Large-Eddy Simulation and Acoustic Analysis of a Swirled Staged Turbulent Combustor

Transcription

1 AIAA JOURNAL Vol. 44, No. 4, April 2006 Large-Eddy Simulation and Acoustic Analysis of a Swirled Staged Turbulent Combustor Charles E. Martin, Laurent Benoit, and Yannick Sommerer Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, Toulouse, France Franck Nicoud Université de Montpellier II, Montpellier Cedex 5, France and Thierry Poinsot Institut de Mécanique de Fluides de Toulouse, 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 e source term due to combustion (equivalent to e Rayleigh criterion) is balanced by e acoustic fluxes at e boundaries when reaching e limit cycle. Additionally, an Helmholtz-equation solver including flame acoustics interaction modeling is used to predict e stability characteristics of e system. Feeding e flame-transfer function from e LES into is solver allows to predict an amplification rate for each mode. The unstable mode encountered in e LES compares well wi e mode of e highest amplification factor in e Helmholtz-equation solver, in terms of mode shape as well as in frequency. Nomenclature [A] = square matrix of size N c = sound velocity, m/s D k = k species diffusion coefficient, m 2 /s E = efficiency function E a = activation energy, cal/mol E 1 = instantaneous global acoustic energy term, J e 1 = acoustic energy, J/m 3 F = flame ickening factor F 1 = instantaneous global acoustic fluxes, W f = frequency, Hz i = square root of 1 N = number of nodes of e grid n = magnitude of e flame transfer function, Pa/m n = outward normalized normal vector [ ˆP] = column vector of size N associated to an eigenmode p = pressure, Pa R = perfect gas constant, cal/mol K Sc k = k species Schmidt number S 1 = instantaneous global Rayleigh term, W sl 0 = laminar flame speed, m/s s T = turbulent flame speed, m/s T = temperature, K t = time, s u = velocity vector, m/s Received 19 November 2004; revision received 21 April 2005; accepted for publication 31 July Copyright c 2005 by e American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of is paper may be made for personal or internal use, on condition at e copier pay e $10.00 per-copy fee to e Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include e code /06 $10.00 in correspondence wi e CCC. Ph.D. Student, Computational Fluid Dynamics Team, 42 Avenue G. Coriolis. Research Engineer, Computational Fluid Dynamics Team, 42 Avenue G. Coriolis. Professor, Maématique; also at Institut de Modélisation et de Maématiques de Montpellier, Centre National de la Recherche Scientifique, Place Bataillon. Research Director, allée du Professeur Camille Soula; also at Ecoulements et Combustion, INP de Toulouse, Centre National de la Recherche Scientifique. Associate Fellow AIAA. Y k = k species mass fraction Z = local reduced acoustic impedance γ = polytropic coefficient δl 0 = flame ermal ickness, m ν = cinematic viscosity, Pa/s ρ = mass density, kg/m 3 τ = time delay of e n τ model, s φ = phase of e flame transfer function ω = pulsation, rad/s ω k = k species reaction rate, mol/m 3 s ω T = unsteady heat release, J/m 3 Subscripts L = laminar ref = related to e reference point of e n τ model 0 = steady part 1 = fluctuating part Superscripts = ickened quantity ˆ = Fourier transformed Introduction COMBUSTION oscillations are frequently encountered during e development of many combustion chambers for gas turbines. 1 4 These oscillations cannot be predicted at e design stage, and correcting actions can be extremely costly at later stages. Testing burners in simplified combustion chambers is a common meod to verify eir stability but is also an ambiguous approach because most experimentalists know at a given burner can produce unstable combustion in one chamber and not in anoer. Meods providing stability analysis before any tests are, erefore, requested. Large-eddy simulations (LES) are an obvious choice for such studies: They are powerful tools to study e dynamics of turbulent flames. (See recent works on turbulent combustion. 4,5 ) Multiple recent papers have demonstrated e power of ese meods However, an important limitation of LES is its cost: The intrinsic nature of LES (full ree-dimensional resolution of e unsteady Navier Stokes equations) makes it very expensive, even on today s computers. Moreover, even when it is confirmed at a combustor is unstable, LES does not indicate why and how to control it. Therefore, 741

2 742 MARTIN ET AL. tools are needed to analyze LES results but also to provide capacities for optimization and control of ermoacoustic oscillations in chambers. A proper framework to analyze combustion stability is e wave equation in a reacting flow. 4 Such an equation is complex to derive because most assumptions used in classical acoustics must be revisited in a multispecies, nonisoermal, reacting gas. For low Mach numbers, an approximate equation controlling e propagation of pressure ( perturbations in a reacting gas is ) 1 ρ 0 c 2 0 p 1 2 ρ 0 t p 2 1 = (γ 1) ω T 1 ρ 0 c 2 0 t u 1 : u 1 (1) where e subscript 0 refers to mean quantities and e subscript 1 to small perturbations. Here ω T 1 is e local unsteady heat release, and c 0 and ρ 0 are e sound speed and e density, respectively, which can change locally because of changes in temperature and composition due to chemical reactions. These reactions are also e source of e additional right-hand-side (RHS) source term (γ 1) ω T 1 / t at is responsible for combustion noise and instabilities. This equation does not assume a constant polytropic coefficient γ and, us, differs slightly from e one derived in a previous work (Eq. 1.1 in Ref. 13). This approach is better suited to reacting flows where γ can change by 30% from fresh to burnt gases. Equation (1) is difficult to use directly in practice, and multiple meods have been proposed to solve it. 3,14 17 This paper presents a meod where Eq. (1) is used togeer wi LES. First an acoustic solver based on a Helmholtz equation is developed to provide all acoustic modes of a combustion chamber. In is approach, Eq. (1) is solved in e frequency domain by assuming sinusoidal oscillations. This solver uses information given by e LES on e mean temperature field and e flame transfer function. Second, a new analysis tool to analyze e budget of