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Self-accelerating flames in long narrow open channels
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Proceedings of the Combustion Institute 35 (2015) 921–928
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Self-accelerating flames in long narrow open channels Vadim N. Kurdyumov a,⇑, Moshe Matalon b a
Department of energy, CIEMAT, Avda. Complutence 40, 28040 Madrid, Spain b University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Available online 26 June 2014
Abstract In this work we extend our earlier asymptotic one-dimensional analysis of flame propagation in long narrow channels open at both ends to two-dimensional flames. The analysis follows two tracks; a multiscale asymptotic study and a full numerical study of the unsteady propagation. We show that during the early stages of propagation the flame accelerates at a nearly constant rate, independent of the channel height. In sufficiently narrow channels, the flame retains a constant acceleration until it reaches the end of the channel, consistent with our earlier work. In wider channels, however, the flame beyond a certain distance begins to accelerate at a nearly-exponential rate, reaching exceedingly large speeds at the end of the channel. The flame self-acceleration arises from the combined effects of gas expansion and lateral confinement. The gas expansion that results from the heat released by the chemical reactions produces a continuous flow of burned gas directed towards the ignition end of the channel. Due to the frictional forces at the walls and, since the pressure at both ends is maintained constant, the gas motion that develops in the burned gas sets a pressure gradient that drives the fresh unburned gas towards the other end of the channel. Stretching out to reach additional fuel, the flame extends towards the fresh mixture propagating faster. And because of lateral confinement, the gas expansion induces large straining on the elongated flame surface that further increases its propagation speed. The asymptotic approximation properly predicts the initial propagation stage, the location within the channel where the sudden acceleration begins and the early stages of the self-accelerating process. The full numerical study confirms and extends the asymptotic results, showing that in long but finite channels premixed flames could self-accelerate reaching velocities that are ten-to-twenty times larger than the laminar flame speed. Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Premixed flames; Accelerating flames; Thermal expansion; Long channels; DDT
1. Introduction Understanding the propagation of premixed flames in narrow channels and tubes is of practical
⇑ Corresponding author. Fax: +34 91 346 6269.
E-mail addresses: [email protected] (V.N. Kurdyumov), [email protected] (M. Matalon).
interest in the design of micro-propulsion devices, has various safety applications and is of great importance in understanding the deflagration-todetonation transition in gases and condensed energetic materials. The early studies [1–3] have recognized that the boundary conditions imposed at the end of the channel have a significant effect on the flame propagation. The objective of this work is to study the propagation of premixed
http://dx.doi.org/10.1016/j.proci.2014.05.082 1540-7489/Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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flames in sufficiently long but narrow channels open at both ends, that allows for the gas to leave the channel freely. In a recent study, an asymptotic analysis based on the assumption that the channel height is smaller than the flame thickness [4] was presented. This simplification led to a one-dimensional formulation that enabled extracting simple results about the flame position and its propagation speed, as well as the overall travel time within the channel. In this work the aforementioned assumption has been removed and the two-dimensional problem is considered along two tracks: a multi-scale asymptotic study that assumes that the channel height is much smaller than its length, in which case the two-dimensional flame structure propagates quasi-steadily throughout the channel, and a full numerical study of the time-dependent problem that validates the asymptotic solution and extends its validity. We show that in the early stages following ignition the flame accelerates at a nearly constant rate, but after reaching a certain distance down the channel the flame suddenly begins to accelerate rapidly, at a nearly-exponential rate, that continues until it reaches the end of the channel. The self-acceleration process results from the combined effects of thermal expansion and lateral confinement. Numerous theoretical [5,6], numerical [7–10] and experimental studies [11–13] discussing dynamical aspects of flame propagation in channels and tubes have been reported. The work most closely related to the present investigation is the numerical simulations reported in [14–16]. In [14,15] rapid acceleration was observed in tubes/ channels closed at the ignition end. In this configuration, however, the burned gas is trapped between the flame and the closed boundary and the expanded gas acts as a piston on the fresh mixture that must leave the channel at the other end. Thus, despite the similarity with our results, it is not clear how the different boundary conditions and the inclusion of compressibility effects, included in their code but filtered out in our study, affect the dynamics. The simulations in [16] carried out in open channels also show rapid acceleration, but details about the flow and pressure fields and about the imposed conditions at the two ends of the tube, were not provided. Neither of these studies, however, provide details about the transition from a nearly-constant to a rapid exponential-like acceleration, nor do they identify the early-stage quasi-steady nature of the flame propagation. 2. General formulation A combustible mixture is contained in a channel of length L and height h, and ignited at time ~t ¼ 0 at its left end; i.e., at ~x ¼ 0 (see Fig. 1). Upon ignition, the diaphragms containing the mixture in the
Fig. 1. Channel and flow configurations illustrating the various length scales associated with the flame propagation. The extent of the flame zone on the order of the diffusion length is much smaller than the length of the channel L.
