Flame Double Input Describing Function analysis

The Flame Describing Function (FDF) is a useful and relatively cheap approximation of a flame’s nonlinearity with respect to harmonic velocity fluctuations. When embedded into a linear acoustic network, it is able to predict the amplitude and stability of harmonic thermoacoustic oscillations through the harmonic balance procedure. However, situations exist in which these oscillations are not periodic, but their spectrum contains peaks at several incommensurate frequencies. If one assumes that two frequencies dominate the spectrum, these oscillations are quasiperiodic, and the FDF concept can be extended by forcing the flame with two amplitudes and two frequencies. The nonlinearity is then approximated by a Flame Double Input Describing Function (FDIDF), which is a more expensive object to calculate than the FDF, but contains more information about the nonlinear response. In this study, we present the calculation of a non-static flame’s FDIDF. We use a G-equation-based laminar conical flame. We embed the FDIDF into a thermoacoustic network and we predict the nature and amplitude of thermoacoustic oscillations through the harmonic balance method. A criterion for the stability of these oscillations is outlined. We compare our results with a classical FDF analysis and self-excited time domain simulations of the same system. We show how the FDIDF improves the stability prediction provided by the FDF. At a numerical cost roughly equivalent to that of two FDFs, the FDIDF is capable to predict the onset of Neimark-Sacker bifurcations and to identify the frequency of oscillations around unstable limit cycles. At a higher cost, it can also saturate in amplitude these oscillations and predict the amplitude and stability of quasiperiodic oscillations.This project was funded by the European Research Council through Project ALORS 2590620.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Elsevier

Flapping states of an el astically anchored wing in a uniform flow

Linear stability analysis of an elastically anchored wing in a uniform flow is investigated both analytically and numerically. The analytical formulation explicitly takes into account the effect of the wake on the wing by means of Theodorsen's theory. Three different parameters non-trivially rule the observed dynamics: mass density ratio between wing and fluid, spring elastic constant and distance between the wing center of mass and the spring anchor point on the wing. We found relationships between these parameters which rule the transition between stable equilibrium and fluttering. The shape of the resulting marginal curve has been successfully verified by high Reynolds number direct numerical simulations. Our findings are of interest in applications related to energy harvesting by fluid-structure interaction, a problem which has recently attracted a great deal of attention. The main aim in that context is to identify the optimal physical/geometrical system configuration leading to large sustained motion, which is the source of energy we aim to extract.Comment: 10 pages, 11 figures, submitted to J. Fluid. Mec

Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube

In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart-Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations' amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes

Frequency domain and time domain analysis of thermoacoustic oscillations with wave-based acoustics

Many thermoacoustic systems exhibit rich nonlinear behaviour. Recent studies show that this nonlinear dynamics can be well captured by low-order time domain models that couple a level set kinematic model for a laminar flame, the $G$-equation, with a state-space realization of the linearized acoustic equations. However, so far the $G$-equation has been coupled only with straight ducts with uniform mean acoustic properties, which is a simplistic configuration. In this study, we incorporate a wave-based model of the acoustic network, containing area and temperature variations and frequency-dependent boundary conditions. We cast the linear acoustics into state-space form using a different approach from that in the existing literature. We then use this state-space form to investigate the stability of the thermoacoustic system, both in the frequency and time domains, using the flame position as a control parameter. We observe frequency-locked, quasiperiodic and chaotic oscillations. We identify the location of Neimark–Sacker bifurcations with Floquet theory. We also find the Ruelle–Takens–Newhouse route to chaos with nonlinear time series analysis techniques. We highlight important differences between the nonlinear response predicted by the frequency domain and the time domain methods. This reveals deficiencies with the frequency domain technique, which is commonly used in academic and industrial studies of thermoacoustic systems. We then demonstrate a more accurate approach based on continuation analysis applied to time domain techniques.This project was funded by the European Research Council through Project ALORS 2590620.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2015.13

G-equation modelling of thermo-acoustic oscillations of partially-premixed flames

