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Aerodynamics of a slender axisymmetric Bunsen flame with large gas expansion
Combustion and Flame 156 (2009) 1063–1067
Contents lists available at ScienceDirect
Combustion and Flame www.elsevier.com/locate/combustflame
Aerodynamics of a slender axisymmetric Bunsen flame with large gas expansion F.J. Higuera ∗ E.T.S. Ingenieros Aeronáuticos, UPM, Pza. Cardenal Cisneros 3, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 20 August 2008 Received in revised form 16 October 2008 Accepted 27 October 2008 Available online 13 November 2008 Keywords: Laminar flames Bunsen flames
a b s t r a c t The flow around a stationary axisymmetric premixed flame attached to a plug flow Bunsen burner and propagating with a constant burning velocity relative to the fresh gas is analyzed for large values of the ratio of fresh gas injection velocity to flame burning velocity and small values of the burnt-to-fresh gas density ratio. The shape of the flame is close to a cone of small semiangle, and its presence induces only a small perturbation in the fresh gas. The velocity of the burnt gas is the superposition of a uniform axial velocity and the velocity induced by a line of sources on the axis of the burner. The pressure of the burnt gas on the flame is the sum of a uniform part that does not change the velocity of the fresh gas and a small variable part that causes a vertical acceleration of this gas and a slight deformation of the flame. Only a weak vorticity is generated at the flame. The analysis breaks down in small regions around the tip and the base of the flame where the assumption of a constant burning velocity is not valid. © 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Bunsen flames are among the most familiar and widely studied configurations for the stabilized combustion of premixed gases. In a comprehensive investigation of the effectively bidimensional Bunsen flame above a vertical slot burner with a Poiseuille velocity profile, Lewis and von Elbe [1] ascertained the conditions of stability of the flame between flash back and blow off; measured the velocity fields of the fresh and burnt gases from the traces of stroboscopically illuminated particles carried by the flow, showing that the presence of the flame has a remarkably small effect on the velocity of the fresh gas except in the vicinity of the burner rim; and determined the local burning velocity of the flame, showing that it is uniform over most of the flame surface, except in a small region around the tip, where it increases, and in the region of attachment to the burner, where it decreases to zero. Lewis and von Elbe [2] reviewed the early research in this area. Uberoi et al. [3] analyzed the generation of vorticity at a flame, idealized as a discontinuity in an inviscid flow, and derived general relations between the flame curvature and the change in the rate of expansion of the streamtubes that cross it. From their analysis and their experiments with bidimensional Bunsen flames in a plug flow slot burner, these authors conclude that a Bunsen flame can exist only when the burning velocity increases around its tip, and that the tip flips and becomes convex toward the fresh gas when its curvature is not enough to cause the required increase of the burning velocity; see also Ref. [4].
*
Fax: +34 91 336 3295. E-mail address: [email protected].
Bunsen flames have been used to measure the burning velocity of planar flames by means of a variety of methods, local or global in character, that rely on the fact that the structure and normal velocity of a tilted Bunsen flame attached to a burner are nearly identical over most of the flame surface to those of a planar flame propagating in the same mixture; see Refs. [5–9]. Andrews and Bradley [6] critically reviewed the early methods used to determine the velocity of the gas and the position and area of the flame. More recently, Bunsen flames have also been used to investigate the effects of the front curvature and strain rate of the flow on the burning velocity in the vicinity of the tip [10–12]. Rigorous theoretical accounts of these effects exist in the asymptotic limit of weakly curved infinitely thin flames [13–15], and the manner in which these theoretical results can be used for real flames of finite thickness has been clarified by Davis et al. [16,17]. The shape of a Bunsen flame in a given flow has been computed by Lewis and von Elbe [1] for a constant burning velocity and by Sivashinsky [18] and Matalon et al. [19] taking into account the variation of the burning velocity with the stretch of the flame due to its curvature and to the strain rate of the flow. Uberoi and Zimmerman [20] approximately computed the shape of a bidimensional flame with constant burning velocity taking into account the perturbation that the flame induces in the flow but leaving out the effect of the vorticity generated by the flame. Buckmaster and Crowley [21] examined the slender flame that appears in the asymptotic limit when the velocity of the fresh gas is large compared with the burning velocity of the planar flame. They used a Oseen linearization of the momentum equations in the region where the burning velocity of the flame is constant and carried out a separate analysis of the structure of the tip.
