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Effects of thermal expansion on the stabilization of an edge-flame in a mixing-layer model
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Proceedings of the
Proceedings of the Combustion Institute 32 (2009) 1107–1115
Combustion Institute www.elsevier.com/locate/proci
Effects of thermal expansion on the stabilization of an edge-flame in a mixing-layer model V. Kurdyumov a,*, M. Matalon b a b
Modeling and Numerical Simulation Group, CIEMAT, Avda. Complutense 22, 20A, 28040, Madrid, Spain Mechanical Sciences and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract This paper examines the influence of thermal expansion on the stabilization of an edge-flame in a mixing layer. We find, similar to the earlier predictions based on a constant-density model, two modes of flame stabilization: a steady mode at low injection velocities and an oscillatory mode at higher velocities. The gas expansion has an effect on the flame standoff distance: as a result of the reduced density in the preheat zone the flow accelerates when crossing the flame which consequently forces its edge to relocate at an upstream position where its propagation speed balances the gas velocity. The onset of oscillations at relatively high flow rates is predicted with or without invoking the constant-density approximation; the critical conditions of the onset, however, are affected by density variations. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Edge flame; Diffusion flame; Thermal expansion; Oscillations; Instabilities
1. Introduction This study is concerned with edge-flames formed in combustion of initially non-premixed gases. Edge-flames are relevant to various applications: In large industrial furnaces the diffusion flame is often lifted off the burner and the lifted flame has an edge that is stabilized further downstream within the jet [1]. The diffusion flame sustained between the gaseous hydrogen and liquid oxygen streams in a liquid rocket engine depends on the stabilization of an edge-flame near the injector lip [2]. Edge-flames also occur near holes that are formed in wrinkled turbulent diffusion flames. They can propagate towards the pocket of premixed gas that develops as fuel and oxidizer *
Corresponding author. E-mail address: [email protected] (V. Kurdyumov).
come in contact re-establishing burning throughout the hole, or propagate backwards spreading extinction to other parts of the flame. There have been several theoretical studies of edge-flames in non-premixed systems. Buckmaster and co-workers [3,4] used a model problem of a flat diffusion flame along an axis, with fuel and oxidizer supplied at two opposing ends in the transverse direction. An edge results by cuttingoff the fuel supply arbitrarily at a finite position. Their primary objective has been to illustrate the occurrence of oscillation in mixtures with Lewis number significantly larger than one. The more realistic problem of an edge-flame stabilized in the mixing layer of two streams, one of fuel and the other of oxidizer, was studied by Kurdyumov and Matalon. The earlier studies [5,6] assumed that the streams are of equal and constant velocities, but more recently [7,8] the flow field was calculated based on the Navier–Stokes equations and
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.06.042
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kept frozen when examining the flame dynamics. The results identify conditions for flame stabilization near the injector, including the dependence of the standoff distance on the physico-chemical parameters; conditions for the onset of instabilities in the form of oscillations, with the edge moving back and forth but remaining near the plate; conditions leading to flame re-stabilization, now in a position further downstream where the gas velocity is larger and can balance the edge-flame propagation speed, and conditions associated with blowoff. The flame structure, that resembles the observed flames, and its position are found to depend on the mixture composition (fuel-tooxidizer ratio), the mobility of the reactants (Lewis numbers) and radiative heat losses. The latter affect not only the flame shape and size, but also the onset of oscillations. Furthermore, it has been established [5] that heat losses alone can trigger flame oscillation, even in unity Lewis number flow mixtures. The aforementioned studies have all adopted the constant-density approximation, which conveniently decouples the hydrodynamics from the transport equations of heat and mass. As such it does not elucidate the influence of thermal expansion on flame characteristics. In [9] the effect of gas expansion has been added as a perturbation, but instead of solving the appropriate free-boundary problem, the analysis assumed the shape of the flame without identifying conditions under which such a structure exist. The significance of gas expansion was also noted in [10] through analysis of the results of a numerical study, and in [11–14] based on direct numerical simulations for specific conditions. A full parametric study of the problem has not yet been given and this paper marks the first such attempt. Solving the time-dependent coupled Navier–Stokes and transport equations with detailed chemistry for a wide range of parameters and over a sufficiently long time that permits addressing the dynamical behavior of interest is a formidable task. Stripping off some of these details reduces the number of parameters involved and enables a more complete description of the time-dependent problem with deeper physical understanding. The model under investigation assumes an overall chemical reaction scheme and constant gas properties. Further simplifications are associated with the boundary conditions at the supply end, where the gas properties are prescribed, and in the transverse direction where lateral boundaries are assumed to coincide with flow streamlines. These simplifications will be relaxed in future studies. 2. Formulation Two separate streams, one containing fuel and the other oxidizer, are flowing through porous
plates located at x ¼ 0, in a bounded domain L0 < y < L0 . The gas emerging from the plates is uniform with speed U 0 . At the exit of the plugs the fractional mass flux of each of the two reactants is specified. This, however, depends on the thickness of the plug and its porosity as discussed in [15]. For thin plates these conditions reduce to specifying the mass fractions at the exit of the plug, a limit that we adopt for simplicity. The mass fractions at x ¼ 0 are therefore assumed given: Y F0 for the fuel ðy > 0Þ and Y O0 for the oxidizer ðy < 0Þ. The thermal conductivity of the plate is assumed sufficiently high so as to maintain the gas temperature at the exit uniform and equal to T 0 . A mixing layer is produced beyond the station x ¼ 0 where the fuel and oxidizer interdiffuse. When the mixture is ignited, an edge-flame forms as shown in Fig. 1. The governing equation describing the mixing and combustion processes are (in dimensionless form) qoh=ot þ qv rh ¼ r2 h þ ð1 þ /Þx;
ð1Þ
2 qoY F =ot þ qv rY F ¼ Le1 F r Y F x;
ð2Þ
qoY O =ot þ qv rY O ¼ qð1 þ chÞ ¼ 1; oq=ot þ r qv ¼ 0;
2 Le1 O r YO
/x;
ð3Þ ð4Þ ð5Þ
qov=ot þ ðqv rÞv ¼ rp þ Prðr2 v þ rðr vÞ=3Þ: ð6Þ In writing these equations the transverse distance L0 was chosen as a unit of length, the characteristic velocity Dth =L0 as a unit of speed and L20 =Dth as a unit of time; here Dth is the thermal diffusivity of the mixture. The mixture density q and the mass fractions Y F ; Y O have been normalized with respect to their values in the supply streams, q0 and Y F0 ; Y O0 , and a non-dimensional temperature h ¼ ðT T 0 Þ=ðT a T 0 Þ was introduced, where T a ¼ T 0 þ QY F0 =cp ð1 þ /Þ is the adiabatic, or stoichiometric, temperature1 with Q the heat release (per unit mass of fuel) and / ¼ mY F0 =Y O0 the initial mixture strength (m is the mass-weighted stoichiometric coefficient and cp the specific heat). The parameters in these equations are the Prandtl number Pr, the dimensionless velocity m ¼ U 0 L0 =Dth or injection speed, the Lewis numbers associated with the fuel and oxidizer LeF ¼ Dth =DF and LeO ¼ Dth =DO , where DF and DO are the molecular diffusivities, respectively, and the heat release or thermal expansion parameter c ¼ ðT a T 0 Þ=T 0 . Assuming a global one-step overall chemical reaction, the reaction rate is given by 1
The temperature T a is approached far downstream along the stoichiometric surface, where the fuel and oxidizer are completely consumed under adiabatic conditions.
