This study investigates the effect of partial acoustic reflection at inlet or outlet of a combust... more This study investigates the effect of partial acoustic reflection at inlet or outlet of a combustor on thermoacoustic stability. Parametric maps of the thermoacoustic spectrum are utilized for this purpose, which represent frequencies and growth rates of eigenmodes for a wide range of model parameters. It is found that a decrease of the acoustic reflection at the boundaries does not always imply an increase in the stability margin of the thermoacoustic system. As a matter of fact, a reduction in the acoustic reflection may sometimes destabilize a thermoacoustic mode. Additionally, we show that perturbed passive thermoacoustic modes may become intrinsic thermoacoustic (ITA) modes in the fully anechoic case. We briefly discuss the mode definitions “acoustic” and “intrinsic” commonly found in the literature. The computational analysis is based on a state-space formulation of the linearized Navier–Stokes equations (LNSEs) with discontinuous Galerkin discretization. This approach allows to describe the thermoacoustic system as a linear combination of internal acoustics, flame dynamics, and acoustic boundaries. Such a segregation grants a clear analysis of the respective effects of the individual subsystems on the general stability of the system, expressed in terms of adjoint-based eigenvalue sensitivity. The state-space formulation of the LNSE proposed in this paper offers a powerful and flexible framework to carry out thermoacoustic studies of combustors with arbitrary geometry and acoustic boundary conditions.
Abstract A hybrid approach for thermoacoustic stability analysis is formulated in a state-space f... more Abstract A hybrid approach for thermoacoustic stability analysis is formulated in a state-space framework. The approach distinguishes between regions of the computational domain with and without important interactions between acoustics and mean flow or unsteady heat release, respectively. The former regions are modeled by a discontinuous Galerkin finite element method (DG-FEM) for the linearized Navier-Stokes equations in conservative form. The latter are represented by reduced-order models of acoustic wave propagation or dissipation, and provide complex-valued, frequency dependent impedance boundary conditions for the DG-FEM domain. The flow-flame coupling is modeled by a flame transfer function that governs a volumetric source term for the fluctuating heat release rate. The respective (sub-)models are formulated and interconnected in a state-space framework, which facilitates the monolithic formulation of hybrid thermoacoustic models. Moreover, the state-space interconnect framework makes it possible to formulate thermoacoustic stability analysis as a linear eigenvalue problem – even if flame transfer function or acoustic boundary conditions depend in a non-trivial manner on frequency. The approach is first verified against analytical solutions for a duct with mean flow across a thin heat source, similar to a Rijke tube. Then the thermoacoustic eigenmodes of a premixed, swirl-stabilized combustor are computed in order to validate the method against experimental data for a configuration of applied interest. For this second validation case, a detailed comparison against predictions of a low-order network-model is also presented.
Model order reduction can play a pivotal role in reducing the cost of repeated computations of la... more Model order reduction can play a pivotal role in reducing the cost of repeated computations of large thermoacoustic models required for comprehensive stability analysis and optimization. In this proof-of-concept study, acoustic wave propagation is modeled with a 1D network approach, while acoustic-flame interactions are modeled by a flame transfer function. Three reduction techniques are applied to the acoustic subsystem: firstly modal truncation based on preserving the acoustic eigenmodes, and then two approaches that strive to preserve the input-output transfer behavior of the acoustic subsystem, i.e., truncated balanced realization and iterative rational Krylov algorithm. After reduction, the reduced-order models (ROMs) are coupled with the flame transfer function. Results show that the coupled reduced system from modal truncation accurately captures thermoacoustic cavity modes with weak influence of the flame, but fails for cavity modes strongly
The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear an... more The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear analysis, based on the compressible flow equations with a two-step reaction scheme for methane combustion. The flame transfer function (FTF) is computed from this linear model, in excellent agreement with reference nonlinear calculations. The input-output gain between externally applied forcing and the global heat release rate response is computed, and peaks in the gain are related to intrinsic thermoacoustic (ITA) modes. The receptivity of the flame to arbitrary flow forcing is characterised by the resulting amplitude of global heat release rate fluctuations. Linear resolvent analysis is used to identify optimal forcing structures and their associated flame response, leading to a discussion of the dominant mechanisms for the amplification of flow perturbations, which trigger flame oscillations. These seem to involve a resonance with ITA instability modes. Keywords: combustion instability; linear instability; premixed laminar flame; flame transfer function; resolvent analysis; intrinsic thermoacoustic mode its supporting flow field [2], or they may involve an acoustic resonance between the flame and its surrounding chamber [3]. Chamber resonance effects, which underpin the classical thermoacoustic instability, are not investigated in this paper. Flame instability has been approached in weakly and fully nonlinear frameworks [4, 5], but linear analysis still plays an important role in fundamental studies and industrial applications, such as sensitivity analysis, optimisation and control [6]. The linear flame transfer function (FTF), which relates fluctuations of the global heat release rate to local fluctuations of the velocity in a chosen measurement point, is widely used for the modelling of the flame response to acoustic forcing. Classically, the flame transfer function is obtained from experiments or numerical simulations via empirical system identification methods based on time-series of forcing and response data. This strategy is data-driven, and represents a black-box approach [7]. We advocate here an alternative strategy, based on the linear flow equations, which may be described as a white-box method. To this end, we linearize the governing equations and calculate the Jacobian matrix around a steady base flow state, which is obtained as a fixed point of the nonlinear system of equations. The resulting linear system governs the dynamics of small-amplitude fluctuations, and it can be accessed with a large variety of linear tools for a physics-based analysis. Note that our objective is not to create analytical models of the FTF, as one might do for instance by using a G-equation [8]. The single hypothesis that we employ here is that we work with fluctuations of small amplitudes in the linear limit. The complete set of linearized governing equations and the information of each flow variable in the base flow is preserved. This method allows efficient calculation of the FTF, because no time-stepping is required. A pioneering attempt in this direction was made by van Kampen et al. [9] to reduce the computational burden of FTF calculations by use of a "linear coefficient method", based on the response of flow variables to small-amplitude perturbations. The response of global heat release rate to the fluctuation of inlet equivalence ratio was investigated for a turbulent swirling flame. The response obtained via the linear coefficient method was reported to be in good agreement with reference calculation results. Nevertheless, this method does not include some essential characteristics of the flame dynamics, such as the effect of equivalence ratio fluctuations on the flame front movement. The approach used in the present study fully encompasses those effects. Another important feature is that the white-box method gives spontaneously a multi-input, multi-output (MIMO) system. While the classical flame transfer function is represented as a single-input, single-output (SISO) system associating the heat release rate with velocity disturbances, one may also want to explore other physical quantities in flames. A MIMO formulation is required if one wants to, for instance, compute the flame responses to incoming fluctuations of the mixture composition [10, 11, 