0010-2180/$ – see front matter © 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2008.10.023
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Conditions usually met in the analysis of Bunsen flames can be summarized as follows. The size of the burner is large compared with the thickness of the flame, which therefore can be viewed as a discontinuity. The thermal expansion that the gas experiences on crossing the flame induces a flow whose Reynolds number is large of the order of the ratio of the size of the burner to the thickness of the flame. Transport effects are therefore confined to the interior of the flame and to a shear layer between the burnt gas and the quiescent ambient gas, and can be neglected in the rest of the flow field. The curvature of the flame front is small compared to the inverse of the flame thickness on most of the flame, with the exception of a small region around the flame tip. The strain rate of the flow around the flame is also small compared to the typical velocity gradient inside the flame. In these conditions, the internal structure of the flame is nearly identical to that of a planar flame and, away from its tip, the normal velocity of the flame relative to the fresh gas nearly coincides with the burning velocity of a planar flame in this gas. In this paper an analysis is reported of the flow around an axisymmetric Bunsen flame attached to the rim of a nozzle through which a plug flow of fresh gas is discharged. The velocity of the fresh gas is assumed to be large compared to the burning velocity of the flame, and the burnt-to-fresh gas density ratio is assumed to be small. Attention is confined to the region where the burning velocity can be assumed to be constant, which excludes a small region around the flame tip. The results provide an explanation for the observed nearly uniform flow between the burner and the flame and the nearly conical shape of the flame. The small departure of the flame from a slender cone and the weak generation of vorticity at its surface are computed. 2. Analysis and results 2.1. Problem setup Consider a stream of a flammable gas mixture of density ρu issuing upward with a uniform velocity U 0 from a vertical pipe fitted with a convergent nozzle of exit radius a into a quiescent inert atmosphere. The gas burns in a stationary premixed flame that extends from the rim of the nozzle and advances with a constant burning velocity U L relative to the gas. The ratio U 0 /U L will be assumed to be large. The velocity of the gas is taken to be uniform at the exit of the nozzle on the assumption that the boundary layer around the nozzle wall is thin compared to the nozzle radius. Pressure variations of order ρu U 02 would occur in the fresh gas if velocity variations of order U 0 existed upstream of the flame. However, different experiments suggest that the velocity of the fresh gas is nearly uniform in many cases [3,4,10,22]. The velocity U 0 must be in a finite range bounded by a certain minimum U 0min U L below which flash back occurs, and a certain maximum above which the flame is blown off. The blow off value depends on the burner and may be large compared to U L . Within its allowed range, the ratio U 0 /U L is a free parameter whose value determines the aspect ratio of the flame. The height of the flame above the burner is of order L = (U 0 /U L )a, which is large compared to the radius of the nozzle if U 0 /U L is large. The assumption of a constant burning velocity is expected to fail in a small region around the tip of the flame where the effect of flame stretch is important, and also in a small region around the base of the flame. These regions will not be analyzed here. The density of the gas decreases across the flame due to the localized heat release, but it is uniform at each side of the flame. Subscripts u and b are used in what follows to denote conditions in the fresh and burnt gases. The density ratio = ρb /ρu is a fairly small number, in the range of 0.1–0.2 for typical flames.
The mass and momentum conservation conditions across a stationary flame read
ρu u un = ρb ubn , pu + ρ
2 u u un
(1) 2 b u bn ,
= pb + ρ
(2)
u u t = u bt ,
(3)
where subscripts n and t denote the components of the velocity normal and tangent to the flame. If u un = U L constant, then u bn = U L / , which is fairly large compared to U L . Similarly, the term ρb ub2 in the normal momentum condition (2) is larger than the n
term
ρu u 2un by a factor 1/ , so that the pressure jump across the
flame is p b − p u = −(1 − )ρb u b2 ≈ −ρu u 2un / , which is a negative n constant if the normal velocity of the flame is U L . The effect of buoyancy forces on the dynamics of the burnt gas will be neglected on the assumption that the Froude number F = ρb ub2n /ρu ga is large. This Froude number may be of the order of 25 for typical methane- or propane-air flames and a = 1 cm (but see additional comments at the end of Section 2.2 below). Velocity variations of order u bn = U L / are to be expected in the burnt gas as the flow turns to become nearly vertical. This leads to pressure variations of order ρb u b2 = ρb U L2 / 2 . Since the n pressure jump across the flame is a constant, pressure variations originating in the burnt gas are transmitted across the flame and affect the flow of the fresh gas. The importance of these pressure variations is measured by the ratio of dynamic pressures ρb ub2n /ρu U 02 = (U L /U 0 )2 / . The ratio of dynamic pressures is small if U 0 is large compared to U L / 1/2 . Then the pressure variations originating in the burnt gas cannot noticeably influence the flow of the fresh gas, which is nearly uniform. In these conditions, the flame is nearly conical, of equation
rf =a 1−
UL x U0 a
for 0
UL x U0 a
1,
(4)
to satisfy the condition u un = U L . Here x is the vertical distance above the nozzle and r f (x) is the radius of the section of the flame by a horizontal plane at a height x. It is shown at the end of Section 2.2 that (4) is also valid when U 0 is not large compared to U L / 1 / 2 . 2.2. Flow of the burnt gas The vertical (u b , axial) and horizontal (v b , radial) components of the velocity of the gas emerging from the flame are U 0 and u bn = U L / , respectively, from conditions (1) and (3) and the condition that the flame surface is nearly vertical. Continuity requires that the radial velocity of the burnt gas decrease as the inverse of the distance r to the symmetry axis as the gas moves away from the flame. This is an important difference with the case of a bidimensional flame, which was already noted by Lewis and von Elbe [1]; see the discussion in page 93 of their paper. From Bernoulli’s equation, the deceleration of the burnt gas is accompanied by an increase of pressure, which is equal to 12 ρb u b2 at n distances from the flame large compared to its radius r f = O (a). Since the angle of the flame to the vertical is small, this increase of pressure occurs nearly uniformly along the flame and causes little variation of the axial velocity of the burnt gas. The streamlines of the burnt gas bend upward in a layer of characteristic radius rb = (u bn /U 0 )a around the flame (from the condition that v b ∼ u b with v b ∼ u bn a/rb and u b ∼ U 0 ), and the flow becomes quasi-unidirectional beyond this layer. The radius of the region occupied by the quasi-unidirectional flow is of order R b = a/ 1/2 , from the condition that the mass flux of burnt gas
F.J. Higuera / Combustion and Flame 156 (2009) 1063–1067
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= 0.1 in (a) and 0.2 in (b).
