handbook of combustion (online) || control of acoustically coupled combustion instabilities

16Control of Acoustically Coupled Combustion InstabilitiesS�ebastien Ducruix, Thierry Schuller, Daniel Durox, and S�ebastien Candel

16.1Introduction

Among the many issues that need to be considered in the design and operation ofcombustion systems those associated with combustion dynamics are probably themost difficult. The process involves a resonant coupling between the flame and thesystem acoustics [1, 2]. This has many undesirable effects like high levels of noiseradiation, structural vibrations, fatigue of the mechanical components, intensifica-tion of the heat fluxes to the walls, flame blow-off, and in extreme cases failures andirreversible damages. Combustion dynamics is a central issue in key technologicaldevelopments in energy and propulsion but they are also of considerable importancein various industrial and practical applications. For industrial gas turbines, therequirement for lower emission of nitric oxides has led to the development ofpremixed burners providing an important reduction of these emissions. However,this has also resulted in reduced stability associated with an increased susceptibilityto acoustic disturbances [3, 4]. Near the lean blow-off limit where the combustor isdesigned to operate, pressure fluctuations may take sizable amplitudes and reach, insome cases, levels of the order of a small percent of themean chamber pressure withRMS (rootmean square) values of the order of 0.1MPa ormore. Such levels cannot besustained under normal conditions and for extensive periods of time and they mustbe reduced to avoid incidents or restrictions in the operational domain of themachine. The dynamics of lean premixed combustion and control of instabilitieshas been the subject of intensive research. This has led to the identification of drivingprocesses, analyses of the unsteady flame motion, description of system dynamics,and related developments of passive and active control methods. Much of this workcan be used to deal with many other configurations of technical interest.

In jet engine combustors, the trend has also been to devise advanced systemsoperating in the fully or partially premixed modes but progress has been slowed byproblems of the same nature as those found in industrial gas turbines (see the recentpaper by Cazalens et al. [5]). In this field, the reliability constraints aremore stringent.It is also less easy in this case to adopt geometrical variations of the type devised for

Handbook of Combustion Vol.5: New TechnologiesEdited by Maximilian Lackner, Franz Winter, and Avinash K. AgarwalCopyright � 2010 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-32449-1

j403

terrestrial gas turbines, due to size and weight constraints. Combustion dynamicshave been a problem in rocket engine developments. Early efforts in this field havebeen hampered by many spectacular failures associated with various types ofinstabilities (see in particular the review by Yang and Anderson [6]). Coupled lowfrequency instabilities involving the propellant feed system and the combustionchamber have beenmainly solved by decoupling these components by incorporatinglarge head losses in the injection units. A value of Dp/p� 10% is typically used inliquid rocket engines. Themost difficult problems are associatedwith high frequencyinstabilities involving transverse modes of the thrust chamber.

Combustion dynamics phenomena are also encountered in many practical appli-cations like large-scale boilers used in power plants, radiant heaters employed invarious industrial processes, smaller scale domestic boilers, and so on. There aremany common processes in this broad range of applications but also many differ-ences, which justify specific investigations carried out in the past [7, 8]. The topic ofinstability control constitutes a point of common interest, which can be usefullyconsidered by sharing analysis, solutions, and systems. One objective of this chapteris to review fundamental issues in combustion dynamics and discuss the variousmethods devised to counteract combustion dynamics and alleviate oscillations.

This chapter begins with a review of acoustics fundamentals for reactive flows(Section16.2).Attention ismainly focusedonsituationswhere thewavelength is largecompared to the transverse dimension of the systemso that propagation only involvesplane waves and modes. Derivations lead to the acoustic energy balance equation,whichcanbeusedconveniently todescribethethreemaincontrolstrategies thatcanbeenvisaged for combustion instabilities: (i) passive control methods (PCM), (ii) flamedynamic control methods (FDCM), and active control methods (ACM). Passivecontrol methods (PCM) are reviewed in Section 16.3. In essence, these methods aredirected at augmenting acoustic damping, thus reducing the sharpness of resonance,a goal that can be fulfilled by various systems. Other methods, called flame dynamicscontrol methods (FDCM) in the present chapter, consist in modifying the injectorandflame dynamics to reduce their sensitivity to perturbations. They are described inSection 16.4. Section 16.5 is focused on active control methods (ACM), a topic thathas been extensively investigated recently. This subject is reviewed and methods areevaluated from a practical point of view. We try to sort out progress made in this areaand review problems of application of these methodologies.

16.2Fundamentals of Acoustics for Reacting Flows

Fundamental aspects for acoustics analysis of reacting flows are discussed in thissection. The study is restricted to lowMach number flow (M � 1) for simplicity andalso because in most combustion applications the Mach number remains relativelysmall. There are, however, regions of high speedflowswithin a combustion chamber,for example at the injector outlets where the fuel is often delivered at high pressureor at the chamber exhaust were hot combustion products are accelerated through

404j 16 Control of Acoustically Coupled Combustion Instabilities

a nozzle or a distributor blade row. These regions of high velocities principallymodifythe acoustic response at the chamber termination and their specific dynamicalresponse needs to be considered in a more advanced acoustic analysis. The theo-retical framework presented in what follows is intended to be an overview, highlight-ing themain physicalmechanismsgoverning acoustics in reactingflows.Derivationspresented belowdonot encompass all the complexities or the state of art and are oftensubjected to restrictions and simplifying assumptions. References cited in thischapter provide further details and derivations including various complexities ora more rigorous approach.

Consider a perfect gas obeying a state law r¼r(p,s) where r is the density, p thepressure, and s the entropy. One considers flow perturbations in pressure (p1),velocity (v1) and entropy (s1). While standard acoustic waves are isentropic, it isimportant to consider entropy fluctuations, which can arise in reacting systems [9].These perturbations may take several forms. Velocity fluctuations can, for example,result from large coherent structures originating in shear layers [10, 11], bulk flowoscillations [12, 13], or acoustic waves propagating in the system [14, 15]. Entropyfluctuations result primarily from heat release rate perturbations within the reactionzone, but other mechanisms like mixture composition or temperature inhomoge-neities in the fresh reactants or the hot products also constitute alternative sources ofentropy perturbations, which can modify the acoustics of the system [16–18].

These flow perturbations must comply with mass, momentum, and energytransport equations. It is here preferable to use a transport equation for entropyinstead of the more standard energy balance. The same results can, however, bederived using a transport equation for the enthalpy. Using the conservative form,these equations write:

qrqt

þr � rv ¼ 0 ð16:1Þ

qrvqt

þr � rvv ¼ �rp ð16:2Þ

qrsqt

þr � rvs ¼ _qT

ð16:3Þ

This set of expressions is completed by the state equation for a perfect gas:

p=rc ¼ C expðs=cvÞ ð16:4Þwhere C is a real constant and c¼ cp/cv designates the ratio of constant pressure toconstant volume specific heats cp and cv respectively. Thermal diffusion and viscousstresses are not included in these equations. They essentially induce damping ofacoustic waves, an effect that can be treated separately [19]. The detailed set oftransport equations for species is also not included for simplicity. These equationscan be important if one wishes to study noise source terms associated to non-isomolar combustion [20, 21]. The main effect of the heat released by the chemicalconversion of the fresh reactants into hot products appears in the right-hand side of

