MEASUREMENTS OF INCOHERENT ACOUSTIC WAVE SCATTERING FROM TURBULENT PREMIXED FLAMES

Proceedings of the Combustion Institute, Volume 29, 2002/pp. 1809 1815 MEASUREMENTS OF INCOHERENT ACOUSTIC WAVE SCATTERING FROM TURBULENT PREMIXED FLAMES TIM LIEUWEN, RAJESH RAJARAM, YEDIDIA NEUMEIER and SURAJ NAIR School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332-0150, USA This paper presents measurements of acoustic wave scattering from turbulent, premixed flames. These measurements were obtained by directing highly coherent 5 15 khz waves at a rim-stabilized, premixed flame and measuring the scattered waves. Specific attention is focused on the incoherent wave characteristics. These measurements were motivated by theoretical work suggesting that the frequency content of the scattered waves are Doppler shifted by flame front motion, resulting in a spectrally broadened incoherent acoustic field. The measured acoustic spectra show a narrow, coherent spectral peak at the driving frequency, with lower amplitude, incoherent sidebands that decay exponentially with spectral distance from the driving frequency, f d. The spectral dependence of these sidebands is found to approximately satisfy the relationship C f f /f e d d, where C is a coefficient with a value of 10 3. This observed dependence of the spectral bandwidth upon incident wave frequency, f d, is in excellent agreement with theoretical predictions. Theoretical predictions also suggest that the value of C is inversely proportional to the root mean square (rms) value of the flame front velocity, V f. Using the measured value of C leads to an estimate of V f 46 cm/s, an apparently reasonable estimate considering that the measured rms velocity of the unburned mixture at the burner centerline is 28 cm/s. Some asymmetry in the upper and lower frequency sidebands is observed, suggesting that the flame front moves at a higher velocity when it is traveling toward the unburned gas than the burned gas. These data also show, in agreement with theoretical predictions, that the total power in the scattered incoherent field initially increases exponentially and then saturates with increases in frequency. Introduction This paper describes measurements of acoustic wave interactions with turbulent flames. Such interactions play an important role in a number of fundamental and applied problems, such as the characteristic unsteadiness of practical combustion processes. They also play a central role in the problem of combustion instabilities, where acoustic waves and flames interact in a self-exciting manner that is detrimental to hardware. The basic problem of acoustic waves impinging upon a turbulent flame is depicted in Fig. 1. Upon reaching the flame front, the waves are scattered and potentially amplified because of the significant change in sound speed and density at the flame front, the response of the flame front position and mass burning rate to the perturbations, and interactions with intrinsic flame and flow instabilities. The objective of this study is to obtain fundamental measurements of the spectral characteristics of these scattered acoustic waves that are excited by coherent, harmonically oscillating acoustic disturbances. Acoustic-wave flame interactions have been considered in a number of theoretical and experimental Fig. 1. Schematic of investigated configuration. Flame illustration adapted from Dinkelacker et al. [1]. 1809

1810 TURBULENT COMBUSTION Measurements in Premixed Turbulent Flames studies. Several experimental studies have visualized flame-acoustic interactions in unstable combustors using phase-locked schlieren, CH or OH planer laser-induced fluorescence, or chemiluminescence measurements (e.g., see Ref. [2] or [3]). Quantitative measurements of the flame s response to acoustic perturbations have also been reported; see, for example, Poinsot et al. [4] or Harper et al. [5]. These measurements are heavily influenced by the overall combustor system and are more properly characterized as combustion-system acoustic interaction measurements. No fundamental experimental studies of these interactions appear to have been performed, though, such as measurements of the dependence of the scattered acoustic field upon the length/timescales of flame front wrinkling. Several fundamental theoretical treatments of the acoustic-flame interaction problem have been reported. The first analysis was performed by Chu [6], who treated the flame as a temperature discontinuity. This work has been extended in several recent analyses, such as those of McIntosh and coworkers [7,8], Peters and Ludford [9], and Ledder and Kapila [10], which analyzed the unsteady flame structure. Analyses incorporating these acoustic burningrate coupling effects have also been reported by McIntosh [11], Lieuwen [12], Clavin et al. [13], and others. Another important process in these interactions, the periodic acceleration and/or convection of the flame front by the