Publications
93 results found
Doan NAK, Polifke W, Magri L, 2020, Physics-informed echo state networks, Journal of Computational Science, Vol: 47, ISSN: 1877-7503
© 2020 Elsevier B.V. We propose a physics-informed echo state network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional loss function during the training, which is based on the system’s governing equations. The additional loss function penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system and a truncation of the Charney–DeVore system. Compared to the conventional ESNs, the physics-informed ESNs improve the predictability horizon by about two Lyapunov times. This approach is also shown to be robust with regard to noise. The proposed framework shows the potential of using machine learning combined with prior physical knowledge to improve the time-accurate prediction of chaotic dynamical systems.
Huhn F, Magri L, 2020, Learning ergodic averages in chaotic systems, International conference on computational science, Pages: 124-132, ISSN: 0302-9743
© Springer Nature Switzerland AG 2020. We propose a physics-informed machine learning method to predict the time average of a chaotic attractor. The method is based on the hybrid echo state network (hESN). We assume that the system is ergodic, so the time average is equal to the ergodic average. Compared to conventional echo state networks (ESN) (purely data-driven), the hESN uses additional information from an incomplete, or imperfect, physical model. We evaluate the performance of the hESN and compare it to that of an ESN. This approach is demonstrated on a chaotic time-delayed thermoacoustic system, where the inclusion of a physical model significantly improves the accuracy of the prediction, reducing the relative error from 48% to 1%. This improvement is obtained at the low extra cost of solving a small number of ordinary differential equations that contain physical information. This framework shows the potential of using machine learning techniques combined with prior physical knowledge to improve the prediction of time-averaged quantities in chaotic systems.
Doan NAK, Polifke W, Magri L, 2020, Learning hidden states in a chaotic system: A physics-informed echo state network approach, International conference on computational science, Pages: 117-123, ISSN: 0302-9743
© Springer Nature Switzerland AG 2020. We extend the Physics-Informed Echo State Network (PI-ESN) framework to reconstruct the evolution of an unmeasured state (hidden state) in a chaotic system. The PI-ESN is trained by using (i) data, which contains no information on the unmeasured state, and (ii) the physical equations of a prototypical chaotic dynamical system. Non-noisy and noisy datasets are considered. First, it is shown that the PI-ESN can accurately reconstruct the unmeasured state. Second, the reconstruction is shown to be robust with respect to noisy data, which means that the PI-ESN acts as a denoiser. This paper opens up new possibilities for leveraging the synergy between physical knowledge and machine learning to enhance the reconstruction and prediction of unmeasured states in chaotic dynamical systems.
Magri L, Doan NAK, 2020, First-principles machine learning modelling of COVID-19
The coronavirus disease 2019 (COVID-19) has changed the world since the World Health Organization declared its outbreak on 30th January 2020, recognizing the outbreak as a pandemic on 11th March 2020. As often said by politicians and scientific advisors, the objective is "to flatten the curve", or "push the peak down", or similar wording, of the virus spreading. Central to the official advice are mathematical models and data, which provide estimates on the evolution of the number of infected, recovered and deaths. The accuracy of the models is improved day by day by inferring the contact, recovery, and death rates from data (confirmed cases). A data-driven model trained with both data and first principles is proposed. The model can quickly be re-trained any time that new data becomes available. The method can be applied to more detailed epidemic models with virtually no conceptual modification.
Huhn F, Magri L, 2020, Optimisation of chaotically perturbed acoustic limit cycles, Nonlinear Dynamics, Vol: 100, Pages: 1641-1657, ISSN: 0924-090X
© 2020, Springer Nature B.V. 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.