12, 13, 14], or represent acoustic scattering [15, 16]. MIMO systems can in principle be identified with black-box methods, but special technical considerations are necessary, for instance so as to ensure statistical independence among different signal channels [7, 17]. A linear approach, as presented in this study, inherently preserves the causal coupling between all signals, and all mappings can be extracted from one linear system. The applicability of linear analysis to flame dynamics has already been demonstrated in several recent studies, for premixed as well as diffusion flames. Throughout this paper, the linear analysis is carried out in a global setting, i.e. on a spatially developing base flow, in contrast to local analysis which is based on the assumption of a streamwise-invariant base flow. For reacting flows, local analysis has indeed been successfully applied in reacting bluff-body wakes that can be regarded as being weakly non-parallel [18]. This assumption however does not hold for most flames, so we use global analysis where the word "global" means that the variations in the base flow are resolved both in the streamwise and in the cross-stream direction. A brief state-of-the-art survey is given for such studies on premixed flames. The reader may want to refer to [19, 20, 21, 22, 23] for work on diffusion flames. Investigating the role of precessing vortex core (PVC) dynamics in combustion instability, Oberleithner et al. [24, 25] performed modal analysis on experimental data of turbulent flames in a swirl combustor. A solver based on the linearized incompressible Navier-Stokes equations with an eddy-viscosity model was applied. The predicted frequency of the instability mode was found to be in excellent agreement with the oscillation frequency measured in the experiment. For other linear calculations with regard to the PVC, the reader may refer to [26, 27]. Blanchard et al. [28] developed a linear solver based on the compressible Navier-Stokes equations with a one-step
Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analys... more Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analysis that combines the Linearized Navier–Stokes Equations (LNSE) with a global Flame Transfer Function (FTF), generates spurious entropy waves when used to model acoustically forced premixed flames. The inability of the global FTF to account for the effects of flame movement is identified as the root cause of this unphysical behavior. Utilization of a local FTF, which resolves unsteady heat release on scales comparable to the reaction zone of the flame, suppresses the spurious entropy perturbations. This affirms that fine-grained resolution of the spatio-temporal distribution of heat release rate fluctuations in the combustion zone is required to model the movement of the flame front, even for acoustically and convectively compact flames. As an alternative to hybrid models, a Linearized Reactive Flow (LRF) approach is employed, which extends the LNSE by the linearized species transport equations as well as the reaction mechanism. Such a monolithic approach inherently accounts for the locally resolved flame dynamics, including the movement of the flame front, and does not require an external model for the flame-flow interaction. Thus the LRF eliminates the need for the cumbersome identification of a local FTF. Two configurations of lean premixed methane-air flames, i.e. a freely propagating 1D flame and a 2D flame anchored in a duct, are considered for validation. All results obtained with linearized modeling approaches and conclusions deduced thereof are validated against high resolution CFD results with excellent quantitative accuracy.
Linear,low-order network models, constructed from aset of interconnected acoustic subsystem eleme... more Linear,low-order network models, constructed from aset of interconnected acoustic subsystem elements, are an established tool in acoustic analysis. This paper introduces ageneralized linear modeling framework for acoustic systems that is based on as tate space formalism and linear system theory.I ti ss hown hows ubsystem models retrievedf rom 1D lowo rder modeling, 3D linearized perturbation equations as well as computational fluid dynamics or experiment can be translated into state space models. This translation does not involveany approximations beyond those made by the acoustic modeling in the first place. An algorithm to interconnect state space models is presented, which results in asingle, joint state space model for the complete acoustic system. The resulting eigenvalue problem is linear,thus eigenvalues and eigenvectors can be determined for small and moderate problem sizes with very little effort, using standard numerical routines. In order to makepossible the solution of larger problem sizes, as theyresult from 3D models, the framework exploits the sparse structure of the models. The methods and algorithms are exemplified and validated using simplistic configurations, buta lso ag eneric annular combustor test case with non trivial topology.
In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed... more In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed using a linearized reactive flow (LRF) solver. The LRF solver is based on linearized compressible Navier-Stokes and reacting species transport equations and thereby includes a model for the dynamic response of the flame to flow perturbations in an inherent manner. The equations are discretized using the discontinuous Galerkin finite element method. By way of example, thermoacoustic characteristics of attached and lifted laminar premixed flames are investigated. First, the respective flame transfer functions (FTFs) are computed in the frequency domain with the LRF solver. The results are in agreement with reference FTFs identified from CFD time-series. Secondly, the LRF solver is employed for thermoacoustic stability analysis, i.e. computation of shape, frequency, and growth rate of eigenmodes. Results are compared to established hybrid methods that couple FTFs with a low-order thermoacoustic network-model or a linearized Navier-Stokes equations solver. All solvers capture the dominant thermoacoustic mode, but only the LRF resolves local flow-flame interaction, revealing e.g. the onset of the flame movement and the propagation of distortions along the flame.
Equivalent reflection coefficient u [m s −1 ] Acoustic velocity ϕ [m 2 s −1 ] Velocity potential ... more Equivalent reflection coefficient u [m s −1 ] Acoustic velocity ϕ [m 2 s −1 ] Velocity potential γ [−] Heat capacity ratio ρ [kg m −3 ] Density ω [rad s −1 ] Complex frequency DoE Design of Experiments LOM Low-Order Model NRMSE Normalized Root-Mean-Square Error INTRODUCTION Can-annular combustors are commonly found in heavy-duty land-based gas turbines. In this application, N identical combustor cans are aligned along an annulus, hence the name. On the upstream side, the cans are acoustically connected to the compressor plenum. Downstream, a transition duct, as the name indicates, allows the cross-sectional area of the can to transform from a circular to an annular shape, in order to properly feed the turbine. Single can burners have often been considered to be a good approximation of the full system, since the combustion takes place in individual cans. However, a small annular gap is present just in front of the turbine inlet guide vanes, and this cross-talk area allows acoustic communication between neighboring cans. Recent studies showed that the can-to-can communication cannot be neglected when investigating thermoacoustic stability. While annular combustors have been extensively studied in the last decades, can-annular configurations have received less attention. Bethke et al. [1] and Kaufmann et al. [2] were the first to numerically study can-annular configurations. They showed that accounting for the cross-talk area in a full system gives rise to new eigenmodes that were not observed in single can configurations. Panek et al. [3], based on experimental evidence and by means of a modal analysis, arrived at the same conclusion. Modes with mode shapes that involve multiple cans were observed for the full configuration but could not exist in a single can approximation. Farisco et al. [4] numerically investigated the effect of the geometry of the gap on the acoustic interaction between neighboring cans and showed that the cross-talk effect cannot be neglected. Ghirardo et al. [5] demonstrated numerically, by means of 2D Helmholtz simulations, and gave experimental evidence that modes of various azimuthal order arise due to the weak coupling between cans. It was shown that these modes come in "clusters", i.e. collections of several distinct modes with very close frequencies, but different growth rates.
