Session M06: Thermoacoustics

Exceptional points in the thermoacoustic spectrum of longitudinal configurations

Exceptional points (EPs) are found in the spectrum of a longitudinal thermoacoustic system as the parameters of the flame transfer function are varied. EPs occur when two eigenvalue trajectories with different physical nature collide: One trajectory originates from an acoustic mode, whereas the other trajectory originates from an intrinsic thermoacoustic mode. The system's sensitivity to changes in the parameters is infinite because the EP is a square-root branch point of the determinant. Traditional iterative methods to find eigenvalues are not robust in the vicinity of EPs. A contour-integration-based approach is deployed to enable the robust computation of the thermoacoustic spectrum. The appropriate expansion at the EP, which can be used to calculate sensitivities to the system’s parameters for passive control, is in fractional powers (Puiseux series) of the parameters. The large sensitivity at an EP may help design new control schemes to mitigate thermoacoustic instabilities with small changes in the design variables. The existence of EPs in thermoacoustic systems has implications for physical understanding, computing, modelling and control.
  

Higher-order degenerate points in annular thermoacoustic systems

Annular combustors, such as those typically found in gas turbines for power generation and aeroengines, nominally feature discrete rotational symmetry. Most of the eigenvalues associated with azimuthal thermoacoustic modes in these systems are therefore twofold degenerate, with a corresponding two-dimensional eigenspace. Moreover, as we have recently shown, exceptional points, which are manifestations of defective eigenvalues, are generic features of thermoacoustic systems and emerge from the collision of an acoustic and an intrinsic mode. In a longitudinal, single-flame system, the exceptional eigenvalue has algebraic and geometric multiplicity two and one, respectively. However, in an annular system with discrete rotational and reflection symmetry, lower-order azimuthal eigenvalues -- acoustic as well as intrinsic -- are already twofold degenerate. Consequently, at an exceptional point in an annular system, the defective eigenvalue corresponds to a higher-order degeneracy with algebraic multiplicity four and geometric multiplicity two. We illustrate the formation of exceptional points in annular thermoacoustic systems and discuss implications for modeling and stability analysis.

  

Data assimilation and parameter estimation of thermoacoustic instabilities in a ducted premixed flame

Thermoacoustic instabilities are a persistent challenge in the design of jet and rocket engines. The time-accurate calculation of thermoacoustic instabilities is challenging due to the presence of both aleatoric and epistemic uncertainties, as well as the extreme sensitivity to small changes in certain parameters. Our thermoacoustic system is a vertical Rijke tube containing a premixed Bunsen flame. We conduct experiments and high-fidelity numerical simulations. We then perform data assimilation and parameter estimation using the ensemble Kalman filter to estimate the state and parameters of a simple G-equation model of the flame. Data assimilation provides an optimal estimate of the true state of a system, and improves the predicted shape and location of the flame. Parameter estimation uses the data to find a maximum-likelihood set of parameters for the model while simultaneously quantifying their uncertainty. At the same time, we identify deficiencies in the model with this approach. This process renders the G-equation model quantitatively accurate over the tested range of conditions. As a physics-based model, we expect the G-equation model to extrapolate smoothly. This is going to be tested in future work.   

A Bayesian Approach for Predicting Thermoacoustic Oscillations in an Electrically-Heated Rijke Tube

Predicting and eliminating thermoacoustic oscillations is a significant challenge in gas turbine design. Here we combine a thermoacoustic experiment with a thermoacoustic model and use data assimilation to infer the parameters of the model, rendering it predictive. The experiment is a vertical Rijke tube containing an electric heater (up to 300 Watts). The heater drives a base flow via natural convection, and thermoacoustic oscillations via velocity-driven heat release fluctuations. The growth/decay rates and frequencies of these oscillations are measured every few seconds. There are two models: one for the base flow and one for the acoustics. Both are unsteady. The parameters of the base flow model (Nusselt numbers and pressure loss coefficients) are estimated from many thousand measurements using an ensemble Kalman filter that accounts for both experimental and state and parameter errors. The parameters of the acoustic model are inferred by regression. This study shows that, with thorough Bayesian inference, a simple model with a few parameters can become a predictive model. The process reveals deficiencies in the model and, when combined with physical insight, shows how to improve it. This study proves the concept for small systems and prepares the ground for complex systems.   

Experimentation with Porous Inert Material to Control Thermoacoustic Instabilities in Lean Direct Injection Combustion

Stringent environmental regulations require more efficient and cleaner jet engines for civil aviation applications.  An interesting development has been the introduction of lean direct injection (LDI) combustion techniques. Previous research has proven that the LDI combustion yields lower nitrogen oxides and carbon monoxide emissions in gas-turbine jet engines. However, thermoacoustic instabilities remain an issue with LDI combustion. Previous research conducted at the University of Alabama has indicated that the insertion of a Porous Inert Material (PIM) can reduce such instabilities. Positioned on the dump plane, the PIM interferes with the flow field in the corner and central recirculation regions. In this study, a single LDI injector was tested without the PIM to ascertain the equivalence ratio at which a strong instability occurs and establish a baseline case. Then, experiments at several test conditions of interest are performed to determine the effect of the PIM. Measurements include acoustic probes to record the sound spectra and OH*- chemiluminescence imaging to characterize the reaction zone. PIM is shown to improve the acoustic performance although geometric optimization would be necessary to achieve the best results.