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AIAA JOURNALVol. 44, No. 4, April 2006

Large-Eddy Simulation and Acoustic Analysisof a Swirled Staged Turbulent Combustor

Charles E. Martin, Laurent Benoit, and Yannick Sommerer

Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique, 31057 Toulouse, FranceFranck Nicoud

Universite de Montpellier II, 34095 Montpellier Cedex 5, France

and

Thierry Poinsot

Institut de M ecanique de Fluides de Toulouse, 31400 Toulouse, France

The analysis of self-excited combustion instabilies encountered in a laboratory-scale,swirl-stabilized combustion

system is presented. The instability is successfully captured by reactive large-eddy simulation (LES) and analyzed

by using a global acoustic energy equation. This energy equation shows how the source term due to combustion

(equivalent to the Rayleigh criterion) is balanced by the acoustic fluxes at the boundaries when reaching the limit

cycle. Additionally, an Helmholtz-equation solver including flameacoustics interaction modeling is usedto predict

the stability characteristics of the system. Feeding the flame-transfer function from the LES into this solver allows

to predict an amplification rate for each mode. The unstable mode encountered in the LES compares well withthe mode of the highest amplification factor in the Helmholtz-equation solver, in terms of mode shape as well as in

frequency.

Nomenclature

[A] = square matrix of size Nc = sound velocity, m/sDk = kth species diffusion coefficient, m

2/sE = efficiency functionEa = activation energy, cal/molE1 = instantaneous global acoustic energy term, Je1 = acoustic energy, J/m

3

F = flame thickening factor

F1 = instantaneous global acoustic fluxes, Wf = frequency, Hzi = square root of1N = number of nodes of the gridn = magnitude of the flame transfer function, Pa/mn = outward normalized normal vector[ P] = column vector of size N associated to an eigenmodep = pressure, PaR = perfect gas constant, cal/mol KSkc = kth species Schmidt numberS1 = instantaneous global Rayleigh term, Ws0L = laminar flame speed, m/ssT = turbulent flame speed, m/sT = temperature, Kt = time, su = velocity vector, m/s

Received 19 November 2004; revision received 21 April 2005; acceptedfor publication 31 July 2005.Copyright c 2005 by theAmerican InstituteofAeronautics and Astronautics, Inc. All rights reserved. Copies of this papermay be made for personal or internal use, on condition that the copier paythe $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose-wood Drive, Danvers, MA 01923; include the code 0001-1452/06 $10.00 incorrespondence with the CCC.

Ph.D. Student, Computational Fluid Dynamics Team, 42 Avenue G.Coriolis.

Research Engineer, Computational FluidDynamics Team, 42 Avenue G.Coriolis.

Professor, Mathematique; also at Institut de Modelisation et deMathematiques de Montpellier, Centre National de la Recherche Scien-

tifique, Place Bataillon.Research Director, allee du Professeur Camille Soula; also at Ecoule-ments et Combustion, INP de Toulouse, Centre National de la RechercheScientifique. Associate Fellow AIAA.

Yk = kth species mass fractionZ = local reduced acoustic impedance = polytropic coefficient0L = flame thermal thickness, m = cinematic viscosity, Pa/s = mass density, kg/m3

= time delay of the n model, s = phase of the flame transfer function = pulsation, rad/s

k = kth species reaction rate, mol/m3

sT = unsteady heat release, J/m3

Subscripts

L = laminarref = related to the reference point of the n model0 = steady part1 = fluctuating part

Superscripts

th = thickened quantity

= Fourier transformed

Introduction

C OMBUSTION oscillations are frequently encountered dur-ing the development of many combustion chambers for gasturbines.14 These oscillations cannot be predicted at the designstage, and correcting actions can be extremely costly at later stages.Testing burners in simplified combustion chambers is a commonmethod to verify their stability but is also an ambiguous approachbecause most experimentalistsknow that a given burner can produceunstable combustion in one chamber and not in another. Methodsprovidingstabilityanalysisbefore anytests are, therefore,requested.

Large-eddy simulations (LES) are an obvious choice for suchstudies: They are powerful tools to study the dynamics of turbu-lent flames. (See recent works on turbulent combustion.4,5) Multiplerecent papers have demonstrated the power of these methods.612

However, an important limitation of LES is its cost: The intrinsicnature of LES (full three-dimensional resolution of the unsteadyNavierStokes equations) makes it very expensive, even on todayscomputers. Moreover, even when it is confirmed that a combustor isunstable,LESdoesnot indicate whyandhow tocontrol it.Therefore,

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742 MARTIN ET AL.

tools are needed to analyze LES results but also to provide capac-ities for optimization and control of thermoacoustic oscillations inchambers.

A proper framework to analyze combustion stability is the waveequation in a reacting flow.4 Such an equation is complex to derivebecause most assumptions used in classical acoustics must be revis-ited in a multispecies, nonisothermal, reacting gas. For low Machnumbers, an approximate equation controlling the propagation ofpressure perturbations in a reacting gas is

0c20

1

0p1

2

t2p1

= ( 1)T1

t 0c

20 u1 : u1 (1)

where the subscript 0 refers to mean quantities and the subscript 1 tosmall perturbations. Here T1 is the local unsteady heat release, andc0 and0 are the sound speedand the density, respectively, whichcanchange locally because of changes in temperature and compositiondue to chemical reactions. These reactions are also the source of theadditional right-hand-side (RHS) source term ( 1)T1/t thatis responsible for combustion noise and instabilities. This equationdoes not assume a constant polytropic coefficient and, thus, differsslightly from the one derived in a previous work (Eq. 1.1 in Ref. 13).

