vortex studied in [2,20,25–27,29]. To check this possibility we simulated flame interaction with a single vortex for the same vortex intensity and the same flame parameters as in Fig. 5a. The simulation results are shown in Fig. 5b. The results of Fig. 5b demonstrate not only qualitative, but even good quantitative agreement with the first pulse in Fig. 5a. We also stress noticeable quantitative difference between the single-vortex case and the main sequence of pulsations in Fig. 5a. By this reason, previous studies of flame interaction with a single vortex may not be used for calculations of the statistically stationary turbulent flame speed. After collision with a single vortex, flame velocity goes down to Uw = 1.28Uf , determined by the DL instability [31,32,38]. To be sure that the central four vortices do reproduce a piece of a statistically stationary sequence, we performed also simulations for a longer tube L = 16D, shown in Fig. 5c. In that case the central sequence of flame pulsations contains eight maxima instead of four of Fig. 5a. However, the flame pulsations have the same properties in Figs. 5a and 5c. Similar to Fig. 5a, the starting and final flame pulsations are noticeably different from the main sequence. As a result, Fig. 5 demonstrates that the tube length L = 10D is sufficient to obtain several flame pulsations reproducing statistically stationary burning through a vortex array. In the following we will use only tube length L = 10D. During the statistically stationary parts of Figs. 5a and 5c the turbulent burning rate varies considerably from Uw,min /Uf = 1.18 up to Uw,max /Uf = 2.45, with the average flame velocity Uw /Uf = 1.75 shown by the horizontal dashed line. The pulsation period shows the time needed for the flame front to cross a vortex; in Fig. 5 this time is about 0.5D/Uf . The

relative increase in the flame length Dw /D − 1 is shown in Fig. 5 by another dashed line, which practically coincides with the solid line for the burning rate. Therefore, both methods of calculating the flame velocity work very well. It is also interesting that the flame speed Uw remains always larger than Uf , which means that the flame front never becomes planar. This is different from the early theories of turbulent flame velocity employing weak nonlinearity [3,22,37] and similar to more recent theoretical results [17,18]. This effect may be explained in two ways: (1) either the approach of weak nonlinearity does not hold in that case; or (2) the DL instability influences flame dynamics, as it was suggested in [17,18], since D > Dc in Fig. 5. Both factors acting together are also possible. Figs. 6a, 6b correspond to the same tube as Fig. 5a with low vortex intensity; we chose initially Urms,0 /Uf = 0.05 in Fig. 6a and Urms,0 /Uf = 0.5 in Fig. 6b. The most interesting point in Fig. 6a is a noticeable increase in the rms-velocity because of the flame propagation. Instead of the initial value Urms,0 /Uf = 0.05, the vortex velocity was 2–3 times larger during the simulations, Urms /Uf = 0.11–0.16. These vortices, though rather weak, led to considerable oscillations of the flame velocity. Still, the average flame velocity Uw /Uf ≈ 1.29 is close to the value expected because of the DL instability in the case of zero turbulence [31,32,38]. Fig. 6b looks qualitatively similar to Fig. 5a; one observes a sequence of five almost identical flame pulsations, corresponding to the flame crossing five vortices. However, the average turbulent flame velocity in Fig. 6b is noticeably smaller than in Fig. 4a; it is Uw /Uf ≈ 1.33, which is also close to the velocity produced by the DL instability [31,32,38]. Another curious feature of Fig. 6b is markedly strong first and final pulsations, which demonstrates one more time that flame collision with a single vortex may be quite different from statistically stationary burning. Similar to Figs. 5 and 6a, the flame front in Fig. 6b never becomes completely planar, with Uw /Uf − 1 always noticeably larger than zero. Again, it makes one suspect some influence of the DL instability. To check the guess, we performed simulations for a narrow tube D/Dc = 1 with Urms,0 /Uf = 1, see Fig. 6c. In that case the DL instability is not possible, and we can see

460

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

(a)

(b) Fig. 5. The instantaneous rms-velocity at the flame front Urms /Uf (solid), relative increase in the flame velocity Uw /Uf − 1 (dashed), and relative increase in the flame length Dw /D − 1 (solid) versus the scaled time Uf t/D for Urms,0 /Uf = 1, the tube width D/Dc = 2, and the tube lengths L/D = 10 (a) and L = 16D (c). The horizontal dashed line shows average increase in the turbulent flame velocity Uw /Uf − 1. (b) is a counterpart of (a) for the case of flame interaction with a single vortex (with the same vortex intensity and the same flame parameters).