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
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Fig. 1. Edge-flames for two distinct values of the injection speed: m ¼ 3 for the figures on the left and m ¼ 10 for the figures on the right. The dark solid curves represent reaction-rate contours and the light curves are streamlines. The color shades in (a) and (b) show temperature variations, and those in (c) and (d) the vorticity field; note however, that the scales in (a) and (c) and different than those in (b) and (d). (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)
x ¼ Dq2 Y F Y O exp½b0 =ð1 þ chÞ where b0 ¼ E=RT 0 is the activation energy parameter and D ¼ q0 L20 BY O0 =Dth the Damko¨hler number. Here B is an appropriate pre-exponential factor, E the overall activation energy and R the gas constant. The boundary conditions at the plate, x ¼ 0, are: q ¼ 1; h ¼ 0; u ¼ m; v ¼ 0 1; y > 0 0; YF ¼ ; YO ¼ 0; y < 0 1;
y>0 : y<0
The lateral boundaries are assumed to coincide with the flow streamlines, implying that the boundary conditions along y ¼ 1 are: oh=oy ¼ oY F =oy ¼ oY O =oy ¼ ou=oy ¼ v ¼ 0: The preheated zone ahead of the edge-flame is characterized by the diffusion-length, dT ¼ Dth =S L , where S L is the propagation speed of a premixed flame in the corresponding stoichiometric mixture. The ratio L0 =dT D1=2 is an important parameter in this model, whose magnitude determines the influence of the lateral boundaries on the solution. The physically most interesting limit, L0 ! 1, requires small enough spatial resolution, and for time-dependent calculations also temporal resolution with excessively large computation time. To determine a reasonable value of L0 , a series of
computations were carried out for steady conditions and a wide range of velocities m, with D increased systematically within 0.5 1012– 2 1012. The results were found to converge rapidly as D approached its upper limit; the flame standoff distance, for example, for the two extreme values of D was found to be within few percents for the whole range of m computed. This implies that the influence of the boundaries imposed at finite locations is reduced when L0 , or equivalently D, increases. In the calculations presented below we have thus chosen L0 =dT 20 corresponding to D ¼ 1:6 1012 . We have also assigned the fixed value b0 ¼ 72, corresponding to a Zel’dovich number b EðT a T 0 Þ=RT 2a ¼ 10 with c ¼ 5, which is a typical value for hydrocarbon oxidation. Since the focus in this work is on the influence of thermal expansion, we have limited the discussion to stoichiometric conditions, / ¼ 1, and equal but nonunity Lewis numbers LeF ¼ LeO Le. This choice implies that the flame remains symmetric with respect to the axis, even in the presence of thermal expansion. Finally, the values Le ¼ 1:5 and Pr ¼ 0:72 were adopted for the Lewis and Prandtl numbers. 3. Numerical approach Steady as well as time-dependent solutions of Eqs. (1)–(6) are reported in this study. It is advantageous for numerical calculations to eliminate the
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pressure from the momentum equation by introducing the vorticity f ¼ vx uy where subscripts, here and thereafter, denote partial differentiation. Taking the curl of Eq. (6) one finds that f satisfies q ft þ qv rf ¼ Prr2 f þ J
ð7Þ
where the vorticity production term J (primarily baroclinic generation) is given by J ¼ ðqy ut qx vt Þ þ ½ðquÞy ux ðquÞx vx þ ½ðqvÞy uy ðqvÞx vy : For steady solutions, the counterpart of Eqs. (1)– (4) and (7) obtained by setting o=ot ¼ 0 are considered. The continuity equation is satisfied automatically by introducing a streamfunction w, defined from qu ¼ wy ; qv ¼ wx , which satisfies ðwx =qÞx þ ðwy =qÞy ¼ f:
ð8Þ
This elliptic system (7)–(11) and (1)–(4) for f; w and the remaining state variables was solved using a Gauss-Seidel iteration with successive overrelaxation. We note that the steady solutions obtained using this approach may or may not be stable. For the unsteady calculations, it is convenient to rewrite the continuity equation in the form 2
r v ¼ c½r h þ ð1 þ /Þx
ð9Þ
where use has been made of Eqs. (1) and (4). Based on Helmholtz theorem, the velocity field can, in general, be decomposed into irrotational and solenoidal components, namely v ¼ mi þ r/ þ w, where / is a potential and w is divergence-free; here, i is a unit vector in the axial direction. The irrotational component can be obtained from Eq. (9) and the solenoidal part by ~ that satisfies introducing a stream-like function w r w ¼ 0 automatically. The velocity components then take the form ~y; u ¼ m þ /x þ w
~ x ¼ / , which should be taken into account that w y in evaluating the vorticity equation along the plate. 4. Results The two cases presented in Fig. 1 correspond to distinct values of m, or (steady) flames stabilized at two different locations. Profiles of the axial velocity, temperature and reaction rates along the x-axis are shown in Fig. 2. The reaction-rate contours represented by solid dark curves in Fig. 1 clearly identify the shape of the flame and its tribrachial structure. There are two premixed segments, one lean and the other rich, and a diffusion flame trailing behind where the excess reactants are consumed. The light curves
1
20
0.8 15
0.6
θ
u 10 0.4
ω
5
0.2
0
0
1
2
3
x
4
b 40
1
~x v ¼ /y w
0.8 30
where r2 / ¼ c½r2 h þ ð1 þ /Þx ~ ¼ f: r2 w
0
ð10Þ ð11Þ
~ and the Equations. 7, 11 and (1)–(4) for f; /; w remaining state variables were discretized using a second-order finite-difference approximation and an explicit marching method was used with firstor second-order discretization to advance the solution in time. Due to the exponential nonlinearity of the reaction rate, a sufficiently small time step was chosen to ensure numerical stability. The boundary conditions associated with the ~ ¼ / ¼ 0 at newly introduced variables are: w x ~ ¼ f ¼ / ¼ 0 at y ¼ 1. Far downx ¼ 0 and w y ~ x ¼ fx ¼ 0. It stream we require that /xx ¼ w should be also noted that v ¼ 0 at x ¼ 0 implies
0.6
θ
u 20 0.4
ω
10
0.2
0
0
1
2
x
3
4
0
Fig. 2. Axial velocity, temperature and reaction-rate profiles along the axis x ¼ 0 for the two cases illustrated in Fig. 1; the reaction rate peaks at x ¼ 59:68 and x ¼ 28:07, respectively.
