Combustion Science and Technology Characteristics of ...ronney.usc.edu/AME513b/Lecture6/Papers/LibbyCST1989-G...To cite this Article Libby, Paul A.(1989) 'Characteristics of Laminar

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [University of Southern California]On: 18 March 2011Access details: Access Details: [subscription number 911085157]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Combustion Science and TechnologyPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713456315

Characteristics of Laminar Lifted Flames in a Partially Premixed JetPaul A. Libbya

a University of California San Diegi, La Jolla, California 92093, U.S.A.

To cite this Article Libby, Paul A.(1989) 'Characteristics of Laminar Lifted Flames in a Partially Premixed Jet', CombustionScience and Technology, 68: 1, 15 — 33To link to this Article: DOI: 10.1080/00102208908924067URL: http://dx.doi.org/10.1080/00102208908924067

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713456315

http://dx.doi.org/10.1080/00102208908924067

http://www.informaworld.com/terms-and-conditions-of-access.pdf

Combust. Sci. and Tech., 1989, Vol. 68, pp. 15-33Reprints available directly from the publisherPhotocopying permitted by license only

© Gordon and Breach Science Publishers, Inc.Printed in Great Britain

Theoretical Analysis of the Effect of Gravity onPremixed Turbulent Flames

PAUL A. LIBBY University of California San Diego, La Jolla,California 92093, U.S A.

(Received June 5, /989: in final form September l l , /989)

Abstract-The Bray-Moss-Libby model of premixed turbulent combustion supplemented with a model forthe mean rate of creation of product is used to analyze the influence of gravity on horizontal premixedturbulent flames propagating either upward or downward into reactants. A parameter determining theextent of the influence of gravity involving various aerothermochemical quantities is identified. Earlierstudies of gravity free flames are extended to include a prediction of turbulent flame speed including theinfluence of gravity by adoption of the KPP-condition of a minimum speed. A linearized treatment of theeffect of gravity on the structure of premixed turbulent flames is carried out.

NOMENCLATURE

A" BI

cDrFG,G,GhH110

koK10LP

!XI

Po, PI' 1'0' 1'1Xu, Xu('Av4>.

constants arising in the expansions about c 0 (cf. Eq. (2\))progress variableDarnkohler parameter (cf. Eqs, (10) and (II))dimensionless turbulent flux variable (cf, Eq. (12))gravity parameters (cf. Eqs, (16), (19) and (21))h = h(c) (cf. Eq. (15))gravity function (cf. Eq. (16))dimensionless velocity intensity (cf Eq. (12))turblent flame speed parameterturbulent kinetic energy in the reactant streamdimensionless third moment function (cf, Eqs, (13) and (14»)integral length scale in the reactant streamdimensionless third moment function (cf, Eqs. (13) and (14))pressurex-wise velocity componentlaminar flame speedturbulent flame speedtemperatureintegral time scale of flamelet crossingsmean rate of creation of productcoordinate normal to the flame

Greek

exponent in the expansion about ccoefficients in the expansions about c 0, Idissipation termsempirical constant A = KI - 4>.flamelet crossing frequencyparameter arising from the velocity-chemical creation correlation(cf. Eq, (9»)

15

Downloaded By: [University of Southern California] At: 00:44 18 March 2011

16

(l

T

oOCJ

rp

PAUL A. LIBBY

mass densityheat release parameter (cf. Eqs. (I) and (2))

Subscripts

reactant streamproduct streamreactantsproducts

INTRODUCTION

It can be expected that horizontal premixed turbulent flames propagating upward intodense reactants can under some circumstances behave considerably differently fromsuch flames propagating downward into those same reactants. In one case thegravitational vector is codirectional with the mean density gradient while in thesecond case it is oppositely directed. Recent work (Bray et al., 1981: Masuya andLibby, 1981; and Libby, 1985) showing that mean force fields interact with densityfluctuations to influence significantly the transport processes and the turbulenceproduction in premixed turbulent flames suggests that the body force due to gravityshould likewise have a similar influence under appropriate circumstances. Thepurpose of the present study is to analyze the effect of gravity on premixed turbulentflames, their rate of propagation and structure and thereby to determine the circum-'stances under which that effect is significant.

The literature concerning acceleration effects in general and gravitational effects inparticular on such flames appears to be exceedingsly sparse. Bray and Moss (1981) usea classical gradient transport analysis to show that a premixed turbulent flame canpropagate at a considerably higher than normal velocity if subjected to an acceleration which is codirectional with the mean density gradient. Motivation for theirstudy is provided by observations of extraordinary propagation speeds for turbulentflames associated with industrial accidents. An extensive study of such flames isprovided by Gugan (1978).

Strehlow and coworkers (see Strehlow, 1983) report the results of experimentsinvolving the unconfined propagation of flames in premixed gases. A hemisphericalbag made of a thin film is anchored on the ground and filled with reactants which atthe beginning of the test are centrally ignited on the ground plane. A roughlyhemispherical flame propagates into the reactants and soon tears the confining bagfrom its constraints. Motion pictures permit the mean flame trajectory and flamecharacter to be analyzed. In a series of tests the flames are found to be weakly turbulentand to exhibit gravitational effects. The horizontal portion of the flame on the topof the hemisphere propagates into the reactants at from 1.3 to 1.7 times the speedof the vertical portion of the flame along the ground plane. Thus these data areindicative of gravitational effects but not adequate for quantitative analysis.

