How fast can we burn?

Twenty-Fourth Symposium (International) on Combustion/The Combustion Institute, 1992/pp. 247-262

INVITED LE(YI'URE

H O W F A S T C A N W E B U R N ?

DEREK BRADLEY Department of Mechanical Engineering

University of Leeds" Leeds LS2 9JT, United Kingdom

The roots of our present understanding of turbulence, flame chemistry and their interaction are traced. Attention is then focused on premixed turbulent combustion and the different analyses of stretch--free theories of turhulent burning velocity, u , are reviewed and new work presented. The various expressions for u, are very different and this becomes more important when allowance must be nmde for flame stretch. On the basis of previous asymptotic analyses the appropriate dimeusionless groups emerge for the correlation of experimental values of u, and a correlation based on these is presented save that, 'pro tern', the Lewis rather than the Markstein uumbcr is employed.

The presented correlations are used as a test bed for a stretched flame, Reynolds stress, laminar flamelet model of turbulent combustion. Computed laminar flame data on flame quenching by stretch are generalised and used in the model. The problems of appropriate probability density fimctions of hoth stretch and temperature are discussed. There is good agreement between model predictions and experiment over a wider range than would be expected from the Williams criterion for flamelet modelling: namely, that the Kohnogorov distance scale should be greater than the laminar flame thickness. The reasons for this wider applicahility are discussed, particularly in relation to recent direct numerical simulations.

Some of the concepts diseussed are applied to spherical explosions and continuous swirling combustion. There is wide diversity in burning rates because of widely different circum- stances, but our knowledge is becoming adequate enough to develop reasonably accurate inathematical models fur the wide range of engineering applications.

Introduction

Table I co,npares a variety of mean euerg'y release rates. The answer to the question posed by the paper's title is, not surprisingly, "it all depends"; it depends on how last we want to burn, where and why?, on whether combustion is re- quired for heat transfer or power production. Prob- ably the fluidised bed gives the [lest reconciliation of burning and heat transfer rates. Is combustion steady or time varying? What are the material properties of the containing wall? What are the en- vironmental limitations? To respond to the wide range of possible requirements demands a high de- gree of control of diverse processes, the complexi- ties of which require aecurate, computer based, mathematical models. These rest upon the funda- mental understandings revealed by comparatively simple burners, explosion bombs, shock tubes and clear minds.

�9 Burke-Schumann fast chemistry Historicallv, the " 1 theory of laminar diffusion flame burning (1928) predates, by ten years, the Zeldovich, Frank-

Kamenetskii z3 theory of premixed burning. Ill tile wake of the understanding of chain reactions in combustion, acquired through the researches of tlinshelwood and Semenov ill tile thirties, the years 1938-1942 were years of important advances. In 1938 Zeldovich and Frank-Kamenctskii produced the first satisfactory analysis of the laminar burning velocity, to show how it was controlled by chemical reaction rates. In 1940 Damk6hler, 4 unaware of this work and using Prandtl mixing length ideas, showed the influence of turbulent length scale and laminar fame thickness on the ratio of turbulent to laminar burning velocity, uffut, whilst in 1941 Kolmogorov 5'6 published his seminal ideas on the structure of turbulence. Spalding 7 in his translation and commen- tary has shown how in 1942 Kolmogorov, remark- ably, also anticipated the two equation, k-E, model of turbulence. Two equations are the minilnum necessary for the prediction of the r.m.s, turbulent velocity, u', and the integral length scale, L. Early understanding of turbulent combustion was ham- pered by failure to appreciate the full significance of the latter and to measure it.

247

248 INVITED LECTURE

TABLE I Energy release intensities (GW m -3)

Electric spark

Hydrocarbon atmospheric laminar flame (peak)

Dissipation in vehicle brake

Aero gas turbine (take off) primary, zone

whole combustor

Gas central heating boiler

Gasoline engine

Pressurised water nuclear reactor

Gas-fired fluidised bed (atmospheric)

Pulverised coal (cyclone combustor)

10

1.5

0.2

0.1

0.1

0.04

0.01

In the 1960s laminar flame structures and burning velocities were computed on the basis of com- prehensive chemical kinetics, first for the hydrazine flame, 8'9 then for the hydrogen I~ and methane H flames. Simultaneously, in computational fluid dy- namics the first order, k-E method, pioneered by Harlow, 12 was able to predict some turbulent flow fields reasonably accurately. In it an isotropic eddy viscosity enables second moment to be determined from first moment quantities. The laminar diffusion flame analysis of Burke and Schumann i stimulated the concept of a mixing controlled burning rate, and in both the eddy break-up theory, of Spalding 13 and the eddy dissipation model of Magnussen and Hjer-

14 tager, the rate determining step is a turbulence decay rate.

Synthesis of chemical reaction and turbulence was addressed in the Bray-Moss-Libby ~ premixed turbulent flame model. This introduced a reaction progress variable, c, in the forln of an assumed double delta function probability density function (pdf) of c for unburnt and burnt gas, and second order closure of moment equations. Laminar diffusion flame analysis invoked a conserved scalar, ~:, usually the mass fraction of all material originating in the fuel stream. Bilger 16 showed how gradients of it yield a scalar dissipation function, from which reaction rates can be obtained. In turbulent flames 17 an assumed pdf in ~:, usually a beta function, can yield mean values of scalars and reaction rates.

The most common pdf assumption is that other flame parameters have the same relationship to either a reaction progress variable or a conserved scalar as they have in a laminar flame: the turbulent flame is an array of laminar flames.

The assumed pdf has been widely used but, not

only might it be inaccurate, it also becomes intrac- table when a joint pdf of several dimensions is in- volved. For such cases Pope 18'19 has pioneered the computation, by Monte-Carlo methods, of pdfs without any assumption as to their shape. Both reaction and convective transport can be treated ex- actly without modelling assumptions. Weaknesses arise in the modelling of molecular diffusion. This approach is more rigorous, but demands more computer power.

The present paper concentrates on premixed gas- eous combustion, in the absence of strong pressure waves. The first theories of turbulent burning suggested an "onwards and upwards" indefinite increase of ugul with u'/ut. However, in 1953 Kar- lovitz et al. introduced the concept of flame stretch to describe flame extinction and in 1963 Klimov 21 showed theoretically the influence of stretch upon laminar burning velocity. Flame extinction, due to stretch in controlled experiments has been dem- onstrated for both laminar ~'z~ and turbulent z4 flames. Quantification of this effect is vital to an understanding of turbulent combustion but, before discussing this, it is necessary to discuss some fun- dament',d aspects of turbulence.

Length Scales and the Frequency Spectrum

Although the integral length scale is a convenient, measurable parameter, more than this is re- quired. The r.m.s, strain rate, u'/A, as defined by Taylor z~ relates the Taylor microscale, A, to the mass specific rate of energy dissipation, e, for homogeneous, isotropic, turbulence and the kinematic viscosity, v:

The dissipation occurs at the highest frequencies of the turbulent energy spectrum and

CD k 1"5 e = ~ , (2)

L

where k is the turbulent energy = 3u'2/2 and CD an assumed numerical constant. The relationship between A and L then involves the constant, A:

A 2 A v (3)

L u'

Experimental measurements ~'27 suggest A = 40.4 and consequently, from these equations, Co = 0.202. ~

In 1941 Kolmogorov 5'6 developed the concept of

HOW FAST CAN WE BURN? 2 4 9

the dissipative energy flux flowing from the larger to smaller scales of turbulence, before complete dissipation to molecular motion on the Kolmogorov distance scale, given by

(4)

with a Kolmogorov time scale given by

(5)

The reciprocal of rn the rate of strain on the Kol- mogorov scale, is sometimes employed. Equation (1) shows this to be I5 ~ times the r.m.s, value u' / ,s Kolmogorov's second similarity hypothesis is that at intermediate scales, in the inertial range, the distribution of energy over the wavelengths depends only upon ~, which flows from larger to smaller scales, with ultimate dissipation. For eddies of size I, their rate of dissipation of energy is proportional to the energy divided by a time scale given by In divided by the square root of the energy. As e is constant throughout this range, it follows that the energy must scale as ~/3 lnZ73

