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Flame-Intrinsic Kelvin-Helmholtz Instability of Flickering Premixed Flames Young Tae Guahk,† Dae Keun Lee,‡ Kwang Chul Oh,§ and Hyun Dong Shin*,† Department of Mechanical Engineering, Korea AdVanced Institute of Science and Technology, 373-1, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea, Korea Institute of Energy Research, 102 Gajeong-ro, Yuseong-gu, Daejeon 305-343, Republic of Korea, and EnVironmental Parts Research and DeVelopment Center, Korea AutomotiVe Technology Institute, 74, Yongjung-ri, Pungse-myun, Chonan, Cheungnam 330-912, Republic of Korea ReceiVed February 19, 2009. ReVised Manuscript ReceiVed June 15, 2009
The flame-intrinsic Kelvin-Helmholtz (KH) instability, a self-oscillation of laminar flame fronts extending to the streamwise direction, was experimentally investigated using inverted conical premixed flames, and the governing parameters of flame flickering were deduced. We supposed that the flame flickering is caused by the KH instability, and the feature may be determined by which is dominant between the buoyancy-driven one taking place in the shear layer between hot products and ambient air and the flame-intrinsic destabilization because of the vorticity generation of flame fronts. For the distinction between the two, the former effect was excluded using a cylindrical quartz tube preventing the interaction with the ambient air and the latter was characterized by the flickering frequencies and wavelengths obtained by chemiluminescence signals and instantaneous images, respectively. Besides, laminar burning velocities were measured from the velocity normal to the flame front by the particle image velocimetry (PIV) method. From the dimensional analysis and physical considerations on an oblique plane flame front, it was estimated that the frequency will be determined by the flame structure and described by the modified Richardson number Rif involved with the flame thickness and laminar burning velocity. From the experiments, it was revealed that the Strouhal number, dimensionless frequency, is proportional to Rif0.869 rather than the conventional Richardson number within the experimental conditions. This means that the flickering motions could be influenced by the flame-intrinsic KH instability described by the flame structure, in addition to the buoyancy-driven KH instability determined by the mixture velocity and length scale of burner geometry.
Introduction It has been known that a flame flickering, a convective instability of laminar flame front, is produced by the KelvinHelmholtz (KH) instability taking place at a shear layer of combustion flow fields. For buoyant diffusion flames, flickering frequency ranging from 10 to 20 Hz was observed for a wide variety of burner sizes, flow rates, and compositions.1,2 According to Buckmaster and Peters,3 these flickering phenomena are mainly caused by the buoyancy-driven KH instability at the shear layer via a buoyancy-induced velocity field surrounding jet flow, and they predicted a low-frequency oscillation around 17 Hz using the stability analysis of the infinite candle model, an ideal plane diffusion flame, in which the flow field is induced solely by buoyancy. Chen et al.4 visualized two distinct vortices for flickering diffusion flames, large toroidal vortices outside the luminous flame and small roll-up vortices inside the luminous flame, by the Mie-scattering method. For these diffusion flames, the flame is the stoichiometric surface estab* To whom correspondence should be addressed. Telephone: 82-42-3508821. Fax: 82-42-350-8820. E-mail:
[email protected]. † Korea Advanced Institute of Science and Technology. ‡ Korea Institute of Energy Research. § Korea Automotive Technology Institute. (1) Kimura, I. Int. Symp. Combust. 1965, 10, 1295–1300. (2) Toong, T. Y.; Salant, R. F.; Stopford, J. M.; Anderson, G. Y. Int. Symp. Combust. 1965, 10, 1301–1313. (3) Buckmaster, J.; Peters, N. Int. Symp. Combust. 1986, 21, 1829–1836. (4) Chen, L. D.; Seaba, J. P.; Roquemore, W. M.; Goss, L. P. Int. Symp. Combust. 1988, 22, 677–684.
lished by the fuel and ambient air diffusions and acts merely like a heat source on the flow fields. Hot buoyant products accelerate the axial velocity of gas flow, and then, shear layers are generated between the flame and ambient air or fuel flow. Thus, the flickering frequency of diffusion flames is closely related to the hydrodynamic KH instability taking place at the cylindrical shear surface of the order of burner diameter D, so that the ratio of the buoyancy force acting on hot gas to the inertia of issuing jet of velocity U becomes important. As a result, there has been many studies relating the dimensionless frequency, Strouhal number fD/U, to the Richardson number gD/U2.5,6 The flame flickering can also be found in premixed flames; however, the instability mechanism is somewhat different, and complex and fewer works has been performed.7-9 Kostiuk and Cheng8 and Cheng et al.9 reported flickering phenomena in conical premixed flames and V-shaped premixed flames, respectively. Although the experimental conditions of their works are similar in the range of equivalence ratio and Reynolds number based on D and U, the flickering frequencies of (5) Hamins, A.; Yang, J. C.; Kashiwagi, T. Int. Symp. Combust. 1992, 24, 1695–1702. (6) Sato, H.; Amagai, K.; Arai, M. Combust. Flame 2000, 123, 107– 118. (7) Durox, D.; Baillot, F.; Scouflaire, P.; Prud’homme, R. Combust. Flame 1990, 82, 66–74. (8) Kostiuk, L. W.; Cheng, R. K. Combust. Flame 1995, 103, 27–40. (9) Cheng, R. K.; Bedat, B.; Kostiuk, L. W. Combust. Flame 1999, 116, 360–375.
