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Behaviors of a flame ignited by a hot spot in a combustible vortex (vortex-bursting initiation revisited)
Proceedings of the Combustion Institute, Volume 29, 2002/pp. 1729–1736
BEHAVIORS OF A FLAME IGNITED BY A HOT SPOT IN A COMBUSTIBLE VORTEX (VORTEX-BURSTING INITIATION REVISITED) SHOICHI TAKAMORI and AKIRA UMEMURA Department of Aerospace Engineering Nagoya University Nagoya 464-8603, Japan
Ignition characteristics in turbulent flow are different from those in stagnant gas, because strong vortices may significantly modify microscopic flame propagation processes. To gain theoretical insight into misfiring in strong turbulence, we examine behaviors of a flame ignited by a hot spot in a combustible Rankine’s vortex. The fate of the flame is classified by the relationship between the vortex strength (magnitude of swirl velocity) and the input heat (including heat of combustion) effects on the azimuthal vorticity wave formation. If the input heat is large enough to realize steady flame propagation along the vortex, the flame front moves at the same constant speed as the propagation speed of azimuthal vorticity wave, which is proportional to the swirl velocity of the vortex. This flame front is stable to any small disturbance because it’s convected by the flow induced by the azimuthal vorticity distribution and situated at an equilibrium position in the negative strain rate field. On the other hand, when the vortex is too strong, the flame front is located behind the center of azimuthal vorticity distribution to be elongated in the axial direction and eventually extinguishes cooled from sides by the surrounding cold gas. The propagation speed of azimuthal vorticity wave is an intrinsic property to the vortex and independent on the way of ignition, whereas the intensity of azimuthal vorticity wave, represented by the maximum value of azimuthal vorticity or axial velocity induced at the vortex centerline, increases with the amount of effective input heat. Steady flame propagation is realized only when the latter exceeds the former. Otherwise, the flame extinguishes.
Introduction The vortex breakdown phenomena [1] appear in a variety of fluid [2–6] and combustion problems. Concerning combustion problems, the phenomena have attracted combustion researchers’ attention in the studies of swirl combustors [7] and enhanced flame propagation speed in turbulent combustion [8–15], in particular. However, the recognition that these different problems are governed by the same fluid dynamic principle of vortex breakdown is rare in the past studies. For example, the swirl is used to hold flame stably in a high-speed jet using the occurrence of reverse flow at large swirl ratio. This flow property is identical with the vortex breakdown phenomena. The similar thing can be said for the study of rapid flame propagation in a vortex tube, which is considered to be one of the candidate mechanisms achieving enhanced flame propagation speed in turbulent combustion. Except for Chomiak’s pioneering work [9], this problem has not been studied on proper fluid dynamic considerations. Recently, Umemura and Tomita [16] characterized this phenomenon in the framework of vortex breakdown phenomena and showed that the two problems are a common phenomenon, observed simply on different scales and in a different frame of reference. Once this phenomenon is recognized as a kind of
vortex breakdown phenomena, it becomes easier to understand the underlying physics. The following are found from our recent studies on the flame propagation in a vortex tube [17]. (1) An azimuthal vorticity wave, which propagates at a constant speed proportional to the swirl velocity of vortex, is produced by thermal expansion of combustion within the vortex tube. (2) A flame ignited at a point in a combustible vortex propagates with the same speed as the azimuthal vorticity wave. However, no firm analysis is made on the flame stability. For the realization of a steady flame propagation state, there must be a mechanism that anchors the flame at a certain position relative to the surrounding fluid flow. Furthermore, the extinction of ignited flame has been observed for strong vortices [11]. This phenomenon is especially interesting in connection with ignition characteristics in strong turbulence [18]. The ignition characteristics in turbulent flow are different from those in a stagnant gas, because the presence of strong vortices may significantly modify the microscopic flame propagation process. In addition, it’s not rare that an injected fuel is ignited in turbulent flow before complete mixing of fuel and oxidizer. Therefore, in the present study, extending our previous work on wave characterization on vortex bursting [17], we examine the behavior of a flame ignited by a hot spot in a combustible
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r
LAMINAR FLAMES—Flame-Vortex Interactions
r Calculation domain Inert gas
Azimuthal velocity distribution
Ignition kernel
a Flame front
a o
Combustible gas
z
Rankines vortex tube Fig. 1. Model configuration.
Rankine’s vortex as a model to explore the microscopic flame propagation process. New information about the stability of vortex-bursting flame and extinction in strong vortices will be derived from the analysis. Analysis Model The same physical model and numerical calculation method as Ref. [17] are used in the present study. A combustible Rankine’s vortex (radius a, angular velocity X) immersed in an infinite extent of inert gas is ignited at a central portion. After the formation of a flame kernel, two flame fronts are produced and propagate along the axis of the vortex tube in opposite directions (Fig. 1). The initial temperature T is uniform throughout the whole space. The ignition is made with a Gaussian heat source, which is characterized by the heat-source strength (heating rate) q and heating period tig. The axis-symmetric evolution of this reacting flow is pursued numerically by solving the compressible Navier-Stokes equations and scalar conservation equations in the Semi-implicit Predictor-Corrector Method [19]. The size of heating region is smaller than the vortex radius. Combustion obeys an Arrhenius type of onestep reaction. Since the Lewis number of reactant gas is assumed to be unity, local burning rate is not affected by flame curvature or stretching [20]. Therefore, possible flame extinction will occur because of the effect of heat loss behind the flame. By symmetry, we only consider the phenomena on the right-hand side of the plane of symmetry, z ⳱ 0. Obviously, non-unity Lewis number effect, three-dimensionality, and full kinetics affect the flame-kernel formation process and flame-front stability, but such effects may be satisfactorily identified only after the dominant physics are revealed in the simplest model. In the following, all calculation results will be presented in dimensionless form. The cylindrical coordinates (r, z), time t, and velocity v ⳱ (u, w) are
