Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification

Monte Carlo and Active Subspace Identification methods are combined with first- and second-order adjoint sensitivities to perform (forward) uncertainty quantification analysis of the thermo-acoustic stability of two annular combustor configurations. This method is applied to evaluate the risk factor, i.e., the probability for the system to be unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by up to $\sim\mathcal{O}(M)$, as many as the Monte Carlo samples.European Research Council (Project ALORS 2590620), Royal Academy of Engineering (Research Fellowships Scheme

Adjoint Methods as Design Tools in Thermoacoustics

In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.</jats:p

Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity

We present an adjoint-based method for the calculation of eigenvalue perturbations in nonlinear, degenerate and non-self-adjoint eigenproblems. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem. We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurations are analysed. The benchmark sensitivities are obtained by finite difference, which involves solving the nonlinear eigenproblem at least as many times as the number of parameters. By solving only one adjoint eigenproblem, we obtain the sensitivities to any thermo-acoustic parameter, which match the finite-difference solutions at much lower computational cost.The authors are grateful to the 2014 Center for Turbulence Research Summer Program (Stanford University) where the ideas of this work were born. L.M. and M.P.J acknowledge the European Research Council – Project ALORS 2590620 for financial support. L.M gratefully acknowledges the financial support received from the Royal Academy of Engineering Research Fellowships scheme. The authors thank Prof. Franck Nicoud for fruitful discussions. Fig. 1 was adapted from the article of S.R. Stow and A.P. Dowling, A time-domain network model for nonlinear thermoacoustic oscillations, ASME Turbo Expo, GT2008-50770 [9] with permission of the original publisher ASME

Methods for the calculation of thermoacoustic stability margins and monte carlo-free uncertainty quantification

Thermoacoustic instabilities are a major threat for modern gas turbines. Frequency-domain based stability methods, such as network models and Helmholtz solvers, are common design tools because they are fast compared to compressible CFD computations. Frequency-domain approaches result in an eigenvalue problem, which is nonlinear with respect to the eigenvalue. Nonlinear functions of the frequency are, for example, the n–τ model, impedance boundary conditions, etc. Thus, the influence of the relevant parameters on mode stability is only given implicitly. Small changes in some model parameters, which are obtained by experiments with some uncertainty, may have a great impact on stability. The assessment of how parameter uncertainties propagate to system stability is therefore crucial for safe gas turbine operation. This question is addressed by uncertainty quantification. A common strategy for uncertainty quantification in thermoacoustics is risk factor analysis. It quantifies the uncertainty of a set of parameters in terms of the probability of a mode to become unstable. One general challenge regarding uncertainty quantification is the sheer number of uncertain parameter combinations to be quantified. For instance, uncertain parameters in an annular combustor might be the equivalence ratio, convection times, geometrical parameters, boundary impedances, flame response model parameters etc. Assessing also the influence of all possible combinations of these parameters on the risk factor is a numerically very costly task. A new and fast way to obtain algebraic parameter models in order to tackle the implicit nature of the eigenfrequency problem is using adjoint perturbation theory. Though adjoint perturbation methods were recently applied to accelerate the risk factor analysis, its potential to improve the theory has not yet been fully exploited. This paper aims to further utilize adjoint methods for the quantification of uncertainties. This analytical method avoids the usual random Monte Carlo simulations, making it particularly attractive for industrial purposes. Using network models and the open-source Helmholtz solver PyHoltz it is also discussed how to apply the method with standard modeling techniques. The theory is exemplified based on a simple ducted flame and a combustor of EM2C laboratory for which experimental validation is available.</jats:p

Degenerate perturbation theory in thermoacoustics: High-order sensitivities and exceptional points