oscillatory flow field, has been discussed by Markstein [14], Searby and Rochwerger [15], and Fleifel et al. [16]. The primary emphasis of the theoretical work above discussed is to model the laminar flame acoustic-wave interaction problem. Several more recent studies, however, have used these developed analysis approaches to analyze the turbulent-flame acoustic-wave interaction problem [17,18]. These studies analyzed the scattering of acoustic waves from singly connected, wrinkled flame fronts by modeling the flame as a dynamically evolving, corrugated temperature discontinuity. The results suggest that several qualitative differences exist between the characteristics of waves scattered from laminar and turbulent flames and are discussed in more detail below. The result of most relevance to this study is the prediction that a coherent, harmonically oscillating acoustic wave, p drive, incident upon a turbulent flame generates both coherent, p s,c, and incoherent scat- tered waves, p s,i, where p s,i p drive 0, p s,c p drive 0 and angle brackets denote the expected value operator. The dynamic, wrinkled characteristics of turbulent flames decrease the energy in the coherent acoustic field by transferring it to the incoherent field. As such, the power in the scattered coherent and incoherent fields decreases and increases, respectively, as the scale of flame wrinkling, r, increases and/or the acoustic wavelength, k, decreases. This coherent energy damping source is particularly significant for disturbances whose wavelengths are smaller than the characteristic scales of flame wrinkling; that is, r/k 1. The random movement of the flame front serves as the mechanism for this coherent field incoherent field energy transfer. Specifically, the flame s movement causes the scattered field to oscillate over a range of Doppler-shifted frequencies from that of the incident wave. It has been shown [18] that the incoherent field frequency spectrum has sidebands about the frequency of the incident wave whose spectral characteristics are a function of the flame front movement spectrum; that is, p s,i ( f ) g(f( f f drive )), where p s,i ( f ), f( f ), and f drive denote the power spectra of the scattered incoherent acoustic field, the power spectra of the flame front position, and the frequency of the incident wave, respectively. In cases where the statistics of the flame front position are stationary (not necessarily a good assumption; see Ref. [19]), this predicted dependence takes the approximate form: p s,i( f ) 2pi( f f )s (kr) 2( f( f )(1 e 2pif s drive )df e e ds s (1) where k 2p/k 2pf/c is the wave number. It should be emphasized that equation 1 is a simplification of the general expression derived in Ref. [18], which is significantly more complicated because of its simultaneous spatiotemporal dependence upon flame front wrinkling characteristics. In general, it can be seen that the relationship between the acoustic and flame front spectrum is frequency dependent and quite complex. This relationship attains a simple form in two limiting cases. When r/k K 1, the acoustic spectrum is directly related to the incident frequency shifted flame front spectra by the relation 2 P ( f ) f f( f f ) (2) s,i drive If r/k k 1, the acoustic spectra is described by: 2 2 (( f f ) )/(2(kr) f 2 drive f( f )df ) p ( f ) e (3) s,i Such involved expressions are not required, however, to understand the basic physics of the predicted phenomenon. Assume a harmonically oscillating acoustic wave (generated by a stationary source) is incident upon a surface moving with a Mach wave of M. The reflected wave oscillates at the Dopplershifted frequency (measured by a stationary observer) frefl f drive(1 M n) (4) where n denotes the unit normal direction of the incident wave. This expression shows that the spectral bandwidth of the reflected waves is given by:

INCOHERENT WAVE SCATTERING FROM TURBULENT FLAMES 1811 2 1/2 2 1/2 ( f f drive) (Df ) 2 1/2 fdrive (M n) (5) Prior direct measurements of the flame front velocity (e.g, see Refs. [19 21]) have shown that its characteristics are not symmetric; that is, the flame velocity characteristics are different when it is moving toward the unburned mixture than when it is moving toward the burned gas (note that our discussion of the flame velocity is referring to the front s movement in an absolute frame and not the flame speed relative to that of the unburned gases). These results suggest, then, that the distribution of the incoherent acoustic spectral sidebands should not be symmetric about the incident wave frequency. To summarize, these results suggest that acoustic interactions with turbulent flames result in narrowband noise generation whose bandwidth is determined by that of the flame front movement and center frequency by that of the incident wave. The authors are not aware of prior experimental work noting the existence of these predicted acoustic spectral sidebands or