Rijke Tube Research Articles

Hopf Bifurcation Analysis of the Combustion Instability in a Liquid Rocket Engine

The bifurcation process of self-sustained combustion instability pressure perturbations in a liquid rocket combustor is investigated based on the Helmholtz equations and a pressure dependent flame describing function. The modal frequency and growth rates are numerically resolved by the commercial software COMSOL multiphysics. Validation of the numerical approach is firstly conducted on a Rijke tube combustor, and a supercritical bifurcation for the first longitudinal mode is observed. The bifurcation diagrams for the first transverse mode for different time delays and gain index of the flame describing function are analyzed. Only the supercritical bifurcation presents for this configuration. The trajectory of Hopf points and the bifurcation diagram feature period motions with increasing the time delay. The effect of flame length distributions on the bifurcation diagrams is analyzed by considering a non-uniform flame length distribution model. Results show that the distribution has a large impact on the bifurcation process, e.g., the first transverse mode is more unstable for the non-uniform distribution. Finally, a subcritical bifurcation is found when a more complicated flame describing function is considered; the bistable region presents and the condition for this is discussed.

Effect of helmholtz resonator shape on suppression of thermo-acoustic instability

In combustion chambers thermo-acoustic instabilities arise due to the coupling between the unsteady heat release and pressure fluctuations. The instabilities created cause serious structural vibration, flame blow out, increase in heat transfer and damage, thereby reducing the life span and performance of jet engines, gas turbines and industrial burners. Therefore, there is need to develop a simple technique for effective control of these instabilities. The present work involves experimental study of thermo-acoustic instabilities occurring in a Rijke tube and their suppression by using Helmholtz resonators of different shape (a passive method of suppression). The combustor model is replaced by Rijke tube which is an experimental setup extensively employed for research purpose. Rijke tube is provided with a co-axial pre-mixed gas burner as the source of heat which could be placed at any desired position. When a heat source with proper air to fuel ratio is placed at about one fourth of the tube length, 3rd mode of instability is obtained and a loud noise of frequency 681 Hz is produced. A comparative study of thermo-acoustic instability suppression using cylindrical, spherical and conical Helmholtz resonators has been performed. Spherical resonator has been found to be the most effective one imparting a maximum damping effect of 20 dB and an average damping effect of 10.5 dB.

Occasional coupling enhances amplitude death in delay-coupled oscillators.

This paper aims to study amplitude death in time delay coupled oscillators using the occasional coupling scheme that implies intermittent interaction among the oscillators. An enhancement of amplitude death regions (i.e., an increment of the width of the amplitude death regions along the control parameter axis) can be possible using the occasional coupling in a pair of delay-coupled oscillators. Our study starts with coupled limit cycle oscillators (Stuart-Landau) and coupled chaotic oscillators (Rössler). We further examine coupled horizontal Rijke tubes, a prototypical model of thermoacoustic systems. Oscillatory states are highly detrimental to thermoacoustic systems such as combustors. Consequently, a state of amplitude death is always preferred. We employ the on-off coupling (i.e., a square wave function), as an occasional coupling scheme, to these coupled oscillators. On monotonically varying the coupling strength (as a control parameter), we observe an enhancement of amplitude death regions using the occasional coupling scheme compared to the continuous coupling scheme. In order to study the contribution of the occasional coupling scheme, we perform a detailed linear stability analysis and analytically explain this enhancement of the amplitude death region for coupled limit cycle oscillators. We also adopt the frequency ratio of the oscillators and the time delay between the oscillators as the control parameters. Intriguingly, we obtain a similar enhancement of the amplitude death regions using the frequency ratio and time delay as the control parameters in the presence of the occasional coupling. Finally, we use a half-wave rectified sinusoidal wave function (motivated by practical reality) to introduce the occasional coupling in time delay coupled oscillators and get similar results.

Generation of low frequency flame oscillations for the development of passive acoustic flame arresters

