Stabilization mechanisms of lifted laminar flames in axisymmetric jet flows

Stabilization Mechanisms of Lifted Laminar Flames in Axisymmetric Jet Flows YUNG-CHENG CHEN and ROBERT W. BILGER

Department of Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, N.S.W. 2006, Australia Stabilization mechanisms of lifted laminar propane flames are investigated in an axisymmetric jet flow configuration. Detailed mixing and flow fields upstream of the flame lift-off heights measured by Chung and coworkers [28 –30] are calculated on a nonreacting flow basis. The local stoichiometric axial velocity, U st , and scalar dissipation rate, ␹ st , are obtained at points that are upstream of the stabilization locations by a redirection region length. Variation is investigated with jet exit velocity, nozzle diameter, coflow air velocity as well as partial jet premixing with air. It is found that U st decreases consistently with increasing ␹ st when the centerline mixture fraction is higher than that of the rich flammability limit. Beyond this threshold, a rich-joined flame is formed at downstream locations with U st independent of the local scalar dissipation rate. Instead, U st decreases with increasing ␰ st , the stoichiometric mixture fraction of the jet flow. For lifted flames stabilized close to the burner exit with an edge-flame appearance, the maximum attainable mixture fraction gradient approaches that calculated for extinction of stretched diffusion flames in the counterflow geometry. The flame propagation velocity remains positive. The trend of decreasing U st with increasing ␹ st is also found for numerical simulations with a unity Lewis number [18] and for experimental data of lifted methane flames [19]. These results corroborate the triple flame stabilization concept. An empirical formula for the triple flame propagation velocity is proposed with a nonlinear dependency on local scalar dissipation rate. In addition, different flame stabilization mechanisms previously proposed for lifted turbulent diffusion flames are reconciled for lifted laminar flames, depending on the local flow/mixing conditions. Appropriate flame stability and blow-out criteria are derived and these predict the flame lift-off height and blow-out velocity accurately. Implications for flame stabilization in turbulent jet flows with such triple flame structures are also discussed. © 2000 by The Combustion Institute

INTRODUCTION The mechanism of stabilization of lifted flames is an important topic of both fundamental interest as well as practical application. Several proposed stabilization criteria for lifted turbulent jet diffusion flames have been reviewed and assessed by Pitts [1] in terms of local flow field/fuel–air premixing conditions at the flame stabilization location. He concluded that all the available theories are more or less limited in explaining the existing experimental data. The theories were based on the assumption that the flame structure in the stabilization region is either fuel/air fully premixed [2] or diffusioncontrolled [3]. For the former, a stability criterion can be defined as a balance between the flow velocity and turbulent burning velocity at the flame lift-off height, U st ⫽ S T ; while for the latter it is set by the reaching of a quenching scalar dissipation rate, ␹ st ⫽ ␹ q . The subscript *Corresponding author. E-mail: [email protected]. edu.au COMBUSTION AND FLAME 122:377–399 (2000) © 2000 by The Combustion Institute Published by Elsevier Science Inc.

st represents the stoichiometric condition. Recently, a refined model incorporating the above two criteria has been proposed [4] to emphasize the importance of both premixed flame propagation and diffusion flamelet quenching in a partially premixed environment. A triple flame structure [5–7] formed in a laminar mixing layer with a mixture fraction gradient is suggested as the characteristic flame stabilization structure. Particle imaging velocimetry (PIV) measurements [8 –10] have recently been made near the flame stabilization region of turbulent lifted jet flames for a range of flow Reynolds numbers up to 20,000. The conditionally measured local flow velocities are shown to be on the order of the laminar burning velocity of a stoichiometric o premixed flame S L . However, because of the unsteady nature of the leading-edge flame fronts, these experiments do not provide direct measurements of the instantaneous flame propagation velocity. Although the relatively low flow velocity seems to be consistent with the triple flame concept, no triple flame structures have been observed from the OH- [9] or CH0010-2180/00/$–see front matter PII S0010-2180(00)00120-6

378 laser-induced fluorescence [LIF] [10] images at the lifted flame base, where a ring-shaped structure is typically expected. Rather, an edge-flame with a well-defined propagation velocity [11, 12] appears to be a more appropriate description of the leading-edge structures of lifted jet flames stabilized on straight tubes [13, 14]. On the other hand, isolated flame fronts with an arrowheaded structure, similar to triple flames, have been observed in flame luminescence images of lifted nonpremixed flames in the hysteresis region on a bluff-body [15]. Large-scale vortex shedding from the recirculation zone dominates the fuel/air mixing process in these flames and creates a cluster of fragmented flame fronts at the lifted flame base. This provides a favorable condition for development of isolated triple flames propagating in a partially premixed turbulent flow. The observed distorted flame fronts have indeed quite similar appearance to those from direct numerical simulations (DNS) of vortex/triple flame interactions [16]. In view of this emerging evidence, this work is intended to investigate in greater detail the stabilization mechanisms of lifted laminar jet flames, particularly in relation to the triple flame concept. Although the unsteady nature of turbulence is excluded in laminar flows, local velocity and mixing fields responsible for flame stabilization can be accurately determined to quantify the flame propagation ability. Moreover, validation of the triple flame concept in laminar jet flows constitutes a necessary step in its application to the much more complicated turbulent lifted flames where interaction of the turbulent strain field with the triple flame can be significant. The structure of laminar triple flames assuming equal diffusivity has been investigated in a strained planar mixing-layer geometry without considering heat release [7, 17]. At large Damko ¨hler numbers or small scalar dissipation rates, the flame propagation velocity approaches a constant value. With increasing scalar dissipation rates or strain rates, the propagation velocity decreases monotonically and becomes negative at sufficiently high strain rates [7, 17]. The main effect of heat release is to diverge the streamlines in front of triple flames [18] so that the far-field flow velocity, U p , exceeds the laminar burning velocity of a planar

Y.-C. CHEN AND R. W. BILGER o stoichiometric premixed flame, S L . At the limit of large mixing-layer thickness, or small scalar dissipation rate, the velocity ratio reaches an asymptotic value that is proportional to the square root of density ratio between unburnt and burnt gases,

冑

Up ␳u , o ⬀ ␳b SL

(1)

assuming constant pressure across the triple flame along the streamtube wrapping around the flame. Thus, flame stability is enhanced as the flow velocity at the triple point, the joint point of the premixed flame front and the o trailing diffusion flame, remains close to S L . This behavior is further corroborated in numerical simulations of lifted methane flames [19, 20]. The effects of nonunity Lewis numbers, Le, have been studied [21] by applying asymptotic analysis in a one-dimensional model of edge flames. At the large Damko ¨hler number limit such that the reaction rate of the edge flame is much higher than that of the trailing diffusion flame, the edge speed decreases nonlinearly with a decreasing mixing length scale which characterizes heat and mass transport in the transverse direction. It is also found that edge flames of the same mixing length but larger Le numbers have higher edge speeds. In a special case of unity Le number [21], the edge speed U F decreases with increasing scalar dissipation rate ␹ (assuming inversely proportional to the square of the mixing length) in a linear form of UF ␹ o ⫽ 1 ⫺ ␹o SL

␹ o is the scalar dissipation rate when U F is zero. This formula is shown in Ref. 21 (cf. Fig. 3) to be consistent with results [7, 17] obtained from a two-dimensional mixing-layer only at large Damko ¨hler numbers. In a planar mixing-layer geometry, other studies [22, 23] have found that the unequal diffusivities between fuel and oxidizer streams modify the propagation of triple flames in a similar way as the stretched premixed flames; e.g., see Ref. [24]. This is also supported by numerical calculations [23], in which the flame-front velocity U F increases with decreasing fuel Lewis numbers. Overshoot of

LAMINAR FLAME STABILIZATION U F occurs at Le ⬍ 1. It is noted that these analyses on the Le number effects [21–23] used a constant-density assumption. In the absence of heat release, the far-field flow velocity U p remains the same as the flame-front velocity U F at the triple point. Recent DNS studies of methanol–air [25] and hydrogen–air [26] triple flames have shown that the Lewis number effect is almost negligible in the enhancement of triple flame speed. Despite these many analyses, there has not been any attempt yet to quantify the triple flame propagation velocity, U p , as well as its parametric dependency. In continuation of preliminary work [27], this paper is intended to address these unresolved issues with a combined experimental/numerical method. The lift-off characteristics of propane triple flames have been investigated in an axisymmetric laminar jet flow configuration in a series of well-controlled experiments [28 –30]. The variation of flame liftoff height with jet exit velocity, nozzle diameter, coflow air velocity as well as partial premixing with air has been reported. A tribrachial, or triple, flame structure is observed at the base of these lifted flames over a range of initial conditions. The flow/mixing field between nozzle exit and the flame stabilization positions is shown to be the same as that in nonreacting jet flows up to a redirection region of 1 ⬃ 2 mm ahead of stabilized triple flames. Thus, computational demands can be drastically reduced to solve only the mass and momentum conservation equations as these are sufficient to determine the upstream flow/mixing structure responsible for flame stabilization. It is expected that from a better understanding of the flame stabilization conditions the lift-off and blow-out phenomena can be predicted more accurately. This paper begins with a brief summary of the boundary-layer approximations for axisymmetric laminar jet flows. The numerical methodology used in this work is then introduced. Comparisons of computational results with analytical predictions as well as experimental data are also presented to validate the accuracy of numerical solutions. The scaling laws derived by Ruetsch et al. [18] are further extended for asymmetric triple flames in an axisymmetric geometry. The propagation ability of lifted propane flames is investigated by showing the relationship be-