Fig. 1. Conical slender flame and streamlines of the flow for two values of the burnt-to-fresh gas density ratio;
(π R b2 ρb U 0 ) should be equal to the mass flux issuing from the pipe (π a2 ρu U 0 ); see, e.g., Ref. [22]. The radial velocity of the burnt gas is of order u bn a/ R b = U L / 1/2 at distances of order R b from the symmetry axis, leading to pressure variations of order ρb U L2 / in the region of quasi-unidirectional flow, whose axial gradient is too small to noticeably change the axial velocity. This velocity is thus nearly equal to U 0 everywhere in the burnt gas, and the mass and momentum conservation equations can be simplified to
∂ ub 1 ∂ + (r v b ) = 0, ∂x r ∂r ∂u ∂u ρb U 0 b + ρb v b b = 0, ∂x ∂r ∂ pb ∂ vb ∂ vb ρb U 0 + ρb v b =− , ∂x ∂r ∂r
(5) (6) (7)
where the term ρb v b ∂ v b /∂ r in (7) [and similarly ρb v b ∂ u b /∂ r in (6)] must be retained for any value of the ratio U 0 /U L , and gives the dominant contribution to the radial acceleration of the burnt gas in the vicinity of the flame. This is in contrast to the case of a bidimensional flame with U 0 /U L 1 analyzed by Buckmaster and Crowley [21]. The solution of Eqs. (6) and (5) with the boundary conditions ub = U 0
and
v b = U L /
at r = r f (x)
and
vb =
UL r f
r
pb − pa = − (9)
,
with r f (x) given by (4). Some streamlines of this velocity field are shown in Fig. 1, which may be compared to the experimental flame shape and streamlines in Figs. 117 and 121 of Lewis and von Elbe [2]. The comparison, however, is only qualitative because the experiments of Ref. [2] were carried out with a slot burner that gives a nearly planar flow. The shear layer between the burnt gas and the quiescent ambient gas coincides with the streamline issuing from the base of the flame, which is given by r = r s (x)
1/2 1 UL x UL x UL x 2− for 0 1. =a 1+
U0 a
U0 a
U0 a
(10)
= 0.1 (solid) and 0.2 (dashed).
Finally, the pressure distribution computed from (7) with the condition p b = pa at r = r s (x), where pa is the pressure of the ambient gas, is
(8)
is ub = U 0
Fig. 2. Pressure of the burnt gas on the flame for
1 2
ρb
2 U L2 r f r s2 − r 2
2
r s2
r2
+ ρb
U L2
ln
r rs
.
(11)
As was advanced before, for r f /r s = O ( 1/2 ) and 1 the pressure increases by the constant amount 12 ρb U L2 / 2 in an inner layer around the flame [first term on the right-hand side of (11)] and has smaller changes in the region of quasi-unidirectional flow. The pressure acting on the flame [p f = p b (r = r f )] is p f (x) − pa = −
1 2
ρu U L2
1
1−
r 2f r s2
+ ln
, 2
r s2 rf
(12)
which is shown in Fig. 2. This pressure is the sum of a constant, pa − 12 ρu U L2 / , plus a variable part of order ρu U L2 which decreases from the base to the tip of the flame. The variable depression is transmitted to the fresh gas, but it induces only small relative changes of its velocity for any U 0 U L . The solution computed here is not valid in a small region around the tip of the flame. The solution should also fail in a
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F.J. Higuera / Combustion and Flame 156 (2009) 1063–1067
certain region around the base of the flame where the gas must emerge from the flame at a pressure nearly equal to pa and therefore cannot undergo the predicted pressure increase 12 ρb U L2 / 2 . However, Fig. 2 shows that the pressure at the flame (12) adjusts itself to pa when r s tends to r f for x → 0. The flow emerging from the flame is nearly horizontal for any U 0 U L / . When U 0 becomes of order U L / 1/2 , the radial extent rb of the region where the streamlines bend upward becomes of the order of R b and both are of the order of the height of the flame, so that the flow of the burnt gas ceases to be quasiunidirectional. However, the largest decrease of the radial velocity, leading to the pressure rise 12 ρb U L2 / 2 , occurs at distances from the flame of order a, where the flow is still nearly horizontal. This pressure rise is now of the order of the dynamic pressure of the fresh gas. However, it is nearly uniform along the flame and therefore does not change the velocity of the fresh gas. As a consequence, the flame remains nearly conical when U 0 = O ( U L / 1 / 2 ) . The effect of the buoyancy can be taken into account by adding a term (ρu − ρb ) g to the right-hand side of (6) and letting p in (7) denote the modified pressure p + ρu gx. For a slender flame, buoyancy causes variations of the vertical velocity of the burnt gas of order u b = (1 − ) ga/ U L [from the balance ρb U 0 u b / L ∼ (ρu − ρb ) g in the modified form of (6), with L ∼ (U 0 /U L )a]. The ratio u b /U 0 = (1 − )(U L /U 0 )/ 2 F may be of order unity when U 0 ∼ U L / 1/2 for typical values of the density ratio and the Froude number. Buoyancy forces change then the flow of the burnt gas computed above, but they do not change the main pressure rise discussed in the previous paragraph. The flame would cease to be slender when U 0 becomes of the order of U L . However, it seems unlikely that a stationary flame with a small value of may exist in these conditions, for the following reasons. First, since the component of the velocity tangent to the flame is continuous while the component normal to the flame increases by a factor 1/ on crossing the flame, the burnt gas would emerge almost normally to the flame with a uniform velocity U L / . Second, since pressure variations of order ρb U L2 / 2 are not admissible in the fresh gas, the surface of the flame should nearly coincide with the isobar p = pa from the point of view of the burnt gas. The total pressure would then be nearly uniform in the burnt gas, the vorticity would be nearly zero, and the flow cannot apparently proceed away from the flame without causing inadmissible pressure and velocity variations.