Lewis (1973) describes the results of an experiment involving premixed turbulentflames advancing radially inward in a tube which is rotating in a horizontal planeabout a vertical axis through its mid-point. By varying the rotational speed of the tubethe flames are subjected to various degrees of acceleration codirectional with theirmean density gradient. No turbulence is generated prior to ignition so the turbulenceis generated by the flow induced by the expansion of the product gases. Results arepresented for several hydrogen-air and propane-air mixtures in terms of observed


PREMIXED TURBULENT FLAMES 17

flame speed relative to the tube versus the centrifugal force. No significant influenceof accerleration is found until the centrifugal force is roughly 102 g. Increases beyondthis value result in apparent quadratic increases in flame speed until extinction occursbut there is sufficient scatter in the data that a linear increase in flame speed withaccerlatation can be rationalized."

Several comments concerning the results of Lewis (1973) are indicated. Theobserved flame speed is not the usual turbulent flame speed because the reactants arecompressed and moved radially inward as a consequence of the lower density of theproducts. Thus the former speed is a combination of inward reactant flow and theturbulent flame speed. In addition turbulent flame speeds are usually expressed interms of the ratio uo/k~/2 or a related quotient where Uo is the flame speed and ko is theturbulent kinetic energy in the reactant stream approaching the flame.j In the experiments of Lewis this energy is uncertain; it is not zero because of the compression ofreactants noted earlier results in boundary layers on the walls of the tube and becauseupstream, pressure-induced fluctuations arise in the reactant stream from the turbulence in the products. It is thus reasonable to suggest that the insensitivity of theobserved flame speed for low to moderate centrifugal forces may be associated withnearly constant values of the quotient uo/k~/2 while at least part of the observedincrease of that speed for high forces is due to increases in ko. The implication fromthese comments is that the data from Lewis (1973) regarding the influence of acceleration on turbulent flame speed are quantitatively ambiguous but are qualitatively inaccord with expectations.

The central theoretical problem involved in assessing the influence of gravity onpremixed turbulent flames can be focused on the following question: Given a flameundergoing a constant gravitational acceleration and propagating into reactants witha specified turbulent kinetic energy and turbulent length scale determine the speed andstructure of the flame. It is this problem which we consider here by applying theBray-Moss-Libby (BML) model of premixed combustion (cf. Bray et al., 1981 andLibby, 1985) to horizontal planar flames subject to gravity.

It benefits our subsequent exposition to set forth in a general way the regime ofpremixed turbulent combustion amenable to treatment by the BML-theory. A widelyaccepted means for delimiting those regimes is the Borghi diagram (cf., e.g., Peters(1986)) which considers the quantities identifying the turbulence and combustion: thecharacteristic velocities, u;, and UL, the rms intensity in the reactant stream and thelaminar flame speed respectively, and the characteristic lengths, 10 and IL , the turbulence integral scale and the laminar flame thickness respectively. These quantities canbe used to define several dimensionless parameters: a turbulence Reynolds numberRe == u;'/o/v; a turbulence Damkohler number Da == uJo/u;'IL and a turbulenceKarlovitz number Ka == Rel

/2IDa. In the Borghi diagram the quotients U;'IUL and 101lL

are related by lines of constant Re, Da and Ka, lines which are conveniently straightin a log-log plane. The flarnelet regime in which the BML theory applies is identifiedby Re ~ I and Da ~ I subject to the restriction that Ka < I. This restriction impliesthat the flame thickness must be smaller than the Kolmogoroff length, i.e., that theKlimov-Williarns criterion applies, and that the stretch to which the flamelets aresubjected must be suitably weak. The implication is that loosely the BML-theoryapplies to flows involving vigorous turbulence and fast chemistry but with a restraintsuch that the flamelets are not excessively stretched.

'The author is indebted 10 K.N.C. Bray for pointing out this possibility.'[But see our later discussion regarding the variability of uo/k~12.


18 PAUL A. LIBBY

The BM L-model has been successfully applied to infinite normal (Bray et al., 1981)and oblique (cf Masuya and Libby, 1981) flames and has exposed conceptuallyimportant phenomena such as countergradient and nongradient diffusion and turbulence production due to the interaction between density fluctuations and mean forcefields arising from gradients of mean pressure and Reynolds shear stress. A shortcoming of these applications is the inability of the analysis to predict turbulent flamespeeds with the consequence that a parameter, denoted 10 and related to the quotientuolkodiscussed earlier, is assumed given. Clearly this shortcoming must be remediedbefore one objective of the present study, viz to predict the influence of gravity onturbulent flame speed, can be achieved.

