Publications
93 results found
Magri L, 2017, On indirect noise in multicomponent nozzle flows, Journal of Fluid Mechanics, Vol: 828
A one-dimensional, unsteady nozzle flow is modelled to identify the sources of indirect noise in multi-component gases. First, from non-equilibrium thermodynamics relations, it is shown that a compositional inhomogeneity advected in an accelerating flow is a source of sound induced by inhomogeneities in the mixture (i) chemical potentials and (ii) specific heat capacities. Second, it is shown that the acoustic, entropy and compositional linear perturbations evolve independently from each other and they become coupled through mean-flow gradients and/or at the boundaries. Third, the equations are cast in invariant formulation and a mathematical solution is found by asymptotic expansion of path-ordered integrals with an infinite radius of convergence. Finally, the transfer functions are calculated for a supersonic nozzle with finite spatial extent perturbed by a methane-air compositional inhomogeneity. The proposed framework will help identify and quantify the sources of sound in nozzles with relevance, for example, to aeronautical gas turbines.
Nastac G, Labahn JW, Magri L, et al., 2017, Lyapunov exponent as a metric for assessing the dynamic content and predictability of large-eddy simulations, Physical Review Fluids, Vol: 2
Metrics used to assess the quality of large-eddy simulations commonly rely on a statistical assessment of the solution. While these metrics are valuable, a dynamic measure is desirable to further characterize the ability of a numerical simulation for capturing dynamic processes inherent in turbulent flows. To address this issue, a dynamic metric based on the Lyapunov exponent is proposed which assesses the growth rate of the solution separation. This metric is applied to two turbulent flow configurations: forced homogeneous isotropic turbulence and a turbulent jet diffusion flame. First, it is shown that, despite the direct numerical simulation (DNS) and large-eddy simulation (LES) being high-dimensional dynamical systems with O(107) degrees of freedom, the separation growth rate qualitatively behaves like a lower-dimensional dynamical system, in which the dimension of the Lyapunov system is substantially smaller than the discretized dynamical system. Second, a grid refinement analysis of each configuration demonstrates that as the LES filter width approaches the smallest scales of the system the Lyapunov exponent asymptotically approaches a plateau. Third, a small perturbation is superimposed onto the initial conditions of each configuration, and the Lyapunov exponent is used to estimate the time required for divergence, thereby providing a direct assessment of the predictability time of simulations. By comparing inert and reacting flows, it is shown that combustion increases the predictability of the turbulent simulation as a result of the dilatation and increased viscosity by heat release. The predictability time is found to scale with the integral time scale in both the reacting and inert jet flows. Fourth, an analysis of the local Lyapunov exponent is performed to demonstrate that this metric can also determine flow-dependent properties, such as regions that are sensitive to small perturbations or conditions of large turbulence within the flow field. Finally, i
Aguilar JG, Magri L, Juniper MP, 2017, Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach, Journal of Computational Physics, Vol: 341, Pages: 163-181
Strict pollutant emission regulations are pushing gas turbine manufacturers to develop devices that operate in lean conditions, with the downside that combustion instabilities are more likely to occur. Methods to predict and control unstable modes inside combustion chambers have been developed in the last decades but, in some cases, they are computationally expensive. Sensitivity analysis aided by adjoint methods provides valuable sensitivity information at a low computational cost. This paper introduces adjoint methods and their application in wave-based low order network models, which are used as industrial tools, to predict and control thermoacoustic oscillations. Two thermoacoustic models of interest are analyzed. First, in the zero Mach number limit, a nonlinear eigenvalue problem is derived, and continuous and discrete adjoint methods are used to obtain the sensitivities of the system to small modifications. Sensitivities to base-state modification and feedback devices are presented. Second, a more general case with non-zero Mach number, a moving flame front and choked outlet, is presented. The influence of the entropy waves on the computed sensitivities is shown.
Rigas G, Esclapez L, Magri L, 2017, Symmetry breaking in a 3D bluff-body wake
The dynamics of a three-dimensional axisymmetric blu -body wake are examined at low Reynolds regimes where transitions take place through spatio-temporal symmetry breaking. A linear stability analysis is employed to identify the critical Reynolds num- ber associated with symmetry breaking, and the associated eigenmodes, known as global modes. The analysis shows that the axisymmetric stable base ow breaks the rotational symmetry through a pitchfork m = 1 bifurcation, in agreement with previously reported results for axisymmetric wakes. Above this threshold, the stable base ow is steady and three-dimensional with planar symmetry. A three-dimensional global stability analysis around the steady re ectionally symmetric base ow, assuming no homogeneous direc- tions, predicts accurately the Hopf bifurcation threshold, which leads to asymmetric vortex shedding. DNS simulations validate the stability results and characterize the ow topology during the early chaotic regime.