Fig. 3. Conical flame in a uniform fresh gas stream (solid) and upward-curved flame in the stream accelerated by the depression (12) (dashed) for U L /U 0 = 0.2, 0.3, and 0.5, increasing from top to bottom, with = 0.1 in (a) and 0.2 in (b). (A small uniform strain has been applied to the vertical coordinate x in (12) to make the singularity of p f coincide with the tip of the curved flame.)
2.3. Flame shape and vorticity generation Under the action of the pressure distribution (12) (plus the constant pressure jump across the flame), the vertical velocity of the fresh gas changes slightly to become u u (x) = U 0 [1 − 2( p f (x) − pa )/ ρu U 02 ]1/2 (from Bernoulli’s equation), and a small radial veloc-
ity v u = − 12 r du u /dx appears (from the continuity equation). The shape of the flame in this quasi-unidirectional flow can be computed by a simple geometrical construction and is shown in Fig. 3 (dashed curves) for some values of and U L /U 0 . The curved flame is more tilted near its base than the conical flame, due to the inward curvature of the streamlines of the accelerated flow (v u < 0), and then bends upward due to the increase of u u (x). The computed shapes are in agreement with experimental visualizations [4,22]. Uberoi et al. [3] derived the expression ω = [(1 − )/2U L ] × ∂ ut2 /∂ s for the vorticity immediately downstream of a flame propagating with constant burning velocity in an irrotational flow. Here ut is the component of the velocity tangent to the flame and s is the arc length on the meridional section of the flame, measured
Fig. 4. Vorticity of the burnt gas as a function of radial distance in the horizontal plane x = a(U 0 /U L ) through the tip of the flame for = 0.1 (solid) and 0.2 (dashed).
from its tip. For a slender flame ∂ ut2 /∂ s ≈ −∂ u 2u /∂ x with an error of order (U L /U 0 )2 and
ω U L /a
= −(1 − 2 )
UL U0
1+
r s2 r 2f
a3 r f
2 r s4
.
(13)
Since ω/r is conserved on the streamlines, the vorticity can be easily computed everywhere in the burnt gas. It is shown in Fig. 4 as a function of r in the horizontal plane through the tip of the flame. The vorticity is of order (U 0 / R b )(U L / 1/2 U 0 )2 , with R b = a/ 1/2 , in the region of quasi-unidirectional flow r = O ( R b ). Its presence amounts to a small perturbation to an essentially irrotational flow in this region for any U 0 U L / 1/2 . The singularity of the vorticity at the tip of the flame, as well as that of the pressure (12), is to be taken care of by a different local analysis. The peak of the vorticity in Fig. 4 at the outer boundary of the burnt gas reflects the
F.J. Higuera / Combustion and Flame 156 (2009) 1063–1067
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existence of a large vorticity on the streamlines crossing the flame near its base. This can be traced to the decrease of the denominator 2 r s4 of the last factor of (13), which changes from O (1) to O ( 2 ) at the base of the flame.
Acknowledgments
3. Conclusions
References
The shape of an axisymmetric Bunsen flame attached to the rim of a plug flow burner and propagating with constant normal velocity relative to the gas, as well as the flow that the flame induces in the fresh and burnt gases, have been computed for values of the injection velocity large compared to the normal velocity of the flame and small values of the burnt-to-fresh gas density ratio. In agreement with experiments and previous analyses, the flame in this limit is nearly a slender cone and induces only a small perturbation in the fresh gas. The flow of the burnt gas is that of a line of sources extending along the axis of the burner in a uniform axial stream. The pressure that the burnt gas exerts on the flame consists of a uniform part that does not change the velocity of the fresh gas plus a small variable part that decreases with height above the burner and therefore causes a streamwise acceleration of the fresh gas and a departure of the flame shape from a cone. The vorticity generated by the perturbed flame induces only a small correction to the flow of the burnt gas for large values of the injection velocity. The assumption of a constant burning velocity fails in small regions around the tip and the base of the flame which would require a separate analysis.