379 tween the stoichiometric flow velocity, U st , and scalar dissipation rate, ␹ st , upstream from the measured flame lift-off heights by a redirection region length. The Le number effect is also examined by reanalyzing the numerical data of Le ⫽ 1 simulations [18] as well as calculating the nonreacting flow/mixing fields in front of the lifted laminar methane flames reported in Ref. 19. Consistent with previous observations in lifted turbulent nonpremixed flames [15], different stabilization mechanisms are found to apply in the present laminar jet flames. The appropriate flow/mixing conditions for these mechanisms are discussed. Finally, lifted flame stability and blow-out conditions via triple flame structures are derived and compared with previously proposed criteria [31] for lifted laminar jet flames. LAMINAR JET FLOW THEORY Considering a laminar jet flow in a two-dimensional axisymmetric cylindrical coordinate ( x, r), the jet momentum flux J and fuel mass flow ˙ F are expressed as: rate M J⫽

␲␳ jU j2d 2 3⫹n

˙F⫽ M

共n ⫽ 0 or 1兲

␲␳ jU jd 2Y F, j 4

(2a) (2b)

where U j is mean jet exit velocity based on the volumetric flow rate, d is nozzle diameter, ␳ j is jet fluid density, and Y F, j is fuel mass fraction at nozzle exit. The subscript j is referred to as the jet exit condition. The index n indicates the nozzle exit velocity profile of a Poiseuille (n ⫽ 0) or a uniform (n ⫽ 1) distribution. Boundary-layer theory [32, 33] usually provides good approximations for the jet downstream flow/mixing structure. The axial velocity and fuel mass fraction can be written as follows: U⫽

3J 1 8 ␲␳ ⬁␯ ⬁x 共1 ⫹ ␩ 2/4兲 2

YF ⫽

˙F 共2Sc ⫹ 1兲M 1 8 ␲␳ ⬁␯ ⬁x 共1 ⫹ ␩ 2/4兲 2Sc

(3a) (3b)

where ␯⬁ and ␳⬁ are the kinematic viscosity and density of the ambient fluid, subscript ⬁, which

380

Y.-C. CHEN AND R. W. BILGER

is air throughout this work. Sc ⬅ ␯ ⬁ /D F is the Schmidt number with D F the diffusion coeffi2 cient of the fuel species. ␩ ⬅ 公3J/16 ␲␳ ⬁ ␯ ⬁ (r/x) is the transformed similarity variable. Some dimensionless variables which are frequently used to characterize jet flow development are defined below for later comparison with numerical calculations. These include the normalized centerline axial velocity U CL /U j,o and fuel mass fraction Y F,CL /Y F, j as well as their half-maximum radius, r 0.5,U /d e and r 0.5,Y /d e . Based on Eq. 3, the normalized centerline distributions, subscript CL, are derived as follows: 3 U CL ⫽ U j,o 8共3 ⫹ n兲共2 ⫺ n兲 X

冉

2 ␹ CL 2Sc ⫹ 1 ⫽ 32 D F/d e2 Re 2X 4

(4a) (4b)

冊

2

(4c)

Note Eq. 4 can also be derived from the exact solution of a point momentum source issued into an unbounded quiescent environment [34, 35]. The half-maximum radii are expressed as:

冑

共n ⫽ 0 or 1兲

冑

共n ⫽ 0 or 1兲

3 ⫹ n 1/ 2 r 0.5,U ⫽8 共2 ⫺ 1兲 X de 3 r0.5,Y 3 ⫹ n 1/ 2Sc ⫽8 共2 ⫺ 1兲 X de 3

(5a)

(5b)

In Eqs. 4 and 5, U j,o ⬅ (2 ⫺ n)U j is the centerline axial velocity at nozzle exit (n ⫽ 0 or 1) and d e ⬅ 公␳ j / ␳ ⬁ d is the effective nozzle diameter. Re ⬅ U j d e / ␯ ⬁ is the jet Reynolds number. X ⬅ ( x/d e ) Re ⫺1 is the dimensionless axial distance. ␹ ⬅ 2D F (ⵜ ␰ ) 2 is the scalar dissipation rate with ␰ the mixture fraction based on Bilger’s formula [36]. ␰ equals 1 at the fuel jet exit and 0 in ambient air. In nonreacting flows, ␰ is reduced to:

␰⫽

␥ Y F ⫺ Y O ⫹ Y O,⬁ ␥ Y F, j ⫺ Y O, j ⫹ Y O,⬁

冑 冋冉 冊

共3 ⫹ n兲 Xst rst ⫽8 de 3 X 共n ⫽ 0 or 1兲

共n ⫽ 0 or 1兲

Y F,CL 共2Sc ⫹ 1兲 ⫽ Y F, j 32X

riving Eq. 4c, ␰ CL is taken as Y F,CL /Y F, j . This is true as long as the stoichiometric fuel mass fraction is the same as that of a pure fuel jet and the molecular differential diffusion is neglected, i.e. Y O ⫽ 0.233(1 ⫺ Y F ). Because the triple flame is stabilized along the stoichiometric contour, approximations of the normalized stoichiometric radius, axial velocity, as well as scalar dissipation rate in case of ␰ ⫽ Y F /Y F, j can also be derived as:

(6)

where Y O,⬁ ⫽ 0.233 for air and ␥ is the stoichiometric oxidizer–fuel mass ratio. In de-

⫺1 X

冉 冊

3 U st X ⫽ U j 8共3 ⫹ n兲 X X st

(7a) 1/Sc

3Sc 2␰ 2st ␹ st 2⫽ D F/d e 2共3 ⫹ n兲 X 2 䡠

冋冉 冊 X X st

册

1/ 2Sc

1/ 2Sc

⫺

共n ⫽ 0 or 1兲 (7b)

冉 冊 册 X X st

1/Sc

共n ⫽ 0 or 1兲

(7c)

where X st ⬅ (2Sc ⫹ 1)/(32 ␰ st ) is the normalized axial position where the stoichiometric mixture fraction contour intersects with the jet centerline. For simplicity, only the radial component of mixture fraction derivatives is considered in deriving Eq. 7c. Both components, however, are included in the calculations of ␹ st from numerical solutions. It is noted that the axial decay of ␹ st in Eq. 7c along the stoichiometric contour is slower than ␹ CL in Eq. 4c along the centerline. NUMERICAL CALCULATIONS Numerical Methods Two different burner configurations are calculated in this work. One is a single jet flow of different nozzle diameters, case SINJ [28], or of different air dilution percentages, case SINJA [30]; the other is a double concentric jet flow with different coflow air velocities, U a , case TWINJ [29]. Experimental conditions of these three cases are summarized in Table 1. A

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381

TABLE 1 a

Experimental Conditions for the Three Calculated Cases

d (mm) Y F,j U a (m/s) n

SINJ

SINJA

TWINJ

varied 1.0 0.0 1

0.195 varied 0.0 1

0.37 1.0 variable 0

a

The exact values of variables in each case are given in Fig. 9. The index n specifies the nozzle exit velocity profile. n ⫽ 0 for a Poiseuille exit profile; 1 for a uniform profile.

detailed description of the numerical method is given here for the TWINJ case. An axisymmetric coordinate system of 750 mm in axial ( x⫺) and 150 mm in radial (r⫺) direction is adopted as the computational domain, as shown in Fig. 1. Orthogonal grid points of 100 by 160 are distributed unequally in the axial and radial directions, respectively. The burner exit plane is located at x ⫽ 100 mm so that jet-induced entrainments from upstream of the burner exit can be fully accounted for. Along the radial direction near both the central and outer tubes, the grid size is smallest of 0.017 mm ⬇ 0.046 d.

Fig. 1. Burner geometry for numerical computations in case TWINJ. The applied boundary conditions are also specified.

The grid size along the axial direction increases monotonically from 0.28 mm ⬇ 0.75 d at the burner exit to 20.84 mm at the downstream exit plane. The continuity, momentum (U and V), and species (Y F as well as Y O ) mass conservation equations are solved, as given in Eq. 8. ⭸ ⭸ 共r ␳ U兲 ⫹ 共r ␳ V兲 ⫽ 0 ⭸x ⭸r

冋

1 ⭸ ⭸␬ r ␳ U ␬ ⫺ r⌫ ␬ r ⭸x ⭸x ⫹

冋

册

(8a)

册

1 ⭸ ⭸␬ r ␳ V ␬ ⫺ r⌫ ␬ ⫽ S␬ r ⭸r ⭸r

(8b)

where ␬ represents U, V, Y F , and Y O respectively. ⌫␬ and S ␬ are the generalized transport coefficients and source terms which are given in Table 2. Gas mixture density, ␳, is calculated from the ideal gas law. The energy equation is not considered since the main concern here is to investigate the upstream unreacted flow and mixing fields of a lifted laminar jet flame. Temperatures of both the fuel and air streams are set at the ambient temperature, 293 K. The inlet boundary conditions for the central fuel jet and annular air flows are taken as Y F ⫽ 1, Y O ⫽ 0, V ⫽ 0, ⭸P/⭸ x ⫽ 0, and Y F ⫽ 0, Y O ⫽ 0.233, V ⫽ 0, ⭸P/⭸ x ⫽ 0, respectively. At the Reynolds numbers of interest here, there is little difference between uncontaminated and no net flux boundary conditions [37]. The initial axial velocity profiles are prescribed as a Poiseuille flow for the fuel stream and a uniform flow for the coflow air stream. The other boundary conditions are specified in Fig. 1. The lateral boundary condition ⭸V/⭸r at r ⫽ 150 mm implies that V does not decay with increasing radius outside the boundary. A sufficiently large computational domain is thus needed to ensure that the effects of a constant radial velocity outside the domain can be neglected. This is confirmed by a radial velocity less than 1 mm/s at the chosen boundary. For cases SINJ and SINJA, a large enclosure of 300 ⫻ 300 ⫻ 500 mm3 surrounds the nozzle. The lateral boundary condition of U ⫽ V ⫽ 0 is therefore used. Global mass conservation is enforced in calculating the U component at the downstream exit plane. A comparison case is also run with doubled axial distance to test the downstream

