Session E04: Acoustics: Thermoacoustics (3:10pm - 3:55pm CST)

Suppression of self-excited thermoacoustic oscillations using genetic programming

Genetic programming (GP) is a powerful tool for unsupervised data-driven discovery of closed-loop control laws. In fluid mechanics, it has been used for various purposes, such as to enhance mixing in a turbulent shear layer and to delay flow separation. This model-free control framework is well suited for such complex tasks as it exploits an evolutionary mechanism to propagate the genetics of high-performing control laws from one generation to the next. Here we combine automated experiments with GP to discover model-free control laws for the suppression of self-excited thermoacoustic oscillations in a Rijke tube. Using a GP-based controller linked to a single sensor (a microphone) and a single actuator (a loudspeaker), we rank the performance of all the control laws in a given generation based on a cost function that accounts for the pressure amplitude and the actuation effort. We use a tournament process to breed further generations of control laws, and then benchmark them against conventional periodic forcing optimized via open-loop mapping. We find that, with only minimal input from the user, this GP-based control framework can identify new feedback actuation mechanisms, providing improved control laws for the suppression of self-excited thermoacoustic oscillations.   

Critical Structures in Vortex Dominated Thermoacoustic Systems

In most fluid dynamical systems, large coherent structures are involved in the process of sound production. In addition to sound production due to vortices in the flow-field, vortices also enhance heat production in combustors which in turn amplifies sound production leading to detrimental effects in thermoacoustic systems. The root cause behind this phenomenon can be understood by investigating the spatio-temporal dynamics of coherent structures responsible for sound production. Lagrangian Coherent Structures (LCSs), a sophisticated framework based on the fluid properties, currently gives the best representations of coherent structures in the fluid flow-field. Individual saddle point trajectories referred to as "critical structures", are tracked during thermoacoustic instability regime in a bluff-body combustor as they emerge from the upstream side wall and move towards the top wall of the combustor. These critical trajectories with the informed physics provide an optimal location to implement a passive control strategy. Upon control leading to suppression of tonal sound, we found that the jet pushes the trajectories to divert its path almost parallel to the horizontal axis of the combustor.   

Intermittency route to chaos in a self-excited thermoacoustic system

In nonlinear dynamics, there are three classic routes to chaos, namely the period-doubling route, the Ruelle--Takens--Newhouse route and the intermittency route. The first two routes have previously been observed in self-excited thermoacoustic systems, but the third has not. In this experimental study, we present evidence of the intermittency route to chaos in the self-excited regime of a prototypical thermoacoustic system -- a Rijke tube powered by a laminar premixed flame. We identify the intermittency to be of type II from the Pomeau--Manneville scenario through an analysis of (i) the probability distribution of the quiescent epochs between successive bursts of chaos, (ii) the first return map, and (iii) the recurrence plot. By establishing the last of the three classic routes to chaos, this study strengthens the universality of how strange attractors arise in self-excited thermoacoustic systems, paving the way for the application of generic suppression strategies based on chaos control.   

Physics-informed statistical learning for model comparison and uncertainty quantification of thermoacoustic instability

Thermoacoustic instability remains a persistent challenge in the design of jet and rocket engines. While experiments and high-fidelity simulations are useful for physical understanding, reduced-order models are used in design because high-fidelity simulations are barely feasible, and the acquisition of experimental data is both expensive and difficult. Consequently, when data is assimilated into reduced order-models, these models must be chosen and calibrated carefully. We present a statistical learning framework based on Bayesian regression and Gaussian processes in order to assimilate the data and to evaluate the reduced-order model. The key features of our analysis are: (i) a generative picture of reduced-order models consisting of governing equations, parameters and states; (ii) uncertainty quantification for state predictions and parameters estimates; and (iii) a discussion regarding the role of physics in statistical learning. We apply our statistical learning framework to experimental measurements from a laboratory-scale system and a linear model of its thermoacoustic behavior. This physics-informed statistical learning framework balances the robustness and interpretability of reduced-order models against the expressive and predictive capabilities of machine learning.   

