scholarly journals Bistability and Triggering Analysis of Thermoacoustic Instability in a Horizontal Rijke Tube

2019 ◽  
Vol 37 (1) ◽  
pp. 48-56
Author(s):  
Jianchang Feng ◽  
Wen Ao ◽  
Peijin Liu
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Time Lag ◽  
Damping Coefficient ◽  
Nonlinear Flow ◽  
Velocity Perturbation ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
Heater Power ◽  
Bistable Region

Dynamical systems theory has been often employed to study nonlinear flow and flame dynamics in combustion systems. However, the corresponding studies using nonlinear dynamics to analyze the Rijke tube thermoacoustic system are still occasional. Little study has been performed to elucidate the characteristics of triggering phenomenon in the bistable region of the thermoacoustic system. In this regard, the main objectives of the present research are to contribute analysis for the lack of literature in these areas, especially to study the bistability and triggering properties of a thermoacoustic system. The thermoacoustic model of a horizontal Rijke tube is firstly established. The governing equations are expanded and solved by using Galerkin method. The analysis of the system is carried out by using nonlinear dynamics theory. Linear and nonlinear stability boundaries for the variation of non-dimensional heater power, damping coefficient and the relative heater location are obtained for different values of non-dimensional time lag in the system. Regions of global stability, global instability and bistability are characterized. The bistable region in the relative heater location is distributed symmetrically with xf=0.25. It is observed that the bistable region in the relative heater location firstly decreases with an increase in the non-dimensional time lag, reaching a minimum value at a certain critical value of τ, then increases. The situation for the bistable region in the damping coefficient presents a reverse variation, And the bistable region reach the maximum at τ=0.5. The triggering phenomenon and limit cycle of the system in the bistable region are studied. The critical triggering values are determined with the changes of the non-dimensional heater power, the damping coefficient and the relative heater location. The critical triggering value of velocity perturbation decreases with the increase of non-dimensional heater power, whereas an increasing trend is observed with the increase of damping coefficient. Interestingly, the critical triggering value firstly decreases and then increases with the increase of the relative heater location. The variation of critical triggering value for pressure perturbation is found to correspond with velocity perturbation. In the bistable region, the amplitude and frequency of the steady limit cycle oscillation of the system are independent of the initial perturbation values, but the perturbation value has strong effect on the duration needed to achieve the steady limit cycle, and the time required for the system to reach the limit cycle under the perturbation of U1=0.4 is about 3 times longer than that of U1=0.8.

scholarly journals Thermoacoustic instability – a dynamical system and time domain analysis

2014 ◽  
Vol 753 ◽  
pp. 448-471 ◽  
Author(s):  
Taraneh Sayadi ◽  
Vincent Le Chenadec ◽  
Peter J. Schmid ◽  
Franck Richecoeur ◽  
Marc Massot
Keyword(s):  
Time Series ◽  
Heat Source ◽  
Limit Cycle ◽  
Nonlinear Regime ◽  
Galerkin Projection ◽  
Nonlinear Flow ◽  
Mean Flow ◽  
Retarded Time ◽  
Thermoacoustic Instability ◽  
Linear And Nonlinear

AbstractThis study focuses on the Rijke tube problem, which includes features relevant to the modelling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source and nonlinearities. This thermoacoustic system features both linear and nonlinear flow regimes with complex dynamical behaviour. In order to synthesize accurate time series, we tackle this problem from a numerical point of view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness – in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by using the projection method proposed by Jarlebring (PhD thesis, Technische Universität Braunschweig, 2008), which alleviates the problems arising from linearization with respect to the retarded time. The results are then compared with the analytical solution of the undamped system and with the results obtained from Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We demonstrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case predicted by using Galerkin expansion show spurious oscillations resulting from the Gibbs phenomenon. Finally, time series in the fully nonlinear regime, where a limit cycle is established, are analysed and dominant modes are extracted. By comparing the modes of the linear regime to those of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching, which appear as the nonlinearities become significant and ultimately affect the form of the limit cycle. Analysis of the saturated limit cycles shows the presence of higher-frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not exhibited when the coupling between different frequencies is not accounted for. In conclusion, a dedicated solver for capturing thermoacoustic instability is proposed and methods for analysing linear and nonlinear regions of the resulting time series are introduced.