The development of acoustic-based flame arresters needs a profound understanding of low frequency flame fluctuations. The present study attempts the synthesis and analysis of such low frequency flame oscillations in a Rijke tube combustor subjected to external acoustic forcing. A high frequency (fn = 437±3 Hz) self-excited laminar premixed conical flame is considered. This flame is forced (ff)around its natural frequency (fn) to generate beating (fb=|ff−fn|). The results show that the system falls into lock-in synchronization for a range of forcing frequencies (0.97≤ff/fn≤1.03) at a forcing amplitude (εf) of 10% wherein quenching of natural mode is achieved. A strong flame beating is observed on either side of this synchronization region. The flame transfer function (FTF) at fb is then estimated to delineate the strength of the beating. The FTF gain at fb shows different behavior on either side of the synchronization region. For ff >fn, the gain exhibits a local peak at low beating frequencies (∼20 Hz) which is consistent with the response of a stable laminar conical flame subjected to acoustic forcing at these frequencies. However, for ff<fn, negligible FTF gain is obtained at low beating frequencies, followed by a continuous increase as the beating frequency increases. This asymmetric behavior is further analyzed using proper orthogonal decomposition technique (POD). For ff>fn, the POD results at low fbshow that the flame base undergoes a strong flapping-like oscillation at the beating frequency. However for ff <fn, the flame base oscillates only at the natural mode of the combustor. This different beating behavior of the system can be attributed to the strong and weak thermoacoustic coupling, after and before the synchronization region, respectively.

From Pressure Time Series Data to Flame Transfer Functions: A Framework for Perfectly-Premixed Swirling Flames

Abstract We present a two-step optimization (TSO) framework, which uses the pressure data of an unstable combustion process to estimate the Flame Transfer Function (FTF). From the pressure time series, we obtain the instability frequency and the amplitudes of the pressure fluctuations. The first optimization step is based on an acoustic network model of the combustor: the TSO approach uses the pressure data to find a simplified n-tau model, which reproduces the unstable combustion process. This step has already been validated for the Rijke tube, a laminar and a turbulent flame in Ghani et al. (2020). The major contribution of this work adds a second optimization loop to extend the n-tau model to the FTF: the gain and phase obtained by the n-tau model are used to fit a distributed time delay model. Our proposed method is applied to a turbulent, premixed, swirl-stabilized flame operated at two power ratings and two swirler positions. The model results for the FTFs are compared against experimentally measured FTFs for these four configurations and all agree well. To the best of our knowledge, this is the first attempt to estimate the complex-valued FTF solely based on pressure measurements. Compared to classical methods for FTF determination such as experimental tests or numerical simulations, our TSO approach is fast and accurate. The proposed framework is suitable for perfectly-premixed flames stabilized by a swirling flow field, requires two pressure sensors and is easy to implement.

A unified framework for acoustic instabilities based on the tailored Green’s function

This paper presents a unified framework to analyse the role of thermoacoustic and aeroacoustic sources in a one-dimensional acoustic resonator. We introduce two forms of the tailored Green’s function: one that satisfies the boundary conditions of the acoustic pressure and one that satisfies those of the acoustic velocity. We consider acoustic sources, which can enter into a feedback loop with the acoustic field in the resonator and hence give rise to an instability. We assume that they are compact, and we model them as a monopole, dipole or a combination of the two. The reaction of a source to the acoustic field can be described in the time-domain (e.g. by a relationship between the volume outflow of a monopole to the local acoustic velocity at earlier times) or in the frequency domain (e.g. by a transfer matrix). Governing equations for the complete acoustic system are derived from the 1-D version of the acoustic analogy equation, using a Green’s function approach. They turn out to be two coupled Volterra equations for the acoustic pressure and velocity. Their solution is straightforward and numerically inexpensive. We demonstrate our approach with three diverse examples: a nonlinear Rijke tube, a whistling orifice, and a flame anchored on an orifice plate with bias flow. Our framework can be applied to various configurations, in particular to combustion test rigs, which consist of a duct-shaped resonator housing acoustically active elements, such as a flame and an orifice with bias flow.

Active control of combustion oscillation with active disturbance rejection control (ADRC) method

This research aims to develop a novel method for the active control of combustion oscillation based on active disturbance rejection control (ADRC). The effectiveness of parameter tuning and the controller in suppressing combustion oscillation was studied by simulation and experiments. The control performance was verified using a Rijke tube and turbulent premixed swirling flame within a model combustor. For the former, the results show that the first-order ADRC can rapidly mitigate the limit cycle oscillation and maintain the effectiveness of the control when the oscillation frequency changes. For the latter, a system identification method based on an acoustic network was adopted. Combining the identification results with a thermoacoustic network, the formation process of the limit cycle oscillation and the effectiveness of the first-order ADRC were simulated by SIMULINK. The experimental results show that the first-order ADRC performs better than the proportional–integral–derivative (PID) controller. At the same time, the first-order ADRC and second-order ADRC achieve good outcomes when the equivalence ratio changes. However, the second-order ADRC exhibits a better pressure-suppression effect and smaller controller output voltage.