The wave number, n, for such eddies is 2';'r/l,, and the mass specific turbulent energy per unit wave number, E(n) is proportional to (~13 inZ/3)n-1 and hence E z/3 n -5/3, such that the power spectral density function in this regime is given by

E(n) = C~ e el3 n -5/3

where CK is the Kolmogorov constant. The kinetic energy in the inertial range between n = 2"rr/L and n = 27r/r/is given by

~n ='2"rr/~ ' ~n=2,rr/~

E(n)dn = CK ~ z/3 1 n- 5/3dn (7) Jn=2"rr/L J n = 2 ~ / L

At the frequency associated with the integral length scale the integrated turbulent kinetic energy is only 0.23 k. 29 Evaluation of the right of Eq. (7) and in- vocation of Eq. (2) gives

[(;) +] 0.23 = Ct( Co 2/3 0.441 1 -

It is readily shown that ~/L = 1.28 RL -3/a, where RL = u'L/~', when Co = 0.202. With this same value in Eq. (8) CK = 1.57 if RL = 100 and 1.52 if Rt. = 1000. Chasnov 3~ reviews experimental values of CK, that range between 1.34 and 2.1. He also presents a numerical value of 2.1 obtained for

101 ~ . . . . . . ~'a i I I I

10~ �9 ~ _ % slope

x6.THz , \ \,~, +,3.3 ; \ oz0 i "~ ;\ = r~ 30 I " I I l'n

vso ~ x ~ , , e67 I ! \^ : I-7 ~ :+

, , ! ,

10-2 10-1 10 0 101 10 z

10-1

10-z

10-3

10-~

lO-S 10-3 10 3

FIG. 1. Dimensionless power spectral density function S (F), measured in fan-stirred bomb at different fan speeds.

high Reynolds numbers by large-eddy simulation, about 30% higher than measured in high Reynolds number atmospheric experiments. Such a value would necessitate a downward revision of Co to 0.124 and an upward revision of A to 65.9. This

(6) quantification of turbulence rests entirely upon Kol- mogorov's hypotheses and the experimental or numerical evaluation of but a single constant. How- ever, it can be seen that there is still appreciable uncertainty in the values of some of the constants in widespread use.

It is convenient to represent all parameters in dimensionless form and create a normalised frequency ~' by multiplying the actual frequency f by the integral time scale ra = L/ft where ti is the mean speed. Figure 1 shows an experimental.d!- mensionless power spectral density function S(F) obtained from a fan-stirred bomb by laser doppler velocimetry, 29 plotted against F. Vertical lines indicate frequencies associated with the normalised Kolmogorov strain rate, /Tn = (~_/u)~ the Taylor

(8) strain rate P~ = (u'/~)'r~, and FL = (u'/L)'ra = 1. Figure 2 shows from the same data how the normalised square root of turbulent kinetic energy and the rate of dissipation of that energy develop from the high frequencies to the lower ones, with ~r the rate of dissipation in all frequencies from infinity to /7 and u'k the r.m.s, turbulent velocity associated with the energy in the same frequency range.

The distribution in Fig. 1 is an equilibrium one.

250 INVITED LECTURE

0.8

0.6

0.t,

0.2

I i i

tE ) 4 u ~ lu

. Z 10 ~ 10 2 101 10 0 10 "1 10 -2

FIG. 2. Development of r.m.s, turbulent velocity and square root of rate of turbulent energy dissipation, with dimensionless frequency (reciprocal time).

Direct numerical simulations 31 show a randumised velocity pulse at low frequency to relax to such a distribution. Because it is the high frequencies that influence flame propagation in the early stages, the form of the spectral distribution is important at ig- nition in gasoline engines. Arcoumauis et al. 32 have shown that swirling flow in an engine generates rel- atively stronger high frequency components than does tumbling flow. On the other hand, the tum- ble-generated turbulent energy was higher at all frequencies.

Stretch Free Wrinkled Flames

In non-reacting turbulent flows wrinkling of a constant scalar surface arises from all the self-similar scales in the inertial range. Because of the self- similarity, the surface is fractal in nature between a sharp inner cut-off of 7/and an outer one of, at least, L. The ratio of the area of the fractal surface with -q as the inner cut-off to that with a resolution of L is:

(9)

where D is the fractal dimension of the surface.

Mandelbrot, 33 from the Kolmogorov inertial range n -~/3 law, proposed a value of D of 8/3, but Sreen- ivasan et al.,' on the basis of theory and experiment, proposed a value of 7/3. With the latter value and the previous evaluation of ~/L

A t -- = 0.92 RL 0"25 (10) A

This gives the increase in overall molecular transport that occurs due to turbulent micro-wrinkling of the constant scalar surface. Definition of the outer can be more problematic than of the inner cut-off. If this expression were to apply to reacting flows and the increase in burning velocity due to turbulence were to arise solely from the increase in area due to such wrinkling, it would yield values of u,/ut h~gh enough for large scale atmospheric explosions, where RL may be as high as 10 , to have fearsome consequences. However, a flame surface with molecular processes of transport coupled with those of reaction, to create a burning velocity, is very different from that of a non-reacting scalar front. The flame is of finite thickness and its burning velocity is affected not only by stretch due to aerodynamic strain and the curvature of the front, but also by the Lewis number,

Even apart from such considerations, Peters 3~ ha.s pointed out that the higher frequencies cannot contribute to flame front wrinkling. From Eq. (6), the turbulent energy associated with an eddy size 1, (= 2 7r/n) is given by

E(n)n = Ctr �9 2/Z(l,121r) ~/3 (11)

When this energy is divided by the associated eddy lifetime this expression yields e and it follows that this lifetime is CK e-1/z (lj2vr)~/3. The eddy would be consumed by a laminar flame, in a chemical lifetime of In/ul. When eddy lifetimes are less than this they cannot contribute to flame wrinkling. This regime is defined by

C~ e-1/3(l~121r) 2/3<- l.lul (12)

The limiting size, lc, has been called the Gibson scale by Peters. 36 It must be greater than rl, the ultimate physical limit for the inner cut-off. From Eqs. (2) and (12)

L ( (2'1~1")2 f3CD 1/3 (1.5)~ 3 (13)

With this ratio of inner to outer cut,~ffon the right of Eq. (9), the ratio of.wrinkle~ tonon wrinkled flame area and hence of turbulent to laminar burn-

HOW FAST CAN WE BURN? 251

ing velocity, with D = 7/3, CK = 1.6 and Co = 0.20 becomes:

60

At ut u' . . . . 1. ,52- (14) A Ul Ul

50 The relationship suggests ut is independent of ul:

the turbulent burning velocity is governed entirely by turbulence and not at all by chemical kinetics. The assumption that the sole effect of turbulence is to wrinkle a laminar flame without changing its structure can only be valid if there is no turbulent I /+0 structure within the thickness, 6t, of the laminar flame, or if the influence of that structure is very weak. The first of these conditions is met by the Williams a7 laminar flamelet criterion "q > (~t. Fur- thermore, any stretching of the flame changes its 30 structure, to alter the fractal dimension from the near-constant value of Gouldin. a8

One of the first wrinkled flame expressions for ut/ul , in the absence of stretch, is that derived by Schelkin, 39 who used the wrinkled flame approach 20 of Damk6hler, with no turbulent structure within the laminar flame thickness:

(15) ,[ - - = 1 + - -

ul \ ut / J

This expression and that of Eq. (14), are shown by full line curves D and F, respectively, in Fig. 3.

A more rigorous analysis of stretch-free flames by Yakhot 4~ employed the equation of motion of the scalar field to represent the flame surface, as proposed by Williams. 41 The dynamic renormalisation group method was applied to this equation to give an expression for ut. Such methods, through Kol- mogorov's generalisations concerning the inertial range, avoid characteristic length scales and frequencies. With no adjustable parameters and with the laminar flame thickness 61 < r/, he obtained

2 2

,, [(u,) / (" ' )1 - - = exp u! L k U t / I \ U l / J

This is showu by the full line curve, I, in Fig. 3. At the present Symposium Wirth and Peters 42 also present an analysis based on the same equation of Williams, which yields a relationship in its stretch- free tbrm close to Eq. (14) at the higher values of U'/U I.