Thermoacoustic properties of can-annular combustors are commonly investigated by means of single-... more 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 cancan coupling present in the full engine. However, current approaches either lack a solid theoretical foundation or are not practicable for high-pressure rigs. In the present 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 3D model of a realistic can-annular combustor configuration. For the latter we show that cancan 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.
The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear an... more The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear analysis, based on the compressible flow equations with a two-step reaction scheme for methane combustion. The flame transfer function (FTF) is computed from this linear model, in excellent agreement with reference nonlinear calculations. The input-output gain between externally applied forcing and the global heat release rate response is computed, and peaks in the gain are related to intrinsic thermoacoustic (ITA) modes. The receptivity of the flame to arbitrary flow forcing is characterised by the resulting amplitude of global heat release rate fluctuations. Linear resolvent analysis is used to identify optimal forcing structures and their associated flame response, leading to a discussion of the dominant mechanisms for the amplification of flow perturbations, which trigger flame oscillations. These seem to involve a resonance with ITA instability modes. Keywords: combustion instability; linear instability; premixed laminar flame; flame transfer function; resolvent analysis; intrinsic thermoacoustic mode its supporting flow field [2], or they may involve an acoustic resonance between the flame and its surrounding chamber [3]. Chamber resonance effects, which underpin the classical thermoacoustic instability, are not investigated in this paper. Flame instability has been approached in weakly and fully nonlinear frameworks [4, 5], but linear analysis still plays an important role in fundamental studies and industrial applications, such as sensitivity analysis, optimisation and control [6]. The linear flame transfer function (FTF), which relates fluctuations of the global heat release rate to local fluctuations of the velocity in a chosen measurement point, is widely used for the modelling of the flame response to acoustic forcing. Classically, the flame transfer function is obtained from experiments or numerical simulations via empirical system identification methods based on time-series of forcing and response data. This strategy is data-driven, and represents a black-box approach [7]. We advocate here an alternative strategy, based on the linear flow equations, which may be described as a white-box method. To this end, we linearize the governing equations and calculate the Jacobian matrix around a steady base flow state, which is obtained as a fixed point of the nonlinear system of equations. The resulting linear system governs the dynamics of small-amplitude fluctuations, and it can be accessed with a large variety of linear tools for a physics-based analysis. Note that our objective is not to create analytical models of the FTF, as one might do for instance by using a G-equation [8]. The single hypothesis that we employ here is that we work with fluctuations of small amplitudes in the linear limit. The complete set of linearized governing equations and the information of each flow variable in the base flow is preserved. This method allows efficient calculation of the FTF, because no time-stepping is required. A pioneering attempt in this direction was made by van Kampen et al. [9] to reduce the computational burden of FTF calculations by use of a "linear coefficient method", based on the response of flow variables to small-amplitude perturbations. The response of global heat release rate to the fluctuation of inlet equivalence ratio was investigated for a turbulent swirling flame. The response obtained via the linear coefficient method was reported to be in good agreement with reference calculation results. Nevertheless, this method does not include some essential characteristics of the flame dynamics, such as the effect of equivalence ratio fluctuations on the flame front movement. The approach used in the present study fully encompasses those effects. Another important feature is that the white-box method gives spontaneously a multi-input, multi-output (MIMO) system. While the classical flame transfer function is represented as a single-input, single-output (SISO) system associating the heat release rate with velocity disturbances, one may also want to explore other physical quantities in flames. A MIMO formulation is required if one wants to, for instance, compute the flame responses to incoming fluctuations of the mixture composition [10, 11, 12, 13, 14], or represent acoustic scattering [15, 16]. MIMO systems can in principle be identified with black-box methods, but special technical considerations are necessary, for instance so as to ensure statistical independence among different signal channels [7, 17]. A linear approach, as presented in this study, inherently preserves the causal coupling between all signals, and all mappings can be extracted from one linear system. The applicability of linear analysis to flame dynamics has already been demonstrated in several recent studies, for premixed as well as diffusion flames. Throughout this paper, the linear analysis is carried out in a global setting, i.e. on a spatially developing base flow, in contrast to local analysis which is based on the assumption of a streamwise-invariant base flow. For reacting flows, local analysis has indeed been successfully applied in reacting bluff-body wakes that can be regarded as being weakly non-parallel [18]. This assumption however does not hold for most flames, so we use global analysis where the word "global" means that the variations in the base flow are resolved both in the streamwise and in the cross-stream direction. A brief state-of-the-art survey is given for such studies on premixed flames. The reader may want to refer to [19, 20, 21, 22, 23] for work on diffusion flames. Investigating the role of precessing vortex core (PVC) dynamics in combustion instability, Oberleithner et al. [24, 25] performed modal analysis on experimental data of turbulent flames in a swirl combustor. A solver based on the linearized incompressible Navier-Stokes equations with an eddy-viscosity model was applied. The predicted frequency of the instability mode was found to be in excellent agreement with the oscillation frequency measured in the experiment. For other linear calculations with regard to the PVC, the reader may refer to [26, 27]. Blanchard et al. [28] developed a linear solver based on the compressible Navier-Stokes equations with a one-step
Bulletin of the American Physical Society, Nov 22, 2020
global response to forcing of a 2D laminar premixed slot flame is investigated with a linearized ... more global response to forcing of a 2D laminar premixed slot flame is investigated with a linearized approach, where multiphysics coupling in a non-parallel reacting flow is accounted for. The classical flame transfer functions obtained this way are in good agreement with nonlinear reference calculations, performed with the AVBP code from CERFACS. Optimal forcing input, leading to maximal heat release fluctuations, are identified via resolvent analysis. Compared to similar recent investigations, we use a finite-element discretization on an unstructured mesh, and we employ a reduced two-step scheme for methane-air reaction. Strong sensitivity to flame parameters and chemical modeling is observed. The present linear analysis opens new ways for the physical discussion of instability dynamics in flames.