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

461

(c) Fig. 5. (continued)

(a) Fig. 6. The instantaneous rms-velocity at the flame front Urms /Uf and the relative increase in the flame velocity Uw /Uf − 1 (solid lines) versus the scaled time Uf t/D for D/Dc = 2, Urms,0 /Uf = 0.05 (a); D/Dc = 2, Urms,0 /Uf = 0.5 (b); D/Dc = 1, Urms,0 /Uf = 1 (c). The horizontal dashed line shows the average increase in the flame velocity Uw /Uf − 1.

462

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

(b)

(c) Fig. 6. (continued)

the respective change in the time dependence of the flame velocity. Unlike previous plots, Fig. 6c demonstrates pulsations of the flame velocity down to zero 0 < Uw /Uf − 1 < 0.95. Fig. 6c supports our guess about the role of the DL instability in the flame pulsations.

We have investigated flame speed for different tube widths and vortex intensities. Snapshots of the flame shape and vorticity in the case of strongly turbulent flow are shown in Fig. 7. Fig. 7a presents the temperature distribution for a narrow tube D/Dc = 1 with Urms,0 /Uf = 5.28. The respective vorticity dis-

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

463

(a)

(b)

(c)

(d) Fig. 7. Flame shape/position and the flow with Urms,0 /Uf = 5.28 in a tube of width D/Dc = 1 at the time instant Uf t/D = 0.75 (a, b) and with Urms,0 /Uf = 3.26 in a tube of width D/Dc = 2.36 at the time instant Uf t/D = 0.64 (c, d). The colors in (a, c) show the temperature from the cold gas (blue) to the burnt matter (red). The white curves show the streamlines of the flow. The colors in (b, d) show vorticity in the flow: red for the positive counterclockwise direction; blue for the negative clockwise direction.

tribution is shown in Fig. 7b. Figs. 7a, 7b correspond to the time instant Uf t/D = 0.75. Unlike Fig. 3, we observe noticeable flame bending in that case, with a strong vortex ahead of the flame front and even with a pocket of unburnt gas drifted into the products. Figs. 7c, 7d present the flame shape and vorticity in a tube of width D/Dc = 2.36 with the initial rms-velocity Urms,0 /Uf = 3.26 for the time instant Uf t/D = 0.64. Burning rate is shown in Fig. 8 for Urms,0 /Uf = 0.05, 0.5, and 1 versus the tube width by filled squares, triangles and circles, respectively. The solid lines show the theoretical predictions for the same vortex intensity within the approach of weak nonlinearity [17]. As we can see, only the plot with Urms,0 /Uf = 0.05 reproduces the theoretical curve well taking into account finite accuracy of the theory and the simulations. Keeping in mind that initial vortex intensity Urms,0 /Uf = 0.05 increases to Urms /Uf = 0.1–0.16 because of flame propagation, we can evaluate that the approach of weak nonlinearity holds for Urms /Uf < 0.2. At higher vortex intensities the numerical results are noticeably different from the theoretical predictions. The most interest-

ing feature is increase in the flame velocity with the tube width, with no visible limit or saturation. This feature has been noticed already in numerical simulations [20] for flame interaction with a single vortex. At the same time, it contradicts all theoretical ideas about the turbulent flame velocity proposed so far [3– 6,10,16,17,39]. One of the main tools of the theories is the model of an infinitely thin flame front, which is equivalent to an infinitely wide tube, D  Dc . In that case, assuming universal character of Fig. 8, we obtain an infinitely large speed of an infinitely thin flame front, which makes the model meaningless. Of course, the domain of length scales attainable in direct numerical simulations is limited; Fig. 8 does not guarantee increase in the flame velocity on infinitely large length scales. Still, even within the domain of Fig. 8, the numerical results make a challenge to the theory. Another important feature of the plots is strong decrease of the flame velocity at small scales, close to the critical tube width and below it, D/Dc  1. This property of the plots agrees quite well with the theory [17–19]; see also [22,37]. One can observe even better agreement when recalculating the theoretical