System Parameter Identification of a Colored-Noise-Driven Rijke Tube Simulator

Abstract In this paper, we report an experimental investigation on the influence of colored noise (generated by the Ornstein–Uhlenbeck (OU) process) on thermoacoustic coupling in an electro-acoustic Rijke tube simulator. In the absence of noise, the simulator exhibits subcritical Hopf bifurcation. Although noise in a practical system has a finite correlation time, yet the system identification methods are based on the assumption of white noise. In this study, we investigate the effects of correlation time, and intensity of colored noise on the estimation of the growth rates of acoustic oscillations determined using Fokker–Planck equation in stable, bistable, and linearly unstable regions. Subsequently, we compare the findings against results obtained considering white noise approximations. We report the observed deviation of the estimated growth rates from the actual values as a function of noise intensity and correlation time. We find that with the colored noise model, the deviation in the estimated growth rates lies within the range of 0–10% compared to the deviation of 5–25% observed considering the white noise approximation. We also report that increasing noise amplitudes leads up to a deviation of approximately 30% in the estimated growth rates from the actual values.

Self-coupling: an effective method to mitigate thermoacoustic instability

The presence of undesirable large-amplitude self-sustained oscillations in combustors resulting from thermoacoustic instability can lead to performance loss and structural damage to components of gas turbine and rocket engines. Traditional feedback controls to mitigate thermoacoustic instability possess electromechanical components, which are expensive to maintain regularly and unreliable in the harsh environments of combustors. In this study, we demonstrate the quenching of thermoacoustic instability through self-coupling—a method wherein a hollow tube is used to provide acoustic self-feedback to a thermoacoustic system. Through experiments and modeling, we identify the optimal coupling conditions for attaining amplitude death, i.e., complete suppression of thermoacoustic instabilities, in a horizontal Rijke tube. We examine the effect of both system and coupling parameters on the occurrence of amplitude death. We thereby show that the parametric regions of amplitude death occur when the coupling tube length is close to an odd multiple of the length of the Rijke tube. The optimal location to place the coupling tube for achieving amplitude death is near the antinode of the acoustic pressure standing wave in the Rijke tube. Furthermore, we find that self-coupling mitigates thermoacoustic instability in a Rijke tube more effectively than mutual coupling of two identical Rijke tubes. Thus, we believe that self-coupling can prove to be a simple, cost-effective solution for mitigating thermoacoustic instability in gas turbine and rocket combustors.

Experimental Investigation on Self-Excited Thermoacoustic Instability in a Rijke Tube

The experimental investigations into the thermoacoustic instability in a Rijke tube are presented. In order to capture the dynamics of the temperature, a single-ended tunable diode laser absorption spectroscopy (TDLAS) technique was developed, with a measurement rate of 5 kHz. The temperature was found to fluctuate periodically at a dominant frequency of 230 Hz, corresponding to the fundamental frequency of the Rijke tube used in the experiment. The flame chemiluminescence was detected by a high-speed camera to demonstrate flame response to thermoacoustic instability. It was evident that the flame front stretched regularly and had jagged edges. To quantitate the fluctuations of chemiluminescence intensity, the relative area was defined. According to the result, the intensity also oscillated at 230 Hz. Furthermore, the same feature was found in regard to pressure at the exit of the Rijke tube. Compared with temperature and chemiluminescent intensity, the pressure oscillations presented the most approximate standard waveform, as they suffered the least disruptions. The results indicated that the dominant frequencies of temperature, chemiluminescent intensity and pressure were consistent, in accordance with the fundamental frequency of the Rijke tube in the experiment. In addition, etalon effects on the TDLAS signals were mitigated efficiently by a lowpass filter.

Rijke tube: A nonlinear oscillator.

Dynamical systems theory has emerged as an interdisciplinary area of research to characterize the complex dynamical transitions in real-world systems. Various nonlinear dynamical phenomena and bifurcations have been discovered over the decades using different reduced-order models of oscillators. Different measures and methodologies have been developed theoretically to detect, control, or suppress the nonlinear oscillations. However, obtaining such phenomena experimentally is often challenging, time-consuming, and risky mainly due to the limited control of certain parameters during experiments. With this review, we aim to introduce a paradigmatic and easily configurable Rijke tube oscillator to the dynamical systems community. The Rijke tube is commonly used by the combustion community as a prototype to investigate the detrimental phenomena of thermoacoustic instability. Recent investigations in such Rijke tubes have utilized various methodologies from dynamical systems theory to better understand the occurrence of thermoacoustic oscillations and their prediction and mitigation, both experimentally and theoretically. The existence of various dynamical behaviors has been reported in single and coupled Rijke tube oscillators. These behaviors include bifurcations, routes to chaos, noise-induced transitions, synchronization, and suppression of oscillations. Various early warning measures have been established to predict thermoacoustic instabilities. Therefore, this review article consolidates the usefulness of a Rijke tube oscillator in terms of experimentally discovering and modeling different nonlinear phenomena observed in physics, thus transcending the boundaries between the physics and the engineering communities.