10.1021/ef900147x CCC: $40.75 2009 American Chemical Society Published on Web 07/24/2009
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Figure 1. Schematics of (left) conical flames, (middle) inverted conical flames with both a FSL and BSL, and (right) inverted conical flames only with a FSL.
V-shaped flames decrease with the equivalence ratio, while those of conical flames increase. In other words, the conditions of reactant momentum (Reynolds number) and buoyancy acting on products (equivalence ratio) for both flames are similar to each other, but the trends are opposite. This implies that the flame-geometrical aspect can be a primary cause on their opposite trends, and further, it can be related to the flameflickering mechanism. Cheng et al.9 also mentioned that the opposite trends might be explained by the plume geometry. For the axisymmetric conical flame, the plume always interacts with the ambient air and makes a buoyancy-driven shear layer. However, for the V-shaped flame composed of twin flames, the plume is always inside the two flames that shield the products from the ambient air until it leaves the flame zone. The stability of premixed flames has been investigated with the hydrodynamic effect caused by thermal expansion (DarrieusLandau instability), the diffusive-thermal (Lewis number) effect caused by the preferential diffusion of mass and heat, and the buoyancy effect caused by the density difference of reactants and products (Rayleigh-Taylor instability). These theoretical approaches are restricted to the stability of horizontal planar flames normal to uniform reactant flows, so that the flow fields are straight and parallel, and therefore, the effects of hydrodynamics outside the flame are minimized. However, it is wellknown that stationary curved flames generate vortical motions10,11 and impose them upon the gas flow crossing the flame front. For a simpler geometry, such as inclined planar flames, the vorticity can also be generated by the baroclinic torque. Lee et al.12 has obtained the basic solution of inclined flames, which generate vorticity and induce a tangential velocity jump across the flames. He has also shown that, as the flame is more inclined from the horizontal plane, it shows more unstable characteristics because of not only the decrease of the stabilizing effect of gravity but also the increase of the destabilizing effect of the KH type instability as a result of the tangential velocity jump across the flame front. Furthermore, it was shown that the effect of hydrodynamics dominates the other parameters, such as the effect of the Lewis number, as the flame angle increases. This KH instability of the oblique flame front needs to be distinguished from the buoyancy-driven KH instability. While the latter is generated by the velocity difference between the buoyant flow and the ambient air in the shear layer, the former is generated by the tangential velocity jump across the flame front. In other words, the latter is generated everywhere the density gradient exists even without the flame,5,13 but the former needs (10) Hayes, W. D. J. Fluid Mech. 1959, 2, 595–600. (11) Pindera, M. Z.; Talbot, L. Combust. Flame 1988, 73, 111–125. (12) Lee, D. K.; Kim, M. U.; Shin, H. D. Combust. Theory Model. 2005, 9, 587–615. (13) Yuan, T.; Durox, D.; Villermaux, E. Exp. Fluids 1994, 17, 337– 349.
the flame-producing vorticity. Henceforth, for the distinction between the two, the former will be designated as the flameintrinsic KH instability. The objective of this study is to understand the characteristics of flame-intrinsic KH instability and to find dominant parameters governing them compared to the buoyancy-driven one. For these purposes, the flame with a long inverted cone shape was selected, so that the flame-intrinsic KH instability could be easily generated via a large baroclinic torque because of near perpendicularity between the direction of the density gradient and the buoyancy force. In addition, the flame was surrounded with a tube wall for the exclusion of the effect of the buoyancydriven KH instability. Through the dimensional analysis of the governing equations, we obtained a modified Richardson number that has the information of the flame structure, such as the burning velocity and flame thickness. Experimentally, the burning velocity and flame angle were measured for the modified Richardson number, and flickering frequencies and wavelengths were also measured. Finally, the empirical correlation among the Strouhal number St, modified Richardson number Rif, and Reynolds number Re was obtained. Physics of Flame Flickering and Theoretical Approach Usually the buoyancy-induced shear layer (BSL), formed by the velocity difference between the buoyant hot product and still ambient air, has been considered for the study of flickering phenomena. However, the geometry of flames is another important parameter besides the Richardson number, as seen from the study of Kostiuk and Cheng8 and Cheng et al.9 The difference between velocity fields of conical flames and inverted conical flames with both a BSL and the flame-induced shear layer (FSL), formed by the velocity difference between the cold reactant and the hot product, can be seen in the left and middle panels of Figure 1. In conical flames, the streamline is deflected outward when crossing the flame front and the FSL shifts away from the BSL as they are far from the burner exit. Because the flame front around the perimeter of the burner nozzle is close to the BSL, it can be easily affected by buoyancy-driven KH instability. Thus, the flickering flame wrinkles were observed at the middle part of the conical flame.8 However, in inverted conical flames, the streamline is deflected inward and the FSL shifts close to the BSL as they are far from the burner exit. Because the flame front is not close to the BSL until downstream, the flickering flame wrinkles were observed in the upper part of the inverted conical flame.9 If the flame angle is rather small (less than 60° from the horizontal plane), such as in the experiment conducted by Kostiuk and Cheng8 and Cheng et al.,9 the effect of the FSL is supposed not to be large enough because the flame-generated vorticity will be small. However, as the flame angle increases,