382

Y.-C. CHEN AND R. W. BILGER TABLE 2 Expressions for the Transport Coefficients ⌫␬ and Source Terms S ␬ for Variable ␬ in the Generalized Transport Equation 8b

␬

⌫␬

U

␮

⫺

V

␮

⫺

YF YO

DF DO

DO ⫽

冋

1 ⫺ XF XO 1 ⫺ XF ⫺ XO ⫹ D FO D FN

冋

册

1 ⫺ XO 1 ⫺ XF ⫺ XO XF ⫹ D ON D FO

册

冋 册 冋 冋 册 冋 册

册

⭸P ⭸ 1 ⭸ 2 ⭸ ⭸U ⭸V ⭸U ␮ ⭸共rV兲 ␮ ⫹ r␮ ⫺ ␮ ⫹ ⫹ ⭸x ⭸x ⭸x r ⭸r ⭸x 3 ⭸x ⭸x r ⭸r

boundary condition. The difference in the axial velocity at x ⫽ 650 mm is found to be less than 0.1%, thus the effects are negligible. The differential diffusion effects are considered in numerical calculations by using an effective species diffusion coefficient in the mass conservation equations for propane, D F , and oxygen, D O , respectively: DF ⫽

冋 册

S␬

(9a)

冋

册

V ⭸U ⭸V ⭸U ␮ ⭸共rV兲 ⭸P ⭸ 1 ⭸ 2 ⭸ ⫺ 2␮ 2 ⫹ ⫹ ␮ ⫹ r␮ ⫺ ␮ ⭸r r ⭸x ⭸r r ⭸r ⭸r 3 ⭸r ⭸x r ⭸r 0 0

⑀⫽

max兩 ␬ i,m⫹1 ⫺ ␬ i,mj兩 j max兩 ␬ i,m⫹1 j 兩

⬍ 5 䡠 10 ⫺5

(10)

Here m is the iteration number, and i and j are the grid points. In addition, summation of normalized residual from all the grid points is checked to be less than 0.1%. Tests have also been made to ensure that the numerical solutions are independent of the chosen computational domain and grid distribution. Comparison with Boundary-Layer Approximations

(9b)

where D FO ⫽ 0.109 cm2/s, D FN ⫽ 0.113 cm2/s, and D ON ⫽ 0.204 cm2/s are the binary diffusion coefficients of propane in oxygen, propane and oxygen in nitrogen, respectively. X i is the local mole fraction of species i (i ⫽ F or O). Gas mixture viscosity, ␮, is calculated with Wilke’s approximation [38]. Discretization of Eq. 8 is based on integration over a finite control volume. A staggered grid system is applied to derive the finite-difference equations [39]. The convection– diffusion terms of the transport equations, Eq. 8b, are formulated based on a power-law differencing scheme. The velocity–pressure coupling is resolved using the SIMPLE algorithm [40]. Therefore, the final algebraic equations can be solved with a line-by-line procedure using a tridiagonal matrix algorithm. Sequential iterations of each variable, ␬, are performed until the following convergence criterion is satisfied.

Numerical results are first compared with predictions based on the boundary-layer theory. Figure 2 shows the calculated centerline velocity and mixture fraction distributions for case SINJ with a uniform exit velocity profile. The linear decay of both U CL /U j,o and Y F,CL /Y F, j predicted in Eq. 4 are well reproduced at down-

Fig. 2. Comparison of the normalized centerline axial locity, U CL /U j,o , and fuel mass fraction, Y F,CL /Y F, j , for case SINJ with a uniform exit velocity profile. Lines from boundary-layer approximations in Eq. 4; symbols numerical solutions.

vethe are are

LAMINAR FLAME STABILIZATION

383

Fig. 3. Comparison of the normalized half-maximum radius, r 0.5,U /d e and r 0.5,Y /d e for the case SINJ with a uniform exit velocity profile. Lines are from boundary-layer approximations in Eq. 5; symbols are numerical solutions.

stream axial positions. This is expected as the jet flow becomes self-similar in the far field even without the introduction of a virtual origin. The calculated potential core length agrees also quite well with the theoretical prediction of X ⫽ 0.0307 [41]. A linear jet expansion is verified in Fig. 3 where r 0.5,U /d e and r 0.5,Y /d e increase linearly with the normalized axial distance, X, in accord with Eq. 5. Of particular interest here are the axial velocity and scalar dissipation rate distributions along the stoichiometric contour, ␰ ⫽ ␰ st ⫽ 0.06. Comparisons of numerical results with Eq. 7 are shown in Fig. 4 for case SINJ. The calculated rich flammability contour r rich/d e at ␰ ⫽ ␰rich ⫽ 0.154 for case SINJ is also plotted for later discussion. The normalized stoichiometric radius, r st /d e , is well predicted by Eq. 7a for X ⬎ 0.7. This is consistent with the good agreement in stoichiometric scalar dissipation rate, ␹ st , at downstream locations. However, large discrepancies between calculations and boundary-layer approximations are observed close to the nozzle exit in both the mixing and flow fields. This is attributed to the influence of the ⭸ 2 /⭸r 2 term in Eq. 8b, which is not considered in the boundary layer formulation, as well as the no-slip boundary condition for the burner wall at the inlet (cf. Fig. 1). Particularly, the stoichiometric axial velocity, U st , decreases drastically when X approaches zero, in contrast to Eq. 7b. The discrepancies in U st between numerical and boundary-layer solutions can be reduced by using the virtual origin concept. Thus, the nor-

Fig. 4. Comparison of the normalized stoichiometric radius, r st /d e , axial velocity, U st /U j , and scalar dissipation rate, ␹ st /(D F /d e2 ), for the case SINJ with a uniform exit velocity profile. Lines are from boundary-layer approximations in Eq. 7; symbols are numerical solutions.

malized axial distance X in Eq. 7b is replaced by X ⫹ X ␯ , where X ␯ is the location of the virtual origin upstream of the jet exit. The thick dash line in Fig. 4b shows that the numerical solutions match well with Eq. 7b at downstream locations for X ␯ ⫽ 0.3. Close to the nozzle exit, however, there is still a lack of good agreement between numerical results and boundary-layer solutions. This indicates that the boundary-layer approximations are not able to reproduce the realistic flow/mixing fields accurately, and thus are not used in this work.

384

Fig. 5. Comparison of numerical calculations (lines) and experimental data (symbols) [29] at measured flame stabilization heights, x, for two laminar jet flows in case TWINJ with a coflow air velocity of 0.3 m/s. The solid lines are the axial velocity, U; the dashed lines are the mixture fraction, ␰. The stoichiometric radial position, r st 兩 flow, is indicated with an arrow. The vertical dashed lines show the measured flame stabilization position at the lifted flame base, r st 兩 flame.

Comparison with Experimental Data Velocity measurements in laminar nonreacting jet flows have been conducted for case TWINJ [29] and can be used to validate the present numerical solutions. The good agreement shown in Fig. 5 between experimental and numerical data from near nozzle ( x ⫽ 43 mm) to far downstream ( x ⫽ 175 mm) axial positions indicates that the nonreacting flow field of laminar jets is well predicted. The calculated radial mixture fraction profiles are also shown in Fig. 5. At the radial positions where lifted flames are stabilized, a constant axial velocity of 0.53 m/s has been measured in nonreacting jet flows [29], independent of the nozzle exit velocity U j . These radial positions are marked by a vertical dash line in Fig. 5 for two different exit velocities. A much higher axial velocity is, however, found at the calculated stoichiometric radius, r st兩flow , and it increases with increasing U j.

Y.-C. CHEN AND R. W. BILGER The difference in the radial stoichiometric positions between reacting and nonreacting flows can be explained by the heat release effects [18]. For previous analyses [7, 18, 23] conducted in a planar mixing-layer geometry with an assumed stoichiometric mixture fraction of 0.5, the redirection of streamlines in front of the triple flame is symmetric about the stoichiometric contour, which remains at the same cross-stream position as in nonreacting flows. For the axisymmetric jet flows considered in this work, the stoichiometric contour is not aligned with the symmetric centerline so that the lifted flame is stabilized at an off-center location. The redirection of flow streamlines becomes nonsymmetric because the velocity of approaching flows is higher on the fuel-rich side than that on the fuel-lean side. This deflects the stoichiometric contour to be at a positive angle with the jet centerline. Thus, the triple point of the lifted flame will be located at a larger radial position than the stoichiometric point in the corresponding nonreacting flow. The reported constant flow velocity of 0.53 m/s [29] therefore can not be related to the triple flame propagation velocity because the measurements were taken at the flame stabilization radius, but in nonreacting flow. Without flow redirection, the mixture fraction at these radial locations is less than the stoichiometric value. This effect can be clearly seen in Fig. 9 of Ref. 29, where the radius of the flow streamline going through the stabilization point of a lifted flame increases by approximately 0.7 mm at a U j of 7.9 m/s. A radial shift in the stoichiometric position of this magnitude can change substantially the relevant axial velocity in nonreacting flows, as demonstrated in Fig. 5. The above argument is further examined in Fig. 6, which compares directly the calculated radial mixture fraction distributions in nonreacting case with measurements in the reacting case at the flame stabilization height as well as 5 mm upstream of a lifted laminar methane flame [19]. For these numerical calculations, the initial axial velocity of outer air coflow has been adjusted to match with the measured profile as given in Fig. 2 in Ref. 19. Boundary conditions of no net scalar flux [37] were used, because of the small Reynolds number of only 44 for the central fuel jet. In Fig. 6, the agreement of

LAMINAR FLAME STABILIZATION

385

Fig. 7. Comparison of the axial velocities along the line r ⫽ 3.2 mm between nonreacting flow calculations and PIV measurements (Fig. 11 in Ref. 19) for the flame of the same conditions as in Fig. 6.

