Can-annular Combustion Research Articles

A network model for can-annular combustors using a multidimensional transfer matrix based on the mode matching method

Heavy-duty land-based gas turbines with can-annular combustors typically exhibit a set of thermoacoustic modes with similar frequencies, due to their rotational symmetry and the weak coupling between neighbouring cans. The Bloch boundary condition is often used to simplify these combustors. Because the widely used (quasi-)one-dimensional model cannot naturally consider high-order modal components in the three-dimensional acoustic field, it is valid only when the frequency is sufficiently low and the length of the can-to-can interaction gap is sufficiently short. In this paper, we present a theoretical model using the mode-matching method to consider high-order modal components. A reflection coefficient matrix is used to accurately describe the acoustic characteristics of the gap area. This reflection coefficient matrix is found to depend on the nondimensional cross-sectional area ratio, the Helmholtz number based on the length of the gap area, and the Bloch number. To facilitate a full low-order network modelling of the high-frequency thermoacoustic modes in can-annular combustors, a flame transfer matrix is also used to describe the modal coupling effect across the flame. Based on this model, the thermoacoustic problem of a four-flame can-annular combustor is studied as an example. It is shown that the frequency and growth rate of the thermoacoustic modes can be predicted accurately using the reflection coefficient matrix and flame transfer matrix. The effects of high-order modal components across the flame and in the gap areas are studied separately. The results show that the model containing both of the two coupling effects has the widest range of validity. These couplings can be ignored if only low-frequency plane acoustic waves are considered but need to be carefully considered for higher frequency modes. Moreover, when multiple flames are considered, asymmetric flame distributions are found to more easily lead to modal coupling compared to symmetric flame distributions. All results are validated by comparing with the numerical solutions of the full two-dimensional Helmholtz equation using the finite element method.

Bridging the gap between annular and can-annular acoustic spectra.

In the literature on thermoacoustic instabilities in combustors, a distinction is typically made between annular and can-annular systems because these are the most common gas turbine architectures. In reality, however, annular combustors typically feature discretely symmetric elements, such as burner tubes, and can-annular combustors feature an azimuthally symmetric plenum at the turbine inlet. To better understand the general case in between the annular and can-annular extremes, we analyze the acoustic spectrum of an idealized can-annular combustion chamber with variable geometry, where the length of the axial gap distance beyond the ends of the cans-hence, the coupling strength-may be adjusted. For small gap sizes, the geometry approaches a set of isolated combustor cans, whereas for large gap sizes, it approaches that of pure annular systems. We present two theoretical models based on Bloch wave theory and validate them against finite-element simulations of the Helmholtz equation. We demonstrate that the azimuthal modes transform into the eigenmodes of an annular chamber as the gap is fully opened, and we show that below a certain frequency, all the modes in between the can-annular and annular limits can be classified by their axial and azimuthal mode orders.

Weakly nonlinear analysis of thermoacoustic oscillations in can-annular combustors

Can-annular combustors feature clusters of thermoacoustic eigenvalues, which originate from the weak acoustic coupling between $N$ identical cans at the downstream end. When instabilities occur, one needs to consider the nonlinear interaction between all $N$ modes in the unstable cluster in order to predict the steady-state behaviour. A nonlinear reduced-order model for the analysis of this phenomenon is developed, based on the balance equations for acoustic mass, momentum and energy. Its linearisation yields explicit expressions for the $N$ complex-valued eigenfrequencies that form a cluster. To treat the nonlinear equations semianalytically, a Galerkin projection is performed, resulting in a nonlinear system of $N$ coupled oscillators. Each oscillator represents the dynamics of a global mode that oscillates in the whole can-annular combustor. The analytical expressions of the equations reveal how the geometrical and thermofluid parameters affect the thermoacoustic response of the system. To gain further insights, the method of averaging is applied to obtain equations for the slow-time dynamics of the amplitude and phase of each mode. The averaged system, whose solutions compare very well with those of the full oscillator equations, is shown to be able to predict complex transient dynamics. A variety of dynamical states are identified in the steady-state oscillatory regime, including push–push (in-phase) and spinning oscillations. Notably, the averaged equations are able to predict the existence of synchronised states. These states occur when the frequencies of two (or more) unstable modes with nominally different frequencies lock onto a common frequency as a result of nonlinear interactions.