Huhn F, Magri L, 2020, Stability, sensitivity and optimisation of chaotic acoustic oscillations, JOURNAL OF FLUID MECHANICS, Vol: 882, ISSN: 0022-1120
Yu H, Juniper MP, Magri L, 2020, A data-driven kinematic model of a ducted premixed flame, Proceedings of the Combustion Institute, ISSN: 1540-7489
© 2020 The Combustion Institute. Reduced-order models of flame dynamics can be used to predict and mitigate the emergence of thermoacoustic oscillations in the design of gas turbine and rocket engines. This process is hindered by the fact that these models, although often qualitatively correct, are not usually quantitatively accurate. As automated experiments and numerical simulations produce ever-increasing quantities of data, the question arises as to how this data can be assimilated into physics-informed reduced-order models in order to render these models quantitatively accurate. In this study, we develop and test a physics-based reduced-order model of a ducted premixed flame in which the model parameters are learned from high-speed videos of the flame. The experimental data is assimilated into a level-set solver using an ensemble Kalman filter. This leads to an optimally calibrated reduced-order model with quantified uncertainties, which accurately reproduces elaborate nonlinear features such as cusp formation and pinch-off. The reduced-order model continues to match the experiments after assimilation has been switched off. Further, the parameters of the model, which are extracted automatically, are shown to match the first-order behavior expected on physical grounds. This study shows how reduced-order models can be updated rapidly whenever new experimental or numerical data becomes available, without the data itself having to be stored.
Magri L, Doan NAK, 2020, Physics-informed data-driven prediction of turbulent reacting flows with lyapunov analysis and sequential data assimilation, Data Analysis for Direct Numerical Simulations of Turbulent Combustion: From Equation-Based Analysis to Machine Learning, Pages: 177-196, ISBN: 9783030447175
© Springer Nature Switzerland AG 2020. High-fidelity simulations of turbulent reacting flows enable scientific understanding of the physics and engineering design of practical systems. Whereas Direct Numerical Simulation (DNS) is the most suitable numerical tool to understand the physics, under-resolved and large-eddy simulations offer a good compromise between accuracy and computational effort in the prediction of engineering flows. This compromise speeds up the computations but reduces the space-and-time accuracy of the prediction. The objective of this chapter is to (i) evaluate the predictability horizon of turbulent simulations with chaos theory, and (ii) enable the space-andtime- accurate prediction of rare and transient events using a Bayesian statistical learning approach based on data assimilation. The methods are applied to DNS of Moderate or Intense Low-oxygen Dilution (MILD) combustion. The predictability provides an estimate of the time horizon within which the occurrence of ignition kernels and deflagrative modes, which are considered here as rare and transient events, can be accurately predicted. The accurate detection of ignition kernels and their evolution towards deflagrative structures are well captured on a coarse (under-resolved) grid when data is assimilated from a costly refined DNS. Physically, such an accurate prediction is important to understand the stabilization mechanism of MILD combustion. These techniques enable the space-and-time-accurate prediction of rare and transient events in turbulent flows by combining under-resolved simulations and experimental data, for example, from engine sensors. This opens up new possibilities for on-the-fly calibration of reduced-order models for turbulent reacting flows.
Magri L, Juniper MP, Moeck JP, 2020, Sensitivity of the rayleigh criterion in thermoacoustics, Journal of Fluid Mechanics, Vol: 882, ISSN: 0022-1120
© Cambridge University Press 2019. Thermoacoustic instabilities are one of the most challenging problems faced by gas turbine and rocket motor manufacturers. The key instability mechanism is described by the Rayleigh criterion. The Rayleigh criterion does not directly show how to alter a system to make it more stable. This is the objective of sensitivity analysis. Because thermoacoustic systems have many design parameters, adjoint sensitivity analysis has been proposed to obtain all the sensitivities with one extra calculation. Although adjoint sensitivity analysis can be carried out in both the time and the frequency domain, the frequency domain is more natural for a linear analysis. Perhaps surprisingly, the Rayleigh criterion has not yet been rigorously derived and comprehensively interpreted in the frequency domain. The contribution of this theoretical paper is threefold. First, the Rayleigh criterion is interpreted in the frequency domain with integral formulae for the complex eigenvalue. Second, the first variation of the Rayleigh criterion is calculated both in the time and frequency domain, both with and without Lagrange multipliers (adjoint variables). The Lagrange multipliers are physically related to the system’s observables. Third, an adjoint Rayleigh criterion is proposed. The paper also points out that the conclusions of Juniper (Phys. Rev. Fluids, vol. 3, 2018, 110509) apply to the first variation of the Rayleigh criterion, not to the Rayleigh criterion itself. The mathematical relations of this paper can be used to compute sensitivities directly from measurable quantities to enable optimal design.