acoustic energy in a reacting flow is described. This integral form of Eq. (1) is a generalization of e well-known Rayleigh criterion 18 (also see Ref. 4), which allows an evaluation of all terms of e acoustic energy equation in e LES. The first objective of e present work is to couple ese ree tools (LES, Helmholtz solver, and acoustic energy budget) and show how ey can be combined to understand combustion instabilities. This exercice will be performed on a staged swirled combustion chamber installed at Ecole Centrale de Paris. In is device, e outlet boundary condition will be changed in e LES from nonreflecting to perfectly reflecting (pressure node) to demonstrate e effect of is condition on e burner unsteady activity and prove at e ree tools used roughout e paper provide reasonable explanations for is phenomenon. The presentation starts wi a description of e acoustic energy equation. The LES tool characteristics are recalled before presenting e Helmholtz tool. The configuration is en described before e presentation of e results. Stable and unstable regimes evidenced by LES are discussed. In is last case, a scenario where e combustion instability grows, reaches a limit cycle, and en decays is studied. This control of e instability is obtained by changing e outlet boundary condition, and e budget of acoustic energy during e whole evolution is used to analyze e instability, e mechanisms controlling its limit-cycle amplitude, and its decay. Finally, e Helmholtz solver results are presented: The flame transfer function measurement meodology is described and applied to obtain e frequency as well as e grow rate of e combustor eigenmodes. It is en verified at e most unstable mode matches e LES observations. Acoustic Energy Equation The total acoustic energy equation is an integral form of e wave equation (1), which is quite useful to understand basic mechanisms of combustion instabilites. This equation cannot be used to predict unstable modes like e Helmholtz solver, but is a powerful meod to analyze e results of an LES as done here. The conservation equation for e acoustic energy e 1 = 1 ρ 2 0u p2 1 /(ρ 0c0 2 ) can be written 4 e 1 (γ 1) = s 1 (p 1 u 1 ), s 1 = p 1 ω T 1 (2) t γ p 0 If integrated over e whole volume V of e combustor bounded by e surface A, it yields d e 1 dv = s 1 dv p 1 u 1 n da (3a) dt V V A or d dt E 1 = S 1 F 1 (3b) where n is e surface normal vector. This surface consists of walls or of inlet/outlet sections. In Eq. (3), all terms are time dependent. The RHS source term S 1 corresponds to e Rayleigh criterion 18 : It measures e correlation between unsteady pressure p 1 and unsteady heat release ω T 1 averaged over e whole chamber. It can act as a source or a sink term for e acoustic energy. The oer RHS term F 1 is less studied because it is impossible to measure experimentally. It is an acoustic flux integrated on all of e boundaries. Walls have zero contribution in is term because e velocity perturbations u 1 n vanish on walls. However, F 1 may be large on inlets and outlets where it is usually a loss term. Equation (2) is, erefore, a generalization of e Rayleigh criterion: The total acoustic energy in e chamber E 1 will grow if e coustic gain term S 1 is larger an e acoustic losses F 1. The magnitudes and relative importance of e two terms S 1 and F 1 are controversial issues in e field of combustion instabilities. For example, one important question is to know wheer acoustic losses are important in e determination of limit cycles. For ese limit cycles, e acoustic energy E 1 must remain constant over a period of oscillations, and Eq. (2) shows at such a cycle can be reached for two situations. 1) The limit cycle may be combustion controlled: If e acoustic losses are small (F 1 = 0), e pressure and heat release signals may adjust to give S 1 0. The limit cycle is reached when is phase shift leads to a zero Rayleigh term S 1 as observed in certain experiments. Physically, is is often obtained when e heat release oscillations saturate (because e minimum reaction rate reaches zero at some instant of e cycle) or when e phase between pressure and heat release changes so at combustion itself controls e limit cycle amplitude. 2) The limit cycle may be acoustically controlled: The source term S 1 may be large (pressure and heat relase oscillating in phase) but e acoustic losses F 1 are too large and compensate S 1. In is case, e final amplitude of oscillation is controlled by e acoustic impedances of outlets and inlets. Clearly, ese two solutions lead to very different approaches of combustion instabilities: If e limit cycle is combustion controlled, e acoustic behavior of inlets and outlets has a limited effect on e stability; if it is acoustically controlled, acoustic impedances of inlets and outlets become essential elements of any meod (experimental or numerical). In e present study, e LES results are postprocessed to measure all terms of Eq. (2) and determine wheer e unstable mode is combustion or acoustically controlled. LES for Reacting Flows in Complex Geometries Numerical Meods for Compressible Reacting LES Most academic LES are limited to fairly simple geometries for obvious reasons of cost and complexity reduction. In many cases, experiments are designed using simple two-dimensional shapes 6,19,20 or axisymmetrical configurations 21,22 and simple regimes (lowspeed flows, fully premixed or fully nonpremixed flames) to allow research to focus on e physics of e LES (subgrid scale models, flame/turbulence interaction model) and, more generally, to demonstrate e validity of e LES concept in academic cases. This approach is clearly adequate in terms of modeling development, but it can also be misleading in various aspects when it comes to dealing

