Nonlinear hydrodynamic and thermoacoustic oscillations of a bluff-body stabilised turbulent premixed flame

Turbulent premixed flames often experience thermoacoustic instabilities when the combustion heat release rate is in phase with acoustic pressure fluctuations. Linear methods often assume a priori that oscillations are periodic and occur at a dominant frequency with a fixed amplitude. Such assumptions are not made when using nonlinear analysis. When an oscillation is fully saturated, nonlinear analysis can serve as a useful avenue to reveal flame behaviour far more elaborate than period-one limit cycles, including quasi-periodicity and chaos in hydrodynamically or thermoacoustically self-excited system. In this paper, the behaviour of a bluff-body stabilised turbulent premixed propane/air flame in a model jet-engine afterburner configuration is investigated using computational fluid dynamics. For the frequencies of interest in this investigation, an unsteady Reynolds-averaged Navier–Stokes approach is found to be appropriate. Combustion is represented using a modified laminar flamelet approach with an algebraic closure for the flame surface density. The results are validated by comparison with existing experimental data and with large eddy simulation, and the observed self-excited oscillations in pressure and heat release are studied using methods derived from dynamical systems theory. A systematic analysis is carried out by increasing the equivalence ratio of the reactant stream supplied to the premixed flame. A strong variation in the global flame structure is observed. The flame exhibits a self-excited hydrodynamic oscillation at low equivalence ratios, becomes steady as the equivalence ratio is increased to intermediate values, and again exhibits a self-excited thermoacoustic oscillation at higher equivalence ratios. Rich nonlinear behaviour is observed and the investigation demonstrates that turbulent premixed flames can exhibit complex dynamical behaviour including quasiperiodicity, limit cycles and period-two limit cycles due to the interactions of various physical mechanisms. This has implications in selecting the operating conditions for such flames and for devising proper control strategies for the avoidance of thermoacoustic instability.     

The Rayleigh integral is always positive in steadily operated combustors

The Rayleigh index has been used for decades by a large number of researchers as an indicator to determine if a flame is driving or damping thermoacoustic interaction mechanisms. The use of the Rayleigh criterion has found applications in rocket combustors, gas turbine combustion technology and basic combustion research. The global Rayleigh index or integral is obtained by integrating the product of heat release rate and pressure fluctuations over space and time. Depending on the phase between pressure oscillations and heat release rate response, the oscillations can be enhanced or damped. It is commonly assumed in literature that the sign of the Rayleigh index from steady state data can be used to determine if the thermoacoustic feedback loop is stabilizing or destabilizing. However, we show in this paper that under fairly general conditions, a correctly measured Rayleigh index is always positive if evaluated from statistically stationary data. This proves to be true even if the heat release rate response to pressure fluctuations is in phase opposition to those pressure fluctuations. This is shown in a straightforward manner by substituting the wave equation with a heat release rate source term into the Rayleigh index. This was verified experimentally on a fully premixed combustion system by measuring the flame chemiluminescence using a photo multiplier and pressure fluctuations using a microphone placed sufficiently close to the flame to ensure acoustic compactness for the frequency range of interest. A large range of operating conditions have been tested, spanning linearly stable and unstable stationary thermoacoustic states, respectively corresponding to resonance or a limit cycle driven by the inherent stochastic forcing from the turbulent combustion noise. The experimental results corroborated the analytic finding: the Rayleigh index is found to be positive for all frequencies and all operating conditions.     

Lock-in and quasiperiodicity in hydrodynamically self-excited flames: Experiments and modelling

Hydrodynamically self-excited flames are often assumed to be insensitive to low-amplitude external forcing. To test this assumption, we apply acoustic forcing to a range of jet diffusion flames. These flames have regions of absolute instability at their base and this causes them to oscillate at discrete natural frequencies. We apply the forcing around these frequencies, at varying amplitudes, and measure the response leading up to lock-in. We then model the system as a forced van der Pol oscillator.Our results show that, contrary to some expectations, a hydrodynamically self-excited flame oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, it responds at both frequencies as well as at several nearby frequencies, indicating quasiperiodicity. When forced at high amplitudes, it locks into the forcing. The critical forcing amplitude for lock-in increases both with the strength of the self-excited instability and with the deviation of the forcing frequency from the natural frequency. Qualitatively, these features are accurately predicted by the forced van der Pol oscillator. There are, nevertheless, two features that are not predicted, both concerning the asymmetries of lock-in. When forced below its natural frequency, the flame is more resistant to lock-in, and its oscillations at lock-in are stronger than those of the unforced flame. When forced above its natural frequency, the flame is less resistant to lock-in, and its oscillations at lock-in are weaker than those of the unforced flame. This last finding suggests that, for thermoacoustic systems, lock-in may not be as detrimental as it is thought to be.     

