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Effect of oscillatory stretch on the flame speed of wall-stagnating premixed flame
Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 875–882
EFFECT OF OSCILLATORY STRETCH ON THE FLAME SPEED OF WALLSTAGNATING PREMIXED FLAME TARO HIRASAWA, TOSHIHISA UEDA, AKIKO MATSUO and MASAHIKO MIZOMOTO Department of Mechanical Engineering Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama, Kanagawa, 223-8522, Japan
The effect of unsteady stretch on the wall-stagnating lean methane, premixed flame was investigated experimentally. The influence of the oscillatory stretch on the flame speed was examined when the stagnation wall was oscillated sinusoidally in the direction of the main flow at a constant amplitude. Using LDV, the velocities of unburned gas and of flame in laboratory coordinate were measured to obtain flame speed (propagating velocity relative to unburned gas). The oscillatory characteristic of the flow field is indicated by the Strouhal number, representing the ratio of the angular frequency of the wall times the mean wall location to the flow velocity at the nozzle exit. There exist two factors dominating the oscillatory stretch, that is, the change of the distance between the wall and the nozzle, which is a quasi-steady factor, and the change of the velocity of the wall, which is an unsteady factor. When the flow is oscillated at low Strouhal number (#1), the quasi-steady factor, which is independent of the Strouhal number, is dominant. And the oscillation of the flame speed is not observed. In this case, quasi-steady factor is not large enough to oscillate the flame speed since the ratio of the amplitude of the wall oscillation to the mean distance between the wall and the nozzle is small. On the other hand, when the flow is oscillated at high Strouhal number (#1), the unsteady factor, which is proportional to the Strouhal number, becomes dominant. In this case, the oscillation of the flame speed is observed and the amplitude of the oscillation of flame speed increases with increasing Strouhal number.
Introduction It has recently been recognized that effects of unsteady flow field on flames may be considerable in many practical flames. As reported in several recent studies [1–4], the unsteadiness of the flow field can introduce an additional effect that may significantly modify the flame behavior. The unsteady effect of acoustic flow field on flame instabilities is a serious problem in many kinds of combustors, including rocket motors. Solid propellant rocket motors are susceptible to axial instabilities, which may interfere with engine performance [5]. The flow disturbance caused by interaction between the solid propellant surface and the flame leads to acoustic field oscillation. When the acoustic field is forced to oscillate, the flame oscillatory movement is observed and considered to be caused by oscillatory reaction rate within the flame and the oscillatory nature of the velocity of the combustible mixture, by Sankar et al. [5]. It is also suggested by Im et al. [1] that acoustic instability of flames may be influenced significantly by Lewis number (Le). Turbulent flames are affected by nonuniform unsteady stretch. It is difficult to examine both the effect of nonuniform stretch and the effect of unsteady stretch at the same time. However, it is not so difficult to examine them separately, while it naturally
occurs at the same time in turbulent flames. Recently, the effect of unsteady stretch, which is not clear in many points, is investigated extensively [1– 4,6–14]. As a result, the effect is found to depend on the Lewis number [1], which is similar to that of nonuniform stretch. Also, the flame is found to be weakened even though the mean of stretch rate does not vary and to be quenched below the steady quenching rate due to the effect of unsteady stretch on the flame [3,6]. The quenching caused by unsteady stretch is responsible for undesirable phenomena, for instance, significant ignition delay in a pulse combustor where a large vortical structure is formed [15,16]. It is well known that the flame speed of the mixture with Le ? 1 depends on flame stretch and that the flame speed of the mixture with Le , 1 increases with increasing stretch rate [17]. While studies on unsteady stretched flame abound, investigations on the effect of unsteady stretch on the flame speed are less numerous. However, there are some investigations [4,18] that provide us useful information when examining its effect on the flame speed. Mueller et al. [18] experimentally examined the response of flame strength to the unsteady stretch when the premixed flame was stretched by a counter-propagating toroidal vortex. The peak dilatation rate at each
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Fig. 1. Schematic drawing of wall-stagnation flow burner system. Diameter of the nozzle is 12 mm. Location of the stagnation wall is oscillated sinusoidally along stagnation streamline.
flame segment was used as an indicator of the local flame strength. They found that the flame requires a relatively long time to be weakened by positive stretch, yet it is rapidly strengthened by negative stretch, rising linearly to 240% of the unstretched value. As for flame temperature, Egolfopoulos [4] observed that the flame temperature is oscillated by stretch oscillation. The phase delay and the attenuation of the temperature oscillation is observed when the stretch oscillates at high frequency. Similarly, the stretch oscillation is expected to affect the flame speed significantly, which is examined in this study. In the present study, the effect of unsteady stretch on the wall-stagnating, lean methane premixed flame (which is isolated from the effect of curvature) is investigated experimentally. In order to examine the influence of oscillatory stretch on the flame speed, the flame speed is measured, while the stagnation wall is oscillated sinusoidally at a constant amplitude in the direction of the main flow. Experimental Setup Figure 1 shows the schematic drawing of a burner system. A laminar flat flame is established in the stagnating region when CH4/air mixture is issued from the nozzle with a 12-mm-diameter exit perpendicularly to the stagnation wall at 1.07 m/s. Coflowing air flows surrounding the main flow to minimize the interaction between the main flow and the ambient air. The coordinate y is defined as the downstream direction from the nozzle exit along the center axis of the nozzle. The location of the stagnation wall (yw) is oscillated sinusoidally along the coordinate y; that is, yw 4 y*w sin(xt) ` y¯w
(1)
where y* ¯ w 4 16 mm, x is angular w 4 1.5 mm, y frequency, and t is time. The stagnation wall is a