their characteristics. The measurements reported in this paper were obtained in order to perform such a characterization and to assess the validity of these theoretical predictions. Experimental Setup, Instrumentation, and Data Analysis An illustration of the experiment is shown in Fig. 1. A natural-gas-fueled premixed flame is stabilized on a 35 mm diameter tube. A total of 30 equally spaced 3.2 mm diameter premixed pilot flame holes are located around the perimeter of the burner. Data were obtained at main burner equivalence ratios of 0.79 and 0.93 and at an average flow velocity of U 10.5 m/s. This corresponds to a Reynolds number of Re D 23,000 based on jet diameter. A total of 1.2% of the flow was diverted through the pilot, which was operated at an equivalence ratio of 5.4. These values correspond to total heat release rates of 26 and 30 kw, respectively, for the main burner and 1.5 kw for the pilot. Cold flow turbulence intensity measurements of u rms/u 2.7% were measured at the jet centerline with a Dantec hot wire anemometer. Using the methane flame speed values reported in Andrews and Bradley [22], these correspond to u rms/sl 1.1 and 0.76 for 0.79 and 0.93, respectively. As can be seen from equations 1 3, theoretical analyses predict that the order of magnitude of the parameter kr plays a critical role in determining the dependence of the acoustic spectrum upon the flame front position spectrum [18]. A rough estimate of the scale of flame front wrinkling, r, was made by determining the thickness of the flash brush from time-averaged photographs of the flame front, suggesting that r 1 cm. For the 5 15 khz driving frequencies used in this study, the resulting estimated value of kr range between unity and 3. Acoustic disturbances were generated with 38 mm diameter, type 616341 electrostatic Polaroid transducers. The transducer was located approximately 15 cm from the midpoint of the flame length and oriented at an angle of approximately 45 to the flame front. These transducers were biased with 150 volts DC and driven at 9 volts AC peak-peak. Data were obtained at 5 15 khz driving frequencies and amplitudes of approximately 0.12 Pa, or 76 db (Re 20 lpa), at 1 m. The ratio between the fundamental and first harmonic of the driving frequency was around 40 db, indicating that nonlinear interactions are negligible at these amplitudes. Using a plane wave approximation [23], the estimated acoustic velocity amplitude in the incident wave is 6 10 3 m/s. Thus, the ratio of acoustic to turbulent velocity fluctuations is u acoustic/u turb 0.2%, indicat- ing that acoustic velocity perturbation effects upon the flame front should be negligible relative to that of the background turbulence. The disturbances generated at these 5 15 khz frequencies were estimated to radiate with spreading angles of 8 25 [23]. Acoustic oscillations were measured with Bruel and Kjaer type 4191 0.5 in. microphones, which have a flat frequency response to 40 khz. The microphone was situated roughly 20 cm from the flame front and directed at an angle of 45 to the flame. The microphone output was amplified with a Nexus type 2690 conditioning amplifier and digitized and recorded with a 12-bit National Instruments A/D board. These measurements were obtained in an anechoic chamber in the aerospace combustion lab at Georgia Tech. Its walls are lined with 18 in. wedges that reflect less than 10% of the incident energy at frequencies above 150 Hz. The chamber is fitted with a baffled air intake and exhaust to minimize exterior noise sources. In order to distinguish background flow and flame noise from disturbances due to the excited waves, measurements were obtained of the ambient noise floor, flame on and transducer off, transducer on and flame off, and both flame and transducer on. The ambient noise floor contributed negligibly to the combustion and driven noise in all cases for the frequencies of interest. Power spectra of the measured data were obtained using a standard fast Fourier transform algorithm. Special care was taken in determining the number of data points taken and the sampling frequency in order to minimize bias errors and uncertainty in these spectral estimates, while still retaining good spectral resolution. Bias errors arise from spectral leakage [24], which was a particular concern in this study because of the need to measure potentially low-amplitude, distributed-frequency sidebands in

