Effect of Developing Turbulence and Markstein Number on the ...

M. Z. Haq Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh e-mail: [email protected]

Effect of Developing Turbulence and Markstein Number on the Propagation of Flames in Methane-Air Premixture In spark ignition engines, initial flame kernel is wrinkled by a progressively increasing bandwidth of turbulence length scales until eventually the size of the flame kernel is sufficient for it to experience the entire turbulence spectrum. In the present study, an effective rms turbulence velocity as a function of time, estimated by integrating the nondimensional power spectrum density (psd) function for isotropic turbulence, is utilized to analyze the statistical distribution of flame front curvatures and turbulent burning velocities of flames propagating in methane-air premixtures. The distributions of flame front curvatures show these to become more dispersed as the effective turbulence velocity increases, and result in increased burning of premixtures. A decrease in the Markstein number also results in a further increase in curvature dispersion and enhanced burning, in line with the flame stability analysis. 关DOI: 10.1115/1.2056537兴

Introduction

length scales relative to turbulence. In this situation, the flame is confined to a relatively thin layer and the dominant effect of the turbulence is to wrinkle the flame front and to increase the flame surface area and to strain the local flame 关9兴. Consequently, the effect of strain has been used to correlate the turbulent burning velocity 关2兴. However, flame front curvature strongly influences the distribution of fast diffusing radicals such as H, the concentration of which increases in regions of negative curvature because of flame focusing 关10兴. This results in a further increase in local displacement speed of the flame and plays a significant role in turbulent flame propagation. Therefore, recent analyses of turbulent flame in the flamelet regimes have involved the various spatial statistical properties of the flame surface 关11兴. Hence, the flame surface area and curvature statistics provide a complete geometrical description of the turbulent flame propagation in laminar flamelet regime 关12兴. In this regime, a turbulent flame then can be treated as an ensemble of laminar flamelets whose burning velocities are determined by the local flame stretch. Integration of the local burning rate over the flame surface gives the total burning rates of the turbulent flame 关13兴. Using two-dimensional Laser sheet imaging techniques, it is possible to visualize the turbulent flame front, derive the statistics of curvatures of flame fronts, and profile the distribution of burned and unburned gases at all stages of the flame propagation. The statistics thus obtained are required to model three-dimensional flame propagation and its interaction with the turbulent flow field. The present paper reports the effects of developing turbulence on the propagation of flames in the methane-air premixture at 300 K using images obtained with planar Mie scattering 共PMS兲 technique. The sequences of two-dimensional images are analyzed to derive the statistics of flame front curvatures and turbulent burning velocities for rms turbulence intensities of 0.6 and 1.2 m / s, at two initial pressures of 0.1 and 0.5 MPa. Experimental conditions of the present study, corresponding mixture properties and turbulence parameters are summarized in Table 1.

In a turbulent premixture inside the engine cylinder, spark initiates reaction and a propagating flame front capable of overcoming the high geometric stretch. As the flame kernel grows and the influence of the geometric stretch is superseded by stretch due to aerodynamic strain, the flame is, at first, wrinkled by only the smallest scales of turbulence 关1,2兴. When the flame ball is small relative to the average eddy size, the high frequency smaller eddies are most effective in shearing and wrinkling the flame front, and the eddies larger than the flame kernel mostly propel the flame around 关3兴. The size of the eddy interacting with the initial flame kernel plays an important role in the early flame development. Although most of the mass in the combustion chamber burns in the later stage of the combustion, a large portion of the total combustion time is occupied by the early flame growth 关4兴. It is now widely accepted that the early flame development is an important factor affecting the cycle-to-cycle variations in the engine 关5,6兴. As the flame kernel continues to grow it becomes progressively wrinkled by the larger length scales with an associated increase in the turbulent burning velocity as a result of the contribution of an increasingly larger portion of the turbulence spectrum 关1兴. During this period, turbulent velocity effective in wrinkling the flame is u⬘k 共u⬘k 艋 u⬘兲, where u⬘ is the rms turbulent velocity. After sufficient time, eventually the size of the kernel is sufficient for it to experience the entire turbulence spectrum. The effect of turbulence 共embodied in u⬘k 兲 on flame wrinkling is then fully developed and is equal to u⬘ 关2兴. However, a reverse effect, “de-developing turbulence,” might occur as the flame approaches the wall such that progressively smaller scales of turbulence are available to affect the flame 关7兴. It is now widely recognized that turbulent flames in sparkignited engines can be treated as an array of laminar flamelets with no turbulence structure residing within them, and the flame front is considered as an interface separating cold reactants and the burned combustion products 关8兴. Flamelet combustion corresponds to chemical reaction occurring at fast time scales and short