ceramic disk with a 76 mm diameter, and the thermal condition of the wall is considered to be adiabatic. The stagnation wall is oscillated by a voice coil motor and controlled precisely by the feedback control system with a wall location sensor (precision potentiometer), a function generator and a control device. A laser Doppler velocimetry (LDV) system is used to measure the flame speed of lean CH4/air mixtures at equivalence ratios (f) of 0.77 and 0.95. In this case, the effect of Lewis numbers less than unity is expected as observed by Mizomoto et al. [19], although the Lewis number at room temperature is evaluated at almost unity. To perform LDV measurements, scattering particles (fine silicone oil droplets) are seeded in the premixed mixture. To measure the oscillating flow velocity and the oscillating wall location simultaneously, the analog outputs of an LDV counter and the wall location sensor are recorded on computer memories through the simultaneous sampling AD converter. Results and Discussion Oscillatory Velocity Measurement To measure the oscillatory velocity, the data reduction procedure, that is, ensemble averaging of the randomly arriving LDV data, is performed. This technique, which permits determination of the periodic velocity history by synchronizing the beginning of the sampling interval of the LDV data with respect to the periodic wall oscillations, is similar to those utilized by Sankar et al. [5] and Lepicovsky [20]. Figures 2a and 2b show the history of the flow velocity oscillation at a point along the stagnation streamline (u) when the frequencies of the wall oscillation (f) are 0.5 and 8.0 Hz, plotted against the phase of the wall oscillation (xt). The oscillatory characteristic of the flow field is indicated by the Strouhal number (S), representing the ratio of the characteristic time of the flow to the characteristic time of the oscillation. The Strouhal number is defined as S 4 xy¯w/U0, where y¯w is the mean wall location and U0 is the flow velocity at the nozzle exit. When f are 0.5 and 8.0 Hz, S are 0.047 and 0.75, respectively. The history of the flow velocity oscillation at y 4 8.0 mm, shown by the triangle in Fig. 2a, is correctly sinusoidal waveform, and the plots are obtained at all phases. If the burned gas has reached the point of measurement for some phases, plots will not have been obtained for those phases due to evaporation of the scattering particles. Thus, this history shows that the flame does not reach the point of measurement (y 4 8.0 mm) at any phase. The flow velocity at y 4 9.0 mm decreases when xt increases from 908 to 1808 because the stagnation
OSCILLATORY STRETCH ON THE FLAME SPEED
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Fig. 2. Oscillatory history of the flow velocity and of the unburned gas velocity at the flame front. (a) f 4 0.95, f 4 0.5[Hz] (S 4 0.047). (b) f 4 0.95, f 4 8.0[Hz] (S 4 0.75). Flame moves upstream (I) or downstream (II) and approaches the point of measurement.
wall approaches the point of measurement. After that, as the flame moves upstream and approaches the point of measurement (I), the velocity attains a minimum at xt 4 2108, and then increases due to thermal expansion in the preheat zone. Therefore, that the velocity attains a minimum indicates that the flame front reaches the point of measurement, which is similar to Yu et al. [21]. As a result, the unburned gas velocity at the flame front in laboratory coordinates (uf), shown by a square plot, is obtained. At the same time, xt when the flame reaches the point of measurement is obtained. By changing y, xt at different flame locations is obtained. As a result, yf is derived as a function of xt. In the range of xt 4 2308–3208, the velocity data disappear because of the evaporation of scattering particles. Then, as the flame moves downstream and approaches the point of measurement (II), the velocity data appear, and the velocity attains a minimum at xt 4 3308 shown by a square plot, indicating the arrival of the flame front at the point of measurement. The square plots that are labeled neither I nor II correspond to the measurements at y 4 8.5, 9.5, 10.5, and 11.0 mm, which are not plotted in Fig. 2 to avoid being crowded. Thus, two squares are obtained from one velocity history at a point of measurement. It is noteworthy to mention here that two squares of uf obtained from one velocity history at f 4 0.5 Hz are close to each other; however, they are different from each other at f 4 8.0 Hz. The reason why they are different at f 4 8.0 Hz is the following: When the flame moves upstream (I), uf is smaller than the flame speed. On the other hand, when the flame moves downstream
(II), uf is larger than the flame speed. If the flame speed is constant, it is obvious that the difference increases with an increase in frequency of the flame oscillation. Waveform of Oscillations Figure 3 shows the oscillatory histories of yw, yf, uf, the velocity of the flame motion (dyf /dt) in laboratory coordinate, and the laminar flame speed (Su) (propagating velocity relative to unburned gas), plotted against xt. The history of the flame location yf is well fit with the sine wave, and the frequency agrees correctly with the frequency of the wall oscillation, which are confirmed by high-resolution measurement of the flame location with the video image ([22], Fig. 4). These facts are confirmed in the present study by the LDV measurement as well, although the number of plots is fewer. Thus, yf can be described as yf 4 ¯ f. Each value of y*f , h, and y¯f, is y* f sin(xt ` h) ` y obtained by using the least-squares method to fit the measured yf plots (see Fig. 3). The velocity of the flame motion dyf /dt is, then, obtained as xy*f cos(xt ` h) by the differentiation of yf. uf, which is subject to the effect of flame motion, is obtained by the LDV data when the flame front arrives at a point of measurement. The laminar flame speed Su is, therefore, obtained as Su 4 uf 1 dyf /dt. As shown in Fig. 3a, the flame oscillates following the oscillation of the wall location, and the flame speed is constant at any xt. In other words, the flame oscillates, keeping its propagating speed constant and following the oscillation of the velocity profile
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Fig. 3. Oscillatory histories of stagnation wall location (yw), flame location (yf ), unburned gas velocity at the flame front in laboratory coordinate (uf ), velocity of the flame motion in laboratory coordinate (dyf /dt), and laminar flame speed (Su) (propagating velocity relative to unburned gas). (a) f 4 0.95, f 4 0.5[Hz] (S 4 0.047). (b) f 4 0.95, f 4 8.0[Hz] (S 4 0.75). (c) f 4 0.95, f 4 23[Hz] (S 4 2.2). (d) f 4 0.95, f 4 51[Hz] (S 4 4.8).