Acoustic Boundary Conditions for Can-Annular Combustors

This paper derives and validates an analytical model for acoustic boundary conditions on a can-annular gas turbine combustion system composed of discrete cans connected to an open annulus upstream of a turbine. The analytical model takes one empirical parameter: a connection impedance between adjacent cans. This impedance is extracted from time-marching computations of two-can sectors of representative combustors. The computations show that reactance follows the Rayleigh conductivity, while resistance takes a value of order 0.1 as a weak function of geometry. With a calibrated value of acoustic resistance, the analytical model reproduces can-to-can transfer functions predicted by full-annulus computations to within 0.03 magnitude at compact frequencies. Varying the combustor–turbine gap length, both model and computations exhibit a minimum in reflected energy, which drops by 63% compared to the datum gap. A parametric study yields a design guideline for gap length at the minimum reflected energy, allowing the designer to maximise transmission from the combustion system and reduce damping requirements.

Acoustics of can-annular combustors: Experimental characterisation and modelling of a lab-scale multi-can setup with adjustable geometry

In can-annular combustors, a narrow can-turbine junction leads to an acoustic coupling between the cans, yielding unique features that greatly impact the (thermo)acoustic response of these geometries. To investigate these features, we designed and assembled a modular lab-scale can-annular system equipped with 12 cans connected downstream by a short annulus, creating an annular gap that models the can-turbine junction. The acoustic response of the setup is measured experimentally for various gap heights on a broad range of frequencies. We observe that eigenfrequencies form clusters, confirming recent theory. To aid the analysis of the experimental results, a low-order model is proposed, which has analogies with methods recently developed for studying azimuthal modes in annular combustors using azimuthal averaging. Our model accounts for the fact that the azimuthal structure of the modeshapes is not purely sinusoidal, and for the influence of the can’s first transverse mode. We observe that the eigenfrequencies obtained experimentally are well predicted by the low-order model. The experimental and low-order model results are then compared to 3D Helmholtz calculations, which allow us to visualise the 3D acoustic modeshapes, assess the validity of the assumptions used in the low-order model, and test the low-order model beyond the parameter range achievable in our experiments. Last, we discuss how the behaviour of the low-frequency clusters can be predicted qualitatively without calculations. Our experiments are the first to characterise the presence of clusters of eigenvalues in the acoustic response of a can-annular setup as a function of the can-turbine junction size.

Unified low-order modeling approach to can-annular combustors

We present a unified, physics-based low-order modeling approach to can-annular combustors, combining a linear stability analysis and a nonlinear Langevin approach to unravel the highly complex, multi-physical dynamics in this type of system. In the linear part, Howe's Rayleigh conductivity model is combined with the projected Helmholtz equation and a Bloch wave ansatz to arrive at a single equation for the frequency spectrum of an N-can combustor. Starting from first principles, we illustrate and give a physical explanation for the coupling-induced amplification and suppression of thermoacoustic instabilities. Adding another layer of complexity, we then take into account nonlinearities in the flame response and the aeroacoustic coupling. The nonlinear dynamics of a symmetric model combustor are explored by exploiting the gradient structure of the averaged slow-flow dynamics. We obtain exact analytical expressions for the steady-state statistics, highlighting the connection between different emergent patterns and the resistive coupling between the cans. We leverage our analysis to explain the intermittent energy transfer between Bloch modes observed in real-world gas turbines.

Chimera states in a can-annular combustion system

We experimentally investigate the collective behaviour of four turbulent lean-premixed combustors coupled in a ring network. We adopt a complex systems approach, treating each combustor as an individual self-excited thermoacoustic oscillator and examining how a minimal network of such oscillators can self-organize into synchronous and asynchronous states. Using proven synchronization metrics, we identify a broad range of collective behaviour, including a weak anti-phase chimera, a breathing chimera, intermittent frequency locking, and three-frequency quasiperiodicity. The discovery of chimera states in a minimal network of coupled thermoacoustic oscillators suggests that chimera control techniques could be exploited for the suppression of thermoacoustic oscillations in can-annular combustion systems.