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the effect of the FSL is expected to grow and the flickering motion can be induced even without a BSL. The right panel of Figure 1 shows the inverted conical flame used in the present research, in which the BSL was removed for the evaluation of the FSL effect. The flickering flame wrinkles are expected to be observed in the middle part of the flame because the flameintrinsic KH instability is induced along the whole flame front. For flames with a BSL, the important scales are burner diameter D and mean velocity U because the velocity field downstream of the flame front is similar to that of the thermal plume. However, for flames without a BSL, the flame-intrinsic KH instability is important; thus, the important scales will be flame thickness d, across which the tangential velocity jump occurs, and burning velocity uL, which affects the tangential velocity jump via the flame residence time. Also, it is noted that the gravitational force plays an important role because there would be no jump for the tangential velocity without the acceleration because of the buoyancy. To validate these physical conjectures and obtain the governing parameters of the flameintrinsic KH instability, dimensional analysis was performed and the analysis of Lee et al.12 was adopted, although his analysis is limited to a two-dimensional inclined planar flame of infinite flame length, whose flame angle from the horizontal plane is not too large. A Cartesian coordinate system is chosen so that the x axis is orthogonal to the mean flame surface of the flickering inclined flame. When a one-step overall reaction is adopted, the dimensionless governing equations are written as follows: Continuity St
∂ ∂F + (Fu ) ) 0 ∂t ∂xi i
Navier-Stokes equations
(
F St
)
( )
∂ui ∂ui ∂2ui ∂ ∂uj ∂p + uj )+ Pr1 + Pr2 ∂t ∂xj ∂xi ∂xj∂xj ∂xi ∂xj F
gid uL2
Energy equation
(
F St
)
∂T ∂T ∂2T + ui ) + ΛLF(T, Φ) ∂t ∂xi ∂xi∂xi
Species equation
(
F St
)
∂Φ ∂2Φ ∂Φ + ui ) Le-1 - ΛLF(T, Φ) ∂t ∂xi ∂xi∂xi
rate parameter. When ΛL is given, St is determined as an eigenvalue of the above equations and is a function of Pr1, Pr2, gid/uL2, ΛL, and Le. If we restrict our concern to the stability of the lean premixed flame of the methane/air mixture with an equivalence ratio from 0.4 to 0.9, some parameters can be neglected. Le is 0.903 and 0.917, respectively, at an equivalence ratio of 0.4 and 0.9. The range of Le becomes narrower in the range of the equivalence ratio of the flickering region; thus, the contribution of the Lewis number can be neglected. The effects of Pr1 and Pr2 can also be neglected. After the analysis of Lee et al.,12 Pr2 does not appear in the dispersion relation of an inclined flame with a near-unity Lewis number. For Pr1, while it participates in the dispersion equation, it cannot change the flame stability qualitatively. The Strouhal number is then only a function of the dimensionless buoyancy term gid/uL2 and the reaction term ΛL. With the assumption of no heat loss, ΛL is determined by the mixture and boundary conditions and is as strong of a function of the equivalence ratio as uL. Hence, it is suspected that the flame flickering will be characterized by the burning velocity uL, the flame-geometrical aspect of buoyancy gi, and the flame thickness d. For a more detailed analysis, the solution of Lee et al.12 on the basic flow of an inclined planar flame will be quoted. We have focused on the formation of the flickering phenomena so that the region of our concern is around the leading flickering curvatures of length equal to several wavelengths. The radial location of those was from 10 to 15 mm from the centerline. In a nonreactive flow, the flow starts uniform because of the contraction nozzle when it enters a tube and the area of uniform flow decreases because of the developing shear layer in the wall as the height increases; however, the flow in the region of our concern is nearly uniform. Therefore, the flow was assumed to be uniform and free from the developing shear layer in the wall. To model flames of axisymmetric configuration is difficult because of the existence of singularity at a position of pilot flame. Thus, a 2D configuration was used for the qualitative analysis. Figure 2a is the geometry used for the analysis. A premixed gas is supplied vertically upward and an inclined flame is placed with flame angle θf. The flame of thickness d is approximated as a discontinuity with an incompressible flow of two different densities. The steady solutions (∂/∂t ) 0) are assumed to have a one-dimensional flame structure independent of the y direction (∂/∂y ) 0) and vary in the x direction only. However, the pressure term in the y direction is set to nonzero for the balance with the gravitational force in the y direction (∂p/∂y * 0). Then, each equation can be solved sequentially, and the distribution of the velocities u and V in the x direction are obtained as follows:12
Equation of state γ T 1-γ γ ) 1 - Ff/Fu
(
F) 1+
)
uj )
-1
,
thermal expansion parameter
where i is the summation index, which means x, y, and z directions and all nondimensional scales are the same as those of Lee et al.12 Exceptionally, time t was normalized by the inverse of the flickering frequency f. St ) fd/uL is the Strouhal number representing the nondimensional frequency, which is an unknown. Pr1 and Pr2 are, respectively, Prandtl numbers based on the dynamic and bulk viscosities. gi are the projections of g on each axis. Le is the Lewis number representing the ratio of thermal diffusivity and mass diffusivity. ΛL is the burning
{
{
γ ex (x < 0) 1-γ 1 (x > 0) 1-γ
1+
(1)
Vj ) V1 +
where
gyd uL2
γ[Pr1 + H(x)]ex/Pr1 + V2 +
gyd uL2
1 γ x ln 1 + e (x < 0) γ 1-γ
(
)
γx (x > 0) (2)
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e(1-1/Pr1)ξ dξ, V1 ) tan θf, x 1 - γ + γeξ gyd 1 1 V2 ) V1 + 2 γ Pr1 + ln γ 1 γ uL
H(x) )
∫
0
(
)
(2a)