Anand and Pope 4a mathematically modelled turbulent combustion with a laminar flamelet approach, in terms of a reaction progress variable and a one step reaction model. A modelled transport equation for the joint pdf of velocity and reaction progress variable was solved. Modelling closure procedures were necessary and results were ex-

10

0 10 20 30 U' /U t

FIG. 3. Stretch-flee expressions for turbulent burning velocity. Curve A, Bray,*' Eq. (19), B, Present work, 1), Schelkin, aa Eq. (15), E, Ahdel- Gayed and Bradley 27 (RL = 2,000), F, Eq. (14), G, Klimov, ~ Curve tl, Libby et al.l~ (RL = :r I, Yak- hot, 4~ Eq. (16).

pressed as a function of unburnt to burnt gas density, p,,/pt,. For most practical values of this ratio, in the absence of stretch,

U t {l t

- - = 1.5--, ( 1 7 )

Ul Ul

close to Eq. (14). Bray 44 and co-workers have expressed the mean

vohunetric chemical source term for a reaction progress variable, c, zero at the unburned and unity at the burned side of the flame, as:

~, = p . ut v Io (18)

252 INVITED LECTURE

Here, ~ is the laminar flamelet surface area per unit volume and Io 4~ represents the influence of flame stretch upon ul, discussed in the next Sec- tion. Derivation of Io is from an equation of the form of Eq. (20). Experiments 46 suggest ~ can be expressed in terms of c and an integral length scale of a square wave variation of c. This is expressed in terms of u' /u l on the basis of a correlation obtained from the cellular automation simulations of Said and Borghi. 47 Whereas in classical laminar flame theory (dd/dc)~,o = 0, this is not true for mean values with a turbulent flame. Zeldovich 48 has drawn attention to a solution obtained by Kolmogorov and coworkers in 1937 for the spreading of an advan- tageous gene when this limit is finite. This gives the propagation of a front in terms of such a rate of change and a diffusion coefficient. Zeldovich employed the expression to obtain the laminar burning velocity of a reactive flame, whilst Libby 49 has derived a similar expression for ut, eschewing any assumption of gradient transport. With the same approach and the stretch-free assumption Io = 1, Bray 44 obtains

0.5

u___tut = 2.56 \~uu/ \ ~ / ' (19)

with an uncertainty range of +--50%. With the ratio of burned to unburnt gas temperature (TI,/T,) = 6 this stretch-free relationship, which again suggests a dependence of ut mainly on u', is plotted as curve A in Fig. 3.

Also shown, by broken curves, are the predictions of the earlier stretch free theories of Libby et al. 15 (curve H), Abdel-Gayed and Bradley z7 (curve E) and Klimov 5~ (curve G). All these approaches are different, but are not self-contained. All re- quired recourse to experimental values of ut and these, inevitably, were affected by flame stretch.

The Correlation of Measured Values of Turbulent Burning Velocity

Flame stretch is expressed in terms of the time rate of change of an infinitesimal area of the flame surface, A, as A -1 (dA/dt). It is usually normalised by ut/61, referred to the unstretched flame. With stretch in laminar flames changing Ul to Un, Clavin has shown, for small stretch 51'52

ul - un 1 dA t31

ut A dt Ul Ma (20)

Here, Ma is the Markstein number, the ratio of a length introduced by Markstein ~3 to express stretch including curvature effects, to ~i. It also is an important parameter of flame stability.

Flame stretch can be expressed in a variety of forms. 54'al A convenient one is 55

) . . . . + + VtV (21) A dt R - '

in which R is the total radius of flame curvature, V1 is the fluid velocity component normal to the flame and Vt V denotes the fluid velocity gradient

along the flame surface. This can be applied to a variety of laminar flame configurations, some of which are discussed in Ref. 55. Under turbulent conditions the direct numerical simulations of Bray and Cant 45 suggest a near-gaussian distribution of mean curvature with a zero mean value. With re- gard to the influence of turbulence upon stretch rate, Abdel-Gayed et al. 56 likened the flame surface to a material surface element with line elements tending to lie in the direction of maximum strain rate. This gives an r.m.s, stretch rate close to the eulerian value of u' /A, but with no probability of negative stretch. The direct numerical simulations of Yeung et al. s7 for both material and randomly orientated surfaces give the former an r.m.s, stretch rate about 30% greater than the eulerian value of the latter.

It therefore seems logical to define a Kar|ovitz stretch factor for turbulent flame in terms of u' /A, such that

With the commonly used approximation 6 / = v/ut, which probably provides an underestimation, where all parameters apply to the premixture, and with A = 40.4 in Eq. (3) this yields, for isotropic turbulence,

K = 0.157 (u'/ut) ~ RL -~ (23)

The value of Ma in Eq. (20) depends upon chemical kinetic detail and associated transport properties. For a two reactant mixture and single step reaction the asymptotic analysis of Clavin and Williams 5s yields

Ma = - In Y

/3 (Le - 1) (1 - Y) f ~ / O - ~) In (1_ + x) + !

2 Y Jo x dx (24)

where 3, = (p~ - P~,)/Pu, ~ is the Zeldovich number, Le the Lewis number and x a dummy variable. Over a wide range of mixtures Ma changes near-linearly with Le.

l lOW FAST CAN WE BURN? .2.53

Equation (20) suggests that for laminar flamelet modelling of turbulent combustion, the group (K Ma) expresses the reduction in burning velocity due to stretch. Only recently have reliable values of Ma become available, through the expe~en t s of Searby and Quinard. 50 Peters and Williams have reduced the complex kinetics of methane oxidation to a four- step mechanism. With this, Rogg and Peters 61 ana- lysed weakly strained stoichiometric methane-air flames by asymptotic methods and obtained a value tbr Ma. However, pending the availability of more data, it was decided to correlate experimental values of Ut/U I in terms of the wrinkling factor u'k/ul and the flame stretch factor KLe although, for the last, ultimately it will be more logical to use KMa.

Shown in Fig. 4 are the correlations from about 1650 meastlrements of u,. ~ ' ~ The experimental points, which are not shown, correlate better with KLe than solely with K in the two separate ranges of Le (Le <- 1.3 and Le > 1.3) previously employed, z9 In flame propagation from a point source the stretch rate due to turbulence is developed more rapidly than is the flame wrinkling. For this reason K is based upon u'/A, but for flame wrinkling the value of u' on the x-axis must be that experienced at the appropriate instant by the flame front, u'k, and this shows a development in dimensionless time, 29 suggested by the reciprocal of/7 in Fig. 2. The influence of flame wrinkling in increasing ut is shown by the locus of a curve of constant KLe with increase in u'k/ut. The influence of flame stretch

in decreasing ut is shown by an increase in KLe at constant u'k/ut. These influences are reflected in the appearance of flame schlieren photographs 82 and can be quantified by the flame fractal dimension.

Flame Stretch Quenching in Turbulent Flames

Usually, at sufficiently high stretch rates the flame is quenched. Quenching stretch rates, Sq, for laminar premixed and diffusion flames have been computed for a variety of mixtures and flow geometries, with detailed chemical kinetics. Some of these for premixed flames are shown at different values of equivalence ratio, ~b, in Fig. 5(a), for two circular cross section counterflowing, symmetric streams that produced a fiat flame. Those of Stahl and Warnatz s3 for CH4-air arc shown by curve A, of Stahl et al. ~ with CH4-air and C3Hc~-air by curves B and C, respectively. Kee et at. used two formulations for CH4-air, one in which the stretch field is charac- terised by the potential flow velocity gradient, curve D, and a more accurate representation, curve E. The experimental results of Law et al. ~ are rep- resented by curves F and G for CH4-air and C3Hs- air.