Large eddy simulation of combustion instabilities in multiple combustors densely interacting with each other

Large eddy simulation (LES) of combustion instabilities in the can-annular gas turbine combustors are performed. Naturally occurring thermoacoustic instability is simulated in the can-annular combustors for the first time, and the resulting dynamic flames are investigated. Premixed swirl flames are generated in the two combustors, and the combustor-combustor interaction of pressure wave occurs in the cross-talk connecting each combustor. Both symmetric and asymmetric conditions are considered for explaining their effects on flame perturbation and instability mode during the operation. In the symmetric inlet condition, longitudinal pressure wave induces the flame fluctuation resulting in a push-push mode in the combustion system, which indicates that the instability phases in two combustors are identical. In the asymmetric case, an inlet velocity pulsation with an opposite phase of combustion instability is applied, which results in the suppression of instability and the attenuation of acoustic energy transferred to the whole system through the cross-talk. This work expands the scope of LES together with high-fidelity turbulent combustion models for addressing the combustion instability in the can-annular combustors. The results indicate that asymmetry in flow rate oscillations provides an effective actuation for the attenuation of combustion instabilities.

Interplay of Clusters of Acoustic and Intrinsic Thermoacoustic Modes in Can-Annular Combustors

Abstract Thermoacoustic systems can exhibit self-excited instabilities of two nature, namely cavity modes or intrinsic thermoacoustic (ITA) modes. In heavy-duty land-based gas turbines with can-annular combustors, the cross-talk between cans causes the cavity modes of various azimuthal order to create clusters, i.e., ensembles of modes with close frequencies. Similarly, in systems exhibiting rotational symmetry, ITA modes also have the peculiar behavior of forming clusters. In the present study, we investigate how such clusters interplay when they are located in the same frequency range. We first consider a simple Rijke tube configuration and derive a general analytical low-order network model using only dimensionless numbers. We investigate the trajectories of the eigenmodes when changing the downstream length and the flame position. In particular, we show that ITA and acoustic modes can switch nature and their trajectories are strongly influenced by the presence of exceptional points. We then study a generic can-annular combustor. We show that such configuration can be approximated by an equivalent Rijke tube. We demonstrate that, in the absence of mean flow, the eigenvalues of the system necessarily lie on specific trajectories imposed by the upstream conditions.

Thermoacoustic oscillations in a can-annular model combustor with asymmetries in the can-to-can coupling

Can-annular combustors are equipped with a set of nominally identical cans, circumferentially arranged around the shaft. Adjacent cans are coupled acoustically via a small gap at the downstream end, where the circular cross-section transitions into the annular turbine inlet. Recent experimental and theoretical work has shown that the coupling strongly affects thermoacoustic system stability and, hence, may give rise to damaging pressure oscillations that originate from a constructive interference of pressure waves and heat release rate fluctuations. A laboratory-scale can-annular combustor has been operated with premixed CH4–H2–air mixtures to study these instabilities. The combustor consists of eight identical cans, connected acoustically to their respective neighbours via size-adjustable side branches; the latter allow for a variation of the coupling strength. New experimental results are presented in which five asymmetric sets of coupling strengths are investigated and compared to a previously studied symmetric configuration. The sets are chosen such that the symmetry is gradually reduced until a fully asymmetric configuration is realized. The perturbations affect the observed mode type, frequency of oscillation and the pressure amplitude distribution over the eight cans. The perturbations via the can-to-can couplings do not trigger mode localization. The symmetric configuration considered previously, shows activity in an azimuthal mode of third and fourth order, with unsteady switching between the two. The mode shape is characterised through novel phase-averaged OH* chemiluminescence images capturing all eight cans.

Coupling-induced instability in a ring of thermoacoustic oscillators.

Thermoacoustic instabilities in can-annular combus-tors of stationary gas turbines lead to unstable Bloch modes which appear as rotating acoustic pressure waves along the turbine annulus. The multiscale, multiphysical nature of the full problem makes a detailed analysis challenging. In this work, we derive a low-order, coupled oscillators model of an idealized can-annular combustor. The unimodal projection of the Helmholtz equation for the can acoustics is combined with the Rayleigh conductivity, which describes the aeroacoustic coupling between neighbouring cans. Using a Bloch-wave ansatz, the resulting system is reduced to a single equation for the frequency spectrum. A linear stability analysis is then performed to study the perturbation of the spectrum by the can-to-can interaction. It is observed that the acoustic coupling can suppress or amplify thermoacoustic instabilities, raising the potential for instabilities in nominally stable systems.