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
emanating from the plate are streamlines, which remain nearly parallel when leaving the porous plate but are deflected when approaching the preheat zone of the premixed flame segments. As a result of the lower density the gas accelerates when crossing the flame (see also Fig. 2). The color shades in (a) and (b) show the temperature variations and in (c) and (d) the vorticity field. One observes that the premixed segments of the edge-flame are elongated when the flame is stabilized further downstream (case b) because of the additional mixing that has taken place beyond the plate. As a result the diffusion flame is confined to a thin region centered near the stoichiometric surface (the centerline in this case). Although the reaction rate reaches its maximum near the edge, approximately located at x 1:9, the temperature reaches its maximum value along the diffusion flame sheet. The vorticity generated at the flame forms two vortices: one rotating clockwise (for y > 0) and the other counterclockwise (for y < 0), consistent with the direction of the streamline deflections. At low injection velocities the edge-flame is stabilized very close to the plate (case a). In this case the burner intrudes into the preheat zone and there is significant heat loss to the porous plug. Vorticity is also created near the plate because of the sharp velocity gradient that results from the immediate increase in temperature. These influences have an artificial effect on the standoff distance of the edge-flame, which should be re-examined on a scale comparable to dT . This case has been presented simply to illustrate the effect of the injection velocity on the flame characteristics, while the main focus in this study is on edge-flames that stand well away from the boundary. The profiles plotted in Fig. 2b show the variation in temperature and velocity along the axis. These profiles differ from the familiar profiles across a planar flame. Far upstream the flow is nearly parallel and the axial speed nearly constant. There is a slight drop in velocity just ahead of the edge and then a sharp increase through the reaction zone. Beyond the edge the flow continues to accelerate. The temperature far upstream is nearly uniform; it increases rapidly near the premixed edge, reaches a maximum value, and then drops slightly beyond the reaction zone approaching the diffusion flame stoichiometric temperature far downstream. These characteristics of the edgeflame are associated with its unique tribrachial structure. Close to the edge, the flow diverges and accelerates. Since the premixed segments are highly curved, there are Oð1Þ transverse variations in velocities with lateral expansion that is as significant as the axial expansion. As a result of the transverse velocities there is a drop in axial speed, which is clearly seen in Fig. 3 where u and v velocities are plotted as a function of y for different x-positions. The drop in flame temperature just
x=1.2 1
1111
x=1.4 x=1.6
x=1
0.5
y
0
-0.5
-1
6
8
10
12
u
14
1
0.5
x=1.4 x=1.6
y
0
x=1 x=1.2
-0.5
-1 -3
-2
-1
0
1
2
3
v Fig. 3. Variations of (a) axial velocity u, and (b) transverse velocity v across the channel for several positions x located near the upstream end of the edge flame.
beyond the edge is also well understood. Ahead of the edge there is strong mixing and the reaction rate reaches its maximum value along the axis, which is also the stoichiometric surface. Beyond the edge, complete combustion along the diffusion flame is not reached immediately, which explains the slight drop in temperature before reaching the stoichiometric temperature, or the adiabatic temperature associated with the Burke–Schumann flame sheet aligned with the axis. To identify the influence of thermal expansion on flame stabilization, we show in Fig. 4 a comparison of the standoff distance xw for steady flames, in the presence/absence of thermal expan-
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3
1
constant-density
m=5
xw 0.5 2 0
xw
0
1
2
t
3
4
3
4
5
m=6
1
variable-density
1
xw 0.5 0
0
0
5
10
m
0
1
2
15 2
Fig. 4. Flame standoff distance as a function of the injection speed for D ¼ 1:6 1012 ; the two cases correspond to computations based on variable and constantdensity models.
xw
t m=8
t
5
1 2
1
4 3
sion. The latter was computed by setting q ¼ 1 in order to uncouple the flow field from the transport equations. The standoff distance xw is defined here as the location along x where the reaction rate takes on its maximum value x ¼ xmax . For large values of m, the constant-density model overestimates the standoff distance. The flow accelerates
Le=1.5 γ=4.8
γ=4.5
3
γ=4 γ=5 2
xw γ=5.5
1
0
0
5
m
10
15
Fig. 5. The dependence of the flame standoff distance on m for various values of the thermal expansion parameter. Solid curves correspond to stable flame and dashed curves to oscillatory states. For given m, the triangles identify for the case c ¼ 5 the extent of the oscillation displacement; the triangles pointing up/down correspond to the highest/lowest positions attained in a cycle.
0
0
1
2
t
3
4
5
Fig. 6. Standoff distance as a function of time for three values of m corresponding to a stable state ðm ¼ 5Þ, a state close to the point of exchange of stability ðm ¼ 6Þ and an unstable oscillatory state ðm ¼ 8Þ.
as a result of thermal expansion which forces the flame to adjust its location at a more favorable position further upstream, a location where the gas velocity precisely equal the edge-flame propagation speed. Here the influence of the boundary conditions imposed at x ¼ 0 on the flame are negligible. For small values of m, on the other hand, both models produce nearly identical results; the slight variations in xw are insignificant and result from inaccuracies in the precise determination of xmax . The dependence of the flame standoff distance xw on thermal expansion is shown in Fig. 5. At low speed the edge is close to the porous plug and is affected by heat losses to the plate. The minimum standoff distance seen here is similar to the observation of planar flames stabilized on top of a porous plug burners. The minimum xw is attained at very small values of m as c increases, and is not always shown in the figure. For each value of c the entire curve (solid and dashed parts) was produced using the steady code. Consistent with the earlier discussion the flame, for given m, adjusts its position as c increases to a value xw that is further upstream. In general, changes in c affect the reactivity of the mixture through the heat release Q and the flow field through
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
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Fig. 7. The vorticity field corresponding to four different instants (identified in Fig. 6) within one cycle of oscillations.