Fig. 6. Comparison of the radial mixture fraction, ␰, profiles at flame stabilization height ( x ⫽ 38 mm) as well as 5 mm upstream between a nonreacting (calculations, line) and a lifted reacting (measurements, symbol), nitrogen-diluted, laminar methane jet [19]. The jet exit velocity is 1.178 m/s, Y F, j ⫽ 0.679, Y O,⬁ ⫽ 0.233. Note the increase of stoichiometric radius at x ⫽ 38 mm, but not at x ⫽ 33 mm.

mixture fraction ␰ between reacting and nonreacting cases is greatly improved when moving from the flame stabilization point upstream. This can be attributed to the decline of the heat release effects with increasing distance away from the lifted flame base. At x ⫽ 33 mm, there is fairly good correspondence in ␰ from lean to rich mixtures, except close to the jet centerline. Detailed analyses show that the measured centerline fuel mass fraction is ⬃15% less than the present calculations from x ⫽ 33 to 38 mm. Apart from this discrepancy, the calculated axial velocity in the nonreacting case compares quite well with PIV measurements upstream of the triple flame, as shown in Fig. 7. In comparing the axial coordinates between Figs. 6 and 7, it is to be noted [19] that the flame stabilization height was shifted 6.5 mm downstream when the flow was seeded with particles. In nonreacting flows, Fig. 6 shows that the calculated stoichiometric radial position re-

mains the same at r ⫽ 2.5 mm from x ⫽ 33 to 38 mm, whereas the measured stoichiometric radial position increases by approximately 0.5 mm in reacting flows. This supports the argument that the stoichiometric contour is deflected close to the leading edge of an asymmetric triple flame. The deviation of 0.5 mm is approximately one-third of the radial distance between the lean and rich flammability limits at the flame stabilization height, denoted as w st . w st can be regarded as the characteristic size of the triple flame. The same order of deflection in the stoichiometric radius r st can also be found in Fig. 5 for the x ⫽ 43 mm case. The relatively large deviation of approximately (2/3)w st at the far downstream position of x ⫽ 175 mm is attributed to the merging of two fuel-rich branches at jet centerline, which ultimately limits the growth of the lifted flames with increasing axial distance. To illustrate the redirection effects of the flow/mixing fields, a schematic diagram is plotted in Fig. 8 for a nonsymmetric triple flame in axisymmetric coordinates. The increase of r st in front of the lifted flame has been qualitatively reproduced in numerical calculations of a lifted turbulent methane jet flame based on the triple flame concept [4]. A SCALING LAW In view of Fig. 8, the mixing-layer thickness immediately preceding the flame, ␦2, is larger

386

Y.-C. CHEN AND R. W. BILGER

Fig. 8. Schematic diagram for the redirection region in front of lifted triple flames in the axisymmetric coordinate. Note the increase of stoichiometric radius in the redirection region is not present in the nonreacting flow calculations. Four locations along the stoichiometric line, ␰ ⫽ ␰ st , are numbered: (1) far upstream; (2) immediately preceding the flame; (3) immediately following the flame; (4) far downstream.

than that in the far upstream position, ␦1. Here ␦ i is the annulus width between the streamlines at location i in Fig. 8. Based on scaling laws derived by Ruetsch et al. [18], the ratio ␦1/␦2 scales with the square root of the density ratio of burnt to unburnt gases, 公␳ b / ␳ u , in the planar mixing-layer geometry (see Appendix for the derivation). In a jet flow configuration, the increase of the stoichiometric radius r st modifies this scaling as:

冑

␦1 ␳ b r st兩 2 ⬀ 䡠 ␦2 ␳ u r st兩 1

(11)

The ratio of mixing layer thickness in Eq. 11 is inversely proportional to the ratio of stoichiometric mixture fraction gradient under the assumptions of a linear decrease of mixture fraction with increasing radius near stoichiometric and an unity Schmidt number. Because of the many assumptions involved, the approximation of ⵜ ␰ 兩 st,2 /ⵜ ␰ 兩 st,1 with Eq. 11 is only qualitative. Nevertheless, a quantitative check of its correctness can be made for the flame in Fig. 6. Assuming a realistic density ratio of 1/6, the term on the right-hand side of Eq. 11 is 0.51, which is approximately 20% less than the measured mixture fraction gradient ratio ⵜ ␰ 兩 st,2 / ⵜ ␰ 兩 st,1 of 0.63. One can thus expect that the ratio of mixing layer thickness ␦1/␦2 shows qual-

itatively the same trend as the ratio ⵜ ␰ 兩 st,2 / ⵜ ␰ 兩 st,1 . This implies a decrease of mixture fraction gradients, or scalar dissipation rates, along the stoichiometric line toward the stabilized triple flame, specifically point 2 in Fig. 8. A favorable environment of lower scalar dissipation rates is thus created by the divergence of flow/mixing fields in front of the triple flame i.e. ␹ st,2 ⬍ ␹ st,1 . In extreme situations where ␹ st,1 but not ␹ st,2 can quench a steady diffusion flame, the trailing diffusion flame of the triple flame would still survive. This feature is consistent with the decrease of flow velocity along the stoichiometric line from point 1 to point 2 in Fig. 8, which enhances the flame propagation ability of the premixed front. It is reasonable to assume that the mixture fraction gradient remains the same across the triple flame, i.e. ⵜ ␰ 兩 st,2 ⫽ ⵜ ␰ 兩 st,3 . However, the scalar dissipation rate ␹ st ⫽ 2D F (ⵜ ␰ 兩 st ) 2 can increase substantially because the diffusion coefficient D F is a strong function of temperature, i.e. D F ⬀ T 1.67 . With a temperature ratio of 6, the measured ratio of stoichiometric scalar dissipation rate ␹ st,3 / ␹ st,1 becomes 7.9 for the flame in Fig. 6. In general, a much smaller scalar dissipation rate can always be expected ahead of the lifted triple flames in the upcoming cold flows than that of the trailing diffusion flame, i.e. ␹ st,3 ⬎ ␹ st,1 . This statement does not invalidate the previous argument that triple flame can be stabilized in an environment where a diffusion flame would otherwise be quenched, which corresponds to ␹ st,2 ⬍ ␹ st,1 . It rather indicates that a flame stabilization mechanism based on diffusion flamelet quenching [3] should not compare ␹ st,2 or ␹ st,1 with the quenching scalar dissipation rate ␹ q usually reported for strained diffusion flames. This is because the latter is evaluated at flame temperature and thus can be substantially higher than the scalar dissipation rates in front of the lifted flames at ambient temperature. For a temperature ratio of 6, the scalar dissipation rate can increase by a factor of ⬃20 across the premixed flame-front of the triple flame even though the mixture fraction gradient at stoichiometric remains the same. This provides an explanation of the relatively low scalar dissipation rates measured in the unburnt mixtures where turbulent lifted flames can be stabilized [42, 43].

LAMINAR FLAME STABILIZATION Another important consequence from Fig. 8 is that the calculated nonreacting flow solutions are only valid up to a redirection region in front of lifted flames. The length of the redirection region is estimated to be 2 mm for all the investigated cases SINJ, SINJA, and TWINJ based on velocity measurements [29]. For a stabilized triple flame, the stoichiometric axial velocity in front of the redirection region is analogous to the far-field velocity, U p [18], which is not modified by the heat release effects of triple flames and remains the same as that in nonreacting flows. The axial velocity at the leading edge of the lifted flame base, i.e. at the triple point, is known [18 –20] to be very close to o the stoichiometric laminar burning velocity, S L , and thus is subject mainly to flame chemistry. The deceleration of flow velocity within the redirection region is due to the divergence of streamline tubes across the triple flame [44]. Upstream of the redirection region, the nonreacting flow solutions should still apply, as can be seen in Figs. 6 and 7. It is believed that the stabilization ability of lifted triple flames is directly related to the far-field flow and mixing structures in front of the redirection region, indicated by the “star” symbol in Fig. 8. This argument is discussed in more detail in the next section. RESULTS AND DISCUSSION After ignition at suitable jet downstream positions, a lifted triple flame propagates upstream along the stoichiometric contour and is stabilized at a certain lift-off height if flame attachment does not occur. When this distance from the leading edge of the triple flame to nozzle exit is much longer than the characteristic thickness of the mixing layer, flame stabilization is achieved by the balance between the triple flame propagation velocity U p and the local flow velocity at stoichiometric [45]. Although the triple flame stabilizes itself in the physical location of the triple point, the propagation velocity of the entire triple flame structure is not the flow velocity at the triple point but that at the point in front of the redirection region [18]. Thus, the triple flame stabilization mechanism can be expressed as

387 U p ⫽ U st,

(12)

where U st can be obtained from the nonreacting flow calculations ahead of the lifted flame base by a redirection region length and is not affected by the flow redirection effects. One of the main purposes in this study is to quantify the propagation velocity of triple flames and its parametric dependency based on reported experimental data of flame lift-off heights [28 –30]. Because there are no redirection effects at the point where U st is to be obtained, the much smaller radial velocity can be neglected and U st identifies itself with the axial velocity. It is also considered that U p depends mainly on the local mixing structure, which can be characterized by the stoichiometric scalar dissipation rate, ␹ st [27]. To verify this argument, U st is plotted versus ␹ st in Fig. 9 for all the investigated cases SINJ, SINJA, and TWINJ. Values of U st and ␹ st are taken from nonreacting flow calculations at 2 mm upstream of the measured flame lift-off heights. These data are within 3% of those obtained at the exact flame stabilization positions due to the short redirection region. This 3% uncertainty is also regarded as the maximum error due to the assumption of a constant length of the redirection region. Figure 9 shows some interesting behavior of lifted triple flames that deserve further discussion. For clarity, these issues are first listed here: (1) Why does U p decrease consistently with increasing ␹ st ? (2) Why is the maximum value of ␹ st limited by 1/s? (3) Why is the decrease of U p with ␹ st nonlinear? (4) Why does U p level off at smaller values of ␹ st ? Decrease of Up with Increasing ␹st In case SINJ of Fig. 9, U p decreases consistently with increasing ␹ st for high scalar dissipation rates regardless of the exit nozzle diameter, d. The same trend is also found for the other two cases where the values of Y F, j and U a are changed, respectively. A consistent decrease of U p with increasing ␹ st among these cases indeed supports the argument that U p depends mainly on the local scalar dissipation rate ␹ st immediately upstream of the lifted flame. Different jet initial conditions, such as U j , d, Y F, j , and U a , affect flame stability, or lift-off height, only via