464

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

Fig. 8. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled tube width D/Dc for Urms,0 /Uf = 0.05, 0.5, and 1 (squares, triangles, and circles, respectively). The theoretical predictions for the same rms-velocities [17] are shown by the solid lines. The empty markers show the theoretical points [17] recalculated for Urms instead of Urms,0 .

points for the real Urms instead of the initial one, Urms,0 . The respective points are shown in Fig. 8 by empty markers for Urms,0 /Uf = 0.5 and 1. Fast decay of the flame velocity below the critical tube width D/Dc  1 demonstrates that cutoff of the turbulent flame structure correlates with the cut off of the DL instability, λc . Scaling for the turbulent flame cutoff is one of the most important parameters in the theory of turbulent burning; it has been discussed for a long time as reviewed in [40]. The theory [17–19,21,22, 37] demonstrated strong correlation between the turbulent cutoff and the DL cutoff. The same results have been obtained experimentally [23] and numerically for flames propagating along the vortex axes [41]. Present numerical simulations demonstrate the same property for a flame propagating across the turbulent vortices. This feature holds even for numerical points with negligible role of the DL instability. The reason for such behavior is that the DL cutoff plays the role of an effective flame thickness. At length scales comparable to the cutoff and below, thermal conduction becomes important for the hydrodynamic flow, no matter if the flow was created by the instability or external turbulence. Fig. 9 shows the burning rate versus the vortex intensity for D/Dc = 1 and 2.36 (the later tube width is chosen keeping in mind the experiments [13], see below). Triangles correspond to the initial value Urms,0 ; circles show the flame velocity for the more realis-

tic recalculated value Urms . Empty and filled markers correspond to D/Dc = 1 and D/Dc = 2.36, respectively. Both plots demonstrate almost linear dependence Uw /Uf − 1 ∝ Urms /Uf , which is also different from the theoretical predictions [3–5,10,16,17,22,37]. According to the theory, one should expect quadratic 2 /U 2 at low vortex independence U/Uf ∝ Urms f tensity, and a linear dependence at high intensity. In Fig. 9 we can see only slight traces of a possible quadratic dependence at Urms < 0.5, which indicates limitations on the approach of weak nonlinearity. These limitations agree with what we deduced from Fig. 8. For a tube width above the critical value D/Dc = 2.36 we can also observe influence of the DL instability with Uw /Uf − 1 ≈ 0.3 for small vortex intensity. However, this influence is also noticeable only at Urms < 0.5 when the flame is weakly curved. All simulations above have been performed for the expansion factor Θ = 8 typical for stoichiometric propane–air and methane–air flames. Fig. 10 demonstrates also the case of Θ = 5 usual for lean flames. Fig. 10 is plotted for the tube width D/Dc = 1. Empty and filled markers present the cases of Θ = 5 and Θ = 8, respectively; circles and triangles correspond to Urms and Urms,0 . As we can see, there is almost no difference between the cases with different thermal expansion Θ = 5 and 8 as long as they are realistically large, which agrees quite well with the theoretical predictions [17].

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

465

Fig. 9. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled rms-velocity Urms /Uf for the tube widths D/Dc = 1 and 2.36 (filled and empty markers, respectively). Triangles correspond to Urms,0 ; circles correspond to Urms .

Fig. 10. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled rms-velocity Urms /Uf for the tube of width D/Dc = 1 and the expansion factors Θ = 5 and 8 (filled and empty markers, respectively). Triangles correspond to Urms,0 ; circles correspond to Urms .