caused by the change of the wall location. In addition, the amplitude of dyf /dt is very small at f 4 0.5 Hz, which shows that the flame oscillates quasisteadily. As shown in Fig. 3b, at f 4 8.0 Hz, both the amplitude of dyf /dt oscillation and the amplitude of uf oscillation are larger amplitudes than those of Fig. 3a due to the effect of increasing of the Strouhal number; that is, the characteristic time of the oscillation becomes smaller compared with the characteristic time of the flow. Su is almost constant though small fluctuation is observed. At high frequencies ( f 4 23 and 51 Hz), as shown in Figs. 3c and 3d, Su
clearly oscillates. At both f 4 23 and 51 Hz, Su attains a maximum when the wall and flame are moving downward. This is discussed later with the discussion concerning the amplitude of the flame speed. Oscillation of Flame Location as a Function of the Strouhal Number Figure 4 shows the region of oscillatory yf and the amplitude of yf (y* f ), which is derived from the waveforms shown in Fig. 3, plotted against the Strouhal number. The flame of f 4 0.77 is formed closer to
OSCILLATORY STRETCH ON THE FLAME SPEED
879
Fig. 4. Region of flame oscillation and amplitude of flame oscillation as a function of Strouhal number.
the wall than that of f 4 0.95, owing to smaller Su. Both mean locations of yf are constant regardless of the Strouhal number, while both amplitude of yf decreases with increasing Strouhal number when S . 0.5. The similar qualitative behavior is shown by Im et al. [1], Stahl et al. [2], and Egolfopoulos [4]. Oscillation of the Flame Speed as a Function of the Strouhal Number Figure 5 shows the region of oscillatory Su and amplitude of Su (S* u ) that is derived from the waveforms shown in Fig. 3, as a function of the Strouhal number. Both mean values of Su are constant regardless of the Strouhal number. Two broken lines show the unstretched laminar flame speeds [23] that correspond to each f. At the Strouhal numbers more than around unity, both S*u increase with an increase in the Strouhal number. It is well known that the flame speed of the mixture with Le , 1 increases with increasing stretch. The flow velocity oscillation in the stagnating flow with the wall oscillation can be expressed by two effects [22]: the effect of the oscillatory wall location (which is equal to the effect of the stagnating point oscillation, shown as the dotted and dashed lines in Fig. 6 in Ref. [22]) and the effect of the oscillatory wall velocity (which is equal to the effect of the oscillation of the flow velocity at the stagnating point, shown as the long-dashed line in Fig. 6 in Ref. [22]). Then, when the stagnation wall is oscillated as in equation 1, the flow velocity at the stagnating streamline (u) can be written as
u 4 A(y)x cos(xt) ` B(y) sin(xt) ` C(y)
(2)
The first term is proportional to the oscillatory wall velocity (dyw/dt) and shows the effect of the oscillatory wall velocity because dyw d(y*w sin(xt) ` yw) 4 4 y*w x cos(xt) dt dt
(3)
In addition, the second and third terms show the effect of the oscillatory wall location. A general definition of flame stretch (K) is K 4
1 dAs As dt
(4)
where As is the area of an infinitesimal element of the surface. In terms of flow variables, it can be shown [17,24] that K 4 {¹t • vt ` (V • n)(¹ • n)}s
(5)
where ¹t and vt are the tangential components of ¹ and a velocity of the field (v) evaluated at the surface, V is a velocity of the surface, and n is the unit normal vector of the surface, pointed in the direction of the unburned gas. In the case of a flat flame in a stagnating flow, K can be deduced as K 4
]vr ]r
(6)
at the flame front, where vr is the radial velocity and r is the radial coordinate. From the equation of continuity,
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Fig. 5. Range of flame speed oscillation and amplitude of flame speed oscillation as a function of Strouhal number.
]u K41 ]y
(7)
at the flame front. Substituting equation 2 into equation 7, ]A(y) x cos(xt) ]y ]B(y) ]C(y) ` sin(xt) ` ]y ]y
5
K4 1
6
(8)
The first term indicates that the stretch rate varied proportional to x. In other words, the stretch varies proportional to the oscillatory wall velocity (see equation 3). The second term shows that the amplitude does not depend on x, and the third term shows the mean stretch rate. The second and third terms show the stretch rate that depends on the oscillatory wall location. In other words, those terms represent the variation of stretch rate in the quasi-steady state. The amplitude of K(K*) is
!1 ]y x2 ` 1 ]y 2 ]A(y) ]B(y) ]A(y) 4 ) x `1 ]y )! ]y @ ]y 2
K* 4
]A(y)
2
]B(y)
2
(9)
as the present study (y* ¯ w 4 0.094). This agrees w/y well with experimental results where no oscillation of the flame speed is observed at low x. When x k []B(y)/]y]/[]A(y)/]y], K* is nearly equal to |x ] A(y)/]y|, showing that K* increases linearly with an increase in x. Therefore, the amplitude of the flame speed oscillation is expected to increase with increasing x at high x because the flame speed varies proportionally to the stretch rate [17]. This agrees well with experimental results, as shown in Fig. 5. At high x, when the first term of equation 8 is dominant, the stretch attains a maximum at xt 4 1808. The flame speed should attain a maximum at xt 4 1808 because the flame speed of the mixture with Le , 1 increases with an increase in stretch rate. However, the flame speed attains a maximum approximately at xt 4 2408. The phase difference between the flame speed oscillation and the flame stretch oscillation, which is around 608, may indicate the phase delay of flame speed to the stretch. More investigation is needed to examine this phenomenon. As for the location of flame and the temperature of flame, the existence of the phase delay of those oscillations to the strain rate is shown by Stahl et al. [2], and Egolfopoulos [4].