Dynamical states and bifurcations in coupled thermoacoustic oscillators.

The emergence of rich dynamical phenomena in coupled self-sustained oscillators, primarily synchronization and amplitude death, has attracted considerable interest in several fields of science and engineering. Here, we present a comprehensive theoretical study on the manifestation of these exquisite phenomena in a reduced-order model of two coupled Rijke tube oscillators, which are prototypical thermoacoustic oscillators. We characterize the dynamical behaviors of two such identical and non-identical oscillators by varying both system parameters (such as the uncoupled amplitudes and the natural frequencies of the oscillators) and coupling parameters (such as the coupling strength and the coupling delay). The present model captures all the dynamical phenomena-namely, synchronization, phase-flip bifurcation, amplitude death, and partial amplitude death-observed previously in experiments on coupled Rijke tubes. By performing numerical simulations and deriving approximate analytical solutions, we systematically decipher the conditions and the bifurcations underlying the aforementioned phenomena. The insights provided by this study can be used to understand the interactions between multiple cans in gas turbine combustors and develop control strategies to avert undesirable thermoacoustic oscillations in them.

Thermoacoustic stabilization of combustors with gradient-augmented Bayesian optimization and adjoint models

Bayesian optimization (BO) is a global optimization algorithm well-suited for multimodal functions that are costly to evaluate, e.g. quantities derived from computationally expensive simulations. Recent advances have made it possible to scale BO to high-dimensional functions and accelerate its convergence by incorporating derivative information. These developments have laid the groundwork for a productive interplay between BO and adjoint solvers, a tool to cheaply obtain gradients of objective functions w.r.t. tunable parameters in a simulated physical system. In thermoacoustics, adjoint-based optimization has previously been applied to Helmholtz solvers and low-order network models to find optimally stable combustor configurations. These studies have used conjugate gradient or quasi-Newton optimizers which can get stuck in local optima and may require many evaluations of the underlying model to find a good optimum. In this paper, we propose using gradient-augmented BO to optimize adjoint models. We consider two test cases from the thermoacoustics literature: optimizing design parameters in a 1D adjoint Helmholtz model of a Rijke tube and geometry optimization in a low-order network model of a longitudinal combustor. We show that compared to BFGS, a standard quasi-Newton method, our gradient-enhanced BO arrives at multiple, more optimal configurations using considerably fewer evaluations of the solver. This approach holds great promise for efficient thermoacoustic stabilization when designing using expensive 3D adjoint Helmholtz solvers.

Generating a physics-based quantitatively-accurate model of an electrically-heated Rijke tube with Bayesian inference

We perform 7000 experiments at 175 stable operating points on an electrically-heated Rijke tube. We pulse the flow and measure the acoustic response with eight probe microphones distributed along its length. We assimilate the experimental data with Bayesian inference by specifying candidate models and calculating their optimal parameters given prior assumptions and the data. We model the long timescale behaviour with a 1D pipe flow model driven by natural convection into which we assimilate data with an Ensemble Kalman filter. We model the short timescale behaviour with several 1D thermoacoustic network models and assimilate data by minimizing the negative log posterior likelihood of the parameters of each model, given the data. For each candidate model we calculate the uncertainties in its parameters and calculate its marginal likelihood (i.e. the evidence for that model given the data) using Laplace’s method combined with first and second order adjoint methods. We rank each model by its marginal likelihood and select the best model for each component of the system. We show that this process generates a model that is physically-interpretable, as small as possible, and quantitatively accurate across the entire operating regime. We show that, once the model has been selected, it can be trained on little data and can extrapolate successfully beyond the training set. Matlab code is provided so that the reader can experiment with their own models.

Quenching and amplification of thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling.

We numerically explore the quenching and amplification of self-excited thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling. On applying either type of coupling separately, we find that the presence of nonidentical heater powers can shrink the regions of amplitude death in both oscillators, while producing new regions of amplitude amplification in the weaker oscillator. We find that the magnitude of amplitude amplification grows with the heater power mismatch and with the total power input. These effects are also present when both types of coupling are applied simultaneously. This study highlights the critical role that nonidentical thermal loads can play in determining the amplitude response of coupled thermoacoustic systems, facilitating the design of control strategies for coupled oscillatorlike devices such as gas turbines.