This approach is better suited to reacting flows where can changeby 30% from fresh to burnt gases. Equation (1) is difficult to usedirectly in practice, and multiple methods have been proposed tosolve it.3,1417 This paper presents a method where Eq. (1) is usedtogether with LES.

First an acoustic solver based on a Helmholtz equation is devel-oped to provide all acoustic modes of a combustion chamber. In thisapproach, Eq. (1) is solved in the frequency domain by assumingsinusoidal oscillations. This solver uses information given by theLES on the mean temperature field and the flame transfer function.

Second, a new analysis tool to analyze the budget of acousticenergy in a reacting flow is described. This integral form of Eq. (1)is a generalization of the well-known Rayleigh criterion18 (also seeRef. 4), which allows an evaluation of all terms of the acoustic

energy equation in the LES.The first objective of the present work is to couple these threetools (LES, Helmholtz solver, and acoustic energy budget) and showhow they can be combined to understand combustion instabilities.This exercice will be performed on a staged swirled combustionchamber installedat Ecole Centrale de Paris. In this device, theoutletboundary condition will be changed in the LES from nonreflectingto perfectly reflecting (pressure node) to demonstrate the effect ofthis conditionon theburner unsteady activityand provethat thethreetools used throughout the paper provide reasonable explanations forthis phenomenon.

The presentation starts with a description of the acoustic energyequation. The LEStool characteristics are recalled before presentingthe Helmholtz tool. The configuration is then described before thepresentation of the results. Stable and unstable regimes evidenced

by LES are discussed. In this last case, a scenario where the com-bustion instability grows, reaches a limit cycle, and then decays isstudied. This control of the instability is obtained by changing theoutlet boundary condition, and the budget of acoustic energy duringthe whole evolution is used to analyze the instability, the mecha-nisms controlling its limit-cycle amplitude, and its decay. Finally,the Helmholtz solver results are presented: The flame transfer func-tion measurement methodology is described and applied to obtainthe frequency as well as the growth rate of the combustor eigen-modes. It is then verified that the most unstable mode matches theLES observations.

Acoustic Energy Equation

The total acoustic energy equation is an integral form of the waveequation (1), which is quite useful to understand basic mechanismsof combustion instabilites. This equation cannot be used to predictunstable modes like the Helmholtz solver, but is a powerful methodto analyze the results of an LES as done here. The conservation

equation for the acoustic energy e1 =120u

21 +

12

p21/(0c20) can be

written4

e1

t= s1 (p1u1), s1 =

( 1)

p0p1T1 (2)

If integrated over the whole volume V of the combustor boundedby the surface A, it yields

d

dt

V

e1

dV = V

s1

dV A

p1u

1n dA (3a)

or

d

dtE1 = S1 F1 (3b)

where n is the surface normal vector. This surface consists of wallsor of inlet/outlet sections.

In Eq. (3), all terms are time dependent. The RHS source termS1 corresponds to the Rayleigh criterion

18: It measures the corre-lation between unsteady pressure p1 and unsteady heat release T1averaged over the whole chamber. It can act as a source or a sinkterm for the acoustic energy. The other RHS term F1 is less studiedbecause it is impossible to measure experimentally. It is an acousticflux integrated on all of the boundaries. Walls have zero contribu-

tion in this term because the velocity perturbations u1 n vanish onwalls. However, F1 may be large on inlets and outlets where it isusually a loss term. Equation (2) is, therefore, a generalization of theRayleigh criterion: The total acoustic energy in the chamber E1 willgrowif the coustic gain termS1 is larger than the acoustic lossesF1.The magnitudes and relative importance of the two terms S1 and F1are controversial issues in the field of combustion instabilities. Forexample, one important question is to know whether acoustic lossesare important in the determination of limit cycles. For these limitcycles, the acoustic energy E1 must remain constant over a periodof oscillations, and Eq. (2) shows that such a cycle can be reachedfor two situations.

1) The limit cycle may be combustion controlled: If the acousticlosses are small (F1 = 0), the pressure and heat release signals mayadjusttogive S1 0.Thelimitcycleisreachedwhenthisphaseshiftleads to a zero Rayleigh termS1 as observed in certain experiments.Physically, this is often obtained when the heat release oscillationssaturate (because the minimum reaction rate reaches zero at someinstant of the cycle) or when the phase between pressure and heatrelease changes so that combustion itself controls the limit cycleamplitude.

2) The limit cycle may be acoustically controlled: The sourceterm S1 may be large (pressure and heat relase oscillating in phase)but the acoustic losses F1 are too large and compensate S1. In thiscase, the final amplitude of oscillation is controlled by the acousticimpedances of outlets and inlets.

Clearly, these two solutions lead to very different approaches ofcombustion instabilities: If the limit cycle is combustion controlled,the acoustic behavior of inlets and outlets has a limited effect on the

stability; if it is acoustically controlled, acoustic impedances of inletsand outlets become essential elements of any method (experimentalor numerical). In the present study, the LESresults are postprocessedto measure all terms of Eq. (2) and determine whether the unstablemode is combustion or acoustically controlled.