We also compared the flame velocity obtained for a single-mode vortex array to the multimode case, adopting the Kolmogorov spectrum for the am-

plitudes in Eqs. (14)–(15). The snapshots of flame shape in the multimode flow with N = 5 harmonics in the Kolmogorov spectrum are shown in Fig. 11

466

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

(a)

(b)

(c)

(d)

(e)

(f) Fig. 11. Flame shape/position and the multimode flow with Urms,0 /Uf = 1, N = 5 harmonics in Kolmogorov turbulent spectrum in a tube of width D/Dc = 2 at the time instants Uf t/D = 0.26, 0.52, and 1.03 (a, c, e). The colors show temperature from the cold gas (blue) to the burnt matter (red). The white curves show the streamlines of the flow. The colors in (b, d, f) show vorticity in the flow: red for the positive counterclockwise direction; blue for the negative clockwise direction.

for a tube of width D/Dc = 2 at the time instants Uf t/D = 0.26, 0.52, and 1.03 (Figs. 11a, 11c, and 11e). The respective vorticity distribution are presented in Figs. 11b, 11d, and 11f, respectively. The burning rate is presented in Fig. 12 for the number of modes N = 1, 5, and 10. The plot for a singlemode array shows a noticeably larger velocity of flame propagation, and there is almost no difference between the cases N = 5 and 10. The obtained feature may be understood by keeping in mind the results of Fig. 8. According to Fig. 8, the flame velocity de-

pends on the wavelength of the turbulent vortices: vortices of the same intensity, but of smaller size produce smaller increase in the flame velocity. In the case of N = 1 all energy is stored in the mode of largest wavelength, and we obtain higher velocity of flame propagation. In the cases of N = 5 and 10 some energy is deposited in the modes of small scales, which reduces the flame velocity. Fig. 13 is a counterpart of Fig. 5a for the multimode flow with N = 10 harmonics. Other flame and flow parameters were taken the same as in Fig. 5a. During statistically

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

467

Fig. 12. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled initial rms-velocity Urms,0 /Uf for the tube widths D/Dc = 1 and different number of harmonics in the turbulent spectrum: N = 1, 5, and 10 (triangles, circles, and squares, respectively).

Fig. 13. The instantaneous rms-velocity at the flame front Urms /Uf (solid), relative increase in the flame velocity Uw /Uf − 1 (dashed), and relative increase in the flame length Dw /D − 1 (solid) versus the scaled time Uf t/D for a multimode array of vortices with N = 10, Urms,0 /Uf = 1, the tube width D/Dc = 2, and the tube lengths L/D = 10. The horizontal dashed line shows average increase in the turbulent flame velocity Uw /Uf − 1.

468

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

Fig. 14. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled tube width D/Dc for the rms-velocities Urms,0 /Uf = 0.5 and 1 (triangles and circles, respectively). The dashed lines present Eq. (26) with the parameter C = 0.5.

stationary flame propagation, the turbulent burning rate in Fig. 13 varies from Uw,min /Uf = 1.18 up to Uw,max /Uf = 2.68, with the average flame velocity Uw /Uf = 1.53 presented by the horizontal dashed line. It is quite curios that the maximal burning rate in Fig. 13 exceeds that of Fig. 5a, but the average flame velocity is noticeably smaller for the multimode flow. One of the referees suggested that smaller flame velocity may be related to faster decrease of turbulent intensity in the multimode case. In order to check the possibility, we also plotted variations of Urms with time in the process of burning in Fig. 13. One can see that the turbulent intensity of a multimode flow resembles that of Fig. 5a. Thus the hypothesis of the referee is not confirmed. It is interesting to compare present simulations to the results of the previous papers. Simulations [20,24] considered flame interaction with a single vortex and suggested the following tendency for the burning rate, Uw /Uf − 1 = C

  Urms D 2/3 , Uf D c

(26)

where C is some tuning coefficient. Our numerical results agree with this tendency for sufficiently wide tubes and strong turbulence. Figs. 14 and 15 plotted for C = 0.5 show that Eq. (26) works reasonably well for D/Dc > 2, Urms /Uf > 1. Still, formula (26) is simply a numerical fit valid in a limited domain of parameters. In narrow tubes the flame speed goes noticeably down in comparison with Eq. (26). At D/Dc =