Thermoacoustic properties of can-annular combustors are commonly investigated by means of single-... more 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 cancan coupling present in the full engine. However, current approaches either lack a solid theoretical foundation or are not practicable for high-pressure rigs. In the present 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 3D model of a realistic can-annular combustor configuration. For the latter we show that cancan 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.
Die genaue Modellierung dynamischer Systeme resultiert oft in einer sehr hohen Anzahl an Zustands... more Die genaue Modellierung dynamischer Systeme resultiert oft in einer sehr hohen Anzahl an Zustandsvariablen. Die Systemmatrizen konnen dadurch zu gros werden, um noch Zustandsraummodelle (ss Objekte) in MATLAB definieren zu konnen. In diesem Beitrag werden Toolboxen vorgestellt, welche durch Einfuhrung sog. sparse state space Objekten (sss) die Analyse hochdimensionaler Systeme ermoglichen. Mittels Modellordnungsreduktion (sssMOR) kann die wesentliche Dynamik in Modellen deutlich niedrigerer Ordnung abgebildet werden.
Journal of Engineering for Gas Turbines and Power, 2021
Heavy-duty land-based gas turbines are often designed with can-annular combustors, which consist ... more 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 refl...
Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analys... more Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analysis that combines the Linearized Navier–Stokes Equations (LNSE) with a global Flame Transfer Function (FTF), generates spurious entropy waves when used to model acoustically forced premixed flames. The inability of the global FTF to account for the effects of flame movement is identified as the root cause of this unphysical behavior. Utilization of a local FTF, which resolves unsteady heat release on scales comparable to the reaction zone of the flame, suppresses the spurious entropy perturbations. This affirms that fine-grained resolution of the spatio-temporal distribution of heat release rate fluctuations in the combustion zone is required to model the movement of the flame front, even for acoustically and convectively compact flames. As an alternative to hybrid models, a Linearized Reactive Flow (LRF) approach is employed, which extends the LNSE by the linearized species transport equations as well as the reaction mechanism. Such a monolithic approach inherently accounts for the locally resolved flame dynamics, including the movement of the flame front, and does not require an external model for the flame-flow interaction. Thus the LRF eliminates the need for the cumbersome identification of a local FTF. Two configurations of lean premixed methane-air flames, i.e. a freely propagating 1D flame and a 2D flame anchored in a duct, are considered for validation. All results obtained with linearized modeling approaches and conclusions deduced thereof are validated against high resolution CFD results with excellent quantitative accuracy.
International Journal of Spray and Combustion Dynamics, 2017
The response of a laminar, premixed flame to perturbations of upstream equivalence ratio is inves... more The response of a laminar, premixed flame to perturbations of upstream equivalence ratio is investigated and modelled, with emphasis on the generation of ‘entropy waves’, i.e. entropic inhomogeneities of downstream temperature. Transient computational fluid dynamics simulations of two adiabatic lean methane-air flames of different Péclet numbers provide guidance and validation data for subsequent modelling. The respective entropy transfer functions, which describe the production of temperature inhomogeneities, as well as transfer functions for the variation of the heat release, are determined from the computational fluid dynamics time series data by means of system identification. The processes governing the dynamics of the entropy transfer functions are segregated into two sub-problems: (1) heat release due to chemical reaction at the flame front and (2) advective and diffusive transport. By adopting a formulation in terms of a mixture fraction variable, these two sub-problems can ...
Over many years, scientists and engineers have developed a broad variety of mathematical formulat... more Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suff ers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving...
Journal of Engineering for Gas Turbines and Power, 2019
This study investigates the effect of partial acoustic reflection at inlet or outlet of a combust... more This study investigates the effect of partial acoustic reflection at inlet or outlet of a combustor on thermoacoustic stability. Parametric maps of the thermoacoustic spectrum are utilized for this purpose, which represent frequencies and growth rates of eigenmodes for a wide range of model parameters. It is found that a decrease of the acoustic reflection at the boundaries does not always imply an increase in the stability margin of the thermoacoustic system. As a matter of fact, a reduction in the acoustic reflection may sometimes destabilize a thermoacoustic mode. Additionally, we show that perturbed passive thermoacoustic modes may become intrinsic thermoacoustic (ITA) modes in the fully anechoic case. We briefly discuss the mode definitions “acoustic” and “intrinsic” commonly found in the literature. The computational analysis is based on a state-space formulation of the linearized Navier–Stokes equations (LNSEs) with discontinuous Galerkin discretization. This approach allows ...
In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed... more In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed using a linearized reactive flow (LRF) solver. The LRF solver is based on linearized compressible Navier-Stokes and reacting species transport equations and thereby includes a model for the dynamic response of the flame to flow perturbations in an inherent manner. The equations are discretized using the discontinuous Galerkin finite element method. By way of example, thermoacoustic characteristics of attached and lifted laminar premixed flames are investigated. First, the respective flame transfer functions (FTFs) are computed in the frequency domain with the LRF solver. The results are in agreement with reference FTFs identified from CFD time-series. Secondly, the LRF solver is employed for thermoacoustic stability analysis, i.e. computation of shape, frequency, and growth rate of eigenmodes. Results are compared to established hybrid methods that couple FTFs with a low-order thermoacoustic network-model or a linearized Navier-Stokes equations solver. All solvers capture the dominant thermoacoustic mode, but only the LRF resolves local flow-flame interaction, revealing e.g. the onset of the flame movement and the propagation of distortions along the flame.