Greater generality of the data in Fig. 5(a) is obtained by introducing a Karlovitz flame extinction stretch factor ~

= s o 8 z / u l (25)

20 ' - I ' I 1 6 ~ . +a ' KL+-5.3 -I + O0 /

is [ ~ 1 1 ~ ' _____-_ :~__ ~i'J' _ ~ ~ 3 0 0 r t . . . . . . . . : 12 ' "~ - " o o o . . . . . . . . . . . . /20 25 30 3s ,0

_ i ,,, -~,'11"I/./'~ Y.I;S-F~] ...... / .r6 I ~;91"llf/./~ <",.~'~--"-" _.,~.~"

s l 2~ 2,11flI-Z/A'~ ~.--- ~,o-

' ' '5 0 5 10 1 20 u~,/ut

FIG. 4. Correlation of turbulent burning velocities. Broken curves show Rt./Le, z with RL evaluated for the fillly developed r.m.s, turbulent velocity, u', equal to u'k.

254 INVITED LECTURE

SO00

4000

~- 3oool zooo !

1000

I i i i i I I

lal

0.6 0-8 1.0 1.2~ 1.& 1.6 1-8 2.0

O.S

0.~

~, 0.3

0.2

0.1

0 0.6

(b) ~ ' / C UPPER

G LOWER

F 1 I I I

0-8 1.0 1-2 1./, 1-6

FIG. 5(a). Variation of quenching stretch rate with equivalence ratio, q~, for countertlowing premixed flames. Curve A, Stahl and Warnatz, s3 CH4-air, B, Stahl et al., ~ ClI4-air, C, Stahl et al. ~ C311s-air, D, Kee et al. ~ CIl4-air, E, Kee et al. ~'~ m improved CH4-air, F, Law et al.,r'6 CtI4-air (experimental), G, Law et al., 6s Calls-air (experimental). (b) Same results in dimensionless form.

When values are multiplied by Le, the results of Fig. 5(a) transform to those of Fig. 5(b). This suggests upper and lower bounds for KuLe for the generalisations that will be attempted.

The effects of stretching in turbulent flames are more difficult to generalise. Flame geometries and flows are complex and forever changing and, in addition, there is a distribution of stretch rates. The influence of this on the volumetric heat release rate now is considered in order to derive a quenching stretch rate applicable to premixed turbulent flames. Non-premixed turbulent stretched flamelets, with detailed chemical kinetics, are discussed in refs. 67 and 68. Modelling is on the basis of the laminar flamelet assumption. The turbulent premixed flame is considered as an array of laminar flames, the volumetric heat release in which, qt (0, s), is a function of a reaction progress variable, 0, and the stretch rate, s. The former variable is the actual temperature rise at a point in the flame, normalised by

the adiabatic temperature rise. For the turbulent flame the mean volumetric heat release rate is

fo' q, = qt(O, s) p(0, s) d# ds (26)

where p(O, s) is a joint pdf. It is assumed that 0 and s are independent and p(0, s) becomes the product of the separated pdfs p(O) and p(s). For symmetric counter-flowing laminar streams there is little change in the profile of qt(O, s) with stretch until the flame extinction stretch rate, s o, is at- tained. 69 The distinction is not as clear-cut with asymmetric counter-fluw of the premixture against adiabatic combustion products.

With no change in heat release rate due to stretching until extinction and the assumptions of statistical independence and that the heat release rate is unaffected by negative stretch rates

(h = qt (0) p(O) dO (27)

More needs to be discovered about the influence of negative stretch rate. This is of a convenient form, with its separate evaluation of the stretch rate integral. It is preserved for asymmetric counter-flow, through the introduction of an equivalent turbulent extinction stretch rate, sot, such that s o in Eq. (27) is replaced by sot. This is evaluated numerically by equating Eqs. (26) and (27).

Detailed evaluation was from qt (O, s) values computed for an atmospheric lean methane-air flame, 69 the structure of which had been computed, at different stretch rates and with detailed chemical kinetics, by Dixon-Lewis and co-workers. 70-72 The pdf of stretch rates, p(s), was based upon the numerical simulations of Yeung et al., 57 with a shift in mean and r.m.s, values with KLe to accommodate the transition from the flame as a randomly orientated flame surface at KLe = 0 to a material surface when KLe > 0.258. Values of p(0) were found from experimentally generalised 73 values of first and second moments of temperature and an assumed beta function. Girimaji 7~ has estab- lished by direct numerical simulations that the beta pdf can model two-scalar mixing in stationary, isotropic combustion, through evolutionary stages from a double delta to a gaussian pdf. Values of sqt were generalised as KqtLe, following Eq. (25). Fuller de- tails are given in ref. 55. It was found that KqtLe was close to KqLe at low values of KLe, but decreased with increase in KLe. The differences between the values of KqtLe for the symmetric and asymmetric laminar flow conditions were signifi-


t >-

m

-J

rn

o

Q_

p (b).~

1

0 STRETCH RATE -

independent of KLe, up to the limit of tolerable measurements at KLe = 1, supports the validity of flamelet analysis up to this limit.

Laminar Flamelet Modelling

A laminar flamelet approach with turbulent flames comprising an array of laminar flames, with no turbulent structure residing within the flamelets, re- quires r/ -> 61. From Eqs. (1), (4) and 6t = U/Ul, an alternative expression for K is

K = (6t/~) 2 15 -0.5 (29)

FIG. 6. Stretch rate p.d.f. The shaded area represents the probability of burning, p(b).

cantly less than the spread in KqLe values shown in Fig. 5(b).

These findings are applied to the correlation of experimentally measured values of ut. In the above binary model, with either a full or a zero rate of flame propagation at any point at any instant, only a proportion of the mean flame front area will be propagating. Those parts of the flame brush with a stretch rate greater than Sqt will quench, those with one less tha n Sqt will propagate at the normal rate. Figure 6 represents the pdf of stretch rates, p(s), with 8qt the quench rate. For a given value of KLe (upon which the pdf depends) the ratio of the actual turbulent burning velocity to that which would occur if there were only flame wrinkling without quenching, Utl/Uto, must be equal to the shaded area, p(b), the stretch rate integral of Eq. (27) with 8qt as the upper limit. Values of Uto/Ut were found in this way for all the different values of u'k/ul and KLe on Fig. 4, with the previously described generalisations of Sqt evaluated numerically as ~

gqtLe = 0.166 - 0.014 KLe (28)

Values of Uto/lUl against u' /ul were generated numerically from this expression and the stretch rate integral. These are shown by curve B in Fig. 3. Although the computations ranged over a wide range of flame stretch rates, corresponding to a range of KLe from zero to unity, there was no influence of KLe upon the stretch-free burning velocity ratio Uto/ Ul. At the higher values of KLe the burning velocity becomes more difficult to measure as flame quenching makes definition of the flame front difficult. Curve B has the merit of being experimentally based and lies close to curve A, corresponding to the stretch-free theory of Bray. That the experimental burning velocities can be processed in this way to give values of Uto/Ul that are largely

so that the Williams criterion becomes K -< 0.258. It is of interest to note, in passing, that the necessary condition for the Gibson sCale that lc ~ 71 implies K -< 0.057. That the flamelet treatment of the experimental values of Ut/U I gives consistent values of Uto/Ul for K > 0.258 suggests that this criterion might be overly restrictive.

The work of Poinsot and co-workers 75'76 is of interest in this context. Direct numerical simulation of the interaction of a premixed laminar flame front with a vortex pair, with one step chemistry, showed the Kolmogorov scales to have no effects on the flame front, notwithstanding the associated high strain rates, mainly because their lifetime is too short. As a result, the authors suggested that the flamelet regime extended beyond K = 0.258 and that the boundary of the domain where distributed reaction zones are to be expected was increased by at least an order of magnitude on the u'/ul axis of the Borghi 77 diagram. This limit is shown by the curve for KLe = 5.3 on the inset portion of Fig. 4. The domain of all the curves on the main figure lies entirely within a flamelet regime, as defined by these researchers.