2
2
When x K []B(y)/]y]/[]A(y)/]y], K* is nearly equal to |] B(y)/]y|, showing that K* is constant regardless of x. |]B(y)/]y| is small when the ratio of the amplitude of the wall to the mean distance between the wall and the nozzle exit (y*w /y¯w) is as small
Concluding Remarks The effect of unsteady stretch on the wall-stagnating, lean methane premixed flame is investigated experimentally. The influence of oscillatory stretch on the flame speed is examined, when the location
OSCILLATORY STRETCH ON THE FLAME SPEED
of stagnation wall (yw) is oscillated as yw 4 y* w sin(xt) ` y¯w in the direction of the main flow, where ¯ w 4 16 mm. The Strouhal numy* w 4 1.5 mm and y ber (S) defined as S 4 xy¯w/U0 represents the ratio of the characteristic time of the flow to the characteristic time of the oscillation, where U0(41.07 m/ s) is flow velocity at the nozzle exit. In the present study, the Strouhal number is varied from 0.047 to 4.8. The flame speed is obtained as the propagating velocity relative to unburned gas. The history of flow velocity measured by the LDV system is used in order to obtain the unburned gas velocity at the flame front and the oscillatory flame location. The velocity of the flame motion is calculated from differentiation of the oscillatory flame location. At the Strouhal numbers less than around unity, the oscillation of the flame speed is not observed. On the other hand, at the Strouhal numbers more than around unity, the oscillation of the flame speed is observed, and the amplitude of flame speed oscillation increases with an increase in the Strouhal number. It is well known that the flame speed of lean methane flame increases with increasing stretch rate. Since the axial flow velocity is described as u 4 A(y)x cos(xt) ` B(y) sin(xt) ` C(y), the flame stretch is shown as ]B(y) ]C(y) x cos(xt) ` sin(xt) ` 5]A(y) ]y ]y ]y 6
K4 1
When the wall is oscillated at low frequency and the flow is oscillated at low Strouhal number, the quasi-steady factor of oscillatory stretch represented by the second and third terms, which are independent of the frequency, is dominant. The stretch rate does not vary significantly because the quasi-steady factor is not so large when the amplitude of the wall oscillation is as small as the present study (y*w/y¯w 4 0.094). As a result, at the Strouhal numbers less than around unity, the oscillation of the flame speed is not observed. On the other hand, when the wall is oscillated at high frequency and the flow is oscillated at high Strouhal number, the unsteady factor of oscillatory stretch represented by the first term becomes dominant and the amplitude of stretch oscillation increases proportional to the Strouhal number. Therefore, at the Strouhal numbers more than around unity, the oscillation of the flame speed is observed, and the amplitude of the flame speed oscillation increases with an increase in the Strouhal number. Nomenclature f S Su
frequency of the stagnation wall oscillation Strouhal number (4xy¯w/U0) laminar flame speed (propagating velocity relative to unburned gas)
S*u U0 u uf yf y*f yw y*w y¯w
881
amplitude of flame speed flow velocity at the nozzle exit flow velocity at the stagnation streamline unburned gas velocity at the flame front in laboratory coordinates flame location amplitude of flame oscillation stagnation wall location amplitude of the stagnation wall oscillation mean wall location
Greek Symbols x
angular frequency of the stagnation wall oscillation REFERENCES
1. Im, H. G., Bechtold, J. K., and Law, C. K., Combust. Flame 105:358–372 (1996). 2. Stahl, G. and Warnatz, J., Combust. Flame 85:285–299 (1991). 3. Ghoniem, A. F., Soteriou, M. C., Knio, O. M., and Cetegen, B., in Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 223–230. 4. Egolfopoulos, F. N., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 1365–1373. 5. Sankar, S. V., Jagoda, J. I., and Zinn, B. T., Combust. Flame 80:371–384 (1990). 6. Pearlman, H. G. and Sohrab, S. H., Combust. Sci. Technol. 105:19–31 (1995). 7. Im, H. G., Law, C. K., Kim, J. S., and Williams, F. A., Combust. Flame 100:21–30 (1995). 8. Kistler, J. S., Sung, C. J., Kreutz, T. G., Law, C. K., and Nishioka, M., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 113–120. 9. Fleifil, M., Annaswamy, A. M., Ghoneim, Z. A., and Ghoneim, A. F., Combust. Flame 106:487–510 (1996). 10. Petrov, C. A. and Ghoniem, A. F., Combust. Flame 102:401–417 (1995). 11. Takagi, T., Yoshikawa, Y., Yoshida, K., Komiyama, M., and Kinoshita, S., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1103–1110. 12. Nguyen, Quang-Viet and Paul, Phillip H., in TwentySixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 357–364. 13. Roberts, William L. and Driscoll, James F., Combust. Flame 87:245–256 (1991). 14. Rolon, J. C., Aguerre, F., and Candel, S., Combust. Flame 100:422–429 (1995). 15. Barr, P. K. and Keller, J. O., Combust. Flame 99:43–52 (1994). 16. Keller, J. O., Barr, P. K., and Gemmen, R. S., Combust. Flame 99:29–42 (1994).
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17. Law, C. K., in Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 1381–1402. 18. Mueller, C. J., Driscoll, J. F., Reuss, D. L., and Drake, M. C., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, 1996, pp. 347–355. 19. Mizomoto, M., Asaka, Y., Ikai, S., and Law, C. K., in Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, pp. 1933– 1939. 20. Lepicovsky, J., AIAA J. 24:27–31 (1986).
21. Yu, G., Law, C. K., and Wu, C. K., Combust. Flame 63:339–347 (1986). 22. Hirasawa, T., Ueda, T., Matsuo, A., and Mizomoto, M., First Asia-Pacific Conference on Combustion, the Japanese Section of the Combustion Institute, 1997, pp. 326–329. 23. Vagelopoulos, C. M., Egolfopoulos, F. N., and Law, C. K., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 1341–1347. 24. Chung, S. H. and Law, C. K., Combust. Flame 55:123– 125 (1984).