made dimensionless using vortex radius a, angular velocity X, and swirl velocity Xa, respectively. For a given combustible gas, the values of laminar burning velocity SL and flame thickness d are determined. Hence, the calculation parameters, specifying the vortex size a and strength Xa, are made dimensionless as ␣ ⳱ a/d and b ⳱ Xa/SL. Since the combustible vortex decays with time due to dissipative effects, only the phenomena occurring in the lifetime of the combustible vortex will be considered in the following. Steady Flame Propagation State We start with the description of steady flame propagation state to note that the characteristic feature of vortex breakdown with combustion is the wave nature of the flow. Figure 2a shows the contours of temperature (color) and azimuthal vorticity (line) at an instant in steady flame propagation state. A distribution of positive azimuthal vorticity, which induces a positive axial velocity at the flame-tip location, is created at the shoulder part of the parabola-like flame front. Thick lines in Fig. 3 show the temporal changes in the axial moving speeds of a flame tip, the maximum azimuthal vorticity point and a fluid particle at the maximum azimuthal vorticity point together with the temporal change in the maximum velocity at the vortex centerline (called ‘‘maximum vortex centerline velocity’’ hereafter) for the same reacting case as Fig. 2a. It is found that the moving speed of the maximum azimuthal vorticity point coincides with the flame-tip propagation speed but not with the material velocity at the maximum vorticity point. Thin lines in Fig. 4 show the axial velocity distribution along the vortex centerline at several instants. It is found that a torus of azimuthal vorticity distribution moves with a constant speed as a wave in a way that the axial vortex filaments of original vortex spiral to produce positive azimuthal vorticity and then resolve downstream. This wave nature of vortex breakdown [17] will play an important role in the following discussion on flame stability and extinction. Diagram The realization of steady flame propagation depends on the conditions of the original vortex and heat source. Fig. 5 shows the calculation results for various a(␣) and Xa(b). The heating condition is fixed. We marked a solid circle for the realization of the steady flame propagation state. Otherwise, we marked a cross or a solid triangle. In the figure, the conditions used in other investigators’ turbulent combustion simulations and vortex ring or tube experiments are also plotted for reference. The figure indicates that the flame quenches due to thermal dissipation for such a weak fine vortex that the vortex
Radial distance r
STABILITY OF VORTEX-BURSTING FLAME
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t=10
Radial distance r
Axial distance z (a) Steady burning case ( =15, =75) t=10
Radial distance r
Axial distance z (b) Quenching case ( =5, =100) t=10
Axial distance z (c) Non-reacting case (Re1125) Fig. 2. Contours of temperature and azimuthal vorticity. (a) Steady burning case (␣ ⳱ 15, b ⳱ 75); (b) quenching case (␣ ⳱ 5, b ⳱ 100); (c) non-reacting case (Re ⳱ 1125). Temperature (color) and vorticity (solid line, positive value; broken line, negative value) are non-dimensionalized by initial gas temperature and angular velocity of vortex, respectively.
Reynolds number Re ⳱ ␣b 200. The existing three-dimensional direct numerical simulations of turbulent combustion [21,22] belong to this flamequenching condition. The figure also indicates that the flame quenches when the swirl velocity is too large (case of Fig. 2b). In the following, we first examine the stability of flame tip for the case when steady flame propagation realizes, and then explore the underlying physics of flame extinction for strong vortices to derive the critical condition b ⳱ 9.6␣ depicted in the figure. Flame Stability It is well known that, in a stagnation-point flow, a premixed flame is stabilized at a plane where the normal velocity to the symmetry plane is, in magnitude, equal to the laminar burning velocity [24,25]. This flame stabilization is caused by the negative strain-rate flow. As seen in Fig. 4, a similar flow is produced by the azimuthal vorticity wave, upstream of the maximum velocity point on the vortex centerline. Hence, the flame tip located there is stable.
This flame-stabilization mechanism is used in swirl combustors to hold flame. (Confine ourselves to the flow field within the deformed vortex tube of our problem. Superimposition of a flame against uniform stream with the same speed as the flame-tip propagation speed describes a stationary flame stabilized in a swirl jet.) This conventional stability argument can be applied to characterize the unsteady flame behavior appearing in our vortex-bursting initiation systematically. To do it, we need to separate out the velocity field which is induced by the azimuthal vorticity wave and then, to clarify the roles of heating and combustion. So, we do these preparations in the next two sections. Velocity Field Induced by Azimuthal Vorticity Wave Using a scalar potential and vector potential A, the velocity v can be decomposed as v ⳱ Ⳮ ⳯ A The divergence of this equation yields
(1)
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Fig. 3. Temporal variations of azimuthal vorticity wave propagation speed (broken line), flame-tip propagation speed (solid line), axial material velocity at maximum vorticity point (dotted line), and maximum velocity at vortex centerline (chain line). Reacting case (thick line): ␣ ⳱ 15, b ⳱ 75, qtig ⳱ 6.0. Non-reacting case (thin line): Re ⳱ 1125, qtig ⳱ 6.0.
Fig. 4. Temporal variation of axial velocity distribution at vortex centerline together with locations of center of azimuthal vorticity wave and flame tip. Thin line, full calculation result; thick line, contribution of azimuthal vorticity wave.
1 Dq ⳱ •v ⳱ ⳮ q Dt 2
(2)
while the curl of equation 1 leads to 2Ah ⳱ ⳮx
(3)
for the azimuthal component Ah. The distributions of volumetric expansion rate and azimuthal vorticity x are obtained from the numerical calculation. Thus, the velocity fields induced by these source distributions can be calculated by solving equations 2 and 3 for and Ah with the null condition at infinity. The thick lines in Fig. 4 show the distributions of
Fig. 5. Diagram of flame behavior after ignition. Present calculation: 䊉 steady-flame propagation; 䉱, quenching for too strong vortex; ⳯, quenching for too weak fine vortex. Turbulent combustion simulation ⵧ, / Tanahashi et al. [21]; , Hasegawa et al. [22]. Experiment: 䉮, quenching condition in Asato’s vortex ring experiment [11]; 䉫, steady flame propagation in Ishizuka’s vortex ring experiment [12]; 䉭, steady flame propagation in Hasegawa’s straight vortex experiment [23].
axial velocity w ⳱ rⳮ1 (rAh)/ r at the vortex centerline, which coincides with the full-calculation curve (thin line) except in a short section (preheat zone) where small dilatation-induced velocity appears. It is notable that, in steady flame propagation state at 10 t 15, the sum of the laminar burning velocity and the azimuthal vorticity-induced velocity evaluated at the flame tip location coincides with the azimuthal vorticity wave speed. Hence, we can analyze the flame stability by regarding the flame as a singular surface, which propagates in the flow field induced by the azimuthal vorticity wave. Vortex Bursting Caused by Heating Alone When the heat of combustion is equated to zero, our problem reduces to the non-reacting case. This case can be utilized to clarify the role of combustion. Fig. 2c shows a result of non-reacting case. A similar azimuthal vorticity wave is formed by the heating alone. First, azimuthal vorticity is created near the boundary of heating region. As the vorticity distribution grows, it moves away from the heating region and a complete azimuthal vorticity wave is established at a distance apart from the heating region. Until this instant t1, the property of azimuthal vorticity wave varies with time. Thin lines in Fig. 3 show the temporal variations of azimuthal vorticity wave speed and maximum vortex centerline velocity in the
STABILITY OF VORTEX-BURSTING FLAME
Fig. 6. Dependence of azimuthal vorticity wave speed on total input heat.