0.5 the flame propagation velocity Uw is practically the same as the planar flame velocity Uf . Numerical simulations cannot guarantee universality of Eq. (26) for large scales either. It may happen that Uw /Uf tends to some limit at D/Dc → ∞, though within the domain of simulations we do not see any sign of the velocity saturation. In a similar way, Eq. (26) works only for sufficiently strong turbulence Urms /Uf > 1. It does not reflect the tendencies expected from the theory in the limit of weak nonlinearity [3–5,10,16– 19,22,37]: it does not include the DL instability or 2 /U 2 . quadratic dependence U/Uf ∝ Urms f Not everything is clear in the tendencies of Eq. (26). It is easier to explain the linear dependence U ∝ Urms , since all theoretical papers [4,5,10, 16–19] predicted Uw ∝ Urms for sufficiently high turbulent intensity. It is much more difficult to understand the flame velocity increase with the vortex size, U/Uf ∝ D 2/3 . Colin et al. [20] tried to explain this feature using properties of the Kolmogorov turbulent spectrum. Obviously, this explanation does not work either for the geometry of a single vortex used in [20] or for a vortex array of the present simulations. Our guess is that the velocity increase U/Uf ∝ D 2/3 is an intermediate tendency, with the flame velocity saturating for sufficiently wide tubes D/Dc → ∞. However, this saturation is not observed in the present numerical simulations. We also stress that tuning coefficient C is not universal. For example, taking multimode turbulence, we obtain another

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

469

Fig. 15. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled rms-velocity Urms /Uf for the tube widths D/Dc = 1 and 2.36 (filled and empty markers, respectively). The solid lines present Eq. (26) with the parameter C = 0.5.

coefficient. Papers [20,24] suggested the third one, which does not come even close to the present numerical results. Thus, we face the question if there is “a turbulent flame speed” in the meaning of Eq. (1). Comparison of the present results to [20,24] suggests a negative answer. There is no universal formula for the flame speed, since it depends on particular properties of the turbulent flow: a single vortex or a vortex array, a single mode or a multimode flow, and, of course, on the spectrum. Still, the same comparison suggests some general tendencies of the turbulent flame speed, which remain similar for any type of turbulence. It is also interesting to compare the simulation results to the experiments; such comparison illustrates the role of large-scale flow in the experimental measurements. We choose experiments [13] on turbulent burning in stoichiometric propane–air mixtures in a tube with both ends open, which is similar to the geometry of the present work. As we mentioned in the Introduction, the tube cross section in [13] was rather large for direct numerical simulations, 3.5 × 9 cm. Still, the turbulent length scale in the experiments was only λT ≈ 0.5 cm, which is comparable to the DL cutoff λc = 0.21 cm calculated in [17] for the stoichiometric propane–air flame. Taking λT as evaluation for the vortex size, we find λT /λc = D/Dc = 2.36, which explains the choice of the tube width made above. The experimental and numerical data for the flame velocity are compared in Fig. 16; the simula-

tion results are shown by filled markers (triangles for Urms,0 ; circles for Urms ); the experimental results are shown by crosses. As we can see, there is no agreement between the numerical simulations and the experiments. However, it would be surprising to find quantitative agreement. First, the numerical results are two-dimensional, while the experimental flow is three-dimensional. According to the theory [5,17], the difference between√2D and 3D turbulent flame velocity may be about 2. Fast burning along the vortex axis may also influence the results [41,42]; this 3D process was not taken into account in the present 2D simulations. Second, as we explained in the Introduction, the experiments [13] involve both small-scale turbulence and large-scale effects, while the numerical simulations reproduce only the small-scale turbulent burning. Adopting the renormalization idea of thickened flames, we should expect that the numerical results differ from the experimental data by a numerical factor produced by the large-scale effects. The empty markers in Fig. 16 show the numerical results multiplied by a factor of f = 4; in that case the numerical data practically coincide with the experimental data. In the present paper we do not analyze the origin of the correction factor. We only point out that, due to the correction, the flame speed is rather large even at zero turbulent intensity; it may be as large as Uw /Uf ≈ 5.2. This comparison indicates that there are other processes except for turbulence, which