Shape Optimization of a 3D Annular Combustor to Eliminate Thermoacoustic Instability

Thermoacoustic instabilities, which arise due to the interaction between confined flames and acoustics, are sensitive to small changes in system parameters. Gas turbines are notoriously susceptible to this instability, which often recurs during final engine tests. The instabilities are usually eliminated by inspired trial and error, making small changes to the geometry or adding passive devices. The aim of this project is to identify these small changes systematically. We do this with adjoint-based shape optimization of a 3D finite element Helmholtz solver. We examine the case of a 30kW laboratory-scale combustor (MICCA), which is azimuthally unstable. First, using the Hadamard theorem, we find the eigenvalue shape derivatives in order to identify the most influential regions of the combustor. Then we apply an optimization algorithm based on the shape gradients to stabilize the azimuthal mode. We only apply shape changes that do not break the discrete rotational symmetry of the annular combustor so that the two-fold degenerate azimuthal mode does not unfold. The ability to handle complex 3D geometries makes this tool a strong candidate for the iterative design of thermoacoustically stable combustors.   

Sound generation by entropy perturbations passing through sudden flow expansions

Entropy perturbations generate sound when accelerated/decelerated by a non-uniform flow. Current analytical models provide a good prediction of this entropy noise when the flow cross-sectional area changes are gradual, as is the case for nozzle flows. However, they typically rely on quasi-1D and isentropic assumptions, and their predictions differ significantly from experimental measurements when sudden area increases are involved. This work uses a theoretical approach to quantitatively identify the main mechanisms responsible for the mismatch. A new form of the acoustic analogy is derived in which the entropy-related source terms are rigorously identified for the first time. The theory includes three-dimensional and non-isentropic effects. The approach is applied to the flow through a sudden area expansion, for which the large-scale flow separation creates a recirculation zone. The derived acoustic analogy is simplified for low Mach numbers and frequencies, and solved using a Green's function method. The results provide the first quantitative evidence that the presence and spatial extent of the recirculation zone, rather than the flow non-isentropicity, is the dominant factor causing deviation from predictions from quasi-1D, isentropic theory.   

A Bayesian approach for predicting and filtering linear and nonlinear thermoacoustic oscillations

The modelling of thermoacoustic oscillations is typically based on qualitative models. We propose a Bayesian approach to make a prototypical thermoacoustic model quantitatively accurate by combining model predictions with data. We investigate both linear (around a fixed point) and nonlinear (limit cycle, frequency locked, quasiperiodic, chaotic) regimes. We design a square-root filter by reformulating the time-delayed problem into a Markov chain. The filter updates the ensemble mean without artificially sampling the measurement error, which overcomes the limitation of standard ensemble Kalman filters. We simulate a multi-microphone experiment where measurements are taken from multiple pressure sensors. Numerical experiments are performed to analyse the filter's performance in linear and nonlinear regimes. We propose sampling strategies based on the Nyquist–Shannon criterion and positive Lyapunov exponent. The filter is robust and capable of predicting the reference state even in the presence of large measurement errors. This work opens up new possibilities for the quantitative prediction of self-excited thermoacoustic oscillations by on-the-fly assimilation of experimental data into reduced-order models.   

Thermoacoustic Response of Counterflow Diffusion Flames

Thermoacoustic instabilities in combustors result from a favorable coupling between the unsteady heat release of the flame and the acoustic pressure field. These instabilities can lead to deleterious effects such as excessive noise and mechanical fatigue. In this work, we study the thermoacoustic response of counter flow diffusion flames. Using a fully compressible formulation and detailed chemistry, we perturb the flame with acoustic oscillations and measure the response in terms of the gain and phase of notable quantities, such as the heat release rate. Compared to typical low Mach formulations, the fully compressible formulation allows for the consideration of the hydrodynamic effects of the spatially varying pressure field in conjunction with the thermodynamic response. A wide range of frequencies are tested, and the behavior of the flame at both ends of the spectrum are characterized. The transmission and reflection of acoustic waves by the flame is also discussed.   

Acoustic Wave Equation and its Boundary Conditions in 1-D Ducts with Inhomogeneous Media and Mean Flow

We derive the generalized Helmholtz equation governing the acoustic pressure field in a quasi 1-D duct with axially varying cross-section and inhomogeneous mean flow properties such as the velocity, temperature, density and pressure. A linearly-exact derivative boundary condition to the Helmholtz equation of the form $\dv{\hat{p}}{x}(x;\omega) = f(\hat{p},\hat{u},\hat{\rho}; \omega)$ is also developed, where $\hat{p}$, $\hat{u}$ and $\hat{\rho}$ are the pressure, velocity and density fluctuation fields, respectively, and $\omega$ is the angular frequency. It is seen that the pressure fluctuation field obtained by solving the Helmholtz equation in conjunction with the derivative boundary condition is identical to that obtained through the solution of the Euler equations. Furthermore, the linearly exact relationship between the density and pressure fluctuations is obtained, which is then compared with the ``classical" relation, $\hat{\rho} = \hat{p}/\bar{c}^2$, where $\bar{c}$ is the mean sound speed. In ducts with inhomogeneous mean properties, the classical $\hat{\rho}-\hat{p}$ relation differs substantially from the exact relation both in amplitude and phase.   