(1)
(1)
1
θ
40
0.5
0 (2)
20
0
1
2
3
4
x
0 (2)
1
θ
(3)
0
1
2
3
4
x
0
1
2
3
4
x
0
1
2
3
4
x
0
1
2
3
4
x
ω
100
0.5
0
ω
50
0
1
2
3
4
x
0 (3)
1
θ
ω 40
0.5 20 0 (4)
0
1
2
3
4
x
1
θ
ω 20
0.5
0
0 (4)
10
0
1
2
3
4
x
0
Fig. 8. Temperature (a) and reaction rate (b) profiles along the axis at four different instants within one cycle; the four values are marked in Fig. 6.
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density variations. The flame is held stationary if the gas velocity affected by density variations balances the edge-flame propagation speed determined by the mixture reactivity. We note that the constant-density model referred to earlier (see Fig. 4) is an ad hoc approximation used only to illustrate the influence of the gas expansion on the flow field and does not coincide with the limit process c ! 0. Strictly speaking, the limit c ! 0 implies that the heat release is very small and the reaction practically frozen. The steady states shown in Fig. 5 were tested for stability using the time-dependent code. The solution was slightly perturbed and its development in time was followed to see whether or not the steady state is recovered after sufficiently long time. Solid segments of the curves were found to correspond to stable states and dashed segments to unstable states. At low speeds the flame is held close to the plate with xw increasing as m increases. Above a critical value of m spontaneous oscillations occur with the edgeflame moving back and forth, relative to the unperturbed (unstable) state xw . Figure 6 illustrates a typical example near the point of exchange of stability. The graph shows the evolution of xw in time for three values of m, starting from arbitrary initial data. The steady state corresponding to m ¼ 5 is stable. For m ¼ 6, which is just beyond criticality, a small amplitude solution centered around xw ¼ 0:9 develops. For m ¼ 8 the solution evolves into a finite-amplitude oscillatory state. Note that the average flame position xw ¼ 1:3 differs from the position of the unperturbed (unstable) state xw ¼ 1:05 implying that, relative to the steady (unstable) solution, the flame spends more time closer to the plate than away from it. The amplitude of oscillations increases as m increases as shown by the triangular symbols on the graph corresponding to c ¼ 5 in Fig. 6; the highest and lowest values of xw in a cycle are denoted by the symbols 4 and 5, respectively. Finally we note that during a cycle the flame moves faster towards the porous plate than away from it. To illustrate the nature of the solution during the cycle of an oscillatory state, we have marked in the graph corresponding to m ¼ 8 four distinct points. Figures 7 and 8 show the vorticity and profiles of temperature and reaction rate at these four instants. As the flame moves downstream the reaction rate diminishes significantly but it is then rejuvenated when the flame moves back towards the plate. The extent of fuel and oxidizer mixing increases during the time when the flame is away from the plate, which explains the relatively elongated premixed flame segments during the remaining part of the cycle. Figure 9 shows the temporal variations of temperature at four locations along the x-axis for the same con-
ditions as the previous two figures. It is evident that it is the edge-flame that oscillates dragging behind the diffusion flame sheet. The oscillations are damped further downstream so that the diffusion flame sheet at large distances is stable and steady. The figure also illustrates the nonlinear nature of the oscillations which differs significantly from the sinusoidal behavior near the onset of the instability. 5. Conclusions This paper constitutes the first attempt to systematically examine the influence of thermal expansion on an edge-flame in a mixing layer. In order to address the time-dependent problem and cover a wide range of parameters we have imposed specific boundary conditions at the injection plane and in the lateral directions. Nevertheless, the results remain qualitatively correct, except perhaps when the flame is very close to the injection plate. The influence of thermal expansion is manifested primarily by the reloca-
x=1, y=0
1
θ
0.5
0
0
0
3
4
5
1
2
t
3
4
5
3
4
5
3
4
5
x=2, y=0
1
0.5
0
0
1
2
t
x=2.5, y=0
1
θ
t
0.5
0
θ
2
x=1.5, y=0
1
θ
1
0.5
0
0
1
2
t
Fig. 9. Temporal variations of the temperature at four different location along the axis, showing a decay of oscillation with increasing position.
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
tion of the edge-flame relative to the plane separating fuel and oxidizer, i.e. where the mixing layer starts developing. Since as a result of gas expansion the flow accelerates, the edge-flame would be held further upstream at a position where its speed balances the upcoming flow. This study shows for the first time that the onset of oscillations at sufficiently high flow rates occurs even when gas expansion is accounted for and is not merely a result of the highly simplified assumption of constant density. Edge-flame stabilization occurs in one or of two ways: in a steady mode at low injection speeds and in an oscillatory mode at high flow rates. In the latter the oscillations are restricted to the edge-flame and are damped further downstream along the diffusion flame sheet.
Acknowledgments The work has been partially supported by the National Science Foundation under Grants CBET-0733146 and DMA-0708588. V.K. also acknowledges the support of the Spanish Government through the Ramo´n y Cajal Program and CM under Project S-0505/ENE/0229 and M.M. acknowledges the support of the US–Israel Bi-National Science Foundation under Grant 2004069.
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References [1] S.H. Chung, B.J. Lee, Combust. Flame 86 (1991) 62–72. [2] M. Juniper, S. Candel, J. Propul. Power 19 (3) (2003) 332–341. [3] J. Buckmaster, Yi Zhang, Combust. Theor. Model. 3 (1999) 547–565. [4] J.D. Buckmaster, A. Hegab, T.L. Jackson, Phys. Fluids 12 (2000) 1592–1600. [5] V. Kurdyumov, M. Matalon, Proc. Combust. Inst. 29 (2002) 45–52. [6] V. Kurdyumov, M. Matalon, Combust. Flame 139 (2004) 29–339. [7] V. Kurdyumov, M. Matalon, Proc. Combust. Inst. 31 (2007) 909–917. [8] V. Kurdyumov, M. Matalon, Proc. Combust. Inst. 31 (2007) 929–938. [9] S. Ghosal, L. Vervisch, J. Fluid Mech. 415 (2000) 227–260. [10] G.R. Ruetsch, L. Vervisch, A. Linan, Phys. Fluids 7 (6) (1995) 1447–1454. [11] V.R. Katta, W.M. Roquemore, Combust. Flame 102 (1–2) (1995) 21–40. [12] I.G. Im, J.H. Chen, Combust. Flame 119 (1999) 436–454. [13] V. Favier, L. Vervisch, Combust. Flame 125 (2001) 788–803. [14] X. Qin, I.K. Puri, S.K. Aggarwal, Proc. Combust. Inst. 29 (2002) 1565–1572. [15] V. Kurdyumov, M. Matalon, Combust. Flame 153 (2008) 105–118.