388

Fig. 9. Log–log plots of the calculated U st and ␹ st taken at 2 mm (redirection region length) upstream of the measured flame stabilization heights for all three investigated cases. At this position, U st equals the triple flame propagation velocity, U p . Data points of the same symbol represent experimental initial conditions of different nozzle exit velocity, U j , with the other variables kept constant.

the modification of local flow/mixing field. In any case, lifted flames will adjust themselves to be stabilized at an axial position where the upstream flow velocity at stoichiometric equals the triple flame propagation velocity, as expressed in Eq. 12. The Eddy Dissipation Concept [46] argues that a lifted flame is stabilized when flame stretch equals the extinction limit of a premixed flame, and the smallest flow time is the critical time scale for flame extinction. However, the attempt to correlate the stoichiometric

Y.-C. CHEN AND R. W. BILGER ᠬ with U p was unsucstrain rate a st ⬅ ⫺nn: ⵜV cessful. Here n ⬅ ([1 ⫹ (dr st /dx) 2 ] ⫺0.5 , [1 ⫹ (dr st /dx) ⫺2 ] ⫺0.5 ) is a unit vector along the ᠬ ⬅ (U, V) is the stoichiometric contour and V flow velocity. This could be due to the fact that scalar dissipation rate does not scale linearly with the strain rate for laminar jets as it does in the counterflow geometry [47]. Thus, it can be concluded that the mixing structure, rather than the flow field structure as represented by a st , is more appropriate for quantifying the triple flame propagation velocity. The physical meaning of the parameter ␹ st used to correlate U p in Fig. 9 is better understood as a measure of the mixing strength in the lateral direction, i.e. the mixture fraction gradient across the triple flames rather than a characteristic diffusion time scale. The larger the mixture fraction gradient, the stronger the lateral diffusion of heat and species away from the premixed front of the triple flame [18]. Thus, the divergence of the flow field in front of the triple flame becomes less effective. This implies that triple flame stabilization mechanism is more related to the fluid dynamic effects of triple flame propagation rather than flame extinction, consistent with Eq. 12. The heat release rate of the trailing diffusion flame, however, may still scale with the stoichiometric scalar dissipation rate in a similar way as that for a one-dimensional laminar diffusion flame. This argument is further supported in discussions below. Maximum Limit of ␹st The decrease of U p with increasing ␹ st shown in Fig. 9 is qualitatively consistent with previous analyses of laminar triple flames [7, 18]. On the other hand, it has been experimentally observed [28] that the premixed fronts of lifted triple flames in case SINJ become weakened or even disappear when the stabilization location moves close to the nozzle exit. The flame appears rather like an edge flame. Thus, it would be interesting to examine whether the triple flame at the limit of large scalar dissipation rates approaches an edge flame, and edge flame extinction may represent the appropriate stabilization mechanism. The maximum scalar dissipation rate reached in case SINJ is on the order

LAMINAR FLAME STABILIZATION of 1/s (cf. Fig. 9). For this flame, the calculated radial distance between rich and lean flammability limit is only 0.58 mm, which is of the same order of the laminar thermal flame thickness of a stoichiometric propane/air flame. At a first look, ␹ st ⬇ 1/s is still far away from the extinction limit of propane diffusion flames where the stoichiometric scalar dissipation rate has been recently calculated to be 27.7/s* using a C-3 mechanism in an opposed jet configuration [48]. However, as discussed previously with Fig. 6, the value of ␹ st in front of the triple flame would be much smaller than that of the trailing diffusion flame due to the strong temperature dependency of diffusion coefficients involved in the definition of scalar dissipation rates. The use of mixture fraction gradient ⵜ ␰ 兩 st to characterize extinction of lifted flames can avoid this dependency. In fact, the square of stoichiometric mixture fraction gradient at the flame extinction limit, (ⵜ ␰ 兩 st ) q2 , is only 3.5/cm2 from counterflow calculations, which is even smaller than 4.59/cm2, obtained from data points of ␹ st ⬇ 1/s in Fig. 9. A continuous diffusion flame surface cannot survive this high mixture fraction gradient ⵜ ␰ 兩 st and would be quenched by the augmented scalar diffusion at flame temperature. For lifted triple flames, ⵜ ␰ 兩 st decreases toward the triple point following qualitatively the scaling law derived in Eq. 11. The trailing diffusion flame can thus remain stabilized when ⵜ ␰ 兩 st at the triple point is less than ⵜ ␰ 兩 st,q . The flame stabilization radius r st 兩 flame is measured as 1.5 mm at the lowest flame lift-off height for case SINJ, d ⫽ 0.195 mm [28]. The calculated radius r st 兩 flow in the nonreacting flow of the same exit conditions is 0.68 mm. Based on an estimated temperature ratio of 6.5, the maximum attainable value of (ⵜ ␰ 兩 st ) 2 at the triple point becomes 3.4/cm2, which is indeed slightly less than the diffusion flame extinction limit in counterflow geometry, 3.5/cm2. This supports the argument that edge flame extinction sets the limit for existence of steady triple flames at high scalar dissipation rates. The calculated a st is, however, still two *This value remains almost unchanged whether all the 31 species are considered in Bilger’s formula [36] or only propane and oxygen mass fractions are considered as is exactly given in Eq. 6.

389 orders of magnitude smaller than the extinction strain rate, a q ⫽ 909.4/s. For data points with ␹ st smaller than 1/s in Fig. 9, the lifted flames move to positions further downstream where the mixing layer thickness is larger. The premixed flame fronts can therefore develop at the leading edge to form a triple flame and support its trailing diffusion flame at higher reaction rates. Thus, triple flame propagation, instead of edge flame extinction, becomes the relevant stabilization mechanism. It is also noted that previous analyses under the constant density assumption [7, 17] have predicted negative flame speeds of triple flames. The present data cannot support the existence of receding triple flames when the stoichiometric scalar dissipation rate reaches the extinction limit. This is because receding edge flames with negative speeds cannot be stabilized in the steady laminar flows investigated in this work. Nonlinear Dependency of Up on ␹st The parametric dependency of triple flame propagation velocity, U p , is the most important quantity for prediction of flame stabilization. Previous analyses [7, 17] and present data both indicate that U p decreases monotonically with increasing scalar dissipation rate, ␹ st . A linear decrease of U p with ␹ st has been proposed in Ref. 19 as well as applied in modeling of turbulent lifted methane flames [4]. A nonlinear dependency is, however, clearly evident in Fig. 9 for lifted propane flames. A recent analysis, based on a one-dimensional model of edge flames [21], has shown that the nonunity Lewis number is involved in the dependency of U p on ␹ st . In view of the asymptotic behavior of U p /S Lo shown in Eq. 1, a general expression of the triple flame propagation velocity is proposed as: Up o ⫽ SL

冑 冋 冉 冊册 ␳u ␹ st 1⫺ ␳b ␹a

m

(13)

where m and ␹ a are additional parameters to be determined. The inclusion of dependency on scalar dissipation in Eq. 13 excludes the assumption of an infinite mixing-layer thickness in the derivation of Eq. 1. Instead, it is based on the experiments [28 –30], where the mixing layer

390

Y.-C. CHEN AND R. W. BILGER

Fig. 10. Log–log plot of the normalized triple flame veloc0 公 0 ity, (S TF /S L ) 1 ⫺ ␣ , versus scalar dissipation rate, ( ␦ L,ref / ␦ M ) 2 , from the numerical simulations in Ruetsch et al. [18]. The same notation is adopted here. 公1 ⫺ ␣ equals 公␳ b / ␳ u with ␣ the gas expansion factor. ␦ M is the stoichiometric 0 mixing layer thickness at the inlet. ␦ L,ref is the reference premixed flame thickness taken as that in Simulation case I. A curve-fit of the data points using Eq. 13 is given.