Support for an extended flamelet regime also is provided from comparisons of the predictions of flamelet models with experimental data. These have been developed by various groups for complex mul- tidimensional flow fields, usually with recircula- tions, where the turbulent burning velocity is not a suitable parameter to express the burning rate. The widely used k-E turbulence model, with its assumptions of isotropic turbulence and gradient transport cannot model accurately swirling and re- circulating flows and it is necessary to replace the turbulent energy, k, equation with separate Reyn- olds stress equations. 78 Furthermore, higher order turbulence closure and flows with high curvature require higher order numerical procedures. 79-sl A current Leeds model 82 uses Reynolds stress turbulence 83-86 modelling and generates Favre first and second moments of temperature fuctuations from the energy equation. Values of qt (0) in Eq.

256 INVITED LECTURE

20 , , ,1,.. ~ , ~ . - ,

r ,o, ,o / . ,7" / _ # / =_,o X X / /

i i i i

0 10 20 u ' k / u I

FIG. 7. Computed (broken curves) and measured (full curves) values of u, for fiat premixed turbulent flames.

(27) are obtained directly from a laminar flame sub- model, or by associated empirical correlations s7 of this function in terms of Ul. The mean volumetric heat release rate is derived from Eq. (27), with s o replaced by a value of Sqt consistent with Eq. (28), and the two pdfs described previously. This approach involves the solution of sixteen simultaneous partial differential equations. These cover 3 velocity components, 1 static pressure, 6 Reynolds stresses, 1 temperature, 3 heat fluxes, 1 temperature fluc- tuations and 1 turbulent dissipation rate. Most generally, solutions are in elliptic form.

This stretched flamelet, Reynolds stress model has been used to compute turbulent burning velocities of uz for a one dimensional, fiat, turbulent, premixed flame of methane-air, equivalence ratio 0.84, under initial atmospheric conditions with isotropic turbulence. For computational stability the time-dependent form of the equations was employed and solutions obtained as temporal change tended to zero. Stable solutions were not possible for low values of RL. Shown in Fig. 7 by the broken curves, are the resulting values of l t t / t t I for different values of KLe (evaluated for the cold reac- tants), while the full line curves are the experimental correlations for the corresponding values of KLe. The agreement is good.

The ability to obtain accurate flame measurements by laser diagnostics s8 is a powerful aid to model validation. Single point measurements now are supplemented by multi-planar, three-dimensional, imaging, in which laser induced fluores- cence from designated species is pre-eminent. Here, attention is directed on the validity of the currently assumed beta function distribution of temperature at a point. Shown in Fig. 8 are experimental and computed pdfs, obtained recently by colleagues Malcolm Lawes, Mike Scott and Xiao Jun Gu at Leeds. The stepped distribution is a typical tem-

10

8

6

t~

2

0

H

. NPu. I I .

500 1000 1500 2500

FIo. 8. Measured and computed p.d.f.s, of temperature (K) in turbulent flame reaction zone. H is qz (0).

perature pdf for an atmospheric CH4-air flame, equivalence ratio = 1.1, at a mean temperature of 1424 K, obtained by the coherent ani-Stokes Ra- man scattering technique s9 at a low value of KLe. The full line distribution is computed from the first and second moments of temperature and the beta function. Curve H is the volumetric heat release rate. Equation (27) shows how accurate evaluation of the mean heat release rate depends crucially upon the accuracy of p(O) when q~(O) is of significant magnitude. On the one hand, although the beta function cannot reflect the true shape of the temperature distribution, on the other hand there are considerable difficulties in accurately measuring temperature pdfs, not least because of the finite measuring volume.

The Validity of Flamelet Modelling

Despite the limitations of'assumed pdfs, flamelet modelling appears to be valid over a wide range. In addition to its compatibility with the range of measured turbulent velocities, it has predicted lift- off heights of turbulent diffusion flames, 6s combustion fields in premixed swirling flow, s2 and flame blow-off. 9~ The viability of the laminar flamelet approach has been discussed by Peters. 36 Bilger 91 has argued that a flamelet criterion appropriate to non- premixed flames is violated in many of the flames of interest.

The relationship between 17 and 8z cannot be evaluated precisely. Even the definition of 8z presents problems. For example, a thickness that included the whole of CO afterburning would be larger than one that included, say, 90% of the heat release. The present model is primarily based on heat release rate, essential for modelling the field. Whilst being suitable for modelling hydrocarbon interme- diates, 9~ it is less so for modelling CO concentra- tions, still less those of NO, where reactions continue beyond the zone of principal heat release.


\

\

/

FIG. 9. Three-dimensional perspective view of vortex lines in a homogeneous turbulent flow from direct numerical simulation. Local vorticity intensity indicated by shading, ranging from light grey (low) to black (high). From Ref. 95, courtesy of Royal Society.

Because there is a distribution of strain rates in the flow field, flame quenching will occur first in the regions of highest strain rates, whilst flamelets will survive in those of lower strain rates. ~ Flame- let burning could exist, because in such regions of reduced strairl rates the Kolmogorov length scale could be greater than ~t. The low Karlovitz turbulent flame extinction stretch factor, Kqt, of Eq. (28) reinforces this view. ~

Further support for flamelet modelling is suggested by direct numerical simulations. Solutions of the Navier-Stokes equation on a fine grid reveal the structure of turbulence. This comprises a weakly correlated random background field, consistent with Kolmogorov's hypotheses, within which are small volumes of strongly correlated, highly localised, vortex tubes, as suggested by the earlier experiments of Kuo and Corrsin, ~3 responsible for inter- mittency. Shown in Fig. 9 is a three dimensional perspective view of the vortex lines that pass through points of high vorticity intensity, computed by She

al 94,9.~ A tangent to a vortex line gives the di- e t" . rection of the vorticity vector, which is normal to the plane of the rotation. The diameter of the tubes is of the order of r /and their length of the order of A, 31 tending to confirm the structure suggested

by Tennekes, ~ in which the tubes are spaced at the order of the Taylor scale.

The rate of strain tensor is a key term in the vorticity equation. Vorticity is concentrated in the vortex tubes and the largest principal rate of strain, which is compressive, tends to align perpendicular to them, while the stretching components tend to align along the axis. 97 It is conceivable that the high strain rates in the vicinity of the strong vortex tubes will quench any flame in that region, thus confining flame propagation to regions of lower stretch rates and lower vorticity, where the flamelet criterion may be satisfied.

How Fast Can We Burn? Some Conclusions

The highest localised volumetric heat release rates occur in unstretched laminar flames. Turbulence stretches laminar flamelets to reduce the localised heat release. With appropriate flow patterns it can, however, redistribute the reaction zone to more ef- fectively utilise the available volume and increase the overall burn rate. Design for this is at the heart of mathematical models and an illustrative example is presented. A second example of a spherically propagating flame shows how stretch effects reduce the mass burning rate to a value lower than is suggested by ut.

Sculpting that creates a more compact combustion zone through swirling of the flow is illustrated in Fig. 10. This comprises flame photographs through a circular silica glass flame tube of 39.2 mm diam. and 0.5 m length, with underneath each photo- graph the contours of volumetric heat release rate 9~ computed from the stretched flamelet Reynolds stress model. The axial flow rate remains un- changed but the swirl number, S, of the CH4-air premixture (q~ = 0.84) is increased progressively from top to bottom. This was achieved by increasing the angular velocity of a matrix of hypodermic tubes within a rotating burner tube of 16 mm diam. Increase in swirl produced a more compact combustion zone.

Computed velocity vectors, not shown, indicate that with S = 0 the flame is stabilised by an elon- gated recirculation zone, resulting from the expansion step. Increase in S increases the flow diver- gence, shortens this zone and causes the flame to shorten and flatten. At S = 0.48 the flame is in contact with the wall. Further increase in swirl creates a second, central, zone of recirculation, to force the reaction zone upstream towards the central region of the matrix, as shown for S = 0.84. Here the flame is stabilised by two recirculation zones.