Hybrid CFD/low-order modeling of thermoacoustic limit cycle oscillations in can-annular configurations

We propose a hybrid strategy for modeling non-linear thermoacoustic phenomena, e.g. limit-cycle (LC) oscillations, in can-annular combustion systems. The suggested model structure comprises a compressible CFD simulation limited to the burner/flame zone of one single can, coupled to a low-order model (LOM) representing the remaining combustor. In order to employ the suggested strategy for modeling non-linear phenomena like LC oscillations, the LOM must capture non-linear flame dynamics in the cans, which are not resolved by CFD. Instead of identifying such non-linear flame models in preliminary simulations, we aim at learning the non-linear dynamics “on-the-fly”, while simulating the self-excited system under consideration. Based on the observation of flame dynamics in the CFD domain, the parameters of the employed non-linear models are estimated during run time. The present study reveals that block-oriented models, which comprise a linear dynamic part followed by a static non-linear function, are well suited for this purpose. The proposed hybrid model is applied to a laminar can-annular combustor. Results agree well with the monolithic CFD simulation of the entire combustor, while the computational cost is drastically reduced. The employed flame models, whose parameters are identified during the simulation of the self-excited LC oscillation, represent well the relevant non-linear dynamics of the considered flame.

Reduced-order modelling of thermoacoustic instabilities in can-annular combustors

Thermoacoustic instabilities in stationary gas turbines may cause high-amplitude limit cycles, leading to damaged components and costly down-time. To better understand the physical origin of such instabilities in a can-annular combustor configuration, we study the properties of the spectrum of a reduced-order can-annular thermoacoustic system. Increased focus is placed on representing the aeroacoustic interaction between the longitudinal eigenmodes of the individual cans with physically relevant models. To represent the acoustic pressure dynamics in the combustor, we combine an analytical, experimentally validated model for the can-to-can impedance with a frequency-dependent model of the flame response in the cans to acoustic perturbations. By using this approach, we perform a parametric study of the linear stability of an atmospheric can-annular thermoacoustic system, and emphasize general features of the structure and properties of the eigenvalues and the eigenvectors of can-annular combustors. Lastly, we emphasize the differences in the can-to-can coupling that arise when considering open-end boundary conditions – representative of atmospheric set-ups – or closed-end boundary conditions – representative of real gas turbine combustors.

An effective impedance for modelling the aeroacoustic coupling of ducts connected via apertures

An effective impedance that models the aeroacoustic coupling between periodic arrays of ducts that communicate via apertures is derived. A lumped description of the interaction between the one-dimensional acoustics in the ducts and the transverse unsteady fluctuations in the connecting apertures is exploited in the derivation of the impedance. This configuration is of relevance for some industrial applications, for example to model the aeroacoustic coupling between cans in gas turbines equipped with can-annular combustors. The derived impedance can be used as a Bloch-type boundary condition to reduce theoretical and computational analysis on the dynamics of these arrays of ducts to a single duct. After the derivation of a general result, simpler limits of interest are discussed. For some of these limits, impedance models are available in the literature. It is shown how the effective impedance derived in this study reduces to already available models in these limits, but more generally embeds new features, including the explicit dependence of the impedance on the grazing flow Mach number and the finite extension of the apertures.

A Non-Compact Effective Impedance Model for Can-to-Can Acoustic Communication: Analysis and Optimization of Damping Mechanisms