Orchini A, Magri L, Silva CF, et al., 2020, Degenerate perturbation theory in thermoacoustics: High-order sensitivities and exceptional points, Journal of Fluid Mechanics, ISSN: 0022-1120
© The Author(s), 2020. 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.
Doan NAK, Polifke W, Magri L, 2019, A physics-aware machine to predict extreme events in turbulence
We propose a physics-aware machine learning method to time-accurately predict extreme events in a turbulent flow. The method combines two radically different approaches: empirical modelling based on reservoir computing, which learns the chaotic dynamics from data only, and physical modelling based on conservation laws. We show that the combination of the two approaches is able to predict the occurrence and amplitude of extreme events in the self-sustaining process in turbulence-the abrupt transitions from turbulent to quasi-laminar states-which cannot be achieved by using either approach separately. This opens up new possibilities for enhancing synergistically data-driven methods with physical knowledge for the accurate prediction of extreme events in chaotic dynamical systems.
Yu H, Juniper MP, Magri L, 2019, Combined state and parameter estimation in level-set methods, Journal of Computational Physics, Vol: 399, ISSN: 0021-9991
© 2019 Elsevier Inc. Reduced-order models based on level-set methods are widely used tools to qualitatively capture and track the nonlinear dynamics of an interface. The aim of this paper is to develop a physics-informed, data-driven, statistically rigorous learning algorithm for state and parameter estimation with level-set methods. A Bayesian approach based on data assimilation is introduced. Data assimilation is enabled by the ensemble Kalman filter and smoother, which are used in their probabilistic formulations. The level-set data assimilation framework is verified in one-dimensional and two-dimensional test cases, where state estimation, parameter estimation and uncertainty quantification are performed. The statistical performance of the proposed ensemble Kalman filter and smoother is quantified by twin experiments. In the twin experiments, the combined state and parameter estimation fully recovers the reference solution, which validates the proposed algorithm. The level-set data assimilation framework is then applied to the prediction of the nonlinear dynamics of a forced premixed flame, which exhibits the formation of sharp cusps and intricate topological changes, such as pinch-off events. The proposed physics-informed statistical learning algorithm opens up new possibilities for making reduced-order models of interfaces quantitatively predictive, any time that reference data is available.
Yu H, Jaravel T, Ihme M, et al., 2019, Data Assimilation and Optimal Calibration in Nonlinear Models of Flame Dynamics, Journal of Engineering for Gas Turbines and Power, Vol: 141, ISSN: 0742-4795
<jats:title>Abstract</jats:title> <jats:p>We propose an on-the-fly statistical learning method to take a qualitative reduced-order model of the dynamics of a premixed flame and make it quantitatively accurate. This physics-informed data-driven method is based on the statistically optimal combination of (i) a reduced-order model of the dynamics of a premixed flame with a level-set method, (ii) high-quality data, which can be provided by experiments and/or high-fidelity simulations, and (iii) assimilation of the data into the reduced-order model to improve the prediction of the dynamics of the premixed flame. The reduced-order model learns the state and the parameters of the premixed flame on the fly with the ensemble Kalman filter, which is a Bayesian filter used, for example, in weather forecasting. The proposed method and algorithm are applied to two test cases with relevance to reacting flows and instabilities. First, the capabilities of the framework are demonstrated in a twin experiment, where the assimilated data are produced from the same model as that used in prediction. Second, the assimilated data are extracted from a high-fidelity reacting-flow direct numerical simulation (DNS), which provides the reference solution. The results are analyzed by using Bayesian statistics, which robustly provide the level of confidence in the calculations from the reduced-order model. The versatile method we propose enables the optimal calibration of computationally inexpensive reduced-order models in real-time when experimental data become available, for example, from gas-turbine sensors.</jats:p>
Yu H, Jaravel T, Ihme M, et al., 2019, Data Assimilation and Optimal Calibration in Nonlinear Models of Flame Dynamics, ISSN: 0742-4795
Yu H, Sengupta U, Juniper M, et al., 2019, Time-Accurate Calibration of a Thermoacoustic Model on Experimental Images of a Forced 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. We extend our previous work (Yu et al., CTR summer program 2018; Yu et al., J. Eng. Gas Turbines Power 2019) by applying our recently published level-set data assimilation framework (Yu et al., J. Comput. Phys. 2019) to experimental images of a forced premixed flame. We force a Bunsen flame with a loudspeaker and record videos at different frequencies and amplitudes. 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 and identifying deficiencies in the model. We demonstrate our level-set data assimilation framework using both the ensemble Kalman filter and smoother. More generally, we take a physics-informed, reduced-order model and use statistical learning techniques to make it quantitatively accurate.
