Turbulent premixed combustion modelling using fractal geometry

Twenty-Third Symposium (International) on Combustion/The Combustion Institute, 1990/pp. 835-842

TURBULENT PREMIXED COMBUSTION MODELLING USING FRACTAL G E O M E T R Y OMER L. GOLDER National Research Council of Canada Division of Mechanical Engineering, M-9 Ottawa, Ontario KIA OR6, Canada

The fractal geometry concept has been applied to the problem of turbulent premixed flame propagation. This approach has been based on the fact that turbulent surfaces are wrinkled over a wide range of scales, and experimental evidence shows that the scales are self-similar. Self-similarity of the scales over a wide range is the distinctive characteristics of fractal objects. A new model, derived using the fractal geometry, for the wrinkled flames (or flamesheets) regime has been proposed. The distinct feature of the present formulation is the adoption of a variable inner cutoff scale as a function of turbulent and molecular diffusivities. With a fractal dimension of 7/3 and an outer cutoff approximating the integral length scale of turbulence, our analysis has yielded udttl

cr [Ur k /I U l/\1/2 n e t ~L4 9

This model suggests a wrinkled flame structure influenced by a turbulent length scale as well as the turbulence intensity. Comparison of the predictions of the model and the experimental data from a variety of rigs shows excellent agreement despite the uncertainties involved in the measurement of the turbulence parameters, propagation velocities, and the intuitive assumptions made in the theories.

Introduction It has been recently shown that the fractal dimension is a useful concept for estimating the area of complex surfaces in turbulent flows, e.g. Sreenivasan et al.1 These surfaces are wrinkled over a wide range of scales and the experimental evidence shows that the scales of wrinkling are self-similar. Self-similarity of the scales over a wide range is the distinctive feature of fractals, z The fractal theory lends itself as a promising means for the analysis of the dynamics of turbulent prcmixed flame surfaces. The concept of laminar flamelets provides a useful tool to describe the turbulent premixed flames. This concept assumes that the combustion within a turbulent flame is confined to asymptotically thin moving laminar flamclets which are imbedded in the turbulent flow.a'4 Since the instantaneous behaviour of these thin layers is the same as those of laminar flames, turbulent burning velocity can be approximated by the product of the flamelets surface area and the laminar burning velocity. Gouldin, 5 in his seminal paper, proposed a model for turbulent burning velocity based on the assumption that chemical reaction and heat release rates can be modeled as occurring in thin flamelets and that the geometry of these thin sheets can be described by the fractal geometry.

In order to formulate a model for turbulent premixed combustion using the fractal approach, two important issues should be resolved. The first one is the determination of the fractal dimension which is a characteristic dimension associated with the roughness of the fractal surfaces. Techniques based on laser tomography have been used to measure the fractal dimension of the turbulent flame surfaces.6-9 The second issue is related to the physical limits to the self-similarity of the scales within the turbulent flame front surface. These limits are referred to as inner and outer cutoffs. Basically, the inner cutoff is the scale where a finer surface convolution does not exist, and for nonreacting turbulent flows Kolmogorov length scale seems to be the logical inner cutoff.I~ The outer cutoff, in a very general sense, can be defined as the size of the experiment. 11 In turbulent flows, it is proposed that the outer cutoff is comparable to the integral length scale of turbulence. In this work, a turbulent premixed combustion model based on fractal theory is presented. The distinctive features of this formulation are the adoption of a variable inner cutoff scale and the use of a mean fractal dimension of 7/3. Resulting model has been compared to available experimental data and good agreement has been obtained.