Effect of Spray-Wall Interaction On Thermoacoustic Instability Prediction by Flame Transfer Function and the Convective Time Delay Method

2021 ◽  
Author(s):  
Sheng Meng ◽  
Man Zhang
Keyword(s):  
Time Delay ◽  
Transfer Function ◽  
Numerical Models ◽  
Time Lag ◽  
Interaction Model ◽  
Phase Lag ◽  
Thermoacoustic Instability ◽  
Spray Flame ◽  
Wall Interaction ◽  
Definition Of

Abstract This study numerically investigates the effect of spray-wall interactions on thermoacoustic instability prediction. The LES-based flame transfer function (FTF) and the convective time delay methods are used by combining the Helmholtz acoustic solver to predict a single spray flame under the so-called slip and film spray-wall conditions. It is found that considering more realistic film liquid and a wall surface interaction model achieves a more accurate phase lag in both of the time lag evaluations compared to the experimental results. Additionally, the results show that a new time delay exists between the liquid film fluctuation and the unsteady heat release, which explains the larger phase value in the film spray-wall condition than in the slip condition. Moreover, the prediction capability of the FTF framework and the convective time delay methodology in the linear regime are also presented. In general, the instability frequency differences predicted using the FTF framework under the film condition are less than 10 Hz compared with the experimental data. However, an underestimation of the numerical gain value leads to requiring a change in the forcing position and an improvement in the numerical models. Due to the ambiguous definition of the gain value in the convective time delay method, this approach leads to arbitrary and uncertain thermoacoustic instability predictions.

Modelling nonlinear thermoacoustic instability in an electrically heated Rijke tube

2011 ◽  
Vol 680 ◽  
pp. 511-533 ◽  
Author(s):  
SATHESH MARIAPPAN ◽  
R. I. SUJITH
Keyword(s):  
Mach Number ◽  
Heat Source ◽  
Asymptotic Analysis ◽  
Heat Release ◽  
Acoustic Field ◽  
Release Rate ◽  
Governing Equations ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
Acceleration Term

An analysis of thermoacoustic instability is performed for a horizontal Rijke tube with an electrical resistance heater as the heat source. The governing equations for this fluid flow become stiff and are difficult to solve by the computational fluid dynamics (CFD) technique, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore, an asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. The unknown variables are expanded in powers of Mach number. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. In this analysis, the coupling between the acoustic field and the unsteady heat release rate from the heater appears from the asymptotic analysis. Furthermore, a non-trivial additional term, referred to as the global-acceleration term, appears in the momentum equation of the hydrodynamic zone, which has serious consequences for the stability of the system. This term can be interpreted as a pressure gradient applied from the acoustic onto the hydrodynamic zone. The asymptotic stability of the system with the variation of system parameters is presented using the bifurcation diagram. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD techniques for the hydrodynamic zone. The results confirm the importance of the global-acceleration term. Bifurcation diagrams obtained from the simulations with and without the above term are different. Acoustic streaming is shown to occur during the limit cycle and its effect on the unsteady heat release rate is discussed.

One-dimensional model for the Rijke tube

1989 ◽  
Vol 202 ◽  
pp. 83-96 ◽  
Author(s):  
C. Nicoli ◽  
P. Pelcé
Keyword(s):  
Heat Transfer ◽  
Transfer Function ◽  
Dimensional Model ◽  
Unsteady Heat Transfer ◽  
Order Unity ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
One Dimensional ◽  
The Stability ◽  
Stability Limits

We develop a simple model in which longitudinal, compressible, unsteady heat transfer between heater and gas is computed in the small-Mach-number limit. This calculation is used to determine the transfer function of the heater, which plays an important role in the stability limits of the thermoacoustic instability of the Rijke tube. The transfer function is determined analytically in the limit of small expansion parameter γ, and numerically for γ of order unity. In the case ρμ/cp = constant, an analytical solution can be found.

scholarly journals Stability and Limit Cycles of a Nonlinear Damper Acting on a Linearly Unstable Thermoacoustic Mode