Mensah GA, Magri L, Moeck JP, 2017, Methods for the calculation of thermoacoustic stability margins and monte carlo-free uncertainty quantification
Copyright © 2017 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 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 nτ 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
Magri L, See YC, Tammisola O, et al., 2017, Multiple-scale thermo-acoustic stability analysis of a coaxial jet combustor, Proceedings of the Combustion Institute, Vol: 36, Pages: 3863-3871, ISSN: 1540-7489
© 2016 by The Combustion Institute. Published by Elsevier Inc. In this paper, asymptotic multiple-scale methods are used to formulate a mathematically consistent set of thermo-acoustic equations in the low-Mach number limit for linear stability analysis. The resulting sets of nonlinear equations for hydrodynamics and acoustics are two-way coupled. The coupling strength depends on which multiple scales are used. The double-time-double-space (2T-2S), double-time-single-space (2T-1S) and single-time-double-space (1T-2S) limits are revisited, derived and linearized. It is shown that only the 1T-2S limit produces a two-way coupled linearized system. Therefore this limit is adopted and implemented in a finite-element solver. The methodology is applied to a coaxial jet combustor. By using an adjoint method and introducing the intrinsic sensitivity, (i) the interaction between the acoustic and hydrodynamic subsystems is calculated and (ii) the role of the global acceleration term, which is the coupling term from the acoustics to the hydrodynamics, is analyzed. For the confined coaxial jet diffusion flame studied here, (i) the growth rate of the thermo-acoustic oscillations is found to be more sensitive to small changes in the hydrodynamic field around the flame and (ii) increasing the global acceleration term is found to be stabilizing in agreement with the Rayleigh Criterion.
Magri L, O’Brien J, Ihme M, 2017, Effects of nozzle helmholtz number on indirect combustion noise by compositional perturbations
Copyright © 2017 ASME. By modeling a multi-component 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 travelling 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 chocked 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.
Magri L, Bauerheim M, Nicoud F, et al., 2016, Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification, Journal of Computational Physics, Vol: 325, Pages: 411-421
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 as much as the Monte Carlo samples.
Silva CF, Magri L, Runte T, et al., 2016, Uncertainty quantification of growth rates of thermoacoustic instability by an adjoint Helmholtz solver, Journal of Engineering for Gas Turbines and Power, Vol: 139
Thermoacoustic instabilities are often calculated with Helmholtz solvers combined with a low-order model for the flame dynamics. Typically, such a formulation leads to an eigenvalue problem in which the eigenvalue appears under nonlinear terms, such as exponentials related to the time delays that result from the flame model. The objective of the present paper is to quantify uncertainties in thermoacoustic stability analysis with a Helmholtz solver and its adjoint. This approach is applied to the model of a combustion test rig with a premixed swirl burner. The nonlinear eigenvalue problem and its adjoint are solved by an in-house adjoint Helmholtz solver, based on an axisymmetric finite-volume discretization. In addition to first-order correction terms of the adjoint formulation, as they are often used in the literature, second-order terms are also taken into account. It is found that one particular second-order term has significant impact on the accuracy of the predictions. Finally, the probability density function (PDF) of the growth rate in the presence of uncertainties in the input parameters is calculated with a Monte Carlo approach. The uncertainties considered concern the gain and phase of the flame response, the outlet acoustic reflection coefficient, and the plenum geometry. It is found that the second-order adjoint method gives quantitative agreement with results based on the full nonlinear eigenvalue problem, while requiring much fewer computations.
Magri L, Bauerheim M, Juniper MP, 2016, Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part I. Sensitivity, Journal of Computational Physics
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.