835

836

TURBULENT COMBUSTION the fractal geometry yields the following relationship$ (Fig. 1)

Fraetal Dimension

One way of measuring the area of a three dimensional surface is to count the number of cubic boxes, N(e), of size ~ essential to cover the whole surface. The area then is of order N(e)" ~. If the surface is a classical one, i.e., can be described by Euclidean geometry, N(e)" ~ approaches to a constant value independent of e. For a .fi'actal surface, which exhibits serf-similarity over a wide range of scales, the measured area estimates ~11 increase with increasing resolution (i.e. decreasing value of e) according to a power law. 2 The variation of the area with the resolution ~ for a f~actal surface, on a loglog scale, exhibits a straight line with the slope 2 - D, where D is the fractal dimension of the surface, Fig. 1. Therefore, the area A(e) is given by

A(~) = ~ - o

(1)

The range of scales over which the power laws of the type Eq. (1) holds is bounded by cutoffs on both ends imposed by physical limits. As originally suggested by Damk6hler, 12 the ratio of the turbulent to laminar flame velocity should be proportional to the ratio of the instantaneous flame surface area of the turbulent flame to the flow cross section area, i.e., ut -

uz

=

At -

(2)

-

Ao

where ut and ul are the turbulent and laminar flame velocities, respectively. At is the area of the turbulent flame, and Ao is the flow cross section. Then,

Inner Cutoff

< ~ S l o p e = 2

- D

Outer utoff

Ei

A, (,o/~-~ Ao

(3)

\ei/

where ~o is the outer cutoff, Ei is the inner cutoff, and D is the fractal dimension. For nonreacting turbulent flows, Mandelbrot 1~ proposed a fraetal dimension of 2(2/3) for GaussKolmogorov turbulence and 2(1/2) for Gauss-Bergen turbulence. Recently, Sreenivasan et al1 showed that 2.36 is a better estimate of the fractal dimension of the surfaces in nonreacting turbulent flows. In his formulation, Gouldin5 used D = 2.37, and later his findings at very low turbulent intensities yielded 2.11. 6 North and Santavicca8 represented their fractal dimension data as a function of u'/uz, i.e., D = 1.19(u'/uz)~176 + 1.01. u' is the turbulent rms velocity for isotropie turbulence, and ut is the laminar burning velocity. For a range of u'/ut values 0.15 to 15, D varies from 2.13 to 2.32. s The fractal dimension data of Murayama and Takeno 7 and Takeno et all3 vary between 2.2 and 2.35, with an average value around 2.26. 7 These turbulent flame data5 - 8 were obtained from measurements on open atmospheric flames. Under the spark i~nition engine running conditions, Mantzaras et al report D = 2.36 - 3% whereas Ziegler et a114 report a value about 2.19 (see Table I). It seems that in low intensity turbulence fractal dimension is lower than 7/3. Possible reasons for this behaviour have been discussed by Gouldin et al.6 Reported experimental and analytical results point to an average value of 2(1/3) for turbulent premixed flames in the wrinkled flame regime. Peters4 and Gouldin5 took the integral length scale of turbulence, L, as the outer cutoff limit in their formulations. Measurements of North and Santaviccas in an open flame and data of Ziegler et a114 obtained in a square cross-section spark ignition engine simulator yielded outer cutoff scales approximating7L, whereas the data of Murayama and Takeno and Gouldin et al,8 obtained from atmospheric open flames, showed outer cutoff scales larger than L. The outer cutoff scale was found7 to be much larger than the integral length scale L, and was about 2/3 of the burner exit diameter. Gouldin5 also notes that outer cutoff scale is expected to change with u'/ut. In our formulation, ~ze assume that the outer cutoff scale approximates .he integral length scale of turbulence.

Eo log

FIG. 1. Graphical description of the area versus measurement resolution relationship for a fraetal surface. D is the fractal dimension.

Proposed Model for Inner Cutoff

As the inner cutoff scale of the turbulent premixed flame surfaces, Peters4 suggested the Gibson

837

MODELLING USING FRACTAL GEOMETRY TABLE I Summary of reported values for fractal dimension D, inner cutoff ~, and outer cutoff Co. Reference

D

e~

~o

"0 Lc fl*/ Lc ~10"O ~

L L L L L f2L f~L

8z max(Sz.~)

L

Mandelbrot ~~ Sreenivasan et al 1 Peters 4 Gouldins Kerstein 15 Gouldin et al6 Murayama & Takeno~