Fig. 11. The calculated U st and ␹ st taken at 5 mm (redirection region length) upstream of the measured flame stabilization heights for the nitrogen-diluted, lifted laminar methane flames [19]. Open circles represent data points of varied U j between 1.168 m/s and 1.601 m/s, open squares represent data points of varied Y CH4,Co-flow from 0.0075 to 0.0149, and diamonds represent data points of varied U Coflow from 0.638 to 0.647 m/s.

thickness in front of the lifted flames is normally small. To test the applicability of Eq. 13, we have first replotted the data of numerical simulations from Ruetsch et al. [18] in Fig. 10. These simulations have Le ⫽ 1 but varied mixing layer o thickness, ␦ M , or laminar burning velocity, S L . A well-correlated nonlinear relationship can be found between the normalized farfield flame 0 公 speed, (S TF /S L ) ␳ b / ␳ u , and the scalar dissipation rate far upstream from the triple flames, 0 ( ␦ L,ref / ␦ M ) 2 . For consistency, the same notation is used here as in Ref. 18. The correlation constants m and ␹ a are determined from a linear regression analysis between log (1 ⫺ 0 公 0 (U p /S L ) ␳ b / ␳ u ) and log ( ␦ L,ref / ␦ M ) 2 . Normal0 ization of the velocity ratio S TF /S L by its asymptotic value, 公␳ u / ␳ b , is necessary to correlate between data points of varied mixing layer thickness and laminar burning velocity. This justifies the use of the 公␳ u / ␳ b term on the right-hand side of Eq. 13. In addition to the above numerical data, test of Eq. 13 is also made for the lifted laminar methane flames (Le ⬇ 1) reported in Ref. 19. The redirection region length is estimated here to be 5 mm based on the PIV measurements (cf. 0 Fig. 7). The data points of U p /S L and ␹ st in Fig. 11 were thus obtained from nonreacting flow

calculations at 5 mm upstream of the measured flame lift-off heights. The laminar burning ve0 locities, S L , used for normalization are taken as those for stoichiometric mixtures upstream from the flame stabilization positions. Determined from CHEMKIN calculations [49] with the C-3 mechanism, the values varied by 5% among the investigated flames due to the slightly different nitrogen dilution percentages. As expected, the data points in Fig. 11 show a consistent trend independent of changes in various global parameters, such as the nozzle or coflow exit velocity or the coflow fuel mass fraction. Due to limited range of data points, two different curve-fits can be made, as shown in Fig. 11. The dash line is made with Eq. 13 and the density ratio ␳ u / ␳ b is taken as 2.72 from laminar flame calculations. This nonlinear correlation is, however, not able to predict flame stability of lifted methane flames (for details, see discussions in the next section). A linear correlation (m ⫽ 1) with a curve-fitted density ratio of 2.36 is instead used. The calculated stoichiometric scalar dissipation rates at flame stabilization for the investigated lifted laminar methane flames in Ref. 19 are found to be much smaller than the quenching scalar dissipation rate, ␹ q ⫽ 30/s. This suggests that these flames are still far away from the edge flame extinction

LAMINAR FLAME STABILIZATION limit. Lift-off heights from triple flames closer to the nozzle exit can help in providing data points of higher scalar dissipation rates to give a better correlation and resolve the issue of its linearity. Such data, however, were not reported. From the data presented in Figs. 10 and 11, it is argued that the dependency of triple flame propagation velocity on scalar dissipation rate is not due to the varied Lewis numbers. Flame chemistry may play an important role in determining the values of m and ␹ a in Eq. 13. An “idealized” chemical scheme is adapted in the numerical simulations of Ref. 18. More investigations are needed to justify Eq. 13 and to relate the parameters m and ␹ a to other physical quantities. Level-Off of Up at Small ␹st For ␹ st less than approximately 0.2/s in case SINJ in Fig. 9, the triple flame propagation velocity does not follow the increasing trend with decreasing scalar dissipation rates but approaches an asymptote of 0.75 m/s. The lack of data points at ␹ st ⬍ 0.01/s is limited by experimental conditions, as extreme care is needed to keep downstream stabilized lifted flames from any laboratory interferences. It has been argued [18] that the triple flame speed will reach an asymptotic value when the scalar dissipation rate approaches zero. The present “asymptotic” behavior, however, is attributed to the weakening of the fuel-rich premixed wings near the jet centerline when lifted flames are stabilized at downstream axial positions. The jet centerline mixture fraction decreases inversely with the normalized axial distance, X, as shown in Fig. 2. The fuel-rich limit is reached at X ⫽ X rich ⬅ (2Sc ⫹ 1)/(32 ␰ rich) for the case SINJ. This corresponds to X/Y F, j ⬇ 0.76 in Fig. 4. Further downstream, the rich premixed branches of lifted triple flames start to merge at the jet centerline due to the axisymmetric geometry (cf. Fig. 19). The higher the triple flame is stabilized, the smaller the mixture fraction at jet centerline, and the shorter the rich premixed flame front becomes. Thus, the increase of U p with decreasing ␹ st can be suppressed due to less heat release from the rich premixed flame front. This is because the difference between U p

391

Fig. 12. Comparison of the normalized radial mixture fraction, ␰ / ␰ st , profiles upstream of the redirection region with ␹ st ⫽ 0.1/s in case SINJA of varied Y F, j . Note that the normalized centerline mixture fraction is decreasing with decreasing Y F, j and is less than the fuel-rich limit ␰rich/ ␰ st , which is 2.56 for propane flames.

and S L at both ends of the redirection region scales with the reaction, or heat release, rates along the premixed wings [16]. To demonstrate this effect, Fig. 12 compares the calculated radial mixture fraction profiles at different liftoff heights but under the condition of constant ␹ st ⫽ 0.1/s for case SINJA. Due to different amounts of air dilution in the fuel jets, the normalized centerline mixture fraction, ␰ CL / ␰ st , decreases with decreasing Y F, j although ␹ st remains the same. A smaller ␰ CL / ␰ st is associated with a shorter premixed flame front on the fuel-rich side, which in turn results in less overall heat release, and a smaller U p . Thus, the reduction of heat release rates for these richjoined flames scales with the decrease of Y F, j , independent of the scalar dissipation rate. This implies that the level-off of data points in Fig. 9 at low scalar dissipation rates does not represent the true dependency of U p on ␹ st for an isolated, complete triple flame. Based on Eq. 1, the maximum U p can be estimated as 1.16 m/s, assuming the density ratio of unburnt to burnt gases to be 7 and a laminar burning velocity of 0.44 m/s for propane as the fuel. This limiting velocity is still much higher than the asymptotic values shown in Fig. 9. The stoichiometric scalar dissipation rate at the axial position X ⫽ X rich can be estimated according to Eq. 7c as

392

Y.-C. CHEN AND R. W. BILGER release rate across the radial mixture fraction distribution. For such flames, according to Fig. 9, U p can be approximated to the first order as a constant and is taken as the same value at which the limit of ␰ CL ⫽ ␰ rich is reached. FLAME STABILITY AND BLOW-OUT CRITERIA Stability Criteria

Fig. 13. Comparison of the stoichiometric scalar dissipation rates, with error bars, read from Fig. 9 (cases SINJ and SINJA) at the positions where the linear log–log relationship stops with predictions based on Eq. 14.

␹ st 共D F/d e2兲 䡠

冏

⫽ Xrich

冋冉 冊 ␰st ␰rich

3共32Sc ␰ st␰ rich兲 2 2共3 ⫹ n兲共1 ⫹ 2Sc兲 2

1/ 2Sc

⫺

冉 冊 册 ␰st ␰rich

1/Sc

共n ⫽ 0 or 1兲 (14)

Equation 14 is only valid for cases SINJ and SINJA without coflow air. Good agreement of the threshold scalar dissipation rates, ␹ st 兩 Xrich, is shown in Fig. 13 between predictions based on Eq. 14 and the experimental data in Fig. 9 at which the deviation of the linear log–log relationship begins. This indicates the importance of the limit ␰ CL ⫽ ␰ rich for laminar flame stabilization in an axisymmetric geometry, as the lifted flame propagation velocity starts to level off when the scalar dissipation rate is smaller than the threshold value, ␹ st 兩 Xrich. The ␰ CL ⫽ ␰ rich limit is reached at X ⫽ (2Sc ⫹ 1)/(32 ␰ rich), the value decreases with increasing air dilution in the fuel stream, i.e. decreasing Y F, j . This explains the relatively lower U p with smaller Y F, j for case SINJA in Fig. 9. A characteristic of these rich-joined flames is that the rich premixed wings are joined together to form a continuous flame front spanning all the way between the two lean flammability limits on opposite sides of jet centerline. The flame propagation velocity is no longer determined solely by the stoichiometric scalar dissipation rate at the triple point as in the case of an isolated triple flame, but depends on the entire heat

As discussed in the previous section, except at the high scalar dissipation rate limit where edge flame extinction occurs, triple flame stabilization is achieved by the velocity balance between U p and U st . Both U p and U st vary along the axial direction for axisymmetric laminar jet flows. U st can be readily obtained from numerical calculations. An example is shown in Fig. 4 for a single propane jet configuration. For triple flames stabilized at an axial position less than X rich, U p depends mainly on the scalar dissipation rate immediately upstream from flame stabilization, irrespective of the burner global variables. All such data points for the three investigated cases are plotted in Fig. 14. The curve-fitted formula based on Eq. 13 is given below: Up 0 ⫽ SL

冑 冋 冉 冊 册 ␳u ␹ st 1⫺ ␳b 56.2

0.159

(15)

Here U p is normalized with the stoichiometric 0 laminar flame speed of propane flames, S L ⫽ 0.44 m/s. The density ratio ␳ u / ␳ b is calculated as the temperature ratio of the adiabatic flame temperature, 2262 K, to the ambient temperature, 293 K. The normalized triple flame prop0 agation velocity U p /S L , shown in Fig. 14, ranges from 0.8 to 0.5 公␳ u / ␳ b and is consistent with the argument that 公␳ u / ␳ b sets the upper bound 0 for U p /S L at the asymptotic limit of small scalar 0 dissipation rates [18]. The values of U p /S L decrease very quickly from 2.22 to 1.66 as ␹ st increases only slightly from near zero to ⬃0.1/s. As ␹ st becomes even larger, i.e. away from the 0 limit of large Damko ¨hler numbers, U p /S L decreases gradually to a lower value of ⬃1.39. To verify the accuracy of Eq. 15, the axial distributions of U st and U p are calculated and shown in Fig. 15 for both near nozzle ( ␰ CL ⬍

LAMINAR FLAME STABILIZATION

393

Fig. 14. Curve-fit of the normalized triple flame propaga0 公 tion velocity, (U p /S L ) ␳ b / ␳ u , and the scalar dissipation rate, ␹ st , from all the investigated cases before the linear log–log relationship in Fig. 9 levels off.