Experimental and Analytical Techniques

Contributed by the Internal Combustion Engine Division of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received January 24, 2004; final manuscript received February 18, 2005. Assoc. Editor: D. Assanis.

Methane-air premixtures were ignited in a fan-stirred combustion vessel shown in Fig. 1. It is a 380 mm diameter spherical stainless steel vessel capable of withstanding the temperatures and pressures generated from explosions at initial pressures of up to 1.5 MPa and initial temperatures of up to 600 K. It has extensive

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Table 1 Initial conditions and mixture properties of methane-air premixtures at 300 K Initial conditions

Mixture properties, from Ref. 关14兴

Turbulence parameters

P 共MPa兲

␾

u⬘ 共m/s兲

共mm兲

␭ 共mm兲

␳u 共Kg/ m3兲

␳b 共Kg/ m3兲

ul 共m/s兲

Mab

0.1 0.1 0.1 0.5 0.5

0.8 1.0 1.0 1.0 1.0

0.6 0.6 1.2 0.6 1.2

0.142 0.142 0.085 0.043 0.025

2.93 2.93 2.07 1.31 0.93

1.12 1.11 1.11 5.55 5.55

0.167 0.148 0.148 0.732 0.732

0.265 0.36 0.36 0.195 0.195

11.8 21.7 21.7 −19.3 −19.3

␩

optical access through three pairs of orthogonal windows of 150 mm diameter and 100 mm thickness, made with Schlieren quality glass. Turbulence was generated using four identical, eight-bladed, independently controlled fans, and these fans were symmetrically disposed in a regular tetrahedron configuration. The turbulence velocity field in the optically accessed central region was calibrated for different fan speeds using laser Doppler velocimetry 共LDV兲 and found to be nearly isotropic with no sig-

nificant mean flow 共further details can be found in Ref. 关15兴兲. The rms turbulent velocity, u⬘, was found to vary linearly with fan speed and the turbulence integral length scale of 20 mm was estimated from the spatial correlation. The arrangement used for flame visualization using the planar Mie scattering 共PMS兲 technique used in the present study is shown in Fig. 2. A copper vapor laser, model CU15-A, made by Oxford Lasers, operating simultaneously at wavelengths of 510.6 nm 共green兲 and 578.2 nm 共dark yellow兲, was used as a light source. The pulse rate was 4.5 kHz, the pulse duration was 15 ns, and the pulse energy was approximately 6 mJ. The light sheet was obtained using a 500 mm spherical bi-convex lens which focused the laser beam, followed by a 1000 mm plano-convex lens to expand it in one plane to form a sheet with an estimated thickness of 0.5 mm. The laser sheet was passed along a vertical plane just in front of the spark plug located in the center of the vessel and was Mie scattered from tobacco smoke particles. The particles evaporated, sublimed, or burned as they crossed the flame front and consequently did not scatter lights, and so provided a planar section of the complex three-dimensional flame front geometry. The scattered lights were imaged perpendicular to the incident light sheet and the images were recorded and stored using a Kodak Ektapro HS Motion Analyzer 共Model 4540兲 at a rate of 4500 images per second. The light sensitivity of the camera was ISO 3000 at the high gain setting. It had a 256⫻ 256 pixel sensor, with an 8 bit level response. A 510.6 nm interference filter was attached to the camera lens to prevent flame-generated light from obscuring the images. The electronic triggering features of the motion analyzer made it possible to store images prior to, after, and at both sides of the