channel are simultaneously removed and both ends remain open and exposed to a constant (atmospheric) pressure, allowing the gas to leave the channel freely. Of particular interest is to examine the flame propagation in sufficiently long but narrow channels, i.e., L h with the height h on the order of the flame thickness dT DT =S L , where DT is the thermal diffusivity of the mixture and S L the laminar flame speed. Hence a h=dT is an Oð1Þ parameter, but ‘ L=dT 1. For simplicity, we consider here the case of adiabatic walls. The chemical reaction is modeled by a global step of the form F þ O ! P , where F denotes the fuel, O the oxidizer and P the products, and ~ ¼ Bð~ proceeds at a rate x qY F Þð~ qY O Þ exp ~ is the density of the mixture, ðE=RT~ Þ where q Y F ; Y O the mass fractions of fuel and oxidizer, E the activation energy, R the gas constant, and B a pre-exponential factor. For a lean mixture, the oxidizer mass fraction is nearly constant and can be absorbed into B, such that the reaction rate depends only on the mass fraction of the fuel, denoted below by Y. We introduce dimensionless variables (when the same symbols are used the one with the accent “tilde” denotes the dimensional quantity) as follows: u=S L ; x ¼ ~x=dT ; y ¼ ~y =h; t ¼ ~t=ðdT =S L Þ; u ¼ ~ ~=qu ; p ¼ a2 ð~ p patm Þ=qu S 2L ; v ¼ ~v=aS L ; q ¼ q Y ¼ Y~ =Y u h ¼ ðT~ T u Þ=ðT a T u Þ; where T a ¼ T u þ QY u =cp is the adiabatic temperature with Q the total heat release and cp the specific heat (at constant pressure) of the mixture. Assuming constant properties and a low-Mach number approximation, the (dimensionless) governing equations are: qt þ ðq uÞx þ ðq vÞy ¼ 0;
1 1 4 1 qðut þ uux þ vuy Þ ¼ 2 px þ Pr 2 uyy þ uxx þ vxy ; a a 3 3 1 1 4 1 qðvt þ uvx þ vvy Þ ¼ 4 py þ Pr 2 vyy þ uxy þ vxx ; a a 3 3 1 qðht þ uhx þ vhx Þ hxx þ 2 hyy ¼ x; a 1 1 Y xx þ 2 Y yy ¼ x; qðY t þ uY x þ vY y Þ Le a q ¼ 1=ð1 þ chÞ;
ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ
V.N. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 35 (2015) 921–928
where x; y are respectively the longitudinal and transverse coordinates (see Fig. 1), u; v are the corresponding velocity components, and p the pressure deviations from the ambient value. The small pressure variations, on the order of the square of the Mach number, are properly neglected in the equation of state. The parameters appearing in these equations include the heat release parameter c ¼ ðT a T u Þ=T u , the Prandtl number Pr ¼ m=DT representing the ratio of the viscous to thermal diffusivities of the mixture (with m the kinematic viscosity), and the Lewis number Le ¼ DT =DF representing the ratio of the thermal diffusivity of the mixture to the mass diffusivity of the fuel DF . The reaction rate x takes the form 2 b2 1þc bðh 1Þ ; x¼ 2 Y exp ð1 þ chÞ=ðc þ 1Þ 2sL Le 1 þ ch where b ¼ EðT a T u Þ=RT 2a is the Zel’dovich number and qb ¼ qu T u =T a is the density of the burned gas. The asymptotic expression for the laminar flame speed qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS L Þasp ¼ 2LeBqu DT =b2 ðqb =qu Þ eE=2RT a ; valid for b ! 1 was introduced as reference and the adjustment factor sL ¼ S L =ðS L Þasp appearing in x is the eigenvalue of the planar adiabatic problem that needs to be computed numerically for any finite b, as discussed in [4]. Equations (1)– (6) must be solved subject to ux ¼ v ¼ hx ¼ Y x ¼ 0 at x ¼ 0; ‘ for 0 < y < 1; u ¼ v ¼ hy ¼ Y y ¼ 0 at y ¼ 0; a for 0 < x < ‘: These are invariably the appropriate conditions when the flame is sufficiently far from the end of the channel. The modifications required when the flame is within a close distance OðdT Þ from either end would, for sufficiently long channels, have a small and negligible effect on the overall propagation. Finally, since following ignition the channel remains open, the pressure at both ends is constant and equal to the ambient pressure, p¼0
at x ¼ 0; ‘ for 0 < y < 1;
ð7Þ
consistent with the assumption that the flow leaving the channel remains parallel to the walls. 3. Multi-scale asymptotic analysis (‘ 1) In addition to the residence time dT =S L introduced above as a unit of time, there exists another time scale L=S L associated with the travel time throughout the entire channel. For long channels, ‘ 1, the variables t and s ¼ t=‘ representing the fast and slow times, respectively, may be treated as independent. It is also convenient to introduce
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a coordinate attached to the flame, n ¼ x ‘xf , with xf ¼ Oð1Þ defined as the location where the reaction rate x is maximum along the symmetry plane y ¼ 1=2. Then @ t in Eqs. (1)–(6) must be replaced by @ t x_ f @ n , where x_ f ¼ dxf =ds. If the flame structure remains quasi-steady, i.e., @ t ¼ 0, then xf ¼ xf ðsÞ and Eqs. (1)–(5) reduce to ½qðu x_ f Þn þ ðqvÞy ¼ 0;
ð8Þ 1 1 4 1 ð9Þ qðu x_ f Þun þ qvuy ¼ 2 pn þ Pr 2 uyy þ unn þ vny ; a a 3 3 1 1 4 1 qðu x_ f Þvn þ qvvy ¼ 4 py þ Pr 2 vyy þ uny þ vnn ; ð10Þ a a 3 3 1 ð11Þ qðu x_ f Þhn þ qvhy hnn þ 2 hyy ¼ x; a 1 ð12Þ qðu x_ f ÞY n þ qvY y Le1 Y nn þ 2 Y yy ¼ x; a
for ‘xf < n < ‘ð1 xf Þ, along with the equation of state (6). The flame zone, which comprises of the zone where heat conduction, mass diffusion and chemical reactions occur spans a relatively small region compare to the entire length of the channel, and separates two hydrodynamic regions at the far right/left, as shown schematically in Fig. 1. The solution of Eqs. (1)–(6) must therefore match, as n ! 1, the induced flow in the hydrodynamic regions. To this end we introduce the scaled coordinate ^ n ¼ n=‘ and the outer (far-field) expansions u þ ; v ¼ ^v þ ; p ¼ ‘^ p0 þ ^ p1 u¼^ ^ b h ¼ h þ ; Y ¼ Y þ : Since axial diffusion becomes negligible in the farfield regions, the state of the gas is constant and uniform and ^v ¼ 0, implying that the flow field is unidirectional and satisfies the equations ^ p0^n ¼ Pr ^ uyy . The resulting Poiseuille p0y ¼ 0 and ^ flows are given by ^ u ¼ 6 U yð1 yÞ
^ p 0^ ¼ 12 Pr U ; n
ð13Þ
where U denote the mean velocity, with the corresponding to ^ n?0, respectively. With the matching conditions uðyÞ as n ! 1 vn 0; un 0; u ^ hn ¼ Y n 0 as n ! 1 h ¼ 0; y 1 as n ! þ1 Equations (8)–(12) constitute an eigenvalue problem for the determination of x_ f , which will be addressed below. A relation exists U þ and U can be obtained by integrating the continuity Eq. (8) throughout the volume comprising of the flame zone, using the boundary conditions along the walls and the matching conditions as n ! 1. One finds
ðc þ 1Þ U þ x_ f ¼ U x_ f ; ð14Þ which is a statement of mass conservation relating the flow ahead and behind the flame characterized
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by U þ and U , respectively. For a given set of parameters, the eigenvalue problem (8)–(12) determines the propagation speed x_ f as a function of U þ , or equivalently U þ ¼ F ð_xf Þ:
ð15Þ
The corresponding one-dimensional problem with U þ ¼ 0 is the classical eigenvalue problem for the determination of the laminar flame speed. Although U þ must be determined by the specific conditions of the problem at hand, the relation (15) effectively determines the propagation speed for a given flow characterized by U þ , which may be considered prescribed, for example by driving the fresh mixture with an appropriate Poiseuille flow. In this sense, the eigenvalue problem (8)– (12) is a generalization of the classical eigenvalue problem for a two-dimensional adiabatic flame. 3.1. Numerical treatment of the eigenvalue problem It is convenient for numerical calculations to adopt a streamfunction-vorticity formulation. With the streamfunction w, defined from qðu x_ f Þ ¼ wy ;
qv ¼ wn ;
and the vorticity f ¼ a2 vn uy , the Navier-Stokes equations reduce to 1 qðu x_ f Þfn þ qv fy ¼ Pr fnn þ 2 fyy þ J ; a a2 ðwn =qÞn þ ðwy =qÞy ¼ f; where J is the vorticity production given by J ¼ ½qðu x_ f y un a2 ½qðu x_ f Þn vn þ ðqvÞy uy a2 ðqvÞn vy : These equations along with (11) and (12), the boundary and matching conditions (properly translated in terms of w and f), were solved numerically to determine, for a given set of parameters, the functional dependence (15). The equations were discretized using a finite difference second-order three-points approximation for space derivatives and solved using a Gauss-Seidel iteration with successive over-relaxation. The calculations reported below were carried out for Le ¼ 1; Pr ¼ 0:72, and with c ¼ 5; b ¼ 10. 3.2. General results of the eigenvalue problem Figure 2 shows the dependence of x_ f on U þ for selected values of channel height a. The interpretation of these results should, at this stage, be regarded in the general context of a flame propagating in a channel with a prescribed flow of mean velocity U þ ; it propagates in the direction of the flow when U þ > 0 and against the flow when
Fig. 2. The propagation speed x_ f for a given U þ , resulting from the solution of the eigenvalue problem Eqs. (8)–(12).