The inability to predict flame speed is a shortcoming of many flame theories. The"cold boundary difficulty" (cf, e.g., Williams 1985, 145-149) in laminar flame theoryis a well known example of the need to bring supplementary information to bearbefore a propagation speed is predicted. However, there is an extensive literaturerelated to these shortcomings and to remedies therefor. Of relevance to the presentanalysis is the Kolomogorov-Petrovsky-Piskunov (KPP) theory (cf, eg., Clavin andLinan 1984, Section II .2.2) which can be interpreted as follows: Of the theoreticallypossible either continuous or discrete spectrum of flame speeds only the lowest isrelevant for the steady propagation of a flame. The KPP-theory shows that for a classof reaction rate functions the speed of a flame is determined by the flow at the cold,i.e., leading, edge of the flame. Given the clear interdependence of the reactant andproduct streams in turbulent flames as suggested by both physical and mathematicalconsiderations, that Uo should be determined by conditions at one edge of the flamemay be disquieting. However, in a study of laminar flame propagation in a mediumthat reacts at its initial temperature Zeldovitch (1980) shows the importance of theleading edge of the flame in determining flame speed. Moreover, we might expect thecold boundary to serve a flame holding function in turbulence. Finally, since thedistribution of the mean rate of creation of product across a turbulent flame is likelyto correspond to that for which the KPP-theory applies, i.e., a relatively broaddistribution of the mean rate of creation with mean temperature, we here examine forthe first time the implications of the theory relative to a BML-description of horizontal premixed turbulent flames. To justify this examination we promtly summarize ourfindings: A prediction of the speed of turbulent flames free of gravitational effects inaccord with some but not all of the available experimental data is obtained. Inaddition a plausible prediction of the effect of gravity on flame speed includingextinction under suitable conditions is given. Thus the present application of theKPP-theory leads to useful results.

Two related works deserve mention. Hakberg and Gosman (1984) apply theKPP-theory to a classical gradient transport description of premixed turbulent flamessimilar to that of Bray and Libby (1976). They show that uolk612 is constant for a widerange of conditions with a value again in agreement with some but not all of theavailable data. Professor K. N. C. Bray has informed the author in a private communication of finding in unpublished work involving characteristic solutions for thepropagation of one-dimensional premixed turbulent flames whose aerothermochemistry is described by the BML-model that flame speed is determined by conditions at the cold boundary.

Our presentation is organized as follows: Employing the BML-model for theaerothermochemistry, we analyze in the next section a horizontal premixed turbulentflame including the influence of gravity. A recently developed model for the mean rateof creation of product is introduced and leads to the identification of a parametercharacterizing the influence of gravity. The describing equations lead naturally to a



(I)

(2)

parameter identifying the circumstances under which gravitational effects are significant. Application of the KPP-theory is shown to determine /0 and to lead to plausibleresults. To simplify the numerical analysis we solve the resulting equations for smallvalues of the gravity parameter and present numerical results for ranges of theparameters of applied interest. We close with a discussion of the implications of ourfindings.

2 ANALYSIS

Figure I shows the significant variables associated with a horizontal premixed turbulent propagating upward into cold reactants and depositing behind hot products, i.e.,with an acceleration vector codirectional with the x-coordinate, Ifwe change the signof g we have the reverse situation, a horizontal flame propagating downward intoreactants.

According to the Bray-Moss (BM) model of the thermochemistry the entire stateof the gas is determined by a progress variable c(x, t) with the value zero in reactants,unity in products and intermediate values in gas possibly undergoing chemical reaction. In this case the equation of state is approximated by

SL = __1_ T = I + rc1], I + rc T,

where the subscript r denotes reactants and r is a heat release parameter whose valueis readily determined from the temperature in the products, i.e., form

T...J!. = I + rT,

Values of r of applied interest are generally about six but can range from three to nine.Central to the BM-model and its derivative, the BML-model, is the assumption that

reaction takes place in thin surfaces such that states corresponding to values of the

CeO

x

FIGURE I Schematic representation of a horizontal premixed turbulent flame.


20 PAUL A. LIBBY

progress variable intermediate between zero and unity are rare and thus that theprobability density of that variable is essentially bimodal. The consequence of thisassumption is that the modeling necessary to achieve closure of a second momentformulation is greatly simplified.

(3)o

2.1 The Conservation Equations

For the flame shown in Figure 1 the describing equations in terms of Favre averagesare as follows (cf, e.g., Libby (1985» .

d __

dx ou

dp _- - + (lg

dx(4)

d (_ __ + ----;;-;;)dx (lUC (lU C (5)

d--2 un.J!.. -xdx U

(6)

--;; dp + -n--c - U W - Xdx uc

(7)

Several comments regarding these equations are indicated. The effect of gravity iscontained in the one term on the right side of Eq. (4). If conventional rather thanFavre averaging is employed, there would appear on the right ;ide of the (lU"2

equation a ou' g term in place of the u" dp/dx. Thus gravitational effects appeardifferently in the two formulations. In these equations the influence of pressurefluctuations is neglected compared with that of the mean pressure gradient; given thethinness of the flame typically on the order of the integral length of the turbulence inthe reactant stream, this influence is believed to be small and in any case no satisfactory model therefor exists. These equations apply to flows involving continuousvariations of density but in the present application the two widely different densities,those of reactants and products, are separted by thin interfaces, the laminar flamelets,which advance into the reactants at the laminar flame speed Uf.. Thus the usualmechanisms of interaction of gravity and density inhomogenieties would appear notto apply. Several of the terms obviously calling for modeling in these equations, u",c" and u"w, are readily obtained from the BM-model. In particular we have

u"ou'c"

r-(I,

e"(lC"2 c(1 - c)

r - = r +--,--------!-(I, . I + rc (8)

u"w_ ou"e" _

- w --(1 -) (c - <Pn)(lC - C



where rP. is a parameter dependent on the description of the instantaneous rate ofcreation of product w(c), taken as in our earlier studies to have the value 0.833. Themodeling of the third moment quantities in Eqs. (6) and (7) is greatly facilitated as willbe shown later by the Bray-Moss-Libby (BML) model of the aerothermochemistryof premixed turbulent combustion. We are thus left to consider the dissipation termsXu and Xu, and the mean rate of creation of product w. The modeling of these termsis also discussed later.