470

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

Fig. 16. Relative increase in the flame velocity Uw /Uf − 1 versus the scaled rms-velocity Urms /Uf for D/Dc = 2.36. Triangles and circles correspond to Urms,0 and Urms . Filled markers show the simulation results; empty markers show the same results multiplied by a correction factor f = 4. The experimental data [13] are shown by crosses.

bend the flame front and increase the flame velocity in the experiments. Among different reasons, the DL instability and flame interaction with nonslip walls are the most probable candidates for the large-scale wrinkling [16,17,35]. Finally, the present modeling shows the validity domain of the analytical renormalization theory of turbulent burning [16,17]. As we pointed out in the Introduction, the theory assumes a wide turbulent spectrum, so that every step in the renormalization process may be described within the approach of weak nonlinearity. According to the present simulations, the approach of weak nonlinearity holds for a sufficiently low intensity of the vortex array, Urms /Uf < 0.2 at the length scales λ λc . Going over to a wide spectrum with λT  λc we have turbulent intensity depending on a length scale urms = urms (λ). For example, we remind the dependence urms ∝ λ1/3 for the Kolmogorov turbulence. Respectively, turbulent intensity at λc may be evaluated as  1/3 λc . urms (λc ) = Urms (27) λT Thus, the renormalization theory may be applied only for the turbulent intensity   Urms λc 1/3 < 0.2. urms (λc )/Uf = (28) Uf λT

The dimensionless combination in Eq. (28) is similar to the Karlovitz number; the relation of these two parameters is discussed in [16].

5. Summary In the present work we perform extensive direct numerical simulations of a flame propagating through an array of turbulent vortices in a two-dimensional tube/channel with adiabatic slip walls and with both tube ends open. The flow imitated statistically stationary turbulent burning for different values of vortex intensity and size controlled by the channel width. The two cases of a single-mode vortex array and a multimode Kolmogorov turbulent spectrum were considered. We have shown that the scaled turbulent flame velocity Uw /Uf depends strongly on the scaled rmsvelocity of the vortex flow Urms /Uf and on the size of the turbulent vortices. The dependence on the vortex rms-velocity is almost linear for sufficiently high vortex intensity, which agrees well with the general belief and the theoretical predictions. Besides, the turbulent flame speed increases with the vortex size approximately as D 2/3 . Such behavior is in agreement with previous simulations [20], but we are not aware of any theoretical work predicting a similar tendency. Thus, the obtained property of the turbulent flame velocity

V. Akkerman et al. / Combustion and Flame 151 (2007) 452–471

makes a challenge for the theory. The main tendencies for the flame speed are put together in an empirical formula, Eq. (26). In the case of Kolmogorov spectrum of vortices we have obtained a noticeably smaller flame speed in comparison with a singlemode array. We show that the flame speed depends weakly on the thermal expansion of the burning matter within the domain of realistic expansion factors Θ = 5–8. We also demonstrated that the flame interaction with a single vortex (or a vortex pair) leads to results quantitatively different from those obtained for the statistically stationary turbulent burning. Comparison to the experimental data [13] demonstrated that small-scale turbulence is not the only effect, which influence flame speed in the experimental flows. Other large-scale processes, such as the DL instability and flame interaction with viscous walls, also make a considerable contribution to the flame speed. Their role may be even stronger than the contribution of external turbulence. On the contrary, if we limit the study by small-scale effects only, then the DL instability is important only in the case of sufficiently small turbulent intensity. Acknowledgments The authors are grateful to Mikhail Ivanov for useful discussions. We also thank Arkady Petchenko and Dmitry Maksimov for help in the presentation of some results. This work has been supported in part by the Swedish Research Council (VR) and by the Kempe Foundation. References [1] F.A. Williams, Combustion Theory, Benjamin, Redwood City, CA, 1985. [2] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, second ed., Edwards, Ann Arbor, MI, 2005. [3] P. Clavin, F.A. Williams, J. Fluid Mech. 90 (1979) 589. [4] V. Yakhot, Combust. Sci. Technol. 60 (1988) 191. [5] A. Pocheau, Phys. Rev. E 49 (1994) 1109. [6] R.C. Aldredge, Combust. Flame 106 (1996) 29. [7] Wm.T. Ashurst, Combust. Theory Modelling 4 (2000) 99. [8] L. Kagan, G. Sivashinsky, Combust. Flame 120 (2000) 222. [9] B. Denet, Combust. Theory Modelling 5 (2001) 85. [10] V. Bychkov, B. Denet, Combust. Theory Modelling 6 (2002) 209. [11] R.G. Abdel-Gayed, D. Bradley, M. Lawes, Proc. R. Soc. London Ser. A 414 (1987) 389.

