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1. Optimizing Thermoacoustic Characterization Experiments for Identifiability Improves Both Parameter Estimation Accuracy and Closed-Loop Controller Robustness Guarantees

   https://doi.org/10.1080/00102202.2020.1858818  

   Chen, Xiaoling; O’Connor, Jacqueline; Fathy, Hosam (January 2021, Combustion Science and Technology)

   null (Ed.)

   This article examines the degree to which optimizing a Rijke tube experiment can improve the accuracy of thermoacoustic model parameter estimation, thereby facilitating robust stability control. We use a one-dimensional thermoacoustic model to describe the combustion dynamics in a Rijke tube. This model contains two unknown parameters that relate velocity perturbations to heat release rate oscillations, namely, a time delay τ and amplification factor β. The parameters are estimated from experiments where the system input is the acoustic excitation from a loudspeaker and the output is the pressure response captured by a microphone. Our work is grounded in the insight that optimizing an experiment’s design for higher Fisher identifiability leads to more accurate parameter estimates. The novel goal of this paper is to apply this insight in the laboratory using a flame-driven Rijke tube setup. For comparison purposes, we conduct a benchmark experiment with a broadband chirp signal as the excitation input. Next, we excite the Rijke tube at two frequencies optimized for Fisher identifiability. Repeats of both experiments show that the optimal experiment achieves parameter estimates with uncertainties at least one order of magnitude smaller than the benchmark. With smaller parameter estimate uncertainties, an LQG controller designed to attenuate combustion instabilities is able to achieve stronger robustness guarantees, quantified in terms of closed-loop structured singular values that account for parameter estimation uncertainty. 

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2. Different perspectives on predicting the thermoacoustic energy conversion response in a Rijke tube

   https://doi.org/10.1063/5.0072193  

   Shelton, Cody_M; Majdalani, Joseph (November 2021, Physics of Fluids)

   In this work, six different perspectives on characterizing the thermoacoustic field in an open-ended Rijke tube are considered and discussed. These begin with a three-pronged approach consisting of theoretical, experimental, and numerical investigations of the Rijke tube's time-dependent field. It is followed by a discussion of effective techniques that rely on either Green's function or differential equation models. Finally, a perturbation expansion is introduced that leverages a naturally occurring small parameter in the open tube configuration. This approach is shown to produce accurate predictions of pressure modal shapes and frequencies for an arbitrary specified temperature distribution. It also leads to a set of linear partial differential equations that can be solved in conjunction with a Green's function expression for the thermoacoustic pressure, velocity, and heat oscillations. In this study, the underlying framework is presented and evaluated for the pressure disturbance only. Another fundamental result includes a similarity parameter, coined the Rijke number, which plays an essential role in driving thermoacoustic oscillations, namely, by relating heat-flux fluctuations to the acoustic velocity and pressure. In this context, we find that the peak value of the energy-flux vector modulus, which stands for the modular product of acoustic velocity and pressure, does indeed occur at the heat source location and increases with the heat power input. 

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3. Synchronization-based model for turbulent thermoacoustic systems

   https://doi.org/10.1007/s11071-023-08368-z  

   Weng, Yue; Unni, Vishnu R.; Sujith, R. I.; Saha, Abhishek (March 2023, Nonlinear Dynamics)

   Abstract We present a phenomenological reduced-order model to capture the transition to thermoacoustic instability in turbulent combustors. Based on the synchronization framework, the model considers the acoustic field and the unsteady heat release rate from turbulent reactive flow as two nonlinearly coupled sub-systems. To model combustion noise, we use a pair of nonlinearly coupled second-order ODEs to represent the unsteady heat release rate. This simple configuration, while nonlinearly coupled to another oscillator that represents the independent sub-system of acoustics (pressure oscillations) in the combustor, is able to produce chaos. Previous experimental studies have reported a route from low amplitude chaotic oscillation (i.e., combustion noise) to periodic oscillation through intermittency in turbulent combustors. By varying the coupling strength, the model can replicate the route of transition observed and reflect the coupled dynamics arising from the interplay of unsteady heat release rate and pressure oscillations. 

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4. Acoustic Pressure Mode Shapes and Frequencies in a Circular Tube for an Arbitrary Temperature Distribution

   https://doi.org/10.2514/6.2023-0768  

   Shelton, Cody; Majdalani, Joseph (January 2023, AIAA SCITECH 2023 Forum)

   In this work, an asymptotic expansion is presented that takes into account a naturally-occurring perturbation parameter in the context of a circular tube with an open-open endpoint configuration. This approach is shown to produce accurate predictions of pressure mode shapes and frequencies for arbitrary temperature distributions that mimic a wide variety of flow heating arrangements including, but not limited to, those associated with a Rijke tube. The underlying formulation consists of two linearly coupled partial differential equations that can be solved simultaneously while using a Green’s function to capture the thermoacoustic pressure. In the present investigation, the strategy leading to an accurate prediction of the unsteady pressure oscillations is fully detailed and then applied to several representative cases. Results pertaining to the pressure oscillations are systematically discussed and compared to other recently developed models in the literature. 

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

   Chen, Xiaoling; Dillen, Evan; Fathy, Hosam; O'Connor, Jacqueline (July 2019, American Controls Conference)

   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. 

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