3 MARTIN ET AL. 743 wi complex flames in complex geometries, especially in real gas turbines for which specific problems arise: 1) Real geometries cannot be meshed easily and rapidly wi structured or block-structured meshes: Until now, most LES of reacting flows have been performed in combustion chambers where structured meshes were sufficient to describe e geometry. This is no longer e case in gas turbines, and is brings additional difficulties. Indeed, on structured meshes, building high-order spatial schemes (typically four to six order in space) is easy and provides very precise numerical meods For complex geometries such structured meshes must be replaced by unstructured grids, on which constructing high-order schemes is a more difficult task. 2) Unstructured meshes also raise a variety of new problems in terms of subgrid-scale filtering: Defining filter sizes on a highly anisotropic irregular grid is anoer open research issue Many LES models, developed and tuned on regular hexahedral grids, may perform poorly on e low-quality unstructured grids required to mesh real combustion chambers. For example, e filtered structure model 24 is difficult to extend to unstructured grids. 3) LES validation is often performed in laboratory low-speed unconfined flames, in which acoustics do not play a role and e Mach number remains small so at acoustics and compressibility effects can be omitted from e equations. 10,21 In most real flames (for example in gas turbines), e Mach number can reach high values and acoustics are important so at taking compressibility effects into account becomes mandatory. This leads to a significantly heavier computational task: Because acoustic waves propagate faster an e flow, e time step becomes smaller and e boundary conditions must handle acoustic wave reflections. 4 Being able to preserve computational speed on a large number of processors en also becomes an issue simply to obtain a result in a finite time. 4) At e present time, it is impossible to perform a true LES everywhere in e flow and it will remain so for a long time. For example, e flow between vanes in swirled burners, inside e ducts feeding dilution jets, or rough multiperforated plates would require too many grid points. Compromises must be sought to offer (at least) robustness in places where e grid is not sufficient to resolve e unsteady flow. In e present work, e full compressible Navier Stokes equations are solved on hybrid (structured and unstructured) grids in a code called AVBP. Subgrid stresses are described by e wall adapting local eddy viscosity model. 30 The flame/turbulence interaction is modeled by e ickened flame (TF) model. 6,31 The numerical scheme is explicit in time and provides ird-order spatial and irdorder time accuracy. 31 TF Model and Chemical Scheme For is study, e standard TF model 31 is used: In is model, preexponential constants and transport coefficients are bo modified to offer icker reaction zones at can be resolved on LES meshes. The fundamental property justifying is approach has been put forward by Butler and O Rourke 32 by considering e balance equation for e k-species mass fraction Y k in a one-dimensional flame of ermal ickness δl 0 and speed s0 L : ρy k t + ρu iy k x i = x i Modifying is equation to have ρy k t + ρu iyk = x i x i ( ρ D k Y k x i (ρ FD k Y k x i ) ) + ω k (Y j, T ) (4) + 1 F ω k( Y j, T ) (5) leads to a TF equation where F is e ickening factor and superscript indicates ickened quantities. Introducing e variable changes X i = x i /F and = t/f leads to ρy k + ρu iy k = X i X i (ρ D k Y k X i ) + ω k ( Y j, T ) (6) which has e same solution as Eq. (4) and propagates e flame front at e same speed sl 0. However, Yk (x, t) = Y k(x/f, t/f) shows at e flame is ickened by a factor F. The ickened flame ickness is δl = Fδ0 L. Choosing sufficiently large values of F allows to obtain a ickened flame at can be resolved on e LES mesh. Typically, if n is e number of mesh points wiin e flame front (n is of e order of 5 10) and x e mesh size, e resolved flame ickness δl is n x so at F must be F = n x/s0 L. Note at F is not an additional parameter of e model but is imposed by e preceding relation as soon as e mesh is created. In e framework of LES, is approach has multiple advantages: When e flame is a laminar premixed front, e TF model propagates it, in e limit of an infinitely in front, at e laminar flame speed exactly as in a G equation approach. However, is flame propagation is due to e combination of diffusive and reactive terms, which can also act independently so at quenching (near walls, for example) or ignition may be simulated. Fully compressible equations may also be used as required to study combustion instabilities. The ickening modification of e flame front also leads to a modified interaction between e turbulent flow and e flame: Subgridscale wrinkling must be reintroduced. This effect can be studied and parameterized using an efficiency function E derived from direct numerical simulation results. 31,33,34 This efficiency function measures e subgrid-scale wrinkling as a function of e local subgrid turbulent velocity u e and e filter wid e. In practice, e diffusion coefficient D k is replaced by EFD k and e preexponential constant A by AE/F so at e conservation equation for species k is ρyk + ρu iyk = ) Yk (ρ EFD k + E t x i x i x i F ω k( Y j, T ) (7) Such an equation propagates e turbulent flame at a turbulent speed s T = EsL 0, while keeping a ickness δ L = Fδ0 L. In laminar regions, E goes to unity, and Eq. (7) simply propagates e front at e laminar flame speed sl 0. The subgrid-scale wrinkling function E was obtained from e initial model of Ref. 31 as a function of e local filter size e, e local subgrid-scale turbulent velocity u e, e laminar flame speed sl 0, and e laminar and e flame icknesses δl 0 and δ L. The TF model uses finite rate chemistry: Here e configuration corresponds to a lean premixed flame so at a one-step Arrhenius kinetics is sufficient. This one-step scheme (called 1sCM1) has been fitted wi a genetic algorim-based tool on a laminar flame structure. The reference mechanism used to fit 1sCM1 is e Peters and Rogg propane scheme. 35 Scheme 1sCM1 takes into account five species (C 3 H 8, O 2, CO 2, H 2 O and N 2 ): C 3 H 8 + 5O 2 3CO 2 + 4H 2 O (8) The rate of e single step reaction is given by q = A ( ρy C3 H 8 / WC3 H 8 ) n C 3 H 8 ( ρyo2 / WO2 ) n O 2 exp( Ea /RT) (9) where e rate parameters are provided in Table 1. The diffusion coefficient D k of species k is obtained as D k = ν/sc k, where ν is e viscosity and Sc k e fixed Schmidt number of Table 1 Rate constants and Schmidt numbers for 1sCM1 scheme a Constants Value Chemical rate constant A 3.29E 10 n C 3H n O E a Schmidt number C 3 H O CO H 2 O N a Activation energy is in calories per moles and e preexponential constants in cgs units.