LES for Reacting Flows in Complex Geometries

Numerical Methods for Compressible Reacting LES

Most academic LES are limited to fairly simple geometries forobvious reasons of costand complexity reduction.In manycases, ex-periments are designed using simple two-dimensional shapes6,19,20

or axisymmetrical configurations21,22 and simple regimes (low-speed flows, fully premixed or fully nonpremixed flames) to allowresearch to focus on the physics of the LES (subgrid scale models,flame/turbulence interaction model) and, more generally, to demon-strate the validity of the LES concept in academic cases. This ap-proach is clearly adequate in terms of modeling development, but itcan also be misleading in various aspects when it comes to dealing


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MARTIN ET AL. 743

with complex flames in complex geometries, especially in real gasturbines for which specific problems arise:

1) Real geometries cannot be meshed easily and rapidly withstructured or block-structured meshes: Until now, most LES of re-acting flows have been performed in combustion chambers wherestructured meshes were sufficient to describe the geometry. This isno longer the case in gas turbines, and this brings additional dif-ficulties. Indeed, on structured meshes, building high-order spatialschemes (typically fourth to sixth order in space) is easy and pro-

vides veryprecise numerical methods.2325

For complex geometriessuch structured meshes must be replaced by unstructured grids, onwhich constructing high-order schemes is a more difficult task.

2) Unstructured meshes also raise a variety of new problems interms of subgrid-scale filtering: Defining filter sizes on a highlyanisotropic irregular grid is another open research issue.2629 ManyLES models, developed and tuned on regular hexahedral grids, mayperform poorly on the low-quality unstructured grids required tomesh real combustion chambers. For example, the filtered structuremodel24 is difficult to extend to unstructured grids.

3) LES validation is often performed in laboratory low-speed un-confined flames, in which acoustics do not play a role and the Machnumber remains small so that acoustics and compressibility effectscan be omitted from the equations.10,21 In most real flames (for ex-ample in gas turbines), the Mach number can reach high values and

acoustics are important so that taking compressibility effects intoaccount becomes mandatory. This leads to a significantly heaviercomputational task: Because acoustic waves propagate faster thanthe flow, the timestep becomes smallerand the boundary conditionsmust handle acoustic wave reflections.4 Being able to preserve com-putational speed on a large number of processors then also becomesan issue simply to obtain a result in a finite time.

4) At the present time, it is impossible to perform a true LESeverywhere in the flow and it will remain so for a long time. For ex-ample, the flow between vanes in swirled burners, inside the ductsfeeding dilution jets, or through multiperforated plates would re-quire too many grid points. Compromises must be sought to offer(at least) robustness in places where the grid is not sufficient toresolve the unsteady flow.

In the present work, the full compressible NavierStokes equa-tions are solved on hybrid (structured and unstructured) grids in acode called AVBP. Subgrid stresses are described by the wall adapt-ing local eddy viscosity model.30 The flame/turbulence interactionis modeled by the thickened flame (TF) model.6,31 The numericalscheme is explicit in time and provides third-order spatial and third-order time accuracy.31

TF Model and Chemical Scheme

For this study, thestandardTF model31 is used: Inthismodel, pre-exponential constants and transport coefficients are both modified toofferthicker reaction zones that canbe resolved on LESmeshes.Thefundamental property justifying this approach has been put forwardby Butler and ORourke32 by considering the balance equation forthe k-species mass fraction Yk in a one-dimensionalflame of thermal

thickness 0

L and speed s0

L :

Yk

t+

ui Yk

xi=

xi

Dk

Yk

xi

+ k(Yj, T) (4)

Modifying this equation to have

Ythkt

+ui Y

thk

xi=

xi

F Dk

Ythkxi

+

1

Fk

Ythj , Tth

(5)

leads to a TF equation where F is the thickening factor and su-perscript th indicates thickened quantities. Introducing the variablechanges Xi =xi/F and = t/F leads to

Ythk

+ui Y

thk

Xi

=

XiDk

Ythk

Xi + kYthj , Tth (6)

which hasthe same solution as Eq.(4) andpropagates theflamefrontat the same speed s0L . However, Y

thk (x, t)= Yk(x/F, t/F) shows that

the flame is thickened by a factor F . The thickened flame thicknessis thL =F

0L . Choosing sufficiently large values of F allows to obtain

a thickened flame that can be resolved on the LES mesh. Typically,ifn is the number of mesh points within the flame front ( n is of theorder of 510) and x the mesh size, the resolved flame thicknessthL is nx so that F must be F = nx/s

0L . Note that F is not an

additional parameter of the model but is imposed by the preced-ing relation as soon as the mesh is created. In the framework ofLES, this approach has multiple advantages: When the flame is a

laminar premixed front, the TF model propagates it, in the limit ofan infinitely thin front, at the laminar flame speed exactly as in aG equation approach. However, this flame propagation is due to thecombination of diffusive and reactive terms, which can also act in-dependently so that quenching (near walls, for example) or ignitionmay be simulated. Fully compressible equations may also be usedas required to study combustion instabilities.

Thethickeningmodificationof theflamefrontalsoleadsto a mod-ified interaction between the turbulent flow and the flame: Subgrid-scale wrinkling must be reintroduced. This effect can be studied andparameterized usingan efficiency function E derivedfrom direct nu-merical simulation results.31,33,34 This efficiency function measuresthe subgrid-scale wrinkling as a function of the local subgrid turbu-lent velocity u e and the filter width e. In practice, the diffusioncoefficient Dk is replaced by E F Dk and the preexponential constant

A by AE/F so that the conservation equation for species k is

Ythkt

+ui Y

thk

xi=

xi

E F Dk

Ythkxi

+

E

Fk

Ythj , Tth

(7)

Such an equation propagates the turbulent flame at a turbulent speedsT =E s

0L , while keeping a thickness

thL =F

0L . In laminar regions,

E goes to unity, and Eq. (7) simply propagates the front at thelaminar flame speed s0L . The subgrid-scale wrinkling function Ewas obtained from the initial model of Ref. 31 as a function of thelocal filter sizee, the local subgrid-scale turbulentvelocity u

e

,thelaminar flame speed s0L , and the laminar and the flame thicknesses0L and

thL .