EFFECT OF OSCILLATORY STRETCH ON THE FLAME SPEED OF WALLSTAGNATING PREMIXED FLAME TARO HIRASAWA, TOSHIHISA UEDA, AKIKO MATSUO and MASAHIKO MIZOMOTO Department of Mechanical Engineering Keio University 3-14-1 Hiyoshi, Kohoku-ku Yokohama, Kanagawa, 223-8522, Japan
The effect of unsteady stretch on the wall-stagnating lean methane, premixed flame was investigated experimentally. The influence of the oscillatory stretch on the flame speed was examined when the stagnation wall was oscillated sinusoidally in the direction of the main flow at a constant amplitude. Using LDV, the velocities of unburned gas and of flame in laboratory coordinate were measured to obtain flame speed (propagating velocity relative to unburned gas). The oscillatory characteristic of the flow field is indicated by the Strouhal number, representing the ratio of the angular frequency of the wall times the mean wall location to the flow velocity at the nozzle exit. There exist two factors dominating the oscillatory stretch, that is, the change of the distance between the wall and the nozzle, which is a quasi-steady factor, and the change of the velocity of the wall, which is an unsteady factor. When the flow is oscillated at low Strouhal number (#1), the quasi-steady factor, which is independent of the Strouhal number, is dominant. And the oscillation of the flame speed is not observed. In this case, quasi-steady factor is not large enough to oscillate the flame speed since the ratio of the amplitude of the wall oscillation to the mean distance between the wall and the nozzle is small. On the other hand, when the flow is oscillated at high Strouhal number (#1), the unsteady factor, which is proportional to the Strouhal number, becomes dominant. In this case, the oscillation of the flame speed is observed and the amplitude of the oscillation of flame speed increases with increasing Strouhal number.
Introduction It has recently been recognized that effects of unsteady flow field on flames may be considerable in many practical flames. As reported in several recent studies [1–4], the unsteadiness of the flow field can introduce an additional effect that may significantly modify the flame behavior. The unsteady effect of acoustic flow field on flame instabilities is a serious problem in many kinds of combustors, including rocket motors. Solid propellant rocket motors are susceptible to axial instabilities, which may interfere with engine performance [5]. The flow disturbance caused by interaction between the solid propellant surface and the flame leads to acoustic field oscillation. When the acoustic field is forced to oscillate, the flame oscillatory movement is observed and considered to be caused by oscillatory reaction rate within the flame and the oscillatory nature of the velocity of the combustible mixture, by Sankar et al. [5]. It is also suggested by Im et al. [1] that acoustic instability of flames may be influenced significantly by Lewis number (Le). Turbulent flames are affected by nonuniform unsteady stretch. It is difficult to examine both the effect of nonuniform stretch and the effect of unsteady stretch at the same time. However, it is not so difficult to examine them separately, while it naturally
occurs at the same time in turbulent flames. Recently, the effect of unsteady stretch, which is not clear in many points, is investigated extensively [1– 4,6–14]. As a result, the effect is found to depend on the Lewis number [1], which is similar to that of nonuniform stretch. Also, the flame is found to be weakened even though the mean of stretch rate does not vary and to be quenched below the steady quenching rate due to the effect of unsteady stretch on the flame [3,6]. The quenching caused by unsteady stretch is responsible for undesirable phenomena, for instance, significant ignition delay in a pulse combustor where a large vortical structure is formed [15,16]. It is well known that the flame speed of the mixture with Le ? 1 depends on flame stretch and that the flame speed of the mixture with Le , 1 increases with increasing stretch rate [17]. While studies on unsteady stretched flame abound, investigations on the effect of unsteady stretch on the flame speed are less numerous. However, there are some investigations [4,18] that provide us useful information when examining its effect on the flame speed. Mueller et al. [18] experimentally examined the response of flame strength to the unsteady stretch when the premixed flame was stretched by a counter-propagating toroidal vortex. The peak dilatation rate at each
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Fig. 1. Schematic drawing of wall-stagnation flow burner system. Diameter of the nozzle is 12 mm. Location of the stagnation wall is oscillated sinusoidally along stagnation streamline.
flame segment was used as an indicator of the local flame strength. They found that the flame requires a relatively long time to be weakened by positive stretch, yet it is rapidly strengthened by negative stretch, rising linearly to 240% of the unstretched value. As for flame temperature, Egolfopoulos [4] observed that the flame temperature is oscillated by stretch oscillation. The phase delay and the attenuation of the temperature oscillation is observed when the stretch oscillates at high frequency. Similarly, the stretch oscillation is expected to affect the flame speed significantly, which is examined in this study. In the present study, the effect of unsteady stretch on the wall-stagnating, lean methane premixed flame (which is isolated from the effect of curvature) is investigated experimentally. In order to examine the influence of oscillatory stretch on the flame speed, the flame speed is measured, while the stagnation wall is oscillated sinusoidally at a constant amplitude in the direction of the main flow. Experimental Setup Figure 1 shows the schematic drawing of a burner system. A laminar flat flame is established in the stagnating region when CH4/air mixture is issued from the nozzle with a 12-mm-diameter exit perpendicularly to the stagnation wall at 1.07 m/s. Coflowing air flows surrounding the main flow to minimize the interaction between the main flow and the ambient air. The coordinate y is defined as the downstream direction from the nozzle exit along the center axis of the nozzle. The location of the stagnation wall (yw) is oscillated sinusoidally along the coordinate y; that is, yw 4 y*w sin(xt) ` y¯w
(1)
where y* ¯ w 4 16 mm, x is angular w 4 1.5 mm, y frequency, and t is time. The stagnation wall is a