Fig. 1 The fan-stirred combustion vessel

Fig. 2 Experimental setup for PMS visualization of turbulent flames

456 / Vol. 128, APRIL 2006

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Fig. 3 Flame images for the propagation in methane-air premixture with an isotropic rms turbulent velocity of 0.6 m / s. Time interval between the images is 0.67 ms and real size of the full frame image is 128 mm.

trigger pulse. In the present study, “center recording mode” was used in which images were recorded until the trigger signal was received and then 1536 additional frames were recorded after the trigger. Prior to triggering the spark, the copper-vapor laser was synchronized to the camera-framing rate at 4.5 kHz. Upon receiving the spark signal through a control switch, the motion analyzer stored digital images in dynamic random access memory 共DRAM兲 and these were available for immediate playback or for subsequent image processing. After a sequence of image processing steps, a clear cross-sectional view of the flame, identified as the frontier between the bright, unburned, and the dark, burned regions in the illuminated zone, were obtained. A front tracking algorithm was used to obtain the front coordinates from the images with a spatial resolution of 0.32 mm/ pixel. Flame front coordinates were analyzed to measure turbulent burning parameters. All other aspects of equipment and the experimental techniques are reported in Ref. 关15兴. Shown in Fig. 3 are the images obtained from a flame propagating in stoichiometric methane-air mixture at an initial pressure of 0.1 MPa and initial temperature of 300 K. The flame front wrinkling caused by turbulence and the persistence of history are clearly visible. Shown in Fig. 4 are flame images obtained for different initial conditions where the effects of effective turbulence and Markstein number are evident. Hence, a propagating flame front is subjected to effects due to aerodynamic strain and curvature which together constitute flame stretch and change the flame surface area. For a laminar flame propagation, a burned gas Markstein length, Lb, is defined to account for the sensitivity of the laminar flame speed, Sn, to stretch, ␣, such that 关14兴 S n = S s − L b␣

共1兲

where Ss is the unstretched laminar flame speed and is obtained as the intercept value of Sn at ␣ = 0 in the plot of Sn against ␣ 关14兴. Journal of Engineering for Gas Turbines and Power

Markstein length is normalized by laminar flame thickness, ␦l, to produce Markstein number, Mab. The influence of kinetics, thermodynamics, thermodiffusion, and activation energy on the flame propagation are embodied in the Markstein number 关15兴 and the

Fig. 4 Flame images at different initial conditions

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Markstein number is now being widely used to analyze turbulent flame propagation 关16兴 and the susceptibility to flame instability 关17兴. A high value of Markstein number is reported to stabilize and counter the perturbation in the flame fronts and lower values are also related to the early onset of instability at small flame radius in flame propagation 关17兴. Estimation of Flame Front Curvature. Analytically the flame front curvature, H, can be calculated using flame front coordinates 共x , y兲, from 关12,18兴 H=

d2y/dx2 关1 + 共dy/dx兲2兴3/2

H=

冋冉 冊 冉 冊 册 d 2x ds2

d2y ds2

+

共2兲

共3兲

Estimation of Turbulent Burning Velocity. For a spherical flame propagation, a mass rate of burning can be expressed in the form dm = − 4␲Ri2␳uut dt

␴共r兲r2dr

共6兲

Ri

Similarly, the mass of burned gas within any general radius Ri , mbi, is

mbi共Ri兲 = 4␲␳b

冕

Ri

关1 − ␴共r兲兴r2dr

共7兲

The mass of burned gas outside the general radius Ri , mbo, is

mbo共Ri兲 = 4␲␳b

冕

Rt

关1 − ␴共r兲兴r2dr

共8兲

Ri

Mass conservation implies that dmui dmuo dmbi dmbo dmu + + + + =0 dt dt dt dt dt

共9兲

In the present study, the value of Ri is defined such that the volume of unburned gas inside the circumference of radius, Rv, is equal to that is burned gas outside that circumference. Hence, the value of Rv is estimated by equating mui共Ri兲 / ␳u from Eq. 共5兲 and mbo共Ri兲 / ␳b from Eq. 共8兲, computing its value by Bisection method 关15兴. A turbulent burning velocity, ut, is defined at Rv such that it measures the mass rate of production of burned gas, which is related to the pressure development in engine cylinders, and is given by