In this study, we connect concepts that have been recently developed in thermoacoustics, specifically, (i) high-order spectral perturbation theory, (ii) symmetry induced degenerate thermoacoustic modes, (iii) intrinsic thermoacoustic modes, and (iv) exceptional points. Their connection helps gain physical insight into the behaviour of the thermoacoustic spectrum when parameters of the system are varied. First, we extend high-order adjoint-based perturbation theory of thermoacoustic modes to the degenerate case. We provide explicit formulae for the calculation of the eigenvalue corrections to any order. These formulae are valid for self-adjoint, non-self-adjoint or even non-normal systems; therefore, they can be applied to a large range of problems, including fluid dynamics. Second, by analysing the expansion coefficients of the eigenvalue corrections as a function of a parameter of interest, we accurately estimate the radius of convergence of the power series. Third, we connect the existence of a finite radius of convergence to the existence of singularities in parameter space. We identify these singularities as exceptional points, which correspond to defective thermoacoustic eigenvalues, with infinite sensitivity to infinitesimal changes in the parameters. At an exceptional point, two eigenvalues and their associated eigenvectors coalesce. Close to an exceptional point, strong veering of the eigenvalue trajectories is observed. As demonstrated in recent work, exceptional points naturally arise in thermoacoustic systems due to the interaction between modes of acoustic and intrinsic origin. The role of exceptional points in thermoacoustic systems sheds new light on the physics and sensitivity of thermoacoustic stability, which can be leveraged for passive control by small design modifications

Optimisation of chaotically perturbed acoustic limit cycles

In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure. The objective of manufacturers is to design stable thermoacoustic configurations. In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation. In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down. First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time--averaged cost functionals exist and can be robustly calculated by a shadowing method. Third, we calculate the sensitivities of the time--averaged acoustic energy and Rayleigh index to small changes to the heat--source intensity and time delay. By embedding the sensitivities into a gradient--update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification

Symmetry breaking of azimuthal thermoacoustic modes: the UQ perspective

Since its introduction in the late 19th century, symmetry breaking has been found to play a crucial role in physics. In particular, it appears as one key phenomenon controlling hydrodynamic and acoustic instabilities in problems with rotational symmetries. A previous paper investigated its desired potential application to the control of circumferential thermo-acoustic modes in one annular cavity coupled with multiple flames (Bauerheim et al. 2014e). The present paper focuses on a similar problem when symmetry breaking appears unintendedly, for example when uncertainties due to tolerances are taken into account. It yields a large Uncertainty Quantification (UQ) problem containing numerous uncertain parameters. To tackle this well known “curse of dimensionality”, a novel UQ methodology is used. It relies on the active subspace approach to construct a reduced set of input variables. This strategy is applied on two annular cavities coupled by 19 flames to determine its modal risk factor, i.e. the probability of an azimuthal acoustic mode to be unstable. Since each flame is modeled by two uncertain parameters, it leads to a large UQ problem involving 38 parameters. An acoustic network model is then derived, which yields a non-linear dispersion relation for azimuthal modes. This non-linear problem, subject to bifurcations, is solved quasi-analytically. Results show that the dimension of the probabilistic problem can be drastically reduced, from 38 uncertain parameters to only 3. Moreover, it is found that the three active variables are related to physical quantities, which unveils underlying phenomena controlling the stability of the two coupled cavities. The first active variable is associated with a coupling strength controlling the bifurcation of the system, while the two others correspond to a symmetry breaking effect induced by the uncertainties. Thus, an additional destabilization effect appear caused by the non-uniform pattern of the uncertainty distribution, which breaks the initial rotating symmetry of the annular cavities. Finally, the active subspace is exploited by fitting the response surface with polynomials (linear, quadratic and cubic). By comparing accuracy and cost, results prove that 5% error can be achieved with only 30 simulations on the reduced space, whereas 2000 are required on the complete initial space. It exemplifies that this novel UQ technique can accurately predict the risk factor of an annular configuration at low cost as well as unveil key parameters controlling the stability

Thermoacoustic stabilization of a longitudinal combustor using adjoint methods

We construct a low order thermoacoustic network model that contains the most influential physical mechanisms of a thermoacoustic system. We apply it to a laboratory-scale longitudinal combustor that has been found to be thermoacoustically unstable in experiments. We model the flame, which is behind a bluff body, by a geometric level set method. We obtain the thermoacoustic eigenvalues of this configuration and examine a configuration in which six eigenmodes are unstable. We then derive the adjoint equations of this model and use the corresponding adjoint eigenmodes to obtain the sensitivities of the unstable eigenvalues to modifications of the model geometry. These sensitivities contain contributions from changes to the steady base flow and changes to the fluctuating flow. We find that these two contributions have similar magnitudes, showing that both contributions need to be considered. We then wrap these sensitivities within a gradient-based optimization algorithm and stabilize all six eigenvalues by changing the geometry. The required geometry changes are well approximated by the first step in the optimization process, showing that this sensitivity information is useful even before it is embedded within an optimization algorithm. We examine the acoustic energy balance during the optimization process and identify the physical mechanisms through which the algorithm is stabilizing the combustor. The algorithm works by, for each mode, reducing the work done by the flame, while simultaneously increasing the work done by the system on the outlet boundary. We find that only small geometry changes are required in order to stabilize every mode. The network model used in this study deliberately has the same structure as one used in the gas turbine industry in order to ease its implementation in practice.Cambridge Trust