Magri L, OBrien J, Ihme M, 2016, Compositional inhomogeneities as a source of indirect noise in subsonic and supersonic nozzles, Journal of Fluid Mechanics, Vol: 799, ISSN: 1469-7645
Engine-core noise in aeronautical gas-turbines is commonly divided into direct and indirect noise (Strahle 1978; Dowling & Mahmoudi 2015; Ihme 2017). Direct combustion noise is a source of self-noise, and describes the generation of acoustic pressure uctuations by unsteady heat release in the combustion chamber (Figure 1). In contrast, indirect combustion noise represents an induced noise-source mechanism that arises from the interaction between non-acoustic perturbations exiting the combustion chamber and downstream engine components. The indirect noise generation by temperature inhomogeneities arising from hot and cold spots is referred to as entropy noise (Marble & Candel 1977a), and indirect noise from vorticity uctuations is referred to as vorticity noise (Cumpsty 1979). Once sound has been generated, its propagation through the engine core depends on mean ow gradients and the geometry, which distort, di ract and re ect the acoustic propagation. Contributions of indirect noise to the overall core-noise emission have been examined theoretically and experimentally. These studies focused on separating out the contributions to noise from the (i) direct transmission and (ii) entropy noise. Di erent techniques have been employed to determine the transfer functions, including compact nozzle theories (Marble & Candel 1977a) and expansion methods (Stow et al. 2002; Goh & Morgans 2011; Moase et al. 2007; Giauque et al. 2012; Dur an & Moreau 2013). These theoretical investigations were supported by experimental studies (Bake et al. 2009; Kings & Bake 2010). These studies showed that indirect combustion noise requires consideration in the analysis of engine-core noise and can exceed the contribution from direct noise under some circumstances (see, e.g., Dowling & Mahmoudi 2015).
Magri L, OBrien J, Ihme M, 2016, Compositional inhomogeneities as a source of indirect combustion noise, Journal of Fluid Mechanics, Vol: 799
© 2016 Cambridge University Press. The generation of indirect combustion noise by compositional inhomogeneities is examined theoretically. For this, the compact-nozzle theory of Marble & Candel (J.Sound Vib., vol.55 (2), 1977, pp.225-243) is extended to a multi-component gas mixture, and the chemical potential function is introduced as an additional acoustic source mechanism. Transfer functions for subcritical and supercritical nozzle flows are derived, and the contribution of compositional noise is compared to entropy noise and direct noise by considering an idealized nozzle downstream of the combustor exit. It is shown that compositional noise is dependent on the local mixture composition and can exceed entropy noise for fuel-lean conditions and supercritical nozzle flows. This suggests that the compositional indirect noise requires potential consideration with the implementation of low-emission combustors.
Silva CF, Runte T, Polifke W, et al., 2016, Uncertainty quantification of growth rates of thermoacoustic instability by an adjoint Helmholtz solver
Copyright � 2016 by ASME. The objective of this paper is to quantify uncertainties in thermoacoustic stability analysis with a Helmholtz solver and its adjoint. Thermoacoustic combustion instability may be described by the Helmholtz equation combined with a model for the flame dynamics. Typically, such a formulation leads to an eigenvalue problem in which the eigenvalue appears under nonlinear terms, such as exponentials related to time delays that result from the flame model. Consequently, the standard adjoint sensitivity formulation should be augmented by first- and second-order correction terms that account for the nonlinearities. Such a formulation is developed in the present paper, and applied to the model of a combustion test rig with a premix swirl burner. The uncertainties considered concern plenum geometry, outlet acoustic reflection coefficient, as well as gain and phase of the flame response. The nonlinear eigenvalue problem and its adjoint are solved by an in-house adjoint Helmholtz solver, based on an axisymmetric finite volume approach. In addition to first-order correction terms of the adjoint formulation, which are often used in literature, second-order terms are also taken into account. It is found that one particular second-order term has significant impact on the accuracy of results. Finally, the Probability Density Function of the growth rate in the presence of uncertainties in input paramters is calculated with Monte Carlo simulations. It is found that the second-order adjoint method, while giving quantitative agreement, requires far less compute resources than Monte Carlo sampling for the full nonlinear eigenvalue problem.
Magri L, Juniper MP, 2014, Adjoint-based linear analysis in reduced-order thermo-acoustic models, International Journal of Spray and Combustion Dynamics, Vol: 6, Pages: 225-246, ISSN: 1756-8277
This paper presents the linear theory of adjoint equations as applied to thermo-acoustics. The purpose is to describe the mathematical foundations of adjoint equations for linear sensitivity analysis of thermo-acoustic systems, recently developed by Magri and Juniper (J. Fluid Mech. (2013), vol. 719, pp. 183-202). This method is applied pedagogically to a damped oscillator, for which analytical solutions are available, and then for an electrically heated Rijke tube with a mean-flow temperature discontinuity induced by the compact heat source. Passive devices that most affect the growth rate/frequency of the electrical Rijke-tube system are presented, including a discussion about the effect of modelling the mean-flow temperature discontinuity.