471

[12] R.C. Aldredge, V. Vaezi, P.D. Ronney, Combust. Flame 115 (1998) 395. [13] T.W. Lee, S.J. Lee, Combust. Flame 132 (2003) 492. [14] S.A. Filatyev, J.F. Driscoll, C.D. Carter, J.M. Donbar, Combust. Flame 141 (2005) 1. [15] V.R. Savarianandam, C.J. Lawn, Combust. Flame 146 (2006) 1. [16] V. Bychkov, Phys. Rev. E 68 (2003) 066304. [17] V. Akkerman, V. Bychkov, Combust. Theory Modelling 9 (2005) 323. [18] V. Akkerman, V. Bychkov, Combust. Expl. Shock Waves 41 (2005) 363. [19] V. Bychkov, A. Petchenko, V. Akkerman, Combust. Sci. Technol. 179 (2007) 137. [20] O. Colin, F. Ducros, D. Veynante, T. Poinsot, Phys. Fluids 12 (2000) 1843. [21] V. Akkerman, V. Bychkov, Combust. Theory Modelling 7 (2003) 767. [22] G. Searby, P. Clavin, Combust. Sci. Technol. 46 (1986) 167. [23] H. Kobayashi, T. Kawahata, K. Seyama, T. Fujimari, J.S. Kim, Proc. Combust. Inst. 29 (1998) 1793. [24] L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.U. Schildmacher, W. Krebs, B. Prade, P. Kaufmann, D. Veynante, Combust. Flame 137 (2004) 489. [25] C. Meneveau, T. Poinsot, Combust. Flame 86 (1991) 311. [26] T. Poinsot, D. Veynante, S. Candel, J. Fluid Mech. 228 (1991) 561. [27] T. Hasegawa, T. Morooka, S. Nishiki, Combust. Sci. Technol. 150 (2000) 115. [28] P.H. Renard, D. Thevenin, J.C. Rolon, S. Candel, Prog. Energy Combust. Sci. 26 (2000) 225. [29] S. Kadowaki, T. Hasegawa, Prog. Energy Combust. Sci. 31 (2005) 193. [30] P. Pelce, P. Clavin, J. Fluid Mech. 124 (1982) 219. [31] V. Bychkov, S. Golberg, M. Liberman, L.E. Eriksson, Phys. Rev. E 54 (1996) 3713. [32] O. Travnikov, V. Bychkov, M. Liberman, Phys. Rev. E 61 (2000) 468. [33] V. Bychkov, M. Liberman, Phys. Rep. 325 (2000) 115. [34] V. Bychkov, A. Petchenko, V. Akkerman, L.E. Eriksson, Phys. Rev. E 72 (2005) 046307. [35] V. Akkerman, V. Bychkov, A. Petchenko, L.E. Eriksson, Combust. Flame 145 (2006) 675. [36] V. Akkerman, V. Bychkov, A. Petchenko, L.E. Eriksson, Combust. Flame 145 (2006) 206. [37] R. Aldredge, F.A. Williams, J. Fluid Mech. 228 (1991) 487. [38] V. Bychkov, Phys. Fluids 10 (1998) 2669. [39] A.R. Kerstein, W.T. Ashurst, F.A. Williams, Phys. Rev. A 37 (1988) 2728. [40] O.L. Gulder, G.J. Smallwood, Combust. Flame 103 (1995) 107. [41] A. Petchenko, V. Bychkov, L.E. Eriksson, A. Oparin, Combust. Theory Modelling 10 (2006) 581. [42] S. Ishizuka, Prog. Energy Combust. Sci. 28 (2002) 477.