Acoustic modal analysis with heat release fluctuations using nonlinear eigensolvers

Closed combustion devices like gas turbines and rockets are prone to thermoacoustic instabilities. Design engineers in the industry need tools to accurately identify and remove instabilities early in the design cycle. Many different approaches have been developed by the researchers over the years. In this work we focus on the Helmholtz wave equation based solver which is found to be relatively fast and accurate for most applications. This solver has been a subject of study in many previous works. The Helmholtz wave equation in frequency space reduces to a nonlinear eigenvalue problem which needs to be solved to compute the acoustic modes. Most previous implementations of this solver have relied on linearized solvers and iterative methods which as shown in this work are not very efficient and sometimes inaccurate. In this work we make use of specialized algorithms implemented in SLEPc that are accurate and efficient for computing eigenvalues of nonlinear eigenvalue problems. We make use of the n-tau model to compute the reacting source terms in the Helmholtz equation and describe the steps involved in deriving the Helmholtz eigenvalue equation and obtaining its solution using the SLEPc library

Inverse problems in thermoacoustics

Thermoacoustics is a branch of fluid mechanics, and is as such governed by the conservation laws of mass, momentum, energy and species. While computational fluid dynamics (CFD) has entered the design process of many applications in fluid mechanics, its success in thermoacoustics is limited by the multi-scale, multi-physics nature of the subject. In his influential monograph from 2006, Prof. Fred Culick writes about the role of CFD in thermoacoustic modeling: The main reason that CFD has otherwise been relatively helpless in this subject is that problems of combustion instabilities involve physical and chemical matters that are still not well understood. Moreover, they exist in practical circumstances which are not readily approximated by models suitable to formulation within CFD. Hence, the methods discussed and developed in this book will likely be useful for a long time to come, in both research and practice. [. . . ] It seems to me that eventually the most effective ways of formulating predictions and theoretical interpretations of combustion instabilities in practice will rest on combining methods of the sort discussed in this book with computational fluid dynamics, the whole confirmed by experimental results. Despite advances in CFD and large-eddy simulation (LES) in particular, unsteady simulations for more than a few selected operating points are computationally infeasible. The ‘methods discussed in this book’ refer to reduced-order models of thermoacoustic oscillations. Whether intentional or not, the last sentence anticipates the advent of data-driven methods, and encapsulates the philosophy behind this work. This work brings together two workhorses of the design process: physics-informed reduced-order models and data from higher-fidelity sources such as simulations and experiments. The three building blocks to all our statistical inference frameworks are: (i) a hierarchical view of reduced-order models consisting of states, parameters and governing equations; (ii) probabilistic formulations with random variables and stochastic processes; and (iii) efficient algorithms from statistical learning theory and machine learning. While leveraging advances in statistical and machine learning, we demonstrate the feasibility of Bayes’ rule as a first principle in physics-informed statistical inference. In particular, we discuss two types of inverse problems in thermoacoustics: (i) implicit reduced-order models representative of nonlinear eigenproblems from linear stability analysis; and (ii) time-dependent reduced-order models used to investigate nonlinear dynamics. The outcomes of statistical inference are improved predictions of the state, estimates of the parameters with uncertainty quantification and an assessment of the reduced-order model itself. This work highlights the role that data can play in the future of combustion modeling for thermoacoustics. It is increasingly impractical to store data, particularly as experiments become automated and numerical simulations become more detailed. Rather than store the data itself, the techniques in this work optimally assimilate the data into the parameters of a physics-informed reduced-order model. With data-driven reduced-order models, rapid prototyping of combustion systems can feed into rapid calibration of their reduced-order models and then into gradient-based design optimization. While it has been shown, e.g. in the context of ignition and extinction, that large-eddy simulations become quantitatively predictive when augmented with data, the reduced-order modeling of flame dynamics in turbulent flows remains challenging. For these challenging situations, this work opens up new possibilities for the development of reduced-order models that adaptively change any time that data from experiments or simulations becomes available.Schlumberger Cambridge International Scholarshi