弱非线性热声理论模型的建立及简化

本文指出非零的周期平均热声效应在本质上是非线性的,应保留到二阶精度。在小振幅条件下,利用摄动方法,以无限大平板流道为例,建立了二阶精度的弱非线性热声理论模型,并在不同条件下对模型做了进一步简化。这一理论为理解热声系统的工作机制以及设计优化热声系统提供了强有力的理论工具。     

Nonlinear simulations of combustion instabilities with a quasi-1D Navier-Stokes code

As lean premixed combustion systems are more susceptible to combustion instabilities than non-premixed systems, there is an increasing demand for improved numerical design tools that can predict the occurrence of combustion instabilities with high accuracy. The inherent nonlinearities in combustion instabilities can be of crucial importance, and we here propose an approach in which the one-dimensional (1D) Navier-Stokes and scalar transport equations are solved for geometries of variable cross-section. The focus is on attached flames, and for this purpose a new phenomenological model for the unsteady heat release from a flame front is introduced. In the attached flame method (AFM) the heat release occurs over the full length of the flame. The nonlinear code with the use of the AFM approach is validated against analytical results and against an experimental study of thermoacoustic instabilities in oxy-fuel flames by Ditaranto and Hals [Combustion and Flame 146 (2006) 493-512]. The numerical simulations are in accordance with the experimental measurements and the analytical results and both the frequencies and the amplitudes of the resonant acoustic pressure modes are reproduced with good accuracy.     

Exploring the dynamics of a class of post-tensioned, moment resisting frames

In this paper, we present an equivalent low-order nonlinear system that describes the dynamics of a generic class of post-tensioned frames. The proposed nonlinear single degree of freedom system is derived from energy considerations. We demonstrate that the equation of motion for the entire, planar, post-tensioned frame is equivalent to the dynamics of a single tied rocking block on an elastic foundation. As validation for this analytical model we present physical tests (1/4 scale) undertaken at Bristol. Quasi-static push-pull-over tests and dynamic frequency sine sweep shake table tests are conducted on the physical model. Comparison of results indicate that the analytical model predicts both quasi-static nonlinear push-over and nonlinear dynamic resonant behaviour very well. Further numerical simulations on the analytical model identify the nonlinear resonant frequency backbone curves for a range of system parameters. We explore catchment basins of both Poincaré phase and system parameter spaces. In addition we describe failure boundaries and system integrity surfaces giving an indication as to likely bounds on forcing amplitudes.     

Instability and mode transition analysis of a hydrogen-rich combustion in a model afterburner

Combustion instabilities were investigated experimentally for a hydrogen-rich combustion in a model afterburner installed at the end of a high-enthalpy wind tunnel. Air was supplied at 0.3 MPa and 950 K. The combustion instabilities were studied with the time-resolved measurements of a near-infrared (NIR) emission from water molecules over 780 nm using a high-speed video camera. Pressure was also measured in the combustor. The pressure and the NIR images were analyzed by data-driven approach, which include the fast Fourier transform (FFT), the wavelet transform, the dynamic mode decomposition (DMD) and the Gaussian process latent variable methods (GP-LVM). Thermoacoustic instability was observed under a rich condition, and the amplitude of the pressure oscillation was the maximum at the overall equivalence ratio of approximately 2.4 or 2.7 as a result of the FFT. The combustion dynamics were investigated in detail for an experimental run at the equivalence ratio of 2.4. A pressure spectrogram indicated a flame–vortex interaction with a Strouhal number of 0.5 (2300 Hz), thermoacoustic instability (560 Hz), and their transitions with the wavelet transform. For NIR images, the same tendency was also observed in the spectrogram of the modes obtained by the Gabor-filtered DMD, which could clearly resolve the high-order harmonic modes of the flame–vortex interaction and the thermoacoustic instability. Furthermore, NIR images were analyzed with GP-LVM to study the evolution of the combustion dynamics in a three-dimensional latent space. Recurrence plots with the Euclidean distance function were used to visualize the evolutions of the combustion dynamics. A limit cycle behavior of the flame–vortex interaction was clearly observed, whereas the limit cycle of the thermoacoustic instability showed more complicated behaviors. The transition behaviors of the instabilities were observed in the recurrence plots in detail, indicating that the flame–vortex interaction excited the fourth harmonic mode of the thermoacoustic instability, followed by the basic mode.     