ceramic disk with a 76 mm diameter, and the thermal condition of the wall is considered to be adiabatic. The stagnation wall is oscillated by a voice coil motor and controlled precisely by the feedback control system with a wall location sensor (precision potentiometer), a function generator and a control device. A laser Doppler velocimetry (LDV) system is used to measure the flame speed of lean CH4/air mixtures at equivalence ratios (f) of 0.77 and 0.95. In this case, the effect of Lewis numbers less than unity is expected as observed by Mizomoto et al. [19], although the Lewis number at room temperature is evaluated at almost unity. To perform LDV measurements, scattering particles (fine silicone oil droplets) are seeded in the premixed mixture. To measure the oscillating flow velocity and the oscillating wall location simultaneously, the analog outputs of an LDV counter and the wall location sensor are recorded on computer memories through the simultaneous sampling AD converter. Results and Discussion Oscillatory Velocity Measurement To measure the oscillatory velocity, the data reduction procedure, that is, ensemble averaging of the randomly arriving LDV data, is performed. This technique, which permits determination of the periodic velocity history by synchronizing the beginning of the sampling interval of the LDV data with respect to the periodic wall oscillations, is similar to those utilized by Sankar et al. [5] and Lepicovsky [20]. Figures 2a and 2b show the history of the flow velocity oscillation at a point along the stagnation streamline (u) when the frequencies of the wall oscillation (f) are 0.5 and 8.0 Hz, plotted against the phase of the wall oscillation (xt). The oscillatory characteristic of the flow field is indicated by the Strouhal number (S), representing the ratio of the characteristic time of the flow to the characteristic time of the oscillation. The Strouhal number is defined as S 4 xy¯w/U0, where y¯w is the mean wall location and U0 is the flow velocity at the nozzle exit. When f are 0.5 and 8.0 Hz, S are 0.047 and 0.75, respectively. The history of the flow velocity oscillation at y 4 8.0 mm, shown by the triangle in Fig. 2a, is correctly sinusoidal waveform, and the plots are obtained at all phases. If the burned gas has reached the point of measurement for some phases, plots will not have been obtained for those phases due to evaporation of the scattering particles. Thus, this history shows that the flame does not reach the point of measurement (y 4 8.0 mm) at any phase. The flow velocity at y 4 9.0 mm decreases when xt increases from 908 to 1808 because the stagnation
OSCILLATORY STRETCH ON THE FLAME SPEED
877
Fig. 2. Oscillatory history of the flow velocity and of the unburned gas velocity at the flame front. (a) f 4 0.95, f 4 0.5[Hz] (S 4 0.047). (b) f 4 0.95, f 4 8.0[Hz] (S 4 0.75). Flame moves upstream (I) or downstream (II) and approaches the point of measurement.
wall approaches the point of measurement. After that, as the flame moves upstream and approaches the point of measurement (I), the velocity attains a minimum at xt 4 2108, and then increases due to thermal expansion in the preheat zone. Therefore, that the velocity attains a minimum indicates that the flame front reaches the point of measurement, which is similar to Yu et al. [21]. As a result, the unburned gas velocity at the flame front in laboratory coordinates (uf), shown by a square plot, is obtained. At the same time, xt when the flame reaches the point of measurement is obtained. By changing y, xt at different flame locations is obtained. As a result, yf is derived as a function of xt. In the range of xt 4 2308–3208, the velocity data disappear because of the evaporation of scattering particles. Then, as the flame moves downstream and approaches the point of measurement (II), the velocity data appear, and the velocity attains a minimum at xt 4 3308 shown by a square plot, indicating the arrival of the flame front at the point of measurement. The square plots that are labeled neither I nor II correspond to the measurements at y 4 8.5, 9.5, 10.5, and 11.0 mm, which are not plotted in Fig. 2 to avoid being crowded. Thus, two squares are obtained from one velocity history at a point of measurement. It is noteworthy to mention here that two squares of uf obtained from one velocity history at f 4 0.5 Hz are close to each other; however, they are different from each other at f 4 8.0 Hz. The reason why they are different at f 4 8.0 Hz is the following: When the flame moves upstream (I), uf is smaller than the flame speed. On the other hand, when the flame moves downstream
(II), uf is larger than the flame speed. If the flame speed is constant, it is obvious that the difference increases with an increase in frequency of the flame oscillation. Waveform of Oscillations Figure 3 shows the oscillatory histories of yw, yf, uf, the velocity of the flame motion (dyf /dt) in laboratory coordinate, and the laminar flame speed (Su) (propagating velocity relative to unburned gas), plotted against xt. The history of the flame location yf is well fit with the sine wave, and the frequency agrees correctly with the frequency of the wall oscillation, which are confirmed by high-resolution measurement of the flame location with the video image ([22], Fig. 4). These facts are confirmed in the present study by the LDV measurement as well, although the number of plots is fewer. Thus, yf can be described as yf 4 ¯ f. Each value of y*f , h, and y¯f, is y* f sin(xt ` h) ` y obtained by using the least-squares method to fit the measured yf plots (see Fig. 3). The velocity of the flame motion dyf /dt is, then, obtained as xy*f cos(xt ` h) by the differentiation of yf. uf, which is subject to the effect of flame motion, is obtained by the LDV data when the flame front arrives at a point of measurement. The laminar flame speed Su is, therefore, obtained as Su 4 uf 1 dyf /dt. As shown in Fig. 3a, the flame oscillates following the oscillation of the wall location, and the flame speed is constant at any xt. In other words, the flame oscillates, keeping its propagating speed constant and following the oscillation of the velocity profile
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PREMIXED TURBULENT COMBUSTION
Fig. 3. Oscillatory histories of stagnation wall location (yw), flame location (yf ), unburned gas velocity at the flame front in laboratory coordinate (uf ), velocity of the flame motion in laboratory coordinate (dyf /dt), and laminar flame speed (Su) (propagating velocity relative to unburned gas). (a) f 4 0.95, f 4 0.5[Hz] (S 4 0.047). (b) f 4 0.95, f 4 8.0[Hz] (S 4 0.75). (c) f 4 0.95, f 4 23[Hz] (S 4 2.2). (d) f 4 0.95, f 4 51[Hz] (S 4 4.8).