Numerical simulations aid combustor design to avoid and reduce thermo-acoustic oscillations. Non-linear heat release rate estimation and its modelling are essential for the prediction of saturation amplitudes of limit cycles. The heat release dynamics of flames can be approximated by a Flame Describing Function (FDF). To calculate an FDF, a wide range of forcing amplitudes and frequencies needs to be considered. For this reason, we present a computationally inexpensive level-set approach, which accounts for equivalence ratio perturbations on flames with arbitrarily-complex shapes. The influence of flame parameters and modelling approaches on flame describing functions and time delay coefficient distributions are discussed in detail. The numerically-obtained flame describing functions are compared with experimental data and used in an acoustic network model for limit cycle prediction. A reasonable agreement of the heat release gain and limit cycle frequency is achieved even with a simplistic, analytical velocity fluctuation model. However, the phase decay is over-predicted. For sophisticated flame shapes, only the realistic modelling of large-scale flow structures allows the correct phase decay predictions of the heat release rate response.This work was conducted within the EU 7th Framework Project Joint Technology Initiatives - Clean Sky (AMEL- Advanced Methods for the Prediction of Lean-burn Combustor Unsteady Phenomena), project number: JTI-CS-2013-3-SAGE- 06-009 / 641453. This work was performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam.ac.uk/), provided by Dell Inc. using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England and funding from the Science and Technology Facilities Council

G-equation modelling of thermo-acoustic oscillations of partially-premixed flames

Numerical simulations aid combustor design to avoid and reduce thermo-acoustic oscillations. Non-linear heat release rate estimation and its modelling are essential for the prediction of saturation amplitudes of limit cycles. The heat release dynamics of flames can be approximated by a Flame Describing Function (FDF). To calculate an FDF, a wide range of forcing amplitudes and frequencies needs to be considered. For this reason, we present a computationally inexpensive level-set approach, which accounts for equivalence ratio perturbations on flames with arbitrarily-complex shapes. The influence of flame parameters and modelling approaches on flame describing functions and time delay coefficient distributions are discussed in detail. The numerically-obtained flame describing functions are compared with experimental data and used in an acoustic network model for limit cycle prediction. A reasonable agreement of the heat release gain and limit cycle frequency is achieved even with a simplistic, analytical velocity fluctuation model. However, the phase decay is over-predicted. For sophisticated flame shapes, only the realistic modelling of large-scale flow structures allows the correct phase decay predictions of the heat release rate response.This work was conducted within the EU 7th Framework Project Joint Technology Initiatives - Clean Sky (AMEL- Advanced Methods for the Prediction of Lean-burn Combustor Unsteady Phenomena), project number: JTI-CS-2013-3-SAGE- 06-009 / 641453. This work was performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam.ac.uk/), provided by Dell Inc. using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England and funding from the Science and Technology Facilities Council

Degenerate perturbation theory in thermoacoustics: High-order sensitivities and exceptional points

In this study, we connect concepts that have been recently developed in thermoacoustics, specifically, (i) high-order spectral perturbation theory, (ii) symmetry induced degenerate thermoacoustic modes, (iii) intrinsic thermoacoustic modes, and (iv) exceptional points. Their connection helps gain physical insight into the behaviour of the thermoacoustic spectrum when parameters of the system are varied. First, we extend high-order adjoint-based perturbation theory of thermoacoustic modes to the degenerate case. We provide explicit formulae for the calculation of the eigenvalue corrections to any order. These formulae are valid for self-adjoint, non-self-adjoint or even non-normal systems; therefore, they can be applied to a large range of problems, including fluid dynamics. Second, by analysing the expansion coefficients of the eigenvalue corrections as a function of a parameter of interest, we accurately estimate the radius of convergence of the power series. Third, we connect the existence of a finite radius of convergence to the existence of singularities in parameter space. We identify these singularities as exceptional points, which correspond to defective thermoacoustic eigenvalues, with infinite sensitivity to infinitesimal changes in the parameters. At an exceptional point, two eigenvalues and their associated eigenvectors coalesce. Close to an exceptional point, strong veering of the eigenvalue trajectories is observed. As demonstrated in recent work, exceptional points naturally arise in thermoacoustic systems due to the interaction between modes of acoustic and intrinsic origin. The role of exceptional points in thermoacoustic systems sheds new light on the physics and sensitivity of thermoacoustic stability, which can be leveraged for passive control by small design modifications

Linear stability and adjoint sensitivity analysis of thermoacoustic networks with premixed flames