Numerical investigations on the beating behavior of self-excited combustion instability in a hydrogen-fueled Rijke type combustor

Beating, a specific oscillation mode of combustion instability, occurred in a premixed hydrogen fueled Rijke tube was investigated. A two-dimensional transient numerical model for the self-excited combustion instability was developed and well validated. In a beating cycle, the amplitude of pressure oscillations increased to 1400 Pa and then decreased to 70 Pa, exhibiting an obvious feature of low frequency modulation. The joint time-frequency analysis showed that the dominant frequency shifted from 230 Hz to 330 Hz over time at each cycle. To shed lights on the inherent mechanisms of beating instability, the coupling processes between the combustion and flow dynamics were investigated. At each beating cycle, the shapes of flame were relatively stable in growing stage, while strongly fluctuated in transition stage and recovered to stable state in decaying stage. Notably, due to the large amplitude of pressure oscillations, an obvious recirculation zone was formed around the flame and sucked the high temperature exhausted gas to the flame root during the transition stage. Accordingly, the high temperature area was formed around the flame and affected the oscillation mode. By employing the proper orthogonal decomposition analysis, the apparent oscillations at 1 Hz were observed in the modes of heat release rate, velocity vectors and temperature. Later, it was found that with the increase of the amplitude of pressure oscillations, the mass flow rate of air supply at tube inlet dramatically decreased, which reduced the oxygen around the flame and changed the fluctuation mode of heat release rate. The coupling process was also identified by the phase difference and Rayleigh index.

Determination of the heat conduction transfer function within the thermoacoustic instability limit cycle in a Rijke tube

In this work, experimental measurements of the heat conduction transfer function (HCTF) within the limit cycle of thermoacoustic instabilities were conducted associated with a feedback controller. The responses of the heat conduction from the heater to the air in the vicinity to the oncoming flow disturbance can be described as the HCTF. When self-excited thermoacoustic instability occurs, it is not possible to measure the HCTF. The measurement experiment is twofold. Firstly, the active feedback control is applied to suppress the self-excited thermoacoustic oscillation in the Rijke tube. Secondly, the two-microphone method is applied to achieve the measurement of the HCTF at different working conditions. Experimental results show that the mean air flow velocity has an effect on the attenuation time of the thermoacoustic oscillation. Both of the mean air flow velocity and the electric-heating power have a significant effect on HCTFs.

Adaptive Neural Network Control of Thermoacoustic Instability in Rijke Tube: A Fully Actuated System Approach

Adaptive Neural Network Control of Thermoacoustic Instability in Rijke Tube: A Fully Actuated System Approach

Minimal energy thresholds for triggering in the Rijke tube: the effect of the time delay

Triggering is the process by which a linearly stable thermoacoustic system can reach self-sustained oscillations. This nonlinear phenomenon is activated only for sufficiently large amplitudes of perturbations to the equilibrium state. In this work, using a nonlinear variational optimisation method coupled with energy bisection, we compute the minimal thresholds for triggering in the Rijke tube. In particular, extending previous works, we take into account the effect of the time delay by optimising not only the perturbations at initial time, but also the velocity at the hot-wire position in the time-delay interval. We found that, for sufficiently large time delays, the nonlinearity linked to the delayed flow velocity bears a strong potential for energy growth, leading to transient amplifications of the energy reaching ${O}(10^{2})$, two orders of magnitude larger than those reported in previous studies. Notably, the gain increases with the time delay, but decreases with the initial energy of the perturbation, thus reaching very high values close to the triggering threshold of the system. The minimal energy for triggering self-sustained oscillations achieves energy values as low as ${O}(10^{-4})$, two orders of magnitude smaller than previous estimates. This indicates that, for thermoacoustic systems characterised by a non-negligible time delay, taking into account the effect of the time-delayed variables, as well as the system nonlinearity, is crucial for correctly evaluating the triggering energy thresholds.

Active Control of Multiple Neural Networks for Oscillating Combustion

A multiple neural network controller is proposed and demonstrated to suppress the pressure oscillation of the Rijke tube acoustic network. This controller consists of three modules including two separate neural networks, i.e., the neural network of controlled object that is pretrained before control and the neural network of controller that is trained in real time during the control process. This controller can identify the characteristics of oscillating combustion, achieve adaptive output, and extend the applicability and expansibility. Results show that multiple neural network controller can suppress the pressure oscillation in different oscillating stages using fuel valve or loudspeaker as actuators. When the exact mathematical model of the controlled object is difficult to obtain, it is effective to take a zero-dimensional simplified model with similar oscillating characteristics for prototype controller, and then actively control the real controlled object through parameter migration. It is further demonstrated that this controller is insensitive to noise. In addition, delay correction is achieved by adding sensor and actuator delay modules to the original controller to offset the effects of system delay.