Showing papers on "Rijke tube published in 2019"

Coherence resonance and stochastic bifurcation behaviors of simplified standing-wave thermoacoustic systems

Abstract: In this work, noise-induced motions (i.e., external fluctuations) in two modelled standing-wave thermoacoustic systems are studied when these systems are close to the deterministic stability boundary. These systems include (1) open-open (i.e., Rijke-type) and (2) closed-open boundary conditions. It is found from the smooth transitions of the stationary probability density function that the thermoacoustic system is destabilized via stochastic P bifurcation, as the external noise intensity is continuously increased. In addition, the increased noise intensity can shift the hysteresis region, which makes the system more prone to quasi-periodic oscillations, but also reduces the hysteresis area. The noise-induced coherence motions are observed numerically in the open-open system, which is denoted by the occurrence of a bell-shaped signal to noise ratio (SNR). The SNR is shown to be applicable as a precursor. It becomes larger and the optimal noise intensity is decreased as the modelled thermoacoustic system approaches the critical bifurcation point. In addition, coherence resonance is observed in the closed-open system. To validate the findings, experimental studies are conducted on an open-open Rijke tube. Good qualitative agreements are obtained. The present study shed lights on the stochastic and coherence behaviors of the standing-wave thermoacoustic systems with different boundary conditions.

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

Abstract: 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.

Optimizing the Design of a Rijke Tube Experiment for Combustion Stability Model Identifiability

Abstract: This paper presents the design of a thermoacoustically unstable combustor experiment for identifiability. We examine the impact of sensor placement, flame location, and acoustic excitation frequency on the Fisher identifiability of a one-dimensional combustion stability model's parameters. The model uses linear delay differential equations to describe both the acoustics and heat release dynamics in a laboratory combustor called a Rijke tube. We derive analytic expressions for the frequency-domain Fisher identifiability of the model's parameters. This leads to two key insights. First, excitation frequency, flame location, and sensor placement all have a significant impact on parameter identifiability. Second, the optimal excitation frequencies for identifiability are not strong functions of sensor placement but change with flame location. Building on these insights, the paper concludes by using a genetic algorithm to optimize the design of a Rijke tube experiment for thermoacoustic model identifiability.

Devil’s staircases in a thermoacoustic system with sinusoidal excitations

Abstract: The influences of sinusoidal excitations on a Rijke tube system through the introduction of nonlinear dynamics are investigated in detail. The system is governed by one-dimensional partial differential equations, which are addressed by means of the Galerkin procedure and analysed from the viewpoint of nonlinear dynamics. The results show a rich variety of behaviours in the system, including subcritical bifurcation, hysteresis and phase locking. In certain parameter ranges, the solution switches between periodic and quasi-periodic. These ranges mainly depend on the intensity of the heat source in the tube. The phase-locking intervals that correspond to periodic solutions compose a devil’s staircase, whose Lebesgue measure increases as the amplitude of the excitation increases.

Data assimilation in a nonlinear time-delayed dynamical system.

Abstract: 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. 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 state estimates (background) to produce optimal initial conditions. A cost functional is defined to measure the statistical distance between the model output and the measurements from experiments; and the distance between the 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.

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

Abstract: 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 x f =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 U 1 =0.4 is about 3 times longer than that of U 1 =0.8.

Suppression of Thermo-Acoustic Instabilities using Helhmoltz Resonator

Abstract: In combustion chambers due to coupling between unsteady heat release and pressure fluctuations the thermoacoustic instabilities are developed. These instabilities create structural vibration and damage. Therefore, there is a need to develop a technique which can effectively control these instabilities. The present work focuses on passive control using Helmholtz Resonator. The Rijke tube setup has been developed for characterization of thermos-acoustic instabilities. The study has been carried out with open-open ended steel Rijke tube of L/D ratio as 1. The preliminary results show that for LPG pre-mixed burner position at $x$ /L = 0.2, maximum thermos-acoustic instabilities were observed. The instability of 2nd and 3rd mode at a frequency of 500 Hz and 800 Hz respectively were observed for the different air-fuel ratio. The Helmholtz Resonator was designed and placed at the different position from one of the end. The maximum suppression of thermo-acoustic instabilities was observed at Helmholtz resonator position $x$ /L = 0.6. The experimental results show that approximately 10 dB reduction for the 2nd mode of instability was achieved whereas, the Helmholtz Resonator was not effective for 3rd mode as there is a change in heat source location and frequency of instability.