The TF model uses finite rate chemistry: Here the configurationcorresponds to a lean premixed flame so that a one-step Arrheniuskinetics is sufficient. This one-step scheme(called 1sCM1)has beenfitted with a genetic algorithm-based tool on a laminar flame struc-ture. The reference mechanism used to fit 1sCM1 is the Peters andRogg propane scheme.35 Scheme 1sCM1 takes into account fivespecies (C3H8,O2,CO2,H2O and N2):

C3H8 + 5O2 3CO2 + 4H2O (8)

The rate of the single step reaction is given by

q = AYC3H8

WC3H8

nC3H8 YO2

WO2

nO2exp(Ea/RT) (9)

where the rate parameters are provided in Table 1.The diffusioncoefficient Dk of species kisobtainedas Dk = /S

kc ,

where is the viscosity and Skc

the fixed Schmidt number of

Table 1 Rate constants and Schmidtnumbers for 1sCM1 schemea

Constants Value

Chemical rate constantA 3.29E 10

nC3H8 0.856

nO2 0.503Ea 31526

Schmidt numberC3H8 1.241O2 0.728CO2 0.941H2O 0.537

N2 0.690

aActivation energy is in calories per moles and thepreexponential constants in cgs units.


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744 MARTIN ET AL.

species k. The Schmidt number values used in the present simu-lations are given in Table 1 and correspond to the PREMIX valuesmeasured in the burnt gases. The Prandtl number is set to 0.68.With this parameter set, the agreement between flame profiles ob-tained using AVBP or PREMIX with the same chemical schemeis good. The agreement between the Peters and Rogg35 and the1sCM1 schemes in terms of laminar flame speed is satisfactory forthelean to stoechiometric mixtures.Note that other formulationsareavailable for LES of partially premixed turbulent flows.5,11 To study

combustion/acoustics coupling, however, the TF model offers thebest compromise. First, the G equation is usually implemented inlow Mach number codes, which do not solve for acoustics, whereasacoustics are fully represented in the TF model. Second, the TFapproach has been now validated in multiple complex geometryswirled burners12,36 making it a proper basis for the present study.

Acoustic Solver for the Helmholtz Equation

The acoustic tool used in this study, called AVSP, solves the eigen-value problem associated to the wave equation (1). When dealingwith thermoacoustic instabilities, it is usual to model thegeometry ofthe combustor by a network of one-dimensional or two-dimensionalaxisymmetric acoustic elements where a simplified form of Eq. (1)can be solved.4,15,3739 Jump relationsare used to connect allof theseelements, and the amplitude of the forward and backward acoustic

waves are determined so that the boundary conditions are satisfied.The main drawback of this approach is that the geometrical detailsof a combustor cannot be accounted for and only the first equiva-lent longitudinal or orthoradial modes are sought for. In the acousticsolver, a finite element strategy is used to discretize the exact ge-ometry of the combustor so that no assumption is made a prioriregarding the shape of the modes. This feature gives the Helmholtzsolver the potential to test the effect of geometrical changes on thestability of the whole system.

Equation (1) is solved in the frequency domain by assuming har-monic variations at frequency f =/(2) for pressure, velocity,and local heat release perturbations:

p1 = [ P(x) exp(it)], u1 = [U(x) exp(it)]

T1 = [T exp(it)] (10)

Introducing Eq. (10) into Eq. (1) and neglecting the turbulent noisep0u1:u1 in front of the combustion term ( 1)T1/ t leadsto a modified form of the Helmholtz equation:

0c20 [(1/0)

P] + 2 P = i( 1)T (11)

where the unknown quantities are the complex amplitude P of thepressure oscillation at frequency f and pulsation . Note that T,the amplitude of the heat release perturbation, is also unknown andmust be modeled. This is obviously the difficult part of the mod-eling, and it remains an open research issue today. In the presentwork, the simplest linear approach, initally proposed by Crocco, 14

was chosen as a first step. A direct extension of the standard n xmodel4,14,40 was used to write T1 nu 1(xref, t ), where u1 isthe axial velocity fluctuations. In one-dimensional approaches, theinteraction index n and time delay are two parameters describingtheacousticbehavior of a compact flame located at theaxial positionxref. In the Helmholtz solver, where the geometry of the combustoris fully described, the flame is distributed and the interaction indexand time delay depend on space. These data can be extracted fromLES results by postprocessing either a self-excited or a forced oscil-lating regime. Once measured in LES, the fields n(x, ) and (x, )are used to model the unsteady heat release in Eq. (11) as

T = n(x, ) exp[i (x, )]U(xref) nref (12)

The linearizedmomentumEuler equation U= P/ i canbe usedto relate T to P and close Eq. (11).

Threetypesof boundaryconditions canbe prescribed forEq. (11),wheren is theoutward unit normalvector to theboundary: 1) Dirich-let condition, which imposes P = 0, on fully reflecting outlets;

2) Neumann condition, which imposes P n= 0, on fully rigidwalls or reflecting inlets; and 3) Robin condition, which imposescZ P n= i P , on general boundaries, where Z is the local re-duced complex impedance Z= P/0c0Un. In this study, the re-duced boundary impedance Z has been obtained using Eq. (16) aswill be described later.

