Modeling premixed combustion-acoustic wave interactions---A review

J OURNAL OF P ROPULSION AND P OWER

V ol.19,No.5,September–October2003

Modeling Premixed Combustion–Acoustic

Wave Interactions:A Review

T.Lieuwen¤

Georgia Institute of Technology,Atlanta,Georgia30332-0150

The interactions between acoustic waves and a premixed combustion process can play an important role in the

characteristic unsteadiness of combustiondevices.In particular,they are often responsible for the occurrence of self-

excited,combustion-driven oscillations that are detrimental to combustor life and performance.A tutorial review

is provided of current understanding of these interactions.First,the mutual interaction mechanisms between the

combustion process and acoustic,vorticity,and entropy waves are described.Then,the acoustic– ame interaction

literature is reviewed,primarily focusing on modeling issues.This literature is essentially organized into four parts,

depending on its treatment of1)linear or2)nonlinear analyses of1) amelets or2)distributed reaction zones.

A sizeable theoretical literature has accumulated to model the unsteady response of the laminar ame structure,

for example,the burning rate response to pressure perturbations.However,essentially no serious experimental

effort has been performed to critically assess these predictions.As such,it is dif cult to determine the state of

understanding in this area.On the other hand,good agreement has been achieved between well-coordinated

experiments and theory describing the interactions between inherent ame instabilities and acoustically induced

ow oscillations.Similarly,both the linear and nonlinear kinematic response of simple laminar ames to acoustic

velocity disturbances appear to be well understood,as evidenced by the agreement between surprisingly simple

theory and experiment.Other than kinematic nonlinearities,additional potential mechanisms that introduce heat

release–acoustic nonlinearities,such as ame holding,or extinction,have been analyzed theoretically,but lack

experimental veri cation.Unsteady reactor models have been used extensively to modelcombustionprocesses in the

distributed reaction zone regimes.None of these predictions appears to have been subjected to direct experimental

scrutiny.It is unlikely that this modeling approach will be useful for quantitativecombustion response calculations,

due to their largely heuristic nature and the dif culty in rationally modeling the key interactions between reaction

rate and the global characteristics of the combustion region,such as its volume.Several areas in need of work are

particularly highlighted.These include nite amplitude effects,modeling approaches for interactions outside of

the amelet regime,turbulent ame wrinkling effects,and unsteady vortex– ame interactions.

I.Introduction

T HE objective of this paper is to address the manner in which unsteady ow elds and,especially,acoustic disturbances in-teract with a premixed combustion process.In particular,it focuses on issues associated with modeling these interactions.Such inter-actions play important roles in the characteristic unsteadiness of turbulentcombustionsystems found in most processing,power gen-erating,and propulsion applications.The discussion is particularly motivated by the problem of combustion instabilities,which rou-tinely plague the development of combustion systems in industrial processing,1solid and liquid rockets,2;3ramjets,4afterburners,and land-based gas turbines.5?13These instabilities arise from interac-tions between oscillatory ow and heat release processes in the combustor and are manifested as large-amplitude,organized os-cillations of the combustor’s ow elds.The unsteady heat release generated by the disturbancesadd energy to the acoustic eld when it is in-phasewith the pressure oscillations.14These oscillationslead to enhancedvibration,reduced part life, ame blowoff or ashback, and even complete system failure.They generally occur at frequen-cies associated with the combustor’s natural longitudinal,radial, azimuthal,or bulk modes.However,they can also be associated with coupled convective–acoustic modes,such as may occur when a convected entropy wave or vortex impinges on the downstream nozzle and excites an acoustic wave.15Similar processes arise in

Received4October2002;revision received25June2003;accepted for publication26June2003.Copyright c°2003by T.Lieuwen.Publishedby the American Institute of Aeronautics and Astronautics,Inc.,with permission. Copies of this paper may be made for personal or internal use,on condition that the copier pay the$10.00per-copy fee to the Copyright Clearance Center,Inc.,222Rosewood Drive,Danvers,MA01923;include the code 0748-4658/03$10.00in correspondence with the CCC.

¤Assistant Professor,School of Aerospace Engineering;tim.lieuwen@ 4bf3f43e67ec102de2bd89be.Member AIAA.

pulse combustors,although the enhanced heat or mass transfer gen-erated by the oscillations is desirable in this case.16;17

The literature devoted to the associated unsteady combus-tion problems of combustion noise,inherent ame instabilities, acoustic– ame interactions,and combustion system instabilities is enormous and can only be properly addressed in its entirety by a large volume.As such,this paper makes no attempt to address all of these issues.Rather,its speci c objective is to focus on the interac-tions of premixed ames and acoustic waves without consideration of the larger system in which they occur.Note,then,that many other important issues are not addressedhere.These include, rst,acous-tic characteristics of the overall combustion system.Discussion of linear and nonlinear combustion chamber acoustics may be found in a companion paper in this issue18and numerous other references, suchas Refs.1and19–21.Second,mechanismsthroughwhich these discussedinteractionscouplewith the overallsystemto becomeself-exciting are not discussed nor are interactions of acoustic waves with solid fuels,22liquid sprays,23and non-premixed gaseous and liquid fueled ames.3;24In addition,this paper provides only lim-ited discussionof vortex–premixed ame interactions.This is not to downplay their signi cance;rather,numerous studies have shown that they play critical roles in the oscillatory behavior of premixed systems.4;9;25?27However,several thorough reviews on the subject alreadyexist,for example,see Schadowand Gutmark28and Coats.29

Note several other books and complementary review articles be-sides those appearing in this issue,which also cover related top-ics in unsteady combustion and combustion instability.Putnam’s1 work is an excellent overview of combustion-drivenoscillations in industrial systems.In addition,see Reynst,30Crocco and Cheng,3 Harrje and Reardon,24Yang and Anderson,31Natanzon,32Zinn,16 Culick et al.,33Markstein,34and Williams.35In addition,several re-view articlesbesidesthose mentionedalso treat relatedaspects,such as that of Oran and Gardner,36which focuses on kinetic–acoustic interaction processes;McManus et al.37and Candel,38;39which 765

766LIEUWEN

discuss mechanisms and active control of instabilities;Culick,40which reviews combustion instabilities in liquid-fueled systems;and Clavin,41which addresses several topics in acoustically cou-pled ame instabilities in the more general context of premixed ame dynamics.The preceding list is by no means complete.

The paper is organized in the following manner.The follow-ing background sections brie y describe the different regimes of combustion and the associated approaches for modeling acoustic wave interactions with the combustion process in these regimes.It also describes the decomposition of low-amplitude oscillations into canonicalacoustic,vortical,and entropy modes.Then,the next subsections summarize the effects of unsteady combustion on the acoustic eld and the effect of acoustic oscillations on the combus-tion process.These sections are meant to be primarily tutorial in nature.Section III,“Literature Review,”summarizes the premixed ame–acousticwave interactionliterature.Consistentwith the focus of this paper,it primarily addressesissues associated with modeling these interactions,but does include a number of experimentalinves-tigations that provide insight into the physical processes that need to be modeled.Section IV ,“Flame Transfer Function Calculations,”summarizes several analyses of the interactions between laminar amelets and acoustic waves.The paper then concludes with a dis-cussion of needs for future research and particularlyemphasizesthe need for more coordinated experimental and theoretical studies of the problem.

II.

Background

A.

Combustion Regimes

Acoustic wave– ame interactions involve the simultaneous in-teractions between unsteady kinetic, uid mechanic,and acoustic processes over a large range of timescales.Fundamentally different physical processes may dominate in different regions of the rele-vant parameter space,depending on the relative magnitudesof vari-ous temporal/spatial scales.The different regimes of interaction be-tween the combustion process and broadband turbulent uctuations can be readily visualizedwith the now familiar combustiondiagram (Fig.1).42Note that differentphysicallocationsof the ame may fall into differentregimes on this diagram.The regionsdenoted by wrin-kled and corrugated amelets correspondto situationswhere the re-actions occur in thin sheets that retain their laminar structure.These sheets become increasingly wrinkled and multiconnected with in-creasing values of u 0=S L ,where u 0is the uctuating velocity and S L is the laminar ame speed.The region denoted by well-stirred reactor corresponds to the limit where mixing occurs much more rapidly than chemical kinetics and reaction occurs homogeneously over a distributed volume.Note that there is some debate about the characteristics of the combustion process in the regions noted by well stirred reactor and distributed reaction zone.43

Consider rst the interactionsof acoustic waves and amelets in the wrinkled or corrugated regime.It is useful to consider the ra-tios of the spatialand temporal scales involved in these interactions.Note rst the following length scales:the thickness of a

laminar

Fig.1Classical turbulent combustion diagram.

methane/air ame at standard conditions varies between ?1and 10mm (Ref.44).On the other hand,the acoustic wavelength of a 100,1000,and 10,000Hz sound wave at standard conditions is 3.3m,33cm,and 3.3cm.At higher temperatures,these val-ues are even larger.Given this disparity of length scales,the ame front essentially appears as a discontinuityto the acoustic wave.As such,the uid dynamics of the ows up-and downstream of the ame can be treated separately from that of the ame structure.The situation is quite different with respect to the relevant timescales.Forming a ame response timescale from the ratio of the laminar ame thickness and ame speed leads to values of ?0.002–0.07s for methane/air ames.These are of similar magnitude as acous-tic perturbations with frequencies between 20and 500Hz.Thus,the interior ame structure and,consequently,quantitiessuch as the ame speed do not respond in a quasi-steady manner to acoustic perturbations.This issue will be further addressed in Sec.III.A.1.The preceding length scale comparison shows that amelet–acoustic wave interactions can be modeled by treating the ame front as a surface of discontinuity that divides two homogeneous regions.An unsteady analysis of the ame structure provides the necessary jump conditions coupling the ow characteristics in the regions up-and downstream of the ame.Note that this approach was previously developed for studies of the mean ow eld around the ame 45?47and ame stability,34well before its more recent ap-plication to the ame–acoustic wave interaction problem.