Proceedings of the
Proceedings of the Combustion Institute 32 (2009) 1107–1115
Combustion Institute www.elsevier.com/locate/proci
Effects of thermal expansion on the stabilization of an edge-flame in a mixing-layer model V. Kurdyumov a,*, M. Matalon b a b
Modeling and Numerical Simulation Group, CIEMAT, Avda. Complutense 22, 20A, 28040, Madrid, Spain Mechanical Sciences and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Abstract This paper examines the influence of thermal expansion on the stabilization of an edge-flame in a mixing layer. We find, similar to the earlier predictions based on a constant-density model, two modes of flame stabilization: a steady mode at low injection velocities and an oscillatory mode at higher velocities. The gas expansion has an effect on the flame standoff distance: as a result of the reduced density in the preheat zone the flow accelerates when crossing the flame which consequently forces its edge to relocate at an upstream position where its propagation speed balances the gas velocity. The onset of oscillations at relatively high flow rates is predicted with or without invoking the constant-density approximation; the critical conditions of the onset, however, are affected by density variations. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Edge flame; Diffusion flame; Thermal expansion; Oscillations; Instabilities
1. Introduction This study is concerned with edge-flames formed in combustion of initially non-premixed gases. Edge-flames are relevant to various applications: In large industrial furnaces the diffusion flame is often lifted off the burner and the lifted flame has an edge that is stabilized further downstream within the jet [1]. The diffusion flame sustained between the gaseous hydrogen and liquid oxygen streams in a liquid rocket engine depends on the stabilization of an edge-flame near the injector lip [2]. Edge-flames also occur near holes that are formed in wrinkled turbulent diffusion flames. They can propagate towards the pocket of premixed gas that develops as fuel and oxidizer *
Corresponding author. E-mail address: [email protected] (V. Kurdyumov).
come in contact re-establishing burning throughout the hole, or propagate backwards spreading extinction to other parts of the flame. There have been several theoretical studies of edge-flames in non-premixed systems. Buckmaster and co-workers [3,4] used a model problem of a flat diffusion flame along an axis, with fuel and oxidizer supplied at two opposing ends in the transverse direction. An edge results by cuttingoff the fuel supply arbitrarily at a finite position. Their primary objective has been to illustrate the occurrence of oscillation in mixtures with Lewis number significantly larger than one. The more realistic problem of an edge-flame stabilized in the mixing layer of two streams, one of fuel and the other of oxidizer, was studied by Kurdyumov and Matalon. The earlier studies [5,6] assumed that the streams are of equal and constant velocities, but more recently [7,8] the flow field was calculated based on the Navier–Stokes equations and
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.06.042
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kept frozen when examining the flame dynamics. The results identify conditions for flame stabilization near the injector, including the dependence of the standoff distance on the physico-chemical parameters; conditions for the onset of instabilities in the form of oscillations, with the edge moving back and forth but remaining near the plate; conditions leading to flame re-stabilization, now in a position further downstream where the gas velocity is larger and can balance the edge-flame propagation speed, and conditions associated with blowoff. The flame structure, that resembles the observed flames, and its position are found to depend on the mixture composition (fuel-tooxidizer ratio), the mobility of the reactants (Lewis numbers) and radiative heat losses. The latter affect not only the flame shape and size, but also the onset of oscillations. Furthermore, it has been established [5] that heat losses alone can trigger flame oscillation, even in unity Lewis number flow mixtures. The aforementioned studies have all adopted the constant-density approximation, which conveniently decouples the hydrodynamics from the transport equations of heat and mass. As such it does not elucidate the influence of thermal expansion on flame characteristics. In [9] the effect of gas expansion has been added as a perturbation, but instead of solving the appropriate free-boundary problem, the analysis assumed the shape of the flame without identifying conditions under which such a structure exist. The significance of gas expansion was also noted in [10] through analysis of the results of a numerical study, and in [11–14] based on direct numerical simulations for specific conditions. A full parametric study of the problem has not yet been given and this paper marks the first such attempt. Solving the time-dependent coupled Navier–Stokes and transport equations with detailed chemistry for a wide range of parameters and over a sufficiently long time that permits addressing the dynamical behavior of interest is a formidable task. Stripping off some of these details reduces the number of parameters involved and enables a more complete description of the time-dependent problem with deeper physical understanding. The model under investigation assumes an overall chemical reaction scheme and constant gas properties. Further simplifications are associated with the boundary conditions at the supply end, where the gas properties are prescribed, and in the transverse direction where lateral boundaries are assumed to coincide with flow streamlines. These simplifications will be relaxed in future studies. 2. Formulation Two separate streams, one containing fuel and the other oxidizer, are flowing through porous
plates located at x ¼ 0, in a bounded domain L0 < y < L0 . The gas emerging from the plates is uniform with speed U 0 . At the exit of the plugs the fractional mass flux of each of the two reactants is specified. This, however, depends on the thickness of the plug and its porosity as discussed in [15]. For thin plates these conditions reduce to specifying the mass fractions at the exit of the plug, a limit that we adopt for simplicity. The mass fractions at x ¼ 0 are therefore assumed given: Y F0 for the fuel ðy > 0Þ and Y O0 for the oxidizer ðy < 0Þ. The thermal conductivity of the plate is assumed sufficiently high so as to maintain the gas temperature at the exit uniform and equal to T 0 . A mixing layer is produced beyond the station x ¼ 0 where the fuel and oxidizer interdiffuse. When the mixture is ignited, an edge-flame forms as shown in Fig. 1. The governing equation describing the mixing and combustion processes are (in dimensionless form) qoh=ot þ qv rh ¼ r2 h þ ð1 þ /Þx;
ð1Þ
2 qoY F =ot þ qv rY F ¼ Le1 F r Y F x;
ð2Þ
qoY O =ot þ qv rY O ¼ qð1 þ chÞ ¼ 1; oq=ot þ r qv ¼ 0;
2 Le1 O r YO
/x;
ð3Þ ð4Þ ð5Þ
qov=ot þ ðqv rÞv ¼ rp þ Prðr2 v þ rðr vÞ=3Þ: ð6Þ In writing these equations the transverse distance L0 was chosen as a unit of length, the characteristic velocity Dth =L0 as a unit of speed and L20 =Dth as a unit of time; here Dth is the thermal diffusivity of the mixture. The mixture density q and the mass fractions Y F ; Y O have been normalized with respect to their values in the supply streams, q0 and Y F0 ; Y O0 , and a non-dimensional temperature h ¼ ðT T 0 Þ=ðT a T 0 Þ was introduced, where T a ¼ T 0 þ QY F0 =cp ð1 þ /Þ is the adiabatic, or stoichiometric, temperature1 with Q the heat release (per unit mass of fuel) and / ¼ mY F0 =Y O0 the initial mixture strength (m is the mass-weighted stoichiometric coefficient and cp the specific heat). The parameters in these equations are the Prandtl number Pr, the dimensionless velocity m ¼ U 0 L0 =Dth or injection speed, the Lewis numbers associated with the fuel and oxidizer LeF ¼ Dth =DF and LeO ¼ Dth =DO , where DF and DO are the molecular diffusivities, respectively, and the heat release or thermal expansion parameter c ¼ ðT a T 0 Þ=T 0 . Assuming a global one-step overall chemical reaction, the reaction rate is given by 1
The temperature T a is approached far downstream along the stoichiometric surface, where the fuel and oxidizer are completely consumed under adiabatic conditions.