Knowing the boundary impedance Z, the sound speed c0 and thedensity 0 distribution, the flame response [n(x,),(x, )], andassuming that Z does not depend on , a Galerkin finite element

method is used to transform Eq. (11) into a nonlinear eigenvalueproblem of size N (the number of nodes in the finite element gridused to discretize the geometry) of the form

[A][ P] + [B][ P] + 2[C][ P] = [D()][ P] (13)

where [ P] is the column vector containing the eigenmode at pul-sation and [A], [B], and [C] are square matrices depending onlyon the discretized geometry of the combustor. (Note that the sameresult holds if 1/Z= 1/Z0 +Z1 +Z2/, where Z0, Z1, and Z2are complex-valued constants.) If the impedances Z change with ,Eq. (13) can be solved iteratively and independently for each eigen-mode, by using a Z value adapted to each eigenfrequency. [D()] isthe unsteady contribution of the flame and depends on the pulsationthrough the combustion term n(x, ) exp[i (x, )]. No efficientnumerical method exists to solve this nonlinear eigenvalue problem.However, in the casewhere the unsteadyflame response is neglected,namely, [D()] = 0, Eq. (13) simplifies into a quadratic eigenvalueproblem dependingonly on and2. A variabletransformation canthen be used to obtain an equivalent linear eigenvalue problem ofsize 2 N (Ref. 41). Several numerical methods can then be usedto assess the eigenmodes. Direct methods like the quadratic regu-lator approach are exact and have the advantage to provide all ofthe eigenmodes. However, they can be expensive to solve for largeproblems (N> 103). Because only the first few frequencies are usu-ally of interest from a physical point of view, it is more appropriateto use an iterative method that can be applied for large problems(N> 105) without difficulty. In the Helmholtz solver, we are us-ing a parallel implementation of the Arnoldi method (see Ref. 42),

whichenables to solvecomplexproblems ofsize N 20000inafewminutes.Setting [D()] = 0 is equivalent to finding the eigenmodes of the

burner, taking into account the presence of the flame through themean temperature field but neglecting the flame effect as an acous-tically active element. The boundary conditions are also acountedfor, and this approximation can provide relevant information on theshape and real frequency of the first few modes of the combustor.However, because there is no coupling between the acoustics andthe flame, there is no hope to discriminate between stable and unsta-ble modes, which is the ultimate objective of this study. When it isassumed that the unsteady flame response creates a small perturba-tion of the modes, a linear expansion technique can be developed toassess the imaginary part of, hence, the stability of the perturbedmodes.43,44 Another path has been followed in this study to handle

cases where the unsteady response of the flame changes the modessignificantly andwhen thelinear expansion is notjustified. The non-linear eigenvalue problem Eq. (13) is then solved iteratively, the kthiteration consisting in solving the quadratic eigenvalue problem ink defined as

([A] [D(k 1)])[ P] + k[B][ P] + 2k[C ][

P] = 0 (14)

A natural initialization is to set [D(0)] = 0 so that the computationof the modes without combustion is in fact the first step of the iter-ation loop. Usually, less than five iterations are enough to convergetoward the complex pulsation and associated mode. This linearizedapproach to describe thestabilityof theburner in terms of modes hasdrawbacks but remains one of the basic tools to study instabilities:

1) The linearization is valid only for small-amplitude perturba-tions, a condition which is obviously not true when limit cycles typ-ical of combustion instabilities are observed in gas turbines. How-ever, this assumption is valid when the instability grows45 and helps


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MARTIN ET AL. 745

to determine the unstable modes: Such modes have to appear andgrow before they reach a limit cycle, and any analysis adapted tothis early phase is of interest.

2) Most acoustic tools work on linear regimes for which each os-cillatory mode is independent of other modes. Many combustion in-stabilities exhibit nonlinear coupling, where high-frequency modescouple with low-frequency oscillations.46 These were also observedin the experiment in Ref. 1, in which a 530-Hz mode (often calledrumble) was systematically accompanied by a high-frequency mode

(called screech) at 3750 Hz. The fact that combustion instabilitiesinvolve more than one mode of oscillation is one of the basis ofthe approach of Yang and Culick.47 The tool presented earlier treatseach mode individually and cannot simulate such phenomena.

3) The description of the coupling between acoustics and com-bustion in such modelsis extremely crude.The response of theflameexcited by an acoustic wave depends on several physical phenom-ena such as chemical reactions, species diffusion, vortex shedding,vortexflame interaction, etc. All of these phenomena are not ne-glected in the present study, but their cumulative effect is modeledthrough the global timescale and index n.

Despite these limitations, such tools are useful because they pro-viderelevant information about the modes triggered by the acousticflame coupling while running fast: For the current configuration,only 8000 grid points were necessary to describe the geometry and

obtain the first four modes. For comparison, half a million nodeswere used to perfom the LES. A typical run for solving the quadraticeigenvalueproblemof type Eq.(14)on this grid lasts 10min byusing15 processors (R14000 500-MHz IP35) on an SGI O3800 parallelmachine. Such a tool can, thus, be used in the design process ofnew gas turbines to characterize their thermoacoustic modes. Bydescribing the whole geometry between the compressor and the tur-bine, includingall of theinjectors dispatched aroundthe combustionchamber, such simulations would give unique information about theswirling modes that sometimes show up in large gas turbines. Thedifficult and computationally expensive task would be to computethe flame transfer function by performing a LES of the turbulentflame. Such a simulation would be performed by considering an an-gular sector corresponding to only one injector, saving grid pointsand CPU resources.