caused by the change of the wall location. In addition, the amplitude of dyf /dt is very small at f 4 0.5 Hz, which shows that the flame oscillates quasisteadily. As shown in Fig. 3b, at f 4 8.0 Hz, both the amplitude of dyf /dt oscillation and the amplitude of uf oscillation are larger amplitudes than those of Fig. 3a due to the effect of increasing of the Strouhal number; that is, the characteristic time of the oscillation becomes smaller compared with the characteristic time of the flow. Su is almost constant though small fluctuation is observed. At high frequencies ( f 4 23 and 51 Hz), as shown in Figs. 3c and 3d, Su
clearly oscillates. At both f 4 23 and 51 Hz, Su attains a maximum when the wall and flame are moving downward. This is discussed later with the discussion concerning the amplitude of the flame speed. Oscillation of Flame Location as a Function of the Strouhal Number Figure 4 shows the region of oscillatory yf and the amplitude of yf (y* f ), which is derived from the waveforms shown in Fig. 3, plotted against the Strouhal number. The flame of f 4 0.77 is formed closer to
OSCILLATORY STRETCH ON THE FLAME SPEED
879
Fig. 4. Region of flame oscillation and amplitude of flame oscillation as a function of Strouhal number.
the wall than that of f 4 0.95, owing to smaller Su. Both mean locations of yf are constant regardless of the Strouhal number, while both amplitude of yf decreases with increasing Strouhal number when S . 0.5. The similar qualitative behavior is shown by Im et al. [1], Stahl et al. [2], and Egolfopoulos [4]. Oscillation of the Flame Speed as a Function of the Strouhal Number Figure 5 shows the region of oscillatory Su and amplitude of Su (S* u ) that is derived from the waveforms shown in Fig. 3, as a function of the Strouhal number. Both mean values of Su are constant regardless of the Strouhal number. Two broken lines show the unstretched laminar flame speeds [23] that correspond to each f. At the Strouhal numbers more than around unity, both S*u increase with an increase in the Strouhal number. It is well known that the flame speed of the mixture with Le , 1 increases with increasing stretch. The flow velocity oscillation in the stagnating flow with the wall oscillation can be expressed by two effects [22]: the effect of the oscillatory wall location (which is equal to the effect of the stagnating point oscillation, shown as the dotted and dashed lines in Fig. 6 in Ref. [22]) and the effect of the oscillatory wall velocity (which is equal to the effect of the oscillation of the flow velocity at the stagnating point, shown as the long-dashed line in Fig. 6 in Ref. [22]). Then, when the stagnation wall is oscillated as in equation 1, the flow velocity at the stagnating streamline (u) can be written as
u 4 A(y)x cos(xt) ` B(y) sin(xt) ` C(y)
(2)
The first term is proportional to the oscillatory wall velocity (dyw/dt) and shows the effect of the oscillatory wall velocity because dyw d(y*w sin(xt) ` yw) 4 4 y*w x cos(xt) dt dt
(3)
In addition, the second and third terms show the effect of the oscillatory wall location. A general definition of flame stretch (K) is K 4
1 dAs As dt
(4)
where As is the area of an infinitesimal element of the surface. In terms of flow variables, it can be shown [17,24] that K 4 {¹t • vt ` (V • n)(¹ • n)}s
(5)
where ¹t and vt are the tangential components of ¹ and a velocity of the field (v) evaluated at the surface, V is a velocity of the surface, and n is the unit normal vector of the surface, pointed in the direction of the unburned gas. In the case of a flat flame in a stagnating flow, K can be deduced as K 4
]vr ]r
(6)
at the flame front, where vr is the radial velocity and r is the radial coordinate. From the equation of continuity,
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Fig. 5. Range of flame speed oscillation and amplitude of flame speed oscillation as a function of Strouhal number.
]u K41 ]y
(7)
at the flame front. Substituting equation 2 into equation 7, ]A(y) x cos(xt) ]y ]B(y) ]C(y) ` sin(xt) ` ]y ]y
5
K4 1
6
(8)
The first term indicates that the stretch rate varied proportional to x. In other words, the stretch varies proportional to the oscillatory wall velocity (see equation 3). The second term shows that the amplitude does not depend on x, and the third term shows the mean stretch rate. The second and third terms show the stretch rate that depends on the oscillatory wall location. In other words, those terms represent the variation of stretch rate in the quasi-steady state. The amplitude of K(K*) is
!1 ]y x2 ` 1 ]y 2 ]A(y) ]B(y) ]A(y) 4 ) x `1 ]y )! ]y @ ]y 2
K* 4
]A(y)
2
]B(y)
2
(9)
as the present study (y* ¯ w 4 0.094). This agrees w/y well with experimental results where no oscillation of the flame speed is observed at low x. When x k []B(y)/]y]/[]A(y)/]y], K* is nearly equal to |x ] A(y)/]y|, showing that K* increases linearly with an increase in x. Therefore, the amplitude of the flame speed oscillation is expected to increase with increasing x at high x because the flame speed varies proportionally to the stretch rate [17]. This agrees well with experimental results, as shown in Fig. 5. At high x, when the first term of equation 8 is dominant, the stretch attains a maximum at xt 4 1808. The flame speed should attain a maximum at xt 4 1808 because the flame speed of the mixture with Le , 1 increases with an increase in stretch rate. However, the flame speed attains a maximum approximately at xt 4 2408. The phase difference between the flame speed oscillation and the flame stretch oscillation, which is around 608, may indicate the phase delay of flame speed to the stretch. More investigation is needed to examine this phenomenon. As for the location of flame and the temperature of flame, the existence of the phase delay of those oscillations to the strain rate is shown by Stahl et al. [2], and Egolfopoulos [4].