Under the assumption that the flame is thin we impose conditions on the solutionsto these equations at ± 00, i.e., we assume that that the flame is between two regionsof essentially uniform mean velocity and uniform turbulence, each subject to a meanpressure gradient resulting in hydrostatic equilibrium with densities Il, and IIp'

2.2 The Mean Rate of Creation of Product

In previous studies of normal premixed turbulent flames by means of the BML-modelthe calculations are carried out with the mean progress variable c(x) as the independent variable and indeed comparison between theory and experiment is convenientlyperformed in terms of this variable. In this case no model is needed for the mean rateof creation of product, i.e., for w(x) and a significant difficulty is avoided. Since thepresence of the gravitational term in Eq. (4) results in a requirement for a model forw(x) in the present analysis, it is fortunate that an approprirate one has recentlybecome avialable. An analysis of the statistical geometry of the idealized progressvariable c(t; x) which can be obtained experimentally from a temperature signal T(t;x) suitably interpreted as a two valued fucntion, T,(t; x) and Tp(t; x), is given by Brayet al. (1984) and Bray and Libby (1986). The connection between this analysis andw(x) resides in the simple relation

(9)

where WI is the mean rate of creation of product per flamelet crossing in units ofdensity and v is the frequency of flamelet crossings, i.e., the number of 0-1 and 1-0changes per unit time. The first factor on the right side depends on the distributionof the rate of strain experienced by the flamelets at various locations within theturbulent flame (cf Bray 1986). For weakly and moderately strained flamelets WF

many be treated as constant since as noted earlier the BML-theory applies when theKarlovitz number is less than unity, i.e., when the flamelets are subject to only modestrates of stretch.

The second factor in Eq. (9) is given in a recent paper by Bray et al. (1988) as

v(x) = (I -:- r)c(l - c)T(I + rc)'

where t(x) is an integral time scale determined by the time autocorrelation of the twovalued progress variable, a quantity which at present must be obtained from experiment. In the cited reference the presently available data relative to t are reviewed andit is concluded that for the normal flames considered here t can be reasonablyassumed to be a constant across the flame. We shall find later that the model for w(x)based on Eqs. (8) and (9) leads in the present application to a Damkohler parameterdefined as

(10)


22 PAUL A. LIBBY

where 10 is the integral scale of the turbulence in the reactant stream approachingthe flame and u~ == (U~2)1/2 is the r.rn.s, of the intensity in that stream, quantitiesintroduced earlier.

It will be convenient in the identification of the parameter characterizing the influenceof gravity to have an estimate for both W F and T. Although there is at present noadequate theory for either of these quantities, reasonable estimates for the normalflames under consideration are given in Bray et al. (1987) as follows:

where Ul. is the classical laminar flame speed in the chemical system under consideration. If these estimates are substituted into Eq. (10), we have

(II)

an equation we use to eliminate the parameter Dr.

2.3 The Describing Equations

The principle dependent variables arising in the analysis of Eqs. (3)-(7) are

(12)

so that 1 is a dimensionless intensity with the value 10 in the reactant stream approaching the flame as mentioned earlier and F is a dimensionless flux of the progressvariable and according to the BM-model directly related to the flux of any statevariable.

Two other auxiliary variables arise, namely

(lU") QUill e"L = ----=1 K == ---2-

Q,uo Q,uo(13)

These are dimensionless forms of the third moment quantities in the second momentequations. With approximations appropriate for the present study these are given byte BML-model as

L (I - 2cWF3

(14)

K (I - 2c)hF2

Equations (14) incorporate two approximations involving the conditional statistics ofthe velocity components within reactants and products. The first is that the conditional probability density functions (pdf's) are Gaussian so that the conditionalthird moments are zero. Recent experimental data from Driscoll and Gulati (1987)and computational results from Anand and Pope (1986) show that this assumptionis not completely correct and that the pdf's in question are skewed toward low



velocities but with unimportant consequences. A second approxiamtion in Eqs. (14)relates to neglect of the differences in the conditional intensities. Various models forthese intensities have been employed in earlier work icf. Libby, 1985)but none appearto be generally valid and in any case the experimental data concerning them is highlyuncertain. Fortunately, the differences in question do not appear to play an importantrole in these third moment quantities. The implication flowing from these twoapproximations is that the unconditional third moments are dominated by fluctuations between the conditional mean values within reactants and products, a dominance considered to be valid for present purposes.

As in earlier studies the dissipation terms are taken to describe processes associatedwith chemical reaction and not with conventional molecular effects which requiredistances in the streamwise direction considerably greater than the thickness of theflames under consideration in order to become operative. Thus

Xu

Xuc =_ (lUff e"

K, W --(I -)(}C - C

where KI and K2 are empirical constants discussed later.With these preliminaries we are able to write the principal equations to be dealt

with, namely

[(1 + re)/]' + L' + 2r/(15)

[(I + rc)F]' + K' + I = lz (I + I' - H) - (1 + F')hF(c + .Ie)

where prime now denotes differentiation with respect to e, taken to be the independentvariable. Here Z == K, - <Pn. Inearlier studies the value of.le is taken to be 0.017 whilethat of K2 is either zero or 0.25. The quantity h is introduced for notational convenience and is defined as

h == 1+ rce(1 - c)