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
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Fig. 1. Edge-flames for two distinct values of the injection speed: m ¼ 3 for the figures on the left and m ¼ 10 for the figures on the right. The dark solid curves represent reaction-rate contours and the light curves are streamlines. The color shades in (a) and (b) show temperature variations, and those in (c) and (d) the vorticity field; note however, that the scales in (a) and (c) and different than those in (b) and (d). (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)
x ¼ Dq2 Y F Y O exp½b0 =ð1 þ chÞ where b0 ¼ E=RT 0 is the activation energy parameter and D ¼ q0 L20 BY O0 =Dth the Damko¨hler number. Here B is an appropriate pre-exponential factor, E the overall activation energy and R the gas constant. The boundary conditions at the plate, x ¼ 0, are: q ¼ 1; h ¼ 0; u ¼ m; v ¼ 0 1; y > 0 0; YF ¼ ; YO ¼ 0; y < 0 1;
y>0 : y<0
The lateral boundaries are assumed to coincide with the flow streamlines, implying that the boundary conditions along y ¼ 1 are: oh=oy ¼ oY F =oy ¼ oY O =oy ¼ ou=oy ¼ v ¼ 0: The preheated zone ahead of the edge-flame is characterized by the diffusion-length, dT ¼ Dth =S L , where S L is the propagation speed of a premixed flame in the corresponding stoichiometric mixture. The ratio L0 =dT D1=2 is an important parameter in this model, whose magnitude determines the influence of the lateral boundaries on the solution. The physically most interesting limit, L0 ! 1, requires small enough spatial resolution, and for time-dependent calculations also temporal resolution with excessively large computation time. To determine a reasonable value of L0 , a series of
computations were carried out for steady conditions and a wide range of velocities m, with D increased systematically within 0.5 1012– 2 1012. The results were found to converge rapidly as D approached its upper limit; the flame standoff distance, for example, for the two extreme values of D was found to be within few percents for the whole range of m computed. This implies that the influence of the boundaries imposed at finite locations is reduced when L0 , or equivalently D, increases. In the calculations presented below we have thus chosen L0 =dT 20 corresponding to D ¼ 1:6 1012 . We have also assigned the fixed value b0 ¼ 72, corresponding to a Zel’dovich number b EðT a T 0 Þ=RT 2a ¼ 10 with c ¼ 5, which is a typical value for hydrocarbon oxidation. Since the focus in this work is on the influence of thermal expansion, we have limited the discussion to stoichiometric conditions, / ¼ 1, and equal but nonunity Lewis numbers LeF ¼ LeO Le. This choice implies that the flame remains symmetric with respect to the axis, even in the presence of thermal expansion. Finally, the values Le ¼ 1:5 and Pr ¼ 0:72 were adopted for the Lewis and Prandtl numbers. 3. Numerical approach Steady as well as time-dependent solutions of Eqs. (1)–(6) are reported in this study. It is advantageous for numerical calculations to eliminate the
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pressure from the momentum equation by introducing the vorticity f ¼ vx uy where subscripts, here and thereafter, denote partial differentiation. Taking the curl of Eq. (6) one finds that f satisfies q ft þ qv rf ¼ Prr2 f þ J
ð7Þ
where the vorticity production term J (primarily baroclinic generation) is given by J ¼ ðqy ut qx vt Þ þ ½ðquÞy ux ðquÞx vx þ ½ðqvÞy uy ðqvÞx vy : For steady solutions, the counterpart of Eqs. (1)– (4) and (7) obtained by setting o=ot ¼ 0 are considered. The continuity equation is satisfied automatically by introducing a streamfunction w, defined from qu ¼ wy ; qv ¼ wx , which satisfies ðwx =qÞx þ ðwy =qÞy ¼ f:
ð8Þ
This elliptic system (7)–(11) and (1)–(4) for f; w and the remaining state variables was solved using a Gauss-Seidel iteration with successive overrelaxation. We note that the steady solutions obtained using this approach may or may not be stable. For the unsteady calculations, it is convenient to rewrite the continuity equation in the form 2
r v ¼ c½r h þ ð1 þ /Þx
ð9Þ
where use has been made of Eqs. (1) and (4). Based on Helmholtz theorem, the velocity field can, in general, be decomposed into irrotational and solenoidal components, namely v ¼ mi þ r/ þ w, where / is a potential and w is divergence-free; here, i is a unit vector in the axial direction. The irrotational component can be obtained from Eq. (9) and the solenoidal part by ~ that satisfies introducing a stream-like function w r w ¼ 0 automatically. The velocity components then take the form ~y; u ¼ m þ /x þ w
~ x ¼ / , which should be taken into account that w y in evaluating the vorticity equation along the plate. 4. Results The two cases presented in Fig. 1 correspond to distinct values of m, or (steady) flames stabilized at two different locations. Profiles of the axial velocity, temperature and reaction rates along the x-axis are shown in Fig. 2. The reaction-rate contours represented by solid dark curves in Fig. 1 clearly identify the shape of the flame and its tribrachial structure. There are two premixed segments, one lean and the other rich, and a diffusion flame trailing behind where the excess reactants are consumed. The light curves
1
20
0.8 15
0.6
θ
u 10 0.4
ω
5
0.2
0
0
1
2
3
x
4
b 40
1
~x v ¼ /y w
0.8 30
where r2 / ¼ c½r2 h þ ð1 þ /Þx ~ ¼ f: r2 w
0
ð10Þ ð11Þ
~ and the Equations. 7, 11 and (1)–(4) for f; /; w remaining state variables were discretized using a second-order finite-difference approximation and an explicit marching method was used with firstor second-order discretization to advance the solution in time. Due to the exponential nonlinearity of the reaction rate, a sufficiently small time step was chosen to ensure numerical stability. The boundary conditions associated with the ~ ¼ / ¼ 0 at newly introduced variables are: w x ~ ¼ f ¼ / ¼ 0 at y ¼ 1. Far downx ¼ 0 and w y ~ x ¼ fx ¼ 0. It stream we require that /xx ¼ w should be also noted that v ¼ 0 at x ¼ 0 implies
0.6
θ
u 20 0.4
ω
10
0.2
0
0
1
2
x
3
4
0
Fig. 2. Axial velocity, temperature and reaction-rate profiles along the axis x ¼ 0 for the two cases illustrated in Fig. 1; the reaction rate peaks at x ¼ 59:68 and x ¼ 28:07, respectively.