We willbrie y summarizethis approachnext.More detailedtreat-ments can be found in Refs.34,45,46,and 48.Consider a ame front of arbitrary shape whose instantaneous surface is described by the parametric equation f .x ;t /D 0.It is assumed that the sur-face is continuous with a uniquely de ned normal at each point.By de nition,the following relation always holds on the surface:

d f D @f @t C r f ￠d x

D 0(1)

With use of this relation,Markstein 34derives the following kine-matic equations relating the ame surface position to the local ow

and ame burning velocities:

@f

C u 1￠r f ?S 1jr f j

D 0(2)@f

C u 2￠r f ?S 2jr f j

D 0(3)

where S and u are the ame speed relative to the gases and ow velocity,respectively.The subscripts 1and 2are the value of each quantity on the up-and downstream side of the ame,respectively.Either of the preceding two expressions is often referred to as the G equation in the ame dynamics literature.The ow elds up-and downstream of the ame are coupled across the front by the follow-ing relations.34

Mass:?1S 1D ?2S 2

(4)Normal momentum:p 1C ?1S 21D p 2C ?2S 2

2

(5)Tangential momentum:.u 1?u 2/￡r f D 0

(6)Energy:

?1S 1[h 1C .u 1￠u 1/=2]D ?2S 2[h 2C .u 2￠u 2/=2]

(7)

where ?and h are density and enthalpy,respectively.The dynamics

of the thermodynamic and ow variables up-and downstream of the ame are then described by the mass,momentum,and energy conservation equations.In Sec.IV ,we provide an example calcu-lation illustrating the manner in which these equations can be used to calculate the transfer function relating ow perturbations to the ame’s heat release response.

It is useful to examine simpli ed forms of these expressions to gain insight into the relationships between small-amplitude

LIEUWEN767

Fig.2Schematic of planar ame disturbed by incident acoustic wave.

perturbationsacross the ame.Assume the ame is a nominally at,

verticallyorientedsurfacein a low Mach number ow whose instan-

taneous position is described by the equation x D .y;t/(Fig.2).It

is disturbed by an acoustic plane wave whose wavelength is much

larger than the ame thickness.Manipulation of Eqs.(2–7)leads to

the following approximate expressions coupling the axial velocity

and pressure across the ame49:

.u02=N c1/?.u01=N c1/D.3?1/M s[.S01=N S1/?.°?1/.?01=N?1/](8)

p0

2D p01(9)

where M s,°,and3are the laminar ame speed Mach number,spe-

ci c heats ratio,and mean temperatureratio across the ame.Terms

of O.M2

s /have been neglected.These expressionsshow that the un-

steady pressure is continuous across the ame,that is,to the order

of this approximation,the ame exerts no unsteadyforce on the gas. However,there is a jump in unsteady velocity across the ame,that

is,the ame looks like an acoustic volume source or a monopole. The terms on the right side of Eq.(8)quantifying this jump are related to the ame’s unsteady rate of heat release.This jump is directly proportional to the temperature ratio across the ame and the ame speed Mach number,which typically has quite low values (?0.001for a stoichiometric methane/air ame).Assume a typical acoustic scaling,that is,p0?N?cu0?N c2?0;then it can be seen that the second uctuating term on the right side and the uctuating ve-locity terms on the left are of similar magnitude.Thus,this second term results in a velocity increment across the ame that is on the order of M s and,thus,quite small.The relative magnitudes of the rst term on the right side of Eq.(8)and the uctuating velocity perturbationquantitieson the left depends on the speci c processes causing the ame speed perturbation.More detailed analyses to be discussed later suggest that ame speed perturbations caused by pressure and/or temperature uctuations are of similar magnitude

S0 1=N S1?O.p0=N p/.Thus,acousticwave ampli cation inducedby the

pressure or temperature sensitivity of the ame speed is nonzero,

but quite weak.In contrast, ame speed perturbations induced by ow or acceleration perturbations,such as through strain or curva-

ture uctuations,are potentiallymuch larger because they will scale

as S0

1=N S1?O.u01=N u1/?.1=M/O.u0=N c1/,that is,they are a factor of

the inverse of the mean ow Mach number larger than the already discussed pressure coupling terms.Thus,a velocity or acceleration coupling mechanism is potentiallya much stronger source of acous-tic ampli cation.

Flame area uctuations are another signi cant source of heat re-lease oscillations,which also arise from a velocity coupling mech-anism.The instantaneousdifferential element of ame surface area

is given by

d A FL D s

1C

@&

@y

′2

d y(10)

This term is not present in Eq.(8)because of the nominally plane geometry(@N&=@y D0/,so that the area perturbation is a second-order quantity in perturbation amplitude.However,as discussed later,it is,in general,a very signi cant source of acoustic ampli -cation,particularly because the uctuations in ame area are often generatedby velocity uctuation,that is,A0FL=N A FL?O.u0=N u/,thus, leading to a velocity coupled mechanism of acoustic ampli cation. Analysis in the turbulentcombustion literatureshows that this same term also plays a leading role in the augmentation of the burning rate in turbulent ows.50

More detailed system analysis reveals that the relative role of velocity vs pressure coupled ampli cation mechanisms is not as straightforward as these arguments might suggest.The reason for this is that ame speed uctuationsinducedby velocity/acceleration oscillations,thoughpotentiallysubstantialin amplitude,may be im-properly phased with the combustor pressure,for example,the pres-sure and velocity in an acoustic standing wave is roughly90deg,so that the resultant generation of acoustic energy is still quite weak.51 This result is in contrast to the phasing of the pressure coupled heat release oscillations.These arguments provide insight into general experimental observations that the ame exerts a relatively small ampli cation of acoustic oscillations in any given cycle of oscilla-tion.Typical measurements suggest ampli cations of1%/cycle.51 Generally,many cycles of oscillation are necessary for self-excited oscillations to achieve signi cant overall ampli cation.

Return to the combustion regime diagram in Fig.1,and consider next the opposite extreme to the amelet regime,the well-stirred reactor(WSR)regime.As will be discussed in more detail,several prior studies have suggested that ame–acoustic interactionsin this regime can be modeled by generalizing the steady WSR equations to include nonsteady effects.These unsteady reactor equations can be derived from a straightforward spatial integration of the conser-vation equations over the WSR region by assuming that all spatial quantities are uniform,35

d M

D P m in?P m(11)

d E

D P m in h in?P mh(12)

d M k

d

D P m in Y k;in?P mY k?P W k(13)

where P m and P W k are the mass ow rate and consumption rate of the k th species,respectively.M;E,and M k are the total mass,total energy,and total mass of the k th species in the reactor,respectively. The subscript“in”denotes the inlet value.The steady-state char-acteristics of the WSR are controlled by the ratio of the chemical kinetic time to the reactor residence time,given by the ratio of the mass ow rate and reactor volume,?res D P m=V.The reactor vol-ume or residence time cannot,in general,be speci ed by simpli ed analysis,becauseit is determinedby reactionand mixing rates;prior studies have used experimental and computational analysis to de-termine these quantities,which are then used as inputs to simpli ed models.52

Note that the preceding equations assume that the perturbations are spatially uniform.If the ame zone is acoustically compact, such an approximation may be adequate to describe acoustic per-turbations,but,as will be described next,entropy and vorticity dis-turbances have much shorter length scales.Fluctuations in these quantities could potentially be of much shorter length scale than the reactor size,indicating that the perturbation variables and ow strain eld are spatially distributed in the reactor.

B.Decomposition of Oscillations into Canonical Modes Consider the characteristicsof the oscillations,p0;T0;?0;u0,etc., in more detail.It is useful to decompose these uctuations into three canonical types of disturbances53?55:vortical,entropy,and acoustic.In other words,each uctuating quantity can be decom-posed as p0D p0a C p0v C p0s,?0D?0a C?0v C?0s;u0D u0a C u0v C u0s, and v0D v0a C v0v C v0s,where the subscripts a;v,and s denoteacous-tic,vortical,and entropy disturbances,respectively.Several charac-teristics of these disturbance modes should be noted.

First,acoustic disturbances propagate with a characteristic ve-locity equal to the speed of sound,whereas vorticity and entropy

768LIEUWEN

disturbances have a characteristic velocity equal to the mean ow velocity with which they are convected.In low Mach number ows, such different characteristic velocities cause these disturbances to have substantiallydifferentlength scales over which propertiesvary. Acoustic properties vary over an acoustic length scale,given by ?a D c=f,whereas entropy and vorticity modes vary over a convec-tive length scale,given by?c D u=f.Thus,the entropy and vortical mode wavelength is shorter than the acoustic wavelength by a fac-tor equal to the mean ow Mach number.This can have important implications on acoustic– ame interactions.For example,a ame whose length L FL is short relative to an acoustic wavelength,that is,L FL??a(acousticallycompact)may be of the same order of,or longer than a convectivewavelength.Thus,a convecteddisturbance, such as an equivalenceratio oscillation,may have substantialspatial variation along the ame frontand,thus,result in heat releasedistur-bances generated at different points of the ame being out-of-phase with each other.Studies often nd that a Strouhal number,de ned as Sr D f L FL=u,is a key parameter that effects the ame response to perturbations;note that Strouhal number Sr is simply a ratio of the ame length and convective wavelength,Sr D L FL=?c.A ame whose length is much less than an acoustic convective wavelength is referred to as acoustically convectively compact.

Second,acoustic disturbances,being true waves,re ect off boundaries,are refracted at property changes,and diffract around obstacles.Entropy and vorticity disturbances,on the other hand, are simply convected by the mean ow and diffuse from regions of high to low concentration.Thus,an acoustic wave impinging on the open end of a pipe may re ect back toward its point of origination with almost equal magnitude as that of the incident wave,whereas the entropy and vorticity disturbances simply convect away.(Note, however,that entropy disturbances incident upon a nozzle do gen-erate acoustic waves.)

Third,acoustic disturbancesare associated with irrotational,vol-umetric uctuations.Thus,the divergence of the acoustic velocity eld is nonzero,whereas its curl is zero,that is,r￠u0a D0and r￡u0a D0.In contrast,vortical uctuations are rotational but in-compressible,that is,r￠u0

v D0and r￡u0v D0.V elocity oscilla-tions due to entropy disturbances are generally negligible;rather,

entropy uctuations are primarily manifested by density and tem-perature disturbances.