4 744 MARTIN ET AL. species k. The Schmidt number values used in e present simulations are given in Table 1 and correspond to e PREMIX values measured in e burnt gases. The Prandtl number is set to Wi is parameter set, e agreement between flame profiles obtained using AVBP or PREMIX wi e same chemical scheme is good. The agreement between e Peters and Rogg 35 and e 1sCM1 schemes in terms of laminar flame speed is satisfactory for e lean to stoechiometric mixtures. Note at oer formulations are available for LES of partially premixed turbulent flows. 5,11 To study combustion/acoustics coupling, however, e TF model offers e best compromise. First, e G equation is usually implemented in low Mach number codes, which do not solve for acoustics, whereas acoustics are fully represented in e TF model. Second, e TF approach has been now validated in multiple complex geometry swirled burners 12,36 making it a proper basis for e present study. Acoustic Solver for e Helmholtz Equation The acoustic tool used in is study, called AVSP, solves e eigenvalue problem associated to e wave equation (1). When dealing wi ermoacoustic instabilities, it is usual to model e geometry of e combustor by a network of one-dimensional or two-dimensional axisymmetric acoustic elements where a simplified form of Eq. (1) can be solved. 4,15,37 39 Jump relations are used to connect all of ese elements, and e amplitude of e forward and backward acoustic waves are determined so at e boundary conditions are satisfied. The main drawback of is approach is at e geometrical details of a combustor cannot be accounted for and only e first equivalent longitudinal or ororadial modes are sought for. In e acoustic solver, a finite element strategy is used to discretize e exact geometry of e combustor so at no assumption is made a priori regarding e shape of e modes. This feature gives e Helmholtz solver e potential to test e effect of geometrical changes on e stability of e whole system. Equation (1) is solved in e frequency domain by assuming harmonic variations at frequency f = ω/(2π) for pressure, velocity, and local heat release perturbations: p 1 =R[ ˆP(x) exp( iωt)], u 1 =R[Û(x) exp( iωt)] ω T 1 =R[ ˆ T exp( iωt)] (10) Introducing Eq. (10) into Eq. (1) and neglecting e turbulent noise γ p 0 u 1 : u 1 in front of e combustion term (γ 1) ω T 1 / t leads to a modified form of e Helmholtz equation: ρ 0 c 2 0 [(1/ρ 0) ˆP] + ω 2 ˆP = iω(γ 1) ˆ T (11) where e unknown quantities are e complex amplitude ˆP of e pressure oscillation at frequency f and pulsation ω. Note at ˆ T, e amplitude of e heat release perturbation, is also unknown and must be modeled. This is obviously e difficult part of e modeling, and it remains an open research issue today. In e present work, e simplest linear approach, initally proposed by Crocco, 14 was chosen as a first step. A direct extension of e standard n x model 4,14,40 was used to write ω T 1 nu 1 (x ref, t τ), where u 1 is e axial velocity fluctuations. In one-dimensional approaches, e interaction index n and time delay τ are two parameters describing e acoustic behavior of a compact flame located at e axial position x ref. In e Helmholtz solver, where e geometry of e combustor is fully described, e flame is distributed and e interaction index and time delay depend on space. These data can be extracted from LES results by postprocessing eier a self-excited or a forced oscillating regime. Once measured in LES, e fields n(x,ω)and τ(x,ω) are used to model e unsteady heat release in Eq. (11) as ˆ T = n(x,ω)exp[iωτ(x,ω)]û(x ref ) n ref (12) The linearized momentum Euler equation Û = ˆP/iωρ can be used to relate ˆ T to ˆP and close Eq. (11). Three types of boundary conditions can be prescribed for Eq. (11), where n is e outward unit normal vector to e boundary: 1) Dirichlet condition, which imposes ˆP = 0, on fully reflecting outlets; 2) Neumann condition, which imposes ˆP n = 0, on fully rigid walls or reflecting inlets; and 3) Robin condition, which imposes cz ˆP n = iω ˆP, on general boundaries, where Z is e local reduced complex impedance Z = ˆP/ρ 0 c 0 Û n. In is study, e reduced boundary impedance Z has been obtained using Eq. (16) as will be described later. Knowing e boundary impedance Z, e sound speed c 0 and e density ρ 0 distribution, e flame response [n(x,ω),τ(x,ω)], and assuming at Z does not depend on ω, a Galerkin finite element meod is used to transform Eq. (11) into a nonlinear eigenvalue problem of size N (e number of nodes in e finite element grid used to discretize e geometry) of e form [A][ ˆP] + ω[b][ ˆP] + ω 2 [C][ ˆP] = [D(ω)][ ˆP] (13) where [ ˆP] is e column vector containing e eigenmode at pulsation ω and [A], [B], and [C] are square matrices depending only on e discretized geometry of e combustor. (Note at e same result holds if 1/Z = 1/Z 0 + Z 1 ω + Z 2 /ω, where Z 0, Z 1, and Z 2 are complex-valued constants.) If e impedances Z change wi ω, Eq. (13) can be solved iteratively and independently for each eigenmode, by using a Z value adapted to each eigenfrequency. [D(ω)] is e unsteady contribution of e flame and depends on e pulsation rough e combustion term n(x,ω)exp[iωτ(x,ω)]. No efficient numerical meod exists to solve is nonlinear eigenvalue problem. However, in e case where e unsteady flame response is neglected, namely, [D(ω)] = 0, Eq. (13) simplifies into a quadratic eigenvalue