The overlines denote the steady-state solutions, and V1 and V2 are, respectively, tangential velocities in the upstream and downstream face of the flame front. Panels b and c of Figure 2 show the distribution of normal and tangential velocities in the x direction, respectively. The dimensionless tangential velocity jump across the flame front can also be obtained using V2-V1, a strong function of gyd/uL2, from eqs 2a. For the methane/air mixtures with an equivalence ratio of 0.52 and 0.56, the corresponding burning velocities are 4.3 and 6.7 cm/s, which were measured in the present research and will be described later. For that case, V2-V1 increases from 3.54 to 13.62, 3.85 times as the equivalence ratio decreases from 0.56 to 0.52, while that changes little with respect to the mean velocity. The results show that the tangential velocity jump increases; i.e., the flameintrinsic KH instability strengthens, drastically even with a small change of the equivalence ratio. In the far upstream, the velocity field is (1, V1), i.e., uniform and irrotational. On the other hand, in the downstream, u is constant and V is a linearly increasing function of x, so that the flow is rotational. From eqs 1 and 2, the vorticity is determined as follows:
ω ¯ )
dVj ) dx
{
gyd uL2
[
]
H(x) x/Pr1 e (x < 0) Pr1
γ 1+
gyd uL
γ (x > 0) 2
(3)
Figure 2d shows the distribution of vorticity in the x direction. The vorticity ω is originally zero in the upstream, increases smoothly as the reactants approach the flame front, and then preserves the constant value Ω ) (gyd/uL2)γ downstream of the flame front. The basic solution shows that both the tangential velocity and vorticity are strong functions of the dimensionless buoyancy term in the tangential direction, gyd/uL2. Associated with the results of the dimensional analysis, it can be known that the Strouhal number will be a function of the tangential component of gid/uL2. The dimensionless number gyd/uL2, a different form of the conventional Richardson number Ri ) gD/U2 used in previous works,5,6,8,9 is named as the modified Richardson number Rif. It should be noted that Rif can be thought of as the tangential velocity jump across the flame front, because the other part, except Rif in the V2-V1 expression, is nearly constant in the following experimental conditions. Here, the difference of two distinct Richardson numbers should be emphasized. The buoyancy-driven KH instability in previous works is formed in the BSL; thus, the important variables are the shear layer scale D (burner diameter), where the axial velocity jump occurs, and fuel exit velocity or mean velocity U, which is related to the axial velocity of products via density expansion. As a result, Ri includes burner information of D and flow information of U. However, the flame-intrinsic KH instability is formed in the FSL; thus, the important variables are flame thickness d and burning velocity uL, as mentioned earlier. As a result, Rif includes the flame structure information of d and uL. Another important parameter not shown in dimensional analysis is the tangential velocity V1 in the far upstream, which is the tangential velocity in the flame front when added with the tangential velocity jump across the flame front. V1 can affect
Figure 2. Basic solution of an inclined planar flame (2D case): (a) streamlines, (b) dimensionless normal velocity, (c) dimensionless tangential velocity, and (d) dimensionless vorticity for the case of θf ) 60°, γ ) 0.8, d ) 0.022 cm, uL ) 10 cm/s, and Pr ) 0.75.
Flame-Intrinsic KelVin-Helmholtz Instability
Figure 3. Schematic diagram of the experimental setup.
the generation frequency by convecting out the flame wrinkles and making an environment for a generation of new ones. The effect of V1 can be considered via the mean velocity U because V1 equals tan θf, where θf is determined by U and equivalence ratio φ. The effect of V1 is also included in Rif via gy, which equals g sin θf, but it has only a minor role in Rif. Thus, the effect of V1 will be considered via U using Reynolds number Re ) UD/ν, based on burner diameter D. Here, it is adopted for convenience, although viscosity does not play an important role in flame-intrinsic KH instability.
Energy & Fuels, Vol. 23, 2009 3879 frequency15 is estimated as 140 Hz based on the mean velocity of 20 cm/s. The two frequencies are far from the flickering frequencies; thus, the flickering phenomena in this study are free from the Helmholtz resonance and vortex shedding of the pilot nozzle. The flame behaviors were investigated varying the equivalence ratio from 0.52 to 0.56 and the mean velocity from 20 to 130 cm/s or the Reynolds number from 625 to 4062 based on a 50 mm nozzle diameter. Here, the difference of the tube and burner diameters is due to the nitrogen nozzle. Flame shapes were observed using a high-speed camera (FASTCAM Ultima APX, Photron Ltd.), with the resolution of 1024 × 1024 pixels. The chemiluminescence signal of the flickering flame was measured by a photomultiplier tube installed 10 cm away from the tube at the height where the flickering motions were most severe, and the frequency was determined by FFT analysis. It was also verified by counting the number of wrinkle passages at a certain height and radial position using the high-speed camera. The high-speed camera and photomultiplier were synchronized by a delay generator (model 555, BNC), and the signal of the photomultiplier tube was acquired by an oscilloscope (DL 1640E, YOKOGAWA). The particle image velocimetry (PIV) system was used for the measurement of burning velocity. A continuous Ar ion laser (Stabilite, Spectra Physics) of 2 W and 514.5 nm was used as a light source, and a planar laser sheet, which was created by a cylindrical lens and a convex lens, was irradiated vertically from above the burner into the center of the burner exit. The mixture was seeded with TiO2 particles, and the Mie scattered images were recorded by the high-speed camera at a rate of 1 kHz. Selected two successive images were cross-correlated for velocity calculation using interrogation windows with a 50% overlap ratio between adjacent windows. Two distinctive camera resolutions and interrogation windows were used for the measurement of burning velocity and the visualization of flow field. For the measurement of burning velocity, the camera resolution and the size of the interrogation window were 0.068 mm/pixel and 12 × 100 pixels, respectively. For the visualization of the flow field, they were 0.125 mm/pixel and 16 × 50 pixels, respectively.