In turbulent flame propagation from a point source initially most of the turbulence spectrum does not wrinkle the flame, but convects it. As the flame propagates, flame wrinkling, which increases the

258 INVITED LECTURE

S=O

S=0.24

S=0.~,8

S=0.8/.

i i i i I

0 SO 100 1SO 200 250

FIG. 10. Changing flame shape and computed mean volumetric heat release rate in rotating matrix swirl tunnel burner. S = swirl number. Mean axial velocity = 15 m s -l. Contour 1:2 MW m -a, contour 2:20 MW m 3, contour 3:200 MW m -a.

burning velocity, develops due to the influence of increasingly lower frequencies. The flame stretch rate due to aerodynamic straining develops more rapidly.

Blizzard and Keck 08 assumed the mass burning rate to be the total mass of unreacted gas behind this front, divided by a turbulent reaction time, ~'. On the basis of the turbulent structure suggested by Tennekes, 96 ~" has been related to ~./U1.27,99 Al- lowance for flame stretch quenching introduces the probability of burning p(b), evaluated from the pdf of stretch rates and Eq. (28), as an additional multiplying factor. Bomb experiments by Emil Mushi at Leeds over a wide range of conditions then give r = 0.28A/ut. Together with the Ut/U I data of Fig. 4 and the temporal development data of Fig. 2, computations gave the dimensionless data of Fig. 11. These show the dimensionless temporal change in the ratio of the mass burned behind the spher-

1 i U' /U I = / * , 0

0 .8

0.6

0.2 ~

i I i

~ =4.5

j .

! I I 4 6 8 10

Dimensionless time ( t u ' l L )

F]c. 11. Spherical turbulent flame propagation from a point source: temporal and spatial variation of fractional mass behind the flame front burned, with KLe, u'/ul = 4, Pu/P~ = 6.88 and Le = 1.

ical flame kernel front to the mass entrained, rob~ me. The proportion of unburned gas can be large and this is confirmed by engine experiments 1~176 and laser sheet measurements. This proportion decreases with time but increases with KLe.

The rate of advance of the mean spherical flame front radius, r, relative to the unburnt gas is the burning velocity, ut, if the flame wrinkling is fully developed and is utk if it is partially developed. As KLe increases, these parameters cease to be a measure of the mass burning rate, as a result of the appreciable burning behind the fragmented mean flame front. The computed mass burning rate can be expressed in terms of a mass burning velocity,

1 0 1 utr, obtained by equating the total mass burning rate to 4rrr ~ Utr Pu. Dimensionless values of this are plotted for r / L = 10 on Fig. 12. These show the effect of increasing u', with all other "parameters re- maining constant. This increases KLe and at u' /u l = 10, KLe = 1.19. Also shown is the no-stretch curve B from Fig. 3. The deficit of the ut/ul curve below this curve is due to flame stretch and the deficit of (utk/Ul)r/L= lo below ut/ul is due to flame wrinkling not being developed fully. Increase in u' produces diminishing returns in terms of the increase in ut with u'. This is even more marked for (Utr/Ul)r/L=lO. For a spherical flame approaching in- finite radius, or a fiat flame, utr "~ utk ~ ut. The highest possible burn rate is indicated by curve B. This is approached as the length scale is increased, to reduce K, and as Ma is decreased,

In an engine the burn rate is controlled by the mixture, as well as the turbulence. Both exhaust gas recireulation and lean-burn in gasoline engines reduce ul. Intended compensation, in burning rate by an increase in u' may become counter-produc- tive because of the increase in KLe. io~ One remedy is partially to pre-react the fuel catalytically to pro-


10

8

6

t,

2

NO STR[TCH

Ut Ut,

i i , l i 1

2 t~ 6 8 10 12 u'/u t

FIG. 12. Spherical turbulent flame propagation from a point source: variation of mass burning and normal turbulent burning velocities with u' /ut at r~ L = 10. Also ut/u~. All parameters fixed except u'.

duce hydrogen, which increases ut, with potentially beneficial decreases in both K and also Le.

REFERENCES

1. BURKE, S. P. AND SC}IUMANN, T. E. W.: First Symposium on Combustion, p. 2, The Com- bustion Institute, 1965.

2. ZELi)OVICIt, Y. B. AND FRANK-KAMENETSKII. D. A.: Dokl. Akad. Nauk SSSR 19, 693 (1938).

3. ZELDOViCH, Y. B. AND FRANK-KAMENETSKII, D. A.: Zh. Fiz. Khim, 12, 100 (1938).

4. DAMKOIILr:R, G.: Z. Elektrochem. angew, phys. Chem, 46, 601 (1940). (English translation: Tech. Memo. natn. advis. Comm. Aeronaut, Wash. 1112 (1947)).

5. KOLMOGOROV, A. N.: Doki. Akad. Nauk SSSR 30, 301 (1941). (English translation: Proc. R. Soc. Lond. A434, 9 (1991)).

6. KOLMOGOnOV, A. N.: Dokl. Akad. Nauk SSSR 32, 16 (1941). (English translation: Prec. B. Soc. Lond. A434, 15 (1991)).

7. SPALDING, D. B.: Proc. R. Soc. Load. A434, 211 (1991).

8. SPALDING, D. B.: Phil. Trans. R. Soc. Lond. A249, 1 (1956).

9. ADAMS, G. K. AND COOK, C. B.: Combust. Flame 4, 9 (1960).

I0. DIXON-LEWiS, G. AND WILLIAMS, A.: Ninth Symposium (International) on Combustion, p. 576, Academic Press, 1963.

11. SMOOT, L. D., HECKER, ~vV. C. AND WILLIAMS, G. A.: Combust. Flame 26, 323 (1976).

12. IIAnLOW, F. II. AND NAr~YAMA, P. I.: Phys. Fluids 10, 2323 (1967).

13. SPALDING, D. B.: Thir teenth Symposium (In- ternational) on Combustion, p. 649, The Com- bustion Institute, 1971.

14. MAGNUSSEN, B. F. AND HIERTAGER, B. H.: Six- teenth Symposium (International) on Combus- tion, p. 719, Combustion Institute, 1977.

15. LIBBY, P. A., BRAY, K. N. C. AND MOSS, J. B.: Combust. Flame 34, 285 (1979).

16. BILC, ER, R. W.: Combust. Sci. Tech. 13, 155 (1976).

17. BILGEn, R. W.: Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Springer, 1980.

18. POPE, S. B.: Prog. Energy Comb. Sci. 11, 119 (1985).

19. POPE, S. B.: Twenty-Third Symposium (Inter- national) on Combustion, p. 591, The Com- bustion Institute, 1991.

20. KARLOVITZ, B., DENNISTON, D. W., KNAP- SCHAEFER, D . 1 I . AND WELLS, F . E . : Fourth Symposium (International) on Combustion, p. 613, Williams and Wilkins, 1953.

21. KLIMOV, A. M.: Zh. prikl. Mech. tekh. Fiz. 3, 49 (1963).

22. TsuJI, H. AND YAMAOKn, I.: First Specialist Meeting (International) of the Combustion In- stitute, p. 111, The Combustion Institute, 1981.

23. LAw, C. K., Zllu, D. L. AND YU. G.: Twenty- First Symposium (International) on Combus- tion, p. 1419, The Combustion Institute, 1988.

24. ABDEL-GAYED, R. G. AND BRADLEY, D.: Com- bust. Flame 62, 61 (1985).

25. TAYLOR, G. I.: Proc. R. Soc. Lond. A151, 421 (1935).

26. ARDEL-GAYED, R. G. AND BRADLEY, D.: J. Fluids Eng. 102, 389 (1980).

27. ABDEL-GAYED, a. G. AND BRADLEY, D.: Phil. Trans. R. Soc. Lond. A301, 1 (1981).

28. ABDALLn, A. Y., ALl, B. B., BRADLEY, D. AND CmN, S. B.: Combust. Flame 43, 13 (1981).

~9. ABDEL-GAYED, R. G., BRADLEY, D. ~ D I~WES, M.: Proc. R. Soc. Lond. A414, 389 (1987).