This study investigates the effect of partial acoustic reflection at inlet or outlet of a combust... more This study investigates the effect of partial acoustic reflection at inlet or outlet of a combustor on thermoacoustic stability. Parametric maps of the thermoacoustic spectrum are utilized for this purpose, which represent frequencies and growth rates of eigenmodes for a wide range of model parameters. It is found that a decrease of the acoustic reflection at the boundaries does not always imply an increase in the stability margin of the thermoacoustic system. As a matter of fact, a reduction in the acoustic reflection may sometimes destabilize a thermoacoustic mode. Additionally, we show that perturbed passive thermoacoustic modes may become intrinsic thermoacoustic (ITA) modes in the fully anechoic case. We briefly discuss the mode definitions “acoustic” and “intrinsic” commonly found in the literature. The computational analysis is based on a state-space formulation of the linearized Navier–Stokes equations (LNSEs) with discontinuous Galerkin discretization. This approach allows to describe the thermoacoustic system as a linear combination of internal acoustics, flame dynamics, and acoustic boundaries. Such a segregation grants a clear analysis of the respective effects of the individual subsystems on the general stability of the system, expressed in terms of adjoint-based eigenvalue sensitivity. The state-space formulation of the LNSE proposed in this paper offers a powerful and flexible framework to carry out thermoacoustic studies of combustors with arbitrary geometry and acoustic boundary conditions.
Abstract A hybrid approach for thermoacoustic stability analysis is formulated in a state-space f... more Abstract A hybrid approach for thermoacoustic stability analysis is formulated in a state-space framework. The approach distinguishes between regions of the computational domain with and without important interactions between acoustics and mean flow or unsteady heat release, respectively. The former regions are modeled by a discontinuous Galerkin finite element method (DG-FEM) for the linearized Navier-Stokes equations in conservative form. The latter are represented by reduced-order models of acoustic wave propagation or dissipation, and provide complex-valued, frequency dependent impedance boundary conditions for the DG-FEM domain. The flow-flame coupling is modeled by a flame transfer function that governs a volumetric source term for the fluctuating heat release rate. The respective (sub-)models are formulated and interconnected in a state-space framework, which facilitates the monolithic formulation of hybrid thermoacoustic models. Moreover, the state-space interconnect framework makes it possible to formulate thermoacoustic stability analysis as a linear eigenvalue problem – even if flame transfer function or acoustic boundary conditions depend in a non-trivial manner on frequency. The approach is first verified against analytical solutions for a duct with mean flow across a thin heat source, similar to a Rijke tube. Then the thermoacoustic eigenmodes of a premixed, swirl-stabilized combustor are computed in order to validate the method against experimental data for a configuration of applied interest. For this second validation case, a detailed comparison against predictions of a low-order network-model is also presented.
Model order reduction can play a pivotal role in reducing the cost of repeated computations of la... more Model order reduction can play a pivotal role in reducing the cost of repeated computations of large thermoacoustic models required for comprehensive stability analysis and optimization. In this proof-of-concept study, acoustic wave propagation is modeled with a 1D network approach, while acoustic-flame interactions are modeled by a flame transfer function. Three reduction techniques are applied to the acoustic subsystem: firstly modal truncation based on preserving the acoustic eigenmodes, and then two approaches that strive to preserve the input-output transfer behavior of the acoustic subsystem, i.e., truncated balanced realization and iterative rational Krylov algorithm. After reduction, the reduced-order models (ROMs) are coupled with the flame transfer function. Results show that the coupled reduced system from modal truncation accurately captures thermoacoustic cavity modes with weak influence of the flame, but fails for cavity modes strongly
The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear an... more The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear analysis, based on the compressible flow equations with a two-step reaction scheme for methane combustion. The flame transfer function (FTF) is computed from this linear model, in excellent agreement with reference nonlinear calculations. The input-output gain between externally applied forcing and the global heat release rate response is computed, and peaks in the gain are related to intrinsic thermoacoustic (ITA) modes. The receptivity of the flame to arbitrary flow forcing is characterised by the resulting amplitude of global heat release rate fluctuations. Linear resolvent analysis is used to identify optimal forcing structures and their associated flame response, leading to a discussion of the dominant mechanisms for the amplification of flow perturbations, which trigger flame oscillations. These seem to involve a resonance with ITA instability modes. Keywords: combustion instability; linear instability; premixed laminar flame; flame transfer function; resolvent analysis; intrinsic thermoacoustic mode its supporting flow field [2], or they may involve an acoustic resonance between the flame and its surrounding chamber [3]. Chamber resonance effects, which underpin the classical thermoacoustic instability, are not investigated in this paper. Flame instability has been approached in weakly and fully nonlinear frameworks [4, 5], but linear analysis still plays an important role in fundamental studies and industrial applications, such as sensitivity analysis, optimisation and control [6]. The linear flame transfer function (FTF), which relates fluctuations of the global heat release rate to local fluctuations of the velocity in a chosen measurement point, is widely used for the modelling of the flame response to acoustic forcing. Classically, the flame transfer function is obtained from experiments or numerical simulations via empirical system identification methods based on time-series of forcing and response data. This strategy is data-driven, and represents a black-box approach [7]. We advocate here an alternative strategy, based on the linear flow equations, which may be described as a white-box method. To this end, we linearize the governing equations and calculate the Jacobian matrix around a steady base flow state, which is obtained as a fixed point of the nonlinear system of equations. The resulting linear system governs the dynamics of small-amplitude fluctuations, and it can be accessed with a large variety of linear tools for a physics-based analysis. Note that our objective is not to create analytical models of the FTF, as one might do for instance by using a G-equation [8]. The single hypothesis that we employ here is that we work with fluctuations of small amplitudes in the linear limit. The complete set of linearized governing equations and the information of each flow variable in the base flow is preserved. This method allows efficient calculation of the FTF, because no time-stepping is required. A pioneering attempt in this direction was made by van Kampen et al. [9] to reduce the computational burden of FTF calculations by use of a "linear coefficient method", based on the response of flow variables to small-amplitude perturbations. The response of global heat release rate to the fluctuation of inlet equivalence ratio was investigated for a turbulent swirling flame. The response obtained via the linear coefficient method was reported to be in good agreement with reference calculation results. Nevertheless, this method does not include some essential characteristics of the flame dynamics, such as the effect of equivalence ratio fluctuations on the flame front movement. The approach used in the present study fully encompasses those effects. Another important feature is that the white-box method gives spontaneously a multi-input, multi-output (MIMO) system. While the classical flame transfer function is represented as a single-input, single-output (SISO) system associating the heat release rate with velocity disturbances, one may also want to explore other physical quantities in flames. A MIMO formulation is required if one wants to, for instance, compute the flame responses to incoming fluctuations of the mixture composition [10, 11, 