Results
Figure 3 is a diagram of the experimental setup used in the present research. The methane (99.999% pure research grade)/air mixture was fed into a burner, and honeycombs and a contraction nozzle were used for laminar flow and uniform axial velocity. At the burner exit, an inverted conical flame was anchored by a pilot flame of hydrogen with 20 cc/min flow rate. The hydrogen flow rate was small compared to the methane flow rate, so that the variation of the overall equivalence ratio because of the consumed air by hydrogen was within 0.0002. Inverted conical premixed flames with two distinctive boundary conditions, i.e., only with a FSL (case 1) and with both a BSL and FSL (case 2), were studied. In case 1, a quartz tube was used to prevent the interaction between the flame and the ambient air and nitrogen gas was supplied into the outside nozzle to prevent static pressure perturbation resulting from the on/off contact of the flame front with the tube inner surface. In case 2, the quartz tube and the nitrogen gas were not used. Helmholtz resonance frequency and shedding frequency were calculated to avoid confusion with the flickering phenomena. The diameter and length of the burner (the volume) are 120 and 360 mm, respectively, and the diameter and length of the quartz tube (the neck) are 60 and 220 mm, respectively. Thus, Helmholtz resonance frequency14 of the burner system is estimated as 98 Hz. The outer diameter of the pilot nozzle is 0.3 mm; thus, the shedding
Stability Limits (Case I). Figure 4 shows the limits of flashback, flickering, and flammability for methane/air premixed flames anchored by a pilot flame in a quartz tube (case 1). For a given Reynolds number, flickering flames (point A in Figure 4) were observed at an equivalence ratio under a critical equivalence ratio φc, over which flames were stable. A stable flame flashed back as the equivalence ratio increases over a flash-back limit, and a flickering flame could not be anchored by a hydrogen pilot flame and was blown off as the equivalence ratio decreases under a flammability limit. At point B, which is slightly above the flammability limit, a cylindrical flame of about 2 mm in diameter and tube length was observed with intermittent flickering phenomena. At point C, which is below the flammability limit, a cylindrical flame of shorter length than the tube without flickering phenomena was attached by a pilot flame. It is interesting that the flickering and flammability limits are nearly constant irrespective of the Reynolds number, while the flash-back limit is an increasing function of the Reynolds number. The monotonically increasing flash-back limit and the nearly constant value of the flammability limit can be trivially attributed to the increase of the laminar burning velocity of the lean mixture with the equivalence ratio and the lean combustible limit of the given methane/air mixture, respectively. However, it should be noted that the flickering limit is nearly determined by the equivalence ratio, because this means that the stability limit is not changed by the variation of hydrodynamic fields
(14) Kinsler, L.; Frey, A.; Coppens, A.; Sanders, J. Fundamentals of Acoustics; John Wiley and Sons: New York, 2000; pp 284-286.
(15) Schlichting, H.; Gersten, K. Boundary Layer Theory; Springer: New York, 2000; pp 21-22.
Experimental Apparatus and Method
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Figure 4. Flash-back, flickering limit, and flammability limit for methane/air premixed flames.
Figure 6. (a) Typical image of flickering inverted conical flame and (b) sequential images of the flame front during the convection of flickering instability at an equivalence ratio of 0.54 and a Reynolds number of 1250.
Figure 5. Contour plot (2D case) of the modified Richardson number Rif (solid curves) and the flame angle θf (degrees, dashed curves).
associated with the variation of the flame angle resulting from the Reynolds number change. The reason might be inferred from the basic solution of inclined planar flames. Figure 5 shows the contour plot of the modified Richardson number Rif and flame angle θf in the domain of the Reynolds number Re and the equivalence ratio φ. Here, Re based on the burner diameter D is used for a comparison to the present research, although the length scale of flame thickness d is used in the dimensional analysis. The flame angle can be varied by changing Re and φ. For example, it can be varied from 80° to 75° by decreasing Re (from point D to point E) and increasing φ (from point D to point F). From point D to point E, Rif (Rif ) gyd/uL2 ) gτ/uL × sin θf, where residence time τ ) d/uL) is a function of sin θf, which varies very little because the flame angle is already large enough, so that sin θf is close to unity. Thus, Rif decreases very little when Re decreases. From point D to point F, Rif (Rif ) gτ sin θf/uL) is a function of τ sin θf/uL, which varies largely because the residence time τ is also inversely proportional to uL. Thus, Rif decreases largely when φ increases. In summary, the flame shows the same change of the flame shape, but the hydrodynamic fields, especially the vorticity fields, are completely different depending upon whether Re or φ changes. Because
the flickering limit curve is very similar to iso-Rif curves, it is conjectured that Rif may determine the transition from a stable flame to a flickering flame. For flames with low Re and high φ in Figure 5, θf is not large enough, so that sin θf is no longer close to unity and Rif also varies when Re is changed. It makes the conjecture more reliable that both the flickering limit curve and iso-Rif curves have slopes in the range of low Re. Although the range of the Reynolds number was wide under which flames flickered, the flickering phenomena became irregular as the mean velocity increased. Thus, the high Reynolds number conditions were not appropriate for the measurement of flickering frequency and wavelength and were excluded from experimental conditions. Features of Flickering Flames. Flickering flames were observed with the equivalence ratio between the flammability limit and φc in case 1. The flickering motion is distinguishable from the V-shaped flame of Cheng et al.9 Flame wrinkles are observed in the middle part in the present research, while they are observed in the upper part in the V-shaped flame. The lower part of the flickering flame has no fluctuation and was leastsquares-fitted to an equation of the first degree. The angle, between the fitted line and the horizontal plane, was defined as the flame angle θf. The flame-intrinsic KH instability generates noticeable flame wrinkles successively with regular wrinkle size and time period at a certain height, as shown in Figure 6a. The amplitude of the wrinkles grows rapidly, and the radius of flame curvature decreases as they are convected to the downstream as shown in Figure 6b, which shows time sequence images of
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Figure 7. Vector plot of the flow velocity in the view of (a) a stationary observer and (b) an observer with a mean velocity of reactants at an equivalence ratio of 0.54 and a Reynolds number of 625.