16.2 Fundamentals of Acoustics for Reacting Flows j405

the entropy balance in the form of a heat release rate per unit volume [22].Considering that the mean state in the system is perturbed by small perturbationsand that the Mach number is small (M� 1), the field variables can be writtenp¼ p0 þ p1, s¼ s0 þ s1, and v¼ v0 þ v1 with p1/p0� 1, s1/s0� 1, and v1� 1 becausev0� 0. A first-order analysis of the set of transport equations (16.1)–(16.3) yields thefollowing linearized equations for the perturbed quantities:

qr1qt

¼ �r0r � v1 ð16:5Þ

qv1qt

¼ � 1r0

rp1 ð16:6Þ

qs1qt

¼ _q1r0T0

ð16:7Þ

Equations (16.5) and (16.3) have been combined to obtain Equation (16.7). Thecorresponding perturbed equation of state (16.4) for density fluctuations is:

r1 ¼ rac1 þ ren1 ¼ 1c20

p1� r0c p

s1 ð16:8Þ

The first contribution rac1 ¼ p1=c20 is associated to acoustic fluctuations, that is, to anisentropic process s¼ s0, where ðqp=qrÞs¼s0

¼ c20 is the square of the speed of sound

c0. The second contribution ren1 ¼ �ðr0=c pÞs1 corresponds to isobaric transforma-tions at p¼ p0 associated with entropy fluctuations s1. This termwould be zero in theabsence of an irreversible process, but must be taken into account when combustionis considered. These acoustic and entropy perturbations respectively satisfy:

Acoustic perturbations

1c20

qp1qt

þ r0r � v1 ¼ 0 ð16:9Þ

r0qv1qt

¼ �rp1 ð16:10Þ

s1 ¼ 0 ð16:11ÞEntropy perturbations

� r0cp

qs1qt

þ r0r � v1 ¼ 0 ð16:12Þ

p1 ¼ 0 ð16:13Þ

r0qs1qt

¼ _q1T0

ð16:14Þ


One can use this set of equations to derive jump conditions for flow perturba-tions across area changes and compact flames [22]. It is, however, first desirable toderive a wave equation for pressure perturbations in the presence of combustionand see how to deal with this equation in confined environments. A first relationbetween pressure, velocity, and heat release rate perturbations can be obtainedby combining Equation (16.8) and the perturbed mass and entropy balanceEquations (16.5) and (16.7):

1c20

qp1qt

þ r0r � v1 ¼ _q1c pT0

ð16:15Þ

The second step is to eliminate velocity fluctuations in this expression. This isdone by taking the difference between the time derivative q/qt of Equation (16.15)and the scalar product!� of Equation (16.6). This yields an inhomogeneous waveequation for pressure fluctuations (see Crighton et al. [19] and Candel et al. [23] foradditional terms that are neglected in the present analysis):

r2p1� 1c20

q2p1qt2

¼ � c�1c20

q _q1qt

ð16:16Þ

where the specific heat is expressed as cp¼ rc/(c� 1) and the square of the speedof sound is given by c20 ¼ crT0. Without entropy perturbations s1¼ 0, that is, fornon-reacting flows or for steady combustion _q1 ¼ 0, one retrieves the classicalwave equation [24]. The inhomogeneous wave equation (16.16) indicates thatunsteady combustion _q1„0 generates entropy perturbations s1 „ 0 that constitutea source of sound proportional to the time derivative of heat release ratefluctuations q _q1=qt. Integration of this equation in unbounded domains usingthe open space Green�s function yields an expression for the sound radiated inthe far-field by compact flames [25]. Simplified versions of this expression havebeen verified in many early experiments on sound radiation from premixedflames (see, for example, Price et al. [26]).

16.2.1Acoustic Modes in Confined Environments

It is more difficult to determine the pressure field in the case of confined systems.There are only a few examples of solutions obtained in generic confined burnerconfigurations based on theGreen�s functions technique (see, for example, the recentwork of Heckl and Howe [27]). There are two difficulties to overcome:

1) The solution must comply with the complex geometry of the combustor and alsowith complex impedances at the boundaries [28].

2) The speed of sound cannot be considered to be constant within the chamberbecause of the presence of hot combustion products [29].

Recent studies have led to developments of numerical tools where the speed ofsound is a function of space within the chamber. The same type of analysis as the onepresented above for a constant sound speed c0 can be carried out by assuming that the


speed of sound is a function of position, c0¼ c0 (x). Temporal fluctuations of c0 areneglected because they are assumed to vary over time scales much larger than theacoustic times scales considered. The sound speedfield is thenmainly determined bythe steady combustion process. Some recent studies indicate, however, that tem-perature fluctuations may influence the resonance properties of a system [30]. It isshown that the effective damping of the combustor cavity is increased in the presenceof temperature fluctuations. This effect observed in the case of high frequencytransverse oscillations characterizing rocket engine instabilities is of lesser impor-tance in the low frequency range, which is of interest inmost industrial applications,andwill not be considered inwhat follows. Under these circumstances, onemay onlyconsider the spatial variations of the speed of sound and the wave equation becomes:

r � ðc20rp1Þ� q2p1qt2

¼ �ðc�1Þ q _q1qt

ð16:17Þ

It is useful to examine this problem in the frequency domain. Assuming thatperturbations are harmonic and feature a common factor exp(�ivt) and introducingthis factor in Equation (16.17), one finds an inhomogeneous Helmholtz equation:

r � ðc20rp1Þþv2p1 ¼ ivðc�1Þ _q1 ð16:18Þ

This may be solved by expanding the pressure field in terms of a modal series:

p1 ¼X¥

n¼0gnynðxÞ

where the coefficients gn denote the modal amplitudes. Each mode yn satisfies thehomogeneous equation:

r � ðc20rynÞþv2nyn ¼ 0 ð16:19Þ

where the angular frequencies vn are eigenvalues of the problem. The modalexpansion has been extensively used to analyze combustion instabilities. There arenow computational tools providing eigenmodes and eigenvalues of the Helmholtzequation in complex geometries, with rather general boundary conditions specifiedin terms of an acoustic impedance p1/(v1�n)¼Z, but it is convenient to usea dimensionless admittance b¼r0c0/Z and the boundary condition takes the form:

ryn �n ¼ ibkyn ð16:20ÞDetermination of modal structures and resonance frequencies including sound

speed non-uniformity provides important information in the analysis of combustioninstability and control methods. Many studies focus on coupling the right-hand sidesource in Equation (16.18) to flow perturbations to assess the combustor stability.This aspect will be only briefly reviewed in Section 16.2.2. Moreover, stability can beconsidered guaranteed if acoustic disturbances produced by heat release rateperturbations (the right-hand side term of Equation (16.18)) decay with time. Thiscan be more easily understood by considering the rate of change of the acousticenergy density in the system. This development is proposed in Section 16.2.3.


16.2.2Acoustic–Flame Interactions

Because combustion instabilities have somany facets, they have been examined witha wide variety of theoretical, experimental, and numerical tools. Recent experimentalevidence and some earlier observations indicate that the instabilities involve largeoscillations of the flame front, pulsations of the injected reactant streams (potentiallyleading to equivalence ratio fluctuations), hydrodynamic instabilities, vortex roll-upprocesses, andmutual flame interactions (see, for example, the early experiments byKeller et al. [31]). As shown in Section 16.2.1, the flamemotion by itself is an essentialaspect of the problem. It is in particular crucial to describe the interaction between theunsteadyflowfield, the reaction front, and the associated heat release. Themain issueis then to measure, model, or simulate the flow dynamics coupled with the flamemotion. This is well discussed in several earlier studies and in review articles (see, forexample, Candel [32], Candel [1], and Ducruix et al. [2] for reviews and Lieuwen andYang [4] for a recent book dedicated to combustion instabilities in gas turbines). In theframework of control of acoustically coupled combustion instabilities, it is interestingto specifically focus on acoustic–flame interactionmodels. These are usefulwhenonewishes to devise flame dynamics control methods (FDCM) and active controlmethods (ACM).