Note that hF is finite as e ..... ±0, I so that the singularities in h are benign.These equations differ from those treated earlier only in the H-functions in the first

terms on the right side of each equation, a function defined as

Gl1/2

H - _o_h (I + F') (16)I + r

where

Gglo glo

-D"72 ULU~rUo


24 PAUL A. LIBBY

is a gravity parameter. The second, more convenient form for G arises from use ofEq. (II) to eliminate Dr. We see that the influence of gravity as reflected in G dependson the large turbulence scales via 10 and on two velocities, namely the laminar flamespeed and the r.m.s. fluctuations of the velocity in the reactant stream. Such aninfluence can thus be expected to be significant for flames with large turbulence scalesand small values of the product ULU~, i.e., for lean and rich gas mixtures in lowintensity turbulence, plausible dependencies.

If the structure of the flame in terms of the space variable x is desired, we must useEq. (5) to compute dcidx and must integrate with respect to c to obtain in dimensionless form

~ = .s: J' (I + rc)2(1 + P) dc10 uLIJ/2 1/2 (I + r)c(1 - c

(17)

where without loss ofgenerality we place the origin at c = 1/2. The appearance of thefactor c( I - c) in the denominator of the integrand assures that the integral isdivergent as c -> 0, I as required in order for x -> ± OC).

2.4 The Boundary Conditions

Equations (15) are to be solved subject to boundary conditions at c = 0, I. Therepeated appearance of the factor c(l - c) in the denominator in various termsimplies that series expansions about the ends of the range of integration is called forin order to initiate the integration at points interior to that range. For c -> 0 we let

(18)F ~ 1'oc + ...

where Po and 1'0 are determined by substitution into Eqs. (15) and by collection of theconstant terms. We find

1'0

where

H-(1 + 6) + ((1 + 6)2 - 4C~ A+ 6))"']

Po -(I + K2)1'~ - K21'~ - 3rlo - 21'0(1 + 1'0)(1 + A)6

rGIJ/26 - (I + r)(I + A)

(19)

As c -> I a more elaborate series is required. In this case we let

F ~ 1',(1 - c) + BI(I - C)"1(20)

where I", and BI are arbitrary but PI' A" 0(, and 1'1 are determined by substitutioninto Eq. (15) and collection of the constant terms and the (I - C)",-I terms.


We find

where

PREMIXED TURBULENT FLAMES

(I + A)(I - 1',)

I + (A - 1)1', - AG

- (K2 - 3)(1 + r)1'~

rG/J /2G - (I + r)A

25

(21)

The solutions proportional to (I - c)"' are eigenfunctions which are subject tobenign restrictions discussed in detail in Libby (1985). In brief for the present seriesto apply 0(, must be within the range one to two.

Equations (18)-(21) are interpreted as follows: If 10 is known, an integration ofEqs. (15) in the direction of increasing ccan be initiated from a small positive valueof c. Similarly, if 100 and B, are known, a second integration can be started for c lessthan, but close to, c = I. This interpretation provides three quantities 10 , 100 and B,to make the two independent variables I and F continuous at the terminal point ofthe two integrations. Clearly, we have an undetermined problem; it is this circumstance which leads in our earlier studies to the specification of 10 by reference toexperimental data. For the purposes of those studies, namely the exposure of countergradient diffusion, nongradient diffusion and turbulence production in premixedturbulent flames, this strategy is appropriate since it assures that the predictionsrelate to experimentally observable flames but for the present investigation an alternative strategy is required.

2.5 The Values of 10

At this juncture it is appropriate to discuss the data relative to 10 • In our earlier studieswe have taken either 10 = 0.22 as suggested by a variety of data (cf e.g., Libby 1985)or 10 = 0.16 obtained in an open flame by Moss (1980). However, a wide range ofvalues of10 has recently appeared in the literature. For example, Cheng and Shepherd(1986) find from their experiments on v-flames stabilized on a rod various values downto 10 = 0.02. In a similar but somewhat different experiment Driscoll and Gulati(1987) obtain values as low as 10 = 0.06. Anand and Pope (1987) calculate by a pdfmethod 10 = 0.44 for values of r of applied interest while Hakberg and Gosman (1984)calculate a range of values depending on the details of their analysis, namely0.35 < 10 < 0.65, and quote further experimental results within this range.* Thus a

"Gunther (1983) provides a broad review of experimental techniques and data relative to turbulentflames and inter alia discusses the significant scatter for 140 .


26 PAUL A. LIBBY

theoretician expecting to compare an a prtort prediction of flame speed withexperimental data and with earlier predictions confronts significant uncertainity.Whether this range is due to differences in flame configuration leading to deviationsfrom one dimensionality, e.g., those obviously prevailing in the fan flames of Chengand Shepher and Driscoll and Gulati, or to other aero thermochemical differences inthe various experiments is unknown. Whether the range of predicted values is due todifferences in the models used is likewise unkown. For present purposes we can onlyassume that 10 = 0.22 is a representative value for planar premixed turbulent flamesfree of gravitational effects.