U þ < 0 (flashback), similar to the discussion reported in [5] for constant density flows. The calculated results for small values of a show that x_ f increases linearly with increasing U þ , which agrees extremely well with the analytical expression x_ f 1 þ U þ derived in [4] under the assumption a 1. It is remarkable that this linear dependence remains valid for a as large as 5, provided U þ is not too large. At higher values of a the dependence of x_ f on U þ is no longer linear and a sudden change in slope occurs when the flame transitions from a concave to a convex shape. Since an increase in speed is associated with an increase in flame surface area, the flame must flatten out before changing concavity with x_ f 1 during the transition. The results of Fig. 2 also show that, when subjected to the same mean flow, the flame in a wider channel propagates faster. 3.3. Propagation in a channel open at both ends We now consider the case of a channel open at both ends, in which case U þ is determined by the axial pressure gradients (13) in the hydrodynamic zones. A direct integration yields ( U ð^ xf < ^ n xf Þ n<0 ^p0 ¼ 12 Pr þ ^ xf þ 1Þ ^ < 1 xf ; 0
ð16Þ
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Fig. 3. The dependence of the propagation speed (in units of S L ) on the flame position for various channel heights.
Equations (14) and (16) together with the eigenvalue relation (15) yields a relation for the flame position in terms of its propagation speed xf ¼
1 F ð_xf Þ : c x_ f F ð_xf Þ
ð17Þ
For small values of a, the linear relation F ð_xf Þ ¼ x_ f 1 deduced from Fig. 2 yields x_ f ¼ 1 þ cxf validating the result derived in [4] under the assumption a 1. Figure 3 shows the dependence of the propagation speed on the flame position based on the asymptotic relation (17), for several values of a. For sufficiently small values, a < ac , where ac ¼ 5:355, the flame accelerates at a constant rate throughout the channel. In wider channels, the constant acceleration rate holds only initially and beyond the threshold position, which decreases with increasing a, the flame undergoes a very rapid exponential-like increase in speed. To understand the physical mechanism responsible for the sudden fast acceleration we examine next the solution in the vicinity of the flame in more details. Figure 4 shows the flow and combustion fields in a channel of height a ¼ 5 for increasing values of U þ , corresponding to increasing values of xf . The flow field is depicted by streamlines, drawn relative to the walls in the top half of the channel and relative to the flame in the bottom half of the channel. The color shades in the top half of the channel correspond to variations in values of the reaction rate x and those in the bottom half of the channel to variations in the scaled temperature h; both illustrating the flame shape and the zone where variations in x and h are most significant.
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Fig. 4. Illustration of the flame (based on reaction rate contours) and flow field (streamlines) in the vicinity of the flame for propagation in a channel of height a ¼ 5 for three values of U þ . The solid curves correspond to streamlines and the color scheme to variations in reaction rate or temperature, as explained in the text.
Far to the right/left of the flame (the hydrodynamic zones) the flow becomes unidirectional and the temperature (and hence the density) takes constant values, T u and T a in the unburned/ burned regions, respectively. Unlike a planar unconfined flame, where the heated gas behind the reaction zone expands and moves uniformly in a direction opposite to the propagating flame, the flow in the vicinity of a curved confined flame is more complex as the streamline pattern indicates. Because of its inability to move along the walls, due to friction, the heated gas behind the flame front expands primarily in the axial direction towards the center of the channel. Indeed, mass conservation implies that near the walls u 0 and vy ðqx =qÞ_xf , which is most significant in the flame zone where the temperature, and hence the density gradients are the largest, and is zero away from the flame zone on both sides. The confined flame, therefore, acts as a source of mass emanating from the walls towards the center of the channel where the flow is redirected along the axis on both sides due to the pressure gradients established to accommodate the atmospheric pressure at both ends. The “source” flow near the flame and the Poiseuille flows in the hydrodynamic zones ahead and behind the flame are clearly seen in the upper part of the channel in Fig. 4. In contrast, an observer moving with the flame always sees an inflow of fresh gas, as shown in the figure in the lower part of the channel.