12, 13, 14], or represent acoustic scattering [15, 16]. MIMO systems can in principle be identified with black-box methods, but special technical considerations are necessary, for instance so as to ensure statistical independence among different signal channels [7, 17]. A linear approach, as presented in this study, inherently preserves the causal coupling between all signals, and all mappings can be extracted from one linear system. The applicability of linear analysis to flame dynamics has already been demonstrated in several recent studies, for premixed as well as diffusion flames. Throughout this paper, the linear analysis is carried out in a global setting, i.e. on a spatially developing base flow, in contrast to local analysis which is based on the assumption of a streamwise-invariant base flow. For reacting flows, local analysis has indeed been successfully applied in reacting bluff-body wakes that can be regarded as being weakly non-parallel [18]. This assumption however does not hold for most flames, so we use global analysis where the word "global" means that the variations in the base flow are resolved both in the streamwise and in the cross-stream direction. A brief state-of-the-art survey is given for such studies on premixed flames. The reader may want to refer to [19, 20, 21, 22, 23] for work on diffusion flames. Investigating the role of precessing vortex core (PVC) dynamics in combustion instability, Oberleithner et al. [24, 25] performed modal analysis on experimental data of turbulent flames in a swirl combustor. A solver based on the linearized incompressible Navier-Stokes equations with an eddy-viscosity model was applied. The predicted frequency of the instability mode was found to be in excellent agreement with the oscillation frequency measured in the experiment. For other linear calculations with regard to the PVC, the reader may refer to [26, 27]. Blanchard et al. [28] developed a linear solver based on the compressible Navier-Stokes equations with a one-step
Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analys... more Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analysis that combines the Linearized Navier–Stokes Equations (LNSE) with a global Flame Transfer Function (FTF), generates spurious entropy waves when used to model acoustically forced premixed flames. The inability of the global FTF to account for the effects of flame movement is identified as the root cause of this unphysical behavior. Utilization of a local FTF, which resolves unsteady heat release on scales comparable to the reaction zone of the flame, suppresses the spurious entropy perturbations. This affirms that fine-grained resolution of the spatio-temporal distribution of heat release rate fluctuations in the combustion zone is required to model the movement of the flame front, even for acoustically and convectively compact flames. As an alternative to hybrid models, a Linearized Reactive Flow (LRF) approach is employed, which extends the LNSE by the linearized species transport equations as well as the reaction mechanism. Such a monolithic approach inherently accounts for the locally resolved flame dynamics, including the movement of the flame front, and does not require an external model for the flame-flow interaction. Thus the LRF eliminates the need for the cumbersome identification of a local FTF. Two configurations of lean premixed methane-air flames, i.e. a freely propagating 1D flame and a 2D flame anchored in a duct, are considered for validation. All results obtained with linearized modeling approaches and conclusions deduced thereof are validated against high resolution CFD results with excellent quantitative accuracy.
Linear,low-order network models, constructed from aset of interconnected acoustic subsystem eleme... more Linear,low-order network models, constructed from aset of interconnected acoustic subsystem elements, are an established tool in acoustic analysis. This paper introduces ageneralized linear modeling framework for acoustic systems that is based on as tate space formalism and linear system theory.I ti ss hown hows ubsystem models retrievedf rom 1D lowo rder modeling, 3D linearized perturbation equations as well as computational fluid dynamics or experiment can be translated into state space models. This translation does not involveany approximations beyond those made by the acoustic modeling in the first place. An algorithm to interconnect state space models is presented, which results in asingle, joint state space model for the complete acoustic system. The resulting eigenvalue problem is linear,thus eigenvalues and eigenvectors can be determined for small and moderate problem sizes with very little effort, using standard numerical routines. In order to makepossible the solution of larger problem sizes, as theyresult from 3D models, the framework exploits the sparse structure of the models. The methods and algorithms are exemplified and validated using simplistic configurations, buta lso ag eneric annular combustor test case with non trivial topology.
In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed... more In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed using a linearized reactive flow (LRF) solver. The LRF solver is based on linearized compressible Navier-Stokes and reacting species transport equations and thereby includes a model for the dynamic response of the flame to flow perturbations in an inherent manner. The equations are discretized using the discontinuous Galerkin finite element method. By way of example, thermoacoustic characteristics of attached and lifted laminar premixed flames are investigated. First, the respective flame transfer functions (FTFs) are computed in the frequency domain with the LRF solver. The results are in agreement with reference FTFs identified from CFD time-series. Secondly, the LRF solver is employed for thermoacoustic stability analysis, i.e. computation of shape, frequency, and growth rate of eigenmodes. Results are compared to established hybrid methods that couple FTFs with a low-order thermoacoustic network-model or a linearized Navier-Stokes equations solver. All solvers capture the dominant thermoacoustic mode, but only the LRF resolves local flow-flame interaction, revealing e.g. the onset of the flame movement and the propagation of distortions along the flame.
Equivalent reflection coefficient u [m s −1 ] Acoustic velocity ϕ [m 2 s −1 ] Velocity potential ... more Equivalent reflection coefficient u [m s −1 ] Acoustic velocity ϕ [m 2 s −1 ] Velocity potential γ [−] Heat capacity ratio ρ [kg m −3 ] Density ω [rad s −1 ] Complex frequency DoE Design of Experiments LOM Low-Order Model NRMSE Normalized Root-Mean-Square Error INTRODUCTION Can-annular combustors are commonly found in heavy-duty land-based gas turbines. In this application, N identical combustor cans are aligned along an annulus, hence the name. On the upstream side, the cans are acoustically connected to the compressor plenum. Downstream, a transition duct, as the name indicates, allows the cross-sectional area of the can to transform from a circular to an annular shape, in order to properly feed the turbine. Single can burners have often been considered to be a good approximation of the full system, since the combustion takes place in individual cans. However, a small annular gap is present just in front of the turbine inlet guide vanes, and this cross-talk area allows acoustic communication between neighboring cans. Recent studies showed that the can-to-can communication cannot be neglected when investigating thermoacoustic stability. While annular combustors have been extensively studied in the last decades, can-annular configurations have received less attention. Bethke et al. [1] and Kaufmann et al. [2] were the first to numerically study can-annular configurations. They showed that accounting for the cross-talk area in a full system gives rise to new eigenmodes that were not observed in single can configurations. Panek et al. [3], based on experimental evidence and by means of a modal analysis, arrived at the same conclusion. Modes with mode shapes that involve multiple cans were observed for the full configuration but could not exist in a single can approximation. Farisco et al. [4] numerically investigated the effect of the geometry of the gap on the acoustic interaction between neighboring cans and showed that the cross-talk effect cannot be neglected. Ghirardo et al. [5] demonstrated numerically, by means of 2D Helmholtz simulations, and gave experimental evidence that modes of various azimuthal order arise due to the weak coupling between cans. It was shown that these modes come in "clusters", i.e. collections of several distinct modes with very close frequencies, but different growth rates.