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amount of input heat for the established azimuthal vorticity wave. This can be done in the following way. We examined the effects of q and tig on the azimuthal vorticity wave for the non-reacting case. Fig. 6 shows the q dependence of the azimuthal vorticity wave speed c for a fixed tig. The dependence of azimuthal vorticity wave speed on total input heat is weak and linear. It is considered that the extrapolated value at q ⳱ 0 corresponds to the propagation speed of azimuthal vorticity wave which appears in the spontaneous vortex breakdown. Thus, the figure indicates that the vortex bursting caused by the heating is the same phenomenon as the spontaneously excited vortex breakdown. Fixing the total input heat (qtig ⳱ const ⳱ 6), we examined the effect of heating period on maximum azimuthal vorticity (Fig. 7). A simple analysis shows that, if the produced azimuthal vorticity does not move with flow, the maximum value of azimuthal vorticity xmax, which is produced at the outer edge of the heating region by the isobaric heating with constant q during a period t, is given by q (Xt)2 (4) 2qcpT where q and cp are density and specific heat at constant pressure, respectively. Hence, assuming from Fig. 3 that, before the wave is established, the wave speed changes quadraticly with time as c ⳱ KXa (t/t1)2 with a constant K of O(1), the condition that the wave travels the same distance as the wave size t (estimated to be 2a), 01 cdt ⳱ 2a leads to xmax ⳱
6 (5) K This condition is satisfied in Fig. 7, which shows that the maximum azimuthal vorticity maximizes at a certain dimensionless time Xt 5 irrespective of the heating period. Substitution of equation 5 into equation 4 and evaluating at time t ⳱ t1 yields Xt1 ⳱
Fig. 7. Effect of heating period on maximum azimuthal vorticity.
non-reacting case, which should be compared with the reacting case. For the established azimuthal vorticity wave, there is no significant difference in the wave speed between the reacting and non-reacting cases. However, the maximum vortex centerline velocity of the reacting case is much larger than that of non-reacting case. Obviously, this velocity increase must be caused by the additional input heat of combustion in the heating region, because both the wave speed and the maximum vortex centerline velocity do not change significantly with time after the wave moves apart from the heating region. Therefore, it is important to know a relationship between the magnitude of azimuthal vorticity and the
1 q q 1 6 2 (Xt1)2 ⳱ (6) 2qcpT tig 2qcpT tig K which indicates that the peak value of maximum azimuthal vorticity is inversely proportional to the heating period, consistent with Fig. 7. The similar things can be said for other total input heats. We calculate the values of azimuthal vorticity wave speed, the maximum vortex centerline velocity, and maximum azimuthal vorticity at a fixed time t ⳱ t1 for various total input heats. The results are shown in Fig. 8. There is no significant dependence of wave speed on total input heat. The linear dependence of maximum vortex centerline velocity on maximum azimuthal vorticity, xmax ⳱
冢冣
wmax ⳱ kxmaxa, k 0.8 (7) implies that the velocity field induced by the azimuthal vorticity wave becomes similar for any heating and vortex condition as seen in Fig. 9. These
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number is unity and heat loss effect is negligible. w denotes the axial velocity induced by the azimuthal vorticity wave. As seen in Fig. 4, this velocity distribution is stationary when observed in the frame of reference moving with the wave speed c. Thus, it can be expressed in the form w(t,z) ⳱ f(z ⳮ ct). Putting f ⳱ z ⳮ ct, equation 8 is transformed to df ⳱ f(f) ⳮ c Ⳮ SL dt
(9)
Two cases can be considered from this equation: (1) the right-hand side has zero points and (2) the righthand side is negative for any f.
Fig. 8. Dependence of azimuthal vorticity wave speed (䊊) and maximum vortex centerline velocity on maximum azimuthal vorticity (䉭) for each heating condition.
Fig. 9. Similarity of axial velocity distribution at vortex centerline.
results are important for the following discussion of flame behaviors. Flame-Tip Behavior Now, we consider the flame-tip behavior in a general way. The flame-tip location Z obeys the following equation. dZ ⳱ w(t, Z) Ⳮ SL (8) dt where SL may be regarded as constant when Lewis
Case 1 Steady flame propagation is possible if the flame is located at the zero point f0 (i.e., f(f0) ⳮ c Ⳮ SL ⳱ 0). The flame location f0 is stable if df/df|f⳱f0 0. Otherwise, steady flame propagation does not realize in reality. Thus, for c k SL in which we are interested, it is found that stable, steady flame propagation is possible only when the maximum vortex centerline velocity is equal to or greater than c and the flame is located on the negative strain rate side of the maximum axial velocity location. One thing to be pointed out here is that, unlike the stagnation-point flow case, the strain rate changes locally. In particular, it vanishes as the flame location approaches the maximum axial velocity point. The flame stability in this case is delicate and not considered in the present study. Case 2 If the right-hand side is negative, the flame is eventually passed by the azimuthal vorticity wave. Once the flame front comes behind the center of azimuthal vorticity distribution, the flame cannot go back to the stable location. The shape of the flame located in the positive-strain rate field is elongated in the axial direction and becomes sharp at the tip as described in Ref. 17 in detail. Then, the thin flame front suffers cooling from sides and extinguishes eventually. Therefore, we may say that flame extinction occurs when the flame tip is located behind the center of azimuthal vorticity wave. Now, it is apparent from this stability argument that the realization of steady flame propagation state is determined from whether the maximum vortex centerline velocity can exceed the azimuthal vorticity wave speed by the heating. Critical Condition Summarizing the calculation results of both reacting and non-reacting cases above, the following are found. The azimuthal vorticity wave has a similar structure in both reaction and non-reacting cases. Its
STABILITY OF VORTEX-BURSTING FLAME