30. CHASNOV, J. R.: Phys. Fluids A3(1), 188 (1991). 31. YAMAMOTO, K. ANt) HOSOKnWA, T.: J. Phys. Soc.

Jpn. 57, 1532 (1988). 32. ARCOUMANiS, C., ENOTIADIS, A. C. AND WHITE-

LAW, J. H.: Proc. Instn. Mech. Engrs. 205, 177 (1991).

33. MANDELBnOT, B. B.: J. Fluid Mech. 72 (2), 401 (1975).

34. SREENIVASAN, K. R., RAMSHANKAR, a. AND ME- NEW:AU, C.: Proc. R. Soc. Lond. A421, 79 (1989).

35. ABDEL-GAYED, R. G. AND BRADLEY, D.: Fuel- Air Explosions (J. II. S. Lee and C. M. Guir-

260 INVITED LECTURE

aco, Eds.), p. 51, University of Waterloo Press, 1982.

36. PETERS, N.: Twenty-First Symposium (Inter- national) on Combustion, p. 1231, The Com- bustion Institute, 1988.

37. WILLIAMS, F. A.: Comhust. Flame 26, 269 (1976).

38. GOULDIN, F. C.: Combust. Flame 68, 0.9,49 (1987).

39. SCHELKIN, K. I.: Zh. Eksp. i Teoret. Fiz. 13, 520 (1943). (English tanslation: NACA TM 1110 (1947)).

40. YAKHOT, V.: Combust. Sci. Tech 60, 191 (1988).

41. WILLIAMS, F. A.: Combustion Theory, Addi- son-Wesley, 1988.

42. WIRTH, M, AND PETERS, N.: Twenty-Fourth Symposium (International) on Combustion, Turbulent Premixed Combustion: A Flamelet Formulation and Spectral Analysis in Theory. and IC-Engine Experiments, The Combustion Institute, in press.

43. ANAND, M. S. ANO POPE, S. B.: Combust. Flame 67, 127 (1987).

44. BRAy, K. N. C.: Proc. R. So<?. Lond. A431, 315 (1990).

45. BFa~r K. N. C. AND CANT, 1~. S.: Proc. R. Soc. Lond. A434, 217 (199l).

46. ChEw, T. C., BRAx, K. N. C. AND BBIrrER, R. E.: Combnst. Flame 80, 65 (1990).

47. SAID, R. AND BORC, HI, R.: Twenty-Second Symposium (International) on Combustion, p. 569, The Combustion Institute, 1989.

48. ZELDOVlCH, Y.: Combnst. Flame 39, 219 (1980). 49. LIBBY, P. A.: Combust. Sci. Tech. 68, 15 (1989). 50. KLIMOV, A. M.: AIAA Prog. Astronaut. Aeron-

aut. 88, 133 (1983). 51. CIAVIN, P.: Prog. Energy Comb. Sci. 11, 1

(1985). 52. CLAVIN, P.: Disorder and Mixing ((;uyon et

al. Ed.), p. 293, Kluwer Academic, 1988. 53. MARKSTEIN, G. H.: Nonsteady Flamc Propa-

gation, Pergamon, 1964. 54. MATALON, M.: Combust. Sci. Tech. 31, 169

(1983). 55. BRADLEY, D., LAU, A. K. C. AND LAWES, M.:

Phil. Trans. R. Soc. Lond. A338, 359 (1992). 56. ABDEL-GAYED, R. G., BRADLEY, D. AND LAU,

A. K. C.: Twenty-Second Symposium (Inter- national) on Combustion, p. 731, The Com- bustion Institute, 1989.

57. YEUNG, P. K., GIRIMAII, S. S. AND POPE, S. B.: Combust. Flame 79, 340 (1990).

58. CLAVIN, P. AND WmLtAMS, F. A.: J. Fluid Mech. 116, 251 (1982).

59. SEABBY, G. AND QUINARD, J.: Combust. Flame 82, 298 (1990).

60. PETERS, N. AND WILI.IAMS, F. A.: Combust. Flame 68, 185 (1987).

61. ROGG, B. AND PETERS, N,: Combust. Flame 79, 402 (1990).

62. ABDEL-GAYED, R. C., BRAI)LEu D. AND LUNG, F. K-K.: Combnst. Flame 76, 213 (1989).

63. STAHL, G. AND WARNATZ, J.: Combust. Flame 85, 285 (1991).

64. STAIII., C., ROGC, B. AND WARNATZ, J.: Prog. Astronaut. Aeronaut. 113, 195 (1988).

65. KEE, R. J., MILLER, J. A. AND EVANS, G. H.: Twenty-First Symposium (International) on Combustion, p. 1479, The Combustion Insti- tute, 1,988.

66. LAw, C. K., ZHU, D. L. AND YU, G.: Twenty- First Symposium (International) on Combus- tion, p. 1419, The Combustion Institute, 1988.

67. I,IEW, S. K., Banx, K. N. C. AND MOSS, J. B.: Combust. Flame 56, 199 (1984).

68. BRADLEY, D., GASKELL, P. H. AND LAU, A. K. C.: Twenty-Third Symposium (International) on Combustion, p. 685, The Combustion In- stitute. 1991.

69. BRADLEL D. AND LAU, A. K. C.: Appl. Chem. 62, 803 (1990).

70. DIXON-LEwiS, G. AND ISLAM, S. M.: Nine- teenth Symposium (International) on Combus- tion, p. 283, The Combustion Institute, 1982.

71. DIXON-LEwIS, G.: Proc. Workshop on Gas Flame Structure: Part 2, p. 3, USSR Academy of Sciences (Siberian Division), 1988.

72. DIXON-LEwIS, G.: Astronaut. Aeronaut. 113, ]66 (1988).

73. ABDALLA, A. Y., BRADLEY, D., CHIN, S. B. AND LAM, C.: Nineteenth Symposium (Interna- tional) on Combustion, p. 495, The Combus- tion Institute, 1982.

74. CIRIMAII, S. S.: Combust. Sci. Tech. 78, 177 (1991).

75. PO]NSOT, T., VEYNANTE, D. AND CANDEL, S.: Twenty-Third Symposium (International) on Combustion, p. 613, The Combustion Insti- tute, 1991.

76. MENEVEAU, C. AND POINSOT, T.: Combust. Flame 86, 311 (1991).

77. BoRclII, R.: Recent Advances in the Aero- space Sciences (C. Bruno and C. Casei, Eds.), Plenum Press, 1985.

78. ABD AL-MAsSEEH, W. A., BRADLEY, D., CAS- KELL, P. H. AND LAU, A. K. C.: Seveclth Sym- posium on Turbulent Shear Flows, p. 21, 1989.

79. BRADLEY, D., CHIN, S. B., GASKELL, P . H., LAU, A. K. C. AND MISSAGHI, M.: Interna- tional Conference on Computers in Engine Technology, p. 315, Institution of Mechanical Engineers, 1987.

80. GASKELL, P. H. AND LAU, A. K. C.: Int. J. Num. Meth. Fluids 8, 617 (1988).

81. BRADLEY, D., MISSAGHI, M. AND CHIN, S. B.: Comp. Meth. App. Meehs. and Eng. 69, 133 (1988).


82. ABD AL-MASSEEH, W. A., BRADLEY, D., GAS- KELL, e. H. AND LAU, A. K. C.: Twenty-Third Symposium (International) on Combustion, p. 825, The Combustion Institute, 1991.

83. LAUNDER, B. E., REECE, G. J. AND RODI, W.: J. Fluid Mech. 68, 537 (1975).

84. GIBSON, M. M. AND LAUNDER, B. E.: J. Fluid Mech. 86, 491 (1978).

85. JONES, W. P.: Prediction Methods for Tur- bulent Flows (W. Kollmann, Ed.), p. 379, Hemisphere, 1980.

86. BRAY, K. N. C., LIBBY, P. A. AND MOSS, J. B.: Combust. Flame 61, 87 (1985).

87. BRADLEY, D., HABIK, S. EL-DIN AND EL-SHERIF, S. A.: Combust. Flame 87, 336 (1991).

88. HANSON, R. K.: Twenty-First Symposium (In- ternational) on Combustion, p. 1677, The Combustion Institute, 1988.

89. BRADLEY, D., LAWES, i . , SCOTT, M. J., SHEP- PARD, C. G. W., GREENHALGH, D. A. AND PORTER, F. M.: Twenty-Fourth Symposium (International) on Combustion, Measurement of Temperature PDFs in Turbulent Flames by the CARS Technique, The Combustion Insti- tute, in press.