One method, which has been extensively exploited, consists in describing theflame as a thin sheet separating cold and hot gases. Poinsot and Candel [33] firstindicated that the non-steady response of the flame could be a possible source of lowfrequency oscillations in afterburners or large utility power plants. Premixed flamedynamics were later explored analytically by Boyer and Quinard [34]. More recentstudies like Fleifil et al. [35], Dowling [36], Ducruix et al. [37], and Schuller et al. [38]propose descriptions of flame front dynamics in the case of perfectly premixedflames, usually represented as flame sheets or discontinuities, and presentinga conical or V shape, depending on the stabilization configuration. Recent progresshas been accomplished with more complex kinematic models, including thermaleffects [39], and influence of the anchoring point [40]. Hathout et al. [41], Mohanrajet al. [42] andmore recently Birbaud et al. [43] focused on themodeling of interactionswith equivalence ratio fluctuations. As indicated in these articles, the thin sheetmodel provides a convenient simplification because the internal structure of theflame is not resolved, making it suitable for control algorithms [44].

However, tracking thin discontinuities is less easily extended in three dimensions,while this seemsmandatory for complex practical applications. It is also important torefine the flame representation and specifically include effects of turbulence andchemical aspects (to describe the flame response to strain and curvature, quenching,etc.). Large eddy simulation (LES) then appears as a natural tool for the analysis ofcombustion instabilities in practical systems. LES can also be used to numerically testpassive or active control techniques. Flows submitted to such instabilities are con-trolledbylargevorticesandLESmaybeeasier inthesesituations thaninmorestandardturbulent reacting flows where an extended range of eddies has to be incorporated todescribe turbulence and chemistry interactions [22]. Control strategies may also be


integrated in thesesimulations, asdemonstrated inMettenleiter etal. [45].This topic isbeyondthescopeof thepresentchapterandwillnotbecoveredinwhatfollows.Wenowexamine a basic tool in combustion instability analysis, which consists in examiningthe rate of change of the acoustic energy density in the system. An acoustic energybalance equation is derived and this is then used to define three complementarystrategies applicable to combustion instability control.

16.2.3Acoustic Energy Balance

Determination of an acoustic energy balance in reacting systems is a challengingproblem. The classical work of Rayleigh [46] provided a criterion for oscillationgrowth but his expression did not account for the different processes defining theperturbed energy in the system.More recently, efforts have focused on the derivationof balance equations for energy flow perturbations in compressible multi-speciesreactingmedia [47, 48]. One difficulty has been to define acoustic energy and acousticfluxes under general flow conditions [49–51]. For vanishingly small Mach numberflows, M� 1, the analysis is easier and these quantities can be obtained directly bycombining the preceding set of equations for first-order flow perturbations. Notably,second-order energy density and flux expressions cannot be defined in general usingequations obtained for such first-order perturbations, and the derivation carried outbelow for zero Mach number conditions cannot be extended to more complex flows.Multiplying Equation (16.15) by p1/r0 and taking the scalar product of Equation (16.6)by r0v1, one obtains two expressions for squared pressure and velocity fluctuations:

qqt

12

p21r0c

20

� �þ p1r � v1 ¼ _q1p1

r0cpT0ð16:21Þ

qqt

12r0v

21

� �þ v1 � rp1 ¼ 0 ð16:22Þ

Adding these relations yields a balance for acoustic energy density:

qeqt

þr �F ¼ S ð16:23Þ

where:

e ¼ 12

p21r0c

20þ 1

2r0v

21 is the acoustic energy density ð16:24Þ

F ¼ p1v1 is the acoustic energy flux ð16:25Þ

S ¼ c�1r0c

20

_q1p1 is the acoustic energy source ð16:26Þ

In the absence of heat release rate fluctuations q_1¼ 0, flow perturbations areisentropic (s1¼ 0), and one retrieves the classical acoustic energy balance [24]:


qe=qtþr �F ¼ 0

The balance equation obtained previously can be used to analyze the stability ofunsteady reacting systems with respect to acoustics. The local energy balance (16.23)should first be integrated over a period of oscillation T¼ 2p/v and over thecombustor volume V delimited by the surfaces S, yielding:

qqt

ðVedV

� �þ

ðS

p1v1 � ndA� �

¼ 1T

ðV

c�1r0c

20

_q1p1dV ð16:27Þ

where:

hai ¼ 1=TðTadt

denotes the time average operator. This relation determines the net rate of change ofacoustic energy:

E ¼�ð

VedV

�

in the combustor volume V. If this net energy rises the system will be acousticallyunstable. In the absence of net acoustic fluxes at the boundaries or unsteadycombustion, E remains constant. In reality, E decays with time because of internaldissipation due to energy exchanges with the boundaries and with relaxationprocesses inside the flow, which are not included in this analysis [52, 53]. Thesource termon the right-hand side of the acoustic energy balance features the productof the pressure by the non-steady heat release rate and can be referred to as theRayleigh source term. This term can have positive or negative signs depending on thephase between pressure and heat release rate oscillations in the reaction region [1, 10,54]. The system is potentially unstable for positive values of the Rayleigh source termbut this is only a necessary condition for instability growth and it is important to takeinto account acoustic fluxes at the boundaries [28]. These energy fluxes modify therate of change of acoustic energy as shown by Equation (16.27). They also mainlydetermine limit cycle levels when the system has reached saturation (i.e., when dE/dtequals zero for a finite level of amplitude fluctuation).

16.2.4Conclusions on Control Strategies

Expression (16.27) indicates that different possibilities can be exploited to hinderacoustically coupled combustion instabilities, leading to the following procedures:

. The first set ofmethods consists inmodifying acoustic fluxes at the boundaries toincrease acoustic dissipation. This is widely used and studied in various config-urations, which will be evoked in what follows. On can imagine passive and activecontrol technologies acting on the system boundaries, to counterbalance theRayleigh source term. We will use the expression �direct control strategies,� as


designated for example in Poinsot et al. [55], since this set of methods directlymodifies the acoustic energy balance equation (16.27).

. The second set aims at reducing the Rayleigh source term. This can be achievedeither by acting on the combustion dynamics itself bymodifying q_1 or by changingthe injector and burner geometries to modify the pressure distribution p1 in thereaction region. This has mainly led to the development of strategies based onmodifications of the air or fuel injection systems, these modifications beingpotentially passive or active. This can be considered as belonging to the �indirectcontrol strategies� [55].

Starting from these definitions, it is possible to elaborate three control schemes:

1) Passive control methods (PCM). In this case, the dissipation of acoustic energy isimproved by damping systems (acoustic liners, Helmholtz resonators, quarterwave cavities, and various other devices), which enhance absorption (directcontrol strategies).

2) Flame dynamics control methods (FDCM). The flame geometry has a profoundeffect on the combustion response to incoming perturbations. The modificationof the flow and flame configurations can be used to control the system response.The dynamics of the injection system itself can also be designed to decouple thesystem from the manifold feeding reactants into the combustor. FDCM areusually indirect control strategies.

3) Active control methods (ACM). ACM aim to modify the feedback loop to hinderthe resonant coupling between unsteady combustion dynamics and systemacoustics. ACM have been developed using various strategies, ranging from thecontrol of injection parameters to the active modulation of acoustic boundaryconditions. This means that most of the ACM for combustion instabilities areindirect control strategies, but a few �anti-sound� methods are direct controlstrategies.

16.3Passive Control Methods

As mentioned in Section 16.2.4, passive methods mainly consist in the optimalplacement of damping devices, which are meant to increase the acoustic absorptionin the energy balance (acoustic fluxes in Equation (16.27)). For practical reasons, thisis usually done near the system boundaries (fuel and air inlets, outlets, walls),excepting perhaps baffles, which can be placed anywhere (see below). Such dampersaremainly designed to absorb acoustic energy at a selected frequency (note, however,that liners feature dissipation over a broad range of frequencies).