propagation speed is proportional to the swirl velocity but almost independent of the way of ignition, while the magnitude of azimuthal vorticity strongly depends on the amount of input heat available for the establishment of the azimuthal vorticity wave. This means that the formation of the wave itself is a purely fluid dynamic problem and that the axial velocity distribution at the vortex centerline is expressible as w ⳱ Kf[k(z ⳮ ct)], where only the scaling factors K and k are affected by the way of ignition. The calculation of the non-reacting case reveals that the maximum vortex centerline velocity is determined from the value of qtig only. On the other hand, in the reacting case, the combustion heat released in the heating region also contributes as an additional input heat HBqY exp(ⳮE/RT), where H, B, Y, E, and R are heat of combustion, collision factor, reactant mass fraction, activation energy, and gas constant, respectively. Therefore, if the input heat of the non-reacting case is replaced by the source term of reacting case, that is, q Ⳮ HBqY exp(ⳮE/RT), then the behavior of the azimuthal vorticity wave of the reacting case may be predicted by the relationship obtained from the non-reacting case. For simplicity, we assume that the reaction term is constant over the heating region, so that the apparent input heat is expressed as
冢
冣
E q Ⳮ rHBqY exp ⳮ RTf
(10)
where Tf, Y, and r are adiabatic flame temperature, initial reactant mass fraction, and stoichiometric ratio, respectively. Hence, we have the following condition from equations 6, 7, and 10 for the realization of steady flame propagation state, that is, the maximum vortex centerline velocity exceeding the azimuthal vorticity wave speed. wmax ⳱
2
冢 冣 冤q Ⳮ rHBqY
ka 6 2qcpT K
冢
冣冥 c kXa
E exp ⳮ RTf
(11)
Using the well-known expressions for SL and d [26], the critical condition is expressible in dimensionless form as b H ␣,
冤qcqT
p
2
冢冣
H ⳱ 2.3
6 K
冥
j H Ⳮ r 2 SL tig cpT
(12)
where j is the thermal diffusivity. The coefficient H can be evaluated using r ⳱ 1.0 and the values of parameters used in the numerical calculation, yielding H ⳱ 9.6. In Fig. 5, this critical line is drawn in a solid line, which is in fairly good agreement with
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the numerical calculation results as well as the flameextinction condition in Asato’s vortex ring experiments. Conclusion Flame extinction in strong turbulence has been conventionally discussed in terms of stretching effect. In the present study, we focused our attention on another extinction mechanism relevant to the ignition of turbulent, partially premixed gas, which may be found by examining the behavior of a flame ignited by a hot spot in a combustible Rankine’s vortex. Steady flame propagation is realized only when the maximum value of axial velocity induced at the vortex centerline, which depends on the input heat, exceeds the propagation speed of azimuthal vorticity wave, which is determined from the vortex strength. Otherwise, the flame extinguishes. The derived criterion gives a clear solution to the unsolved problems regarding flame extinction behaviors observed in the vortex ring experiments and why the existing direct numerical simulations of turbulent combustion have failed to generate vortex-busting phenomena. REFERENCES 1. Peckham, D. H., and Atkinson, S. A., Preliminary Results of Low Speed Wind Tunnel Tests on a Gothic Wing of Aspect Ratio 1.0, A.R.C. Technical Report C.P. No. 508, TN. No. Aero. 2504, 1957. 2. Squire, H. B., Analysis of the Vortex Breakdown Phenomena Part I, Imperial College Aeronautics Department report 102, London, 1960. 3. Benjamin, T. B., J. Fluid Mech. 14:593 (1962). 4. Brown, G. L., and Lopez, J. M., J. Fluid Mech. 221:553 (1990). 5. Lopez, J. M., J. Fluid Mech. 221:533 (1990). 6. Lundgren, T. S., and Ashurst, W. T., J. Fluid Mech. 200:283 (1989). 7. Beer, J. M., and Chigier, N. A., Combustion Aerodynamics, Applied Science, London, 1972. 8. McCormack, P. D., Scheller, K., Mueller, C., and Tisher, R., Combust. Flame 19:297 (1972). 9. Chomiak, J., Proc. Combust. Inst. 16:1665 (1976). 10. Ishizuka, S., Combust. Flame 82:176 (1990). 11. Asoto, K., Wada, H., Hiruma, T., and Takeuchi, Y., Combust. Flame 110:418 (1997). 12. Ishizuka, S., Murakami, T., Hamasaki, T., Koumura, K., and Hasegawa, R., Combust. Flame 113:542 (1998). 13. Hasegawa, T., Nishikado, K., and Chomiak, J., Combust. Sci. Technol. 108:67 (1995). 14. Hasegawa, T., and Nishikado, K., Proc. Combust. Inst. 26:291 (1996). 15. Ashurst, W. T., Combust. Sci. Technol. 112:175 (1996).
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16. Umemura, A., and Tomita, K., Combust. Flame 125:820 (2001). 17. Umemura, A., and Takamori, S., Proc. Combust. Inst. 28:1941 (2000). 18. Warnatz, J., Mass, U., and Dibble, R. W., Combustion, Springer, Berlin, 1999, p. 187. 19. Yee, H. C., NASA technical memorandum 101088, 1989. 20. Matalon, M., and Matkowsky, B. J., J. Fluid Mech. 124:239 (1982). 21. Tanahashi, M., Fujimura, M., and Miyauchi, T., Proc. Combust. Inst. 28:529 (2000).
22. Hasegawa, T., Michikami, S., and Nishiki, S., in ThirtySeventh Japanese Symposium on Combustion, Chiba, Japan, 1999, p. 35. 23. Goto, D., Michikami, S., Nomura, T., and Hasegawa, T., in Thirty-Seventh Japanese Symposium on Combustion, Chiba, Japan, 1999, p. 19. 24. Tsuji, H., Prog. Energy Combust. Sci. 8:93 (1982). 25. Buckmaster, J. D., and Ludford, G. S. S., Theory of Laminar Flames, Cambridge University Press, London, 1982, p. 20. 26. Williams, F. A., Combustion Theory, Addison-Wesley, Reading, MA, 1985, p. 135.
COMMENTS Satoru Ishizuka, Hiroshima University, Japan. You have mentioned that radial gas expansion causes the evolution of the azimuthal vorticity. Thus, there is an interaction between the induced vortex and the flame. Why do you get an almost constant velocity of the maximum azimuthal vorticity position independent of its distance from the flame? In your obtained diagram for steady flame propagation, you have plotted the DNS results by Tanshashi et al. In their case, the coherent eddies are stretched and have large positive and negative axial velocity fluctuations. In your model, there is no axial velocity at the beginning. Do you think that you can apply your criterion directly to their results?
Author’s Reply. The hydrodynamics of rotating fluid is important to understand the physics involved in the phenomena under consideration. The vortex bursting is a kind of vortex breakdown phenomena. Once the vortex breakdown is somehow excited by local heating or combustion, the wave (Rossby wave) propagates at a constant value that partially depends on the structure of azimuthal vorticity distribution modified by combustion. The diagram shows that their calculation conditions are in a range that the vortex radii are too small to maintain the vortex bursting or vortex breakdown phenomena. Therefore, we think that the new quenching mechanism proposed in the present paper does not apply to their calculation case.