2
2
When x K []B(y)/]y]/[]A(y)/]y], K* is nearly equal to |] B(y)/]y|, showing that K* is constant regardless of x. |]B(y)/]y| is small when the ratio of the amplitude of the wall to the mean distance between the wall and the nozzle exit (y*w /y¯w) is as small
Concluding Remarks The effect of unsteady stretch on the wall-stagnating, lean methane premixed flame is investigated experimentally. The influence of oscillatory stretch on the flame speed is examined, when the location
OSCILLATORY STRETCH ON THE FLAME SPEED
of stagnation wall (yw) is oscillated as yw 4 y* w sin(xt) ` y¯w in the direction of the main flow, where ¯ w 4 16 mm. The Strouhal numy* w 4 1.5 mm and y ber (S) defined as S 4 xy¯w/U0 represents the ratio of the characteristic time of the flow to the characteristic time of the oscillation, where U0(41.07 m/ s) is flow velocity at the nozzle exit. In the present study, the Strouhal number is varied from 0.047 to 4.8. The flame speed is obtained as the propagating velocity relative to unburned gas. The history of flow velocity measured by the LDV system is used in order to obtain the unburned gas velocity at the flame front and the oscillatory flame location. The velocity of the flame motion is calculated from differentiation of the oscillatory flame location. At the Strouhal numbers less than around unity, the oscillation of the flame speed is not observed. On the other hand, at the Strouhal numbers more than around unity, the oscillation of the flame speed is observed, and the amplitude of flame speed oscillation increases with an increase in the Strouhal number. It is well known that the flame speed of lean methane flame increases with increasing stretch rate. Since the axial flow velocity is described as u 4 A(y)x cos(xt) ` B(y) sin(xt) ` C(y), the flame stretch is shown as ]B(y) ]C(y) x cos(xt) ` sin(xt) ` 5]A(y) ]y ]y ]y 6
K4 1
When the wall is oscillated at low frequency and the flow is oscillated at low Strouhal number, the quasi-steady factor of oscillatory stretch represented by the second and third terms, which are independent of the frequency, is dominant. The stretch rate does not vary significantly because the quasi-steady factor is not so large when the amplitude of the wall oscillation is as small as the present study (y*w/y¯w 4 0.094). As a result, at the Strouhal numbers less than around unity, the oscillation of the flame speed is not observed. On the other hand, when the wall is oscillated at high frequency and the flow is oscillated at high Strouhal number, the unsteady factor of oscillatory stretch represented by the first term becomes dominant and the amplitude of stretch oscillation increases proportional to the Strouhal number. Therefore, at the Strouhal numbers more than around unity, the oscillation of the flame speed is observed, and the amplitude of the flame speed oscillation increases with an increase in the Strouhal number. Nomenclature f S Su
frequency of the stagnation wall oscillation Strouhal number (4xy¯w/U0) laminar flame speed (propagating velocity relative to unburned gas)
S*u U0 u uf yf y*f yw y*w y¯w
881
amplitude of flame speed flow velocity at the nozzle exit flow velocity at the stagnation streamline unburned gas velocity at the flame front in laboratory coordinates flame location amplitude of flame oscillation stagnation wall location amplitude of the stagnation wall oscillation mean wall location
Greek Symbols x
angular frequency of the stagnation wall oscillation REFERENCES
1. Im, H. G., Bechtold, J. K., and Law, C. K., Combust. Flame 105:358–372 (1996). 2. Stahl, G. and Warnatz, J., Combust. Flame 85:285–299 (1991). 3. Ghoniem, A. F., Soteriou, M. C., Knio, O. M., and Cetegen, B., in Twenty-Fourth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1992, pp. 223–230. 4. Egolfopoulos, F. N., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 1365–1373. 5. Sankar, S. V., Jagoda, J. I., and Zinn, B. T., Combust. Flame 80:371–384 (1990). 6. Pearlman, H. G. and Sohrab, S. H., Combust. Sci. Technol. 105:19–31 (1995). 7. Im, H. G., Law, C. K., Kim, J. S., and Williams, F. A., Combust. Flame 100:21–30 (1995). 8. Kistler, J. S., Sung, C. J., Kreutz, T. G., Law, C. K., and Nishioka, M., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 113–120. 9. Fleifil, M., Annaswamy, A. M., Ghoneim, Z. A., and Ghoneim, A. F., Combust. Flame 106:487–510 (1996). 10. Petrov, C. A. and Ghoniem, A. F., Combust. Flame 102:401–417 (1995). 11. Takagi, T., Yoshikawa, Y., Yoshida, K., Komiyama, M., and Kinoshita, S., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 1103–1110. 12. Nguyen, Quang-Viet and Paul, Phillip H., in TwentySixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1996, pp. 357–364. 13. Roberts, William L. and Driscoll, James F., Combust. Flame 87:245–256 (1991). 14. Rolon, J. C., Aguerre, F., and Candel, S., Combust. Flame 100:422–429 (1995). 15. Barr, P. K. and Keller, J. O., Combust. Flame 99:43–52 (1994). 16. Keller, J. O., Barr, P. K., and Gemmen, R. S., Combust. Flame 99:29–42 (1994).
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17. Law, C. K., in Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 1381–1402. 18. Mueller, C. J., Driscoll, J. F., Reuss, D. L., and Drake, M. C., in Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, 1996, pp. 347–355. 19. Mizomoto, M., Asaka, Y., Ikai, S., and Law, C. K., in Twentieth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1984, pp. 1933– 1939. 20. Lepicovsky, J., AIAA J. 24:27–31 (1986).
21. Yu, G., Law, C. K., and Wu, C. K., Combust. Flame 63:339–347 (1986). 22. Hirasawa, T., Ueda, T., Matsuo, A., and Mizomoto, M., First Asia-Pacific Conference on Combustion, the Japanese Section of the Combustion Institute, 1997, pp. 326–329. 23. Vagelopoulos, C. M., Egolfopoulos, F. N., and Law, C. K., in Twenty-Fifth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1994, pp. 1341–1347. 24. Chung, S. H. and Law, C. K., Combust. Flame 55:123– 125 (1984).