Baffles [56], quarter-wave resonators [57], Helmholtz resonators [58], acousticliners, and perforated panels [28, 59, 60] are among the most common systems usedin PCM. Baffles are mainly used to modify the frequency and modal shapes incombustors and reduce the level of velocity fluctuation in the flame region. Baffleswere successfully used in the case of very strong combustion instabilities encoun-tered during the development of the F-1 motor, the booster engine of the Saturn


rocket [6]. In this case, baffles were installed directly on the injector plate and used todamp transverse velocities associated with specific resonant modes (Figure 16.1).This could be achieved by placing the baffles in the reactive region, thus requiringcooling to avoid their destruction under hot fire conditions. Acoustic liners are usedinmany noise control applications to reduce the level of an incident acoustic wave. Incontrast with other devices, liners feature dissipation over a broadband, which isuseful when the frequency to be damped is not well known or varies with time [61].Liners are also used for thermal control likefilm cooling, but the acoustic and thermalefficiencies may require different arrangements for optimum performance.Helmholtz resonators and quarter-wave cavities have been successfully used todamp combustion instabilities in various situations. However, these devices havea narrow absorption bandwidth, meaning that the frequency to be damped must beprecisely estimated and constant in time. In systems featuring several unstablemodes, or prone to frequency shifting, multiple devices must be incorporated, asshown by Zhao and Morgans [62] who propose to insert a series of Helmholtzresonators.

Searby et al. [63] have recently reviewed the literature in this field and describe theviscous damping mechanism for liners and cavities. They associate absorption withviscous drag and heat transfer at the cavity walls. This linear mechanism induces anacoustic resistance, which is independent of the acoustic amplitude. According toSearby et al. [63], this mechanism dominates at low acoustic amplitudes. The secondmechanism arises from the formation of eddies at the exit from the damping cavity(and also internally in the case of a perforated liner or a Helmholtz cavity). This is anonlinear mechanism, and the associated acoustic resistance increases with thevelocity of the flow at the entrance to the cavity [64–66]. The latter mechanismdominates at high acoustic amplitudes. According to the authors, the relativecontribution of these twomechanisms is described by the acoustic Strouhal number,

Figure 16.1 Schematic view of the baffles used in the case of very strong combustion instabilitiesencountered during the development of the F-1 motor, the booster engine of the Saturn rocket.(From Yang and Anderson [6].))

16.3 Passive Control Methods j413

Stac¼vR/u0, where R is the radius of the cavity exit and u0 is the amplitude of theacoustic velocity at the cavity exit [67]. For Stac� 1, the viscous mechanismsdominate.

16.3.1Sound Absorption by Acoustic Liners

In acoustic liners, sound is absorbed as it travels into the material and the associatedvelocity is reduced by friction. The effectiveness of a liner may be quantified bycomparing the incident acoustic energy flux I1 to the flux emerging from the linedsection I2. This is usually designated as a transmission loss (TL) given in decibels: TL(dB)¼ 10log10(I1/I2). This loss depends on the type ofmaterial, thickness of the liner,and acoustic propagation mode. The liner is usually characterized by its impedanceZ¼R� iX, which is defined as the ratio of pressure to normal velocity fluctuation atthe liner face: Z¼ p/(v�n). It is often more convenient to use a dimensionlessadmittance obtained by dividing the characteristic impedance in the medium bythe liner impedance: b¼ (r0c0)/Z. This complex number is usually written in theformb¼ j þ is. It is then instructive to consider as amodel problem the propagationof plane waves in a two-dimensional duct featuring an absorptive liner on one of itswalls. The transverse size of this channel is a. Considering harmonic wave prop-agation in this channel, the pressure field satisfies the Helmholtz equation:

q2pqx2

þ q2pqz2

þ k2p ¼ 0 ð16:28Þ

The boundary condition at the liner wall can be written (qp/qx)x¼a¼ ikbp. On thelower wall the admittance vanishes as the wall is rigid: (qp/qx)x¼0¼ 0. One thenconsiders modal solutions of this boundary value problem in the form:

pðx; z; tÞ ¼ AyðxÞexpðikjj�ivtÞ

Inserting this expression in theprevious equations one obtains a transverse boundaryvalue problem for y:

d2ydx2

þ k2?p ¼ 0;dydx

� �x¼a

¼ ikb;dydx

� �x¼0

¼ 0 ð16:29Þ

where the transverse wavenumber is such that:

k2? ¼ k2�k2jj

The general solution of this problem can be written as:

y ¼ A cosk?xþB sink?x

The transversewavenumber is the solution of a characteristic equation deduced fromthe boundary conditions at the upper and lower walls:

�k?a sin ðk?aÞþ ika b cosðk?aÞ ¼ 0 ð16:30Þ


This equation can be solved numerically in the general case, providing complex valuesof the transverse wavenumber, k?, which in turn yield values of the longitudinalwavenumber k|| anddetermine the longitudinal attenuation of thepressuremode.Theproblem can be worked out analytically by assuming that the liner admittance nearlyvanishes so that the walls are almost rigid boundaries. It is then possible to expandsolutions of the characteristic equation around the solutions corresponding to a rigidwall duct. Considering only the plane wave mode one finds that:

k2jj ’ �ikb=a

so that the axial wavenumber is given by:

kjj ¼ kþ ib=ð2aÞþOðb2Þ ð16:31Þ

The real part of this wavenumber:

ReðkjjÞ ¼ k�s=ð2aÞ

indicates that the phase velocity in the axial direction is slightly perturbed by thepresence of the liner. Its imaginary part:

ImðkjjÞ ¼ j=ð2aÞ

provides the rate of attenuation inducedby the liner.Onefinds that thepressurefield isdamped exponentially, pffi exp(�az) where the damping rate is a¼ j/(2a). Thetransmission loss per unit length of duct is deduced from this expression:

TLðdBm�1Þ ¼ 10 log10½exp j=ð2aÞ� ¼ 4:34 j=ð2aÞ ð16:32Þ

Damping is directly proportional to the real part of the liner admittance and it isinversely proportional to the duct width a.

Liners are often constituted by a set of small perforations and eventually backed bya resonant cavity [56]. Acoustic damping can also be augmented using a bias flowthrough the perforations [52]. Sound absorption is then increased due to vorticityproduction. In this case, damping is mainly due to a transfer of acoustic energy intohydrodynamical perturbations, which are rapidly dissipated by the flow. This takesthe form of vortices shed from the aperture lips when facing pressure perturbations,which are then conveyed away from the aperture by the bias flow. The absorptioncoefficienta¼ 1� |R|2, whereR is the reflection coefficient from the perforated plate,then takes higher values than without mean flow, but is, however, limited to valueslower than 0.5 under small acoustic perturbations. Absorption properties can begreatly enhanced when a perforated plate traversed by a bias flow is used incombination with a back resonant cavity, as shown by Hugues and Dowling [60].It is then possible to obtain a perfectly absorbing liner for pure tones when the biasflow velocity is adequately chosen. These results are, however, limited to smallacoustic perturbation levels, and one needs to consider the effects of the sound levelfor combustion instability analysis, in which the oscillation amplitude may reach


a sizeable value of the mean pressure. An example is presented in Figure 16.2 for aperforated plate with a bias flow and backed by a cavity featuring good absorptionproperties for small sound levels. These properties are shown to depend on the soundlevel and are smeared out when the oscillation level is increased [68].