The role of Exceptional Points in the thermoacoustic spectrum

It has recently been found that, at specific operating conditions, thermoacoustic eigenvalues are extremely sensitive to small perturbations. We have formally related this high sensitivity to the existence of singularities in the thermoacoustic spectrum known as exceptional points. At exceptional points, two (or more) eigenvalues and their associated eigenvectors coalesce. As demonstrated in recent work, exceptional points naturally arise in longitudinal thermoacoustic systems due to the interaction between modes of acoustic and intrinsic origin. We extend these findings also for annular configurations, which are typical of gas turbines and aeroengines. Starting from a known simple or degenerate eigenvalue, we present general perturbation theory based formulae that can be used to accurately estimate the location of exceptional points in the thermoacoustic spectrum. We demonstrate that knowledge on the location of exceptional points is useful because it explains the strong nonlinear veering of the eigenvalue trajectories, which can be exploited to develop optimization strategies that go beyond current first order gradient descent methods.   

Thermoacoustic modes of intrinsic and acoustic origin and their interplay with exceptional points.

Historically, it has been presumed that thermoacoustic (TA) modes are tightly related to the acoustic modes of the cavity in which combustion takes place. Under this paradigm, a TA mode is seen as an acoustic mode that has been perturbed by the influence of an active flame. It should therefore be possible to track the evolution of acoustic modes into their corresponding TA modes by gradual increments in flame response gain, n.  However, in the past few years it has been shown that this tracking is not always straightforward, due to the existence of thermoacoustic modes of 'intrinsic' origin (ITA), which persist also in anechoic conditions. Ambiguity arises in defining the origin of TA modes because two different parameters, flame strength and reflection coefficient, are usually used to define the acoustic and ITA limits. We show that this distinction is instead unique if only one parameter, n, is used to define their origin. We prove, with the help of analytical expressions, that the sets of TA modes of ITA and acoustic origins are distinct in the limit of zero n. Tracking their trajectory by increasing n is then straightforward. We numerically show that the TA eigenfrequencies follow non-monotonic trajectories, and may coalesce at Exceptional Points (EP).   

On core noise for the typical operating conditions in a realistic gas turbine combustor

Commercial aircraft will continue to transport our society for decades to come. However, engine noise is one of the major issues because it adversely affects human's health and constrains air traffic growth. Engine-core noise consists of two distinct mechanisms, namely direct and indirect noise. Direct combustion noise describes the transmission of pressure fluctuations originating from unsteady heat-release in the combustion chamber. In contrast, indirect combustion noise is caused by the convection of unsteady vortices and entropy variations by temperature hot spots as they propagate from the combustor to the downstream turbine and nozzle. More recently, contributions from mixture inhomogeneities were identified as an additional source for indirect combustion noise that can interact and even exceed entropy noise. The relative contributions of these noise-source mechanisms are strongly dependent on the operating conditions, engine type, and interaction with other noise-source mechanisms. Core noise from a realistic gas-turbine combustor is investigated using a hybrid large-eddy simulation/linearized Euler equation (LES/LEE) framework. The effect of operating conditions on the relative noise-source contributions arising from direct and indirect noise are examined by considering cruise and takeoff conditions. The present work can help with the identification and quantification of the generation, transmission, and conversion of combustion noise.   

Understanding Non-linear Flame Response Triggering through High-Order Analysis

During combustion instabilities, the amplitude of excitation grows due to a self-excited feedback loop between exciting disturbances and flame response. However, as amplitude increases, non-linear processes lead to unique behaviors in the amplitude dependent flame response that are either monotonic with excitation amplitude or non-monotonic. Monotonic flame response trends include those which increase linearly and then saturate or those that have no linear slope (i.e. linear flame response is zero) but increase to non-zero values for higher excitation amplitudes and then saturate. In the latter cases, the flame response curve exhibits inflection points that result in 3 intersection points between the flame response curve and the linear damping curve. In this case, both the zero amplitude and finite amplitude intersection points are stable and thus result in bi-stable behavior, hysteresis and triggering. This results in the destabilization of an otherwise stable system through a disturbance that is large enough. In order to understand such flame response behaviors, this work explores the relationship between higher order terms of non-linearity (upto fifth order) in order to better understand the mechanism by which the higher order flame response saturation occurs. Finally, we apply this analysis to a model premixed flame to explore the relationship between different control parameters that causes bi-stable behavior.