problem depending only on ω and ω 2. A variable transformation can en be used to obtain an equivalent linear eigenvalue problem of size 2 N (Ref. 41). Several numerical meods can en be used to assess e eigenmodes. Direct meods like e quadratic regulator approach are exact and have e advantage to provide all of e eigenmodes. However, ey can be expensive to solve for large problems (N > 10 3 ). Because only e first few frequencies are usually of interest from a physical point of view, it is more appropriate to use an iterative meod at can be applied for large problems (N > 10 5 ) wiout difficulty. In e Helmholtz solver, we are using a parallel implementation of e Arnoldi meod (see Ref. 42), which enables to solve complex problems of size N in a few minutes. Setting [D(ω)] = 0 is equivalent to finding e eigenmodes of e burner, taking into account e presence of e flame rough e mean temperature field but neglecting e flame effect as an acoustically active element. The boundary conditions are also acounted for, and is approximation can provide relevant information on e shape and real frequency of e first few modes of e combustor. However, because ere is no coupling between e acoustics and e flame, ere is no hope to discriminate between stable and unstable modes, which is e ultimate objective of is study. When it is assumed at e unsteady flame response creates a small perturbation of e modes, a linear expansion technique can be developed to assess e imaginary part of ω, hence, e stability of e perturbed modes. 43,44 Anoer pa has been followed in is study to handle cases where e unsteady response of e flame changes e modes significantly and when e linear expansion is not justified. The nonlinear eigenvalue problem Eq. (13) is en solved iteratively, e k iteration consisting in solving e quadratic eigenvalue problem in ω k defined as ([A] [D(ω k 1 )])[ ˆP] + ω k [B][ ˆP] + ω 2 k [C][ ˆP] = 0 (14) A natural initialization is to set [D(ω 0 )] = 0 so at e computation of e modes wiout combustion is in fact e first step of e iteration loop. Usually, less an five iterations are enough to converge toward e complex pulsation and associated mode. This linearized approach to describe e stability of e burner in terms of modes has drawbacks but remains one of e basic tools to study instabilities: 1) The linearization is valid only for small-amplitude perturbations, a condition which is obviously not true when limit cycles typical of combustion instabilities are observed in gas turbines. However, is assumption is valid when e instability grows 45 and helps

5 745 MARTIN ET AL. to determine e unstable modes: Such modes have to appear and grow before ey reach a limit cycle, and any analysis adapted to is early phase is of interest. 2) Most acoustic tools work on linear regimes for which each oscillatory mode is independent of oer modes. Many combustion instabilities exhibit nonlinear coupling, where high-frequency modes couple wi low-frequency oscillations.46 These were also observed in e experiment in Ref. 1, in which a 530-Hz mode (often called rumble) was systematically accompanied by a high-frequency mode (called screech) at 3750 Hz. The fact at combustion instabilities involve more an one mode of oscillation is one of e basis of e approach of Yang and Culick.47 The tool presented earlier treats each mode individually and cannot simulate such phenomena. 3) The description of e coupling between acoustics and combustion in such models is extremely crude. The response of e flame excited by an acoustic wave depends on several physical phenomena such as chemical reactions, species diffusion, vortex shedding, vortex flame interaction, etc. All of ese phenomena are not neglected in e present study, but eir cumulative effect is modeled rough e global timescale τ and index n. Despite ese limitations, such tools are useful because ey provide relevant information about e modes triggered by e acoustic flame coupling while running fast: For e current configuration, only 8000 grid points were necessary to describe e geometry and obtain e first four modes. For comparison, half a million nodes were used to perfom e LES. A typical run for solving e quadratic eigenvalue problem of type Eq. (14) on is grid lasts 10 min by using 15 processors (R MHz IP35) on an SGI O3800 parallel machine. Such a tool can, us, be used in e design process of new gas turbines to characterize eir ermoacoustic modes. By describing e whole geometry between e compressor and e turbine, including all of e injectors dispatched around e combustion chamber, such simulations would give unique information about e swirling modes at sometimes show up in large gas turbines. The difficult and computationally expensive task would be to compute e flame transfer function by performing a LES of e turbulent flame. Such a simulation would be performed by considering an angular sector corresponding to only one injector, saving grid points and CPU resources. Configuration Geometry: Swirled Premixed Combustor The meodologies described in e preceding sections were tested for a swirled combustor shown in Fig. 1. The configuration is typical of swirled combustion: Premixed gases are introduced tangentially into a long cylindrical duct feeding e combustion chamber. The tangential injection creates e swirl required for stabilization. The fuel is propane. The two independant swirler elements