Fourth,in a homogeneous,uniform ow,these three disturbance modes propagate independentlyin the linear approximation.How-ever, nite amplitude disturbances do interact,for example,the interaction of two nite amplitude vortical disturbances generates an acoustic disturbance.53Coupling between small-amplitude per-turbations occurs at boundaries or in regions of inhomogeneity. For example,consider a small-amplitude acoustic wave imping-ing obliquely on a rigid wall.Enforcement of the no-slip boundary condition requires that the tangential vortical and acoustic velocity components sum to zero at the wall.Thus,the incident acoustic wave excites vorticity disturbances near the wall whose magnitude is determined by the requirement that the sum of their tangential componentsis zero.This vorticity wave has a signi cant magnitude over a normal distance associated with the acoustic boundary-layer thickness.

C.Effect of Oscillations on the Flame

Disturbances in the ow and acoustic eld exert in uences on the ame in a variety of ways,each of which constitutes a potential mechanismfor self-excitedoscillations.This sectionbrie y summa-rizes the basic manner in which oscillationsdisturb the combustion process to emphasize the number of different ways through which such disturbancesmay occur.As noted by Clanet et al.,56they can be classi ed into different categoriesdepending on whether they mod-ify the local internal structure of the ame(such as the local burning rate)or its global geometry(such as its area).With reference to the earlier discussion in Sec.II.A,the former and latter categories are often associated with pressure and velocity coupled mechanisms, respectively.

The most basic effect of an acoustic wave on a ame is sim-ply causing a uctuation in the mass ow rate of reactive mixture into the ame.This mass ow uctuation is due to both the velocity oscillations,which carry the mixturetoward the ame,or densityos-cillations,which affect the mass of reactivemixture per unit volume. Also,because the ame’s position and orientation depends on the local burning rate and ow characteristics,velocity perturba-tions cause wrinkling and movement of the ame front.In turn,this modi es its local position and curvature,as well as its overall area or volume.These velocity disturbances can be acoustic or vortical in nature and,thus,propagate at the sound or ow speeds,respec-tively.To illustratethe disturbanceof a ame by an acoustic velocity disturbance,Fig.3shows a photograph from Ducruix et al.57of a simple Bunsen ame disturbed by acoustic ow oscillations gener-ated by a loudspeakerplaced upstreamof the ame.Figure3clearly shows the large distortion of the ame front,which is evidenced by the pronounced cusp in the center of the ame.This ame distur-bance is convected downstream by the mean ow,so that it varies spatially over a convective wavelength.Flame perturbations that are generated by vortical disturbancesare generally associated with large-scalevortical structuresthat are due to hydrodynamic ow in-stabilities.Figure4shows a simulatedresult of suchan interaction,58 where the ame is disturbed by vortex structures that are periodi-cally shed off the rapid expansion.Whether the ame is disturbed by acoustic or vortical velocity disturbances,the distortion of the ame front results in heat release oscillations.These heat release oscillationsare generallyattributed to the variation in ame surface area,59although they could also originate from uctuations in the burningrate due to strain rate uctuationsor

evenlocalextinction.In

Fig.3Photograph of ame disturbances generated by acoustic veloc-ity oscillations(courtesy of S.Ducruix,D.Durox and S.Candel,Centre National de la Recherche Scienti que and Ecole Centrale de Paris).

Fig.4Computation of ame disturbed by vortical structure(courtesy of S.Menon and C.Stone).

LIEUWEN 769

addition to this ame area modulation mechanism,coherent vorti-cal structuresalso cause heat releaseoscillationsthroughlarge-scale entrainment of the reactive mixture,which reacts in a sudden burst,such as when the vortex impinges on a wall.28;60

The ame’s burning rate is also sensitive to the perturbations in pressure,temperature,and strain rate that directly accompany the acoustic wave.As such,the local rate of heat release per unit area of ame or volume of reactor oscillates in time,even if the total surface area or volume of the combustion process remains xed.Furthermore,dynamic effects can cause this response to exceed its quasi-steady value by an order of magnitude.

Besides the direct effect of an acoustic oscillation on the burning rate and/or ame position,acoustic pressure and velocity oscilla-tions can also exert indirect effects.For example,disturbances in the premixing section of the combustor generate oscillations in the fuel/air ratio of the reactivemixture.61This fuel/air ratio disturbance can be generated either by a uctuating fuel ow rate (such as by uctuating the pressure drop across the fuel nozzle)or the uctuat-ing air ow rate due to acoustic velocityperturbations.These fuel/air ratio disturbances are convected by the mean ow (and,thus,have an associated convective wavelength)and disturb the ame.62

D.Effect of Heat Release on Oscillations

Having brie y consideredthe different ways in which ow oscil-lations cause combustionprocessdisturbances,we now considerthe

related problem of the manner in which combustion process distur-bances effect or generate ow oscillations.This mutual interaction is ultimatelywhat is responsiblefor self-excited,combustion-driven oscillations.

Oscillations in heat release generate acoustic perturbations.This sound generation is manifested as the broadband combustion roar of turbulent ames 63;64and,in the context of combustion instabil-ities,by discrete tones.In terms of sound generation,a ame can be thought of as a distribution of monopoles whose local source strength is proportional to the unsteady rate of heat release.The fundamental mechanism for this sound generation is the unsteady gas expansion as the mixture reacts.

To illustrate,consider a combustion process in a free- eld envi-ronment.Equation (14)is an expression for the far- eld pressure radiated by the unsteady heat release.63The effects of temperature gradientsand mean ow,which cause additionalre ection,convec-tion,and refraction of sound,are not included in this expression.p 0.x ;t /D

°?1

4?2Z

x s

1x s x @q 0.x s ;t ?j x s ?x j =N c

/@t

d x s

(14)

where q 0.x s ;t /;x ,and x s are the unsteady rate of heat release per

unitvolume,observerlocation,and combustionregion,respectively.This expression shows that the acoustic pressure at the observation point x is related to the integral of the unsteady heat release over the

combustion region,at a retarded time t ?j x s ?x j =N c

earlier.If the combustionregionis much smaller than an acousticwavelength,that is,acoustically compact,then disturbances generated at different points in the ame arrive at the observer point with essentially the same retarded time.In this case,Eq.(14)takes the form

p 0.x ;t /D °?14?N c

2R o @@t Q 0

t ?R o N c ′

(15)

where

Q 0.t /D

Z

x s

q 0.x s ;t /d x s

(16)

and R o is the average distance between the combustion region and observer.Equation (15)shows that in the compact ame case,the distribution of the heat release is unimportant;what matters is the total,spatially integrated value.

The effect of unsteady heat release in a con ned environment is related,but has some key differences.As earlier,a simpleexpression can be derived in the case where the ame region is acoustically

Fig.5Schematic of ducted ame,illustrating velocity jump induced by unsteady heat release.

compact.It is most convenient to express this effect in terms of jump conditions that relate the acoustic perturbations across the ame region:

u 02?u 01D .1=A d /[.°?1/=°N p ]Q 0

(17)

where Q 0has the same de nition as earlier and A d is the cross-sectional area of the duct.Note that this expressionassumes that the

velocity is evaluated at locations far enough up-and downstream of the ame that it is one dimensional.It shows that unsteady heat release causes a jump in acoustic velocity across the ame (Fig.5).This result could be expected,given the unsteady gas expansion that results from the heat release disturbance.Note that this expres-sion is valid only when the ame region is compact;if it is not,waves originating from different regions of the combustion process may destructively interfere.Incorporating noncompactness effects in either the free- eld or ducted case can be accomplished using a multipole expansion procedure.65;66

Besides generating acoustic waves,unsteady heat release also generatesentropy disturbances.Approximate jump conditions sim-ilar to those already given can also be derived.However,deriving these expressionsrequires the much more restrictive assumption of a convectively compact ame zone and is not pursued here.

Whether these entropy disturbances signi cantly affect the dy-namics of the combustor depends on the downstream con guration.If the combustor area remains relatively constant so that the ow passes out of the system unrestricted,such as an open ended pipe,the disturbance will simply convect out of the system and be dissi-pated in the atmosphere.This behavior is in contrast to the acoustic disturbance,which is usually strongly re ected by such a bound-ary.However,if the ow is accelerated,for example,by passing through a nozzle,the entropy disturbance generates sound.67This sound-generationmechanism is dipole in nature and due to the ac-celeration of the density disturbance,generating an unsteady force on the gas.

Even in the absence of heat release perturbations,the presence of steady heat release introduces important coupling between the acoustic,vortical,and entropy modes.First,an acoustic oscillation incident on a ame generates entropy and vorticity disturbances.34The vorticity disturbance is generated through at least two mecha-nisms.Probablymost signi cantis the baroclinicmechanism,which occurs if the wave is obliquely incident on the ame.It is due to the misalignment of the mean density gradient and the uctuating pres-sure gradient,that is,r N ?

￡r p 0D 0.Also,the unsteady wrinkling of the ame front by the acoustic perturbations causes additional unsteady vorticity generation.45

Entropy and vorticity disturbances impinging on a ame excite acoustic waves if their phase speed along the ame front (not the ow speed)is supersonic.34In low Mach number ows,this can occur if the ame is nearly orthogonal to the ow.

E.

Inherent Premixed Flame Instabilities

Even in the absence of acoustic oscillations,premixed ames may be unsteady because of intrinsic instabilities.These instabili-ties are signi cant becausetheir interactionwith externallyimposed acoustic oscillations can result in qualitative changes in the ame’s dynamics.We brie y introduce these instabilities;detailed discus-sions and analysis may be found by Williams 35or Clavin 68.

Williams 35suggests that intrinsic premixed ame instabilities with one-step chemistry can be grouped into three basic categories:body force,hydrodynamic,and diffusive–thermal.The body force

770LIEUWEN

instability is analogousto the classical buoyant mechanism where a heavy uid restingabovea lighterone is destabilizedby the actionof gravity.In the same way, ames propagatingupward divide a higher and lower densityregion and,thus,are unstable.Similar instabilities can be induced by acceleration of the ame sheet,either through a variation of the burning velocity or an externally imposed ow per-turbation.As will be discussed further in the following section,the latter mechanism plays an important role in certain acoustic– ame interaction phenomenon,where the acceleration is provided by the acoustic velocity eld.