2(1/2) 2.36 2.13-2.35 2.37 7/3 2.11 2.26

Takeno et a113 North & Santaviccas

2.2-2.35 (u~) ~176 1.19 + 1.01

Mantzaras et al9

2.36 - 3%

L

Ziegler et al~4

2.12 2.29 2.19

L

scale, La, whereas North and Santavicca proposed the larger of the laminar flame thickness 8/and the Kolmogorov length_ scale r/. Murayama and Takeno7 and Takeno et a113 determined experimentally that the inner cutoff is the thickness of the laminar flame, which is the smallest dimension that can be disturted by the turbulence. (It should be noted that in their experiments the conditions were such that

8z > ,7.) Both Gouldin et al~ and Murayama and Takeno7 found that the inner cutoff ei is significantly larger than the Kolmogorov length scale 71. It seems that, in reacting flows as compared to non-reacting ones, the inner cutoff sets in at a coarser resolution. This suggests that the turbulent flame surface does not behave like a passive surface, but it exhibits a modified characteristics different from the upstream cold flOW.

For low turbulence intensities, Gouldin et al~ reported that ~i - b'~?, where b is of order 10. Then, for u' ~ Ul, one can take ~-n'

~b'n

(4)

As discussed by Gouldin,5 with increasing u'/ul, the inner cutoff decreases to the Kolmogorov length scale asymptotically. Gouldin5 modeled this variation of the inner cutoff by an exponential function which satisfies the two limiting conditions, namely u ' / u t ~ 0 and u ' / u l ~ oo. Here, we are going to attempt to model this behaviour starting from different considerations.

Description non-reacting theory non-reacting flow data model model, ./'1 "> 1 model open V-flame, f2 > 1 open flame, f3 "> 1. eo ~ burner dia. 0.65 -< u'/uz <- 1.62 open flame, Bi > ~/ open flame, 0.15 --< u'/ul <--- 15 SI engine: 4 < u'/ul <- 50, 0.1 < "O/~, < 1. u'lu, = 0.5, r//~, = 3. u'lu~ = 2, 7116~ = 1.5. SI engine: u'/uz = 2.

It can be argued that the dynamics inside the unburned mixture regions of the turbulent flame brush of thickness 8t are governed by the hierarchy of eddies, from size 8t down to Kolmogorov scale r/, linked by a cascade. This concept of flame front structure assumes that, for the wrinkled flames, turbulent flame brush thickness St is a length interval along the flow streamline where both burned and unburned gases exist. The laminar wrinkled flame front is embedded within the turbulent flame brush, and at a given cross section taken parallel to the flow streamline, imaginary lines of upper and lower bounds of the thickness 8t constitute the loci of the maxima and minima of the large scale wrinkles of the laminar flame front, respectively. The ratio *l/St then can be expected to be effective in the formation of the finer scale structure of wrinkles of the laminar flame front within the turbulent flame brush, leading to the following proposition that ~, ~ f" *1' ~ (~l/St)" *1'

(5)

For the Kolmogorov's statistical equilibrium region, the three dimensional energy spectrum is given as E(k) oc e2/3k ~/3, where e is the rate of energy dissipation, e --- u'a/L, and k is the wave number. The kinetic energy for wave numbers k > l / S t is then / - zo

l'5u'~ -- / E(k) " dk ~ e213~ 13 ,11/~t

(6a)

838

TURBULENT COMBUSTION

Also, it can be shown that, within the turbulent flame brush of thickness 6t, the integral length scale L is

It should be noted that the factor f, Eq. (9), can be shown to be f = (Da n 96 ~ / / L ) a/4 = (Daz./ReL) a/4

L2/a-u'Z/e2/a~

f]s E(k).dk/e2/a~

6~t/a

(6b)

t or simply L ~ ~t- Then we can define a turbulent diffusivity, assuming that 6t is a diffusive-reactive zone analogous to a laminar premixed flame, 15 snch that Dt ~ u'L. In view of Eq. (6b), this can be put into the following form Dt ~ u'6t

Flame Propagation Model

(8)

The assumptions made above regarding D and Co, and the model adopted for the inner cutoff ~i yield the following through the use of Eq. (3)