16.3.2Quarter Wave Tubes

Quarter wave tubes are usually plugged on the feedmanifold or combustion chamberas a side branch. The tube responds to pressure oscillations, which arise in the system.A simple way to understand how quarter wave cavities work is to consider a positivepressure excursion at the tube entrance. This perturbation will produce a pulse in thetube that propagates to the tube end in a time l/c0, where l is the tube length. This pulseis reflected and arrives at the tube entrance after a further time l/c0. If this positivepulse reaches the manifold when the pressure passes through a minimum (i.e., afterone half period) a destructive interference will take place and this will reduce thepressure level. This will happen when 2 l/c0¼T/2 or equivalently when c0/4l¼ f. Thiscondition corresponds to a situation where the tube length is exactly equal to thequarter of a wavelength, l¼ l/4. The quarter wave cavity behaves as a half-openresonator, its eigenfrequency can be tuned by adjusting its length L according to f¼ c0/4l. In particular cases, it is possible to estimate the eigenfrequencies of combustionchambers by the eigenmodes of cylinders (Oschwald et al. [69]). Using usualassumptions, one can derive the frequencies of the transverse eigenmodes of acylinder of radius R and write:

Figure 16.2 Modulus of the reflectioncoefficient (R) of a perforated plate with a biasflow and backed by a cavity as a function offrequency and sound pressure level inside thecavity. Plate thickness t¼ 1mm, holes radiusr¼ 0.5mm, hole interspace d¼ 7mm, back

cavity length l¼ 150mm, bias flow velocityu¼ 3ms�1. The sound pressure level (SPL) ismeasured at the rear side of the back cavity.SPL¼ (*) 110, (–) 128, and (– –) 140 dB(adapted from [68]).


f ¼ anmc

2pRð16:33Þ

whereanm is the abscissa of themth extremumof theBessel functionof thefirst kindoforder n. Tuning the quarter wave cavity means adjusting the cavity length so that bothfrequencies are equal. The length of the cavity is then given by the relation:

Lnm ¼ pR2anm

ð16:34Þ

For the first tangential mode, a11¼ 1.841, resulting in a cavity length given by:

L11 ¼ 0:8532R ð16:35ÞUsing this adjustment, the cavity will respond at its maximum for the first

eigenfrequency of the combustion chamber. This response will dissipate acousticenergy through the processes described in the beginning of this section. However, inpractice, applications of l/4 cavities do not always result in the expected performance.Several reasons can be evoked to explain these difficulties, especially in the case ofdampers used in combustion applications [69]. In these cases, the temperature fieldin the combustor is far from uniform and the corresponding eigenfrequencies aremuch more difficult to predict [29]. The choice of the correct length will then bedifficult, and several cavities with different lengths might be necessary.

16.3.3Helmholtz Resonators

Helmholtz resonators are often used to damp combustion instabilities. They consistof a large cavity connected to a smaller neck and operate at very low frequencies. Thelarge volume is used as a capacity to damp external pressure perturbations. Thefrequency response of a Helmholtz resonator is characterized by a bell shaped curveassociated to a bulk oscillation of theflowwithin the device. Flowperturbations in thissystem can be assumed to be spatially uniform.

16.3.3.1 Helmholtz Resonator EquationsHelmholtz resonators consist of a large volume connected to the environment bya narrow tube.When this system is submitted to an external pressure perturbation itresonates at a specific frequency and energy is absorbed by friction and other lossmechanisms. This can be analyzed as follows. The geometry features a large cavity ofvolumeV1¼S1l1 connected to a small tube of section S2 and length l2. This system issubmitted to external pressure perturbations, p2(t), which generate in turn a pressureoscillation, p1(t), within the volume. Assuming isentropic perturbations (p1 ¼ c20r1)within the volume V1, a mass balance at the volume outlet yields:

dp1dt

¼ � r0c20S2V1

v2 ð16:36Þ

This fixes the mass flux oscillation at the tube inlet with a flow velocity v2(t). Thelength l1 of the neck is small enough to consider that the fluid executes bulk


oscillations. Amomentumbalance for themotion of the air columncomprised insidethe tube gives:

r0S2l2dv2dt

¼ ðp1�p2ÞS2�Rv2 ð16:37Þ

where p1 and p2 denote pressure fluctuations on both sides of the neck and�Rv2 is aresistive force opposed to the flow motion that is assumed to be proportional to thebulk velocity oscillation v2. By combining Equations (16.36) and (16.37) one obtains arelation forflowperturbationswithin theneck drivenby pressure perturbations at theHelmholtz resonator outlet:

Md2v2dt2

þRdv2dt

þKv2 ¼ �S2dp2dt

ð16:38Þ

where M¼r0S2l2 is the mass of the air column set in motion by the externalperturbation, R and K¼ cp0S

22/V1 are associated, respectively, to the resistive force

within the tube and to the restoring force associated to the air capacity comprised inthe volumeV1. It is convenient to divide all components by themass of the air columnM and write Equation (16.38) in the form:

d2v2dt2

þ 2ddv2dt

þv20v2 ¼ � S2

Mdp2dt

ð16:39Þ

where:

2d ¼ R=M is the damping rate ð16:40Þ

v20 ¼ K=M is the Helmholtz resonance frequency ð16:41Þ

This expression for the Helmholtz frequency v0 matches experiments when thephysical length l2 is replaced by an effective length l2 þ e, where e is an end correctionthat accounts for the inertia of themass of air set inmotion outside the tube. Notably,the pressure fluctuation at the system outlet p2ðtÞ ¼ pext2 þ pint2 results from thecontributions of the external pressure perturbation pext2 associated to the systemimposing the external perturbation and of the radiated pressure pint2 associated tothe bulk velocity oscillation v2 at the system exhaust. At resonance these contri-butions can be of the same order – see Durox et al. [70] for a detailed analysis oftheir respective contributions. It is possible to characterize the response of theresonator using twomicrophones and a driver unit. Bymaking use of Equation 16.36and a time integration of Equation (16.39), one finally obtains a relation betweenpressure perturbations p2(t) and pressure oscillations p1(t) induced inside the cavityvolume:

d2p1dt2

þ 2ddp1dt

þv20p1 ¼ v2

0p2 ð16:42Þ


16.3.3.2 Frequency Response CurveIn the frequency domain, one can form the ratio between the internal pressureresponse:

p1ðtÞ ¼ ~p1exp ð�ivtÞdriven by external pressure perturbations p2ðtÞ ¼ ~p2exp ð�ivtÞ:

~p1~p2

¼ 1

1� v

v0

� �2

�i2dv0

� �v

v0

� � ð16:43Þ

It is convenient to examine the square modulus of this ratio, which can be extractedfrom power spectral analysis of microphone signals:

j~p1=~p2j2 ¼1

1� v

v0

� �2" #2

þ 2dv0

� �2 v

v0

� �2ð16:44Þ

This response takes a unit value when v is zero and reaches its maximum at theresonance frequency v0. The amplitude is fixed by the damping rate d:

j~p1=~p2j2ðv ¼ v0Þ ¼ v20

4d2ð16:45Þ

Without losses (d¼ 0) the peak value would be infinite. In practical Helmholtzresonators, damping fixes the resonance peak value. The Helmholtz resonancefrequency v0 can thus be easily determined from experiments by examining thefrequency at which the maximum peak is observed, but it is more difficult todetermine the damping rate d from the peak level. A more precise method is todetermine the damping rate d by examining the frequency response curve Equa-tion (16.44) at half-values of themaximumpeak value. This is obtained by finding theroots of the following equation:

1� v

v0

� �2" #2

þ 2dv0

� �2 v

v0

� �2

¼ 12

2dv0

� �2

ð16:46Þ

Solutions are straightforward and take the general form:

v

v0

� �2

¼ 1� 12

2dv0

� �2

� 2dv0

1þ d

v0

� �2" #1=2

ð16:47Þ

Damping is usually small compared to theHelmholtz resonance frequency d/v0� 1and the previous expression simplifies:

v

v0’ 1� d

v0ð16:48Þ

Ameasure of the frequency widthDv of the squaredmodulus of the response curveat half the maximum value yields the damping rate of the Helmholtz resonator:


Dv ¼ 2d ð16:49ÞThis is, for example, illustrated in Figure 16.3, where the response of three differentburners to an external acoustic excitation without flow or combustion is analyzed.The damping coefficients are deduced from the resonance bandwidth.