共4兲

where Ri is a mean general radius associated with the turbulent burning velocity, ut, ␳u is the unburned gas density, and m is the mass of the gas changing from unburned to burned. However, problems arise in defining burned and unburned, and also in selecting the most appropriate value of Ri. In the present study, a thin planar sheet cutting the mean spherical flame kernel is considered. This is shown in Fig. 5 and the general radius, Ri, lies between a root radius, Rr, and a tip radius, Rt. The circumference of radius Rr embraces the burned gas entirely, while the circumference of radius Rt has nothing but unburned gas outside it. It is also necessary to define the masses of unburned and burned gases associated with the different zones. Let mui be the mass of unburned gas within the general perimeter of radius Ri, muo be the mass of unburned gas outside perimeter Ri but within that of Rt, mbi be the mass of burned gas inside the perimeter of Ri, mbo be the mass of burned gas outside the perimeter of Ri but within Rt, and mu be the mass remaining outside Rt. The definitions of the masses are also shown in Fig. 5. In the flame images, the proportions of circumference occupied by unburned and burned gases were measured around the circumference for all the full range of radii at different instants during the flame propagation. From these flame images, values of the average volume fractions occupied by unburned gas at the different spherical radii r, ␴共r兲, were estimated and the assumption of isotropy enables these values to be expressed solely as a function of r. Hence, the mass of unburned gas within any general radius Ri, mui, is given by

冕

Rt

2 1/2

Here the flame curvature is defined being positive if the flame element is convex to the reactant gases. The flame coordinates obtained from digital images required some smoothing of the flame front edge. In the present study, the Savitzky-Golay algorithm 关19兴 is used for data smoothing and numerical differentiation.

mui共Ri兲 = 4␲␳u

冕

o

However, the use of an independent variable, s, has been reported to provide improvement over this procedure 关12兴, because the flame curvature then may be analyzed regardless of the slope and complexities of the flame surface. Here s is the distance along the flame front from a fixed origin located on the flame front. The curvature, H, can then be obtained from 关12,18兴 2

muo共Ri兲 = 4␲␳u

d 共mbi + mbo兲 = 4␲Rv2␳uut dt

共10兲

Estimation of Effective rms Turbulent Velocity. At any instant, the effective rms turbulence velocity, u⬘k , was estimated by the integral of the dimensional power spectral density function,

Ri

␴共r兲r2dr

共5兲

o

The mass of unburned gas outside this general radius Ri, muo, is 458 / Vol. 128, APRIL 2006

Fig. 5 Definition of reference radii and associated masses for a flame image

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from the highest frequency to a threshold frequency given by the reciprocal of the elapsed time from ignition 关2兴. Scott 关20兴 improved the nondimensional power spectrum by replacing dimensional frequency with a dimensionless wave number, k␩, defined as k␩ = k␩

共11兲

where k is the wave number and ␩ is the Kolmogorov length scale. The wave number provides the link to the time scale. At the elapsed time, tk, measured from ignition, the corresponding value of k is given by k=

2␲ u ⬘t k

共12兲

and Kolmogorov length scale, ␩, is defined as

␩ = 共␯3/␧兲1/4

共13兲

where ␧ is the specific rate of dissipation of turbulent kinetic energy and ␯ is the kinematic viscosity. Kolmogorov length scale is related to the Taylor macro-scale, ␭, as

␩ ␭

Fig. 6 Generalized psd function showing the spectrum of turbulent energy

共14兲

= 15−1/4 Re␭−1/2

where Re␭ is the Reynolds number based on Taylor scale and is related to integral length scale, L, as ␭ A = L Re␭

共15兲

where A is a numeric constant. Scott 关20兴 suggested the value of A = 16± 1.5, and it gives a good fit to the measurements, over a wide range of the data 关21兴. ¯ 兲, meaThe nondimensional turbulence power spectrum, ¯S共k ␩ sured over a wide range of physical situations, reveals that the spectra at the higher wave numbers collapse to a universal form of k−5/3 and, as the Reynolds number decreases, the spectra show shorter ranges of the universal behavior 关21兴. Scott 关20兴 produced ¯ 兲 as a function of k and Re : the following correlation for ¯S共k ␩ ␩ ␭ ¯S共k ¯ 兲 ␩ =