Thermoacoustic properties of can-annular combustors are commonly investigated by means of single-... more 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 cancan coupling present in the full engine. However, current approaches either lack a solid theoretical foundation or are not practicable for high-pressure rigs. In the present 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 3D model of a realistic can-annular combustor configuration. For the latter we show that cancan 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.
The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear an... more The response to forcing of a 2D laminar premixed slot flame is investigated by means of linear analysis, based on the compressible flow equations with a two-step reaction scheme for methane combustion. The flame transfer function (FTF) is computed from this linear model, in excellent agreement with reference nonlinear calculations. The input-output gain between externally applied forcing and the global heat release rate response is computed, and peaks in the gain are related to intrinsic thermoacoustic (ITA) modes. The receptivity of the flame to arbitrary flow forcing is characterised by the resulting amplitude of global heat release rate fluctuations. Linear resolvent analysis is used to identify optimal forcing structures and their associated flame response, leading to a discussion of the dominant mechanisms for the amplification of flow perturbations, which trigger flame oscillations. These seem to involve a resonance with ITA instability modes. Keywords: combustion instability; linear instability; premixed laminar flame; flame transfer function; resolvent analysis; intrinsic thermoacoustic mode its supporting flow field [2], or they may involve an acoustic resonance between the flame and its surrounding chamber [3]. Chamber resonance effects, which underpin the classical thermoacoustic instability, are not investigated in this paper. Flame instability has been approached in weakly and fully nonlinear frameworks [4, 5], but linear analysis still plays an important role in fundamental studies and industrial applications, such as sensitivity analysis, optimisation and control [6]. The linear flame transfer function (FTF), which relates fluctuations of the global heat release rate to local fluctuations of the velocity in a chosen measurement point, is widely used for the modelling of the flame response to acoustic forcing. Classically, the flame transfer function is obtained from experiments or numerical simulations via empirical system identification methods based on time-series of forcing and response data. This strategy is data-driven, and represents a black-box approach [7]. We advocate here an alternative strategy, based on the linear flow equations, which may be described as a white-box method. To this end, we linearize the governing equations and calculate the Jacobian matrix around a steady base flow state, which is obtained as a fixed point of the nonlinear system of equations. The resulting linear system governs the dynamics of small-amplitude fluctuations, and it can be accessed with a large variety of linear tools for a physics-based analysis. Note that our objective is not to create analytical models of the FTF, as one might do for instance by using a G-equation [8]. The single hypothesis that we employ here is that we work with fluctuations of small amplitudes in the linear limit. The complete set of linearized governing equations and the information of each flow variable in the base flow is preserved. This method allows efficient calculation of the FTF, because no time-stepping is required. A pioneering attempt in this direction was made by van Kampen et al. [9] to reduce the computational burden of FTF calculations by use of a "linear coefficient method", based on the response of flow variables to small-amplitude perturbations. The response of global heat release rate to the fluctuation of inlet equivalence ratio was investigated for a turbulent swirling flame. The response obtained via the linear coefficient method was reported to be in good agreement with reference calculation results. Nevertheless, this method does not include some essential characteristics of the flame dynamics, such as the effect of equivalence ratio fluctuations on the flame front movement. The approach used in the present study fully encompasses those effects. Another important feature is that the white-box method gives spontaneously a multi-input, multi-output (MIMO) system. While the classical flame transfer function is represented as a single-input, single-output (SISO) system associating the heat release rate with velocity disturbances, one may also want to explore other physical quantities in flames. A MIMO formulation is required if one wants to, for instance, compute the flame responses to incoming fluctuations of the mixture composition [10, 11, 12, 13, 14], or represent acoustic scattering [15, 16]. MIMO systems can in principle be identified with black-box methods, but special technical considerations are necessary, for instance so as to ensure statistical independence among different signal channels [7, 17]. A linear approach, as presented in this study, inherently preserves the causal coupling between all signals, and all mappings can be extracted from one linear system. The applicability of linear analysis to flame dynamics has already been demonstrated in several recent studies, for premixed as well as diffusion flames. Throughout this paper, the linear analysis is carried out in a global setting, i.e. on a spatially developing base flow, in contrast to local analysis which is based on the assumption of a streamwise-invariant base flow. For reacting flows, local analysis has indeed been successfully applied in reacting bluff-body wakes that can be regarded as being weakly non-parallel [18]. This assumption however does not hold for most flames, so we use global analysis where the word "global" means that the variations in the base flow are resolved both in the streamwise and in the cross-stream direction. A brief state-of-the-art survey is given for such studies on premixed flames. The reader may want to refer to [19, 20, 21, 22, 23] for work on diffusion flames. Investigating the role of precessing vortex core (PVC) dynamics in combustion instability, Oberleithner et al. [24, 25] performed modal analysis on experimental data of turbulent flames in a swirl combustor. A solver based on the linearized incompressible Navier-Stokes equations with an eddy-viscosity model was applied. The predicted frequency of the instability mode was found to be in excellent agreement with the oscillation frequency measured in the experiment. For other linear calculations with regard to the PVC, the reader may refer to [26, 27]. Blanchard et al. [28] developed a linear solver based on the compressible Navier-Stokes equations with a one-step
Bulletin of the American Physical Society, Nov 22, 2020
global response to forcing of a 2D laminar premixed slot flame is investigated with a linearized ... more global response to forcing of a 2D laminar premixed slot flame is investigated with a linearized approach, where multiphysics coupling in a non-parallel reacting flow is accounted for. The classical flame transfer functions obtained this way are in good agreement with nonlinear reference calculations, performed with the AVBP code from CERFACS. Optimal forcing input, leading to maximal heat release fluctuations, are identified via resolvent analysis. Compared to similar recent investigations, we use a finite-element discretization on an unstructured mesh, and we employ a reduced two-step scheme for methane-air reaction. Strong sensitivity to flame parameters and chemical modeling is observed. The present linear analysis opens new ways for the physical discussion of instability dynamics in flames.