16.4Flame Dynamic Control Methods

16.4.1Introduction

As indicated previously, control of combustion instability is most often achieved byaugmenting the damping in the system. An alternative solution, which is less wellexplored but has considerable potential, consists in reducing the level of drivingassociated with the injection system and flame response. One tries to alleviatemechanisms that drive instabilities by modifying the flame and/or injector dynam-ics. It is known for example that large-scale vortices interacting with flames in dumpcombustors constitute an important driving mechanism leading to self-sustainedoscillations [10, 71]. These processes can be reduced by replacing the standardconfiguration with a �staircase� geometry as proposed, for example, by Schadowet al. [72] and Schadow and Gutmark [73]. The aim is to prevent the flame/vortexcoupling by generating smaller scale vortices, which enhance mixing, making theflow less susceptible to self-sustained oscillations. Following similar arguments it isalso proposed to insert tabs protruding in the reactant jets, or small-scale-vortexgenerators to break the largest flow structures. Mixing can also be enhanced by

Figure 16.3 Response of three burners B1, B2,and B3 to an external acoustic excitation withoutflow or combustion; p2 ¼ external pressurefluctuation, p1 ¼ internal pressure fluctuation.

The damping coefficients deduced from theresonance bandwidth are, respectively, 2d¼ 11,8, and 6 s�1 for B1, B2, and B3 [12].


replacing the standard circular injection geometry by a non-circular cross section.This reduces the lifetime of coherent ring vortices generated at the nozzle [74].

Combustion instabilities often appear because the feed system is coupled to thecombustion cavity and senses the pressure oscillations induced by combustion.Pressure fluctuations in the chamber induce a modulation of the flow rate at theinjector, which in turn leads to a perturbation in heat release. This mechanism isoften involved in system instabilities, which aremainly observed in the low frequencyrange. This has been extensively investigated in rocket propulsion. One importantresult obtained in this field is that the feed manifold can be decoupled from thechamber dynamics by including a sufficient amount of pressure drop at the injector.The head loss is typically Dp/p� 10–15%.

Another approach consists in adjusting the fuel injection positions inside thedifferent channels delivering reactants to the combustor, as proposed by Steeleet al. [75] and in Lieuwen and Yang [4]. The method uses the differential responseof the fuel injectors submitted to pressure oscillations propagating in the feedingmanifolds. Equivalence ratio fluctuations resulting from this process are convectedfrom the injector to the flame region at themean flow velocity andmay drive acousticoscillations if their convection delay fulfills some phase conditions [76]. It is thenpossible toprevent thisphenomenonby changing the fuel injector locations tomodifytheconvectivedelaysandescapefromthe instabilityband.Controlmethodsbelongingto this second group focus on decoupling the flame dynamics from the systemacoustics bymodifying the underlying causes. One novelmethod proposed byNoirayet al. [77] to reduce oscillations inmultiple flame configurations relies on this generalprinciple. Control is achieved by inserting a dynamical phase converter (DPC). This isused to diminish the global response of the reaction zone to an acoustic modulation.The flame collection is made to respond in a nearly neutral fashion to externalperturbations. This is achieved by the DPC, which introduces a delay in the perturba-tions that reach the flame. By suitably adjusting this delay it is possible to obtainopposing motions of the flames composing the collection, thus notably reducing thelevel of unsteady heat release, eventually preventing the onset of self-induced oscilla-tions. This concept and its practical implementation are illustrated in Figure 16.4,which shows the flame motion in a system equipped with the DPC. The flames aresubmitted to acousticmodulationsoriginating from theupstreamsideof themultipleinjectors. In the absence of the DPC themotion of the two flames is coherent and thelevelofheatreleasefluctuationisamaximum(Figure16.4a).WhentheDPCisinsertedin the system, the flames respond in opposing ways and the level of unsteady heatrelease is minimized (Figure 16.4b). Experimental results indicate that the methodcan be used to avoid a coherent response of the flames to incident perturbations andthat this enhances the stability of the system.

16.4.2Global Modification of the Flame Geometry

Interestingly, on a global point of view, Equation (16.27) also indicates that the sameeffect can be obtained by reducing the contribution of the source term:

16.4 Flame Dynamic Control Methods j421

S ¼ 1T

ðV

c�1r0c

20

_q1p1dV

This term is aminimum either when the heat release rate fluctuations aremade tovanish or when they are out of phase with respect to the pressure. In premixedsituations, variations of the heat release rate most often correspond to variations ofthe flame surface or to perturbations of the mixture equivalence ratio. To control thesource term, it is then necessary to link heat release fluctuations to acoustic orequivalence ratio perturbations. Flames are barely sensitive to pressure oscillationsin many combustion systems like gas turbine engines, thermal power units, andheating systems (the situation may be different in rocket engines where meanpressure and pressure fluctuations are higher) and the combustion dynamics isgoverned by velocity and equivalence ratiofluctuations. The link between velocity andheat release perturbations has been investigated extensively. This has provided flametransfer function (FTF) data defined as [37]:

Fð f Þ ¼_Q0= _Q

v0=�v¼ Gð f Þeijð f Þ ð16:50Þ

This characterizes unsteady heat release rate fluctuations generated by velocityperturbations; G represents the gain, j the phase difference between the velocityfluctuations and the heat release rate fluctuations, and f is the frequency. Thesequantities are determined experimentally by submitting the flame to an acousticexcitation generated by a driver unit. The response is measured with a photomul-tiplier collecting the light emission by radicals likeOH

.

or CH.

. This radiation in turncan be linked to the heat release rate in the premixed case. An early review on flametransfer functions in the case of premixed laminarflames is presented byMatsui [78].Theoretical expressions of the FTF were also derived in Fleifil et al. [35], Ducruix

Figure 16.4 Illustration of a system equippedwith the DPC (dynamical phase converter).Flames are submitted to strong acousticmodulations. (a)WithoutDPC themotionof thetwo flames is coherent and the level of heat

release fluctuation is maximum. (b) With DPC,the flames respond in opposing ways and thelevel of unsteady heat release is minimized.(From Noiray et al. [77].)


et al. [37], Lee and Lieuwen [79], and Schuller et al. [38] using a kinematic descriptionof the flame treated as a discontinuity of G-equation type. Some experiments havealso been carried out on turbulent flames [80]. The influence of the perturbationamplitude on the flame transfer function is analyzed in some recent studies [36, 81].As shown in this last reference, it is natural to introduce a flame describing function(FDF) concept to describe the dependency of the flame transfer function with respectto the input amplitude level:

Fð f ; u0Þ ¼ Gð f ; u0Þeijð f ;u0Þ ð16:51Þ

The gain is in general similar to that of low-pass filters, meaning that velocityfluctuations (or equivalence ratio fluctuations) do not cause fluctuations of the heatrelease rate at high frequencies. They can fold the reactive front but the contributionof all the wrinkles to the global heat release rate variation remains small at everyinstant of the cycle. In contrast, in the low frequency range, it is possible to observea gain exceeding unity, indicating that the flames behave like amplifiers. The relativeheat release rate fluctuation is then higher than the relative velocity fluctuation. Inmost cases, the phase of the flame transfer function evolves almost linearly with thefrequency. It is then possible to associate a time delay twith this phase difference viathe relation j¼ 2pft. The delay t corresponds to the time necessary for the velocityperturbation (or equivalence ratio perturbation) to reach the reactive front. In the caseof premixedflameswith a constant equivalence ratio, similarity rules are applicable tothe flame transfer functions. They are based on the flame geometry, on themass flowrate, and on the laminar burning velocity [35–38, 79]. In the conical flame case, thetransfer functions essentially collapse when plotted in terms of a reduced angularfrequency:

v ¼ vRSl cosa

ð16:52Þ

where

v is the angular frequencyR a characteristic radius at the base of the flamea is the flame half angle with respect to the burner axisSl the laminar burning velocity.