0.01668 Re␭2.5 + 3.74 Re␭0.9 − 70 Re␭−0.1 1 + 共0.127 Re␭1.5¯k␩兲5/3 + 共1.15 Re␭0.622¯k␩兲4 + 共1.27 Re␭0.357¯k␩兲7 共16兲

Fig. 7

Development of effective rms turbulent velocity

The good agreement between Eq. 共16兲 and the data of other research is seen in Fig. 6, where the symbols are the measured values reported in Ref. 关21兴 and the lines correspond to the values obtained from the correlation. At an elapsed time of tk, the rms turbulent velocity effective in wrinkling the flame front, u⬘k , is given by

冉 冕

u⬘k 150.5 = u⬘ Re␭

⬁

¯k ␩

¯S共k ¯ 兲dk ¯ ␩ ␩

冊

1/2

共17兲

Values of u⬘k / u⬘ were obtained at different values of k␩ by numerical integration of Eq. 共17兲 using Eq. 共16兲. Shown in Fig. 7 are such variations of u⬘k / u⬘ plotted against ¯k−1 ␩ for a range of values of Re␭. Hence, the values of u⬘k / u⬘ for different tk are computed and are plotted in Fig. 8 for the experimental conditions of the present study. It is evident in Fig. 8 that the flames are not fully developed turbulence even after 15 ms elapsed time from ignition and it emphasizes the role of developing turbulence effects in the modeling of turbulent flame propagation in engines.

Experimental Results and Discussions Shown in Fig. 9 are the probability density functions 共pdfs兲 of curvatures of a stoichiometric methane-air flame fronts at three Journal of Engineering for Gas Turbines and Power

Fig. 8 Temporal development of the effective rms turbulent velocity

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Fig. 9 Curvature pdfs for a developing methane-air flame front at three different elapsed times

different elapsed times. The pdfs are symmetric with respect to a very small negative mean value. The widths 共variances兲 are increased with time, commensurate with increased flame wrinkling. From such flame curvature pdfs, reported in Fig. 9, it is possible to compute the positive, negative, and overall mean flame curvatures H+, H−, and Ho, respectively. Shown in Fig. 10共a兲 are the values H+, Ho, and H− plotted against elapsed time for three different explosions at u⬘ = 0.6 m / s. The positive and negative values of curvature clearly increase with time, while the mean remains close to zero. Shown in Fig. 10共b兲 are the values of variance of curvature, obtained from these flame propagation. The influence of the effective rms turbulent velocity, u⬘k , and Markstein number, Mab, on turbulent flame propagation are clearly revealed in Fig. 11, which shows an increasing variance of H as u⬘k is increased. At a given value of u⬘k and similar Markstein numbers, the magnitude of variance of H are similar, however lower values of Markstein numbers are associated with the higher variance of the curvature pdfs. This observation is in line with the

Fig. 10 „a… Positive, negative, and mean flame curvatures. „b… Variance of pdf of curvatures, plotted as a function of elapsed time from ignition.

460 / Vol. 128, APRIL 2006

Fig. 11 Variance of curvature pdfs of methane-air flame fronts at different initial conditions of equivalence ratio, pressure, and rms turbulent velocity

flame stability analysis which reports that flames with lower value of Markstein numbers to show greater propensity to cellular flame to occur at smaller flame radii 关15,17兴. Shown in Fig. 12 are the values of normalized turbulent burning velocities defined at Rv , ut / ul plotted as a function of normalized effective rms turbulent velocity, u⬘k / ul for different initial conditions. Hence, laminar burning velocity, ul, is an intrinsic property of combustible fuel-air premixture and is defined as “the velocity, relative to and normal to the flame front, with which the unburned gas moves into the front and is transformed into products under laminar flow conditions” 关22兴. All these dimensionless data, shown in Fig. 12, suggest that, for same values of u⬘k / ul, values of ut / ul are increased with a decrease in Markstein number, and for the same values of Markstein numbers the values of ut共Rv兲 / ul are found to increase with an increase in the values of u⬘k / ul. These all signify the importance of both the effective rms velocity and Markstein number in the turbulent burning of methane-air premixtures. Measured values of turbulent burning velocities and turbulent flame speeds can be related analytically when both of these values are defined at a reference radius, Rv. Hence, mass of the burned gas at any time can be written as 共4␲ / 3兲R3v␳b. Hence the left-hand side of Eq. 共10兲 can be written as