BEHAVIORS OF A FLAME IGNITED BY A HOT SPOT IN A COMBUSTIBLE VORTEX (VORTEX-BURSTING INITIATION REVISITED) SHOICHI TAKAMORI and AKIRA UMEMURA Department of Aerospace Engineering Nagoya University Nagoya 464-8603, Japan
Ignition characteristics in turbulent flow are different from those in stagnant gas, because strong vortices may significantly modify microscopic flame propagation processes. To gain theoretical insight into misfiring in strong turbulence, we examine behaviors of a flame ignited by a hot spot in a combustible Rankine’s vortex. The fate of the flame is classified by the relationship between the vortex strength (magnitude of swirl velocity) and the input heat (including heat of combustion) effects on the azimuthal vorticity wave formation. If the input heat is large enough to realize steady flame propagation along the vortex, the flame front moves at the same constant speed as the propagation speed of azimuthal vorticity wave, which is proportional to the swirl velocity of the vortex. This flame front is stable to any small disturbance because it’s convected by the flow induced by the azimuthal vorticity distribution and situated at an equilibrium position in the negative strain rate field. On the other hand, when the vortex is too strong, the flame front is located behind the center of azimuthal vorticity distribution to be elongated in the axial direction and eventually extinguishes cooled from sides by the surrounding cold gas. The propagation speed of azimuthal vorticity wave is an intrinsic property to the vortex and independent on the way of ignition, whereas the intensity of azimuthal vorticity wave, represented by the maximum value of azimuthal vorticity or axial velocity induced at the vortex centerline, increases with the amount of effective input heat. Steady flame propagation is realized only when the latter exceeds the former. Otherwise, the flame extinguishes.
Introduction The vortex breakdown phenomena [1] appear in a variety of fluid [2–6] and combustion problems. Concerning combustion problems, the phenomena have attracted combustion researchers’ attention in the studies of swirl combustors [7] and enhanced flame propagation speed in turbulent combustion [8–15], in particular. However, the recognition that these different problems are governed by the same fluid dynamic principle of vortex breakdown is rare in the past studies. For example, the swirl is used to hold flame stably in a high-speed jet using the occurrence of reverse flow at large swirl ratio. This flow property is identical with the vortex breakdown phenomena. The similar thing can be said for the study of rapid flame propagation in a vortex tube, which is considered to be one of the candidate mechanisms achieving enhanced flame propagation speed in turbulent combustion. Except for Chomiak’s pioneering work [9], this problem has not been studied on proper fluid dynamic considerations. Recently, Umemura and Tomita [16] characterized this phenomenon in the framework of vortex breakdown phenomena and showed that the two problems are a common phenomenon, observed simply on different scales and in a different frame of reference. Once this phenomenon is recognized as a kind of
vortex breakdown phenomena, it becomes easier to understand the underlying physics. The following are found from our recent studies on the flame propagation in a vortex tube [17]. (1) An azimuthal vorticity wave, which propagates at a constant speed proportional to the swirl velocity of vortex, is produced by thermal expansion of combustion within the vortex tube. (2) A flame ignited at a point in a combustible vortex propagates with the same speed as the azimuthal vorticity wave. However, no firm analysis is made on the flame stability. For the realization of a steady flame propagation state, there must be a mechanism that anchors the flame at a certain position relative to the surrounding fluid flow. Furthermore, the extinction of ignited flame has been observed for strong vortices [11]. This phenomenon is especially interesting in connection with ignition characteristics in strong turbulence [18]. The ignition characteristics in turbulent flow are different from those in a stagnant gas, because the presence of strong vortices may significantly modify the microscopic flame propagation process. In addition, it’s not rare that an injected fuel is ignited in turbulent flow before complete mixing of fuel and oxidizer. Therefore, in the present study, extending our previous work on wave characterization on vortex bursting [17], we examine the behavior of a flame ignited by a hot spot in a combustible
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r
LAMINAR FLAMES—Flame-Vortex Interactions
r Calculation domain Inert gas
Azimuthal velocity distribution
Ignition kernel
a Flame front
a o
Combustible gas
z
Rankines vortex tube Fig. 1. Model configuration.
Rankine’s vortex as a model to explore the microscopic flame propagation process. New information about the stability of vortex-bursting flame and extinction in strong vortices will be derived from the analysis. Analysis Model The same physical model and numerical calculation method as Ref. [17] are used in the present study. A combustible Rankine’s vortex (radius a, angular velocity X) immersed in an infinite extent of inert gas is ignited at a central portion. After the formation of a flame kernel, two flame fronts are produced and propagate along the axis of the vortex tube in opposite directions (Fig. 1). The initial temperature T is uniform throughout the whole space. The ignition is made with a Gaussian heat source, which is characterized by the heat-source strength (heating rate) q and heating period tig. The axis-symmetric evolution of this reacting flow is pursued numerically by solving the compressible Navier-Stokes equations and scalar conservation equations in the Semi-implicit Predictor-Corrector Method [19]. The size of heating region is smaller than the vortex radius. Combustion obeys an Arrhenius type of onestep reaction. Since the Lewis number of reactant gas is assumed to be unity, local burning rate is not affected by flame curvature or stretching [20]. Therefore, possible flame extinction will occur because of the effect of heat loss behind the flame. By symmetry, we only consider the phenomena on the right-hand side of the plane of symmetry, z ⳱ 0. Obviously, non-unity Lewis number effect, three-dimensionality, and full kinetics affect the flame-kernel formation process and flame-front stability, but such effects may be satisfactorily identified only after the dominant physics are revealed in the simplest model. In the following, all calculation results will be presented in dimensionless form. The cylindrical coordinates (r, z), time t, and velocity v ⳱ (u, w) are