The hydrodynamic,or Darrius–Landau instability,has an under-lyingmechanismthat is purelyhydrodynamicin nature.Speci cally, a front dividing two gases of different density that propagates at a constant velocity normal to itself with respect to the more dense gas is unstable for all wavelengths of perturbation.35This mech-anism is due to gas expansion across the ame,which causes the incident ow streamlines to diverge/converge in front of a ame disturbancethat is convex/concave to the unburned gas.The result-ing ow divergence/convergence causes the ow to locally decel-erate or accelerate,respectively,causing the disturbance to further grow.Whereas initial analyses that assumed constant ame speed and neglected gravity effects found this mechanism to be intrinsi-cally destabilizing,more recent analysis has modi ed this result. Speci cally,the dependence of the local burning velocity on the radius of ame curvature can stabilize short wavelength perturba-tions.Longer wavelength perturbationsare stabilized by gravity for downward propagating ames.

The diffusive–thermal instability is due to the effect of ame front curvature on the rate of diffusion of heat and reactive species. For example,a disturbance that causes the front to bulge toward the unburned gas results in defocusing of conductive heat ux that heats the incoming mixture.In the same way,it results in focusing of the diffusive ux of the de cient reactant into the ame.If the heat conductivityand limiting reactantdiffusivityare equal,that is,a unity Lewis number,Le D D T=D M,then theseeffectsbalanceso that the burning velocity is unaltered.For mixtures with Lewis numbers less than about unity,this mechanism is destabilizing.In addition,in multiple reactant systems,variations in the relative diffusion rates of reactants can introduce variations in mixture composition at the ame,also causing instability.

III.Literature Review

Now that some background on the mutual interactions between acoustic waves and premixed ames has been provided,the fol-lowing section reviews prior work on the subject.The literature on the problem treats a variety of experimental hardware and model-ing approaches that address the full spectrum of very fundamental, idealized con gurationsto realistic geometries.The organizationof these studies is as follows.First,studies of the response of ames to low-amplitude disturbances,such as linear theoretical analyses, are described in Subsecs.III.A and III.B.Flame response analyses and measurements of large amplitude perturbationeffects follow in Subsec.III.C.In addition,the low-amplitude response studies are further grouped into the two sections,“Fundamental Studies”and “Realistic Geometries.”Finally,the“Fundamental Studies”section is further subdivided by the combustion regime of consideration, amelets,or WSRs.

A.Fundamental Studies:Linear Response

1.Flamelet Studies

A large number of studies have considered various fundamental aspectsof the acousticwave–premixed ameletinteractionproblem. Manson69appearsto have rst calculatedthe re ection and trans-mission coef cients of a planar ame.The analysis is quite similar to Rayleigh’s14study of wave re ectionsfrom a temperaturediscon-tinuity.Chu70performed a more detailed investigation of a similar problem;his investigatedgeometry is shown in Fig.2.It consists of an in nitely long,planar front that is disturbed by a normally im-pingingacousticwave.Chu useda linearizedversionof the matching conditionsgivenin Eqs.(4–7)and did not considerthe internal ame structure.He calculated the re ection and transmission coef cients of a planar ame front and also showed that acoustic waves are gen-erated or ampli ed by changes in ame speed,heat of reaction of the reactive mixture,entropy of the incoming mixture,or speci c heats ratio.

This result was generalized by Lieuwen,49who considered the interaction between an obliquely incident acoustic wave on a ame front.This generalization introduces the additional phenomena of ame wrinkling and vorticity production into Chu’s problem.70In addition,he49calculated the net acoustic ux out of the ame, thereby allowing for a calculation of the ame’s ampli cation or damping of the disturbance.Energy added to the acoustic eld by unsteady heat releaseprocessesresultsfrom the unsteady ux of un-burnedreactantsthroughthe ame by uctuationsin either the ame burning velocity or density of the unburned reactants.Acoustic en-ergy is dissipated by the transfer of acoustic energy into vorticity uctuations that are generated at the ame front by the baroclinic vorticity production mechanism.Depending on the temperature ra-tio across the ame,magnitude and phase of the ame burning velocity response,and angle of incidencebetween the wave and the ame,the acoustic disturbance could be damped or ampli ed.

A number of analyses,such as those of McIntosh,71McIntosh et al.,72Peters and Ludford,73Van Hartenet al.,74Kellerand Peters,75 and Ledder and Kapila,76have analyzed the internal structure of a at ame perturbed by an acoustic wave using high activation en-ergy asymptoticsand single-stepkinetics.These results quantify the responseof the ame’s burningvelocityto the unsteadypressureand temperature variations in the incident wave.Many of these results are summarized by McIntosh,77who emphasizes the different char-acteristics of the interaction depending on the relative magnitudes of the length and time scales of the acoustic wave and ame pre-heat and reaction zone.Following McIntosh,77de ne the following ratios of these length scales and timescales:

?′

diffusion time

;N′

acoustic wavelength

(18)

These ratios are related by the Mach number of the ame burning velocity,

M s D S L=c u D1=?N(19) Also,de ne the dimensionless overall activation energy,

μD E a=RT b(20)

where E a is the overall activation energy,R is the gas constant,and T b is the burnedgas temperature.He identi es four differentregimes based on the relative magnitudes of these parameters.

1)Nà1=M s,that is,??1;acoustic wavelengthis much longer than the ame thickness and the ame responds in a quasi-steady manner to acoustic disturbances.

2)N?O(1=M s/,that is,??O(1);acoustic wavelength is much larger than the ame thickness,but acoustic and ame response times are commensurate.

3)N?O(1=μ2M s/,that is,??O(μ2/;fast timescale acoustic oscillations affect inner reaction zone.Spatial pressure gradients are not important in combustion zone.

4)N?O(1),that is,??O.1=M s/;pressure gradientsoccur over same length scale as ame thickness.

The regime of most interest to unstable combustors is likely cases 1and2.For example,a frequency of400Hz roughly corresponds to a??1value in a stoichiometric methane/air ame.For these cases,McIntosh derives the following expression relating the mass burning rate and acoustic pressure perturbation71:

m0

′?

p0

′

′àD2μ.°?1/

°

￡.?i?/.s?1C1=3/(21)

LIEUWEN 771

where we assume an exp(?i !t /time dependence,Le is the Lewis number,and

s D p r D p 1?4i ?(22)Figure 6shows the dependenceof the normalized mass burning rate

response o.It increasesroughlylinearlywith μand as the squareroot of dimensionless frequency ?and ame temperature jump,3.The Lewis number dependence is quite weak for Lewis number values near unity.This result illustratesthat the mass burning rate response is substantiallylargerthan its quasi-steadyvalue in the physicallyin-teresting ??O(1)case.Althoughthese analysesare most relevantto the amelet combustionregimes,McIntosh suggeststhat they could also be applied to the distributedreaction regime,where the laminar ame thickness is replaced by that of the thickened preheat zone.Although not speci cally addressing the acoustic problem,re-lated work to determine the response of the mass burning rate to un-steady stretch has been studied analytically by Joulin 78and Huang et al.79and computationally by Im and Chen.80In the steady case,linearizedanalysissuggeststhat the burningvelocitydependenceon curvature and hydrodynamic stretch combines into a single term.68However,Joulin’s 78analysispredictsthat these two terms have a dif-ferent frequency response in the unsteady case;the unsteady strain effect diminishes with frequency,whereas the unsteady curvature term is independent of frequency.The former prediction is consis-tent with Im and Chen’s 80calculations,which predicted that the ame speed response to strain rate uctuationsattenuatesas the fre-quency exceedsthe inverse of the characteristic ame response time (Figure

7).

Fig.6Normalized mass burning rate response to acoustic pressure perturbations o(adapted from McIntosh 71).

Fig.7Dependence of instantaneous ame consumption speed S c on instantaneous Karlovitz number at several frequencies of oscillation;calculation performed at á=0.4for a hydrogen/air ame,where D ,k ,and S L are the diffusivity of oxygen,instantaneous stretch rate,and un-stretched laminar ame speed,respectively (courtesy of Im and Chen 80).

Experiments showing that ames propagating downward from an open end of a tube emit spontaneous acoustic oscillations 81?83has also motivated a number of analyses,such as those of Clavin et al.,84Markstein,34;85Clanetet al.,56Searby and Rochwerger,86and Pelce and Rochwerger.87Similar behavior has also been reported in a Taylor–Couette combustor by Vaezi and Aldredge.88In a typical experiment,83the reactive mixture is ignited at the open end of a vertical tube.Photographs obtained by Aldredge of the resulting sequence of ame characteristicsare shown in Fig.8.As the ame propagates down the tube,it develops a cellular shape due to the inherent Darrieus–Landau instability (Figure 8a).For suf ciently lean mixtures with low ame speeds,the ame propagates down the tube without generating sound.Searby notes,however,that for ames with burning velocities in the 16–25cm/s range,a primary acoustic instability occurs.83Measurements and analysis indicate that this primary instability is generated by acoustic oscillations with a frequency associated with a natural acoustic mode of the duct that modulates the area of the cellular structures and,thereby,the heat release rate.These oscillationsgrow rapidly and eventually result in a remarkable restabilization of the ame front,where the cellular structure disappears and the ame reverts to a nearly planar front (Fig.8b).Measurements indicate that the ame’s propaga-tion speed slows down substantially due to the reduction in surface area and has a value that is close to the laminar burning velocity.83In addition,the growth rate of the oscillations reduces markedly.Analysis indicates that this behavior is due to stabilization of

the

a)

b)

c)

Fig.8Sequence of ame front characteristics as it propagates down-ward in a tube:a) ame wrinkled by Darrieus–Landau instabilitymech-anism,b)planarization of ame by low-amplitude velocity oscillations,and c)parametric instability induced by large-amplitude acoustic oscil-lations (courtesy of R.C.Aldredge).

772LIEUWEN Darrieus–Landau ame instability by the oscillatory acceleration imposed by the acoustic eld(see Refs.86and89).

For ames with laminar burning velocities greater than about

25cm/s,this primary instability is followed by a more violent sec-ondary instability with an even larger growth rate than observed

in the very initial stages(Fig.8c).The nearly planar ame devel-

ops small,pulsating cellular structures whose amplitude increases rapidly.These cellular structures oscillate at half the period of the acoustic oscillations.This parametric acoustic instability is due to

the periodic accelerationof the ame front by the unsteady velocity

eld,which separates two regions of differing densities.With in-creased amplitudes,these organized cellular structures break down

into a highly disordered,turbulent front.