Thus, two flames presenting identical geometrical shapes feature the same responsefor the same reduced frequency. The flame describing function data can be used todefine a dynamic flame controlmethod. Since the flames behave like low-pass filters,it appears possible tomove the resonance frequency of the system to a value forwhichthe gain is sufficiently low such that losses become dominant and prevent the growthof oscillations. Predicting or measuring the flame describing function provides thesensitive range of frequencies. This range turns out to be strongly dependent onthe flame shape, as shown below in Figures 16.6 and 16.7. It is often possible tolink the source term S of Equation (16.27) given above to the flame describingfunction F (f,u0); the phase evolution brings fundamental information to determine


whether a system is prone to combustion instabilities. In confined flows, when astanding longitudinal eigenmode is established in the combustion chamber, theacoustic velocity is in quadraturewith the pressurefield. As a consequence the sourceterm in thebalance of acoustic energywill be positivewhen the phase differencej liesbetween p and 2pmodulo 2p. This condition is also valid in the case of unconfinedflames of the type investigated by Durox et al. [70]. To damp the instability, one maythen modify the geometry so that the resonance frequency of the system is movedout of the band corresponding to the unstable phase difference ([p-2p] modulo 2p) ofthe flame transfer function.

A technique that can be used to avoid instability consists in radically changing thesteady state flame geometry, inducing a fundamental modification of its response toperturbations. To illustrate this concept, it is interesting to consider the simple andwell-controlled laboratory configurations described in Figure 16.5. The burner isaxisymmetric and it is equipped with a rod on the axis being used as a bluff-body, onwhich the flame can be attached or not. Three types of flames can be stabilized, withthe same conditions of flow and equivalence ratio: a conical flame, an �M�-flame, anda �V�-flame. The shape of the flame depends on the ignition procedure. The bulkvelocity is 2.12m s�1, the laminar burning velocity is equal to 0.385m s�1, the outletdiameter is 22mm and the diameter of the rod 6mm. Flame describing functionshave beenmeasured and are presented in Figures 16.6 and 16.7. More details can befound in Durox et al. [82].

The three flame transfer functions exhibit different shapes. The conical flamehas a cut-off frequency much lower than the two others. Beyond 300Hz, its gain isnull, despite the highly wrinkled flame observed in this frequency range. The flamethat presents the widest response band is the �M�-flame. To obtain a cancellation ofthe gain for the same conditions, the frequency must be higher than in the conicalor �V�-flame cases. The shapes of the gain functions of the various flames aremarkedly different. The �V�-flame presents an important gain value close to100Hz, related to coherent shedding of vortices from the burner lips, whichstrongly interact with the flame front. The phase evolutions with frequency alsofeature different slopes. Knowing the flame transfer function phase evolution with

Figure 16.5 (a) Schematic representation ofthe burner outlet. The outlet diameter is 22mmand the rod diameter 6mm; (b)–(d) flameimages with no excitation. The system is fed

with a mixture of methane and air with anequivalence ratio of 1.08. Bulk velocity (Ub) is2.12ms�1. (b) Conical flame; (c) �V�-flame;(d) �M�-flame (adapted from [70]).


Figure 16.6 Flame describing function for the conical flame: u0 corresponds to the RMS axialvelocity measured at 7mm from the axis and at 0.7mm downward the burner outlet; Ub is the bulkvelocity (adapted from [70]).

Figure 16.7 Flamedescribing function for (a) the �V�-flame and (b) �M�-flame; same conditions asin Figure 16.6.


frequency it is thus possible to choose the best flame geometry to avoid instability.For example, at 140Hz, the phase difference for the �M�-flame is betweenp and 2p,lying, in an unstable band, whereas for the �V�-flame it is compoised between 2pand 3p, lying, in a stable band. These results show that a judicious choice of theflame geometry can be helpful to define a favorable condition for the stability of acombustion chamber.

16.5Active Control Methods

Active control methods were first envisaged in the 1950s during the early develop-ment of high-performance rocket engines [83–85]. At that time, severe combustioninstabilities were encountered during various tests, and it was proposed to activelyinject perturbations into the thrust chamber to decouple the physical processesresponsible for the growth of oscillations. While the active control principle was welldescribed in these early articles, it was not possible to test the concept on real engines,because of technical difficulties. Active control methods for combustion instabilitiesdiffer from �anti-sound� control methods devised to reduce acoustic field intensities(see Reference [86] for a review of this specific subject) because the control system ismeant to act on the source of instability (the flame and the flow) and not only on itseffects (the acoustic field).

Active control can be achieved in many different ways. In general one considersthat thefluidflow, combustion, and acousticwaves propagating in the system formaninstable loop leading to oscillations. This process can be detected by a sensor B, whichsenses one of the state variables characterizing the system. This signal is processedthrough a control algorithm V, and the output signal is sent back to the flow throughan actuator system A. The sensor B, algorithm V, and actuator A can take manydifferent forms [87], as briefly summarized in the next section. Starting from thesegeneral principles, it is convenient to identify various types of methods [1, 87]:

. If the control system does not use a time varying input from the combustionsystem (feedback to determine the control action), the systemwill be called �open-loop.� If, in contrast, information detected by a sensor is returned to the systemthrough a controller, this will be designated as �closed-loop.� This second optionwill be considered in the rest of the section.

. If the controller time-scale is much greater than that of the instability, the systemeffects a �trim adjustment.� If the time-scales associated with the controllerresponse and with combustion instability are of the same order, the controlsystem will be called �fast response.� Here, again, only this second option will beconsidered.

. Controllers with basic response characteristics, or transfer function, that do notchange with time will be called �fixed-parameter� controllers and those that haveinternal algorithms allowing time-varying transfer functions through parameteroptimization will be called �adaptive� or self-tuning controllers.


An illustration of these definitions is proposed in Figure 16.8. The control systemconsists of four elements, including a microphone or a photomultiplier (PMT)(sensor B). Air modulation is obtained through a Moog electromagnetic directlydriven proportional valve (DDPV, actuator A). An identification technique, the leastmean squares (LMS) algorithm, is used to identify the system for control (controllerelements V). This forms a fast-response closed-loop systemwith variable parameters.

First demonstrations on practical systems were achieved during the late 1980s. Atthat time, active control was successfully used to suppress or reduce oscillations inRijke tubes [88], premixed laminar and turbulent burners [89], model-scale reheatchannels [14, 15], and non-premixed turbulent combustors [10]. All these studiesindicated that active control methods could be used to suppress combustioninstabilities. While the initial demonstrations were carried out on small laboratorycombustors, control has been implemented in some larger scale systems and hasbeen put in operation on high power burners [4, 8, 90]. Most of the development ofactive control strategies has relied on experiments on model-scale combustors. Thenumerous publications existing in this field include the recent work of Paschereitand Gutmark [91], Uhm and Acharya [92], Barbosa et al. [93], Morgans and Stow [94],and Tachibana et al. [95]. These studies have revealed the potential of the methodsand the technical limitations of the control system components (sensors B,algorithms V, and actuators A). These issues are successively considered in thenext subsections. Complete reviews can be found in the literature; see among othersMcManus et al. [87], Candel [1], Lieuwen and Yang [4], and Dowling andMorgans [96].

Figure 16.8 Schematic view of an active control experimental setup. The control loop is presentedfor both PMT (photomultiplier tube) and microphone sensors. (Adapted from Bernier et al. [97].)

16.5 Active Control Methods j427

16.5.1Sensors B

The sensor B can be pressure transducers measuring pressure fluctuations or a hotwireprobedetecting local velocityfluctuations locatedupstreamof theflames.Usingaprobe downstream of the flames is not convenient because of the high temperaturefield. An optical sensor like a photomultiplier tube can be used to detect the lightemission signal from the radicals like C

2, CH.

, or OH.

. It is not easy to determinethe most suitable sensor, and a limited number of studies are dedicated to thisquestion (see, for example, Poinsot et al. [55], Bernier et al. [97] or Uhm andAcharya [92]).