Yu H, Jaravel T, Ihme M, et al., 2019, Data assimilation and optimal calibration in nonlinear models of flame dynamics, ASME Turbo Expo
Copyright © 2019 ASME. We propose an on-the-fly statistical learning method to take a qualitative reduced-order model of the dynamics of a premixed flame and make it quantitatively accurate. This physics-informed data-driven method is based on the statistically optimal combination of (i) a reduced-order model of the dynamics of a premixed flame with a level-set method, (ii) high-quality data, which can be provided by experiments and/or high-fidelity simulations, and (iii) assimilation of the data into the reduced-order model to improve the prediction of the dynamics of the premixed flame. The reduced-order model learns the state and the parameters of the premixed flame on the fly with the ensemble Kalman filter, which is a Bayesian filter used, for example, in weather forecasting. The proposed method and algorithm are applied to two test cases with relevance to reacting flows and instabilities. First, the capabilities of the framework are demonstrated in a twin experiment, where the assimilated data is produced from the same model as that used in prediction. Second, the assimilated data is extracted from a high-fidelity reacting-flow direct numerical simulation (DNS), which provides the reference solution. The results are analyzed by using Bayesian statistics, which robustly provide the level of confidence in the calculations from the reduced-order model. The versatile method we propose enables the optimal calibration of computationally inexpensive reduced-order models in real time when experimental data becomes available, for example, from gas-turbine sensors.
Magri L, 2019, Adjoint characteristic decomposition of one-dimensional waves, Journal of Computational Physics, Vol: 388, Pages: 454-461, ISSN: 0021-9991
© 2019 Elsevier Inc. Adjoint methods enable the accurate calculation of the sensitivities of a quantity of interest. The sensitivity is obtained by solving the adjoint system, which can be derived by continuous or discrete adjoint strategies. In acoustic wave propagation, continuous and discrete adjoint methods have been developed to compute the eigenvalue sensitivity to design parameters and passive devices (Aguilar et al., 2017, [1]). In this short communication, it is shown that the continuous and discrete adjoint characteristic decompositions, and Riemann invariants, are connected by a similarity transformation. The results are shown in the Laplace domain. The adjoint characteristic decomposition is applied to a one-dimensional acoustic resonator, which contains a monopole source of sound. The proposed framework provides the foundation to tackle larger acoustic networks with a discrete adjoint approach, opening up new possibilities for adjoint-based design of problems that can be solved by the method of characteristics.
Mensah GA, Magri L, Orchini A, et al., 2019, Effects of Asymmetry on Thermoacoustic Modes in Annular Combustors: A Higher-Order Perturbation Study, Journal of Engineering for Gas Turbines and Power, Vol: 141, ISSN: 0742-4795
Copyright © 2019 by ASME. Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice, the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally nonuniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact on the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz and linearized Euler equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function (FTF). The model equations are solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an "inclination rule" that explains the behavior of degenerate eigenvalues at first order is proven.
