Limit Cycles of Spinning Thermoacoustic Modes in Annular Combustors: A Bloch-Wave and Adjoint-Perturbation Approach

2017 ◽  
Author(s):  
Georg A. Mensah ◽  
Jonas P. Moeck
Keyword(s):  
Helmholtz Equation ◽  
Limit Cycle ◽  
Rotational Symmetry ◽  
Wave Theory ◽  
Bloch Wave ◽  
Generic Model ◽  
Perturbation Approach ◽  
Dependent Manner ◽  
Combustion System ◽  
Linear Character

The most straightforward way to assess the thermoacoustic stability of a combustion system is based on modal approaches. The modes are typically computed from linearized equations in the frequency domain, such as the Helmholtz equation. Due to the linear character, nonlinear saturation effects cannot be computed with such models. Flame describing functions have been suggested to fill this gap. They describe the flame response in an amplitude-dependent manner and have been successfully used in recent work for the prediction of limit-cycle amplitudes in single-burner systems and annular combustors. This paper presents a more efficient approach of computing limit-cycle amplitudes of spinning thermoacoustic modes in an annular combustion chamber. As one important feature, adjoint perturbation theory is utilized for the solution of the thermoacoustic Helmholtz equation associated with a flame describing function. This avoids iterations over different amplitude levels to find the limit cycle amplitude, i.e., the amplitude level at which the modal growth rate is zero, as required in previous approaches. Moreover, based on the discrete rotational symmetry of the system, the computation is also accelerated by means of Bloch-wave theory, which reduces computations for annular combustors to a single burner/flame segment. Results for a generic model and a laboratory-scale annular combustion system are presented and discussed.

Time-Domain Bloch Boundary Conditions for Efficient Simulation of Thermoacoustic Limit Cycles in (Can-)Annular Combustors

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Matthias Haeringer ◽  
Wolfgang Polifke
Keyword(s):  
Boundary Conditions ◽  
State Space ◽  
Time Domain ◽  
Cfd Simulation ◽  
Rotational Symmetry ◽  
Wave Theory ◽  
Bloch Wave ◽  
Cfd Simulations ◽  
Suggested Approach ◽  
Combustion Dynamics

Abstract Thermo-acoustic eigenmodes of annular or can-annular combustion chambers, which typically feature a discrete rotational symmetry, may be computed in an efficient manner by utilizing the Bloch-wave theory. Unfortunately, the application of the Bloch-wave theory to combustion dynamics has hitherto been limited to the frequency domain. In this study, we present a time-domain formulation of Bloch boundary conditions (BBC), which allows to employ them in time domain simulations, e.g., computational fluid dynamics (CFD) simulations. The BBCs are expressed as acoustic scattering matrices and translated to complex-valued state-space systems. In a hybrid approach an unsteady, compressible CFD simulation of the burner-flame zone is coupled via characteristic-based state-space boundary conditions to a reduced order model of the combustor acoustics that includes BBCs. The acoustic model with BBC accounts for cross-can acoustic coupling and the discrete rotational symmetry of the configuration, while the CFD simulation accounts for the nonlinear flow–flame acoustic interactions. This approach makes it possible to model limit cycle oscillations of (can-)annular combustors at drastically reduced computational cost compared to CFD simulations of the full configuration and without the limitations of weakly nonlinear approaches that utilize a flame describing function. In this study, the suggested approach is applied to a generic multican combustor. Results agree well with a fully compressible CFD simulation of the complete configuration.

Low-Order Modeling of Can-Annular Combustors

2021 ◽  
Author(s):  
Guillaume J. J. Fournier ◽  
Max Meindl ◽  
Camilo F. Silva ◽  
Giulio Ghirardo ◽  
Mirko R. Bothien ◽  
...  
Keyword(s):  
Reflection Coefficient ◽  
Gas Turbines ◽  
Characteristic Length ◽  
Rotational Symmetry ◽  
Wave Theory ◽  
Bloch Wave ◽  
Geometrical Parameters ◽  
Flow Parameters ◽  
Annular Gap ◽  
Low Order

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 2D 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 2D 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.

Efficient Computation of Thermoacoustic Modes in Annular Combustion Chambers Based on Bloch-Wave Theory

2015 ◽  
Author(s):  
Georg A. Mensah ◽  
Jonas P. Moeck
Keyword(s):  
Rotational Symmetry ◽  
Three Dimensional ◽  
Wave Theory ◽  
Bloch Wave ◽  
Computational Effort ◽  
Mode Shapes ◽  
Efficient Computation ◽  
Burner Flame ◽  
Flame Response ◽  
Type Boundary

Most annular combustors feature a discrete rotational symmetry so that the full configuration can be obtained by copying one burner–flame segment a certain number of times around the circumference. A thermoacoustic model based on the Helmholtz equation then admits special solutions of the so-called Bloch type that can be obtained by considering one segment only. We show that a significant reduction in computational effort for the determination of thermoacoustic modes can be achieved by exploiting this concept. The framework is applicable even in complex cases including a non-homogeneous temperature field and a frequency-dependent, spatially distributed flame response. A parametric study on a three-dimensional combustion chamber model is conducted using both the full scale chamber simulation and a one-segment model with the appropriate Bloch-type boundary conditions. The results for both computations are compared in terms of mode frequencies and growth rates as well as the corresponding mode shapes. This comparison demonstrates the benefits of the Bloch-wave based analysis. It is further shown that even the effect of circumferential asymmetries can be assessed based on computations of one burner–flame segment only by resorting to spectral perturbation theory.