␰ rich) and far downstream ( ␰ CL ⬎ ␰ rich) axial positions in case TWINJ at a constant coflow air velocity of 0.4 m/s. For the former condition, the flame stabilization point is determined as the axial position where the two velocities are equal. For the latter condition, flame stabilization can still be determined as the cross-over point of U p and U st , if the triple flame propagation velocity above the ␰ CL ⫽ ␰ rich limit is approximated as the value calculated at this limit: its value can be obtained by substituting Eq. 14 into Eq. 15. For both conditions, the predicted flame stabilization positions match quite well with the measured lift-off heights indicated by vertical dashed lines. More importantly, the stability of lifted triple flames is validated by U st ⬎ U p upstream of the flame lift-off height and U st ⬍ U p further downstream. This implies that lifted flames will migrate from downstream axial positions toward the point where U st ⫽ U p and be stabilized there. The calculated triple flame propagation velocity has a minimum value of 0.59 m/s according to Eq. 15. Thus, the solid line representing U p in Fig. 15 is only shown above this limiting velocity. This position corresponds to flame stabilization at edge flame extinction. No steady lifted propane flames can exist below this height, which is consistent with the observation of a minimum lift-off height in Chung’s experiments. The U j ⫽ 3.87 m/s case in Fig. 15 was reported to be in the hysteresis region [29]. Both a nozzle-attached and a lifted flame can exist at the same exit velocity. Stability of the

Fig. 15. The calculated axial profiles of U st (circle) and U p (solid line) for both near nozzle (U j ⫽ 3.87 m/s) and far downstream (U j ⫽ 7.05 m/s) axial positions in case TWINJ with U a ⫽ 0.4 m/s. U p is calculated with Eq. 15. Calculated ␹ st is also indicated by triangles. The flame stabilization position is predicted to be the cross-over point, which corresponds well with the measured lift-off heights shown by the vertical dashed lines. The arrows indicate the ␰ CL ⫽ ␰ rich limit.

lifted flame is clearly demonstrated in Fig. 15 in that the triple flame can not propagate further upstream. For the nozzle-attached flame, the upstream heat loss from the anchored triple flame to the burner plays a dominant role [45] and is not considered in this work. This means that the present theory may not be able to predict the lift-off and flash-back velocity of burner-stabilized flames. The curve-fit formulae in Eq. 15 apply only for lifted propane flames. Figure 11 shows that the curve-fitting parameters in Eq. 13 are different for lifted methane flames. This difference may explain why the measured flame lift-off heights increase nonlinearly with increasing nozzle exit velocity for lifted propane and butane flames [28]; but a linear relationship is observed for lifted methane flames [19]. The predicted axial distributions of U p based on the

394

Fig. 16. The calculated axial profiles of U st and U p for a laminar nitrogen-diluted, methane jet in a three-feed burner [19]. Symbols are the same as used in Fig. 15. The jet exit velocity is 1.178 m/s, Y F, j ⫽ 0.679, Y O,⬁ ⫽ 0.233, and U a ⫽ 0.634 m/s. The flame stabilization position is predicted to be the cross-over point of U st and U p . The linear and nonlinear correlations of U p are given in Fig. 11. Also shown is the correlation proposed by Ko and Chung [50].

two correlations given in Fig. 11 are compared in Fig. 16 with the calculated U st distribution for a laminar triple flame of methane [19]. The predicted flame lift-off height at U p ⫽ U st corresponds well to the experimental value (shown by a vertical dashed line). However, the nonlinear correlation predicts much higher flame propagation velocities close to the burner exit, that violate the flame stabilization condition of U st ⬎ U p at upstream locations. This stability condition is instead met by the linear correlation. The failure of the nonlinear correlation is most likely due to the limited range of scalar dissipation rates available for curve-fitting with Eq. 13. As can be seen in Fig. 11, the maximum scalar dissipation rate ␹ st for the data is only ⬃0.24/s. Extrapolation into higher ␹ st values may not be accurate. More experimental data are definitely needed to finalize the correlation. Also shown in Fig. 16 is the predicted U p axial distribution from a correlation derived in unsteady lifted laminar methane flames [50]. Lower flame propagation velocities and thus much smaller flame lift-off heights are predicted. The experimental observation [28] that no methane lifted flame can be stabilized in a single jet configuration, can also be predicted by the present triple flame stabilization mecha-

Y.-C. CHEN AND R. W. BILGER

Fig. 17. The calculated axial profiles of U st and U p for a laminar methane jet of the same burner geometry used in case SINJ. Symbols are the same as in Fig. 15. Note there is no cross-over between the U st and U p profiles.

nism. Figure 17 shows calculation results for a methane jet in which Up is larger than Ust all along the stoichiometric contour. The decay of centerline mixture fraction is much faster in a methane jet than a propane jet because of x st ⫽ Red e X st ⬀ ␳ j (2Sc ⫹ 1)/ ␰ st among different fuels at the same flow conditions. Thus, for x ⬍ x st the methane triple flame would propagate upstream and attach to the nozzle exit; for x ⬎ x st no triple flame is defined such that the flame cannot be stabilized. A more rigourous flame stability criterion for triple flames is obtained by considering an infinitesimal displacement, ds, along the stoichiometric contour: U p ⫽ U st;

dU p dU st ⱖ ds ds

(16)

Equation 16 is consistent with the criteria proposed in Ref. 31, except that the triple flame propagation velocity, U p , instead of the laminar 0 flame speed, S L , is used here. Thus, both the influence of varying mixing layer thickness and of flow divergence in front of the lifted flame are taken into account. Neutral stability is allowed if dU p /ds ⫽ dU st /ds, which can occur for lifted methane flames. A different triple flame stability criterion proposed in Ref. 45 argues that the flow velocity needs to decrease along the stoichiometric contour downstream from the lift-off height. This is consistent with the assumption of a constant triple flame propagation velocity [29]. For jet flows with a fuel Schmidt number larger than 1, the axial velocity

LAMINAR FLAME STABILIZATION

395

along the stoichiometric contour, U st , decreases for downstream positions (cf. Fig. 15). However, for methane flames (Sc ⫽ 0.74), U st increases along the stoichiometric contour, as shown in Fig. 16. Thus, stabilized methane flames are not predicted by a constant triple flame propagation velocity, contrary to experimental observations. Blow-Out Criteria Based on the above flame stabilization criteria, one would expect that flame blow-out occurs at an axial position where the triple flame structure breaks down at ␰ CL ⫽ ␰ st . Beyond this position, ␰ CL is less than ␰rich, therefore U p can only be estimated as the value where ␰ CL ⫽ ␰ rich. Figure 4 shows that ␰ CL ⫽ ␰ st is reached at X ⫽ (1 ⫹ 2Sc)/(32 ␰ st ) for case SINJ. Thus the stand-off distance at blow-out is

冉

冊

1 ⫹ 2Sc Re BO d e x BO ⫽ 32 ␰ st

(17)

with Re BO ⫽ U j,BO d e / ␯ ⬁ . Using Eq. 4a, the blow-out velocity is derived as U j,BO ⫽

共3 ⫹ n兲共1 ⫹ 2Sc兲 U p,BO 12 ␰ st 共n ⫽ 0 or 1兲

(18)

with U p,BO , the lifted flame propagation velocity at blow-out, is obtained by substituting Eq. 14 into Eq. 15 for propane lifted flames. This term has a rather complicated dependency, which involves both the mixing field distribution via Eq. 14 and the fuel chemistry via the U p versus ␹ st relationship, i.e. Eq. 15. Equation 18 is identical to the criterion proposed by Chung and Lee [28]. However, the value of U p,BO here is considered as the triple flame propagation velocity but not the stoichiometric laminar 0 0 burning velocity, S L . By using S L ⫽ 0.44 m/s for U p,BO in Eq. 18, a blow-out velocity of 8.8 m/s is predicted by Chung and Lee [28] for the case SINJ with d ⫽ 0.195 mm. The current criteria predict U p,BO ⫽ 0.74 m/s and a blow-out velocity of 15.4 m/s, which is much closer to the measured value of 15 m/s. The almost 50% improvement in the prediction of blow-out velocity is due to the higher flame propagation velocity based on triple flame arguments. Figure

Fig. 18. Comparison of the predicted blow-out Reynolds numbers, Re BO , (solid line) and the experimental data (symbols) [28] for case SINJ of uniform exit velocity profiles (n ⫽ 1). d is the nozzle diameter. The predicted Re BO for the case of a Poiseuille exit velocity profile (n ⫽ 0) are also shown (dashed line).

18 demonstrates the good agreement between predicted blow-out Reynolds numbers, Re BO , and experimental data for case SINJ [28]. The current blow-out criteria differ from those proposed by Savas¸ and Gollahalli [31] in two aspects. In their argument, the fuel concentration at x BO can be less than the stoichiometric fuel/air mixture, and the laminar burning velocity is used as the flame propagation velocity in the flow/flame velocity balance. Despite these differences, the linear relationship between Re BO and the nozzle diameter d predicted in Ref. 31 can still be reproduced by the current criteria, as shown in Fig. 18. This is because the functional forms of x BO and U j,BO in Eqs. 17 and 18 remain the same as those of Ref. 31. The proportionality constant in Fig. 18 for case n ⫽ 0 is, however, larger by almost a factor of 2. This is due to the much higher triple flame propagation velocity predicted by the current criteria. Implications for Stabilization of Turbulent Lifted Flames The above stability criteria for lifted laminar flames may contribute to understanding of stabilization mechanisms for lifted turbulent nonpremixed jet flames. One important inference is that stabilization of lifted flames is in general achieved by the flow/flame velocity balance instead of diffusion flamelet quenching [3] or