2.6 Application of the KPP-Theory and the Consequences

We now examine the implications of the KPP-theory to the prediction of flame spreedin the context of the present analysis. As noted earlier the essential conclusion of thetheory is that of the spectrum of predicted flame speeds the minimum is the physicallyrelevant one for steady state flame propogation. A restriction on the maximum valueof 10 is clearly imposed by the first of Eqs. (19), namely that the radicand must benonnegative. Since a minimum flame speed corresponds to a maximum in 10 , denoted10", the KPP-theory readily yields

[G 2 J-2

loa = (I + r~(I + 2) + (I + 2)1/2 (22)

where we have selected the plus sign to obtain a single function for loa over the entirerange of permissible values of G.

Equation (22) clearly calls for comment. If G = 0, this equation gives

1+2loa = -4-

Since our earlier studies we find that 2 ~ I yields satisfactory agreement with theavailable data on the structure of premixed turbulent flames, this result is remarkablyclose to. the value we have accepted as representative for gravity-free flames. If

~ = - 2(1 + 2)1/2I+r

then lOa -> 00 implying that iio -> 0 and that extinction occurs for a premixedturbulent flame propagating downward into reactants if 101uLu~ is suitably large. Thatour theory does not predict degeneration to a laminar flame speed rather than iio = 0is a consequence of our neglect of molecular transport in Eqs. (4)-(7). Finally, asG -> 00, loa -> 0, iiolu~ -> 00, i.e., the turbulent flame speed increases indefinitelywith increases in 10/uLu~. Since increases in flame speed are expected to result inaccompanying increases in the intensity of the velocity fluctuations within the flameand in the rates of strain experienced by the laminar f1amelets, in due course thetreatment of WF as constant will be inappropriate with the consequence that someself-limiting flame speed will develop.

These are all plausible results which unfortunately cannot be compared quantitatively with experimental data, given the absence of such data, but which neverthelessencourage examination of the effect of gravity on the structure of premixed turbulent



flames with speeds determined by Eq. (22).* Despite these encouraging results wemust make the following observation: When converted to a relation for the turbulentflame speed in the usual form of uo/u~, Eq. (22) yields a linear dependence with thegravity parameter G, namely

2 ,G(I + 2)1/2 + (I + ,)(1 + 2)

(23)

Such a variant is in agrrement with neither the theoretical predictions of Bray andMoss (1981) based on a classical gradient transport calculation nor the quadraticeffect of acceleration inferred by Lewis (1973) on the basis of highly scattered data.Moreover, we pointed out earlier that because the propagation of a flame in a closetube is involved, increases in Uo lead to corresponding increases in u~' so that areinterpretation of these data tends to diminish the sensitivity of flame speed toacceleration.

The determination of /0 by Eq. (22) significantly influences details of the solutionsto Eqs. (15) without corresponding changes in the predictions relative to gravity-freeflames. The first of Eqs. (19) implies that

Yo = -t(! + (7) (24)

and as G -+ co, then Yo -+ - I, the limiting value ofYo. With G < 0 another possiblerestriction on the minimum in G arises. If

---.!£ = _ (I + 2)1/21+, (25)

i.e., at a vlaue of G prior to extinction, then Yo = O. For smaller values of Gcountergradient diffusion is predicted to occur at the cold edge of the flame with theconsequence that the flame holding capabilities of the cold edge may be consideredto be lost but whether this additional requirement is significant is unknown. Certainlythe notion that at suitably large values of -,G/(I + r) extinction occurs for premixedturbulent flames is appealing but the value at which extinction occurs is uncertain.Furthermore, if /0 is determind either from experimental data or from Eq. (22), thepresent modeling leads to a simplifed strategy of numerical solution in that theintegration from the neighborhood of e = 0 can be carried out free of unknownparameters provided 2 and K, are specified. Iteration is then confined to the determination of /", and B, so that an integration from the neighborhood of e = I leadsto continuous solutions at the terminal point of the two integrations.

2.6 Linearization of the Effect of Gravity

Equations (15) subject to the boundary conditions given by Eqs. (18)-(21) with 10given by Eq. (22) present a difficult two point boundary value problem involvingconstraints within the 0-1 range of integration. In particular I(e) and the conditionalintensities must be nonnegative for all ewith only 2 and K2 available to satisfy thisconstraint. Although we have not examined in detail the singular point of Eqs. (15),

"It is worth noting that in the course of this study an alternative to the KPP-theory, namely to imposeexplicitly a gradient transport model at the cold edge of the flame, was examined but found to impose arestriction on an empirical constant in that model rather than to determine 10 .


28 PAUL A. LIBBY

our experience with numerical results suggests that for G = 0 there exists a relativelysmall domain in J. - K 2-space yielding solutions satisfying the boundary conditionsat both ends of the integration interval and that with G # 0 this domain changes. Onphysical grounds it would be highly desirable if these parameters were fixed for allvalues of rand G rather than adjusted from case to case. Thus we confront significantnumerical and conceptual difficulties. To simplify the numerical analysis we adopt analternative strategy adequate for present purposes, namely we deal only with suitablysmall values of the gravity parameter G so that it may be considered an expansionparameter and select J. and K2 so that satisfactory results for gravity-free flames areobtained. A thorough study ofEqs. (15) for arbitrary values of G is not indicated untilappropriate experimental data on the influence of gravity on premixed turbulentflames are available.

To this end consider the following series expansions:

ICc; G) ~ (Ol/(e) + (I)/(e)G + .

F(e; G) ~ (OJF(e) + (J)F(e)G + .

and the auxiliary series related to Eqs. (22) and (16), namely

10(G) ~ (OlIo + (1)/oG + ...