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stretched resulting in a continuous increase in propagation speed. The axial velocity u at the mid-plane between the centerline and the wall, shown in Fig. 5, clearly indicates the increase in strain rate (proportional to the slope) experienced by the flame as it propagates down the channel. This mechanism explains the sudden exponential-like increase in speed identified in the response curves of Fig. 3. Thus, in very narrow channels, a < ac , the weakly curved flame accelerates when propagating through the combustible mixture reaching a speed of approximately six times the laminar flame speed at the end of the channel, as predicted in [4]. In wider channels the flame at the early stage of propagation develops a highly curved surface that is constantly stretched by the induced flow resulting from the gas expansion and accelerates rapidly as it propagates towards the end of the channel. As a final comment, we note that each response curve Fig. 3 exhibits a turning point (marked by the symbol o on the various graphs) at a finite position x, marking conditions where the quasi-steady assumption fails, requiring a reevaluation of the solution as discussed next. Fig. 5. Illustration of (i) the flame (based on reaction rate contours) and flow field (streamlines) similar to Fig. 4, but in a channel of height a ¼ 10 at two different positions, and (ii) the axial strain rate experienced. þ
Figure 4 shows that at low values of U the flame is slightly concave towards the fresh mixture and propagates at a speed close to the laminar flame speed (_xf 1). The flame propagates to the right and the combustion products move in the opposite direction with velocity U c_xf . The flame transitions to a convex shape when U þ is slightly larger than one and remains convex towards the fresh mixture when U þ further increases. The three illustrations in this figure correspond to the flame being at xf 0:02; 0:2; 0:9, and propagating with speed x_ f 1:1; 2; 6, respectively. Figure 5 presents similar results but in a channel of height a ¼ 10. The three illustrations correspond to flames at xf 0:2; 0:4; 0:5 propagating at a speed x_ f 2:013; 7:26; 14:1, respectively. (These values are marked by a the symbol M on the corresponding graph of Fig. 2.) Note that at xf ¼ 0:2 the flame has the same shape and travels slightly faster than in the narrower channel. It is significantly faster when it reaches xf ¼ 0:4, and is 15 times larger than laminar flame speed at mid-channel. A careful observation of the figures show that as the convex flame becomes elongated towards the center of the channel, the induced flow resulting from gas expansion (source-like emanating from the walls) is most significant near the flame trailing edge where the density gradients are the largest. The main surface is increasingly
4. Numerical solution In this section we examine the flame evolution in a long but finite channel of length ‘. To this end, the complete two-dimensional timedependent Eqs. (1)–(6) were solved for selected values of the parameters. The time marching calculations were carried out in a similar way as proposed in [17], where the velocity was decomposed into irrotational and solenoidal components and solved respectively by introducing potential and stream-like functions. The pressure has been eliminated by solving the vorticity instead of the momentum equations and conditions (7) satisfied by imposing the constraint Z ‘ fy y¼0 dx ¼ 0 0
obtained when Eq. (2) evaluated along the walls 4 px þ Pr fy þ a2 vxy ¼ 0 3 is integrated along the channel and use is made of the boundary conditions at x ¼ 0; ‘. The initial conditions adopted correspond to a hot spot near the left end of the channel in a mixture that is otherwise at rest. The position of the flame front xf ðtÞ was defined as before as the location where the reaction rate x reaches its maximum value along the mid-plane and the corresponding propagation speed x_ f was subsequently determined by differentiation.
V.N. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 35 (2015) 921–928
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Fig. 8. Numerical solution of a flame propagating in a channel of height a ¼ 5 and length ‘ ¼ 100. Shown are selected reaction rate contours (by different colors) at different locations within the channel.
Fig. 6. The propagation speed (in units of S L ) at different location for two channel heights, a ¼ 5; 10, and various lengths.
Fig. 9. Numerical solution of a flame propagating in a channel of height a ¼ 10 and length ‘ ¼ 200. Shown are selected reaction rate contours (by different colors) at different locations within the channel.
Fig. 7. Pressure variations across a channel of height a ¼ 5 and length ‘ ¼ 100 at different times.
Figure 6 shows the dependence of the propagation speed on flame position for two values of a and several values of ‘. The dashed curve in each case corresponds to the asymptotic results for ‘ 1. The effect of the initial conditions are seen to fade away after a short time, with the flame accelerating during the initial period at a nearly constant rate. When a ¼ 5; x_ f varies almost linearly throughout the channel and there is an excellent agreement between the asymptotic and the exact solutions. When a ¼ 10, the initial period of constant acceleration extends to nearly half of the channel length after which the propagation speed changes in an abrupt way with the flame
accelerating towards the end of the channel at a nearly-exponential rate. The symbol o, which denotes the position where the sudden change in acceleration begins, decreases slightly with increasing ‘ and tends to the asymptotic value when ‘ ! 1. Although the asymptotic solution predicts correctly the flame behavior until the rapid change in propagation speed occurs, the quasi-steady approximation fails thereafter. The prediction of rapid acceleration that leads to propagation speeds ten-to-twenty larger than the laminar flame speed are consistent with the numerical simulations reported in [14,15], carried out for propagation from a closed end of the channel, and with the Shelkin’s proposition [18] that flames accelerate in smooth tubes due to the non-slip boundary conditions at the walls through a consequent increase in the flame surface. Figure 7 shows the pressure variations in a channel of width a ¼ 5 and length ‘ ¼ 100 at different times. One observes the nearly constant pressure gradients in the hydrodynamic zones that are responsible for the Poiseuille flows leaving the channel from both ends, and the slight drop in pressure across the relative thinner flame. Figures 8 and 9 show the propagating flame at different locations for two different cases (due to the symmetry the solution is only shown in the lower half
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part of the channel). Note that the x-axis has been shifted to capture the flame as it propagates to the right end of the channel. In the narrow channel the flame curvature varies only slightly during the entire propagation, while in the wider channel the flame becomes elongated towards the center of the channel. Overall, the numerical results validate and complete the general results based on the multi-scale approach reported above.
Acknowledgments M.M. acknowledges partial support by the National Science Foundation under grant CBET-1067259. V.K. acknowledged the support of Spanish MEC under projects #ENE201127686-C02-01 and #CSD2011-0001.
References 5. Conclusions Premixed flames propagating in narrow open channels, can reach exceedingly large speeds if the channel is sufficiently long. During the early stages of propagation the flame accelerates at a nearly constant rate, but begins to rapidly accelerate when reaching a certain distance down the channel at a nearly-exponentially rate that continues until the flame reaches the end of the channel. The physical mechanism responsible for the flame self-acceleration is the combined effects of gas expansion and lateral confinement. Upon ignition, the heat released by the chemical reactions produces a continuous flow of burned gas towards the ignition end, driven by a pressure gradient directed upstream. Since the pressure at both ends is maintained constant, frictional forces at the walls sets a pressure gradient downstream that drives the fresh unburned gas towards the other end of the channel. Stretching out for additional fuel, the flame extends towards the fresh mixture and propagates faster, and because of lateral confinement the gas expansion induces large straining on the elongated flame surface that further increase its propagation speed. In addition to clarifying the mechanism for self acceleration, the asymptotic analysis identifies the quasi-steady nature of the flame propagation during the initial stages, the location within the channel where the sudden exponential-like acceleration begins and the start of the self-accelerating process. The full numerical study confirms the asymptotic results and extends their validity illustrating that the self-acceleration process proceeds until the flame reaches the far end of the channel.