Configuration

Geometry: Swirled Premixed Combustor

The methodologies described in the preceding sections weretested for a swirled combustor shown in Fig. 1. The configuration istypical of swirled combustion: Premixed gases are introduced tan-gentially into a long cylindrical duct feeding the combustion cham-ber. The tangential injection creates the swirl required for stabiliza-tion. The fuel is propane. The two independant swirler elementsallow fuel staging. The staging parameter is defined as the ratioof fuel flow rate of the first to the second swirler.

The regime studied here corresponds to the parameters given inTable 2. The staging of the burners corresponds to = 0.3.

Fig. 1 Staged swirled combustor configuration.

Table 2 Flow parameters forcombustion cases

Parameter Value

Flow rate

Total 22.103

Axial 4.103

Equivalence ratioa 0.8

Reynolds number 4.67 104

aBurner mouth.

Table 3 Acoustic inlet and outlet boundaries characteristicsfor REF and LEAK

Case

Boundary characteristics LEAK REF

Inlet 1,000 1,000Outlet 1,000 10,000Characteristic Nonreflecting ReflectingOutlet impedance at 380 Hz

Measured in LES 0.040.21iCalculated with Eq. (16) 0.850.36i 0.050.23i

Fig. 2 Mean axial velocity field: white line, iso-ux = 0 and black line:iso-T= 1500 K for stable combustion.

Boundary Conditions

Specifying boundary conditions is a critical issue for compress-ible flows. Here, the NavierStokes characteristic boundary condi-

tion technique4,48

was used at the outlet. The level of reflection ofthis boundary can be controlled by changing the relaxation coeffi-cient of the wave correction,49 which determines the amplitudeof the incoming wave L1 entering the computational domain:

L1 = (p pt) (15)

where pt is the prescribed pressure value at infinity. Equation (15)acts on the flow like a spring mechanism with a stiffness . Theimpedance of the boundary is a function of and , which can beobtained analytically49 for simple cases,2

Z = (i/)/(1 i/) (16)

(This impedance can be taken into account by the Helmholtz solverunder the following form: 1/Z= 1 + i/ .)

For small values of , Eq. (15) keeps the pressure p close toits target value pt while letting acoustic waves go out at the sametime4: The outlet is nonreflecting. When large values of are used,the outlet pressure remains strictly equalto pt andthe outlet becomestotally reflecting. Two sets of computation will be shown (Table 3).Thefirstone, LEAK,correspondsto a case where thespring stiffness is small so that the outlet is nonreflecting and the acoustic wavesare evacuated with very small reflection levels. For the second set,REF, is large and the outlet is reflecting. (The pressure oscillationis almost zero.)

LES Results

Stable Flow

The first computation corresponds to the case where the out-let section is nonreflecting (case LEAK in Table 3): The acousticfeedback is minimized, and the flame does not exhibit any strongunstable movement. Themean velocity and fuel mass fraction fieldsare shown in Figs. 2 and 3. As expected, the downstream part of the


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746 MARTIN ET AL.

Fig. 3 Mean fuel mass fraction field: black lines, isoreaction rate forstable combustion.

Fig. 4 Meannormalizedvalues: ,pressure; , heatrelease;and , phase angle between pressure and heat release.

Fig. 5 Mean values: , Rayleigh criterion S1 and . . . . , acousticfluxes F1.

central recirculation zone is filled by burnt gases and stabilize theturbulent flame.

Instability Sequence

LES also reveals that the combustor can exhibit a strong unstablemode when the outlet is acoustically closed (case REF). In thiscase, soon after ignition, the pressure and the global heat releasestart oscillating (Fig. 4) at 380 Hz. To analyze the behavior of thisinstability, the following sequence is set up:

1) Starting from a stable flame (LEAK), the outlet impedance ischanged to becomereflecting (case REF) at time t= 0.127 s (Fig. 4).The oscillation grows and reaches a limit cycle at a frequency of380-Hz mode.

2)At time t= 0.173 s, the outlet impedance is switched again to anonreflecting condition (case LEAK) and the instability disappears.

Thisscenarioprovides fourphases whichare studied sequentially:1) a linear growth between times 0.127 and 0.145 s, 2) an overshootphase between 0.145 and 0.160 s, 3) a limit cycle between times0.160 and 0.173 s, and 4) a decay phase starting at t= 0.173 s. For

each phase, the instability is analyzed in terms of flame shape, flameoscillation, and phase between heat release and pressure. Moreover,the acoustic energy equation budget is closed, and all terms areanalyzed.

Growth Phase

Once the outlet boundary is acoustically closed (t= 0.127 s), thethermoacoustic instability starts. Figure 5 shows the time variationsof the combustion source term S1 and the acoustic losses F1. Thetotal acoustic energy evolution of the chamber E1 is shown inFig. 6.Figure 7 shows that the budget of Eq. (3) is quite well closed bythe LES data: The difference S1 F1 matches the time derivativeofE1. This validates both the LES results and the acoustic energyequation (3). It is also the first example of such a treatment fora resonating combustor. Because the budget is closed, individualterms can then be analyzed.

First, the phase angle between pressure and heat release is shownin Fig. 4. During the growth phase, it is close to zero and slowly

Fig. 6 Evolution of burner acoustic energyE1.

Fig. 7 Comparison: . . . . , time derivative of the acoustic energyE1/t and , S1 F1.

shifting toward /4, leading to a strong coupling between pressureandheat release, that is, a positive S1 term. Duringthe growth phase,

the source term S1 is large and always positive (Fig. 5) because thephase angle stays in the [/2;/2] range. Figure 5 shows that theacoustic losses balance the reacting term S1 in the acoustic budgetequation. The limit cycle is controlled by acoustic losses and not bycombustion.