Fig. 12 Variation of ut„Rv… / ul with uk⬘ / ul for methane-air premixtures at different initial conditions

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Nomenclature H k␩ Lb Mab mbi

⫽ ⫽ ⫽ ⫽ ⫽

mbo ⫽ mu ⫽ mui ⫽ muo ⫽ P Ri Rr Rt Fig. 13 Relationship between turbulent flame speeds and turbulent burning velocities defined at Rv

冉

冊

dRv d d 4␲ 3 共mbi + mbo兲 = R ␳b = 4␲Rv2␳b dt dt dt 3 v

共18兲

Equating Eqs. 共10兲 and 共18兲, turbulent flame speed, dRv / dt, can be related to turbulent burning velocity, ut共Rv兲, by dRv ␳u = ut dt ␳b

共19兲

Equation 共19兲 also has been verified using experimental data and an example is shown in Fig. 13, where the values of dRv / dt are plotted as a function of ut共Rv兲␳u / ␳b, using data obtained from four different experiments of stoichiometric methane-air premixtures at u⬘ = 0.6 m / s. The experimental results satisfy the relationship quite closely; the observed scatter probably arises from the numerical methods employed to estimate these values.

Conclusions The paper reports the effects of developing turbulence and Markstein number on the propagating flame kernel in isotropic methane-air premixture. Two-dimensional PMS images are analyzed to derive the statistics of flame front curvatures and turbulent burning velocities. The major conclusions of the study are • • • •

•

Flame front curvature distributions become more dispersed with increase in elapsed time from ignition. Consideration of developing turbulence is important in determining the pdf of curvatures for flame propagation in turbulent premixture. For same effective rms turbulent velocity, flame front wrinklings are reasonably similar as long as the corresponding Markstein numbers are similar. A decrease in Markstein number increases the flame front curvature dispersion, in line with the flame front stability studies. A flame with a negative Markstein number is more wrinkled than a flame with a positive Markstein number. Turbulent burning velocities increase with increase in effective rms turbulent velocity and with decrease in Markstein number.

Acknowledgment The author acknowledges the friendship and co-operations of the members of the Combustion Group of the School of Mechanical Engineering, The University of Leeds, Leeds, UK. Journal of Engineering for Gas Turbines and Power

⫽ ⫽ ⫽ ⫽

u⬘ ⫽ ul ⫽ u⬘k ⫽

␣ ⫽ ␦l ⫽ ␩ ⫽

␭ ␳u ␳b ␾

⫽ ⫽ ⫽ ⫽

mean flame curvature dimensionless wave number burned gas Markstein length burned gas Markstein number, Mab = Lb / ␦l mass of burned gas within any sphere of general radius, Ri mass of burned gas outside any sphere of general radius, Ri mass of unburned gas outside radius, Rt mass of unburned gas within any sphere of general radius, Ri mass of unburned gas outside any sphere of general radius, Ri pressure any general radius between Rr and Rt root radius, within which all gas is burned tip radius, beyond which all the gas is unburned rms turbulent velocity unstretched laminar burning velocity effective rms turbulent velocity acting on the flame flame stretch laminar flame thickness Kolmogorov length scale of turbulence Taylor length scale of turbulence unburned gas mixture density burned gas mixture density equivalence ratio