[1] E. Mallard, H.L. Le Chatelier, Ann. Mines (1883) 274–388. [2] W. Mason, R.V. Wheeler, J. Chem. Soc. Trans. 117 (1920) 36–47. [3] H. Gue´noche, in: G. Markstein (Ed.), Unsteady Flame Propagation, Macmillan, 1964, pp. 107–181, Chapter E. [4] V.N. Kurdyumov, M. Matalon, Proc. Combust. Inst. 34 (2013) 865–872. [5] J. Daou, M. Matalon, Combust. Flame 124 (2001) 337–349. [6] J. Daou, M. Matalon, Combust. Flame 128 (2002) 321–339. [7] V.N. Kurdyumov, E. Ferna´ndez-Tarrazo, Combust. Flame 128 (2002) 382–394. [8] V. Bychkov, V. Akkerman, G. Fru, A. Petchenko, L-E. Eriksson, Combust. Flame 150 (2007) 263–276. [9] G. Pizza, C.E. Frouzakis, J. Mantzaras, A.G. Tomboulides, K. Boulouchos, Combust. Flame 152 (2007) 433–450. [10] M. Short, D.A. Kessler, J. Fluid Mech. 638 (2009) 305–337. [11] C.J. Evans, D.C. Kyritsis, Proc. Combust. Inst. 32 (2009) 3107–3114. [12] T.A. Connelly, D.C. Kyritsis, Proceedings of the 8th US National Combustion Meeting, Paper 070LT0193 (2013). [13] Y. Yanez, M. Kuznetsov, V. Bykov, Proc. 24th ICDERS (2013). [14] J.D. Ott, E.S. Oran, J.D. Anderson, AIAA J. 41 (2003) 1391–1396. [15] V.N. Gamezo, E.S. Oran, AIAA J. 44 (2006) 329– 336. [16] V. Akkerman, C.K. Law, V. Bychkov, L-E. Eriksson, Phys. Fluids 22 (2010) 053606. [17] V.N. Kurdyumov, M. Matalon, Proc. Combust. Inst. 32 (2009) 1107–1115. [18] K.I. Shelkin, J. Exp. Theor. Phys. 10 (1940) 823– 827.
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Proceedings of the Combustion Institute 35 (2015) 921–928
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Self-accelerating flames in long narrow open channels Vadim N. Kurdyumov a,⇑, Moshe Matalon b a
Department of energy, CIEMAT, Avda. Complutence 40, 28040 Madrid, Spain b University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Available online 26 June 2014
Abstract In this work we extend our earlier asymptotic one-dimensional analysis of flame propagation in long narrow channels open at both ends to two-dimensional flames. The analysis follows two tracks; a multiscale asymptotic study and a full numerical study of the unsteady propagation. We show that during the early stages of propagation the flame accelerates at a nearly constant rate, independent of the channel height. In sufficiently narrow channels, the flame retains a constant acceleration until it reaches the end of the channel, consistent with our earlier work. In wider channels, however, the flame beyond a certain distance begins to accelerate at a nearly-exponential rate, reaching exceedingly large speeds at the end of the channel. The flame self-acceleration arises from the combined effects of gas expansion and lateral confinement. The gas expansion that results from the heat released by the chemical reactions produces a continuous flow of burned gas directed towards the ignition end of the channel. Due to the frictional forces at the walls and, since the pressure at both ends is maintained constant, the gas motion that develops in the burned gas sets a pressure gradient that drives the fresh unburned gas towards the other end of the channel. Stretching out to reach additional fuel, the flame extends towards the fresh mixture propagating faster. And because of lateral confinement, the gas expansion induces large straining on the elongated flame surface that further increases its propagation speed. The asymptotic approximation properly predicts the initial propagation stage, the location within the channel where the sudden acceleration begins and the early stages of the self-accelerating process. The full numerical study confirms and extends the asymptotic results, showing that in long but finite channels premixed flames could self-accelerate reaching velocities that are ten-to-twenty times larger than the laminar flame speed. Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Premixed flames; Accelerating flames; Thermal expansion; Long channels; DDT
1. Introduction Understanding the propagation of premixed flames in narrow channels and tubes is of practical
⇑ Corresponding author. Fax: +34 91 346 6269.
E-mail addresses: [email protected] (V.N. Kurdyumov), [email protected] (M. Matalon).
interest in the design of micro-propulsion devices, has various safety applications and is of great importance in understanding the deflagration-todetonation transition in gases and condensed energetic materials. The early studies [1–3] have recognized that the boundary conditions imposed at the end of the channel have a significant effect on the flame propagation. The objective of this work is to study the propagation of premixed
http://dx.doi.org/10.1016/j.proci.2014.05.082 1540-7489/Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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flames in sufficiently long but narrow channels open at both ends, that allows for the gas to leave the channel freely. In a recent study, an asymptotic analysis based on the assumption that the channel height is smaller than the flame thickness [4] was presented. This simplification led to a one-dimensional formulation that enabled extracting simple results about the flame position and its propagation speed, as well as the overall travel time within the channel. In this work the aforementioned assumption has been removed and the two-dimensional problem is considered along two tracks: a multi-scale asymptotic study that assumes that the channel height is much smaller than its length, in which case the two-dimensional flame structure propagates quasi-steadily throughout the channel, and a full numerical study of the time-dependent problem that validates the asymptotic solution and extends its validity. We show that in the early stages following ignition the flame accelerates at a nearly constant rate, but after reaching a certain distance down the channel the flame suddenly begins to accelerate rapidly, at a nearly-exponential rate, that continues until it reaches the end of the channel. The self-acceleration process results from the combined effects of thermal expansion and lateral confinement. Numerous theoretical [5,6], numerical [7–10] and experimental studies [11–13] discussing dynamical aspects of flame propagation in channels and tubes have been reported. The work most closely related to the present investigation is the numerical simulations reported in [14–16]. In [14,15] rapid acceleration was observed in tubes/ channels closed at the ignition end. In this configuration, however, the burned gas is trapped between the flame and the closed boundary and the expanded gas acts as a piston on the fresh mixture that must leave the channel at the other end. Thus, despite the similarity with our results, it is not clear how the different boundary conditions and the inclusion of compressibility effects, included in their code but filtered out in our study, affect the dynamics. The simulations in [16] carried out in open channels also show rapid acceleration, but details about the flow and pressure fields and about the imposed conditions at the two ends of the tube, were not provided. Neither of these studies, however, provide details about the transition from a nearly-constant to a rapid exponential-like acceleration, nor do they identify the early-stage quasi-steady nature of the flame propagation. 2. General formulation A combustible mixture is contained in a channel of length L and height h, and ignited at time ~t ¼ 0 at its left end; i.e., at ~x ¼ 0 (see Fig. 1). Upon ignition, the diaphragms containing the mixture in the
Fig. 1. Channel and flow configurations illustrating the various length scales associated with the flame propagation. The extent of the flame zone on the order of the diffusion length is much smaller than the length of the channel L.