Overshoot Phase and Limit Cycle

At t= 0.160 s, the instability reaches a limit cycle at 380 Hz. Be-fore reaching this limit cycle, a large overshoot of acoustic energy isobserved: This is typical of combustion instabilities, and it has beenobserved experimentally in other systems.45,50 Figure 4 shows that,reaching the nonlinear zone, the phase difference between pressureand heat release increases from zero to /4 in the limit-cycle zone.The drift of this phase difference together with increasing acoustic

losses lead to the saturation of the instability.The coupling loop between p1 and T1 can be identified fromLES as follows. The longitudinal mode induces the formation of avortex ring at the dump plane. This vortex ring strongly interacts inphase with the flame. Figure 8 shows the interaction between theacousticallyinduced vortex ring andthe flame brush.Figure 9 allowsto locate LES snapshots in the acoustic periodand displays themeanpressure fluctuation p1 in the flame zone, theheat release fluctuationT1,andthefluctuationofthemeanvelocityinthedumpplane u

dump

1 .At instant 1, a vortexringappears at the dump plane whendu

dump

1 /dtis maximum. The ring structure detaches and is convected throughthe flame by the mean flow (instants 2, 3, and 4). Between instants 1and 3, the flow stretches the flame, increasing its area, whereas theflame wrinkling by the vortex ring remains weak. ConsequentlyT1 increases with a medium slope (Fig. 9). Between points 3 and

5, the vortex ring is stretching the flame and T1 increases faster.Moreover, the vortex ring is gradually destroyed, and its globalcoherence disappears between instants 4 and 5, at a moment whendu

dump

1 /dt is minimum. At instant 5, some coherent structures arestill interacting with the flame, producing (noisy) flame pockets andcusps. After instant 5, the flame burns out the fresh gases present inthe chamber and propagates back to the injection pipe decreasingthe overall flame surface and T1.

Decay Phase

The decay phase is triggered by the sudden change in the acous-tic outlet boundary condition switching to nonreflecting (LEAK) att= 0.173 s. The phase angle p between pressure and heat releaseincreases by a large amount:p >/2 at t= 0.174 s, that is, 1.2 msafter relaxing outlet pressure (Fig. 4).At this time,S1 becomes glob-ally negative for the first time and the instability is rapidly damped.The acoustic losses actually become positive (gain term) during thislast oscillation but the instability engine is broken and the Rayleigh


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MARTIN ET AL. 747

Fig. 8 Vortex ring shedding at six instants (Fig. 9) during the limitcycle: isosurface, Q vortex criterion and black lines, isoreaction rate inthe burner central plane.

Fig. 9 Time signals during limit cycle and snapshots corresponding toFig. 8: , pressure; . . . . , inlet velocity; and , total heat release.

termS1 becomes negative during a half-cycle (0.173< t< 0.175 msin Fig. 5) leading to the immediate decay of all unstable activity.

Acoustic Analysis Results

This section describes the results obtained with the Helmholtz

solver for the configuration of Fig. 1. The impedances are calcu-lated from Eq. (16) using the value of given in Table 3. Equa-tion (16) shows that, for a fixed coefficient, the impedance is afunction of the pulsation . To verify that Eq. (16) is sufficientlyaccurate, Table 3 gives the value of the impedance at 380 Hz (whichis the frequency of the first mode observed in the LES) predictedby Eq. (16) and measured in the LES. For these computations, theflame transfer function is needed. In the present work, n and areobtained by postprocessing LES results as described in the follow-ing section. These results are then used to predict the frequency andgrowth rates of all modes using the Helmholtz equation (11) and tocompare them to the LES results of the preceding section.

Measurement of Flame Transfer Function

The key mechanism to predict combustion instabilities inHelmholtz codes is the flame transfer function (FTF). In the modelchosen in this study [Eq. (12)], the heat release fluctuations are re-lated to the velocity fluctuations at a reference point through the

Fig. 10 Flame transfer function magnitude (n parameter) field incentral plane evaluated at f= 432 Hz: black line, iso-n= 2.5106 Pa m1 and +, reference point location.

magnitude n and the phase of the FTF defined as

n(x, ) =

T(x)U(xref) nref

(17)

(x, ) = (x, ) = arg

T(x)

U(xref) nref

(18)

Because the flame is more prone to interact with longitudinaloscillations, the reference normal vector nref is set collinear to theaxial direction so that the scalar product U(xref) nref correspondsto the axial velocity fluctuation at the reference point. This pointis located following the Crocco approach,14 that is, in the fresh gasnear the inlet combustor. As shown in Fig. 2, a recirculation zonelocated at the entrance of the dump combustor is induced by theswirling flow. Consequently, the reference point is located upstreamof this zone3 at 113 mm from the inlet of the device (Fig. 10). Notethat other positions in the vicinity of this location have been testedleading to very similar results with the Helmholtz solver.

Flame transfer functions are usually measured for the wholecombustor39,51,52 or by zones.53 In the present approach, n and must be obtained locally at each point of the combustor, and thiscan be done by analyzing snapshot series from LES. Moreover, the

frequency dependence of these parameters should be taken into ac-count. To do so, it is assumed that FTF depends mostly on the realpart of the frequency and not of the growth rate of the mode. Thus,at each subset k of the algorithm [Eq. (14)], the FTF parametersare evaluated at [k/(2)], and the first guess corresponds to theeigenmodes determined without active flame (where n is set to zeroeverywhere). (Further studies are needed to assess this assumptionused for practical reasons.) From a practical point of view, n and can be extracted from LES results by Fourier processing the localheat release and the unsteady velocity at the reference point andusing Eqs. (17) and (18). Two methodologies can be considered forsuch an analysis: 1) use a stable regime and force the flow (usuallyby introducing acoustic waves through the inlet39,40,54) or 2) use aself-excited regime.