Using the definition of the Kolmogorov's length scale, 7/ = va/4e -1/4, and assuming that the molecular diffusivity D l ~ 1), the factor f in Eq. (5) can be expressed as follows f = ~/6t ~ (Dt/Dl) -a/4

where Da n is the Damk6hler number based on Kolmogorov eddy time, which is the shortest time scale involved in the dynamics of turbulence. Either pair of these parameters, (DaL, ReL) or (Da n, L/6l) in Eq. (12), has been used by most of the investigators to describe the turbulent premixed combustion regimes. 17-z3

(7)

Combining Eqs. (6a) and (7) yields D, ~ el/a~/a

At

Noting that 24 L/~7 ~ Re a/4, we obtain

A,=b,(u']l/2

(9)

If Dt and DI are taken as the relative diffusivities associated with turbulence and the laminar chemical reaction rate over a characteristic time interval of r, such as Dt ~ tt'2r and DI ~ u~'r, we have then the following as the inner cutoff scale*

_ \Ul/

_

Ao

b,/u'\l/Z

Ut

ei, n = Sc-3/4"q = f ' r l

. Re~/4

(14)

Substituting Eq. (14) into Eq. (2), and noting the boundary condition that ut --+ ul as u' ---> 0, we get

(10)

In nonreaeting turbulent flows, when the Schmidt number is less than unity, the inner cutoff gets larger than the Kolmogorov scale ~7, and is given b y 116 '

(13)

So = t ',7,/

_

~i ~ (u'/ul) -3/2 71'

(12)

nl

1=

~//)

"an

(15)

where R n is the Reynolds number based on Kolmogorov length scale, Re 1/4 ~ R n, and b' = 0.46 if one takes b - 10.

(11)

where Sc = v/D! is the Schmidt number. It is possible to claim an analogy between Eqs. (9) and (11) that at low turbulence intensity levels the inner cutoff of a turbulent flame surface is modified by (Dt/Dl) -3/4 in a similar manner as in the nonreaeting flows in which (u/D1) -a/4 modifies the inner cutoff. Unfortunately, it is not possible to make a similar analogy at high and medium turbulence intensities. *A reviewer pointed out that in reacting flows, the inner cutoff length is not merely controlled by the momentum of the flow. How the flame surface reacts to the momentum perturbations is described also by the Prandtl number, especially at moderate to high turbulence intensities. For wrinkled flame regime, however, this effect of Prandtl number is assumed to be small.

Discussion Most of the available experimental data sources for which the turbulent length scales as well as the turbulence intensity were measured by the original investigators (except for one set of data, see the caption of Table II) are summarized in Table II. Data from sources indicated in Table II have been used to test the Eq. (15). Wrinkled flame region is taken as the zone bounded by R n >- 1.5(u'/ul)

(16)

ReL <-- 3200.

(17)

and

Equation (16) is equivalent to the criterion "q -> 1.5& R n is the turbulent Reynolds number based

MODELLING USING FRACTAL GEOMETRY

839

TABLE II Turbulent premixed burning velocity data sources. Data from the paper of Kozachenko and Kuznetsov~s are taken as interpreted by Abdel-Gayed and Bradley. eav is the kinematic viscosity, 10 s m~/s, and u~ is the laminar flame speed, m/s. Key

Reactant mixture

t,

ut

Ref.

A B C D E

CH4 + 202 + 4.5N2 CH4 + 20= + 2.79N2 CH4 + 202 + 1.96N2 2C2H2 + 502 + 27.15N2 2C=H= + 50~ + 18.48Nz

15.91 16. 16.07 15.22 15.09

0.89 1.55 1.93 0.9 1.55

25 25 25 25 25

F G H J K

2CzHz + 5Oz + 14.88N~ Calls/Air; Hz/Air Calls + 50z + 28.5N= CaHs + 50~ + 18Ar Calls + 50~ + 18He

15.01 14.37-20.75 54.31 13.63 42.77

1.98 0.27-2.42 0.74 0.8 1.4

25 26 27 27 27

L M N 0 Z

HJAir Natural Gas/Air CH~/Air CH~/Air; C~Hm/Air CH~/Air

15.7-24.1 15.5 15.5 15.7 16.