I + J. r(1 + J.)'/2~ -4- - 4(1 + r) G + ...

(0)l'/2H(e; G) ~ -1-0 - h(1 + (O)F~)G +

+ r

~ H(e)G + ...

(26)

(27)

If these series are substituted into Eqs. (14) and (15), the lowest order solutions for(0) ICc) and (OlF(C) are given by these equations with the H-functions eliminated fromthe latter equations. Moreover, the boundary conditions of Eqs. (18)-(21) apply withGand {J set to zero.

The next order solutions, those for (I)/(C) and (llF(C), are given by the followingequations after some rearrangement:

(I + re - 2r(OlFY'l/, + (h(OlF)2[3(1 - 2e) + K2j<')F'

= - 3r(lll + [6(h(0)F)2 - (1 - 2e)2h(0)F(2h'(OlF + 3h(OlF')

(28)

- .: (ll/, + [I + re + h(0)F(2(1 - 2e) + e + J.)](I)F'h

- h(e + J.)(I + (O)F'W'lF -li H,


PREMIXED TURBULENT FLAMES

Again expansions about c = 0, 1 are called for; thus as c -+ 0 we take

29

where (1)/0 is known from Eq. (27). Expressions for (I)po and (I)yo are obtained byapplying the expansion in G to Eqs. (19). The corresponding expansions about c = Iare similarly obtained from Eqs. (21). Thus again the solutions for (I)I(c) and for (I)F(c)are found by means of integration from the neighborhood of c = 0, 1 with (1)81 and(1)/

00selected to achieve continuity at a common terminal point.

Finally, the infleunce of gravity on the spatial distributions of the variables isobtained from Eq. (17) with expansions in G. We obtain

x

To

U~ J' (I + ,W(1 + (O)F') -

UL

(0)/d/2 1/2 (1 + ,)c(1 - c) dc

[" .-' (I -)2 ( ((1)1)) ]uo' + ,c (I)' I (0) 0 -

+ uL(01/d/2 L2 (I + ,)c(1 _ c) F - ,(1 + F) (0)/0

dc G +

We are now prepared to discuss the results of this analysis.

6,--~--~-~--~-~--~-~

4

_0::I

<,o

1::1

2

o'---_~__~_~_._~_~__~_ ___.J

-2 o 2 4

FIGURE 2 The variation of turbulent flame speed with the gravity parameter of G: r = 6.


30 PAUL A. LIBBY

/u

128(Oll y,

X uL 0

10 Uo

OL--..d:.L"-------~---~----___:__----_:'::_----J

20

10

-4 (Oll V.X UL 0

louo

3 NUMERICAL RESULTS

We confine our attention to r = 6, a value of the heat release parameter close to thatof applied interest for most premixed turbulent flames. The parametrs Aand K2 areassigned the values - 0.133 and zero respectively. The latter is one of the two values



20r---~--------------~---------,

10

u,

o

-4 o (0', y,X uL 0

'0 Uo

8 12

FIGURE 3 The spatial distributions for various values of G: r = 6. (a) The mean progress variable. (b)The normalized intensity of the velocity fluctuations. (c) The mean turbulent flux.

used in earlier studies. The smaller, value of Ais dictated by the following considerations: The solutions require F"(O) > O. Determination of the next terms, i.e., thoseproportional to c2

, in Eqs. (18) leads to the requirement that for G = 0 A < 0, arequirement confirmed by numerical experimentation. With these values the solutionsfor G = 0 are essentially those given in our earlier studies, in particular /0 = 0.217and the distributions of (O)/(c) and (O)F(c) are little changed.

In Figure 2 we show graphically the linear variation of the turbulent flame speedin the form uo/u~ with the gravity parameter G given by Eq. (23). Extinction isindicated by either of two values depending on the criterion used: at G = - 2.17 ifUo = 0 at extinction or at G = - 1.09 if Yo = 0 at extinction.

In Figures 3a-e we show the spatial distributions of the progress variable, thenormalized intensity QU"2 /QU~2 and the mean turbulent flux F(c) for G = 0,0.2, - 0.2as given by the linearized analysis. Note that the latter two sets of distributions forG # 0 deviate from those for G = 0 in both the ordinate and the abscissa. We seefrom Figure 3a that a flame thickness defined in terms of values of cclose to zero andunity, e.g., c = 0.1 and 0.9, is increased by positive values of G. The most significantquantitative influence of gravity is seen in Figure 3b which indicates a significantincrease in the intensity of the velocity fluctuations throughout the flame and inthe product stream for G > O. This result is consistent with out intuitive expectation of the influence of buoyancy with unstable density distributions. Finally,Figure 3c suggests that there are countervailing influences of gravity on the meanflux; in the center of the flame F is reduced for G > 0 but increased elsewherein the flame. Note that since Yl < 0, gradient transport applies in the final approachto the product stream but that this occurs at distances beyond those shown inthis figure. .


32 PAUL A. LIBBY

4 CONCLUDING REMARKS

An analysis of the influence of gravity on premixed turbulent flames based on theBray-Moss-Libby model of the aerothermochemistry of such flames and on arecently developed model for the mean rate of creation of product results in thefollowing observations:

I. There is identified a gravity parameter G '" gloluLu~ which determines the extentof the influence of gravity on the properties of turbulent premixed flames. Horizontalflames propagating upward into cold reactants correspond to G > 0 and to unstabledensity distributions while those propagating downward correspond to G < 0 and tostable distributions. The quantities appearing in the definition of G indicate thatflames involving large turbulent scales in weak turbulence and in reactants with lowlaminar flame speeds are susceptible to gravitational effects.