made dimensionless using vortex radius a, angular velocity X, and swirl velocity Xa, respectively. For a given combustible gas, the values of laminar burning velocity SL and flame thickness d are determined. Hence, the calculation parameters, specifying the vortex size a and strength Xa, are made dimensionless as ␣ ⳱ a/d and b ⳱ Xa/SL. Since the combustible vortex decays with time due to dissipative effects, only the phenomena occurring in the lifetime of the combustible vortex will be considered in the following. Steady Flame Propagation State We start with the description of steady flame propagation state to note that the characteristic feature of vortex breakdown with combustion is the wave nature of the flow. Figure 2a shows the contours of temperature (color) and azimuthal vorticity (line) at an instant in steady flame propagation state. A distribution of positive azimuthal vorticity, which induces a positive axial velocity at the flame-tip location, is created at the shoulder part of the parabola-like flame front. Thick lines in Fig. 3 show the temporal changes in the axial moving speeds of a flame tip, the maximum azimuthal vorticity point and a fluid particle at the maximum azimuthal vorticity point together with the temporal change in the maximum velocity at the vortex centerline (called ‘‘maximum vortex centerline velocity’’ hereafter) for the same reacting case as Fig. 2a. It is found that the moving speed of the maximum azimuthal vorticity point coincides with the flame-tip propagation speed but not with the material velocity at the maximum vorticity point. Thin lines in Fig. 4 show the axial velocity distribution along the vortex centerline at several instants. It is found that a torus of azimuthal vorticity distribution moves with a constant speed as a wave in a way that the axial vortex filaments of original vortex spiral to produce positive azimuthal vorticity and then resolve downstream. This wave nature of vortex breakdown [17] will play an important role in the following discussion on flame stability and extinction. Diagram The realization of steady flame propagation depends on the conditions of the original vortex and heat source. Fig. 5 shows the calculation results for various a(␣) and Xa(b). The heating condition is fixed. We marked a solid circle for the realization of the steady flame propagation state. Otherwise, we marked a cross or a solid triangle. In the figure, the conditions used in other investigators’ turbulent combustion simulations and vortex ring or tube experiments are also plotted for reference. The figure indicates that the flame quenches due to thermal dissipation for such a weak fine vortex that the vortex
Radial distance r
STABILITY OF VORTEX-BURSTING FLAME
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t=10
Radial distance r
Axial distance z (a) Steady burning case ( =15, =75) t=10
Radial distance r
Axial distance z (b) Quenching case ( =5, =100) t=10
Axial distance z (c) Non-reacting case (Re1125) Fig. 2. Contours of temperature and azimuthal vorticity. (a) Steady burning case (␣ ⳱ 15, b ⳱ 75); (b) quenching case (␣ ⳱ 5, b ⳱ 100); (c) non-reacting case (Re ⳱ 1125). Temperature (color) and vorticity (solid line, positive value; broken line, negative value) are non-dimensionalized by initial gas temperature and angular velocity of vortex, respectively.
Reynolds number Re ⳱ ␣b 200. The existing three-dimensional direct numerical simulations of turbulent combustion [21,22] belong to this flamequenching condition. The figure also indicates that the flame quenches when the swirl velocity is too large (case of Fig. 2b). In the following, we first examine the stability of flame tip for the case when steady flame propagation realizes, and then explore the underlying physics of flame extinction for strong vortices to derive the critical condition b ⳱ 9.6␣ depicted in the figure. Flame Stability It is well known that, in a stagnation-point flow, a premixed flame is stabilized at a plane where the normal velocity to the symmetry plane is, in magnitude, equal to the laminar burning velocity [24,25]. This flame stabilization is caused by the negative strain-rate flow. As seen in Fig. 4, a similar flow is produced by the azimuthal vorticity wave, upstream of the maximum velocity point on the vortex centerline. Hence, the flame tip located there is stable.
This flame-stabilization mechanism is used in swirl combustors to hold flame. (Confine ourselves to the flow field within the deformed vortex tube of our problem. Superimposition of a flame against uniform stream with the same speed as the flame-tip propagation speed describes a stationary flame stabilized in a swirl jet.) This conventional stability argument can be applied to characterize the unsteady flame behavior appearing in our vortex-bursting initiation systematically. To do it, we need to separate out the velocity field which is induced by the azimuthal vorticity wave and then, to clarify the roles of heating and combustion. So, we do these preparations in the next two sections. Velocity Field Induced by Azimuthal Vorticity Wave Using a scalar potential and vector potential A, the velocity v can be decomposed as v ⳱ Ⳮ ⳯ A The divergence of this equation yields
(1)
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LAMINAR FLAMES—Flame-Vortex Interactions
Fig. 3. Temporal variations of azimuthal vorticity wave propagation speed (broken line), flame-tip propagation speed (solid line), axial material velocity at maximum vorticity point (dotted line), and maximum velocity at vortex centerline (chain line). Reacting case (thick line): ␣ ⳱ 15, b ⳱ 75, qtig ⳱ 6.0. Non-reacting case (thin line): Re ⳱ 1125, qtig ⳱ 6.0.
Fig. 4. Temporal variation of axial velocity distribution at vortex centerline together with locations of center of azimuthal vorticity wave and flame tip. Thin line, full calculation result; thick line, contribution of azimuthal vorticity wave.
1 Dq ⳱ •v ⳱ ⳮ q Dt 2
(2)
while the curl of equation 1 leads to 2Ah ⳱ ⳮx
(3)
for the azimuthal component Ah. The distributions of volumetric expansion rate and azimuthal vorticity x are obtained from the numerical calculation. Thus, the velocity fields induced by these source distributions can be calculated by solving equations 2 and 3 for and Ah with the null condition at infinity. The thick lines in Fig. 4 show the distributions of
Fig. 5. Diagram of flame behavior after ignition. Present calculation: 䊉 steady-flame propagation; 䉱, quenching for too strong vortex; ⳯, quenching for too weak fine vortex. Turbulent combustion simulation ⵧ, / Tanahashi et al. [21]; , Hasegawa et al. [22]. Experiment: 䉮, quenching condition in Asato’s vortex ring experiment [11]; 䉫, steady flame propagation in Ishizuka’s vortex ring experiment [12]; 䉭, steady flame propagation in Hasegawa’s straight vortex experiment [23].
axial velocity w ⳱ rⳮ1 (rAh)/ r at the vortex centerline, which coincides with the full-calculation curve (thin line) except in a short section (preheat zone) where small dilatation-induced velocity appears. It is notable that, in steady flame propagation state at 10 t 15, the sum of the laminar burning velocity and the azimuthal vorticity-induced velocity evaluated at the flame tip location coincides with the azimuthal vorticity wave speed. Hence, we can analyze the flame stability by regarding the flame as a singular surface, which propagates in the flow field induced by the azimuthal vorticity wave. Vortex Bursting Caused by Heating Alone When the heat of combustion is equated to zero, our problem reduces to the non-reacting case. This case can be utilized to clarify the role of combustion. Fig. 2c shows a result of non-reacting case. A similar azimuthal vorticity wave is formed by the heating alone. First, azimuthal vorticity is created near the boundary of heating region. As the vorticity distribution grows, it moves away from the heating region and a complete azimuthal vorticity wave is established at a distance apart from the heating region. Until this instant t1, the property of azimuthal vorticity wave varies with time. Thin lines in Fig. 3 show the temporal variations of azimuthal vorticity wave speed and maximum vortex centerline velocity in the
STABILITY OF VORTEX-BURSTING FLAME
Fig. 6. Dependence of azimuthal vorticity wave speed on total input heat.