1.01-3.48 0.45 0.25-0.44 0.08-0.4 0.45

28 29 30 31 32

*

CH4/Hz/Air, 6 mixt.

14.74-15.54

0.18-0.72

33

on Kolmogorov length scale ~7, and 3t is the laminar flame front thickness calculated as v/ut. Turbulent premixed flame propagation data, which are within the wrinkled flame region defined by Eqs. (16) and (17), are plotted as (ut - ut)/Ul versus (u'/ Ul)~ R,7 in Fig. 2 (note that R , ~ Re~/4). The agreement seen between many sets of experimental data is significant, especially when one considers the variety of rigs on which these data were obtained and sources of potential errors in the measured parameters. Figure 2 includes more than 200 data points for 15 mixtures reported by eight different research groups. The solid line in Fig. 2 represents

10

Rv 1.5{u/u t} 8 I WRINKLED FLAME I 6

0 ~ ^ ~ %" ~ A ~

|174 O ~ O O"

.~,~.vm..o ~

|

LEGEN[

0.~. ~)

O~B

~

+-z

9 .;'~1.~.~ ~

0

(ut/ul) - 1 = 0.62(u'/ut) 1/2 Rn

| ~ ~ |/ , ~

v- o

'(+ + T"

e=L @--N EB--O |

(18)

The slope 0.62 which resulted from experimental data is comparable to the value 0.46 in Eq. (15)~a A recently reported set of data by Liu and Lenze is not included in Fig. 2, although their data are within the wrinkled flame regime. This data set gives an exceptionally good fit to the empirical expression ttt/u I = 1 + 5.3(u'/u~5). Since the data of Liu and Lenze aa are well defined, it is compared to Eq. (18) in Fig. 3. The height of the bars in Fig. 3 corresponds to twice the standard deviation of the data points of Fig. 2 within ---1 x-axis value around the bar location. Data points of Liu and Lenze aa can be represented by Eq. (18), but with a slope near unity rather than 0.62, Fig. 3. Although it is difficult to explain this discrepancy, it should be noted that, as stated by the authors themselves in the pa-

, , ,I 0

2

....

I ....

J ....

I ....

4

6

8

,

[u/u,]

05

"

I ,,

10

, , li,

12

, ,

14

R,,

FIG. 2. Comparison of the wrinkled flame model, Eq. (18), to experimental data. Letters in the legend box correspond to the reactant mixtures and data sources in Table II. Note that Re[/4 = R,. per, the data of Liu and Lenze aa display higher turbulent burning velocity values as compared to the data reported by Cho et a135 obtained on a stagnation flame burner which was very similar to the burner used by Liu and Lenze. Since Cho et ala5 did not report any turbulence length scale measurements, a direct comparison of their data to Eq.

840

TURBULENT COMBUSTION 10

-

Rn>l.5lu'/u / } 8

~..'..

I WBINKLEDFLAMEI

[WRINKLEDFLAMEI

10

~,'r

.'~00"'~

7Y/' -oo 4

' / 1 L7

.@ 2

88: ' ~ - -

- -

Equation

(18)

0

.... 0

i .... 2

i .... 4

i .... 6

i .... 8

i .... 10

i .... 12

0 14

[u'/ul]O'SRn FIG. 3. Comparison of the wrinkled flame model, Eq. (18), to the data of Liu and Lenze. ~ Error bars on the solid curve correspond to twice the standard deviation of the experimental data points within ---1 x-axis value.

(18) is not possible. However, the approximate Taylor-scale Reynolds number of 135 they quoted (comments section of Ref. 35) for maximum turbulence intensity values (u' = 0.6-0.7 m/s), measured turbulent burning rates of 1.3-1.4 m/s for lean (~b = 0.74) methane-air flames agree reasonably well with Eq. (18). Figure 4, which is a plot of Eq. (18), illustrates the influences of the turbulent Reynolds number ReL and the nondimensional turbulence intensity u'/ut on the turbulent premixed flame propagation speed ut/ul within the wrinkled flame regime. Equation (2) assumes that the change in the laminar burning velocity of the wrinkled flame front with local stretch and curvature is negligible. This is justifiable for low to moderate levels of turbulence as discussed by Gouldin. 5 There is good evidence that the laminar flame speed is rather insensitive to stretch for Lewis numbers near to unity, a6 Therefore, as a first approximation for the wrinkled flame regime, the propagation rate can be assumed to be independent of the flame straining. However, for higher levels of turbulence, and for non-unity Lewis numbers, effect of flame stretch can be significant. Although definition of the effective flame-straining rate under turbulent conditions is not very clear, a7 proposed approximate correction schemes"sz3 for flame stretch effects can be used in the models. The present simple analysis ignores the effects of combustion on turbulence. Volume expansion in-