1812 TURBULENT COMBUSTION Measurements in Premixed Turbulent Flames Fig. 2. Typical measured power spectrum of acoustic data with flame and transducer on, f drive 10 khz. Fig. 3. Detail of measured power spectrum of acoustic data with the flame on and transducer off, flame off and transducer on, and flame and transducer both on, f drive 7.5 khz. the immediate vicinity (i.e., within 1.2 Hz) of largeamplitude, narrowband coherent oscillations. Because of leakage, spectral estimates of these narrowband oscillations may also contain slowly decaying non-physical sidebands, which can easily swamp out the incoherent sidebands of interest. Leakage-induced bias errors can be reduced by increasing the data record length or by multiplying the data record by window functions [24]. A competing requirement, however, is the need for ensemble averaging to minimize the variance of the spectral estimate. This procedure reduces the standard deviation by the square root of the number of ensembles. However, it also decreases the size of each data record, thereby reducing spectral resolution and increasing bias error. With these considerations in mind, the following data reduction procedure was employed. A total of N T 2 21 2,097,152 data points were taken in each test. The data from each test were divided into N e 64 non-overlapping ensembles, which resulted in a spectral resolution of N e f sample /N T 1.22 Hz and a standard deviation of the spectral estimate of approximately 1/ 64 12% at each point. Bias errors due to spectral leakage of the nearly discrete frequency, coherent peak were made negligible by carefully aligning the sampling and driving frequencies such that fdrive jne f sample/n T (6) where j is an integer. For example, spectral estimates of data taken with only the transducer on at f drive 5 khz indicated that the amplitude of the power spectral density (PSD) at 4998.8 Hz was 52 db lower than at 5000 Hz. In comparison, the relative amplitudes of the PSDs at these two frequencies differ by 22 db when the flame and transducer are both on, indicating that bias errors due to spectral leakage from the coherent peak are on the order of 0.1%. Because leakage from the coherent peak is negligible, the resultant dominant source of bias error arises through leakage from one part of the distributed-frequency, incoherent component to others. We estimate the resulting bias error to be less than 10%, a number determined by comparing spectral estimates using several data reduction procedures that have substantially different leakage characteristics (N e 16 and 64 with a rectangular window, and N e 64 with a Hanning window). Consequently, the total uncertainty in the spectral estimate at each frequency is estimated to 2 be (0.1) 1/N 16%. e Results We next present typical results. Fig. 2 presents a typical acoustic power spectrum measured when the flame and transducer are both on, with a transducer driving frequency f drive 10 khz. Fig. 3 and Fig. 4 show a detail of the spectrum when f drive 7.5 and 15 khz for three cases: (1) flame on and transducer off, (2) transducer off and flame on, and (3) both flame and transducer on. Several items should be noted from these results. First, note the extremely narrow bandwidth of the spectrum when only the transducer is on. It is dominant only at the driving frequency and negligible at all others. This result illustrates, as noted earlier, that spectral leakage from the driving frequency is a negligible source of error. Second, when the transducer is on, the power spectrum has substantially higher values than the background combustion noise in the frequency interval centered around the driving frequency; for example, between f 14.6 and 15.4 khz in Fig. 4.

INCOHERENT WAVE SCATTERING FROM TURBULENT FLAMES 1813 Fig. 4. Detail of measured power spectrum of acoustic data with the flame on and transducer off, flame off and transducer on, and flame and transducer both on, f drive 15 khz. Fig. 6. Characteristics of high- (solid line) and low-frequency (symbol) spectral sidebands at driving frequencies of f drive 5, 7.5, 10, 12.5, and 15 khz and 0.93. The contribution of ambient combustion and flow noise was subtracted out using spectral data taken when only the flame was on. Fig. 5. Characteristics of high- (solid line) and low-frequency (symbol) spectral sidebands at driving frequencies of f drive 5, 7.5, 10, 12.5, and 15 khz and 0.79. Contribution of ambient combustion and flow noise subtracted out using spectral data taken when only the flame was on. For example, at f 14.95 khz, it is 3 orders of magnitude higher than the background combustion noise. This result illustrates that the incoherent sidebands of interest can be clearly distinguished from background noise over a range of frequencies. In this case, the scattered acoustic waves can be distinguished from background noise over a bandwidth of about 800 Hz. In this same frequency interval, approximately 200 times more power is contained in the incoherent sidebands than in the background combustion noise. Thus, the dominant noise source in the vicinity of the driving frequency is due to incoherent waves excited by the incident, coherent disturbance and not turbulent combustion and flow noise. Figure 5 and Fig. 6 plot the dependence of these spectral sidebands upon f f drive for the 0.79 and 0.93 case. The contribution of the background combustion noise has been subtracted out using data taken at the same conditions with only