[1] B. Lewis, G. von Elbe, J. Chem. Phys. 11 (1943) 75–97. [2] B. Lewis, G. von Elbe, Combustion, Flames and Explosions of Gases, Academic Press, Orlando, 1987, chapter 5. [3] M.S. Uberoi, A.M. Kuethe, H.R. Menkes, Phys. Fluids 1 (1958) 150–158. [4] M.S. Uberoi, J. Chem. Phys. 22 (1954) 1784. [5] R.M. Fristrom, J. Chem. Phys. 24 (1956) 888–894. [6] G.E. Andrews, D. Bradley, Combust. Flame 18 (1972) 133–153. [7] G. Dixon-Lewis, S.M. Islam, Proc. Combust. Inst. 19 (1982) 283–291. [8] D.D.S. Liu, R. MacFarlane, Combust. Flame 49 (1983) 59–71. [9] T. Wagner, C.R. Ferguson, Combust. Flame 59 (1985) 267–272. [10] T. Echekki, M.G. Mungal, Proc. Combust. Inst. 23 (1990) 455–461. [11] T. Poinsot, T. Echekki, M.G. Mungal, Combust. Sci. Technol. 81 (1992) 45–73. [12] M.G. Mungal, L.M. Lourenco, A. Krothapalli, Combust. Sci. Technol. 106 (1995) 239–265. [13] P. Clavin, F.A. Williams, J. Fluid Mech. 116 (1982) 251–282. [14] M. Matalon, B.J. Matkowsky, J. Fluid Mech. 124 (1982) 239–259. [15] P. Clavin, G. Joulin, J. Phys. Lett. 44 (1983) L-1. [16] S.G. Davis, J. Quinard, G. Searby, Combust. Flame 130 (2002) 112–122. [17] S.G. Davis, J. Quinard, G. Searby, Combust. Flame 130 (2002) 123–136. [18] G.I. Sivashinsky, J. Chem. Phys. 62 (1975) 638–643. [19] M. Matalon, C. Cui, J.K. Bechtold, J. Fluid Mech. 487 (2003) 179–210. [20] M.S. Uberoi, A. Zimmerman, Phys. Fluids 8 (1965) 1628–1630. [21] J. Buckmaster, A.B. Crowley, J. Fluid Mech. 131 (1983) 341–361. [22] M.J. Remie, M.F.G. Cremers, K.R.A.M. Schreel, L.P.H. de Goey, Combust. Flame 147 (2006) 163–170.
This work was supported by the Comunidad de Madrid through project COMLIMAMS, S-0505/ENE-229.
Contents lists available at ScienceDirect
Combustion and Flame www.elsevier.com/locate/combustflame
Aerodynamics of a slender axisymmetric Bunsen flame with large gas expansion F.J. Higuera ∗ E.T.S. Ingenieros Aeronáuticos, UPM, Pza. Cardenal Cisneros 3, 28040 Madrid, Spain
a r t i c l e
i n f o
Article history: Received 20 August 2008 Received in revised form 16 October 2008 Accepted 27 October 2008 Available online 13 November 2008 Keywords: Laminar flames Bunsen flames
a b s t r a c t The flow around a stationary axisymmetric premixed flame attached to a plug flow Bunsen burner and propagating with a constant burning velocity relative to the fresh gas is analyzed for large values of the ratio of fresh gas injection velocity to flame burning velocity and small values of the burnt-to-fresh gas density ratio. The shape of the flame is close to a cone of small semiangle, and its presence induces only a small perturbation in the fresh gas. The velocity of the burnt gas is the superposition of a uniform axial velocity and the velocity induced by a line of sources on the axis of the burner. The pressure of the burnt gas on the flame is the sum of a uniform part that does not change the velocity of the fresh gas and a small variable part that causes a vertical acceleration of this gas and a slight deformation of the flame. Only a weak vorticity is generated at the flame. The analysis breaks down in small regions around the tip and the base of the flame where the assumption of a constant burning velocity is not valid. © 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Bunsen flames are among the most familiar and widely studied configurations for the stabilized combustion of premixed gases. In a comprehensive investigation of the effectively bidimensional Bunsen flame above a vertical slot burner with a Poiseuille velocity profile, Lewis and von Elbe [1] ascertained the conditions of stability of the flame between flash back and blow off; measured the velocity fields of the fresh and burnt gases from the traces of stroboscopically illuminated particles carried by the flow, showing that the presence of the flame has a remarkably small effect on the velocity of the fresh gas except in the vicinity of the burner rim; and determined the local burning velocity of the flame, showing that it is uniform over most of the flame surface, except in a small region around the tip, where it increases, and in the region of attachment to the burner, where it decreases to zero. Lewis and von Elbe [2] reviewed the early research in this area. Uberoi et al. [3] analyzed the generation of vorticity at a flame, idealized as a discontinuity in an inviscid flow, and derived general relations between the flame curvature and the change in the rate of expansion of the streamtubes that cross it. From their analysis and their experiments with bidimensional Bunsen flames in a plug flow slot burner, these authors conclude that a Bunsen flame can exist only when the burning velocity increases around its tip, and that the tip flips and becomes convex toward the fresh gas when its curvature is not enough to cause the required increase of the burning velocity; see also Ref. [4].
*
Fax: +34 91 336 3295. E-mail address: [email protected].
Bunsen flames have been used to measure the burning velocity of planar flames by means of a variety of methods, local or global in character, that rely on the fact that the structure and normal velocity of a tilted Bunsen flame attached to a burner are nearly identical over most of the flame surface to those of a planar flame propagating in the same mixture; see Refs. [5–9]. Andrews and Bradley [6] critically reviewed the early methods used to determine the velocity of the gas and the position and area of the flame. More recently, Bunsen flames have also been used to investigate the effects of the front curvature and strain rate of the flow on the burning velocity in the vicinity of the tip [10–12]. Rigorous theoretical accounts of these effects exist in the asymptotic limit of weakly curved infinitely thin flames [13–15], and the manner in which these theoretical results can be used for real flames of finite thickness has been clarified by Davis et al. [16,17]. The shape of a Bunsen flame in a given flow has been computed by Lewis and von Elbe [1] for a constant burning velocity and by Sivashinsky [18] and Matalon et al. [19] taking into account the variation of the burning velocity with the stretch of the flame due to its curvature and to the strain rate of the flow. Uberoi and Zimmerman [20] approximately computed the shape of a bidimensional flame with constant burning velocity taking into account the perturbation that the flame induces in the flow but leaving out the effect of the vorticity generated by the flame. Buckmaster and Crowley [21] examined the slender flame that appears in the asymptotic limit when the velocity of the fresh gas is large compared with the burning velocity of the planar flame. They used a Oseen linearization of the momentum equations in the region where the burning velocity of the flame is constant and carried out a separate analysis of the structure of the tip.