Abstract Can-annular combustors can feature azimuthal instabilities even if the acoustic coupling between the individual cans is weak. Recently, various studies have focused on modeling the acoustic communication between adjacent cans in can-annular systems. In this study, a coupling model is presented that, in contrast to previous models, includes the effect of density fluctuations, mean flow, and dissipative effects at the connection gaps. By assuming plane acoustic waves inside each can and exploiting the discrete rotational symmetry of the can-annular system, the acoustic can-to-can interaction can be represented by an effective Bloch-type impedance. A single can modeled with the effective impedance at the downstream end emulates the acoustic response of the entire can-annular arrangement. We then propose the idea of installing a liner just upstream of the first turbine stage to damp azimuthal instabilities. By using the proposed can-to-can coupling model, we discuss in detail the effect that the impedance of the liner has on the effective reflection coefficient for different Bloch wavenumbers. In the low-frequency limit, we derive an analytical condition for achieving maximum damping at a specific Bloch-number. We show that the damping of azimuthal modes depends on the porosity of the liner, mean flow parameters and the Bloch-structure of the mode. These results suggest the possibility of targeting the damping of modes of certain azimuthal order by geometric variations of the liner or of the connection gap. As an exemplary application of the theory, we setup a network model of a generic industrial 12-can combustor and investigate a cluster of acoustic and thermoacoustic eigenvalues for a varying liner porosity. The findings of this study provide a deeper understanding of the mechanisms that drive the can-to-can acoustic communication, and open the path for devising passive damping strategies aimed at stabilizing specific modes in can-annular combustors.

Low-Order Modeling of Can-Annular Combustors

Abstract Heavy-duty land-based gas turbines are often designed with can-annular combustors, which consist of a set of identical cans, acoustically connected on the upstream side via the compressor plenum, and, downstream, with a small annular gap located at the transition with the first turbine stage. The modeling of this cross-talk area is crucial to predict the thermo-acoustic modes of the system. Thanks to the discrete rotational symmetry, Bloch wave theory can be exploited to reduce the system to a longitudinal combustor with a complex-valued equivalent outlet reflection coefficient, which models the annular gap. The present study reviews existing low-order models based purely on geometrical parameters and compares them to two-dimensional Helmholtz simulations. We demonstrate that the modeling of the gap as a thin annulus is not suited for can-annular combustors and that the Rayleigh conductivity model only gives qualitative agreement. We then propose an extension for the equivalent reflection coefficient that accounts not only for geometrical but also flow parameters, by means of a characteristic length. The proposed model is in excellent agreement with two-dimensional simulations and is able to correctly capture the eigenfrequencies of the system. We then perform a Design of Experiments study that allows us to explore various configurations and build correlations for the characteristic length. Finally, we discuss the validity limits of the proposed low-order modeling approach.

Experimental investigation of lean-premixed hydrogen combustion instabilities in a can-annular combustion system

Thermoacoustic interactions in a circumferential network of lean-premixed combustors have a substantial impact on engine-level dynamics in a can-annular gas turbine combustion system. Previous experimental and numerical studies have focused on identifying the formation of large-scale interaction patterns and the modal dynamics of multiple eigenmodes. Since those investigations were primarily concerned with low-frequency interactions between adjacent combustors, there are currently no experimental observations that enable decisive discrimination between low- and high-frequency can-annular combustion instabilities. Here, we use pure hydrogen-air flame ensembles to trigger higher acoustic modes in four-coupled lean-premixed combustors, ultimately to understand the potential influence of self-excited instabilities on the spatiotemporal evolution of a can-annular system. The use of lean-premixed hydrogen-air flames enables measurements of previously unidentified phenomena, particularly in association with the excitation of high acoustic modes up to approximately 1.3 kHz. We demonstrate that self-excited standing azimuthal modes can be excited in the annular cross-talk section, particularly when the phase dynamics of the upstream flame tube sections are defined by alternating anti-phase oscillations. In this case, the temporal evolution of the can-annular system is governed by twofold degeneracy, incorporating an alternating push-pull mode in the longitudinal direction and a standing azimuthal mode in the circumferential direction at the same frequency. Based on experimental observations and Helmholtz simulations, we also show that a mixed state of synchronization and desynchronization can arise simultaneously as a result of symmetry breaking. The coexistence of coherent and incoherent motions is observed to be controlled by interactions between two closely spaced, but slightly misaligned, localized in-phase modes; this observation demonstrates experimentally the existence of a chiral state in can-annular thermoacoustics. The present results, for the first time, reveal a variety of phenomena involved in the response of a can-annular combustion system to higher frequency acoustic perturbations.