flame wrinkles, rotated parallel to the horizontal plane, with a 4 ms time interval. Each wrinkle at 0 ms is convected approximately 2 cm in 24 ms, i.e., with a convection velocity of 83 cm/s. The temporal change of radius of the curvature can be described with the Darrieus-Landau mechanism16,17 in the hydrodynamic region of reactants. Figure 7a shows a vector plot at a Reynolds number of 625 and an equivalence ratio of 0.54. The vector whose axial velocity is greater than 0.8 m/s was left blank for the clear view in the reactants region. Once a flame wrinkle of small amplitude is formed by flame-intrinsic KH instability, unburned mixtures are redistributed and converge to the flame front concave toward the reactants (flame hollow). On the contrary, the flow velocity of the reactants around the flame front convex toward the reactants (flame bulge) decreases lower than the laminar burning velocity because a vacancy is formed because of reactant redistribution to the flame hollow. As a result, the flame bulge moves upstream and the flame hollow moves downstream further, with the radius of curvature decreasing. Because of the redistribution and slowdown of reactants, the flow field shows vortex motion in the view of an observer traveling with mean velocity of reactants, as in Figure 7b. The flickering phenomena according to the equivalence ratio and mean velocity could be characterized by flickering frequencies and wavelengths obtained by chemiluminescence signals and instantaneous images, respectively. The frequency decreases monotonically with the equivalence ratio and increases with the mean velocity, as shown in Figure 8. It is noted that the (16) Darrieus, G. Communication presented at La Technique Moderne, 1938. (17) Landau, L. Acta Physicochim. URSS 1944, 19, 77–85.
Figure 8. Flickering frequencies of premixed flame: (open symbols) case 1 with a Reynolds number of 625, 938, 1250, and 1610 and (solid symbols) case 2 with a Reynolds number of 1610 as a function of the equivalence ratio.
flickering frequency decreases about 6 Hz with a 0.04 increment of φ, while that of the V-shaped flame of Cheng et al.9 decreases about 4 Hz with a 0.2 increment of φ, which implies that different types of KH instability dominate the flow field. The flickering wavelength decreases with the equivalence ratio, but the wavelength normalized by the flame thickness varies little, which could be explained by the result of Lee et al.12 The measurement method and characteristics of flickering wavelength will be discussed more in the Appendix. Measurement of the Burning Velocity. The modified Richardson number Rif (Rif ) gyd/uL2) is very sensitive to the
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3.1, the entrainment effect is small and the normal velocity remains constant in the upstream of the flame front. Therefore, this constant normal velocity is defined as the burning velocity for a given equivalence ratio. In Figure 9b, the measured burning velocity is plotted against equivalence ratios. The results show a reasonable value compared to the experimental data of Yamaoka and Tsuji19 and the numerical data of Tsatsaronis20 and Konnov.21 Discussion Effect of the Modified Richardson Number on FlameIntrinsic KH Instability. From the experimental results of burning velocity uL and flame angle θf, the modified Richardson number Rif ) gyd/uL2, where gy ) g sin θf and d ) FDth/FuuL, was calculated for each experimental condition. Here, adiabatic temperatures for given equivalence ratios were calculated from the EQUIL Program of the CHEMKIN Collection,22 and FDth were evaluated at the mean temperature of reactants and products and 1 atm. As discussed earlier, the modified Richardson number Rif and Reynolds number Re are important parameters determining the Strouhal number St; thus, regression of the form St ∝ RifaReb was considered. As shown in Figure 10a, the most satisfactory correlation was St ) 0.0105Rif0.869Re0.361
Figure 9. (a) Normal velocity along the normal axis with respect to the flame front at an equivalence ratio of 0.55 and (b) burning velocity results compared to experimental data of Yamaoka and Tsuji19 and numerical data of Tsatsaronis20 and Konnov.21
burning velocity and relatively insensitive to other variables. gy ) g sin θf varies very little because θf is already large enough around 80°, so that sin θf is close to unity. Flame thickness d is related to burning velocity uL via d ) FDth/FuuL. As a result, the measurement of the burning velocity is most important to calculate the modified Richardson number. The flame angle method using uL ) U cos θf is not appropriate because the streamlines of reactants in the upstream of the flame front are not straight upward at low mean velocity conditions; rather, they are deflected inward to the conical flame. As an alternative approach, the normal component of the gas velocity just ahead of the flame, relative to the flame, was measured by the PIV method and defined as the burning velocity.18 Normal velocities at three different Reynolds numbers and an equivalence ratio of 0.55 are plotted against the normal coordinate in Figure 9a. In cases of a Reynolds number of 625 and 938, which are respectively corresponding to a conventional Richardson number of 12.3 and 5.5, buoyancy forces of hot buoyant products are 12.3 and 5.5 times stronger than the inertia force of reactants and, thus, the reactants are entrained toward the flame. Because the normal velocity increases in the upstream of the flame front, these cases are not adequate for burning velocity determination. In the case of a Reynolds number of 1250, corresponding to a conventional Richardson number of (18) Tien, J.; Matalon, M. Combust. Flame 1991, 84, 238–248.