0010-2180/$ – see front matter © 2008 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2008.10.023
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Conditions usually met in the analysis of Bunsen flames can be summarized as follows. The size of the burner is large compared with the thickness of the flame, which therefore can be viewed as a discontinuity. The thermal expansion that the gas experiences on crossing the flame induces a flow whose Reynolds number is large of the order of the ratio of the size of the burner to the thickness of the flame. Transport effects are therefore confined to the interior of the flame and to a shear layer between the burnt gas and the quiescent ambient gas, and can be neglected in the rest of the flow field. The curvature of the flame front is small compared to the inverse of the flame thickness on most of the flame, with the exception of a small region around the flame tip. The strain rate of the flow around the flame is also small compared to the typical velocity gradient inside the flame. In these conditions, the internal structure of the flame is nearly identical to that of a planar flame and, away from its tip, the normal velocity of the flame relative to the fresh gas nearly coincides with the burning velocity of a planar flame in this gas. In this paper an analysis is reported of the flow around an axisymmetric Bunsen flame attached to the rim of a nozzle through which a plug flow of fresh gas is discharged. The velocity of the fresh gas is assumed to be large compared to the burning velocity of the flame, and the burnt-to-fresh gas density ratio is assumed to be small. Attention is confined to the region where the burning velocity can be assumed to be constant, which excludes a small region around the flame tip. The results provide an explanation for the observed nearly uniform flow between the burner and the flame and the nearly conical shape of the flame. The small departure of the flame from a slender cone and the weak generation of vorticity at its surface are computed. 2. Analysis and results 2.1. Problem setup Consider a stream of a flammable gas mixture of density ρu issuing upward with a uniform velocity U 0 from a vertical pipe fitted with a convergent nozzle of exit radius a into a quiescent inert atmosphere. The gas burns in a stationary premixed flame that extends from the rim of the nozzle and advances with a constant burning velocity U L relative to the gas. The ratio U 0 /U L will be assumed to be large. The velocity of the gas is taken to be uniform at the exit of the nozzle on the assumption that the boundary layer around the nozzle wall is thin compared to the nozzle radius. Pressure variations of order ρu U 02 would occur in the fresh gas if velocity variations of order U 0 existed upstream of the flame. However, different experiments suggest that the velocity of the fresh gas is nearly uniform in many cases [3,4,10,22]. The velocity U 0 must be in a finite range bounded by a certain minimum U 0min U L below which flash back occurs, and a certain maximum above which the flame is blown off. The blow off value depends on the burner and may be large compared to U L . Within its allowed range, the ratio U 0 /U L is a free parameter whose value determines the aspect ratio of the flame. The height of the flame above the burner is of order L = (U 0 /U L )a, which is large compared to the radius of the nozzle if U 0 /U L is large. The assumption of a constant burning velocity is expected to fail in a small region around the tip of the flame where the effect of flame stretch is important, and also in a small region around the base of the flame. These regions will not be analyzed here. The density of the gas decreases across the flame due to the localized heat release, but it is uniform at each side of the flame. Subscripts u and b are used in what follows to denote conditions in the fresh and burnt gases. The density ratio = ρb /ρu is a fairly small number, in the range of 0.1–0.2 for typical flames.
The mass and momentum conservation conditions across a stationary flame read
ρu u un = ρb ubn , pu + ρ
2 u u un
(1) 2 b u bn ,
= pb + ρ
(2)
u u t = u bt ,
(3)
where subscripts n and t denote the components of the velocity normal and tangent to the flame. If u un = U L constant, then u bn = U L / , which is fairly large compared to U L . Similarly, the term ρb ub2 in the normal momentum condition (2) is larger than the n
term
ρu u 2un by a factor 1/ , so that the pressure jump across the
flame is p b − p u = −(1 − )ρb u b2 ≈ −ρu u 2un / , which is a negative n constant if the normal velocity of the flame is U L . The effect of buoyancy forces on the dynamics of the burnt gas will be neglected on the assumption that the Froude number F = ρb ub2n /ρu ga is large. This Froude number may be of the order of 25 for typical methane- or propane-air flames and a = 1 cm (but see additional comments at the end of Section 2.2 below). Velocity variations of order u bn = U L / are to be expected in the burnt gas as the flow turns to become nearly vertical. This leads to pressure variations of order ρb u b2 = ρb U L2 / 2 . Since the n pressure jump across the flame is a constant, pressure variations originating in the burnt gas are transmitted across the flame and affect the flow of the fresh gas. The importance of these pressure variations is measured by the ratio of dynamic pressures ρb ub2n /ρu U 02 = (U L /U 0 )2 / . The ratio of dynamic pressures is small if U 0 is large compared to U L / 1/2 . Then the pressure variations originating in the burnt gas cannot noticeably influence the flow of the fresh gas, which is nearly uniform. In these conditions, the flame is nearly conical, of equation
rf =a 1−
UL x U0 a
for 0
UL x U0 a
1,
(4)
to satisfy the condition u un = U L . Here x is the vertical distance above the nozzle and r f (x) is the radius of the section of the flame by a horizontal plane at a height x. It is shown at the end of Section 2.2 that (4) is also valid when U 0 is not large compared to U L / 1 / 2 . 2.2. Flow of the burnt gas The vertical (u b , axial) and horizontal (v b , radial) components of the velocity of the gas emerging from the flame are U 0 and u bn = U L / , respectively, from conditions (1) and (3) and the condition that the flame surface is nearly vertical. Continuity requires that the radial velocity of the burnt gas decrease as the inverse of the distance r to the symmetry axis as the gas moves away from the flame. This is an important difference with the case of a bidimensional flame, which was already noted by Lewis and von Elbe [1]; see the discussion in page 93 of their paper. From Bernoulli’s equation, the deceleration of the burnt gas is accompanied by an increase of pressure, which is equal to 12 ρb u b2 at n distances from the flame large compared to its radius r f = O (a). Since the angle of the flame to the vertical is small, this increase of pressure occurs nearly uniformly along the flame and causes little variation of the axial velocity of the burnt gas. The streamlines of the burnt gas bend upward in a layer of characteristic radius rb = (u bn /U 0 )a around the flame (from the condition that v b ∼ u b with v b ∼ u bn a/rb and u b ∼ U 0 ), and the flow becomes quasi-unidirectional beyond this layer. The radius of the region occupied by the quasi-unidirectional flow is of order R b = a/ 1/2 , from the condition that the mass flux of burnt gas
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= 0.1 in (a) and 0.2 in (b).