In the case where the ambient ow eld is highly turbulent,V aezi

and Aldredge90did not nd any evidence of occurrenceof a primary acoustic instability.However,they did note that the parametric in-stability still appeared.In addition,they noted that for suf ciently

high-turbulence levels,the appearance of the parametric instabil-

ity did not result in additional acceleration of the ame front.This

is in contrast to the case where the ambient ow eld is quiescent, where the parametric instability results in substantial ame front acceleration.

Markstein34 rst recognized that the period doubling behavior occurring during the parametric instability was indicative of a para-metrically pumped oscillator.He showed that this behavior could

be described by the following parametric oscillator equation which

he derived on phenomenologicalgrounds:

A d2y.k;t/

d2

C B d y.k;t/

d

C[C0?C1cos.!t/]y.k;t/D0(23)

where A;B,and C are coef cients de ned in Ref.86,k is the wave number of the perturbation,and!is the frequencyof imposed oscil-lation.This equation has subsequentlybeen derived from a rigorous application of laminar ame theory.86The damping coef cient B is always positive,whereas the coef cient C0is negative if the planar ame front is nominally unstable.In the case where C0is nega-tive,this equation has the properties that the solution is unstable in the absence of imposed oscillations,that is,C1D0,is stabilized in the presence of small but nite amplitude perturbations and is destabilized in the presence of large-amplitude parametric oscilla-tions.Searby and Rochwerger86and Bychkov89have shown that the predictions of these analyses are in excellent agreement with measurements.

The cited analyses primarily focused their attention on nominally at,laminar ames.Other studies have investigated the character-istics of wave interactions with wrinkled,turbulent ames91;92with randomly moving fronts.The existing theory is restricted to high-frequency disturbances because it requires that the radii of ame wrinkling be large relative to the acoustic wavelength.It predicts that a coherent,harmonically oscillating wave incident on a ame generates scattered coherent and incoherent disturbances.The in-coherent disturbances have a distributed spectrum that is roughly symmetric about the incident wave frequency f i.These incoherent disturbances are due to the randomly moving ame front,result-ing in Doppler shifted scattered waves.These predicted qualita-tive characteristicshave been con rmed in subsequentexperiments (Figure9),which reveal that the scattered wave spectra has a nar-row band peak at the incident wave frequency with distributed side bands.93

It is the characteristics of the coherent re ected and transmitted waves that are of most interestto this paper.The theory predicts that the wrinkled characteristicsof the ame act as a source of damping of coherent acoustic energy.This energy is fed from the coherent to the incoherent eld.The mechanism for this damping is primarily kinematic in nature as the phase of the scattered waves differ from point to point along the ame front because of differences in dis-tance the wave travels before impinging on the ame and re ecting. Phase mismatch between disturbances originating from different points of the ame results in destructive interferencebetween these different waves.In general,the theory predicts that the character-istics of the scattered eld depend on the statistical distribution

of Fig.9Measured spectra of scattered eld excited by7.5and15kHz incident waves(adapted from Lieuwen93).

Fig.10Dependence of scattered incoherent power on ratio of ame brush thickness?andacousticwavelength?(adapted from Lieuwen93).

the ame front about its average position.In the limit where the scales of ame wrinkling are much smaller than a wavelength,only the variance of the ame position,that is,the turbulent ame brush thickness,is important and the coherent scattered eld is damped by the amount1?2.k?cosμi/2,where k;?,andμi are the acous-tic wave number, ame brush thickness,and relative angle between the incident wave and average ame position.Besides the reduc-tion in amplitude,the coherent eld has a phase offset relative to its smooth surface value if the ame position is asymmetricallydis-tributedabout its mean position.This result is of lower orderthan the amplitude effect,however,because it is proportionalto.k?cosμi/3. Although this predicted damping of coherent oscillations has not been experimentallyveri ed,experimentshave con rmed the com-plementary theoretical prediction that the scattered power in the incoherent eld increases with decreasing acoustic wavelength for k?values less than or near unity and saturates for large k?values,(Fig.10).

We conclude this section with several summary remarks.The in-teractions of an acoustic wave with a planar ame has received ex-tensive theoretical attention(e.g.,Refs.49and69–80),particularly in regard to the ame’s chemicalkinetic,pressurecoupledresponse. However,no complementary experimental investigations to assess these predictions critically appear to have been undertaken.This is due to the dif culty of actually assembling an experiment that emu-lates the simple,fundamental geometries that are most amenable to analytical attack,for example,analyses typically assume in nitely long at ames or that re ected and transmitted waves propagate away without subsequent re ections.Thus,in any real experiment, wave diffractionfrom ame edges or wave re ection from hardware must be accounted for.Given the very slight ampli cation,on the orderof1%,that an acousticwave may experiencedue to ame inter-actions,small errors in correcting for re ection,diffraction or other effects can easily render any quanti cation of wave ampli cation completely useless.Nonetheless,these inherent dif culties do not

LIEUWEN773

detract from the reservation that must be placed on a fairly mature body of theory that has not been subjectedto experimental scrutiny. Researchers and theoreticians need to devise experiments to cir-cumvent the dif culties highlighted,as well as theory that is more amenable to realization in the laboratory.Some resolution to this problem could also be achieved by detailed computational analy-ses,which could circumvent many of the dif culties noted earlier. On the other hand,the problem of acoustic wave interaction with inherent instabilities of a nominally at ame appears to be well understood.This is one of the few problems in the eld where good quantitativeagreement has been achieved between experiments and theory.

2.Distributed Reaction Models

We next consider combustion–acoustic interactions in the WSR regime(Fig.1).Unsteadyreactormodels are routinelyused to study kinetically driven instabilities in multistep chemical mechanisms and extinction and ignition phenomenon.94;95The interactions be-tween acoustic oscillationsand a distributed reaction zone has been similarly modeled.Richards et al.96and Janus and Richards97used such models to describe the dynamics of a pulse combustor and a lean,4bf3f43e67ec102de2bd89being the unsteady WSR equations,they determined the response of the unsteady heat release to perturba-tions in pressure,reactants mass ow rate,or equivalenceratio.The model predicted the dependenceof combustor stability behavior on mean operating parameters,such as inlet temperature or ow rate. Using a similar model,Lieuwen et al.98calculatedthe unsteadyheat release response to equivalence ratio perturbations.These analyses assumed that the reactor residence time was xed,that is,oscilla-tions in reactive mixture composition or temperature did not affect the residence time.The effects of neglecting these variations could be substantial in certain cases,however,as will be discussed here and in Sec.IV.

Park et al.99studied the dynamics of a reactor with a uctuating residencetime.They show that the phase between reactor residence time and heat release oscillations qualitatively changes above and below the point of maximum reactor heat release.They present an intuitive method of explainingthis result(Fig.11).The steady-state reactor conditions are determined by the point where the rate of heat release by chemical reaction equals the net rate of convection of energy out of the reactor.The dependenceof these two rates upon reactor temperature are indicated by the solid and dashed lines in Fig.11,respectively.Energy convection rate curves are drawn at several residence times.Note that maximum reaction rate occurs at a certain value of residence time.Consider the effect of small perturbations in residence time at mean residence times above and below this maximum value.As indicated in Fig.11,these perturba-tions result in reaction rate oscillationsthat are out-of-and in-phase with the residencetime perturbation.Thus,Park et al.99note that,as the combustion process approaches the blowout point,it will pass through this point of phase reversal.It can be anticipated that a qualitativechange in combustion dynamics will result.Whereas the authors original analysis utilized one-step kinetics,they have also

Fig.11Steady-state reactor solution occurs at high-temperature in-tersection of two points;Dependence of——,rate of reaction and----, convection on reactor temperature(after Park et al.99).performed a similar analysis on a four-step reduced mechanism and obtained similar results.100

To extend the reactor approach to situations where the ame was convectively non-compact and,thus, ow disturbances varied sub-stantially over the ame region,Lieuwen et al.62treated the com-bustion zone as a distribution of in nitesimal,independent reac-tors whose input conditions were given by that of the local ow at the associated location.Although this heuristic treatment allowed for a consideration of important noncompactness effects,its basic assumption of reactor independence is questionable.For example, amelet studies show that disturbancesgeneratedat one point of the ame convect downstream and affect its dynamics at other points. However,as is the more general problem with these reactor-based approaches,it is not clear how to incorporate these interaction ef-fects in a rational manner.

None of these reactor-based models has been subjected to di-rect experimental veri cation.(By direct veri cation,we preclude qualitative comparisons between experiments and a larger stability model within which the reactor serves as a submodel.)As such,the accuracyof these models is questionable.Furthermore,the heuristic nature of reactor models makes it unlikely that they can be used for quantitativepredictionsof combustion process response to acoustic perturbations.Two major conceptual issues associated with reac-tor models should be emphasized.First,reactor models are known to be useful for correlating the steady-state blowout characteristics of high-intensity ames.52It is likely that the intense recirculation regions that stabilize these ames have distributedreactorlike prop-erties,hence,the success in reactor models in correlating blowout behavior.However,in many cases,it is also possible that the ma-jority of the ame has amelet-like properties.In these situations,a model that is only valid in a small region of the combustionprocess will not accurately describe its overall dynamics;that is,unsteady reactor models do not necessarily describe the overall dynamics of a combustion region,even if its blowout limits can be satisfactorily correlated with a steady state prediction.

Second,as already alluded to,it will be very dif cult to model rationally the interactions between separate reactorlike regions in space that see different disturbance values(such as mixture com-position),as well as the response of the reactor residence time to perturbations.This secondpointseems particularlysevere,as can be illustrated by the following points.Consider two identical reactors fed by the same fuel ow rates,but at differentpressures,p1and p2. Under the assumptionthat the all of the fuel is reacted to form prod-ucts,it is clear that the total heat release of both reactors is also the same.Now,assume that the pressure in either reactor oscillates in time between the two values,that is,p.t/D p1C.p2?p1/sin!t. Under the assumption that the frequency is low enough,the reac-tor will respond in a quasi-steady manner,implying that the total rate of heat release remains constant.A similar argument can be made for a reactor disturbed by other uctuations,such as temper-ature.Only uctuations in the heat content of the inlet fuel stream will cause a quasi-steady uctuation in reactor heat release.What is happening?Clearly,the changes in pressure or temperaturein u-ence the volumetric reaction rate,for example,heat release/volume. However,in the quasi-steady case,increases in reaction rate are accompanied by reductions in overall reaction volume,that is,the same amount of heat is released,but over a smaller volume.This discussion shows that quasi-steady perturbations that do not affect the heating content of the inlet stream do not introduce uctuations in overall heat release.Thus,any uctuation in heat release that does occur is a dynamic effect,that is,the pressure perturbations referred to could potentially cause heat release oscillationsat suf -ciently large frequency!.In this case,it is then necessary to model the dynamics of the global reaction zone response to the perturba-tion.We will show in Sec.IV that an analogous situation occurs in the amelet regime,where quasi-steady uctuations in ame speed do not cause the global heat release to oscillate because of the ac-companying oscillations in ame area.The difference is that these coupled dynamics can be reasonably modeled from rst principles in the amelet case;how this would be done in the reactor case is uncertain.