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amount of input heat for the established azimuthal vorticity wave. This can be done in the following way. We examined the effects of q and tig on the azimuthal vorticity wave for the non-reacting case. Fig. 6 shows the q dependence of the azimuthal vorticity wave speed c for a fixed tig. The dependence of azimuthal vorticity wave speed on total input heat is weak and linear. It is considered that the extrapolated value at q ⳱ 0 corresponds to the propagation speed of azimuthal vorticity wave which appears in the spontaneous vortex breakdown. Thus, the figure indicates that the vortex bursting caused by the heating is the same phenomenon as the spontaneously excited vortex breakdown. Fixing the total input heat (qtig ⳱ const ⳱ 6), we examined the effect of heating period on maximum azimuthal vorticity (Fig. 7). A simple analysis shows that, if the produced azimuthal vorticity does not move with flow, the maximum value of azimuthal vorticity xmax, which is produced at the outer edge of the heating region by the isobaric heating with constant q during a period t, is given by q (Xt)2 (4) 2qcpT where q and cp are density and specific heat at constant pressure, respectively. Hence, assuming from Fig. 3 that, before the wave is established, the wave speed changes quadraticly with time as c ⳱ KXa (t/t1)2 with a constant K of O(1), the condition that the wave travels the same distance as the wave size t (estimated to be 2a), 01 cdt ⳱ 2a leads to xmax ⳱
6 (5) K This condition is satisfied in Fig. 7, which shows that the maximum azimuthal vorticity maximizes at a certain dimensionless time Xt 5 irrespective of the heating period. Substitution of equation 5 into equation 4 and evaluating at time t ⳱ t1 yields Xt1 ⳱
Fig. 7. Effect of heating period on maximum azimuthal vorticity.
non-reacting case, which should be compared with the reacting case. For the established azimuthal vorticity wave, there is no significant difference in the wave speed between the reacting and non-reacting cases. However, the maximum vortex centerline velocity of the reacting case is much larger than that of non-reacting case. Obviously, this velocity increase must be caused by the additional input heat of combustion in the heating region, because both the wave speed and the maximum vortex centerline velocity do not change significantly with time after the wave moves apart from the heating region. Therefore, it is important to know a relationship between the magnitude of azimuthal vorticity and the
1 q q 1 6 2 (Xt1)2 ⳱ (6) 2qcpT tig 2qcpT tig K which indicates that the peak value of maximum azimuthal vorticity is inversely proportional to the heating period, consistent with Fig. 7. The similar things can be said for other total input heats. We calculate the values of azimuthal vorticity wave speed, the maximum vortex centerline velocity, and maximum azimuthal vorticity at a fixed time t ⳱ t1 for various total input heats. The results are shown in Fig. 8. There is no significant dependence of wave speed on total input heat. The linear dependence of maximum vortex centerline velocity on maximum azimuthal vorticity, xmax ⳱
冢冣
wmax ⳱ kxmaxa, k 0.8 (7) implies that the velocity field induced by the azimuthal vorticity wave becomes similar for any heating and vortex condition as seen in Fig. 9. These
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LAMINAR FLAMES—Flame-Vortex Interactions
number is unity and heat loss effect is negligible. w denotes the axial velocity induced by the azimuthal vorticity wave. As seen in Fig. 4, this velocity distribution is stationary when observed in the frame of reference moving with the wave speed c. Thus, it can be expressed in the form w(t,z) ⳱ f(z ⳮ ct). Putting f ⳱ z ⳮ ct, equation 8 is transformed to df ⳱ f(f) ⳮ c Ⳮ SL dt
(9)
Two cases can be considered from this equation: (1) the right-hand side has zero points and (2) the righthand side is negative for any f.
Fig. 8. Dependence of azimuthal vorticity wave speed (䊊) and maximum vortex centerline velocity on maximum azimuthal vorticity (䉭) for each heating condition.
Fig. 9. Similarity of axial velocity distribution at vortex centerline.
results are important for the following discussion of flame behaviors. Flame-Tip Behavior Now, we consider the flame-tip behavior in a general way. The flame-tip location Z obeys the following equation. dZ ⳱ w(t, Z) Ⳮ SL (8) dt where SL may be regarded as constant when Lewis
Case 1 Steady flame propagation is possible if the flame is located at the zero point f0 (i.e., f(f0) ⳮ c Ⳮ SL ⳱ 0). The flame location f0 is stable if df/df|f⳱f0 0. Otherwise, steady flame propagation does not realize in reality. Thus, for c k SL in which we are interested, it is found that stable, steady flame propagation is possible only when the maximum vortex centerline velocity is equal to or greater than c and the flame is located on the negative strain rate side of the maximum axial velocity location. One thing to be pointed out here is that, unlike the stagnation-point flow case, the strain rate changes locally. In particular, it vanishes as the flame location approaches the maximum axial velocity point. The flame stability in this case is delicate and not considered in the present study. Case 2 If the right-hand side is negative, the flame is eventually passed by the azimuthal vorticity wave. Once the flame front comes behind the center of azimuthal vorticity distribution, the flame cannot go back to the stable location. The shape of the flame located in the positive-strain rate field is elongated in the axial direction and becomes sharp at the tip as described in Ref. 17 in detail. Then, the thin flame front suffers cooling from sides and extinguishes eventually. Therefore, we may say that flame extinction occurs when the flame tip is located behind the center of azimuthal vorticity wave. Now, it is apparent from this stability argument that the realization of steady flame propagation state is determined from whether the maximum vortex centerline velocity can exceed the azimuthal vorticity wave speed by the heating. Critical Condition Summarizing the calculation results of both reacting and non-reacting cases above, the following are found. The azimuthal vorticity wave has a similar structure in both reaction and non-reacting cases. Its
STABILITY OF VORTEX-BURSTING FLAME