2. To remove an ambiguity in the theoretical prediction of turbulent flame speedin general and of the influence' of gravity thereon in particular we examine theimplications of the Kolmogorov-Petrovsky-Piskunov theory and show that it leadsto plausible results for that speed for both gravity-free flames and for the influence inquestion. Extinction is predicted at either of two negative values of G; one valuecorresponds to a zero flame speed while a larger value related to the onset ofcountergradient diffusion at the cold edge of the flame and thus to the possible lossof f1ameholding. The speed of upward propagating flames is predicted to increaseindefinitiely with increases in G. Calculation of a self-limiting flame speed requiresmodification of the model used for the mean rate of creation of product.

3. A linearized treatment of the influence of gravity indicates that the thickness ofpremixed turbulent flames and the intensity of the velocity fluctuations are increasedby gravity, i.e., by G > O. However, gravity affects only slightly the mean flux of theprogress variable and thus of the various state variables.

4. Although physically plausible the absence of definite experimental data ongravitational effects on premixed turbulent flames precludes quantitative assessmentof most of the predictions resulting from this analysis.

ACKNOWLEDGEMENTS

This work was supported by the National Aeronautics and Space Administration under Grant NAG 3-654.We are indebted to Ms. lie Lin for her assistance and to Professors A. Berlad, K. N. C. Bray, R. A.Strehlow. F. A. Williams and C. W. van Alta for helpful discussions.

REFERENCES

Anand, M. S. and Pope, S. B. (1987). Calculation of premixed turbulent flame by PDF methods,Combustion and Flame 67, 127.

Bray, K. N. C., Libby, P. A., Masuya, G., and Moss, J. 8. (1981). Turbulence production in premixedturbulent flames. Combustions Science and Technology 25. 127.

Bray, K. N. C. and Libby, P. A. (1976). Interaction effects in turbulent premixed flames, Physics of FluidsIt, 1687.

Bray, J. N. C. and Moss J. B. (1981). Spontaneous acceleration of unconfined flames. First SpecialistsMeeting (International) of the Combustion Institute, The Combustion Institute (Section Francaise), 7.

Bray, K. N. c.,Libby, P. A., and Moss, J. B. (1984). Flamelet crossing frequencies and mean reaction ratesin premixed turbulent combustion. Combustion Science and Technology 41, 143.

Bray, K. N. C. and Libby, P. A. (1986). Passage times and flamelet crossing frequencies in premixedturbulent combustion. Combustion Science and Technology 47, 253.

Bray, K. N. C. (1986). Scales and burning rates in premixed turbulent flames, Proceedings NinthAustralasian Fluid Mechanics Conference. Auk/and. New Zealand.



Bray, K. N. C., Champion, M., and Libby, P. A. (1988). Mean reaction rats in premixed turbulent flames,Twenty-Second (Internotional) on Combustion. The Combustion Institute, 763.

Cheng, R. K. and Shepherd, I. G. (1986). Interpretation ofconditioned statistics in open oblique premixedturbulent flames, Combustion Science and Technology 49, 17.

Clavin, P. and Linan, A. (1984). Theory of gaseous combustion, in Ed. M. G. Verlarde, NonequilibriumCooperative Phenomena in Physics and Related Fields, Plenum Press, New York, 291-338.

Driscoll, J. F. and Gulati, A. (1987). Measurement of various terms in the TKE balance with a flame andcomparison with theory, Combustion and Flame (To appear).

Gugan, K. (1978). Unconfined Vapor Cloud Explosion, George Goodwin Ltd., London.Gunther, R. (1983). Turbulence properties of flames and their measurement, Prog, Energy and Combust.

Sci. 9, LOS.Hakberg, B. and Gosman, D. (1984). Analytical determination of turbulent flame speed from combustion

models, Twentieth Symposium (International) on Combustion, The Combustion Institute, 225.Lewis, G. D. (1973). Centrifugal-force effects on combustion, Fourteenth Symposium (International) on

Combustion, The Combustion Institute, 413.Libby, P. A. (1985). Theory of normal premixed turbulent flames revisited, Prog, Energy Combust. Sci. II,

83.Masuya, G. and Libby, P. A. (1981). Nongradient theory for oblique turbulent flames with premixed

reactants, AIAA J. 19, 1590.Moss, J. B. (1980). Simultaneous measurements ofconcentration and velocity in a open premixed turbulent

flame, Combustion Science and Technology 22, 119.Peters, N. (1986). Laminar flamelet concepts in turbulent combustion. Twenty-first Symposium

(International) on Combustion, The Combustion Institute, 1231.Strehlow, R. A. (1983). Explosion Hazards and Evaluation, Elsevier Scientific Publishing Company

Amsterdam.Williams, F. A. (1985). Combustion Theory, The Benjamin- Cummings Publishing Company, Menlo Park.Zeldovitch, Y. (1980). Flame propogation in a substance reacting at initial temperature, Combustion and

Flame 39, 219.


Documents

Combustion Science and Technology Characteristics of ...ronney.usc.edu/AME513b/Lecture6/Papers/LibbyCST1989-G...To cite this Article Libby, Paul A.(1989) 'Characteristics of Laminar