allow fuel staging. The staging parameter α is defined as e ratio of fuel flow rate of e first to e second swirler. The regime studied here corresponds to e parameters given in Table 2. The staging of e burners corresponds to α = 0.3. Table 2 Flow parameters for combustion cases Parameter Value Flow rate Total Axial Equivalence ratioa Reynolds number a Table Burner mou. Acoustic inlet and outlet boundaries characteristics for REF and LEAK Case Boundary characteristics Inlet σ Outlet σ Characteristic Outlet impedance at 380 Hz Measured in LES Calculated wi Eq. (16) LEAK REF 1,000 1,000 Nonreflecting 1,000 10,000 Reflecting i i i 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 compressible flows. Here, e Navier Stokes characteristic boundary condition technique4,48 was used at e outlet. The level of reflection of is boundary can be controlled by changing e relaxation coefficient σ of e wave correction,49 which determines e amplitude of e incoming wave L 1 entering e computational domain: L 1 = σ ( p pt ) (15) where pt is e prescribed pressure value at infinity. Equation (15) acts on e flow like a spring mechanism wi a stiffness σ. The impedance of e boundary is a function of σ and ω, which can be obtained analytically49 for simple cases,2 Z = ( iω/σ )/(1 iω/σ ) (16) (This impedance can be taken into account by e Helmholtz solver under e following form: 1/Z = 1 + iσ/ω.) For small values of σ, Eq. (15) keeps e pressure p close to its target value pt while letting acoustic waves go out at e same time4 : The outlet is nonreflecting. When large values of σ are used, e outlet pressure remains strictly equal to pt and e outlet becomes totally reflecting. Two sets of computation will be shown (Table 3). The first one, LEAK, corresponds to a case where e spring stiffness σ is small so at e outlet is nonreflecting and e acoustic waves are evacuated wi very small reflection levels. For e second set, REF, σ is large and e outlet is reflecting. (The pressure oscillation is almost zero.) LES Results Stable Flow Fig. 1 Staged swirled combustor configuration. The first computation corresponds to e case where e outlet section is nonreflecting (case LEAK in Table 3): The acoustic feedback is minimized, and e flame does not exhibit any strong unstable movement. The mean velocity and fuel mass fraction fields are shown in Figs. 2 and 3. As expected, e downstream part of e

6 746 MARTIN ET AL. Fig. 3 Mean fuel mass fraction field: black lines, isoreaction rate for stable combustion. Fig. 6 Evolution of burner acoustic energy E 1. Fig. 4 Mean normalized values:, pressure;, heat release; and, phase angle between pressure and heat release. Fig. 5 Mean values:, Rayleigh criterion S 1 and..., acoustic fluxes F 1. central recirculation zone is filled by burnt gases and stabilize e turbulent flame. Instability Sequence LES also reveals at e combustor can exhibit a strong unstable mode when e outlet is acoustically closed (case REF). In is case, soon after ignition, e pressure and e global heat release start oscillating (Fig. 4) at 380 Hz. To analyze e behavior of is instability, e following sequence is set up: 1) Starting from a stable flame (LEAK), e outlet impedance is changed to become reflecting (case REF) at time t = s (Fig. 4). The oscillation grows and reaches a limit cycle at a frequency of 380-Hz mode. 2) At time t = s, e outlet impedance is switched again to a nonreflecting condition (case LEAK) and e instability disappears. This scenario provides four phases which are studied sequentially: 1) a linear grow between times and s, 2) an overshoot phase between and s, 3) a limit cycle between times and s, and 4) a decay phase starting at t = s. For each phase, e instability is analyzed in terms of flame shape, flame oscillation, and phase between heat release and pressure. Moreover, e acoustic energy equation budget is closed, and all terms are analyzed. Grow Phase Once e outlet boundary is acoustically closed (t = s), e ermoacoustic instability starts. Figure 5 shows e time variations of e combustion source term S 1 and e acoustic losses F 1. The total acoustic energy evolution of e chamber E 1 is shown in Fig. 6. Figure 7 shows at e budget of Eq. (3) is quite well closed by e LES data: The difference S 1 F 1 matches e time derivative of E 1. This validates bo e LES results and e acoustic energy equation (3). It is also e first example of such a treatment for a resonating combustor. Because e budget is closed, individual terms can en be analyzed. First, e phase angle between pressure and heat release is shown in Fig. 4. During e grow phase, it is close to zero and slowly Fig. 7 Comparison:..., time derivative of e acoustic energy E 1 / t and, S 1 F 1. shifting toward π/4, leading to a strong coupling between pressure and heat release, at is, a positive S 1 term. During e grow phase, e source term S 1 is large and always positive (Fig. 5) because e phase angle stays in e [ π/2; π/2] range. Figure 5 shows at e acoustic losses balance e reacting term S 1 in e acoustic budget equation. The limit cycle is controlled by acoustic losses and not by combustion. Overshoot Phase and Limit Cycle At t = s, e instability reaches a limit cycle at 380 Hz. Before reaching is limit cycle, a large overshoot of acoustic energy is observed: This is typical of combustion instabilities, and it has been observed experimentally in oer systems. 