(4)
indicating that St is a strong function of Rif, which supports the conclusion of the dimensional analysis. The correlation states that the Strouhal number St ) f/(uL/d) ) f/(1/τ), the ratio of generation frequency of flame wrinkles and the inverse of flame residence time, is proportional to the 0.869 power of Rif, representing V2-V1, and the 0.361 power of Re, representing V1. When Re increases with fixed φ, Rif varies little and τ is unchanged; thus, the empirical correlation (eq 4) implies that the flame wrinkles are generated more frequently (f ) (1/τ) × 0.0105Rif0.869Re0.361) as the wrinkles are convected out faster along the flame front. When φ decreases with fixed Re, the correlation (eq 4) implies that the flame wrinkles are generated more frequently (f ∼ gy0.869uL-0.607) by the combined effect of a flame angle increase and a burning velocity decrease. It is noted that the power of Rif is close to 1, which shows a large difference compared to the value around 0.5 in previous works.5,6,8 This means that the effect of buoyancy is more dominant in flame-intrinsic KH instability than in buoyancydriven KH instability. Also, we can say that the modified Richardson number is more influential on the instability mechanism than the Reynolds number by comparing the powers of the two dimensionless numbers. In Figure 10b, the data of the present research were plotted with the empirical equation of Kostiuk and Cheng,8 St*2/Ri ) 0.00018Re2/3. St* ) fD/U/(σ + 1) is the reduced Strouhal number (heat release parameter σ ) (Fr/Fp - 1), where F is the gas density with subscripts p and r referring to products and reactants) and Ri ) (gD/U2)σ/(σ + 1) is the conventional Richardson number. As can be seen in the figure, our results (19) Yamaoka, I.; Tsuji, H. Int. Symp. Combust. 1984, 20, 1883–1892. (20) Tsatsaronis, G. Combust. Flame 1978, 33, 217–239. (21) Konnov, A. A. http://homepages.vub.ac.be/∼akonnov/science/ mechanism/worksheets/testG1.html, 2000. (22) Kee, R. J.; Rupley, F. M.; Miller, J. A.; Coltrin, M. E.; Grcar, J. F.; Meeks, E.; Moffat, H. K.; Lutz, A. E.; Dixon-Lewis, G.; Smooke, M. D.; Warnatz, J.; Evans, G. H.; Larson, R. S.; Mitchell, R. E.; Petzold, L. R.; Reynolds, W. C.; Caracotsios, M.; Stewart, W. E.; Glarorg, P.; Wang, C.; Adigum, O.; Houf, W. G.; Chou, C. P.; Miller, S. F. Chemkin Collection, Release 3.7, Reaction Design, Inc., San Diego, CA, 2002.
Flame-Intrinsic KelVin-Helmholtz Instability
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Figure 8 shows flickering frequencies obtained by FFT analyses of the photomultiplier outputs. In case 2, if the momentum of reactants is not large enough, the flow field is susceptible to the perturbation from the ambient air. Thus, the experiments were conducted for a high Reynolds number of 1610; otherwise, the flame can easily lose its symmetry and slop around, leading to irregular flickering. In case 1, the flickering frequency varies largely because of the significant increase of the modified Richardson number, even with a small decrement of the equivalence ratio. In case 2, there is only a slight increase (thermal expansion increases just 5% from 1517 to 1592 K) of the axial velocity of the hot buoyant product when the equivalence ratio changes from 0.52 to 0.56. Thus, it seems that the flow field around the BSL changes very little and as well as the flickering frequency. Conclusion
Figure 10. (a) Empirical correlation between the Strouhal number St, modified Richardson number Rif, and Reynolds number Re in case 1 and (b) its comparison to the empirical correlation of Kostiuk and Cheng.8
show a large deviation from the empirical equation. This shows that the buoyancy-driven KH instability no longer determines the flickering phenomena for flames without a BSL, meaning that the conventional Richardson number is not valid. Our results show that the modified Richardson number that has flame structure information of flame thickness and burning velocity should be used to consider the flame-intrinsic KH instability in the FSL. Effect of the Boundary Conditions on Flickering Features. The flickering characteristics of inverted conical flames without a tube wall (case 2) were also investigated. It depends upon the competition between the BSL and FSL, because they coexist. Several distinct flickering characteristics were observed compared to flames only with a FSL (case 1). In case 2, the premixed flames flicker in every equivalence ratio condition between the flammability limit and flash-back limit; i.e., the flickering limit does not exist. In case 1, as the equivalence ratio increases, the flame-intrinsic KH instability weakens because of the decrease of the modified Richardson number and, finally, the flickering phenomena stop at the flickering limit. However, in case 2, the flickering behavior in the upper part of the flame is inevitable because of the buoyancy-driven KH instability in the BSL. Therefore, there is no flickering limit. In case 2, the flickering frequency changes very little around 10 Hz, but in case 1, it varies from 14 to 24 Hz with equivalence ratios from 0.52 to 0.56.