2018 ◽  
Vol 141 (5) ◽  
Author(s):  
Claire Bourquard ◽  
Nicolas Noiray
Keyword(s):  
Growth Rate ◽  
Combustion Chamber ◽  
Limit Cycle ◽  
Nonlinear Effects ◽  
Coupled System ◽  
High Amplitude ◽  
Van Der Pol Oscillator ◽  
Mean Velocity ◽  
Thermoacoustic Instability ◽  
Numerical Predictions

The resonant coupling between flames and acoustics is a growing issue for gas turbine manufacturers, which can be reduced by adding acoustic dampers on the combustion chamber walls. Nonetheless, if the engine is operated out of the stable window, the damper is exposed to high-amplitude acoustic levels, which trigger unwanted nonlinear effects. This work provides an overview of the dynamics of this coupled system using a simple analytical model, where a perfectly tuned damper is coupled to the combustion chamber. The damper, crossed by a purge flow in order to prevent hot gas ingestion, is modeled as a nonlinearly damped harmonic oscillator. The combustion chamber featuring a linearly unstable thermoacoustic mode is modeled as a Van der Pol oscillator. Analyzing the averaged amplitude equations gives the limit cycle amplitudes as function of the growth rate of the unstable mode and the mean velocity through the damper neck. Experiments are also performed on a simple rectangular cavity, where the thermoacoustic instability is mimicked by an electro-acoustic instability. A feedback loop is built, through which the growth rate of the instability can be controlled. A Helmholtz damper is added to the cavity and tuned to the mode of interest. The stabilization capabilities of the damper and the amplitude of the limit cycle in the unstable cases are in good agreement between the experiments and the analytical and numerical predictions, underlining the potentially dangerous behavior of the system, which should be taken into account for real engine cases.

LIMIT CYCLE OSCILLATION PREDICTION AND CONTROL DESIGN METHOD FOR AEROELASTIC SYSTEM BASED ON NEW NONLINEAR REDUCED ORDER MODEL

2011 ◽  
Vol 08 (01) ◽  
pp. 77-90 ◽  
Author(s):  
GANG CHEN ◽  
YUE-MING LI ◽  
GUI-RONG YAN
Keyword(s):  
Limit Cycle ◽  
Active Control ◽  
Reduced Order Model ◽  
Nonlinear Flow ◽  
Limit Cycle Oscillation ◽  
Control Law ◽  
Order Model ◽  
Reduced Order ◽  
Large Perturbation ◽  
Cycle Oscillation

When the amplitude of the oscillation of the unsteady flow is large or there is large perturbation relative to the mean background flow, the traditional proper orthogonal decomposition/reduced order model (POD/ROM) based on linearized time or frequency domain small disturbance solvers cannot capture the main nonlinear features well such as limit cycle oscillation (LCO), which is very dangerous for the structure. Therefore, the traditional linear ROMs are not good enough for limit cycles prediction and active control law design. A new nonlinear ROM based on dynamically nonlinear flow equation NPOD/ROM was investigated. The nonlinear second-order snapshot equation in time domain for POD basis construction is obtained from the Taylor series expansion of the flow solver. The simulation results indicate that the NPOD/ROM can capture LCO very well and is also very convenient for active control law design, while the traditional POD/ROM lose effectiveness.

Nonlinear Heat-Release/Acoustic Model for Thermoacoustic Instability in Lean Premixed Combustors

1999 ◽  
Vol 121 (3) ◽  
pp. 415-421 ◽  
Author(s):  
A. A. Peracchio ◽  
W. M. Proscia
Keyword(s):  
Heat Release ◽  
Limit Cycle ◽  
Gas Turbines ◽  
Pressure Fluctuations ◽  
Parametric Model ◽  
Operating Conditions ◽  
Cycle Frequency ◽  
Thermoacoustic Instability ◽  
Lean Premixed ◽  
Industrial Gas Turbines