References 关1兴 Ting, D. S., Checkel, M. D., and Johnson, B., 1995, “The Importance of High-Frequency, Small-Eddy Turbulence in Spark Ignited, Premixed Engine Combustion,” SAE Paper No. 952409. 关2兴 Abdel-Gayed, R. G., Bradley, D., and Lawes, M., 1987, “Turbulent Burning Velocities: A General Correlation in terms of Straining Rates,” Proc. R. Soc. London, Ser. A, 414, pp. 389–413. 关3兴 Abdel-Gayed, R. G., Al-Khishali, K, J., and Bradley, D., 1984, “Turbulent Burning Velocities and Flame Straining in Explosions,” Proc. R. Soc. London, Ser. A, 391, pp. 393–414. 关4兴 Checkel, M. D., and Ting, D. S., 1993, “Turbulence Effects on Developing Turbulent Flames in a Constant Volume Chamber,” SAE Paper No. 930867. 关5兴 Shen, H., Hinze, P. C., and Ting, J. B., 1996, “A Study of Cycle-to-Cycle Variations in SI Engines Using a Modified Quasi-Dimensional Model,” SAE Paper No. 961187. 关6兴 Ting, D. S., and Checkel, M. D., 1997, “The Importance of Turbulence Intensity, Eddy Size and Flame Size in Spark Ignited, Premixed Flame Growth,” Proc. Inst. Mech. Eng., Part D 共J. Automob. Eng.兲, 221, pp. 83–86. 关7兴 Merdjani, S., and Sheppard, C. G. W., 1993, “Gasoline Engine Cycle Simulation Using Leeds Turbulent Burning Velocity Correlations,” SAE Paper No. 932640. 关8兴 Bradley, D., 1992, “How Fast Can We Burn?,” Proc. Combust. Inst., 24, 247–262. 关9兴 Williams, F. A., 2000, “Progress in Knowledge of Flamelet Structure and Extinction,” Prog. Energy Combust. Sci., 26, pp. 657–682. 关10兴 Echekii, T., and Chen, J. H., 1996, “Brief Communication: Unsteady Strain Rate and Curvature Effects in Turbulent Premixed Methane-Air Flames,” Combust. Flame, 106, pp. 184–202. 关11兴 Haq, M. Z., Sheppard, C. G. W., Woolley, R., Greenhalgh, D., and Locket, R. D., 2002, “Wrinkling and Curvature of Laminar and Turbulent Premixed Flames,” Combust. Flame, 131, pp. 1–15. 关12兴 Lee, T. W., North, G. L., and Santavicca, D. A., 1993, “Surface Properties of Turbulent Premixed Propane/Air Flames at Various Lewis Numbers,” Combust. Flame, 93, pp. 275–288. 关13兴 Shepherd, I. G., and Cheng, R. K., 2001, “The Burning Rate of Premixed Flames in Moderate and Intense Turbulence,” Combust. Flame, 127, pp. 2066–2075. 关14兴 Gu, X. J., Haq, M. Z., Lawes, M., and Woolley, R., 2000, “Laminar Burning Velocity and Markstein Lengths of Methane-Air Mixtures,” Combust. Flame, 121, pp. 41–58. 关15兴 Haq, M. Z., 1998, “Fundamental Studies of Premixed Combustion,” Ph.D. thesis, The University of Leeds, UK.

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关16兴 Bradley, D., Lau, A. K. C., and Lawes, M., 1992, “Flame Stretch Rate as a Determinant of Turbulent Burning Velocity,” Philos. Trans. R. Soc. London, Ser. A, 338, pp. 359–387. 关17兴 Haq, M. Z., 2005, “Correlations of the Onset of Instabilities of Spherical Laminar Premixed Flames,” ASME J. Heat Transfer, 127, pp. 1410–1415. 关18兴 Haworth, D. C., and Poinsot, T. J., 1992, “Numerical Simulations of Lewis Number Effects of Turbulent Premixed Flames,” J. Fluid Mech., 224, pp. 405–436. 关19兴 Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., 1992,

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Numerical Recipes in Fortran 77: The art of scientific computing, Cambridge Univ. Press, Cambridge, UK. 关20兴 Scott, M. J., 1992, “Distributions of Strain Rate and Temperature in Turbulent Combustion,” Ph.D. thesis, The University of Leeds, UK. 关21兴 McComb, W. D., 1990, The Physics of Fluid Turbulence, Oxford University Press, Oxford, UK. 关22兴 Heywood, J. B., 1988, Internal Combustion Engine Fundamentals, McGrawHill, New York.

Transactions of the ASME

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