These different sensors are used in Poinsot et al. [55] to control combustioninstabilities in a turbulent combustor. Using the reaction rate or the inlet pressuresignal as sensor seems roughly equivalent while the air velocity signal yields poorerresults.Thereasonevoked in that article is that thehotfilmprobesignal containsmanyfrequency components corresponding to local turbulent fluctuations that are notrelated to the instability. These frequency components are amplified by the controlsystemwithout contributing to the control of the instability. The acoustic pressure andtotal C

2 emission(which is integratedovera largevolume)aremoredirectly related tothe process and hence more adequate as input signals to the control loop.

16.5.2Actuators A

Actuation is theweak point inmost active control schemes. Active control requires aneffective actuator, which can be integrated in the system and will change some of thephysical mechanisms involved in the combustion process (vaporization, combus-tion, vortex shedding, etc.). Actuator effectiveness is the most critical issue andintroduces a complicated trade-off between bandwidth and power. One can imagineseveral examples of typical dynamic components used in active control system,including acoustic driver units fitted to an experimental combustor to excite acousticwaves, and servo-valves, which are used to control flow rates to produce a timevariation in its operating state.

An important choice in these devices iswhether to excite the airflow rate (amethodcalled direct control in Poinsot et al. [55]) or the fuelflow rate (amethod called indirectcontrol). While the apparatus needed for direct and indirect control are the same, thephysical principles of these systems are different:

. Direct control devices affect the acoustic field in the duct and can be viewed asa way of modifying the end impedances of the duct where the flame is stabilized.For direct control, driver units mounted on the air inlet duct can be envisaged.These units are excited and this mode of operation favors the longitudinal (ratherthan transverse) acoustic modes in the duct.

. Indirect control acts on the fuel flow rate only. The advantage of indirect control isthat themodulated flow rate is smaller (typically a small percent of the air volume


flow rate) and therefore requires much less energy. For indirect control, a singledriver unit plugged on the fuel supply line can be used. For gaseous fuel, it ispossible tomodulate the flow using direct drive valves (DDV), but the bandwidthis usually limited to about 400Hz [97]. Liquid injection is more difficult tomodulate [98, 99]. Fuel injection timing is an interesting method of depositingenergy at the right moment during an instability cycle [100].

16.5.3Active Control Algorithms V

From the beginning, efforts have beenmade to use concepts originating from controlsystems to describe the active control methods and evaluate the influence of thealgorithm on the controller efficiency. Block diagrams have been used extensively toformalize the control problem. Two general illustrations are proposed in Figure 16.9,adapted from Candel [1]. Figure 16.9a is a schematic representation of the acous-tic–combustion coupling and how active control can be used as an external controlloop on this coupling. As mentioned in Candel [1], this representation is lessstraightforward than it may seem.

Figure 16.9 (a) Schematic representation ofacoustic–combustion coupling and activecontrol using an external control loop; (b) blockdiagram representation of the system shown in

(a); (c) general principle of adaptive instabilitycontrol; (d) block diagram representation ofadaptive control of combustion [1].

16.5 Active Control Methods j429

Indeed, the controller acts on the acoustics of the system but the representationimplies that the path between actuator and sensor is also the path that induces thecoupledmotion in the unstable operation of the system. This is not always the case, aswill be seen in the next paragraph. The actuator modulates in many circumstancesa secondary fuel injection and it is the chemical conversion of the injected fuel thatinduces an acoustic wave. This wave then combines with the acoustic motionassociatedwith the instability in the system.Still, this representation canbe translatedintocontrolsystemsterminology,usingsymbols introducedpreviously(Figure16.9b).Asimplecontrollercanbededucedfromthesensitive timelagtheory,asalreadyshowninFigure16.8. In this situation, a timedelaygeneratorandanaudio-amplifier areusedto introduceamodulation inthevicinityof theflame,which is inphaseoppositionwiththe naturally coupled perturbations. Gain and phase shift control schemes have beenusedwith some success by Lang et al. [89], Bloxsidge et al. [14, 15],Hantschk et al. [98],Yu et al. [100], and Paschereit et al. [101]. In its simplest form an appropriate sensorsignal is delayed and the resulting signal drives the actuator. This technique needsagoodknowledgeofthesystemcharacteristicdelaysanditis limitedtoanarrowdomainofoperation.Low-ordermodeling isexploited inmanystudies, tobuild a controller thatis suitable for a given configuration (see among others Annaswamy et al. [102]).

However, V is expected to bemodified as the operating conditions of the combustorare changed, thereby changing the fluid mechanics and chemical time scales(Figure 16.9c). Work is currently in progress to develop more general and efficientprocessingalgorithmsbasedonoptimizationandadaptiveconcepts,makingitpossibleto automatically optimize the transfer function as the operating regime is changed. InthiscaseHrepresentsthecombustionprocess,coupledbyacousticfeedbackG.W is theLMSfilter,S1,S2 represent theactuator transfer functionandpartof thesecondarypathbetween actuator and sensor S¼S1S2, S is a filter representing the secondary pathS. Adaptive techniques can be envisaged to adjust the filter coefficients. Figure 16.10

Figure 16.10 Microphone signal (a,b) and DDV command signal (c,d) during transition fromuncontrolled to controlled modes (a,c) and from controlled to uncontrolled modes (b,d). Verticallines at t¼ 0.2 and 1.65 s materialize the transition. (Adapted from Bernier et al. [97].)


shows an example of adaptive control of combustion instabilities encountered in theexperimental setup shown in Figure 16.8. Much work has focused on self-tuningadaptive regulatorswith adaptivecontrollersdevisedon thebasis of aphysicalmodel ofthe system [42, 103]. Other schemes employ a direct identification of the burnerdynamics [97, 102,104–108]. In this last representation, thecombustor is the �plant� tobe controlled and its dynamical behavior is obtained from system identificationmethods (Figure 16.9d).

16.6Conclusions

This chapter has reviewed the many routes that have been explored to controlacoustically coupled combustion instabilities. A considerable amount of knowledgehas been accumulated on the application of passivemethods relying on quarter wavecavities, liners, and resonators. Passive control is often used in practice but requiresconsiderable testing and adjustments, leading to a long and costly process. Recentefforts have been made to devise dynamical control methods, which attempt toreduce the driving source, by modifications of the flame geometry or by suitablereduction of the flame sensitivity to incoming perturbations. It is also shown thatknowledge of the flame describing function can be employed tomove the combustoroperating point away from a band of instability or into a region where the flame gainis low. Much research has concerned active control methods with considerabletheoretical investigations and limited demonstrations on practical devices. Onechallenge for the future will be to apply theoretical strategies (state feedback, robustcontrol, model based adaptive control, etc.) to real systems.

One important technical aspect of the problem is related to the performance ofactuators and integration of these units into practical injectors. This is a criticaltechnology and some progress has been made, essentially by exploiting high-performance valves to modulate the flow of reactants (Figure 16.11). There are,however, bandwidth and amplitude level limitations. Advances are clearly needed to

Figure 16.11 Recent example of control of gas turbine combustion instabilities by active meanscarried out by Siemens AG, using methods worked out by IfTA GmbH.

16.6 Conclusions j431

augment the operating range, develop advanced concepts, and overcome currentobstacles. Integration of the actuator into the system also constitutes a centralchallenge. Another critical aspect of concern is that of sensors. Requirements interms of sensors are not often fulfilled by current technology. In many areas ofapplication, the technology does notmatch the level of accuracy and reliability neededfor practical usage. Further developments are also required in control algorithms forincreased performance and reduced sensitivity to noise and parameter variability.Finally, research efforts and systematic testing are needed to fill the gap and allowclosed loop control of combustion.

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handbook of combustion (online) || control of acoustically coupled combustion instabilities

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