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
emanating from the plate are streamlines, which remain nearly parallel when leaving the porous plate but are deflected when approaching the preheat zone of the premixed flame segments. As a result of the lower density the gas accelerates when crossing the flame (see also Fig. 2). The color shades in (a) and (b) show the temperature variations and in (c) and (d) the vorticity field. One observes that the premixed segments of the edge-flame are elongated when the flame is stabilized further downstream (case b) because of the additional mixing that has taken place beyond the plate. As a result the diffusion flame is confined to a thin region centered near the stoichiometric surface (the centerline in this case). Although the reaction rate reaches its maximum near the edge, approximately located at x 1:9, the temperature reaches its maximum value along the diffusion flame sheet. The vorticity generated at the flame forms two vortices: one rotating clockwise (for y > 0) and the other counterclockwise (for y < 0), consistent with the direction of the streamline deflections. At low injection velocities the edge-flame is stabilized very close to the plate (case a). In this case the burner intrudes into the preheat zone and there is significant heat loss to the porous plug. Vorticity is also created near the plate because of the sharp velocity gradient that results from the immediate increase in temperature. These influences have an artificial effect on the standoff distance of the edge-flame, which should be re-examined on a scale comparable to dT . This case has been presented simply to illustrate the effect of the injection velocity on the flame characteristics, while the main focus in this study is on edge-flames that stand well away from the boundary. The profiles plotted in Fig. 2b show the variation in temperature and velocity along the axis. These profiles differ from the familiar profiles across a planar flame. Far upstream the flow is nearly parallel and the axial speed nearly constant. There is a slight drop in velocity just ahead of the edge and then a sharp increase through the reaction zone. Beyond the edge the flow continues to accelerate. The temperature far upstream is nearly uniform; it increases rapidly near the premixed edge, reaches a maximum value, and then drops slightly beyond the reaction zone approaching the diffusion flame stoichiometric temperature far downstream. These characteristics of the edgeflame are associated with its unique tribrachial structure. Close to the edge, the flow diverges and accelerates. Since the premixed segments are highly curved, there are Oð1Þ transverse variations in velocities with lateral expansion that is as significant as the axial expansion. As a result of the transverse velocities there is a drop in axial speed, which is clearly seen in Fig. 3 where u and v velocities are plotted as a function of y for different x-positions. The drop in flame temperature just
x=1.2 1
1111
x=1.4 x=1.6
x=1
0.5
y
0
-0.5
-1
6
8
10
12
u
14
1
0.5
x=1.4 x=1.6
y
0
x=1 x=1.2
-0.5
-1 -3
-2
-1
0
1
2
3
v Fig. 3. Variations of (a) axial velocity u, and (b) transverse velocity v across the channel for several positions x located near the upstream end of the edge flame.
beyond the edge is also well understood. Ahead of the edge there is strong mixing and the reaction rate reaches its maximum value along the axis, which is also the stoichiometric surface. Beyond the edge, complete combustion along the diffusion flame is not reached immediately, which explains the slight drop in temperature before reaching the stoichiometric temperature, or the adiabatic temperature associated with the Burke–Schumann flame sheet aligned with the axis. To identify the influence of thermal expansion on flame stabilization, we show in Fig. 4 a comparison of the standoff distance xw for steady flames, in the presence/absence of thermal expan-
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3
1
constant-density
m=5
xw 0.5 2 0
xw
0
1
2
t
3
4
3
4
5
m=6
1
variable-density
1
xw 0.5 0
0
0
5
10
m
0
1
2
15 2
Fig. 4. Flame standoff distance as a function of the injection speed for D ¼ 1:6 1012 ; the two cases correspond to computations based on variable and constantdensity models.
xw
t m=8
t
5
1 2
1
4 3
sion. The latter was computed by setting q ¼ 1 in order to uncouple the flow field from the transport equations. The standoff distance xw is defined here as the location along x where the reaction rate takes on its maximum value x ¼ xmax . For large values of m, the constant-density model overestimates the standoff distance. The flow accelerates
Le=1.5 γ=4.8
γ=4.5
3
γ=4 γ=5 2
xw γ=5.5
1
0
0
5
m
10
15
Fig. 5. The dependence of the flame standoff distance on m for various values of the thermal expansion parameter. Solid curves correspond to stable flame and dashed curves to oscillatory states. For given m, the triangles identify for the case c ¼ 5 the extent of the oscillation displacement; the triangles pointing up/down correspond to the highest/lowest positions attained in a cycle.