Time Domain Bloch Boundary Conditions for Efficient Simulation of Thermoacoustic Limit-Cycles in (Can-)Annular Combustors

2019 ◽  
Author(s):  
Matthias Haeringer ◽  
Wolfgang Polifke
Keyword(s):  
Boundary Conditions ◽  
State Space ◽  
Time Domain ◽  
Cfd Simulation ◽  
Rotational Symmetry ◽  
Wave Theory ◽  
Bloch Wave ◽  
Cfd Simulations ◽  
Suggested Approach ◽  
Combustion Dynamics

Abstract Thermo-acoustic eigenmodes of annular or can-annular combustion chambers, which typically feature a discrete rotational symmetry, may be computed in an efficient manner by utilizing the Bloch-wave theory. Unfortunately, the application of the Bloch-wave theory to combustion dynamics has hitherto been limited to the frequency domain. In this study we present a time domain formulation of Bloch boundary conditions (BBC), which allows to employ them in time domain simulations, e.g. CFD simulations. The BBCs are expressed as acoustic scattering matrices and translated to complex-valued state-space systems. In a hybrid approach an unsteady, compressible CFD simulation of the burner-flame zone is coupled via characteristic-based state-space boundary-conditions to a reduced order model of the combustor acoustics that includes BBCs. The acoustic model with BBC accounts for cross-can acoustic coupling and the discrete rotational symmetry of the configuration, while the CFD simulation accounts for the nonlinear flow-flame-acoustic interactions. This approach makes it possible to model limit cycle oscillations of (can-)annular combustors at drastically reduced computational cost compared to CFD simulations of the full configuration, and without the limitations of weakly nonlinear approaches that utilize a flame describing function. In the current study the suggested approach is applied to a generic multi-can combustor. Results agree well with a fully compressible CFD simulation of the complete configuration.

Low-order Modeling of Can-annular Combustors

2021 ◽  
Author(s):  
Guillaume Jean Jacques Fournier ◽  
Maximilian Meindl ◽  
Camilo Silva ◽  
Giulio Ghirardo ◽  
Mirko R. Bothien ◽  
...  
Keyword(s):  
Reflection Coefficient ◽  
Gas Turbines ◽  
Characteristic Length ◽  
Rotational Symmetry ◽  
Wave Theory ◽  
Bloch Wave ◽  
Geometrical Parameters ◽  
Flow Parameters ◽  
Annular Gap ◽  
Low Order

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 2D 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 2D 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.

Efficient Computation of Thermoacoustic Modes in Industrial Annular Combustion Chambers Based on Bloch-Wave Theory

2016 ◽  
Vol 138 (8) ◽  
Author(s):  
Georg A. Mensah ◽  
Giovanni Campa ◽  
Jonas P. Moeck
Keyword(s):  
Rotational Symmetry ◽  
Three Dimensional ◽  
Wave Theory ◽  
Bloch Wave ◽  
Computational Effort ◽  
Mode Shapes ◽  
Efficient Computation ◽  
Spatially Distributed ◽  
Flame Response ◽  
Type Boundary

Most annular combustors feature a discrete rotational symmetry so that the full configuration can be obtained by copying one burner-flame segment a certain number of times around the circumference. A thermoacoustic model based on the Helmholtz equation then admits special solutions of the so-called Bloch type that can be obtained by considering one segment only. We show that a significant reduction in computational effort for the determination of thermoacoustic modes can be achieved by exploiting this concept. The framework is applicable even in complex cases including an inhomogeneous temperature field and a frequency-dependent, spatially distributed flame response. A parametric study on a three-dimensional combustion chamber model is conducted using both the full-scale chamber simulation and a one-segment model with the appropriate Bloch-type boundary conditions. The results for both computations are compared in terms of mode frequencies and growth rates as well as the corresponding mode shapes. The same is done for a more complex industrial configuration. These comparisons demonstrate the benefits of the Bloch-wave based analysis.

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely-spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

Abstract It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase of the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of Complexification-Averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes) and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

scholarly journals Whispering Bloch modes

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160103 ◽  
Author(s):  
B. Maling ◽  
R. V. Craster
Keyword(s):  
Helmholtz Equation ◽  
Eigenvalue Problems ◽  
Open Systems ◽  
Rotational Symmetry ◽  
Field Enhancement ◽  
High Order ◽  
Whispering Gallery Modes ◽  
Whispering Gallery ◽  
Field Patterns ◽  
Bloch Modes

We investigate eigenvalue problems for the planar Helmholtz equation in open systems with a high order of rotational symmetry. The resulting solutions have similarities with the whispering gallery modes exploited in photonic micro-resonators and elsewhere, but unlike these do not necessarily require a surrounding material boundary, with confinement instead resulting from the geometry of a series of inclusions arranged in a ring. The corresponding fields exhibit angular quasi-periodicity reminiscent of Bloch waves, and hence we refer to them as whispering Bloch modes (WBMs). We show that if the geometry of the system is slightly perturbed such that the rotational symmetry is broken, modes with asymmetric field patterns can be observed, resulting in field enhancement and other potentially desirable effects. We investigate the WBMs of two specific geometries first using expansion methods and then by applying a two-scale asymptotic scheme.

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

2020 ◽  
Author(s):  
Matthias Haeringer ◽  
Guillaume J. J. Fournier ◽  
Max Meindl ◽  
Wolfgang Polifke
Keyword(s):  
Wave Theory ◽  
Bloch Wave ◽  
Reflection Coefficients ◽  
Test Rig ◽  
Helmholtz Resonators ◽  
Flame Dynamics ◽  
Annular Combustor ◽  
Passive Acoustic ◽  
Thermoacoustic Properties ◽  
Dominant Frequencies

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 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 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.

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