Magri L, 2019, Adjoint Methods as Design Tools in Thermoacoustics, Applied Mechanics Reviews, Vol: 71, ISSN: 0003-6900
© 2019 by ASME. 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 ca
Silva CF, Yong KJ, Magri L, 2019, Thermoacoustic Modes of Quasi- One-Dimensional Combustors in the Region of Marginal Stability, Journal of Engineering for Gas Turbines and Power, Vol: 141, ISSN: 0742-4795
© 2019 by ASME. It may be generally believed that the thermoacoustic eigenfrequencies of a combustor with fully acoustically reflecting boundary conditions depend on both flame dynamics and geometry of the system. In this work, we show that there are situations where this understanding does not strictly apply. The purpose of this study is twofold. In the first part, we show that the resonance frequencies of two premixed combustors with fully acoustically reflecting boundary conditions in the region of marginal stability depend only on the parameters of the flame dynamics but do not depend on the combustor’s geometry. This is shown by means of a parametric study, where the time delay and the interaction index of the flame response are varied and the resulting complex eigenfrequency locus is shown. Assuming longitudinal acoustics and a low Mach number, a quasi-1D Helmholtz solver is utilized. The time delay and interaction index of the flame response are parametrically varied to calculate the complex eigenfrequency locus. It is found that all the eigenfrequency trajectories cross the real axis at a resonance frequency that depends only on the time delay. Such marginally stable frequencies are independent of the resonant cavity modes of the two combustors, i.e., the passive thermoacoustic modes. In the second part, we exploit the aforementioned observation to evaluate the critical flame gain required for the systems to become unstable at four eigenfrequencies located in the marginally stable region. A computationally efficient method is proposed. The key ingredient is to consider both direct and adjoint eigenvectors associated with the four eigenfrequencies. Hence, the sensitivity of the eigenfrequencies to changes in the gain at the region of marginal stability is evaluated with cheap and accurate calculations. This work contributes to the understanding of thermoacoustic stability of combustors. In the same manner, the understanding of the nature of distinc
Doan NAK, Polifke W, Magri L, 2019, Physics-Informed Echo State Networks for Chaotic Systems Forecasting, Pages: 192-198, ISBN: 9783030227463
© 2019, Springer Nature Switzerland AG. We propose a physics-informed Echo State Network (ESN) to predict the evolution of chaotic systems. Compared to conventional ESNs, the physics-informed ESNs are trained to solve supervised learning tasks while ensuring that their predictions do not violate physical laws. This is achieved by introducing an additional loss function during the training of the ESNs, which penalizes non-physical predictions without the need of any additional training data. This approach is demonstrated on a chaotic Lorenz system, where the physics-informed ESNs improve the predictability horizon by about two Lyapunov times as compared to conventional ESNs. The proposed framework shows the potential of using machine learning combined with prior physical knowledge to improve the time-accurate prediction of chaotic dynamical systems.
Giusti A, Magri L, Zedda M, 2019, Flow inhomogeneities in a realistic aeronautical gas-turbine combustor: formation, evolution, and indirect noise, Journal of Engineering for Gas Turbines and Power: Transactions of the ASME, Vol: 141, ISSN: 0742-4795
Indirect noise generated by the acceleration of combustion inhomogeneities is an important aspect in the design of aero-engines because of its impact on the overall noise emitted by an aircraft and the possible contribution to combustion instabilities. In this study, a realistic rich-quench-lean (RQL) combustor is numerically investigated, with the objective of quantitatively analyzing the formation and evolution of flow inhomogeneities and determining the level of indirect combustion noise in the nozzle guide vane (NGV). Both entropy and compositional noise are calculated in this work. A high-fidelity numerical simulation of the combustion chamber, based on the large-eddy simulation (LES) approach with the conditional moment closure (CMC) combustion model, is performed. The contributions of the different air streams to the formation of flow inhomogeneities are pinned down and separated with seven dedicated passive scalars. LES-CMC results are then used to determine the acoustic sources to feed an NGV aeroacoustic model, which outputs the noise generated by entropy and compositional inhomogeneities. Results show that non-negligible fluctuations of temperature and composition reach the combustor's exit. Combustion inhomogeneities originate both from finite-rate chemistry effects and incomplete mixing. In particular, the role of mixing with dilution and liner air flows on the level of combustion inhomogeneities at the combustor's exit is highlighted. The species that most contribute to indirect noise are identified and the transfer functions of a realistic NGV are computed. The noise level indicates that indirect noise generated by temperature fluctuations is larger than the indirect noise generated by compositional inhomogeneities, although the latter is not negligible and is expected to become louder in supersonic nozzles. It is also shown that relatively small fluctuations of the local flame structure can lead to significant variations of the nozzle transfer functi
Traverso T, Magri L, 2019, Data Assimilation in a Nonlinear Time-Delayed Dynamical System with Lagrangian Optimization, Pages: 156-168, ISBN: 9783030227463
© 2019, Springer Nature Switzerland AG. 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.