0

1

2

3

4

5

6

U'/UI

FIC. 4. Dependence of ut/u, on turbulent Reynolds number ReL and the turbulence intensity u'/ ut in the wrinkled flame regime.

duced pressure field and the large density difference between reactants and products result in countergradient transport and production of turbulent energy. These effects seem to be the major areas of uncertainty in turbulent premixed combustion. The Bray-Moss-Libby model and the pdf approach, which are probabilistic field theories for turbulent premixed flames, are able to account for countergradient diffusion and energy production to a certain extent. Merits of these formulations, and other models using asymptotic methods are discussed by Popez3 and Williams. ~ It seems that the experimental studies of controlled perturbations of the wrinkled laminar flames are necessary for an improved understanding of these phenomena. It should be noted that the model introduced for the inner cutoff is preliminary and based on intuition, and is subject to confirmation by the experimental results. However, using the laser sheet tomography technique, the resolution of the smaller scales is limited mainly by the minimum thickness of the laser sheet obtainable. As the improved experimental techniques for flame front imaging become available, it would be possible to resolve the inner cutoff problem as well as others involved in fractal analysis of the turbulent premixed flames.

Concluding Remarks A turbulent premixed flame propagation model for the wrinkled flame regime has been developed

M O D E L L I N G USING FRACTAL GEOMETRY based on the assumption that the flame front surface exhibits a fractal character. The fractal dimension D of the turbulent flame surface has been taken as 7/3 in accordance with the recent values reported in the literature. The outer cutoff has been assumed to be approximated by the integral length scale of turbulence. A variable inner cutoff scale, dependent on the ratio of a turbulent diffusivity to a molecular diffusivity, has been introduced. The developed model for the wrinkled flames proposes a turbulent flame speed proportional to ReI/4 and (u'/ut) 1/z. This model suggests a wrinkled flame structure dependent on a turbulent length scale as well as the turbulence intensity. Predictions of the model agree exceptionally well with the experimental data obtained on different rigs ranging from atmospheric circular burners to constant volume combustion bombs. In view of the good experimental agreement, fraetal geometry description of turbulent flames seems to be a promising means of treating this complex problem.

Acknowledgments The assistance of Mr. G. Burton with data handling is gratefully acknowledged. The work described herein was supported by National Research Council's internal funds (program manager Mr. L. Gardner) and by the Canadian Government's EMR/ PERD program (program manager Mr. P. ReillyRoe). I thank Mr. L. Gardner and Mr. J. Ploeg for their continuous support and trust in my research group's work.

REFERENCES 1. SREENIVASAN,K. R., RAMSHANKAR,R., AND MENEVEAU, C.: Proc. R. Soc. Lond. A421, 79 (1989). 2. MANDELBROT, B. B.: The Fractal Geometry of Nature, Freeman, 1982. 3. BRAY, K. N. C.: Complex Chemical Reactive Systems (J. Warnatz and W. J~iger, Eds.), p. 356, Springer-Verlag, 1987. 4. PETERS, N.: Twenty-First Symposium (International) on Combustion, p. 1231, The Combustion Institute, 1986. 5. GOULDIN, F. C.: Comb. Flame, 68, 249 (1987). 6. GOULDIN, F. C., HILTON, S. M., AND LAMB.: Twenty-Second Symposium (International) on Combustion, The Combustion Institute, p. 541, 1988. 7. MURAYAMA,M., AND TAKENO, T.: Twenty-Second Symposium (International) on Combustion, The Combustion Institute, p. 551, 1988.