774LIEUWEN

To conclude,although reactor models are attractive due to their simplicity,they are not currently useful as a quantitative predictive tool because the relevant dynamic interactions cannot be modeled in a rational manner.As such,for the foreseeable future their utility will at best be restricted to semi-empirical correlations.

B.

Realistic Geometries:Linear Response

This section considers the interaction of ames in more realistic geometries,such as ducted ames stabilized at rapid expansions or bluff bodies.The principle distinction between this problem and those already considered is the introduction of a length scale asso-ciated with the physical size of the combustion region.

Numerous phenomenologicalmodels of the ame transfer func-tion have been reported,for example,Becker and Gunther,101Merk,102or Matsui.103Marble and Candel 104appear to have per-formed the rst physics-based calculation of the ame sheet’s dy-namic response to acoustic perturbations.Their work was followed with similar studies by Subbaiah 105and Poinsot et al.106and incor-porated into a combustion instability model by Yang and Culick.107The geometry considered by Yang and Culick is shown in Fig.12.These investigationsall considered the interaction of a ame stabi-lized in a combustor where the ame was inclined to the ow so that the resultant ow eld was two dimensional.

Marble and Candel’s 104analysis was motivated by the fact that self-excited combustor oscillations in many systems,such as large boilers or aircraft engine afterburners,occur at low frequencies where chemical kinetic processes likely exert minimal in uence on the dynamicsof the interaction.They note,however,that distortions of the overall ame geometry,such as can be seen in Fig.3,are prop-agated along the front by the mean ow speed,that is,the ame’s response depends primarily on uid mechanic adjustments.Thus,the relaxation time of the overall ame front occurs on a time-scale given by the ratio of the ame length and mean ow speed.

The essential approach used in this study 104is similar in most respects to the amelet studies described in the preceding section.The ame dynamics are describedby the kinematic equationsgiven in Eqs.(1–3).Acoustic oscillations,assumed to be one dimensional,are matched using the jump conditions,such as in Eq.(4–7).Free-boundary problems such as these are extremely dif cult to handle analytically.To proceed,these studies 104?107used an integral tech-nique where the equations were integrated in the transverse direc-tion between the ame and combustor wall.Calculations of the acoustic pressure re ection and transmission coef cient showed well-de ned peaks at certain values of the ame Strouhal num-ber,de ned earlier as Sr D f L FL =u .Because the quantity L FL =u refers to the amount of time required for a disturbance to con-vect along the ame at the ow speed,these results showed that the ame ampli ed disturbances with characteristic times T D 1=f that matched particular multiples of this characteristic convective time.Thus,these studies clearly showed the importance of con-sidering a nite sized ame region in accounting for the interac-tions between acoustic disturbances and the ame front.The sig-ni cance of this Strouhal number has been well validated in several experimentalstudies of the ame transfer function,for example,see Refs.57and

108.

Fig.12Interaction of acoustic waves with inclined,ducted ame sheet,such as considered by Yang and Culick.96

Further progress has been made in several more recent studies using similar approaches by Boyer and Quinard,109Baillot et al.,110Fleifel et al.,111and Ducruix et al.57These studies circumvented the analytical dif culties encountered in the described analyses by ne-glecting the coupling of ow perturbationsacross the ame.Rather,they calculated the response of the ame from an imposed velocity disturbance of given amplitude and phase upstream of the ame.In essence,this approximation neglects the density change across the ame front.Nonetheless,the substantially reduced complexity of their approach facilitates much more transparent analysis;more-over,their results appear to give good agreement with experiments.The theoretical results of Boyer and Quinard 109and Baillot et al.110focused on prediction of the ame shape and were in good agreement with their experimental results.The analyses of Fleifel et al.111and Ducruix et al.57focused on prediction of the overall ame front response,characterized by the overall ame front area.Similar to the analyses described earlier,they found that a Strouhal number based on ame length and ow velocity (or,roughly equiv-alent,the ame speed and duct radius)was the dominant parameter controlling the amplitude and phase of the ame response for a given ame shape.Their analysis will be presented in more detail in Sec.IV .More recent measurements of Ferguson et al.112suggest,however,that the ame’s heat release response cannot be uniquely characterized by its instantaneous area,as assumed in Refs.57and 111.This study 112found nonnegligible phase differences be-tween instantaneous measurements of ame surface area and OH ¤chemiluminescence.

The analyses in Refs.57,86,87,and 104–111show that the ame exhibits a strong response to acoustic velocity perturbations,that is,velocity coupling.These studies all assumed that these local velocity perturbations were one dimensional.Basic acoustic con-siderations suggest that such a description of the ame’s acoustic near eld is unrealistic,however,given the multidimensionalitiesin the geometric and ame front con guration.A planar incidentwave impinging on a ame front not only generates planar re ected and transmitted waves,but also multidimensional disturbances that are generally evanescent for the frequencies of interest.65Because the ame is disturbedby the local,multidimensionalacoustic eld,Lee and Lieuwen 113argued that calculations of its interactions with the acoustic eld must account for these multidimensional character-istics.They presented computational results of calculations of the ame’s acoustic near eld showing that the acoustic velocity could exhibit substantial multidimensional characteristics (e.g.,Fig.13).They then substitutedthese acoustic eld calculationsinto the ame dynamic calculationsand compared the results to those obtained by Fleifel et al.111and Ducruix et al.57Surprisingly,they found that the two results were qualitativelysimilar over a wide range of parame-ters,although they could quantitively differ by factors up to two or three.

Numerous experimentalinvestigations(e.g.,Refs.25,37,43,51,58–60,and 144)have demonstrated the substantial interactions be-tween premixed ames with intrinsically unstable or acoustically forced coherent vortical structures.Two primary mechanisms of heat release modulation by these structures have received the most

Fig.13Dump stabilized combustor geometry for case where the ame was excited from upstream:——,instantaneous pressure contours and velocity vectors and average ame location (adapted from Lee and Lieuwen 113).

LIEUWEN775 attention in the literature: ame area modulation59or large-scale

entrainment of reactive mixture that combusts in a rapid burst.60

The former interaction is similar in nature to the heat release per-

turbations induced by acoustic velocity perturbations.Because of

the substantial literature that has appeared on this subject alone,

we refer the reader to the thorough review papers of Coats29and

Schadow and Gutmark28for a more extensive citation of prior work

and discussion of the interaction phenomenology.

To conclude,the kinematics of laminar ame–acoustic wave in-

teractions appears to be well understood,in both the linear and

nonlinear(as will be discussed in the next section)case.Excellent

agreement has been achieved between surprisingly simple theory

and experiments.Because most of the models included purely kine-

matic effects,this agreementdoes not imply that the inner dynamics

of the reaction zone itself or of the ame holding characteristicsare

understood:As noted in Sec.III.A.1,the reaction zone dynamics

have been subjected to extensive theoretical investigation with few

supporting experiments.These points do suggest,however,that the

ame response(at least over the investigated parameter space)is

dominated by the velocity coupled kinematic response of the ame

sheet consistent with the arguments of Clanet et al.56

C.Nonlinear Response

The discussion in Secs.III.A and III.B focused attention on the

linear response of premixed ames to small-amplitude perturba-

tions.Understanding the nonlinear response of ames to nite am-

plitude perturbationsis criticalto predictinglimit-cycle amplitudes.

Also,in certain situations,it is conceivable that the ame–acoustic

problem is inherently nonlinear,for example,in the case where

acoustic disturbances are exciting a ame in the presence of back-

ground turbulence that has suf cient intensity to cause a nonlinear

ame response or even cause local extinction of the ame.Further-

more,recent pressure–heat release transfer function measurements

of Lieuwen and Neumeier115indicate that the nonlinear dynamics

of combustorsare dominated by these heat release–acoustic nonlin-

earities,(Fig.14).

Consider rst these interactionswith a region of distributed reac-

tion.The basic unsteadyWSR equations(11–15)consistof a system

of nonlinear, rst-order equations.Thus,incorporating nite am-

plitude effects into unsteady WSR calculations is straightforward,

although it may require numerics for time stepping.The model used

by Richards et al.,96Rhode et al.,116and Lieuwen et al.98incorporate

these nonlinear equations and,in certain cases, nd complex,and

even chaotic,combustor dynamics.As in the linear studies,stipu-

lating the manner in which the reactor residence time depends on

the other perturbations was not attempted.

Similarly,the nite amplitude response of a ame sheet to acous-

tic perturbations is described by the nonlinear,partial differential

equations given in Eqs.(1–3).In practice,treatment of these equa-

tions is made dif cult because of the tendency of the ame position

to become a multivalued function of the radial or axial

coordinate,

Fig.14Measured amplitude relationship between pressure and CH*0

radical chemiluminescence in a premixed combustor.115

for pockets to pinch off and form islands and for sharp cusps to

appear.This problem has been treated in several analyses,which,

though not directly aimed at the acoustic ame interaction problem,

did analyze the ame front kinematics in an unsteady and/or peri-

odic ow eld.117?120These studieswere able to reproducea number

of ame front phenomenon that are observed in experiments such

as cusp and pocket formation or ame area ampli cation.

Bourehela and Baillot,121Baillot et al.,122and Durox et al.123

performed a systematic experimental and theoretical study of the

response of a Bunsen ame to velocity perturbations of varying

amplitude and frequency.Although their principal observationsare

quite similarto those previouslyobservedby Blackshear,124they ap-

pear to be the rst systematic characterizationof the ame response

as a function of perturbation amplitude.121?123Similar to Fig.3,

they found that,at low frequencies.f<200Hz)and velocity am-

plitudes.u0=N u<0:3/,the ame front wrinkles symmetricallyabout

the burner axis due to a convected wave traveling from the burner

base to its tip.At higher frequencies,but similar low amplitudes,

they observed a phenomenon they refer to as“ ltering,”where the

effect of the oscillations on the ame front is only evident at the

ame base and becomes strongly damped at axial locations farther

downstream.This behavior does not appear to be due to the low-

pass ltering of the ame front to velocity disturbances described,

for example,by Ducruix et al.57or Saitoh and Otsuka.125It may be

due to the increased importance of the ame’s curvature dependent

burning velocity and the very short convective wavelengths of the

imposed disturbances at these higher frequencies.