channel are simultaneously removed and both ends remain open and exposed to a constant (atmospheric) pressure, allowing the gas to leave the channel freely. Of particular interest is to examine the flame propagation in sufficiently long but narrow channels, i.e., L h with the height h on the order of the flame thickness dT DT =S L , where DT is the thermal diffusivity of the mixture and S L the laminar flame speed. Hence a h=dT is an Oð1Þ parameter, but ‘ L=dT 1. For simplicity, we consider here the case of adiabatic walls. The chemical reaction is modeled by a global step of the form F þ O ! P , where F denotes the fuel, O the oxidizer and P the products, and ~ ¼ Bð~ proceeds at a rate x qY F Þð~ qY O Þ exp ~ is the density of the mixture, ðE=RT~ Þ where q Y F ; Y O the mass fractions of fuel and oxidizer, E the activation energy, R the gas constant, and B a pre-exponential factor. For a lean mixture, the oxidizer mass fraction is nearly constant and can be absorbed into B, such that the reaction rate depends only on the mass fraction of the fuel, denoted below by Y. We introduce dimensionless variables (when the same symbols are used the one with the accent “tilde” denotes the dimensional quantity) as follows: u=S L ; x ¼ ~x=dT ; y ¼ ~y =h; t ¼ ~t=ðdT =S L Þ; u ¼ ~ ~=qu ; p ¼ a2 ð~ p patm Þ=qu S 2L ; v ¼ ~v=aS L ; q ¼ q Y ¼ Y~ =Y u h ¼ ðT~ T u Þ=ðT a T u Þ; where T a ¼ T u þ QY u =cp is the adiabatic temperature with Q the total heat release and cp the specific heat (at constant pressure) of the mixture. Assuming constant properties and a low-Mach number approximation, the (dimensionless) governing equations are: qt þ ðq uÞx þ ðq vÞy ¼ 0;
1 1 4 1 qðut þ uux þ vuy Þ ¼ 2 px þ Pr 2 uyy þ uxx þ vxy ; a a 3 3 1 1 4 1 qðvt þ uvx þ vvy Þ ¼ 4 py þ Pr 2 vyy þ uxy þ vxx ; a a 3 3 1 qðht þ uhx þ vhx Þ hxx þ 2 hyy ¼ x; a 1 1 Y xx þ 2 Y yy ¼ x; qðY t þ uY x þ vY y Þ Le a q ¼ 1=ð1 þ chÞ;
ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ
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where x; y are respectively the longitudinal and transverse coordinates (see Fig. 1), u; v are the corresponding velocity components, and p the pressure deviations from the ambient value. The small pressure variations, on the order of the square of the Mach number, are properly neglected in the equation of state. The parameters appearing in these equations include the heat release parameter c ¼ ðT a T u Þ=T u , the Prandtl number Pr ¼ m=DT representing the ratio of the viscous to thermal diffusivities of the mixture (with m the kinematic viscosity), and the Lewis number Le ¼ DT =DF representing the ratio of the thermal diffusivity of the mixture to the mass diffusivity of the fuel DF . The reaction rate x takes the form 2 b2 1þc bðh 1Þ ; x¼ 2 Y exp ð1 þ chÞ=ðc þ 1Þ 2sL Le 1 þ ch where b ¼ EðT a T u Þ=RT 2a is the Zel’dovich number and qb ¼ qu T u =T a is the density of the burned gas. The asymptotic expression for the laminar flame speed qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðS L Þasp ¼ 2LeBqu DT =b2 ðqb =qu Þ eE=2RT a ; valid for b ! 1 was introduced as reference and the adjustment factor sL ¼ S L =ðS L Þasp appearing in x is the eigenvalue of the planar adiabatic problem that needs to be computed numerically for any finite b, as discussed in [4]. Equations (1)– (6) must be solved subject to ux ¼ v ¼ hx ¼ Y x ¼ 0 at x ¼ 0; ‘ for 0 < y < 1; u ¼ v ¼ hy ¼ Y y ¼ 0 at y ¼ 0; a for 0 < x < ‘: These are invariably the appropriate conditions when the flame is sufficiently far from the end of the channel. The modifications required when the flame is within a close distance OðdT Þ from either end would, for sufficiently long channels, have a small and negligible effect on the overall propagation. Finally, since following ignition the channel remains open, the pressure at both ends is constant and equal to the ambient pressure, p¼0
at x ¼ 0; ‘ for 0 < y < 1;
ð7Þ
consistent with the assumption that the flow leaving the channel remains parallel to the walls. 3. Multi-scale asymptotic analysis (‘ 1) In addition to the residence time dT =S L introduced above as a unit of time, there exists another time scale L=S L associated with the travel time throughout the entire channel. For long channels, ‘ 1, the variables t and s ¼ t=‘ representing the fast and slow times, respectively, may be treated as independent. It is also convenient to introduce
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a coordinate attached to the flame, n ¼ x ‘xf , with xf ¼ Oð1Þ defined as the location where the reaction rate x is maximum along the symmetry plane y ¼ 1=2. Then @ t in Eqs. (1)–(6) must be replaced by @ t x_ f @ n , where x_ f ¼ dxf =ds. If the flame structure remains quasi-steady, i.e., @ t ¼ 0, then xf ¼ xf ðsÞ and Eqs. (1)–(5) reduce to ½qðu x_ f Þn þ ðqvÞy ¼ 0;
ð8Þ 1 1 4 1 ð9Þ qðu x_ f Þun þ qvuy ¼ 2 pn þ Pr 2 uyy þ unn þ vny ; a a 3 3 1 1 4 1 qðu x_ f Þvn þ qvvy ¼ 4 py þ Pr 2 vyy þ uny þ vnn ; ð10Þ a a 3 3 1 ð11Þ qðu x_ f Þhn þ qvhy hnn þ 2 hyy ¼ x; a 1 ð12Þ qðu x_ f ÞY n þ qvY y Le1 Y nn þ 2 Y yy ¼ x; a
for ‘xf < n < ‘ð1 xf Þ, along with the equation of state (6). The flame zone, which comprises of the zone where heat conduction, mass diffusion and chemical reactions occur spans a relatively small region compare to the entire length of the channel, and separates two hydrodynamic regions at the far right/left, as shown schematically in Fig. 1. The solution of Eqs. (1)–(6) must therefore match, as n ! 1, the induced flow in the hydrodynamic regions. To this end we introduce the scaled coordinate ^ n ¼ n=‘ and the outer (far-field) expansions u þ ; v ¼ ^v þ ; p ¼ ‘^ p0 þ ^ p1 u¼^ ^ b h ¼ h þ ; Y ¼ Y þ : Since axial diffusion becomes negligible in the farfield regions, the state of the gas is constant and uniform and ^v ¼ 0, implying that the flow field is unidirectional and satisfies the equations ^ p0^n ¼ Pr ^ uyy . The resulting Poiseuille p0y ¼ 0 and ^ flows are given by ^ u ¼ 6 U yð1 yÞ
^ p 0^ ¼ 12 Pr U ; n
ð13Þ
where U denote the mean velocity, with the corresponding to ^ n?0, respectively. With the matching conditions uðyÞ as n ! 1 vn 0; un 0; u ^ hn ¼ Y n 0 as n ! 1 h ¼ 0; y 1 as n ! þ1 Equations (8)–(12) constitute an eigenvalue problem for the determination of x_ f , which will be addressed below. A relation exists U þ and U can be obtained by integrating the continuity Eq. (8) throughout the volume comprising of the flame zone, using the boundary conditions along the walls and the matching conditions as n ! 1. One finds
ðc þ 1Þ U þ x_ f ¼ U x_ f ; ð14Þ which is a statement of mass conservation relating the flow ahead and behind the flame characterized
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by U þ and U , respectively. For a given set of parameters, the eigenvalue problem (8)–(12) determines the propagation speed x_ f as a function of U þ , or equivalently U þ ¼ F ð_xf Þ:
ð15Þ
The corresponding one-dimensional problem with U þ ¼ 0 is the classical eigenvalue problem for the determination of the laminar flame speed. Although U þ must be determined by the specific conditions of the problem at hand, the relation (15) effectively determines the propagation speed for a given flow characterized by U þ , which may be considered prescribed, for example by driving the fresh mixture with an appropriate Poiseuille flow. In this sense, the eigenvalue problem (8)– (12) is a generalization of the classical eigenvalue problem for a two-dimensional adiabatic flame. 3.1. Numerical treatment of the eigenvalue problem It is convenient for numerical calculations to adopt a streamfunction-vorticity formulation. With the streamfunction w, defined from qðu x_ f Þ ¼ wy ;
qv ¼ wn ;
and the vorticity f ¼ a2 vn uy , the Navier-Stokes equations reduce to 1 qðu x_ f Þfn þ qv fy ¼ Pr fnn þ 2 fyy þ J ; a a2 ðwn =qÞn þ ðwy =qÞy ¼ f; where J is the vorticity production given by J ¼ ½qðu x_ f y un a2 ½qðu x_ f Þn vn þ ðqvÞy uy a2 ðqvÞn vy : These equations along with (11) and (12), the boundary and matching conditions (properly translated in terms of w and f), were solved numerically to determine, for a given set of parameters, the functional dependence (15). The equations were discretized using a finite difference second-order three-points approximation for space derivatives and solved using a Gauss-Seidel iteration with successive over-relaxation. The calculations reported below were carried out for Le ¼ 1; Pr ¼ 0:72, and with c ¼ 5; b ¼ 10. 3.2. General results of the eigenvalue problem Figure 2 shows the dependence of x_ f on U þ for selected values of channel height a. The interpretation of these results should, at this stage, be regarded in the general context of a flame propagating in a channel with a prescribed flow of mean velocity U þ ; it propagates in the direction of the flow when U þ > 0 and against the flow when
Fig. 2. The propagation speed x_ f for a given U þ , resulting from the solution of the eigenvalue problem Eqs. (8)–(12).
U þ < 0 (flashback), similar to the discussion reported in [5] for constant density flows. The calculated results for small values of a show that x_ f increases linearly with increasing U þ , which agrees extremely well with the analytical expression x_ f 1 þ U þ derived in [4] under the assumption a 1. It is remarkable that this linear dependence remains valid for a as large as 5, provided U þ is not too large. At higher values of a the dependence of x_ f on U þ is no longer linear and a sudden change in slope occurs when the flame transitions from a concave to a convex shape. Since an increase in speed is associated with an increase in flame surface area, the flame must flatten out before changing concavity with x_ f 1 during the transition. The results of Fig. 2 also show that, when subjected to the same mean flow, the flame in a wider channel propagates faster. 3.3. Propagation in a channel open at both ends We now consider the case of a channel open at both ends, in which case U þ is determined by the axial pressure gradients (13) in the hydrodynamic zones. A direct integration yields ( U ð^ xf < ^ n xf Þ n<0 ^p0 ¼ 12 Pr þ ^ xf þ 1Þ ^ < 1 xf ; 0
ð16Þ
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Fig. 3. The dependence of the propagation speed (in units of S L ) on the flame position for various channel heights.
Equations (14) and (16) together with the eigenvalue relation (15) yields a relation for the flame position in terms of its propagation speed xf ¼
1 F ð_xf Þ : c x_ f F ð_xf Þ
ð17Þ
For small values of a, the linear relation F ð_xf Þ ¼ x_ f 1 deduced from Fig. 2 yields x_ f ¼ 1 þ cxf validating the result derived in [4] under the assumption a 1. Figure 3 shows the dependence of the propagation speed on the flame position based on the asymptotic relation (17), for several values of a. For sufficiently small values, a < ac , where ac ¼ 5:355, the flame accelerates at a constant rate throughout the channel. In wider channels, the constant acceleration rate holds only initially and beyond the threshold position, which decreases with increasing a, the flame undergoes a very rapid exponential-like increase in speed. To understand the physical mechanism responsible for the sudden fast acceleration we examine next the solution in the vicinity of the flame in more details. Figure 4 shows the flow and combustion fields in a channel of height a ¼ 5 for increasing values of U þ , corresponding to increasing values of xf . The flow field is depicted by streamlines, drawn relative to the walls in the top half of the channel and relative to the flame in the bottom half of the channel. The color shades in the top half of the channel correspond to variations in values of the reaction rate x and those in the bottom half of the channel to variations in the scaled temperature h; both illustrating the flame shape and the zone where variations in x and h are most significant.
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Fig. 4. Illustration of the flame (based on reaction rate contours) and flow field (streamlines) in the vicinity of the flame for propagation in a channel of height a ¼ 5 for three values of U þ . The solid curves correspond to streamlines and the color scheme to variations in reaction rate or temperature, as explained in the text.
Far to the right/left of the flame (the hydrodynamic zones) the flow becomes unidirectional and the temperature (and hence the density) takes constant values, T u and T a in the unburned/ burned regions, respectively. Unlike a planar unconfined flame, where the heated gas behind the reaction zone expands and moves uniformly in a direction opposite to the propagating flame, the flow in the vicinity of a curved confined flame is more complex as the streamline pattern indicates. Because of its inability to move along the walls, due to friction, the heated gas behind the flame front expands primarily in the axial direction towards the center of the channel. Indeed, mass conservation implies that near the walls u 0 and vy ðqx =qÞ_xf , which is most significant in the flame zone where the temperature, and hence the density gradients are the largest, and is zero away from the flame zone on both sides. The confined flame, therefore, acts as a source of mass emanating from the walls towards the center of the channel where the flow is redirected along the axis on both sides due to the pressure gradients established to accommodate the atmospheric pressure at both ends. The “source” flow near the flame and the Poiseuille flows in the hydrodynamic zones ahead and behind the flame are clearly seen in the upper part of the channel in Fig. 4. In contrast, an observer moving with the flame always sees an inflow of fresh gas, as shown in the figure in the lower part of the channel.