396 premixed flame extinction [46]. The triple flame propagation velocity depends mainly on the mixing condition at the stoichiometric position. A wide range of stoichiometric scalar dissipation rates ␹ st can be present at the lifted flame base. Due to the strong temperature dependency of diffusion coefficients, realistic values of ␹ st in the cold flow ahead of lifted flames are far below the calculated extinction scalar dissipation rates for diffusion flames in a counterflow geometry. Among all the cases studied in this work, ␹ st is less than 1/s for propane flames and less than 0.25/s for methane flames. This is consistent with the relatively low scalar dissipation rates measured at the base of a lifted turbulent methane flame [43]. Higher values of ␹ st would be expected if the lifted flame is stabilized closer to the nozzle exit. Thus, diffusion flame extinction may occur as a trigger mechanism during the inception of flame liftoff, as is argued for a bluff-body burner [15]. A second inference is that the flame propagation velocity levels off at far downstream positions, where the centerline mixture fraction is less than that of the rich flammability limit. In turbulent flows, coalescence or collapse of triple flames can happen due to the merging of fuelrich premixed wings of several individual triple flames. Flame stability can thus be reduced as the overall flow divergence effects is reduced. A level-off of the measured flow velocity at the lifted turbulent flame base is indeed reported at downstream axial positions [9]. The important parameter for determining the triple flame propagation velocity for these rich-joined flames is changed from the mixing structure ahead of lifted flames, ␹ st in Eq. 15, to the global mixing parameters, ␰ st and ␰rich, in Eq. 14. Finally, it should be noted that at large Kaslovitz numbers, strain and mixing fields are at a scale finer than that present in laminar flames, and concepts and conclusions based on laminar flamelet principles are likely to be invalid. CONCLUSIONS The stabilization mechanism of lifted laminar jet flames is investigated in an axisymmetric geometry with combined numerical/experimental data. The flame structure at stabilization

Y.-C. CHEN AND R. W. BILGER

Fig. 19. Schematic diagram of flame contours and conditions for flame stability and blow-out criteria of a lifted laminar flame in an axisymmetric geometry. A necessary condition for these criteria is the balance between U p and U st at the stoichiometric triple points, which are indicated by solid circles.

position is considered to be a triple flame. Thus, flame stabilization can be enhanced by flow deceleration ahead of the triple flame due to the flow/mixing field divergence within a redirection region. Three flame stabilization mechanisms are observed, corresponding to previously proposed theories: edge flame extinction, triple flame propagation, and flame front propagation. The appropriate conditions for these different mechanisms are illustrated in Fig. 19. For an axisymmetric jet geometry, different mechanisms are found for the near nozzle ( ␰ CL ⬎ ␰ rich) and far downstream ( ␰ CL ⬍ ␰ rich) axial positions. The triple flame propagation mechanism applies for ␰ CL ⬎ ␰ rich. Flame stability is achieved when the propagation velocity of triple flames, U p , is less than the upstream stoichiometric flow velocity, U st . The magnitude of U p depends mainly on the mixing structure immediately in front of the stabilized flame, irrespective of nozzle global variables. In general, U p decreases nonlinearly with increasing

LAMINAR FLAME STABILIZATION

397

␹ st , which characterizes the mixing strength in the transverse direction in front of the triple flame. An empirical formula is proposed to describe the dependency of U p on ␹ st with two curve-fitting parameters. These parameters are found to be different between propane, Eq. 15, and methane (Fig. 11) triple flames. However, this difference is considered to be not only due to the Lewis number effect but also to flame chemistry. At minimum flame lift-off height, the stoichiometric mixture fraction gradient reaches the extinction limit calculated for stretched diffusion flames in a counterflow geometry. Edge flame extinction thus becomes the relevant stabilization mechanism. It is noted that a positive flame speed is still associated with the stabilized lifted flame at this limit. The flame front propagation mechanism occurs when ␰ CL ⬍ ␰ rich. At such downstream axial positions, U p approaches an asymptotic value independent of ␹ st . For a first-order approximation, U p can be estimated as the same value at which the limit of ␰ CL ⫽ ␰ rich is reached. Beyond this limit, the net heat release of triple flames depends on the entire mixture fraction distribution along the premixed flame front. This is because the rich premixed wings of the lifted triple flames merge at jet centerline above the axial position, X ⫽ X rich ⫽ (2Sc ⫹ 1)/(32 ␰ rich). A continuous partially premixed flame front is then formed between the two lean flammability limit locations on opposite sides of jet centerline, with the centerline mixture fraction below the rich flammability limit. As shown in Fig. 19, the flame blow-out limit is reached when the triple flame structure breaks down. This happens if the centerline mixture fraction is less than the stoichiometric value at X ⫽ X st ⫽ (2Sc ⫹ 1)/(32 ␰ st ) for case SINJ. Predictions of the blow-out velocities are much improved due to the triple flame propagation velocity being higher than the stoichiometric laminar burning velocity.

Valuable comments from Prof. A. Masri and Dr. N. Swaminathan during preparation of this paper are also acknowledged. This work is supported by the Australian Research Council.

We are grateful to Prof. S. H. Chung in the Department of Mechanical Engineering, Seoul National University, Korea, who provided the lift-off height data and the burner configurations for the numerical calculations. Profs. N. Peters and M. S. Mansour are acknowledged for providing their data and a preprint of their paper.

o to solve for U 1 /U 2 , which is the ratio of U p /S L , we get

APPENDIX: SCALING LAWS IN AXISYMMETRIC COORDINATE Equation 1 describes the asymptotic triple flame propagation velocity in the plane mixing layer geometry at large mixing thickness and can be derived based on conservation relations [18]. The same methodology and assumptions are adopted here, except they are now applied to the axisymmetric geometry, as schematically shown in Fig. 8. The subscripts 1, 2, 3, and 4 are referred to conditions at the corresponding stations along the stoichiometric line. The Rankine-Hugoniot relations for mass and momentum conservation still hold across the flame between stations 2 and 3 in Fig. 8 if U 2 and U 3 are referring to the local flow velocities perpendicular to the premixed fronts of the triple flame. Thus, we have

␳ 2U 2 ⫽ ␳ 3U 3

(A1)

P 2 ⫹ ␳ 2U 22 ⫽ P 3 ⫹ ␳ 3U 23

(A2)

Equations A3 and A4 can be derived from the Bernoulli equations along the stoichiometric contour, ␰ ⫽ ␰st. A Sc ⫽ 1 assumption is needed to ensure that it coincides with a flow streamline. P1 ⫹

1 1 ␳ 1U 21 ⫽ P 2 ⫹ ␳ 2U 22 2 2

(A3)

P3 ⫹

1 1 ␳ U 2 ⫽ P 4 ⫹ ␳ 4U 24 2 3 3 2

(A4)

Thus, the same set of conservation equations, Eqs. A1 to A4, are maintained between the plane mixing layer and the axisymmetric geometry. Using these equations and the density identity

␳ 1 ⫽ ␳ 2;

冉冊 Up o SL

␳4 ⫽ ␳3

2

⫽

(A5)

␳1/␳4 ⫺ 1 1 ⫹ 共P1 ⫺

P4兲/共12 ␳1U21兲

⫺ 共␳4/␳1兲共U4/U1兲2 (A6)

398

Y.-C. CHEN AND R. W. BILGER

o The equality of U2 and SL is ensured because U2 is defined as perpendicular to the premixed front in Eqs. A1 and A2, but not the axial velocity. To express the velocity ratio in Eq. A6 in terms of only densities, we need to apply again the mass and momentum conservations in a control volume along the streamline “ring” wrapping around the flame between stations 1 to 4. To the first-order approximation, the velocities of points 1 and 4 are used instead of integrating along the radial direction from the inner streamline r ⫽ r i to the outer streamline r ⫽ r o . Additionally, we have assumed r o ⫽ r st ⫹ ␦ / 2 and r i ⫽ r st ⫺ ␦ / 2 to arrive at

␳ 1U 1␦ 1r 1 ⬃ ␳ 4U 4␦ 4r 4

(A7)

␦1 ⬃ ␦2

␦1 ⬃ ␦2

1.

2.

⫹ P 1共 ␦ 2r 2 ⫺ ␦ 1r 1兲 ⬃ P 4␦ 4r 4

3.

⫹

␳ 4U 24␦ 4r 4

⫹ P 4共 ␦ 3r 3 ⫺ ␦ 4r 4兲

4.

␦1 ␳4 r4 ⬃ ␦4 ␳1 r1

U 1 ⬃ U 4;

5.

6. 7. 8. 9. 10.

(A9)

11.

Substituting Eq. A9 into Eq. A6 along with the constant pressure assumption, we arrive at

12.

U1 Up ⫽ o⬃ U2 SL

冑

␳1 ␳4

(A10)

which is identical to Eq. 1 derived in the planar mixing layer geometry. However, not all the scaling laws remain the same between the two different geometries. This is particularly related with the ratio of mixing layer thickness given in Eq. 11. Similar to Eq. A7, we can apply the integral mass conservation law between stations 1 and 2 to get U 1␦ 1r 1 ⬃ U 2␦ 2r 2

13. 14. 15. 16.

17.

(A11)

Substituting Eqs. A9 and A10 into Eq. A11 gives the following scaling law:

(A12)

冑

␳4 ␳1

(A13)

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P 1␦ 1r 1 ⫹

where r is the radius, and ␦ is the mixing layer thickness. Equation A2 has been used in deriving Eq. A8. Also the average pressure in the control volume ahead of and after the flame is approximated to P 1 and P 4 , respectively. This requires that sudden pressure changes from P 1 to P 2 and P 3 to P 4 are confined within narrow regions close to the premixed flame front. One additional relation is needed to close the overall conservation equations. We use the same assumption from Ruetsch et al. [18], i.e. P 4 ⫽ P 1 . This finally leads to

␳4 r2 . ␳1 r1

which is Eq. 11. In the planar mixing layer geometry with a symmetric triple flame as in Ref. 18, the radii r 2 and r 1 disappear in Eq. A11, and we have directly

␳ 1U 21␦ 1r 1

(A8)

冑

18.

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Received 9 November 1998; revised 22 April 1999; accepted 16 February 2000