90. ABD AL-MASSEEH, W. A., BRADLEY, D., GAS- KELL, P. H., ISHIKAWA, A. AND LAU, A. K. C.: Eurotech Direct, European Engineering Re- search and Technology Transfer Congress,

C413/043. Institute of Mechanical Engineers, 1991.

91. BILGER, R. W.: Twenty-Second Symposium (International) on Combustion, p. 475, The Combustion Institute, 1989.

92. BRADLEY, D., KWA, L. K., LAU, A. K. C., MISSAGHI, M. AND CHIN, S. B.: Combust. Flame 71, 109 (1988).

93. Kuo, A. Y-S. AND CORRSIN, S.: J. Fluid Mech. 56, 447 (1979.).

94. SHE, Z-S., JACKSON, E. AND ORSZAG, S. A.: Nature 344, 226 (1990).

95. SHE, Z-S., JACKSON, E. AND ORSZAr S. A.: Proc. R. Soc. Lond. A434, 101 (1991).

96. TENNEKES, H.: Phys. Fluids 11, 669 (1968). 97. KERR, R. M.: J. Fluid Mech. 153, 3, (1985). 98. BLIZZARD, N. C. AND KECK, J. C.: SAE paper

740191 (1974). 99. TABACZYNSKI, R. J. T., TRINKER, F. H. AND

SHANNON, B. A. S.: Combust. Flame 39, l l l (1980).

100. BERETrA, G. P., RASHIDI, M. AND KECK, J. C.: Combust. Flame 52, 217 (1983).

101. ABDEL-GAYED, R. G., BRADLEY, D., LAWES, M. AND LUNG, F. K-K.: Twenty-First Symposium (International) on Combustion, p. 497, The Combustion Institute, 1988.

102. BRADLEY, D., HYNES, J., LAWES, M. AND SHEP- PARD, C. G. W.: International Conference on Combustion in Engines, p. 17, Institution of Mechanical Engineers, 1988.

COMMENTS

F. A. Williams, University o f California, San Diego, USA. What is your opinion concerning whether we will ever completely sort out the dependence of the turbulent burning velocity on the turbulence intensity in reaction-sheet regimes? The linearity is clearly there in the middle, there are various explanations for bending at high intensity, and a quadratic or 4/3-power dependence is present at low intensity when the density is constant, but a paper at this Symposium by G. Joulin suggests curvature in the opposite direction at low intensity when density changes are considered. I have suggested that, before long, all of these questions will be resolved. Do you agree?

Author's Reply. There is some confirmation of Joulin's suggestion in the experimental results (Fief. 55). The full resolution of the dependencies must embrace laminar flame stability and instability and the transition to turbulence.

S. B. Pope, Cornell University, USA. Some of your correlations show Ur "~ 4 u' and even Ur • 6 u'.

What physical process carl be responsible for propagating the front at these speeds? (For a normal distribution of velocity there is a negligible amount of fluid with velocity greater than 4 standard de- viations. )

Author's Reply. Curves A and B in Fig. 3 cer- tainly give high proportionality constants to relate ut and u'. If the lower limits on the error bands for these curves are taken this constant becomes about 3.1 for curve A and 2.7 for curve B. These compare with a value of about 1.5 given by the fractal Eq. 14, joint pdf (Eq. 17), and G equation (Ref. 42) analyses. The spectrum of turbulence relates the fractal surface area (e.g., Eq. 9) to the inner and outer cut-offs. If an inner cut-off of 1c is retained (Eq. 13) but the outer cut-off is increased from L to 5.6 L, then the proportionality constant would become 2.7. This increase in outer cut-off corresponds to a decrease in the dimensionless frequency, #, to 0.18 /TL, which reference to Figs. 1 and 2 suggests is not unreasonable. The effective frequency band for flame propagation is not known with certainty.

262 INVITED LECTURE

Prof. Toshisuke Hirano, University of Tok~jo, Ja- pan. In experiments on turbulent flames, the flame shape and turbulent reaction zone thickness de- pend not only on the turbulence characteristics of the approach flow but also on the size and/or con- figuration of apparatus. How do we apply the re- lations representing turbulent burning velocity presented in your lecture to predict the shape and reaction zone thickness of turbulent flames in practical systems?

Author's Reply. I agree that the turbulent burning velocity can be apparatus dependent. Both spatial and temporal development of the flame are important and are discussed in Refs. 27 and 29. The data of Fig. 4 allow for temporal development in that u~ in the r.m.s, turbulent velocity experienced by the flame front at a particular instant. For a steady burner flame there is an outer cut-off for the length scale.

N. Peters, RWTII Aachen, Germany. You have mentioned the effect of increasing the turbulent length scale in order to decrease flame stretch. What about decreasing the laminar flame thickness by increasing the pressure?

Author's Reply. A decrease in laminar flame thickness with increase in pressure, P, would help, but the burning velocity probably would decrease and it is the ratio 8t/u~ that is important. In addition, the relationship between )t and L (Eq. 3) is important and pressure dependent. These influences are embodied in Eq. (23). If u~ varies as P-', from this equation K varies as the (2 m -- 0.5) power of pressure. Whether pressure decreases or increases the Karlovitz stretch factor depends on the value of m. The Lewis (and Markstein) numbers will ~e independent of pressure.

Author's Reply. 1. The scatter to which you refer is becoming better understood and there is con- vergence to improved theory. I believe the turbulent burning velocity is a useful parameter in some well defined configurations. It is also most usefhl in qualitative discussion of the effects of the various parameters on burning rate. For complex flows and configurations, computational models of turbulent combustion are necessary, but it is useful to vali- date these against measured turbulent burning velocities.

2. We have no such experimental evidence. The- oretically, my paper argues that high strain rates in the vicinity of strong vortex tubes might quench the flame, confining its propagation to regions where the vortex tubes are much weaker.

J. Chomiak, Chalmers University of Technology, Sweden. In the consideration of the geometry of continuous laminar flames in turbulent flows the heat release and gas expansion effects are not accounted for. The heat release depending on the density ratio and flame inclination angle relative to the mean flow will decrease or increase the vorticity in the burned gases and this in turn will change the flame surface perturbation and flame speed. Generally the perturbation will be strongly reduced, especially for normal flames.~ The effects may explain many of the discrepancies in the experimental data, as after ac- counting for them the turbulent flame speeds in the wrinkled flame regime become dependent on the density ratio across the flame and the inclination angle of the flame relative to the mean flow vector.

REFERENCE

1. CnoMI^~:, J.: Prog. Energy and Combustion Sci- ence 5, 207 (1979).

A. Ghoniem, Massachusetts Institute of Technol- ogy, USA. 1. With the enormous scatter between predictions of different theories on the Ur/U~ vs. u ' / uL, does it still make sense to continue using the concept of turbulent burning velocity as it was orig- inally defined by Damkohler?

2. It is difficult to believe that the Kolmogurov length can be smaller than the laminar flame thickness. On the other hand, high turbulence intensity may lead to "interacting flames" which appear to behave as "'distributed reaction" zones. Do you have evidence that the Kolmogorov scale can be smaller than the laminar flame thickness?

Author's Reply. I agree with your comments about these influences. The joint pdf analysis of Ref. 43, the flamelet analysis in Ref. 44 and the expression for Markstein number, Eq. (24), all show an influence of density ratio.

Dr. Ralph Aldredge, California Institute of Tech- nology, USA. Have you looked at the relation between turbulent burning velocity and turbulence intensity at constant turbulence Reynolds number?

Author's Reply. Such loci are traced by the broken curves in Fig. 4.

Documents

How fast can we burn?