Mensah GA, Magri L, Silva CF, et al., 2018, Exceptional points in the thermoacoustic spectrum, Journal of Sound and Vibration, Vol: 433, Pages: 124-128, ISSN: 0022-460X
© 2018 Elsevier Ltd Exceptional points are found in the spectrum of a prototypical thermoacoustic system as the parameters of the flame transfer function are varied. At these points, two eigenvalues and the associated eigenfunctions coalesce. The system’s sensitivity to changes in the parameters becomes infinite. Two eigenvalue branches collide at the exceptional point as the interaction index is increased. One branch originates from a purely acoustic mode, whereas the other branch originates from an intrinsic thermoacoustic mode. The existence of exceptional points in thermoacoustic systems has implications for physical understanding, computing, modeling and control.
Giusti A, Magri L, Zedda M, 2018, FLOW INHOMOGENEITIES IN A REALISTIC AERONAUTICAL GAS-TURBINE COMBUSTOR: FORMATION, EVOLUTION AND INDIRECT NOISE, Proceedings of ASME Turbo Expo 2018: Turbomachinery Technical Conference & Exposition, Publisher: ASME, Pages: V04BT04A024-V04BT04A024
Indirect noise generated by the acceleration of combustion inhomogeneities is an important aspect in the design of aeroengines because of its impact on the overall noise emitted by an aircraft and the possible contribution to combustion instabilities. In this study, a realistic rich-quench-lean combustor is numerically investigated, with the objective of quantitatively analyzing the formation and evolution of flow inhomogeneities and determine the level of indirect combustion noise in the nozzle guide vane (NGV). Both entropy and compositional noise are calculated in this work. A high-fidelity numerical simulation of the combustion chamber, based on the Large-Eddy Simulation (LES) approach with the Conditional Moment Closure (CMC) combustion model, is performed. The contribution of the different air streams to the formation of flow inhomogeneity is identified and separated through seven dedicated passive scalars. This pins down the individual contributions of the air streams to combustion inhomogeneity at the combustor’s exit. LES-CMC results are then used to determine the acoustic sources to feed an NGV aeroacoustic model, which outputs the noise generated by entropy and compositional inhomogeneities. Results show that non-negligible fluctuations of temperature and composition reach the combustor’s exit. Combustion inhomogeneities originate both from finite-rate chemistry effects and incomplete mixing. In particular, the role of mixing with dilution and liner air flows on the level of com-bustion inhomogeneities at the combustor’s exit is highlighted. The species that most contribute to indirect noise are identified and the transfer functions of a realistic NGV are computed. The noise level indicates that indirect noise generated by temperature fluctuations is larger that the indirect noise generated by compositional inhomogeneities, although the latter is not negligible and is expected to become louder in supersonic nozzles. It is also shown that
Yu H, Juniper MP, Magri L, 2018, Physics-informed data-driven prediction of premixed flame dynamics with data assimilation, Stanford University Centre of Turbulence Research Summer Program
We propose an on-the-fly statistical learning method to make a qualitative reduced-order model of the dynamics of a premixed flame quantitatively accurate. This physics- informed data-driven method is based on the statistically optimal combination of (i) a reduced-order model of the dynamics of a premixed flame with a level-set method, (ii) high-quality data, which can be provided by experiments and/or high-fidelity simulations, and (iii) assimilation of the data into the reduced-order model to improve the prediction of the dynamics of the premixed flame. The reduced-order model learns the state and the parameters of the premixed flame on the fly with the ensemble Kalman filter, which is a Bayesian filter used in the data assimilation of high-dimensional dynamical systems, e.g., in weather forecasting. The proposed method and algorithm are applied to two test cases with relevance to reacting flow and instability. First, the capabilities of the framework are demonstrated in a twin experiment, where the assimilated data are produced from the same model as that used in prediction. Second, the assimilated data are extracted from a high-fidelity reacting-flow direct numerical simulation (DNS). The results are analyzed by using Bayesian statistics, which provide the uncertainties of the calculations. This method opens up new possibilities for on-the-fly optimal calibration of computationally cheap reduced-order models when experimental data become available, for example, from sensors.