Fig. 1. Conical slender flame and streamlines of the flow for two values of the burnt-to-fresh gas density ratio;
(π R b2 ρb U 0 ) should be equal to the mass flux issuing from the pipe (π a2 ρu U 0 ); see, e.g., Ref. [22]. The radial velocity of the burnt gas is of order u bn a/ R b = U L / 1/2 at distances of order R b from the symmetry axis, leading to pressure variations of order ρb U L2 / in the region of quasi-unidirectional flow, whose axial gradient is too small to noticeably change the axial velocity. This velocity is thus nearly equal to U 0 everywhere in the burnt gas, and the mass and momentum conservation equations can be simplified to
∂ ub 1 ∂ + (r v b ) = 0, ∂x r ∂r ∂u ∂u ρb U 0 b + ρb v b b = 0, ∂x ∂r ∂ pb ∂ vb ∂ vb ρb U 0 + ρb v b =− , ∂x ∂r ∂r
(5) (6) (7)
where the term ρb v b ∂ v b /∂ r in (7) [and similarly ρb v b ∂ u b /∂ r in (6)] must be retained for any value of the ratio U 0 /U L , and gives the dominant contribution to the radial acceleration of the burnt gas in the vicinity of the flame. This is in contrast to the case of a bidimensional flame with U 0 /U L 1 analyzed by Buckmaster and Crowley [21]. The solution of Eqs. (6) and (5) with the boundary conditions ub = U 0
and
v b = U L /
at r = r f (x)
and
vb =
UL r f
r
pb − pa = − (9)
,
with r f (x) given by (4). Some streamlines of this velocity field are shown in Fig. 1, which may be compared to the experimental flame shape and streamlines in Figs. 117 and 121 of Lewis and von Elbe [2]. The comparison, however, is only qualitative because the experiments of Ref. [2] were carried out with a slot burner that gives a nearly planar flow. The shear layer between the burnt gas and the quiescent ambient gas coincides with the streamline issuing from the base of the flame, which is given by r = r s (x)
1/2 1 UL x UL x UL x 2− for 0 1. =a 1+
U0 a
U0 a
U0 a
(10)
= 0.1 (solid) and 0.2 (dashed).
Finally, the pressure distribution computed from (7) with the condition p b = pa at r = r s (x), where pa is the pressure of the ambient gas, is
(8)
is ub = U 0
Fig. 2. Pressure of the burnt gas on the flame for
1 2
ρb
2 U L2 r f r s2 − r 2
2
r s2
r2
+ ρb
U L2
ln
r rs
.
(11)
As was advanced before, for r f /r s = O ( 1/2 ) and 1 the pressure increases by the constant amount 12 ρb U L2 / 2 in an inner layer around the flame [first term on the right-hand side of (11)] and has smaller changes in the region of quasi-unidirectional flow. The pressure acting on the flame [p f = p b (r = r f )] is p f (x) − pa = −
1 2
ρu U L2
1
1−
r 2f r s2
+ ln
, 2
r s2 rf
(12)
which is shown in Fig. 2. This pressure is the sum of a constant, pa − 12 ρu U L2 / , plus a variable part of order ρu U L2 which decreases from the base to the tip of the flame. The variable depression is transmitted to the fresh gas, but it induces only small relative changes of its velocity for any U 0 U L . The solution computed here is not valid in a small region around the tip of the flame. The solution should also fail in a
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certain region around the base of the flame where the gas must emerge from the flame at a pressure nearly equal to pa and therefore cannot undergo the predicted pressure increase 12 ρb U L2 / 2 . However, Fig. 2 shows that the pressure at the flame (12) adjusts itself to pa when r s tends to r f for x → 0. The flow emerging from the flame is nearly horizontal for any U 0 U L / . When U 0 becomes of order U L / 1/2 , the radial extent rb of the region where the streamlines bend upward becomes of the order of R b and both are of the order of the height of the flame, so that the flow of the burnt gas ceases to be quasiunidirectional. However, the largest decrease of the radial velocity, leading to the pressure rise 12 ρb U L2 / 2 , occurs at distances from the flame of order a, where the flow is still nearly horizontal. This pressure rise is now of the order of the dynamic pressure of the fresh gas. However, it is nearly uniform along the flame and therefore does not change the velocity of the fresh gas. As a consequence, the flame remains nearly conical when U 0 = O ( U L / 1 / 2 ) . The effect of the buoyancy can be taken into account by adding a term (ρu − ρb ) g to the right-hand side of (6) and letting p in (7) denote the modified pressure p + ρu gx. For a slender flame, buoyancy causes variations of the vertical velocity of the burnt gas of order u b = (1 − ) ga/ U L [from the balance ρb U 0 u b / L ∼ (ρu − ρb ) g in the modified form of (6), with L ∼ (U 0 /U L )a]. The ratio u b /U 0 = (1 − )(U L /U 0 )/ 2 F may be of order unity when U 0 ∼ U L / 1/2 for typical values of the density ratio and the Froude number. Buoyancy forces change then the flow of the burnt gas computed above, but they do not change the main pressure rise discussed in the previous paragraph. The flame would cease to be slender when U 0 becomes of the order of U L . However, it seems unlikely that a stationary flame with a small value of may exist in these conditions, for the following reasons. First, since the component of the velocity tangent to the flame is continuous while the component normal to the flame increases by a factor 1/ on crossing the flame, the burnt gas would emerge almost normally to the flame with a uniform velocity U L / . Second, since pressure variations of order ρb U L2 / 2 are not admissible in the fresh gas, the surface of the flame should nearly coincide with the isobar p = pa from the point of view of the burnt gas. The total pressure would then be nearly uniform in the burnt gas, the vorticity would be nearly zero, and the flow cannot apparently proceed away from the flame without causing inadmissible pressure and velocity variations.