The flame-intrinsic KH instability, a self-oscillation of laminar flame fronts extending to the streamwise direction, was investigated. The flame with a long inverted cone shape was selected, so that the flame-intrinsic KH instability could be easily generated. The effect of buoyancy-driven KH instability was excluded by surrounding the flame with a tube wall. From the dimensional analysis and physical considerations on an oblique plane flame front, it was shown that the Strouhal number St ) fd/uL depends upon the modified Richardson number Rif ) gyd/uL2 rather than the conventional Richardson number. From the experimental result of flickering frequencies, it was shown that St is a strong function of Rif, which supports the conclusion of the dimensional analysis. The similarity between the flickering limit curve and the iso-Rif curve also shows the important role of Rif on flickering characteristics. The large deviation between the flickering data in this study and the empirical correlation of Kostiuk and Cheng8 shows that the buoyancy-driven KH instability no longer determines the flickering phenomena for flames without a BSL, meaning that the conventional Richardson number is not valid. The flickering characteristics of inverted conical flames without a tube wall were also investigated. Because the BSL coexists with the FSL, the flickering characteristics depend upon the competition between the two shear layers. Several observations, such as non-existence of the flickering limit and negligible variation of the flickering frequency with an increment of the equivalence ratio, suggest that the effect of the BSL dominates that of the FSL. Acknowledgment. This research was supported by the Combustion Engineering Research Center (CERC) of the Korea Advanced Institute of Science and Technology (KAIST) by the Korea Science and Technology Foundation (KOSEF).
Appendix: Characteristics of the Flickering Wavelength The flickering wavelength, the distance between successive undulations, was measured to characterize the flickering phenomena. The instability wave exists in the tangential direction of the flame front but not in the circumferential direction. Only concentric circles of flame can be seen in the top view of the flickering flame. Therefore, the shape of instability wave in the front view is enough to measure the flickering wavelength. Generally, a small-amplitude wave can be expressed as an exponential function of kx - ωt, where k and ω represent wavenumber and frequency, respectively, in the form of complex numbers. Using the spatial approach of wave propagation, the real part of the frequency ωR is set to zero and the real
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Figure A1. Rotated images of the flame front according to various equivalence ratios, from 0.52 (top) to 0.56 (bottom), at a Reynolds number of 1250.
Figure A3. Empirical correlation between the dimensionless wavelength λ/d, modified Richardson number Rif, and Reynolds number Re in case 1.
Figure A2. Flickering wavelength λ as a function of the equivalence ratio for various Reynolds numbers.
part of the wavenumber kR is set to nonzero. Then, the instability wave can be least-squares-fitted to the following eq A1, and the flickering wavelength is defined as 2π/kI experimentally. f(x, t) ) exp(kx - ωt) ) exp(kRx)exp{i(kIx - ωIt)} (A1) where k and ω equal kR + ikI and iωI, respectively. i represents the imaginary unit, and subscripts R and I stand for real and imaginary parts of complex numbers, respectively. The procedure of the flickering wavelength measurement was as followings. First, flame images in the front view were recorded by a high-speed camera, and the flame sheets at a certain time were marked using the “extract” function of the Photoshop software. Then, the whole pixels had values from 0 to 255, with flame sheets having values lower than 255 and other parts having values of 255. The flame edges were defined by the pixel locations with the lowest value in each horizontal plane using the algorithm scanned from the left. Second, the flame edges were deformed in the axial or radial direction to
be shaped into a sine function. Then, flame edges were rotated in the clockwise direction as much as the π radian was subtracted with the flame angle, so that the left side of the flame front was located parallel to the horizontal plane. Finally, the surface was fitted to the instability wave function A1, and the flickering wavelength 2p/kI was obtained. In the upstream of the flame front, the amplitude of flame wrinkles generated by flame-intrinsic KH instability is small; however, as the wrinkles travel downstream, they cannot be fitted to a sine function anymore. Therefore, the fitting was limited to the instability wave of length equal to two or three wavelengths. Figure A1 presents rotated images of the flame front according to various equivalence ratios, from 0.52 to 0.56, showing that the flickering wavelength decreases as the equivalence ratio increases. Using the fitting method described above, the flickering wavelength was plotted as a function of the equivalence ratio for various Reynolds numbers in Figure A2. From the experimental results of the flickering wavelength λ, the dimensionless flickering wavelength λ/d was calculated for each experimental condition and plotted in Figure A3. Similar to the Strouhal number, regression of the form λ/d ∝ RifaReb was considered. The most satisfactory correlation was λ/d ) 25.8Rif-0.079Re0.068
(A2)
The correlation A2 states that λ/d only weakly depends upon the modified Richardson number and Reynolds number. In other words, λ/d was nearly constant with a constant Lewis number and various flame angle conditions. This result coincides with the results of the instability analysis of Lee et al.,12 stating that the most unstable wavelength is represented by a function only of the Lewis number irrespective of the flame angle and supports the validity of the theory. EF900147X