Lean premixed combustors, such as those used in industrial gas turbines to achieve low emissions, are often susceptible to the thermoacoustic combustion instabilities, which manifest themselves as pressure and heat release oscillations in the combustor. These oscillations can result in increased noise and decreased durability due to vibration and flame motion. A physically based nonlinear parametric model has been developed that captures this instability. It describes the coupling of combustor acoustics with the rate of heat release. The model represents this coupling by accounting for the effect of acoustic pressure fluctuations on the varying fuel/air ratio being delivered to the flame, causing a fluctuating heat release due to both fuel air ratio variations and flame front oscillations. If the phasing of the fluctuating heat release and pressure are proper, an instability results that grows into a limit cycle. The nonlinear nature of the model predicts the onset of the instability and additionally captures the resulting limit cycle. Tests of a lean premixed nozzle run at engine scale and engine operating conditions in the UTRC single nozzle rig, conducted under DARPA contract, exhibited instabilities. Parameters from the model were adjusted so that analytical results were consistent with relevant experimental data from this test. The parametric model captures the limit cycle behavior over a range of mean fuel air ratios, showing the instability amplitude (pressure and heat release) to increase and limit cycle frequency to decrease as mean fuel air ratio is reduced.

scholarly journals A New Megastable Chaotic Oscillator with Blinking Oscillation terms

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Dhinakaran Veeman ◽  
Hayder Natiq ◽  
Nadia M. G. Al-Saidi ◽  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Chaotic Systems ◽  
High Sensitivity ◽  
Sample Entropy ◽  
Chaotic Oscillator ◽  
Hidden Attractors ◽  
Coexisting Attractors ◽  
Initial Values

Recently, megastable systems have grabbed many researchers’ interests in the area of nonlinear dynamics and chaotic systems. In this paper, the oscillatory terms’ coefficients of the simplest megastable oscillator are forced to blink in time. The forced system can generate an infinitive number of hidden attractors without changing parameters. The behavior of these hidden attractors can be chaotic, tori, and limit cycle. The attractors’ topology of the system seems unique and looks like picture frames. Besides, the existence of different coexisting attractors with different kinds of behaviors reflects the system's high sensitivity. Using the sample entropy algorithm, the system’s complexity for different initial values is assessed. In addition, the circuit of the introduced forced system is designed, and the possibility of implicating the system with analog elements is investigated.

Transfer Learning for Detection of Combustion Instability via Symbolic Time Series Analysis

2021 ◽  
Author(s):  
Chandrachur Bhattacharya ◽  
Asok Ray
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
Transfer Learning ◽  
Sufficient Information ◽  
Target Domain ◽  
Pressure Oscillations ◽  
Rijke Tube ◽  
Thermoacoustic Instability ◽  
Series Analysis ◽  
Symbolic Time Series Analysis

Abstract Transfer learning (TL) is a machine learning (ML) tool where the knowledge, acquired from a source domain, is 'transferred' to perform a task in a target domain that has (to some extent) a similar setting. The underlying concept does not require the ML method to analyse a new problem from the beginning, and thereby both the learning time and the amount of required target-domain data are reduced for training. An example is the occurrence of thermoacoustic instability (TAI) in combustors, which may cause pressure oscillations, possibly leading to flame extinction as well as undesirable vibrations in the mechanical structures. In this situation, it is difficult to collect useful data from industrial combustion systems, due to the transient nature of TAI phenomena. A feasible solution is the usage of prototypes or emulators, like a Rijke tube, to produce largely similar phenomena. This paper proposes symbolic time series analysis (STSA)-based transfer learning, where the key idea is to develop a capability of discrimination between stable and unstable operations of a combustor, based on the time series of pressure oscillations from a data source that contains sufficient information, even if it is not the target regime, and then transfer the learnt models to the target regime. The proposed STSA-based pattern classifier is trained on a previously validated numerical model of a Rijke-tube apparatus. The knowledge of this trained classifier is 'transferred' to classify similar operational regimes in: (i) an experimental Rijke-tube apparatus and (ii) an experimental combustion system apparatus. Results of the proposed transfer learning have been validated by comparison with those of two shallow neural networks (NN)-based TL and another NN having an additional long-short-term-memory (LSTM) layer, which serve as benchmarks, in terms of classification accuracy and computational complexity.

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