Thermoacoustic properties of can-annular combustors are commonly investigated by means of single-... more 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 cancan coupling present in the full engine. However, current approaches either lack a solid theoretical foundation or are not practicable for high-pressure rigs. In the present 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 3D model of a realistic can-annular combustor configuration. For the latter we show that cancan 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.
Die genaue Modellierung dynamischer Systeme resultiert oft in einer sehr hohen Anzahl an Zustands... more Die genaue Modellierung dynamischer Systeme resultiert oft in einer sehr hohen Anzahl an Zustandsvariablen. Die Systemmatrizen konnen dadurch zu gros werden, um noch Zustandsraummodelle (ss Objekte) in MATLAB definieren zu konnen. In diesem Beitrag werden Toolboxen vorgestellt, welche durch Einfuhrung sog. sparse state space Objekten (sss) die Analyse hochdimensionaler Systeme ermoglichen. Mittels Modellordnungsreduktion (sssMOR) kann die wesentliche Dynamik in Modellen deutlich niedrigerer Ordnung abgebildet werden.
Journal of Engineering for Gas Turbines and Power, 2021
Heavy-duty land-based gas turbines are often designed with can-annular combustors, which consist ... more 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 refl...
Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analys... more Abstract This work demonstrates that a hybrid approach for linear thermoacoustic stability analysis that combines the Linearized Navier–Stokes Equations (LNSE) with a global Flame Transfer Function (FTF), generates spurious entropy waves when used to model acoustically forced premixed flames. The inability of the global FTF to account for the effects of flame movement is identified as the root cause of this unphysical behavior. Utilization of a local FTF, which resolves unsteady heat release on scales comparable to the reaction zone of the flame, suppresses the spurious entropy perturbations. This affirms that fine-grained resolution of the spatio-temporal distribution of heat release rate fluctuations in the combustion zone is required to model the movement of the flame front, even for acoustically and convectively compact flames. As an alternative to hybrid models, a Linearized Reactive Flow (LRF) approach is employed, which extends the LNSE by the linearized species transport equations as well as the reaction mechanism. Such a monolithic approach inherently accounts for the locally resolved flame dynamics, including the movement of the flame front, and does not require an external model for the flame-flow interaction. Thus the LRF eliminates the need for the cumbersome identification of a local FTF. Two configurations of lean premixed methane-air flames, i.e. a freely propagating 1D flame and a 2D flame anchored in a duct, are considered for validation. All results obtained with linearized modeling approaches and conclusions deduced thereof are validated against high resolution CFD results with excellent quantitative accuracy.
International Journal of Spray and Combustion Dynamics, 2017
The response of a laminar, premixed flame to perturbations of upstream equivalence ratio is inves... more The response of a laminar, premixed flame to perturbations of upstream equivalence ratio is investigated and modelled, with emphasis on the generation of ‘entropy waves’, i.e. entropic inhomogeneities of downstream temperature. Transient computational fluid dynamics simulations of two adiabatic lean methane-air flames of different Péclet numbers provide guidance and validation data for subsequent modelling. The respective entropy transfer functions, which describe the production of temperature inhomogeneities, as well as transfer functions for the variation of the heat release, are determined from the computational fluid dynamics time series data by means of system identification. The processes governing the dynamics of the entropy transfer functions are segregated into two sub-problems: (1) heat release due to chemical reaction at the flame front and (2) advective and diffusive transport. By adopting a formulation in terms of a mixture fraction variable, these two sub-problems can ...
Over many years, scientists and engineers have developed a broad variety of mathematical formulat... more Over many years, scientists and engineers have developed a broad variety of mathematical formulations to investigate the propagation and interactions with flow of flow-induced noise in early-stage of product design and development. Beside established theories such as the linearized Euler equations (LEE), the linearized Navier–Stokes equations (LNSE) and the acoustic perturbation equations (APE) which are described in an Eulerian framework, Galbrun utilized a mixed Lagrange–Eulerian framework to reduce the number of unknowns by representing perturbations by means of particle displacement only. Despite the advantages of fewer degrees of freedom and the reduced effort to solve the system equations, a computational approach using standard continuous finite element methods (FEM) suff ers from instabilities called spurious modes that pollute the solution. In this work, the authors employ a discontinuous Galerkin approach to overcome the difficulties related to spurious modes while solving...
Journal of Engineering for Gas Turbines and Power, 2019
This study investigates the effect of partial acoustic reflection at inlet or outlet of a combust... more This study investigates the effect of partial acoustic reflection at inlet or outlet of a combustor on thermoacoustic stability. Parametric maps of the thermoacoustic spectrum are utilized for this purpose, which represent frequencies and growth rates of eigenmodes for a wide range of model parameters. It is found that a decrease of the acoustic reflection at the boundaries does not always imply an increase in the stability margin of the thermoacoustic system. As a matter of fact, a reduction in the acoustic reflection may sometimes destabilize a thermoacoustic mode. Additionally, we show that perturbed passive thermoacoustic modes may become intrinsic thermoacoustic (ITA) modes in the fully anechoic case. We briefly discuss the mode definitions “acoustic” and “intrinsic” commonly found in the literature. The computational analysis is based on a state-space formulation of the linearized Navier–Stokes equations (LNSEs) with discontinuous Galerkin discretization. This approach allows ...
In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed... more In this paper, the dynamics and thermoacoustic stability of a laminar premixed flame are analyzed using a linearized reactive flow (LRF) solver. The LRF solver is based on linearized compressible Navier-Stokes and reacting species transport equations and thereby includes a model for the dynamic response of the flame to flow perturbations in an inherent manner. The equations are discretized using the discontinuous Galerkin finite element method. By way of example, thermoacoustic characteristics of attached and lifted laminar premixed flames are investigated. First, the respective flame transfer functions (FTFs) are computed in the frequency domain with the LRF solver. The results are in agreement with reference FTFs identified from CFD time-series. Secondly, the LRF solver is employed for thermoacoustic stability analysis, i.e. computation of shape, frequency, and growth rate of eigenmodes. Results are compared to established hybrid methods that couple FTFs with a low-order thermoacoustic network-model or a linearized Navier-Stokes equations solver. All solvers capture the dominant thermoacoustic mode, but only the LRF resolves local flow-flame interaction, revealing e.g. the onset of the flame movement and the propagation of distortions along the flame.
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Papers by Max Meindl