The choice of the methodology raises other fundamental issues,

which are still open today:1) When it is assumed that Eq. (12) is valid for both forced and

self-excited cases, both methodologies are expected to provide thesame n and fields. This is a theoretical argument that has not beenchecked yet and is left for further work.

2) The possible dependence of the transfer function on the acous-tics wave amplitude raises an additional difficulty. Ifn and dependon the wave amplitude, then both methodologies have drawbacks:The forcing method will provide different n and fields when theforcing amplitude changes, whereas the self-excited method uses anonlinear limit cycle to evaluate n and .

These questions are beyond the scope of the present study. Herethe fields of n and were measured by postprocessing the self-excited regime between t= 0.145 and t= 0.177 s shown in Fig. 4.Results given in the next section prove the validity of this method,but further studies are obviously needed. The resulting flame trans-fer function parameters, shown only within the flame zone (whenn 2.5 106 Pa m1), are shown in Figs. 10 and 11.


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748 MARTIN ET AL.

Table 4 Helmholtz solver results

No active flame Active flame

Mode F , Hz , rad s1 F , Hz , rad s1

1 312 48 334 5882 431 32 432 6003 841 4.5 825 64 1140 63 1116 363

Fig. 11 Flame transfer function phase ( parameter) field in thecentral plane evaluated at f= 432 Hz: black line, iso-n =2.5106 Pa m1 and +, reference point location.

Fig. 12 Acoustic pressure modulus for the first four modes calculatedby the Helmholtz solver with active acoustic flame in REF case, ||P||isolines.

Helmholtz Solver Results

Using the mean fields given by LES, the Helmholtz solver isapplied to obtain the thermoacoustic eigenmodes of the burner. Thistool can be run using either a nonactive flame, that is, setting n tozero everywhere, or an active flame, using the postprocessed n and

fields displayed. Moreover, the impedance values are determinedfrom the LES boundary conditions settings (Table 3).

First, when the outlet impedance corresponds to the LEAK case,the Helmholtz solver predicts that all modes are damped. This con-firms the LES result of Fig. 2 for which no acoustic mode was foundfor this regime.

Second, when the outlet impedance corresponds to the REF case,Table 4 presents the modes that are obtained for both the nonactiveand the active flames. These four modes in the REF case are alllongitudinal except in the vicinity of the swirler: The structure ofthe modes in the central plane of the burner is shown in Fig. 12,whereas a one-dimensional cut of the rms pressure field is given inFig. 13. The real frequencies of the active and nonactive cases arevery similar: Typically, taking into account the active effect of theflame shifts theeigenmode frequency by a few percent. Howevertheeffect on thegrowth rate is more dramatic:All modes computed withthe nonactive flame are damped, whereas the modes computed withan active flame areexcited formodes2, 3, and4. Thefastest growing

Fig. 13 Longitudinal structure of first four modes obtained fromHelmholtz solver: , normalized prms

1evolution along burner axis

with acoustics/flame coupling and , without.

Fig. 14 RMS pressure fluctuations prms1

along burner axis; , LES

and , solvers.

modeis mode2 at432Hz, which is close tothe modeobserved intheLES (380 Hz). To verify that the mode 2 obtained by the Helmholtzsolver is indeed the mode appearing in the LES, Fig. 14 shows acomparison of the | P| profiles on the axis and confirms the goodagreement of LES and Helmholtz solver. The difference between thefrequency observed in theLES (380 Hz)and thefrequency predictedby the Helmholtz solver (432 Hz) is probably due to the zero Machnumber approximation for the Helmholtz solver.

Conclusions

Three tools have been used to analyse flameacoustics couplingmechanisms in a staged swirled combustor: Full compressible LES,

Helmholtz analysis, and budget of acoustic energy. The two lattermethods are based on the wave equation in reacting flows. Theyuse LES results but provide essential new elements: The Helmholtzresults allow to predict the stability of the combustor and the exactidentification of modes appearing during the instability, whereas thebudget of acoustic energy demonstrates that the Rayleigh criterion isnot the only or even the largest term in the acoustic energy equation.Acoustic lossesat the outlet of the combustorcontribute significantlyto the budget of acoustic energy and determine the levels of oscil-lation amplitudes as well as their appearance. More generally, thisstudy confirms the need of coupling classical computational fluiddynamics (here LES) and acoustic analysis to understand combus-tion instabilities. It also demonstrates the crucial effect of acousticboundary conditions on the stability of combustors. This has a prac-tical implication. The stability of a given combustion chamber iscontrolled by acoustic impedances upstream and downstream of thecombustor. Removing a combustor section from a full gas turbine toinstall it in a laboratory setup obviously becomes very dangerous to


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MARTIN ET AL. 749

study the stability of the burner. Indeed, the combustion may proveto be stable in one case and unstable in the other one. This resultcalls for the development of more coupled acoustics analysis of thewhole turbine.

Acknowledgments

Certain numerical simulations have been conducted on the com-puters of the Centre Informatique National de lEnseignementSuperieur and Institut du Developpement et des Ressources en In-

formatique Scientifique French national computing centers. Sim-ulations have been supported partly by Alstom Power and by theEuropean Community Program (WP2) FUELCHIEF.

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