A Strategy to Tune Acoustic Terminations of Single-Can Test-Rigs to Mimic Thermoacoustic Behavior of a Full Engine

Abstract Thermoacoustic properties of can-annular combustors are commonly investigated by means of single-can test-rigs. To obtain representative results, it is crucial to mimic can–can coupling present in the full engine. However, current approaches either lack a solid theoretical foundation or are not practicable for high-pressure rigs. In this study, we employ Bloch-wave theory to derive reflection coefficients that correctly represent can–can coupling. We propose a strategy to impose such reflection coefficients at the acoustic terminations of a single-can test-rig by installing passive acoustic elements, namely straight ducts or Helmholtz resonators. In an iterative process, these elements are adapted to match the reflection coefficients for the dominant frequencies of the full engine. The strategy is demonstrated with a network model of a generic can-annular combustor and a three-dimensional (3D) model of a realistic can-annular combustor configuration. For the latter, we show that can–can coupling via the compressor exit plenum is negligible for frequencies sufficiently far away from plenum eigenfrequencies. Without utilizing previous knowledge of relevant frequencies or flame dynamics, the test-rig models are adapted within a few iterations and match the full engine with good accuracy. Using Helmholtz resonators for test-rig adaption turns out to be more viable than using straight ducts.

Analysis of Thermoacoustic Modes in Can-Annular Combustors Using Effective Bloch-Type Boundary Conditions

Abstract Heavy-duty gas turbines are commonly designed with can-annular combustors, in which all flames are physically separated. Acoustically, however, the cans communicate via the upstream located compressor plenum or at the downstream gaps found at the transition to the turbine inlet. In this study, a coupling condition that is based on a Rayleigh conductivity and acoustic flux conservation is derived. It enables acoustic communication between adjacent cans, in which one-dimensional acoustic waves propagate. In addition, because can-annular systems commonly feature a discrete rotational symmetry, the acoustic field can be expressed as a Bloch-periodic wave in the azimuthal direction. We demonstrate how the coupling conditions resulting in a combustion system with N cans can be expressed as an effective impedance for a single can. By means of this Bloch-type boundary condition, the thermoacoustics of a can-annular system can be analyzed considering only one can, thus reducing the size of the problem by a factor of N. Using this method, we investigate in frequency domain the effect of the coupling strength of a generic can-annular combustor consisting of 12 identical cans, which are connected at the downstream end. We describe generic features of can-annular systems that can be efficiently addressed with this framework and derive results on the frequency response of the cans at various Bloch numbers in the low-frequency and high-frequency limits. Furthermore, the formation of eigenvalue clusters with eigenvalues of close frequency and growth rate, but very different mode shapes is discussed.

Influence of rotational asymmetry on thermoacoustic instabilities in a can-annular lean-premixed combustor

Low-frequency combustion instabilities, and ultimately, engine-level dynamics in can-annular gas turbine combustion systems, are driven by acoustic interactions between adjacent combustors through an annular cross-talk section upstream of first stage turbine nozzles. This is an area of substantial technological challenges, because experimental and numerical complexities scale exponentially with the number of combustors, restricting most investigations to a single combustor or a coupled two-combustor configuration. We recently reported that in a can-annular combustion system consisting of four identical lean-premixed combustors pressure oscillations occur in many combinations of characteristic patterns and frequencies, including symmetric (push–push) and asymmetric (push–pull) interactions, when the system is subjected to rotationally symmetric operating conditions. The impact of rotational asymmetry on can-annular thermoacoustics is still a largely open subject; here we use the same experimental setup to perform multiple independent analyses of the effect of broken rotational symmetry on large-scale pattern formations and modal dynamics of multiple eigenmodes. We demonstrate that the presence of rotational asymmetry can lead to fundamentally different dynamic states, including pairwise push-pull modes, spinning azimuthal instabilities, superposition modes, and strong mode localization, which are totally absent under rotationally symmetric conditions. Of particular importance is the experimental observation of spinning azimuthal instabilities in the annular cross-talk section, which are acoustically separate from flame-dynamics-driven standing wave motion in the flame tube sections. This leads to the coexistence of traveling and standing waves in the can-annular system. In conjunction with FEM-based Helmholtz simulations, we determine the origin of the azimuthal modes to be the simultaneous excitation of two different pairwise push-pull interaction modes at the same frequency – a phenomenon known as degenerate eigenmodes.