841

8. NORTH, G. L., AND SANTAVICCA, D. A.: The Combustion Institute/Eastern States Section, Fall Technical Meeting, p. 1-1, 1988. 9. MANTZARAS,J., FELTON, P. G., AND BRACCO, F. V. : Comb. Flame, 77, 295 (1989). 10. MANDELBBOT, B. B.: J. Fluid Mech. 72, 401 (1975). 11. MANDELBROT, B. B.: NATO Advanced Study Institute on Fractal Geometry and Analysis, University of Montreal, July 3-21, 1989. 12. DAMKOHLER,G. : Z. Elektroehem. 46, 601 (1940), (English Translation NACA TM 1112, 1947). 13. TAKENO, T., BABA, N., KUSHIDA, G., AND MURAYAMA, M.: Joint International Conference, Australia/New Zealand and Japanese Sections of The Combustion Institute, p. 145, 1989. 14. ZIEGLER, G. F. W., et al. Joint International Conference, Australia/New Zealand and Japanese Sections of The Combustion Institute, p. 281, 1989. 15. KERSTEIN, A. R.: Combust. Sci. Technol. 60, 441 (1988). 16. CORRSlN, S.: Physics Fluids 7, 1156 (1964). 17. WILLIAMS, F. A.: Combustion Theory, 2nd edition, p. 412, Benjamin/Cummings, 1985. 18. ABRAHAM,J., WILLIAMS, F. A., AND BRACCO, F. V.: SAE Paper No. 850 345, 1985. 19. BRAY, K. N. C. : Turbulent Reactive Flows, (P. A. Libby and F. A. Williams, Eds.), SpringerVerlag, 1980. 20. BORGm, R.: J. Chim. Physique 81, 361 (1984). 21. GOKALP, I.: Comb. Flame 67, 111 (1987). 22. WILLIAMS, F. A.: AIAA J. 24, 867 (1986). 23. POPE, S. B.: Ann. Rev. Fluid Mech. 19, 237 (1987). 24. TENNEKES, H., AND LUMLEY, J. L.: A First Course in Turbulence, ch. 1, The MIT Press, 1970. 25. VINCKIER, J., AND VAN TIGGELEN, A.: Comb. Flame 12, 561 (1968). 26. KOZACHENKO, L. S., AND KUZNETSOV, I. L.: Combust. Expl. Shock Waves 1, 22 (1965). 27. SOKOLIK, S. A., KARPOV, V. P., AND SEMENOV, E. S.: Combust. Expl. Shock Waves 3, 36 (1967). 28. ABDEL-GAYED, R. G., AND BRADLEY, D.: Sixteenth Symposium (International) on Combustion, p. 1725, The Combustion Institute, 1976. 29. KARLOVrrz, B.: Selected Combustion Problems (AGARD), p. 248, Butterworths, 1954. 30. SMITH, K. O., AND GOULDIN, F. C.: Prog. in Astronautics and Aeronautics 58, 37 (1978). 31. WOHL, K., AND SHORE, L.: Ind. Engr. Chem. 47, 828 (1955). 32. SHET, U. S. P., SRIRARMULU,V., AND GUPTA, M. C.: Eighteenth Symposium (International) on Combustion, p. 1073, The Combustion Institute, 1980.

842

TURBULENT COMBUSTION

33. LIU, Y., AND LENZE, B.: Twenty-Second Symposium (International) on Combustion, p. 747, The Combustion Institute, 1988. 34. ABDEL-GAYED, R. G., AND BRADLEY, D.: Phil. Trans. R. Soc. London A301, 1 (1981). 35. CHO, P., Law, C. K., HERTZBERG, J. R., AND CHENG, R. K.: Twenty-First Symposium (In-

ternational) on Combustion, p. 1493, The Combustion Institute, 1986. 36. Wu, C. K., AND LAW, C. K.: Twentieth Symposium (International) on Combustion, p. 1941, The Combustion Institute, 1984. 37. ABDEL-GAYED, R. G., BRADLEY, D., AND LAWES, M.: Proc. R. Soc. London A414, 389 (1987).