Reduced fidelity analysis of combustion instabilities using flame transfer functions in a nonlinear euler solver

Abstract: Combustion instability, a complex phenomenon observed in combustion chambers is due to the coupling between heat release and other unsteady flow processes. Combustion instability has long been a topic of interest to rocket scientists and has been extensively investigated experimentally and computationally. However, to date, there is no computational tool that can accurately predict the combustion instabilities in full-size combustors because of the amount of computational power required to perform a high-fidelity simulation of a multi-element chamber. Hence, the focus is shifted to reduced fidelity computational tools which may accurately predict the instability by using the information available from the high-fidelity simulations or experiments of single or few-element combustors. One way of developing reduced fidelity computational tools involves using a reduced fidelity solver together with the flame transfer functions that carry important information about the flame behavior from a high-fidelity simulation or experiment to a reduced fidelity simulation. To date, research has been focused mainly on premixed flames and using acoustic solvers together with the global flame transfer functions that were obtained by integrating over a region. However, in the case of rockets, the flame is non-premixed and distributed in space and time. Further, the mixing of propellants is impacted by the level of flow fluctuations and can lead to non-uniform mean properties and hence, there is a need for reduced fidelity solver that can capture the gas dynamics, nonlinearities and steep-fronted waves accurately. Nonlinear Euler equations have all the required capabilities and are at the bottom of the list in terms of the computational cost among the solvers that can solve for mean flow and allow multi-dimensional modeling of combustion instabilities. Hence, in the current work, nonlinear Euler solver together with the spatially distributed local flame transfer functions that capture the coupling between flame, acoustics, and hydrodynamics is explored. In this thesis, the approach to extract flame transfer functions from high-fidelity simulations and their integration with nonlinear Euler solver is presented. The dynamic mode decomposition (DMD) was used to extract spatially distributed flame transfer function (FTF) from high fidelity simulation of a single element non-premixed flame. Once extracted, the FTF was integrated with nonlinear Euler equations as a fluctuating source term of the energy equation. The time-averaged species destruction rates from the high-fidelity simulation were used as the mean source terms of the species equations. Following a variable gain approach, the local species destruction rates were modified to account for local cell constituents and maintain correct mean conditions at every time step of the nonlinear Euler simulation. The proposed reduced fidelity model was verified using a Rijke tube test case and to further assess the capabilities of the proposed model it was applied to a single element model rocket combustor, the Continuously Variable Resonance Combustor (CVRC), that exhibited self-excited combustion instabilities that are on the order of 10% of the mean pressure. The results showed that the proposed model could reproduce the unsteady behavior of the CVRC predicted by the high-fidelity simulation reasonably well. The effects of control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size are also studied. The reduced fidelity model could reproduce the limit cycle amplitude within a few percent of the mean pressure. The successful constraints on the model include good spatial resolution and FTF with all modes up to at least one dominant frequency higher than the frequencies of interest. Furthermore, the reduced fidelity model reproduced consistent mode shapes and linear growth rates that reasonably matched the experimental observations, although the apparent ability to match growth rates needs to be better understood. However, the presence of significant heat release near a pressure node of a higher harmonic mode was found to be an issue. This issue was rectified by expanding the pressure node of the higher frequency mode. Analysis of two-dimensional effects and coupling between the local pressure and heat release fluctuations showed that it may be necessary to use two dimensional spatially distributed local FTFs for accurate prediction of combustion instabilities in high energy devices such as rocket combustors. Hybrid RANS/LES-FTF simulation of the CVRC revealed that it might be necessary to use Flame Describing Function (FDF) to capture the growth of pressure fluctuations to limit cycle when Navier-Stokes solver is used. The main objectives of this thesis are: 1. Extraction of spatially distributed local flame transfer function from the high fidelity simulation using dynamic mode decomposition and its integration with nonlinear Euler solver 2. Verification of the proposed approach and its application to the Continuously Variable Resonance Combustor (CVRC). 3. Sensitivity analysis of the reduced fidelity model to control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size. The goal of this thesis is to contribute towards a reduced fidelity computational tool which can accurately predict the combustion instabilities in practical systems using flame transfer functions, by providing a path way for reduced fidelity multi-element simulation, and by defining the limitations associated with using flame transfer functions and nonlinear Euler equations for non-premixed flames.