the flame on. This plot allows for a convenient comparison of sideband characteristics at different driving frequencies. The figure shows, first, that the amount of scattered incoherent power increases substantially with driving frequency (note that the y-axis is a logarithmic scale). This issue is addressed further below. The figure also shows that these sidebands exhibit an almost perfect exponential decay, as evidenced by the near linear dependence of the logarithm of the PSD upon f f drive (calculations show correlation coefficient values of 0.995 between these quantities for the low-frequency sidebands in the f drive 7.5 15 khz cases). With the exception of the 15 khz driving, all the results exhibit this near exponential decay over up to 3 orders of magnitude, until reaching background combustion noise levels. As can be seen in Fig. 4, the sidebands in the 15 khz case exhibit a similar exponential decay up to f f drive 150 Hz. At higher f f drive values, the spectrum also decays exponentially, but with a different slope. It may be that similar behavior occurs in all cases,

1814 TURBULENT COMBUSTION Measurements in Premixed Turbulent Flames Fig. 7. Least squares best fit and variance of coefficient, C f f /f C, in expression e drive drive with spectral sidebands shown in Fig. 6 (x, high-frequency sideband, 0.93;, low-frequency sideband, 0.93; high-frequency sideband, 0.79; o low-frequency sideband, 0.79). Symbols are plotted at slightly offset frequencies so that error bars are visible. Fig. 8. Dependence of power, normalized by value at f drive 15 khz, in incoherent spectral sidebands upon driving frequency (o, 0.93, x, 0.79). but simply cannot be observed because of poor signal-to-noise ratios. The reasons for this functional dependence of the PSD upon frequency are unclear. The figure also shows that the high- and low-frequency spectral sidebands are nearly symmetric, although there are some differences in decay rate that are discussed further below. The primary exception is the peak at f f drive 40 Hz that only occurs in the high- frequency sideband. This peak apparently corresponds to a flame front or hydrodynamic instability that was observed in high-speed flame images. These images revealed the periodic formation of a flame front disturbance near the burner lip at approximately 40 Hz that convected downstream. No such spectral feature was detected in cold flow velocity measurements. The reason this peak is not also observed in the low-frequency sideband is unclear, however. The figure also shows that the spectral decay rate monotonically decreases with increasing driving frequency; that is, the bandwidth of the scattered field increases with driving frequency. The dependence of the decay rate of these sidebands upon frequency is quantified in Fig. 7, which plots the dependence of the coefficient, C, in the expression C f f drive /fdrive e upon driving frequency. Several trends are evident. First, note that the coefficient, C, has a nearly constant value, implying that the spectral decay rate decreases roughly linearly with frequency and that the spectral bandwidth has the form Df f drive /C. This observed dependence of spectral bandwidth upon the incident wave frequency is in excellent agreement with the discussion in the introduction (equation 5). In addition, equation 5 suggests that the root mean square (rms) flame front velocity and C are inversely related; that is, (M n) 2 1/2 1/C. Using a value of C 1000 (Fig. 7), assuming a sound speed of c 330 m/s, and using an angle of 45 for the relative angle between the incident wave and mean flame front position (given in the prior section) leads to an estimate of V f 46 cm/s for the rms flame front velocity. This result lies within a factor of 2 of the measured velocity fluctuations at the centerline of the Burner exit, u rms 28 cm/s. Second, note that the high-frequency sideband consistently decays faster than the low-frequency sideband by up to 20%, suggesting a comparable asymmetry in flame front velocity. This conclusion is consistent with direct flame front velocity measurements [20,21]. Doppler-shift arguments then suggest that the flame front velocity is larger when it propagates toward the unburned gas (i.e., away from the microphone and transducer) than toward the burned gas. Third, note that the sidebands in the 0.79 case decay faster than in the 0.93 case by up to about 10%. This result appears to be in disagreement with the predictions of equations 1 5, if it is assumed that the lower flame speed in the 0.79 case results in increased flame front wrinkling and thus a broader flame front position spectral bandwidth. Figure 8 plots the dependence of the scattered power in the incoherent sidebands upon the driving frequency. The incoherent scattered power was determined by calculating the area under the PSD in the relevant spectral region and neglecting the contribution from the coherent component at f f drive.