High-amplitude thermoacoustic effects in a single pore

Nonlinear effects on thermoacoustic gain in a single pore are investigated experimentally. By creating a sharp temperature gradient in a uniform cross-section pore, the effect of high displacement amplitudes relative to the stack length is isolated from other high amplitude effects and also from effects due to geometrical discontinuities. The experiment probes displacement amplitudes which lie beyond the range of validity of the linear theory. The complex compressibility of nitrogen gas in the pore is measured for displacement amplitudes ranging from 2.5% to 60% of the stack length. No changes in the thermoacoustic response are observed over this range. Extending the upper limit to 175%, the power flow, as a function of the squared ratio of the displacement amplitude and the stack length, behaves linearly over the entire range.     

Impact of turbulence on flame brush development of acoustically excited rod-stabilized flames

Coherent structures, such as those arising from hydrodynamic instabilities or excited by thermoacoustic oscillations, can significantly impact flame structure and, consequently, the nature of heat release. The focus of this work is to study how coherent oscillations of varying amplitudes can impact the growth of the flame brush in a bluff-body stabilized flame and how this impact is influenced by the free stream turbulence intensity of the flow approaching the bluff body. We do this by providing external acoustic excitation at the natural frequency of vortex shedding to simulate a highly-coupled thermoacoustic instability, and we vary the in-flow turbulence intensity using perforated plates upstream of the flame. We use high-speed stereoscopic particle image velocimetry to obtain the three-component velocity field and we use the Mie-scattering images to quantify the behavior of the flame edge. Our results show that in the low-turbulence conditions, presence of high-amplitude acoustic excitation can cause the flame brush to exhibit a step-function growth, indicating that the presence of strong vortical structures close to the flame can suppress flame brush growth. This impact is strongly dependent on the in-flow turbulence intensity and the flame brush development in conditions with higher levels of in-flow turbulence are minimally impacted by increasing amplitudes of acoustic excitation. These findings suggest that the sensitivity of the flow and flame to high-amplitude coherent oscillations is a strong function of the in-flow turbulence intensity.     

A direct approach to generalised multiple mapping conditioning for selected turbulent diffusion flame cases

This work presents a direct and transparent interpretation of two concepts for modelling turbulent combustion: generalised Multiple Mapping Conditioning (MMC) and sparse-Lagrangian Large Eddy Simulation (LES). The MMC approach is presented as a hybrid between the Probability Density Function (PDF) method and approaches based on conditioning (e.g. Conditional Moment Closure, flamelet, etc.). The sparse-Lagrangian approach, which allows for a dramatic reduction of computational cost, is viewed as an alternative interpretation of the Filtered Density Function (FDF) methods. This work presents simulations of several turbulent diffusion flame cases and discusses the universality of the localness parameter between these cases and the universality of sparse-Lagrangian FDF methods with MMC.     

Dynamic behavior of a freely-propagating turbulent premixed flame under global stretch-rate oscillations

Dynamic features of a freely propagating turbulent premixed flame under global stretch rate oscillations were investigated by utilizing a jet-type low-swirl burner equipped with a high-speed valve on the swirl jet line. The bulk flow velocity, equivalence ratio and the nominal mean swirl number were 5 m/s, 0.80 and 1.23, respectively. Seven velocity forcing amplitudes, from 0.09 to 0.55, were examined with a single forcing frequency of 50 Hz. Three kinds of optical measurements, OH-PLIF, OH^* chemiluminescence and PIV, were conducted. All the data were measured or post-processed in a phase-locked manner to obtain phase-resolved information. The global transverse stretch rate showed in-phase oscillations centering around 60 (1/s). The oscillation amplitude of the stretch rate grew with the increment of the forcing amplitude. The turbulent flame structure in the core flow region varied largely in axial direction in response to the flowfield oscillations. The flame brush thickness and the flame surface area oscillated with a phase shift to the stretch rate oscillations. These two properties showed a maximum and minimum values in the increasing and decreasing stretch periods, respectively, for all the forcing amplitudes. Despite large variations in flame brush thickness at different phase angles, the normalized profiles collapse onto a consistent curve. This suggests that the self-similarity sustains in this dynamic flame. The global OH^* fluctuation response (i.e. response of global heat-release rate fluctuation) showed a linear dependency to the forcing velocity oscillation amplitudes. The flame surface area fluctuation response showed a linear tendency as well with a slope similar to that of the global OH^* fluctuation. This indicated that the flame surface area variations play a critical role in the global flame response.     