In Refs.121–123,with an increaseof amplitudeof low-frequency

velocity perturbations,the authors found that the ame exhibited a

varietyof transient ame holdingbehavior,such as ashback,asym-

metric blowoff,unsteady lifting and reanchoring of the ame.In

addition,they note that its response is asymmetric and extremely

disordered.Finally,at high frequencies and forcing amplitudes,

they found that the ame collapses and no longer has a sharp tip.

Rather,the tip region becomes rounded off and,for suf ciently

high forcing intensities(u0=N u>1/,the ame’s mean shape becomes

hemispherical.121This latter phenomenon was more extensively in-

vestigated by Durox et al.,123who showed that the ame’s surface

area was unmodi ed when the ame attens.They note that this

ame attening is equivalent to the phenomenon of acoustic stabi-

lization of cellular ames.

Baillot et al.122also reported a theoretical study of these ame

fronts to predict their unsteady evolution,particularly at larger am-

plitudes of forcing,where the ame front becomes strongly cusped.

They reduced the ame dynamic equation[Eq.(2)],to a Hamilton–

Jacobi type equation,which they solved by the method of charac-

teristics.They found that the predicted and measured ame shapes

were in good agreement.

Dowling126introduced a phenomenological model for the nite

amplituderesponseof the ame to velocityperturbations.The model

is dynamic in nature,but the essential nonlinearity is introduced

from a quasi-steady relation between ow velocity and heat re-

lease rate.Speci cally,it assumes a linear relation between the

heat release Q and velocity perturbation when the total velocity

(u D N u C u0)lies between0and2N u.When u<0,the heat release

goes to zero,and when it is greater than2N u it saturates at2N Q.This

relationship is shown in Fig.15.

Fig.15Relation between quasi-steady heat release and velocity per-

turbation used by Dowling.126

776LIEUWEN In a later study,Dowling127presented a more physics-based model,based on the ame kinematics approaches described ear-

lier(e.g.,Refs.57,104,109,and111.To treat nite amplitude oscillations,Dowling utilized a nonlinearboundaryconditionat the

ame anchoring point.Speci cally,she assumed that the ame re-mained anchored at the bluff body if the total gas velocity exceeded

the ame speed.If the gas velocity fell below the ame speed,the former conditionwas replacedby one that allowed the ame to prop-agate upstream.Dowling notes that the predicted ame behavior is consistent with direct visual observations from prior experiments. Peracchio and Proscia128developed a quasi-steady nonlinear model to describe the response of the ame to equivalence ratio perturbations.First,they assumed the following relationshipfor the response of the instantaneousmixture compositionleaving the noz-

zle exit to velocity perturbations:

á.t/D Ná=[1C ku0.t/=N u](24) where k is a constant with a value near unity.They also utilized

a nonlinear relationship relating the heat release per unit mass of mixture to the instantaneous equivalence ratio.

To conclude,note that investigationsof nonlinearcombustiondy-namics have been primarily theoretical.The reactor-based models suffer the same shortcomings already noted in Sec.III.A.2.Their

utility is primarily restricted toward serving as interesting model problems to study complex,chaotic dynamics.In the amelet case,

good agreement has been achieved between theory and experiment

for simple laminar ames.As already noted,this agreement sug-gests that the linear and nonlinear kinematics of ame’s respond-

ing to acoustic disturbances is well understood.Other analyses of nonlinearity mechanisms,such as ame holding,the equivalence

ratio–velocity relationship,the equivalence ratio–heating value re-lationship,and others,seem reasonablebut haveonlybeencompared qualitatively with measurements.

IV.Flame Transfer Function Calculations

Many of the concepts and conclusions discussed can be rein-forced through analysis of a model problem that explicitly allows

for calculationof the ame responseto perturbations.In this section,

we present a model for the kinematic response of a ame sheet to

two types of perturbations:velocity and fuel/air ratio.This analysis closely follows that of Fleifel et al.111and Ducruix et al.57for the velocity perturbation and Cho and Lieuwen129for the fuel/air ratio perturbation.This analysis speci cally focuses on the global heat release response and is,thus,most relevant to acousticallycompact

ames(Sec.II.D).

The studied geometry is axisymmetric and shown in Fig.16.The

ame’s instantaneous axial position x is given by the following function of the radial and temporal coordinates,r and t:

x D?.r;t/(25) Thus,we de ne the ame position surface f.x;t/[Eqs.(1–3)],

as f.x;t/D x??.r;t/D0.Note that this de nition requires the

ame’s axial position x to be a single-valued function of the radial coordinate.Substituting this expression into Eq.(2)yields the fol-lowing kinematicrelationdescribingthe ame positionas a function

of the ow velocity and ame speed:

@?@t D u?v@?@r?S1

s

@?

@r

′2

C1(26)

Fig.16Schematic of model problem considered in Sec.IV.where u,v,and S1are axial velocity,radial velocity,and ame speed relativeto the unburnedgases,respectively.We assumethat the mean velocity is uniform and purely axial,that is,N v D0,and that the mean ame speed is constant.Although these assumptions are not nec-essary to proceed with the analysis,they do yield more transparent results that retain many of the basic phenomena of interest.We next decompose the variables into their mean and uctuating parts and retain only linear terms in perturbations.This procedure yields the following equations for the mean and uctuating variables:

N u D N S1

s

1C

d N?′2(27)

@?0

D u0?v0d N??N S1

μ

N

p

N

?

?S01

s

1C

d N?′2(28)

As in Ref.111,we simplify Eq.(28)with the additionalassumption that N uàN S1,which implies that the ame length is much greater than its diameter,L FLàR,and that

s

1C

d N?′2??d N?

As such,the mean ame position is given by

N?.r/=L FL D1?.r=R/(29)

Note that L FL=R?N u=N S1.

Assuming harmonic oscillations at an angular frequency ![exp(?i!t/]Eq.(28)becomes an ordinary differential equation that can be solved for the ame position.When it is assumed that the ame remains anchoredat its base,that is,?0.r D R;!/D0,the solution takes the following form:

?0

FL

D exp.?i Srr=R/

Z

r=R

1

μ

S0

1

.′/

N S1?

u0

n

.′/

N S1

′

exp.i Sr′/

?

d′

(30) where the Strouhal number de nition is slightly different than used before,Sr D!L FL=N u,and u0n is the unsteady velocity component normal to the mean ame position.Within the assumptions of this analysis,this expressionshows that a perturbationin normalvelocity or ame speed exerts an identical effect on the ame’s position.As will be shown,however,the overall effects of these perturbationson the heat release are different because the ame speed perturbation effects both the ame front area and the local consumption rate, whereas the velocity perturbation only effects the former. Further progress in analyzing Eq.(30)requires specifying the spatial distributionof the velocity and ame speed perturbation.We assume that the spatial distribution of these two perturbations are acoustic and convective in nature,to illustrate the effects that the different disturbancesmodes have on the ame’s response.As such, u0

n

is assumed to be spatially uniform,a reasonable assumption if the ame is acoustically compact.The ame speed perturbation is assumed to be generated by a convected perturbation in mixture stoichiometry,given by

S0

1D

d S1

′

Ná

￠á0(31)

whereá0is convected by the mean ow and,thus,has an axial distribution given by

á0.x;t/Dá00exp[?i!.t?x=N u/]

Dá00exp.?i!t/exp[i Sr.1?r=R/](32)

LIEUWEN 777

Following Flei l et al.111and Ducruix et al.,57the global heat release rate of the ame is written as

Q .t /D

Z

S

?S 11h R d A FL (33)

where the integral is performed over the ame surface,?is the density,and 1h r is the heat of reaction.We assume that the density of the reactive mixture is constant.Fluctuations in the remaining quantities contribute to heat release oscillations as

Q 0N Q D R 1h 0R d N A FL R 1N h

R d N A FL C R

S 01d N A FL R N S 1d N A FL C A 0

FL N A FL (34)

where A FL is the ame surface area whose instantaneous value is

given by

A FL D

Z

R

2?r

s

1C

@?@r

′2

d r

(35)

Decomposing A FL into its mean and uctuating components yields the following linearized result for the area uctuation term:A 0FL

N A

FL D 2Z

1

exp .?i Srr /

μZ 1r

u 0n .′/N S 1?S 0

1.′/

N S

1′exp .i Sr ′/d ′?

d r (36)

Pulling the results from Eqs.(34)and (36)together,we arrive at

the following expression that decomposes the heat release pertur-bation into its contributionsfrom the velocity and equivalenceratio perturbation:

Q 0=N Q D Q 0=N Q j u 0C Q 0=N Q j á0(37)De ne the following ame transfer functions to perturbations in

velocity,F u ,and equivalence ratio,F á:

F u D Q 0=N Q j u 0u 0n =N S 1D 2

[1C i Sr ?exp .i Sr /]

(38)F áD

Q 0=N Q

j á0á0b

=N á

D F H C F S D F H C .F S ;dir C F A /(39)

where

F H D d .1h R =1N h R /d .á=N á/-

---N á

22f 1C i Sr ?exp .i Sr /g F S ;dir D d .S 1=N S 1/d .á=N á/-

---N á2f 1C i Sr ?exp .i Sr /g F A D

d .S 1=N S 1/d .á=N á/----

N á

22f 1?.1?i Sr /exp .i Sr /g and á0

b in Eq.(39)is the perturbationin equivalenceratioat the ame base.Note that these transfer functions are solely a function of the Strouhal number and the sensitivities of the heat of reaction and ame speed to the equivalence ratio.The amplitude and phase de-pendence of these transfer functions [Eqs.(38)and (39)]are shown in Fig.17.These characteristicsare described in more detail later.The equivalence ratio transfer function F áhas three contributing terms [Eq.(39)].The rst term,F H ,is due to perturbations in heat of reaction,that is,the heat content of the reactive mixture.The second term,F S ,is due to perturbations in ame speed.Note that perturbationsin ame speed are again divided into two factors;one is directly generated by the ame speed sensitivity to equivalence ratio,F S ;dir ,and the other is due to the subsequent uctuation in ame surface area,F A .Our explicit calculations assume a quasi-steady relationshipbetween equivalenceratio and ame speed,that

is,that d .S l =N S

l /=d .á=N á/is independent of frequency.Incorporat-ing the additional dynamics of this relationship can be added in a straightforward manner if necessary,however.

a)

b)

Fig.17Amplitude dependence of ame’s heat release response to

ame speed and acoustic velocity oscillations on Strouhal number .