INCOHERENT WAVE SCATTERING FROM TURBULENT FLAMES 1815 The results show that the scattered power increases substantially at lower frequencies and then appears to saturate to a limiting value. This result is in good agreement with equations 1 and 3 which also suggest an initial exponential increase in scattered power that levels off with increasing frequency. It should be pointed out that part of the reason for this result may be the decreased wave spreading in the higher frequency cases. Quantifying this diffraction effect is difficult, but based on rough transducer beam angle calculations, it could potentially reduce the difference in scattered power between the 5 and 15 khz cases from the value of 50 plotted in Figure 8 by a factor of 2 3. Discussion and Concluding Remarks This paper describes the spectral characteristics of acoustic waves scattered from a turbulent premixed flame in the 5 15 khz range. Most significantly, these measurements confirm theoretical predictions that coherent acoustic waves impinging upon a turbulent flame generate narrowband incoherent oscillations about the incident wave frequency [18]. They also confirm theoretical predictions that the spectral power in the incoherent field initially increases with frequency and eventually saturates. The theory also correctly predicts the dependence of the spectral bandwidth upon driving frequency. Several observations from these data warrant closer investigation. The observed asymmetry (by up to 20%) in the spectral sidebands suggests that the flame front velocity characteristics are not symmetric. Furthermore, analysis of the data predicts an rms flame front velocity that is within a factor of 2 of the measured rms flow velocity. More direct measurements of the dynamic flame front characteristics are needed to directly compare these predictions with measurements. Next, the change in spectral decay rate with equivalence ratio does not appear to be consistent with the theory. Also, the reasons for the substantial asymmetry observed at f f drive 40 Hz as well as the exponential dependence of the power spectra upon frequency are unclear. In addition, results are needed for a broader range of turbulence intensities, equivalence ratios, and frequencies to better understand the dependence of the scattered acoustic field characteristics upon these parameters. Acknowledgments This research was supported by the National Science Foundation under contract CTS-0092535 (Dr. Farley Fisher, technical monitor). REFERENCES 1. Dinkelacker, F., Buschmann, A., Schafer, M., and Wolfrum, J., Spatially Resolved Joint Measurements of OH- and Temperature Fields in a Large Premixed Turbulent Flame, Proceedings of the Joint Meeting of the British and German Sections of the Combustion Institute, Queens College, Cambridge, 1993, p. 295. 2. Samaniego, J. M., Yip, B., Poinsot, T., and Candel, S., Combust. Flame 94:363 380 (1993). 3. Broda, J. C., Seo, S., Santoro, R. J., Shirhattikar, G., and Yang, V., Proc. Combust. Inst. 27:1998. 4. Poinsot, T., Le Chatelier, C., Candel, S. M., and Esposito, E., J. Sound Vibr. 107(2):265 278 (1986). 5. Harper, J., Johnson, C. J., Neumeier, Y., Lieuwen, T., and Zinn, B. T., AIAA paper 2001-0486. 6. Chu, B. T., Proc. Combust. Inst. 4:1952. 7. McIntosh, A. C., Combust. Sci. Technol. 75:287 309 (1991). 8. McIntosh, A. C., and Wilce, S. A., Combust. Sci. Technol. 79:141 155 (1991). 9. Peters, N., and Ludford, G. S. S., Combust. Sci. Technol. 76:21 44 (1991). 10. Ledder, G., and Kapila, A. K., Combust. Sci. Technol. 34:331 344 (1983). 11. McIntosh, A. C., and Wilce, S. A., Combust. Sci. Technol. 54:217 236 (1987). 12. Lieuwen, T., J. Fluid Mech. 435:289 303 (2001). 13. Clavin, P., Pelce, P., and He., L., J. Fluid Mech. 216:299 322 (1990). 14. Markstein, G. H., in Nonsteady Flame Propagation (G. H. Markstein, ed.), Pergamon Press, New York, 1964. 15. Searby, G., and Rochwerger, D., J. Fluid Mech. 231:529 543 (1991). 16. Fleifel, M., Annaswamy, A. M., Ghoniem, Z. A., and Ghoniem, A. F., Combust. Flame 106:487 (1996). 17. Lieuwen, T., Coherent Acoustic Wave Scattering by Turbulent Premixed Flames, paper 171, Proceedings of the Second Joint Meeting of the U.S. Sections of the Combustion Institute, Oakland, CA, 2001. 18. Lieuwen, T., Combust. Flame 126(1 2):1489 1505 (2001). 19. Boyer, L., Clavin, P., and Sabathier, F., Proc. Combust. Inst. 18:1980. 20. Suzuki, T., and Hirano, T., Proc. Combust. Inst. 20:(1984). 21. Suzuki, T., Kudo, N., Kawamata, M., and Hirano, T., Proc. Combust. Inst. 21:(1986). 22. Andrews, G., and Bradley, D., Combust. Flame 19:275 288 (1972). 23. Peirce, A., Acoustics: An Introduction to Its Physical Principles and Applications, Acoustic Society of America, New York, 1991. 24. Bendat, J., and Piersol, A., Random Data: Analysis and Measurement Procedures Wiley, New York, 1986.