Fig. 3. Conical flame in a uniform fresh gas stream (solid) and upward-curved flame in the stream accelerated by the depression (12) (dashed) for U L /U 0 = 0.2, 0.3, and 0.5, increasing from top to bottom, with = 0.1 in (a) and 0.2 in (b). (A small uniform strain has been applied to the vertical coordinate x in (12) to make the singularity of p f coincide with the tip of the curved flame.)
2.3. Flame shape and vorticity generation Under the action of the pressure distribution (12) (plus the constant pressure jump across the flame), the vertical velocity of the fresh gas changes slightly to become u u (x) = U 0 [1 − 2( p f (x) − pa )/ ρu U 02 ]1/2 (from Bernoulli’s equation), and a small radial veloc-
ity v u = − 12 r du u /dx appears (from the continuity equation). The shape of the flame in this quasi-unidirectional flow can be computed by a simple geometrical construction and is shown in Fig. 3 (dashed curves) for some values of and U L /U 0 . The curved flame is more tilted near its base than the conical flame, due to the inward curvature of the streamlines of the accelerated flow (v u < 0), and then bends upward due to the increase of u u (x). The computed shapes are in agreement with experimental visualizations [4,22]. Uberoi et al. [3] derived the expression ω = [(1 − )/2U L ] × ∂ ut2 /∂ s for the vorticity immediately downstream of a flame propagating with constant burning velocity in an irrotational flow. Here ut is the component of the velocity tangent to the flame and s is the arc length on the meridional section of the flame, measured
Fig. 4. Vorticity of the burnt gas as a function of radial distance in the horizontal plane x = a(U 0 /U L ) through the tip of the flame for = 0.1 (solid) and 0.2 (dashed).
from its tip. For a slender flame ∂ ut2 /∂ s ≈ −∂ u 2u /∂ x with an error of order (U L /U 0 )2 and
ω U L /a
= −(1 − 2 )
UL U0
1+
r s2 r 2f
a3 r f
2 r s4
.
(13)
Since ω/r is conserved on the streamlines, the vorticity can be easily computed everywhere in the burnt gas. It is shown in Fig. 4 as a function of r in the horizontal plane through the tip of the flame. The vorticity is of order (U 0 / R b )(U L / 1/2 U 0 )2 , with R b = a/ 1/2 , in the region of quasi-unidirectional flow r = O ( R b ). Its presence amounts to a small perturbation to an essentially irrotational flow in this region for any U 0 U L / 1/2 . The singularity of the vorticity at the tip of the flame, as well as that of the pressure (12), is to be taken care of by a different local analysis. The peak of the vorticity in Fig. 4 at the outer boundary of the burnt gas reflects the
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existence of a large vorticity on the streamlines crossing the flame near its base. This can be traced to the decrease of the denominator 2 r s4 of the last factor of (13), which changes from O (1) to O ( 2 ) at the base of the flame.
Acknowledgments
3. Conclusions
References
The shape of an axisymmetric Bunsen flame attached to the rim of a plug flow burner and propagating with constant normal velocity relative to the gas, as well as the flow that the flame induces in the fresh and burnt gases, have been computed for values of the injection velocity large compared to the normal velocity of the flame and small values of the burnt-to-fresh gas density ratio. In agreement with experiments and previous analyses, the flame in this limit is nearly a slender cone and induces only a small perturbation in the fresh gas. The flow of the burnt gas is that of a line of sources extending along the axis of the burner in a uniform axial stream. The pressure that the burnt gas exerts on the flame consists of a uniform part that does not change the velocity of the fresh gas plus a small variable part that decreases with height above the burner and therefore causes a streamwise acceleration of the fresh gas and a departure of the flame shape from a cone. The vorticity generated by the perturbed flame induces only a small correction to the flow of the burnt gas for large values of the injection velocity. The assumption of a constant burning velocity fails in small regions around the tip and the base of the flame which would require a separate analysis.
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This work was supported by the Comunidad de Madrid through project COMLIMAMS, S-0505/ENE-229.