In general,the relationship between unsteady heat release rate

and the velocity or equivalence ratio perturbation has a complex dynamic.However,for Sr ?1,that is,a convectivelycompact ame,the Q 0u relationship can be put in terms of a simple n –?model,

Q 0u

.t /=N Q D n u u 0n

.t ??u /(40)

where n u D 1=N S

1and ?u D L FL =3N u .The dynamics of Q 0

ácannot be generally described by an n –?model,even in the Sr ?1limit.This is due to the possible negative phase dependenceof F áon Strouhal number when Sr ?1(Fig.18)that is,the ame can not respond before the equivalence ratio per-turbation reaches it.The low Sr dynamics of Q 0áis given by

Q 0u .t /N Q

D n H á0

b .t ??H /C n S d á0b .t /d (41)

where n H D d .1h R =1N h R /-

---N á

;?H D L F

;

n S D

1L F d .S 1=N S 1/-

---N á

Equation (40)indicates that the time response of the heat release

rate to perturbations in acoustic velocity,Q 0u .t /,is delayed by a retarded time,?u ,where ?u represents the time taken for the mean ow to convecta distanceof one-thirdof the ame length,which can be taken as the effective position of concentrated heat release,that is,L eff ?L F =3.However,the heat release response to equivalence ratio perturbations,Q 0á.t /,is delayed or advanceddepending on the combined effect of ?H and a temporal rate of change of ame speed perturbations as shown in Eq.(41).

To quantify the dependence of heat of reaction and ame speed on equivalenceratio,we use the following correlationfrom Abu-Off and Cant 130for methane at 300K and atmospheric pressure (also used by Hubbard and Dowling 131for a similar calculation):

S 1.á/D A áB e ?C .á?D /

2

(42)1h R .á/D

2:9125￡106minimum .1;á/

(43)

778LIEUWEN a)

b)

Fig.18Phase dependence of ame’s heat release response to ame speed and acoustic velocity oscillations upon Strouhal number.

where the followingvalues were used for the constants:A D0:6079, B D?2:554,C D7:31,and D D1:230.

These correlations were used to generate the results in Figs.17–19.Refer to Figs.17and19,and note that F u and F H are nearly identical and decrease monotonically from their maximum response at Sr D0.In contrast,the heat release response to ame speed perturbation,F S,vanishes at Sr D0.This is due to the exact cancellation of the ame speed and area perturbation terms,which have equal magnitudes,but opposite phases.This zero response at Sr D0can be understood from quasi-steady arguments,that is,the ame area uctuateswith the same magnitudeand oppositephase as the ame speed oscillations.This zero response can also be under-stood from the fact that the ame speed and area perturbationterms account for the ame’s response to a mixture with constant heat of reaction.For example,two substoichiometric ames with the same ow of fuel,but differing amounts of air,release the same amount of heat for quasi-steady states,although the ame’s have different areas.As such,slow timescale perturbations may affect the ame’s local consumption rate,but the resultant heat release perturbationis exactly balanced by the resultant variations in ame area.

The latter transfer function F S increases with Strouhal number from zero because of changes in the relative phase of the terms F S;dir and F A.It reaches a global maxima at Sr?4.5,where the two perturbationsreinforceeachother.As the Strouhal number increases further,F S decreases in an oscillatory pattern due to the alternating phase relationship between F S;dir and F A.The total heat release response,Fá,increases until Sr?4and decreases in an oscillatory manner.

Figure17b shows the dependence of the phase of the transfer functionson Strouhal number.As in Eqs.(38)and(39),the acoustic velocity term F u and the heat of reaction term F H have exactly the same Strouhal number dependence.They start at0deg at Sr D0 and increase monotonically to and oscillate around90deg.On the other hand,the ame speed term F S starts at a phase of?90deg. This is the average of F S;dir and F A,which start at0and?180deg, respectively.The phase of Fálies between F H and F S.This phase

of

a)

b)

Fig.19Polar plot of ame transfer functions,where Strouhal number Sr is a parameter,á=1.

Fáinitially increases slightly and then increases more rapidly along with the phase of F S toward90deg.

Figure18shows the effect of the mean equivalence ratio on the ame transferfunctionusing the correlationsin Eqs.(42)and(43).It shows that mixture stoichiometry exhibits little effect on the trans-fer function magnitude for Sr?1and at subsequent minima.In most cases,however,the ame response increases with decreas-ing equivalence ratio.This is due to the increased sensitivity of the ame speed to equivalence ratio for lean mixtures.Note that the Strouhal number value of maximum response,Sr?5,remains essentially constant.Figure18b shows the phase-dependence on Strouhal number at several equivalence ratios.Note that the low Strouhal number heat release response can either lead or lag theáperturbation,depending on meanávalue.

The predictedamplitude and phase characteristicsof the velocity transfer function F u have been shown by Ducruix et al.57to agree well with experiments up to Sr?8(Substantial phase deviations between theory and experiment were observed for larger Strouhal numbers.)This good agreement provides con dence in the basic kinematic modeling approach.Note that no measurements of the equivalence ratio transfer function Fáhave been made.

An important conclusion to be drawn from these analyses is the importance of both the local and global effect of a perturbation on the overall ame response.For example,a ame speed perturbation causes both a local change in heat release rate/unit area,as well as in the overall ame area.The transfer function results illustrated in Fig.17show that inclusionof both effects is crucial in modeling the overall ame response.

This comparison also illustrates the dif culties associated with using unsteady WSRs to model ame–acoustic interactions.For example,consider the Lieuwen et al.62analysis of the response of a WSR to equivalence ratio perturbations,where the WSR volume was assumed xed.Analogous to the area effect in the preceding

LIEUWEN779

amelet calculation,this assumption of xed reactor volume likely causestheir resultsto be erroneous,that is,increasesin the mixture’s local reaction rate result in reductionsin reaction region volume.At low enoughfrequencies,these two effects(reactionrate and volume) must cancel.

V.Conclusions

From a combustor designersviewpoint,the goal of this work is to develop a model that can rapidly predict the qualitative,and prefer-ably quantitative,dependence of combustor stability on geometric and fuel compositionparameters.Accuratelymodelingthe combus-tion process response to ow perturbations is a critical component of such a capability.Reasonable,quantitativepredictions of ame–acoustic interaction phenomenon has been demonstrated for a few simple con gurations,such as the laminarBunsen burnerof Ducruix et al.57or Baillot et al.,110or the nominally at ame of Searby and Rochwerger.86These successes are illustrative of the rapid progress being made in modeling kinematic processes in ame–acoustic in-teractions.In addition,progress is being made in developing hybrid models that use computational simulations to determine various componentsof the combustorsystem– ame interactions.132The de-velopment of accurate,predictive combustion response models for realistic,that is,turbulent,con gurations has not been achieved, however,and remains a key challenge for future workers.The dis-cussion below suggests some requirements needed to achieve this capability.

First and most generally,it seems critical that better coordina-tion between models and experiments be achieved.At present,a signi cant part of the relevant literature consists of essentially de-coupled theoretical models or experimental studies,even in rather fundamental con gurations.For example,a substantial number of fundamental studies have theoretically investigated the response of at,laminar ames to pressure perturbations.71?77No serious effort appears to have been initiated to subject these predictions to exper-imental scrutiny.Similarly,although equivalence ratio oscillations are known to play an importantrole in drivinginstabilities,no exper-imental work appears has been performed to examine the accuracy of models that relate them to the resultant heat release oscillations. Even though these highly fundamental studies may be far removed from practical ames,they are prerequisite building blocks toward modeling realistic systems.

Second,work is needed to develop simpli ed models of vortex– ame interactions.The existing theoretical work on this subject is largely numeric.Analytic methodologies for modeling unstable, reactingshear ows have been developed133and need to be extended to incorporate the unsteady ow effects on the ame.

Third,predictingthe response of ames to nite amplitude waves is immature.The few existing studies are largely phenomenological in nature.For example,it is not presently clear whether limit-cycle amplitudes in lean,premixed gas turbines are controlled by nonlin-ear ame front dynamics,the nonlinear response of the equivalence ratio to velocity perturbations in the premixer,chemical kinetic ef-fects,or some other process.Substantial progress could be made by a set of experiments that isolate the key nonlinear processes that need to be focused upon by modelers.Interpretive guidance of these results can be achieved by parallel systematic studies of potential nonlinearities.In addition,effects such as the stabilization or parametric destabilization of ames discussed in Sec.III.A may cause nite amplitude acoustic oscillations to change substantially the characteristicsof the turbulent ame it is interactingwith.These effects merit further investigation.

Fourth, ame–acoustic wave interactions in realistic environ-ments occur in a very noisy atmosphere,where the ame is a highly perturbed front,even in the absence of coherent acoustic oscilla-tions,and executes large oscillations about its mean position.Any model of the response of laminar ame fronts to velocity,equiv-alence ratio,or vortical disturbances needs to be generalized to include the fact that the“average” ame is highly unsteady.For example,the successful work performed to date on laminar,Bun-sen ames should be extended to turbulent ow situations.Fun-damental issues,such as how the transfer functions measured by Ducruix et al.57change with increasingturbulencelevels,need to be addressed.

Fifth,the interactions of ames with thickened amelets,dis-tributed reaction zones,or well-stirred reactions needs to be con-sidered.As already emphasized,current WSR models are largely phenomenological and have a number of signi cant conceptual problems.Progress in this area will requireclari cation of the nature of the combustionprocessin this regime by the turbulentcombustion community.

Acknowledgments

The author gratefully acknowledges the support of the National Science Foundation,General Electric,the U.S.Department of En-ergy,and Georgia Institute of Technology for past and ongoing sup-port of his work on problems related to those described here.

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