© 2015 The Combustion Institute. We analyse the linear response of laminar conical premixed flames modelled with the linearised front-track kinematic G-equation. We start by considering the case in which the flame speed is fixed, and travelling wave velocity perturbations are advected at a speed different from the mean flow velocity. A previous study of this case contains a small error in the Flame Transfer Function (FTF), which we correct. We then allow the flame speed to depend on curvature. No analytical solutions for the FTF exist for this case so the FTF has to be calculated numerically as its parameters - aspect ratio, convection speed and Markstein length - are varied. Then we consider the stability and sensitivity of thermoacoustic systems containing these flames. Traditionally, the stability of a thermoacoustic system is found by embedding the FTF within an acoustic network model. This can be expensive, however, because the FTF must be re-calculated whenever a flame parameter is varied. Instead, we couple the linearised G-equation directly with an acoustic network model, creating a linear eigenvalue problem without explicit knowledge of the FTF. This provides a simple and quick way to analyse the stability of thermoacoustic networks. It also allows us to use adjoint sensitivity analysis to examine, at little extra cost, how the system's stability is affected by every parameter of the system.This project was funded by the European Research Council through Project ALORS 2590620

Data Assimilation in a Nonlinear Time-Delayed Dynamical System with Lagrangian Optimization

When the heat released by a flame is sufficiently in phase with the acoustic pressure, a self-excited thermoacoustic oscillation can arise. These nonlinear oscillations are one of the biggest challenges faced in the design of safe and reliable gas turbines and rocket motors [7]. In the worst-case scenario, uncontrolled thermoacoustic oscillations can shake an engine apart. Reduced-order thermoacoustic models, which are nonlinear and time-delayed, can only qualitatively predict thermoacoustic oscillations. To make reduced-order models quantitatively predictive, we develop a data assimilation framework for state estimation. We numerically estimate the most likely nonlinear state of a Galerkin-discretized time delayed model of a horizontal Rijke tube, which is a prototypical combustor. Data assimilation is an optimal blending of observations with previous system’s state estimates (background) to produce optimal initial conditions. A cost functional is defined to measure (i) the statistical distance between the model output and the measurements from experiments; and (ii) the distance between the model’s initial conditions and the background knowledge. Its minimum corresponds to the optimal state, which is computed by Lagrangian optimization with the aid of adjoint equations. We study the influence of the number of Galerkin modes, which are the natural acoustic modes of the duct, with which the model is discretized. We show that decomposing the measured pressure signal in a finite number of modes is an effective way to enhance state estimation, especially when nonlinear modal interactions occur during the assimilation window. This work represents the first application of data assimilation to nonlinear thermoacoustics, which opens up new possibilities for real-time calibration of reduced-order models with experimental measurements

Flame Double Input Describing Function analysis

© 2016 The Combustion InstituteThe Flame Describing Function (FDF) is a useful and relatively cheap approximation of a flame's nonlinearity with respect to harmonic velocity fluctuations. When embedded into a linear acoustic network, it is able to predict the amplitude and stability of harmonic thermoacoustic oscillations through the harmonic balance procedure. However, situations exist in which these oscillations are not periodic, but their spectrum contains peaks at several incommensurate frequencies. If one assumes that two frequencies dominate the spectrum, these oscillations are quasiperiodic, and the FDF concept can be extended by forcing the flame with two amplitudes and two frequencies. The nonlinearity is then approximated by a Flame Double Input Describing Function (FDIDF), which is a more expensive object to calculate than the FDF, but contains more information about the nonlinear response. In this study, we present the calculation of a non-static flame's FDIDF. We use a G-equation-based laminar conical flame model. We embed the FDIDF into a thermoacoustic network and we predict the nature and amplitude of thermoacoustic oscillations through the harmonic balance method. A criterion for the stability of these oscillations is outlined. We compare our results with a classical FDF analysis and self-excited time domain simulations of the same system. We show how the FDIDF improves the stability prediction provided by the FDF. At a numerical cost roughly equivalent to that of two FDFs, the FDIDF is capable to predict the onset of Neimark–Sacker bifurcations and to identify the frequency of oscillations around unstable limit cycles. At a higher cost, it can also saturate in amplitude these oscillations and predict the amplitude and stability of quasiperiodic oscillations