Mensah GA, Magri L, Moeck JP, 2018, Methods for the Calculation of Thermoacoustic Stability Boundaries and Monte Carlo-Free Uncertainty Quantification, Journal of Engineering for Gas Turbines and Power, Vol: 140
© 2018 by ASME. 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 flow computations. They result in an eigenvalue problem, which is nonlinear with respect to the eigenvalue. Thus, the influence of the relevant parameters on mode stability is only given implicitly. Small changes in some model parameters, 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. 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. A new and fast way to obtain algebraic parameter models in order to tackle the implicit nature of the problem is using adjoint perturbation theory. This paper aims to further utilize adjoint methods for the quantification of uncertainties. This analytical method avoids the usual random Monte Carlo (MC) 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 data are available.
Magri L, OBrien J, Ihme M, 2018, Effects of Nozzle Helmholtz Number on Indirect Combustion Noise by Compositional Perturbations, Journal of Engineering for Gas Turbines and Power, Vol: 140
By modeling a multicomponent gas, a new source of indirect combustion noise is identified, which is named compositional indirect noise. The advection of mixture inhomogeneities exiting the gas-turbine combustion chamber through subsonic and supersonic nozzles is shown to be an acoustic dipole source of sound. The level of mixture inhomogeneity is described by a difference in composition with the mixture fraction. An n-dodecane mixture, which is a kerosene fuel relevant to aeronautics, is used to evaluate the level of compositional noise. By relaxing the compact-nozzle assumption, the indirect noise is numerically calculated for Helmholtz numbers up to 2 in nozzles with linear velocity profile. The compact-nozzle limit is discussed. Only in this limit, it is possible to derive analytical transfer functions for (i) the noise emitted by the nozzle and (ii) the acoustics traveling back to the combustion chamber generated by accelerated compositional inhomogeneities. The former contributes to noise pollution, whereas the latter has the potential to induce thermoacoustic oscillations. It is shown that the compositional indirect noise can be at least as large as the direct noise and entropy noise in choked nozzles and lean mixtures. As the frequency with which the compositional inhomogeneities enter the nozzle increases, or as the nozzle spatial length increases, the level of compositional noise decreases, with a similar, but not equal, trend to the entropy noise. The noisiest configuration is found to be a compact supersonic nozzle.
Mensah GA, Magri L, Orchini A, et al., 2018, Effects of asymmetry on thermoacoustic modes in annular combustors: A higher-order perturbation study
Copyright © 2018 ASME Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally non-uniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact to the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz- and linearized Euler-equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function, which is solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal mode order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an “inclination rule” that explains the behavior of degenerate eigenvalues at first order is proven.
Silva CF, Yong KJ, Magri L, 2018, Thermoacoustic modes of quasi-1D combustors in the region of marginal stability
Copyright © 2018 ASME. It may be generally believed that the thermoacoustic eigenfrequencies of a combustor with fully acoustically reflecting boundary conditions depend on both flame dynamics and geometry of the system. In this work, we show that there are situations where this understanding does not strictly apply. The purpose of this study is twofold. In the first part, we show that the resonance frequencies of two premixed combustors with fully acoustically reflecting boundary conditions in the region of marginal stability depend only on the parameters of the flame dynamics, but do not depend on the combustor’s geometry. This is shown by means of a parametric study, where the time delay and the interaction index of the flame response are varied and the resulting complex eigenfrequency locus is shown. Assuming longitudinal acoustics and a low Mach number, a quasi-1D Helmholtz solver is utilized. The time delay and interaction index of the flame response are parametrically varied to calculate the complex eigenfrequency locus. It is found that all the eigenfrequency trajectories cross the real axis at a resonance frequency that depends only on the time delay. Such marginally stable frequencies are independent of the resonant cavity modes of the two combustors, i.e. the passive thermoacoustic modes. In the second part, we exploit the aforementioned observation to evaluate the critical flame gain required for the systems to become unstable at four eigenfrequencies located in the marginally stable region. A computationally-efficient method is proposed. The key ingredient is to consider both direct and adjoint eigenvectors associated with the four eigenfrequencies. Hence, the sensitivity of the eigenfrequencies to changes in the gain at the region of marginal stability is evaluated with cheap and accurate calculations. This work contributes to the understanding of thermoacoustic stability of combustors. In the same manner, the understanding of the nature of
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