0
0
1
2
t
3
4
5
Fig. 6. Standoff distance as a function of time for three values of m corresponding to a stable state ðm ¼ 5Þ, a state close to the point of exchange of stability ðm ¼ 6Þ and an unstable oscillatory state ðm ¼ 8Þ.
as a result of thermal expansion which forces the flame to adjust its location at a more favorable position further upstream, a location where the gas velocity precisely equal the edge-flame propagation speed. Here the influence of the boundary conditions imposed at x ¼ 0 on the flame are negligible. For small values of m, on the other hand, both models produce nearly identical results; the slight variations in xw are insignificant and result from inaccuracies in the precise determination of xmax . The dependence of the flame standoff distance xw on thermal expansion is shown in Fig. 5. At low speed the edge is close to the porous plug and is affected by heat losses to the plate. The minimum standoff distance seen here is similar to the observation of planar flames stabilized on top of a porous plug burners. The minimum xw is attained at very small values of m as c increases, and is not always shown in the figure. For each value of c the entire curve (solid and dashed parts) was produced using the steady code. Consistent with the earlier discussion the flame, for given m, adjusts its position as c increases to a value xw that is further upstream. In general, changes in c affect the reactivity of the mixture through the heat release Q and the flow field through
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
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Fig. 7. The vorticity field corresponding to four different instants (identified in Fig. 6) within one cycle of oscillations.
(1)
(1)
1
θ
40
0.5
0 (2)
20
0
1
2
3
4
x
0 (2)
1
θ
(3)
0
1
2
3
4
x
0
1
2
3
4
x
0
1
2
3
4
x
0
1
2
3
4
x
ω
100
0.5
0
ω
50
0
1
2
3
4
x
0 (3)
1
θ
ω 40
0.5 20 0 (4)
0
1
2
3
4
x
1
θ
ω 20
0.5
0
0 (4)
10
0
1
2
3
4
x
0
Fig. 8. Temperature (a) and reaction rate (b) profiles along the axis at four different instants within one cycle; the four values are marked in Fig. 6.
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density variations. The flame is held stationary if the gas velocity affected by density variations balances the edge-flame propagation speed determined by the mixture reactivity. We note that the constant-density model referred to earlier (see Fig. 4) is an ad hoc approximation used only to illustrate the influence of the gas expansion on the flow field and does not coincide with the limit process c ! 0. Strictly speaking, the limit c ! 0 implies that the heat release is very small and the reaction practically frozen. The steady states shown in Fig. 5 were tested for stability using the time-dependent code. The solution was slightly perturbed and its development in time was followed to see whether or not the steady state is recovered after sufficiently long time. Solid segments of the curves were found to correspond to stable states and dashed segments to unstable states. At low speeds the flame is held close to the plate with xw increasing as m increases. Above a critical value of m spontaneous oscillations occur with the edgeflame moving back and forth, relative to the unperturbed (unstable) state xw . Figure 6 illustrates a typical example near the point of exchange of stability. The graph shows the evolution of xw in time for three values of m, starting from arbitrary initial data. The steady state corresponding to m ¼ 5 is stable. For m ¼ 6, which is just beyond criticality, a small amplitude solution centered around xw ¼ 0:9 develops. For m ¼ 8 the solution evolves into a finite-amplitude oscillatory state. Note that the average flame position xw ¼ 1:3 differs from the position of the unperturbed (unstable) state xw ¼ 1:05 implying that, relative to the steady (unstable) solution, the flame spends more time closer to the plate than away from it. The amplitude of oscillations increases as m increases as shown by the triangular symbols on the graph corresponding to c ¼ 5 in Fig. 6; the highest and lowest values of xw in a cycle are denoted by the symbols 4 and 5, respectively. Finally we note that during a cycle the flame moves faster towards the porous plate than away from it. To illustrate the nature of the solution during the cycle of an oscillatory state, we have marked in the graph corresponding to m ¼ 8 four distinct points. Figures 7 and 8 show the vorticity and profiles of temperature and reaction rate at these four instants. As the flame moves downstream the reaction rate diminishes significantly but it is then rejuvenated when the flame moves back towards the plate. The extent of fuel and oxidizer mixing increases during the time when the flame is away from the plate, which explains the relatively elongated premixed flame segments during the remaining part of the cycle. Figure 9 shows the temporal variations of temperature at four locations along the x-axis for the same con-
ditions as the previous two figures. It is evident that it is the edge-flame that oscillates dragging behind the diffusion flame sheet. The oscillations are damped further downstream so that the diffusion flame sheet at large distances is stable and steady. The figure also illustrates the nonlinear nature of the oscillations which differs significantly from the sinusoidal behavior near the onset of the instability. 5. Conclusions This paper constitutes the first attempt to systematically examine the influence of thermal expansion on an edge-flame in a mixing layer. In order to address the time-dependent problem and cover a wide range of parameters we have imposed specific boundary conditions at the injection plane and in the lateral directions. Nevertheless, the results remain qualitatively correct, except perhaps when the flame is very close to the injection plate. The influence of thermal expansion is manifested primarily by the reloca-
x=1, y=0
1
θ
0.5
0
0
0
3
4
5
1
2
t
3
4
5
3
4
5
3
4
5
x=2, y=0
1
0.5
0
0
1
2
t
x=2.5, y=0
1
θ
t
0.5
0
θ
2
x=1.5, y=0
1
θ
1
0.5
0
0
1
2
t
Fig. 9. Temporal variations of the temperature at four different location along the axis, showing a decay of oscillation with increasing position.
V. Kurdyumov, M. Matalon / Proceedings of the Combustion Institute 32 (2009) 1107–1115
tion of the edge-flame relative to the plane separating fuel and oxidizer, i.e. where the mixing layer starts developing. Since as a result of gas expansion the flow accelerates, the edge-flame would be held further upstream at a position where its speed balances the upcoming flow. This study shows for the first time that the onset of oscillations at sufficiently high flow rates occurs even when gas expansion is accounted for and is not merely a result of the highly simplified assumption of constant density. Edge-flame stabilization occurs in one or of two ways: in a steady mode at low injection speeds and in an oscillatory mode at high flow rates. In the latter the oscillations are restricted to the edge-flame and are damped further downstream along the diffusion flame sheet.
Acknowledgments The work has been partially supported by the National Science Foundation under Grants CBET-0733146 and DMA-0708588. V.K. also acknowledges the support of the Spanish Government through the Ramo´n y Cajal Program and CM under Project S-0505/ENE/0229 and M.M. acknowledges the support of the US–Israel Bi-National Science Foundation under Grant 2004069.
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