45,50 Figure 4 shows at, reaching e nonlinear zone, e phase difference between pressure and heat release increases from zero to π/4 in e limit-cycle zone. The drift of is phase difference togeer wi increasing acoustic losses lead to e saturation of e instability. The coupling loop between p 1 and ω T 1 can be identified from LES as follows. The longitudinal mode induces e formation of a vortex ring at e dump plane. This vortex ring strongly interacts in phase wi e flame. Figure 8 shows e interaction between e acoustically induced vortex ring and e flame brush. Figure 9 allows to locate LES snapshots in e acoustic period and displays e mean pressure fluctuation p 1 in e flame zone, e heat release fluctuation ω T 1, and e fluctuation of e mean velocity in e dump plane u dump 1. At instant 1, a vortex ring appears at e dump plane when du dump 1 /dt is maximum. The ring structure detaches and is convected rough e flame by e mean flow (instants 2, 3, and 4). Between instants 1 and 3, e flow stretches e flame, increasing its area, whereas e flame wrinkling by e vortex ring remains weak. Consequently ω T 1 increases wi a medium slope (Fig. 9). Between points 3 and 5, e vortex ring is stretching e flame and ω T 1 increases faster. Moreover, e vortex ring is gradually destroyed, and its global coherence disappears between instants 4 and 5, at a moment when du dump 1 /dt is minimum. At instant 5, some coherent structures are still interacting wi e flame, producing (noisy) flame pockets and cusps. After instant 5, e flame burns out e fresh gases present in e chamber and propagates back to e injection pipe decreasing e overall flame surface and ω T 1. Decay Phase The decay phase is triggered by e sudden change in e acoustic outlet boundary condition switching to nonreflecting (LEAK) at t = s. The phase angle φ pω between pressure and heat release increases by a large amount: φ pω >π/2att = s, at is, 1.2 ms after relaxing outlet pressure (Fig. 4). At is time, S 1 becomes globally negative for e first time and e instability is rapidly damped. The acoustic losses actually become positive (gain term) during is last oscillation but e instability engine is broken and e Rayleigh

7 747 MARTIN ET AL. Fig. 10 Flame transfer function magnitude (n parameter) field in central plane evaluated at f = 432 Hz: black line, iso-n = Pa m 1 and +, reference point location. magnitude n and e phase φ of e FTF defined as n(x, ω) = U (xref ) nref ˆ T (x) φ(x, ω) = ωτ (x, ω) = arg Fig. 8 Vortex ring shedding at six instants (Fig. 9) during e limit cycle: isosurface, Q vortex criterion and black lines, isoreaction rate in e burner central plane. Fig. 9 Time signals during limit cycle and snapshots corresponding to Fig. 8:, pressure;...., inlet velocity; and, total heat release. term S1 becomes negative during a half-cycle (0.173 < t < ms in Fig. 5) leading to e immediate decay of all unstable activity. Acoustic Analysis Results This section describes e results obtained wi e Helmholtz solver for e configuration of Fig. 1. The impedances are calculated from Eq. (16) using e value of σ given in Table 3. Equation (16) shows at, for a fixed σ coefficient, e impedance is a function of e pulsation ω. To verify at Eq. (16) is sufficiently accurate, Table 3 gives e value of e impedance at 380 Hz (which is e frequency of e first mode observed in e LES) predicted by Eq. (16) and measured in e LES. For ese computations, e flame transfer function is needed. In e present work, n and τ are obtained by postprocessing LES results as described in e following section. These results are en used to predict e frequency and grow rates of all modes using e Helmholtz equation (11) and to compare em to e LES results of e preceding section. Measurement of Flame Transfer Function The key mechanism to predict combustion instabilities in Helmholtz codes is e flame transfer function (FTF). In e model chosen in is study [Eq. (12)], e heat release fluctuations are related to e velocity fluctuations at a reference point rough e ˆT (x) U (xref ) nref (17) (18) Because e flame is more prone to interact wi longitudinal oscillations, e reference normal vector nref is set collinear to e axial direction so at e scalar product U (xref ) nref corresponds to e axial velocity fluctuation at e reference point. This point is located following e Crocco approach,14 at is, in e fresh gas near e inlet combustor. As shown in Fig. 2, a recirculation zone located at e entrance of e dump combustor is induced by e swirling flow. Consequently, e reference point is located upstream of is zone3 at 113 mm from e inlet of e device (Fig. 10). Note at oer positions in e vicinity of is location have been tested leading to very similar results wi e Helmholtz solver. Flame transfer functions are usually measured for e whole combustor39,51,52 or by zones.53 In e present approach, n and φ must be obtained locally at each point of e combustor, and is can be done by analyzing snapshot series from LES. Moreover, e frequency dependence of ese parameters should be taken into account. To do so, it is assumed at FTF depends mostly on e real part of e frequency and not of e grow rate of e mode. Thus, at each subset k of e algorim [Eq. (14)], e FTF parameters are evaluated at [ωk /(2π)], and e first guess corresponds to e eigenmodes determined wiout active flame (where n is set to zero