propagation speed is proportional to the swirl velocity but almost independent of the way of ignition, while the magnitude of azimuthal vorticity strongly depends on the amount of input heat available for the establishment of the azimuthal vorticity wave. This means that the formation of the wave itself is a purely fluid dynamic problem and that the axial velocity distribution at the vortex centerline is expressible as w ⳱ Kf[k(z ⳮ ct)], where only the scaling factors K and k are affected by the way of ignition. The calculation of the non-reacting case reveals that the maximum vortex centerline velocity is determined from the value of qtig only. On the other hand, in the reacting case, the combustion heat released in the heating region also contributes as an additional input heat HBqY exp(ⳮE/RT), where H, B, Y, E, and R are heat of combustion, collision factor, reactant mass fraction, activation energy, and gas constant, respectively. Therefore, if the input heat of the non-reacting case is replaced by the source term of reacting case, that is, q Ⳮ HBqY exp(ⳮE/RT), then the behavior of the azimuthal vorticity wave of the reacting case may be predicted by the relationship obtained from the non-reacting case. For simplicity, we assume that the reaction term is constant over the heating region, so that the apparent input heat is expressed as
冢
冣
E q Ⳮ rHBqY exp ⳮ RTf
(10)
where Tf, Y, and r are adiabatic flame temperature, initial reactant mass fraction, and stoichiometric ratio, respectively. Hence, we have the following condition from equations 6, 7, and 10 for the realization of steady flame propagation state, that is, the maximum vortex centerline velocity exceeding the azimuthal vorticity wave speed. wmax ⳱
2
冢 冣 冤q Ⳮ rHBqY
ka 6 2qcpT K
冢
冣冥 c kXa
E exp ⳮ RTf
(11)
Using the well-known expressions for SL and d [26], the critical condition is expressible in dimensionless form as b H ␣,
冤qcqT
p
2
冢冣
H ⳱ 2.3
6 K
冥
j H Ⳮ r 2 SL tig cpT
(12)
where j is the thermal diffusivity. The coefficient H can be evaluated using r ⳱ 1.0 and the values of parameters used in the numerical calculation, yielding H ⳱ 9.6. In Fig. 5, this critical line is drawn in a solid line, which is in fairly good agreement with
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the numerical calculation results as well as the flameextinction condition in Asato’s vortex ring experiments. Conclusion Flame extinction in strong turbulence has been conventionally discussed in terms of stretching effect. In the present study, we focused our attention on another extinction mechanism relevant to the ignition of turbulent, partially premixed gas, which may be found by examining the behavior of a flame ignited by a hot spot in a combustible Rankine’s vortex. Steady flame propagation is realized only when the maximum value of axial velocity induced at the vortex centerline, which depends on the input heat, exceeds the propagation speed of azimuthal vorticity wave, which is determined from the vortex strength. Otherwise, the flame extinguishes. The derived criterion gives a clear solution to the unsolved problems regarding flame extinction behaviors observed in the vortex ring experiments and why the existing direct numerical simulations of turbulent combustion have failed to generate vortex-busting phenomena. REFERENCES 1. Peckham, D. H., and Atkinson, S. A., Preliminary Results of Low Speed Wind Tunnel Tests on a Gothic Wing of Aspect Ratio 1.0, A.R.C. Technical Report C.P. No. 508, TN. No. Aero. 2504, 1957. 2. Squire, H. B., Analysis of the Vortex Breakdown Phenomena Part I, Imperial College Aeronautics Department report 102, London, 1960. 3. Benjamin, T. B., J. Fluid Mech. 14:593 (1962). 4. Brown, G. L., and Lopez, J. M., J. Fluid Mech. 221:553 (1990). 5. Lopez, J. M., J. Fluid Mech. 221:533 (1990). 6. Lundgren, T. S., and Ashurst, W. T., J. Fluid Mech. 200:283 (1989). 7. Beer, J. M., and Chigier, N. A., Combustion Aerodynamics, Applied Science, London, 1972. 8. McCormack, P. D., Scheller, K., Mueller, C., and Tisher, R., Combust. Flame 19:297 (1972). 9. Chomiak, J., Proc. Combust. Inst. 16:1665 (1976). 10. Ishizuka, S., Combust. Flame 82:176 (1990). 11. Asoto, K., Wada, H., Hiruma, T., and Takeuchi, Y., Combust. Flame 110:418 (1997). 12. Ishizuka, S., Murakami, T., Hamasaki, T., Koumura, K., and Hasegawa, R., Combust. Flame 113:542 (1998). 13. Hasegawa, T., Nishikado, K., and Chomiak, J., Combust. Sci. Technol. 108:67 (1995). 14. Hasegawa, T., and Nishikado, K., Proc. Combust. Inst. 26:291 (1996). 15. Ashurst, W. T., Combust. Sci. Technol. 112:175 (1996).
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16. Umemura, A., and Tomita, K., Combust. Flame 125:820 (2001). 17. Umemura, A., and Takamori, S., Proc. Combust. Inst. 28:1941 (2000). 18. Warnatz, J., Mass, U., and Dibble, R. W., Combustion, Springer, Berlin, 1999, p. 187. 19. Yee, H. C., NASA technical memorandum 101088, 1989. 20. Matalon, M., and Matkowsky, B. J., J. Fluid Mech. 124:239 (1982). 21. Tanahashi, M., Fujimura, M., and Miyauchi, T., Proc. Combust. Inst. 28:529 (2000).
22. Hasegawa, T., Michikami, S., and Nishiki, S., in ThirtySeventh Japanese Symposium on Combustion, Chiba, Japan, 1999, p. 35. 23. Goto, D., Michikami, S., Nomura, T., and Hasegawa, T., in Thirty-Seventh Japanese Symposium on Combustion, Chiba, Japan, 1999, p. 19. 24. Tsuji, H., Prog. Energy Combust. Sci. 8:93 (1982). 25. Buckmaster, J. D., and Ludford, G. S. S., Theory of Laminar Flames, Cambridge University Press, London, 1982, p. 20. 26. Williams, F. A., Combustion Theory, Addison-Wesley, Reading, MA, 1985, p. 135.
COMMENTS Satoru Ishizuka, Hiroshima University, Japan. You have mentioned that radial gas expansion causes the evolution of the azimuthal vorticity. Thus, there is an interaction between the induced vortex and the flame. Why do you get an almost constant velocity of the maximum azimuthal vorticity position independent of its distance from the flame? In your obtained diagram for steady flame propagation, you have plotted the DNS results by Tanshashi et al. In their case, the coherent eddies are stretched and have large positive and negative axial velocity fluctuations. In your model, there is no axial velocity at the beginning. Do you think that you can apply your criterion directly to their results?
Author’s Reply. The hydrodynamics of rotating fluid is important to understand the physics involved in the phenomena under consideration. The vortex bursting is a kind of vortex breakdown phenomena. Once the vortex breakdown is somehow excited by local heating or combustion, the wave (Rossby wave) propagates at a constant value that partially depends on the structure of azimuthal vorticity distribution modified by combustion. The diagram shows that their calculation conditions are in a range that the vortex radii are too small to maintain the vortex bursting or vortex breakdown phenomena. Therefore, we think that the new quenching mechanism proposed in the present paper does not apply to their calculation case.