Experiments and low-order modelling of intermittent transitions between clockwise and anticlockwise spinning thermoacoustic modes in annular combustors

Annular combustion chambers of gas turbines and aircraft engines are subject to unstable azimuthal thermoacoustic modes leading to high amplitude acoustic waves propagating in the azimuthal direction. For certain operating conditions, the propagating direction of the wave switches randomly. The strong turbulent noise prevailing in gas turbine combustors is a source of random excitation for the thermoacoustic modes and can be the cause of these switching events. A low-order model is proposed to describe qualitatively this property of the dynamics of thermoacoustic azimuthal modes. This model is based on the acoustic wave equation with a destabilizing thermoacoustic source term to account for the flame’s response and a stochastic term to account for the turbulent combustion noise. Slow-flow averaging is applied to describe the modal dynamics on times scales that are slower than the acoustic pulsation. Under certain conditions, the model reduces formally to a Fokker-Planck equation describing a stochastic diffusion process in a potential landscape with two symmetric wells: One well corresponds to a mode propagating in the clockwise direction, the other well corresponds to a mode propagating in the anticlockwise direction. When the level of turbulent noise is sufficient, the stochastic force makes the mode jump from one well to the other at random times, reproducing the phenomenon of direction switching. Experiments were conducted on a laboratory scale annular combustor featuring 12 hydrogen-methan flames. System identification techniques were used to fit the model on the experimental data, allowing to extract the potential shape and the intensity of the stochastic excitation. The statistical predictions obtained from the Fokker–Planck equation on the mode’s behaviour and the direction switching time are in good agreement with the experiments.     

Experimental evaluation of DC electric field effect on the thermoacoustic behaviour of flat premixed flames

One promising approach to eliminate thermoacoustic instabilities in combustion appliances is the use of adaptive control of the flame/burner acoustic transfer function (TF). Application of a DC electric field (EF) as a spatially distributed, easily and quickly adjustable and low-energy method to affect the flame behaviour may be considered as a possible actuation method to control the flame TF. Experimental evaluation of such a possibility is the main goal of the present study. The effect of a DC EF on the acoustic TF of premixed flat burner-surface stabilized flames is studied systematically as a function of the following parameters: flow velocity, equivalence ratio, applied voltage and burner geometry. It is established that the response of the flame TF on the DC EF can be characterized as a TF shift towards higher frequencies. The mechanism of TF alteration is related to the decrease of the flame stand-off distance and the related increase of the burner deck surface temperature. From a practical point of view, the efficiency of the EF control of the flat flame TF is restricted to a relatively narrow frequency range around the position of the TF resonance peak. To get insight into the physics of the EF–flame interaction, the method of [1] to measure the EF effect on the adiabatic flame speed is improved and the measurement range is extended. The new measurements allow a revision of previous results and allow an explanation for the ambiguity in the old measurements.     

Detection of global and local parameter variations using nonlinear feedback auxiliary signals and system augmentation

Recently, system augmentation has been combined with nonlinear feedback auxiliary signals to provide sensitivity enhancement in both linear and nonlinear systems. Augmented systems are higher dimensional linear systems that follow trajectories of a nonlinear system one at a time. These augmented systems are subject to a specialized augmented forcing which enforces the augmented system to exactly reproduce the trajectory of the nonlinear system when projected onto the lower dimensional (physical) system. Augmented systems have additional benefits outside of handling nonlinear systems, which makes them more desirable than regular linear systems for sensitivity enhancing control. One of the key advantages of augmented systems is the complete control over the augmented degrees of freedom, and the additional sensor-type knowledge from the augmented variables. These sensing and actuation features are very useful when only few physical actuators and sensors can be placed. Such restrictions are common in most applications, and they severely limit the usefulness of traditional linear sensitivity enhancing feedback approaches. Another benefit of the augmentation is that the control exerted on the augmented degrees of freedom does not require any physical energy, rather it is just signal processing. In this work, these benefits are refined to improve the robustness of detection using sensitivity enhancement. Also, the benefits of system augmentation are explored by using few actuators and sensors. An optimization algorithm is employed not only to maximize the sensitivity of resonant frequencies to added mass at particular locations, but also to detect uniform changes in mass and stiffness. In addition to increased sensitivity for both global and local parameter changes, a study of increasing the sensitivity of local changes, while decreasing the sensitivity of global changes is conducted. Additionally, a methodology is presented to accurately extract augmented frequencies from displacement and forcing data corrupted by noise. Numerical simulations of cantilevered beams are used to validate the approach and discuss the effects of noise.     