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stretched resulting in a continuous increase in propagation speed. The axial velocity u at the mid-plane between the centerline and the wall, shown in Fig. 5, clearly indicates the increase in strain rate (proportional to the slope) experienced by the flame as it propagates down the channel. This mechanism explains the sudden exponential-like increase in speed identified in the response curves of Fig. 3. Thus, in very narrow channels, a < ac , the weakly curved flame accelerates when propagating through the combustible mixture reaching a speed of approximately six times the laminar flame speed at the end of the channel, as predicted in [4]. In wider channels the flame at the early stage of propagation develops a highly curved surface that is constantly stretched by the induced flow resulting from the gas expansion and accelerates rapidly as it propagates towards the end of the channel. As a final comment, we note that each response curve Fig. 3 exhibits a turning point (marked by the symbol o on the various graphs) at a finite position x, marking conditions where the quasi-steady assumption fails, requiring a reevaluation of the solution as discussed next. Fig. 5. Illustration of (i) the flame (based on reaction rate contours) and flow field (streamlines) similar to Fig. 4, but in a channel of height a ¼ 10 at two different positions, and (ii) the axial strain rate experienced. þ
Figure 4 shows that at low values of U the flame is slightly concave towards the fresh mixture and propagates at a speed close to the laminar flame speed (_xf 1). The flame propagates to the right and the combustion products move in the opposite direction with velocity U c_xf . The flame transitions to a convex shape when U þ is slightly larger than one and remains convex towards the fresh mixture when U þ further increases. The three illustrations in this figure correspond to the flame being at xf 0:02; 0:2; 0:9, and propagating with speed x_ f 1:1; 2; 6, respectively. Figure 5 presents similar results but in a channel of height a ¼ 10. The three illustrations correspond to flames at xf 0:2; 0:4; 0:5 propagating at a speed x_ f 2:013; 7:26; 14:1, respectively. (These values are marked by a the symbol M on the corresponding graph of Fig. 2.) Note that at xf ¼ 0:2 the flame has the same shape and travels slightly faster than in the narrower channel. It is significantly faster when it reaches xf ¼ 0:4, and is 15 times larger than laminar flame speed at mid-channel. A careful observation of the figures show that as the convex flame becomes elongated towards the center of the channel, the induced flow resulting from gas expansion (source-like emanating from the walls) is most significant near the flame trailing edge where the density gradients are the largest. The main surface is increasingly
4. Numerical solution In this section we examine the flame evolution in a long but finite channel of length ‘. To this end, the complete two-dimensional timedependent Eqs. (1)–(6) were solved for selected values of the parameters. The time marching calculations were carried out in a similar way as proposed in [17], where the velocity was decomposed into irrotational and solenoidal components and solved respectively by introducing potential and stream-like functions. The pressure has been eliminated by solving the vorticity instead of the momentum equations and conditions (7) satisfied by imposing the constraint Z ‘ fy y¼0 dx ¼ 0 0
obtained when Eq. (2) evaluated along the walls 4 px þ Pr fy þ a2 vxy ¼ 0 3 is integrated along the channel and use is made of the boundary conditions at x ¼ 0; ‘. The initial conditions adopted correspond to a hot spot near the left end of the channel in a mixture that is otherwise at rest. The position of the flame front xf ðtÞ was defined as before as the location where the reaction rate x reaches its maximum value along the mid-plane and the corresponding propagation speed x_ f was subsequently determined by differentiation.
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Fig. 8. Numerical solution of a flame propagating in a channel of height a ¼ 5 and length ‘ ¼ 100. Shown are selected reaction rate contours (by different colors) at different locations within the channel.
Fig. 6. The propagation speed (in units of S L ) at different location for two channel heights, a ¼ 5; 10, and various lengths.
Fig. 9. Numerical solution of a flame propagating in a channel of height a ¼ 10 and length ‘ ¼ 200. Shown are selected reaction rate contours (by different colors) at different locations within the channel.
Fig. 7. Pressure variations across a channel of height a ¼ 5 and length ‘ ¼ 100 at different times.
Figure 6 shows the dependence of the propagation speed on flame position for two values of a and several values of ‘. The dashed curve in each case corresponds to the asymptotic results for ‘ 1. The effect of the initial conditions are seen to fade away after a short time, with the flame accelerating during the initial period at a nearly constant rate. When a ¼ 5; x_ f varies almost linearly throughout the channel and there is an excellent agreement between the asymptotic and the exact solutions. When a ¼ 10, the initial period of constant acceleration extends to nearly half of the channel length after which the propagation speed changes in an abrupt way with the flame
accelerating towards the end of the channel at a nearly-exponential rate. The symbol o, which denotes the position where the sudden change in acceleration begins, decreases slightly with increasing ‘ and tends to the asymptotic value when ‘ ! 1. Although the asymptotic solution predicts correctly the flame behavior until the rapid change in propagation speed occurs, the quasi-steady approximation fails thereafter. The prediction of rapid acceleration that leads to propagation speeds ten-to-twenty larger than the laminar flame speed are consistent with the numerical simulations reported in [14,15], carried out for propagation from a closed end of the channel, and with the Shelkin’s proposition [18] that flames accelerate in smooth tubes due to the non-slip boundary conditions at the walls through a consequent increase in the flame surface. Figure 7 shows the pressure variations in a channel of width a ¼ 5 and length ‘ ¼ 100 at different times. One observes the nearly constant pressure gradients in the hydrodynamic zones that are responsible for the Poiseuille flows leaving the channel from both ends, and the slight drop in pressure across the relative thinner flame. Figures 8 and 9 show the propagating flame at different locations for two different cases (due to the symmetry the solution is only shown in the lower half
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part of the channel). Note that the x-axis has been shifted to capture the flame as it propagates to the right end of the channel. In the narrow channel the flame curvature varies only slightly during the entire propagation, while in the wider channel the flame becomes elongated towards the center of the channel. Overall, the numerical results validate and complete the general results based on the multi-scale approach reported above.
Acknowledgments M.M. acknowledges partial support by the National Science Foundation under grant CBET-1067259. V.K. acknowledged the support of Spanish MEC under projects #ENE201127686-C02-01 and #CSD2011-0001.
References 5. Conclusions Premixed flames propagating in narrow open channels, can reach exceedingly large speeds if the channel is sufficiently long. During the early stages of propagation the flame accelerates at a nearly constant rate, but begins to rapidly accelerate when reaching a certain distance down the channel at a nearly-exponentially rate that continues until the flame reaches the end of the channel. The physical mechanism responsible for the flame self-acceleration is the combined effects of gas expansion and lateral confinement. Upon ignition, the heat released by the chemical reactions produces a continuous flow of burned gas towards the ignition end, driven by a pressure gradient directed upstream. Since the pressure at both ends is maintained constant, frictional forces at the walls sets a pressure gradient downstream that drives the fresh unburned gas towards the other end of the channel. Stretching out for additional fuel, the flame extends towards the fresh mixture and propagates faster, and because of lateral confinement the gas expansion induces large straining on the elongated flame surface that further increase its propagation speed. In addition to clarifying the mechanism for self acceleration, the asymptotic analysis identifies the quasi-steady nature of the flame propagation during the initial stages, the location within the channel where the sudden exponential-like acceleration begins and the start of the self-accelerating process. The full numerical study confirms the asymptotic results and extends their validity illustrating that the self-acceleration process proceeds until the flame reaches the far end of the channel.
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