everywhere). (Furer studies are needed to assess is assumption used for practical reasons.) From a practical point of view, n and φ can be extracted from LES results by Fourier processing e local heat release and e unsteady velocity at e reference point and using Eqs. (17) and (18). Two meodologies can be considered for such an analysis: 1) use a stable regime and force e flow (usually by introducing acoustic waves rough e inlet39,40,54 ) or 2) use a self-excited regime. The choice of e meodology raises oer fundamental issues, which are still open today: 1) When it is assumed at Eq. (12) is valid for bo forced and self-excited cases, bo meodologies are expected to provide e same n and φ fields. This is a eoretical argument at has not been checked yet and is left for furer work. 2) The possible dependence of e transfer function on e acoustics wave amplitude raises an additional difficulty. If n and τ depend on e wave amplitude, en bo meodologies have drawbacks: The forcing meod will provide different n and τ fields when e forcing amplitude changes, whereas e self-excited meod uses a nonlinear limit cycle to evaluate n and τ. These questions are beyond e scope of e present study. Here e fields of n and τ were measured by postprocessing e selfexcited regime between t = and t = s shown in Fig. 4. Results given in e next section prove e validity of is meod, but furer studies are obviously needed. The resulting flame transfer function parameters, shown only wiin e flame zone (when n Pa m 1 ), are shown in Figs. 10 and 11.

8 748 MARTIN ET AL. Table 4 No active flame Helmholtz solver results Active flame Mode F, Hz α, rad s 1 F,Hz α, rad s Fig. 11 Flame transfer function phase (φ parameter) field in e central plane evaluated at f = 432 Hz: black line, iso-n = Pa m 1 and +, reference point location. Fig. 13 Longitudinal structure of first four modes obtained from Helmholtz solver:, normalized p rms evolution along burner axis 1 wi acoustics/flame coupling and, wiout. Fig. 14 RMS pressure fluctuations p rms along burner axis;, LES 1 and, solvers. Fig. 12 Acoustic pressure modulus for e first four modes calculated by e Helmholtz solver wi active acoustic flame in REF case, ˆP isolines. Helmholtz Solver Results Using e mean fields given by LES, e Helmholtz solver is applied to obtain e ermoacoustic eigenmodes of e burner. This tool can be run using eier a nonactive flame, at is, setting n to zero everywhere, or an active flame, using e postprocessed n and φ fields displayed. Moreover, e impedance values are determined from e LES boundary conditions settings (Table 3). First, when e outlet impedance corresponds to e LEAK case, e Helmholtz solver predicts at all modes are damped. This confirms e LES result of Fig. 2 for which no acoustic mode was found for is regime. Second, when e outlet impedance corresponds to e REF case, Table 4 presents e modes at are obtained for bo e nonactive and e active flames. These four modes in e REF case are all longitudinal except in e vicinity of e swirler: The structure of e modes in e central plane of e burner is shown in Fig. 12, whereas a one-dimensional cut of e rms pressure field is given in Fig. 13. The real frequencies of e active and nonactive cases are very similar: Typically, taking into account e active effect of e flame shifts e eigenmode frequency by a few percent. However e effect on e grow rate is more dramatic: All modes computed wi e nonactive flame are damped, whereas e modes computed wi an active flame are excited for modes 2, 3, and 4. The fastest growing mode is mode 2 at 432 Hz, which is close to e mode observed in e LES (380 Hz). To verify at e mode 2 obtained by e Helmholtz solver is indeed e mode appearing in e LES, Fig. 14 shows a comparison of e ˆP profiles on e axis and confirms e good agreement of LES and Helmholtz solver. The difference between e frequency observed in e LES (380 Hz) and e frequency predicted by e Helmholtz solver (432 Hz) is probably due to e zero Mach number approximation for e Helmholtz solver. Conclusions Three tools have been used to analyse flame acoustics coupling mechanisms in a staged swirled combustor: Full compressible LES, Helmholtz analysis, and budget of acoustic energy. The two latter meods are based on e wave equation in reacting flows. They use LES results but provide essential new elements: The Helmholtz results allow to predict e stability of e combustor and e exact identification of modes appearing during e instability, whereas e budget of acoustic energy demonstrates at e Rayleigh criterion is not e only or even e largest term in e acoustic energy equation. Acoustic losses at e outlet of e combustor contribute significantly to e budget of acoustic energy and determine e levels of oscillation amplitudes as well as eir appearance. More generally, is study confirms e need of coupling classical computational fluid dynamics (here LES) and acoustic analysis to understand combustion instabilities. It also demonstrates e crucial effect of acoustic boundary conditions on e stability of combustors. This has a practical implication. The stability of a given combustion chamber is controlled by acoustic impedances upstream and downstream of e combustor. Removing a combustor section from a full gas turbine to install it in a laboratory setup obviously becomes very dangerous to

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