Premixed flames for arbitrary combinations of strain and curvature

Many modeling strategies for combustion rely on laminar flamelet concepts to determine structure and properties of multi-dimensional and turbulent flames. Using flamelet tabulation strategies, the user anticipates certain aspects of the combustion process prior to the simulation and selects a flamelet model which mimics local flame conditions in the more complex configuration. Flame stretch, which can be decomposed into contributions from strain and curvature, is one of the conditions influencing a flame’s properties, structure, and stability. The objective of this work is to study premixed flame structures in the strain-curvature space using a recently published composition space model (CSM) and three physical space models for canonical flame configurations (stagnation flame, spherical expanding flame and inwardly propagating flame). Flames with effective Lewis numbers both smaller and larger than unity are considered. For canonical laminar flames, the stretch components are inherently determined through boundary conditions and their specific flame configuration. Therefore, canonical flames can only represent a certain sub-set of stretch effects experienced by multi-dimensional and turbulent flames. On the contrary, the CSM allows arbitrary combinations of strain and curvature to be prescribed for premixed flames exceeding the conditions attainable with the canonical flame setups. Thereby, also influences of negative strain effects and large curvatures can be studied. A parameter variation with the CSM shows that flame structures still significantly change outside the region of the canonical flame configurations. Furthermore, limits in the strain-curvature space are discussed. The present paper highlights advantages of composition space modeling which is achieved by detaching the representation of the flame structure from a specific canonical flame configuration in physical space.     

Cardiac arrhythmias and circle maps-A classical problem

The periodic forcing of nonlinear oscillations can often be cast as a problem involving self-maps of the circle. Consideration of the effects of changes in the frequency and amplitude of the periodic forcing leads to a problem involving the bifurcations of circle maps in a two-dimensional parameter space. The global bifurcations in this two-dimensional parameter space is described for periodic forcing of several simple theoretical models of nonlinear oscillations. As was originally recognized by Arnold, one motivation for the formulation of these models is their connection with theoretical models of cardiac arrhythmias originating from the competition and interaction between two pacemakers for the control of the heart.     

A consistent LES/filtered-density function formulation for the simulation of turbulent flames with detailed chemistry

A hybrid large-Eddy simulation/filtered-density function (LES–FDF) methodology is formulated for simulating variable density turbulent reactive flows. An indirect feedback mechanism coupled with a consistency measure based on redundant density fields contained in the different solvers is used to construct a robust algorithm. Using this novel scheme, a partially premixed methane/air flame is simulated. To describe transport in composition space, a 16-species reduced chemistry mechanism is used along with the interaction-by-exchange with the mean (IEM) model. For the micro-mixing model, typically a constant ratio of scalar to mechanical time-scale is assumed. This parameter can have substantial variations and can strongly influence the combustion process. Here, a dynamic time-scale model is used to prescribe the mixing time-scale, which eliminates the time-scale ratio as a model constant. Two different flame configurations, namely, Sandia flames D and E are studied. Comparison of simulated radial profiles with experimental data show good agreement for both flames. The LES–FDF simulations accurately predict the increased extinction near the inlet and re-ignition further downstream. The conditional mean profiles show good agreement with experimental data for both flames.     

Physics-informed recurrent neural networks for linear and nonlinear flame dynamics

This paper demonstrates the ability of recurrent neural networks (RNNs) to predict the linear and the nonlinear response of a premixed laminar flame to incoming velocity perturbations. We develop data-driven models, which require the velocity and heat release rate fluctuations as input data. Both time series are obtained from Direct Numerical Simulations (DNS) of a laminar flame. The length of the signals, and, hence, the cost of the simulation, is comparable to those used in the linear framework of System Identification. A more robust type of RNNs, namely long short term memory (LSTM), is employed to reduce the dependency on large datasets. The LSTM framework is modeled as a time series regression problem and four models are trained with decreasing data set lengths. All purely data-driven models accurately predict the unsteady time series of the heat release rate and, hence, the Flame Transfer Functions (FTFs). We further improve the model accuracy by incorporating a physical constraint, namely the low-frequency limit for perfectly-premixed flames, into the LSTM model. This step reduces the required data length compared to the purely data-driven approach. The proposed model, called PI-LSTM, is able to reproduce the linear and the nonlinear FTFs for amplitudes up to 50% of the laminar flame based on one numerical simulation, where the length of the time series is 100 ms.