Magri L, Juniper MP, 2014, Global modes, receptivity, and sensitivity analysis of diffusion flames coupled with duct acoustics, Journal of Fluid Mechanics, Vol: 752, Pages: 237-265, ISSN: 0022-1120
© 2014 Cambridge University Press. In this theoretical and numerical paper, we derive the adjoint equations for a thermo-acoustic system consisting of an infinite-rate chemistry diffusion flame coupled with duct acoustics. We then calculate the thermo-acoustic system’s linear global modes (i.e. The frequency/growth rate of oscillations, together with their mode shapes), and the global modes’ receptivity to species injection, sensitivity to base-state perturbations and structural sensitivity to advective-velocity perturbations. Some of these could be found by finite difference calculations but the adjoint analysis is computationally much cheaper. We then compare these with the Rayleigh index. The receptivity analysis shows the regions of the flame where open-loop injection of fuel or oxidizer will have the greatest influence on the thermo-acoustic oscillation. We find that the flame is most receptive at its tip. The base-state sensitivity analysis shows the influence of each parameter on the frequency/growth rate. We find that perturbations to the stoichiometric mixture fraction, the fuel slot width and the heat-release parameter have most influence, while perturbations to the Péclet number have the least influence for most of the operating points considered. These sensitivities oscillate, e.g. positive perturbations to the fuel slot width either stabilizes or destabilizes the system, depending on the operating point. This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. The structural-sensitivity analysis shows the influence of perturbations to the advective flame velocity. The regions of highest sensitivity are around the stoichiometric line close to the inlet, showing where velocity models need to be most accurate.
Magri L, Juniper MP, 2013, A novel theoretical approach to passive control of thermo-acoustic oscillations: Application to ducted heat sources
In this paper, we develop a linear technique that predicts how the stability of a thermo-acoustic system changes due to the action of a generic passive feedback device or a generic change in the base state. From this, one can calculate the passive device or base state change that most stabilizes the system. This theoretical framework, based on adjoint equations, is applied to two types of Rijke tube. The first contains an electrically-heated hot wire and the second contains a diffusion flame. Both heat sources are assumed to be compact so that the acoustic and heat release models can be decoupled. We find that the most effective passive control device is an adiabatic mesh placed at the downstream end of the Rijke tube. We also investigate the effects of a second hot wire and a local variation of the cross-sectional area but find that both affect the frequency more than the growth rate. This application of adjoint sensitivity analysis opens up new possibilities for the passive control of thermo-acoustic oscillations. For example, the influence of base state changes can be combined with other constraints, such as that the total heat release rate remains constant, in order to show how an unstable thermo-acoustic system should be changed in order to make it stable. Copyright © 2013 by ASME.
Galvanetto U, Magri L, 2013, On the use of the theory of dynamical systems for transient problems: A preliminary work on a simple model, Nonlinear Dynamics, Vol: 74, Pages: 373-380, ISSN: 0924-090X
This paper is a preliminary work to address the problem of dynamical systems with parameters varying in time. An idea to predict their behavior is proposed. These systems are called transient systems, and are distinguished from steady systems in which parameters are constant. In particular, in steady systems the excitation is either constant (e.g.; nought) or periodic with amplitude, frequency, and phase angle which do not vary in time. We apply our method to systems, which are subjected to a transient excitation that is neither constant nor periodic. The effect of switching-off and full-transient forces is investigated. The former can be representative of switching-off procedures in machines; the latter can represent earthquake vibrations, wind gusts, etc.; acting on a mechanical system. This class of transient systems can be seen as the evolution of an ordinary steady system into another ordinary steady system, for both of which the classical theory of dynamical systems holds. The evolution from a steady system to the other is driven by a transient force, which is regarded as a map between the two steady systems. © 2013 Springer Science+Business Media Dordrecht.
Magri L, Juniper MP, 2013, Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach, Journal of Fluid Mechanics, Vol: 719, Pages: 183-202, ISSN: 0022-1120
We apply adjoint-based sensitivity analysis to a time-delayed thermo-acoustic system: a Rijke tube containing a hot wire. We calculate how the growth rate and frequency of small oscillations about a base state are affected either by a generic passive control element in the system (the structural sensitivity analysis) or by a generic change to its base state (the base-state sensitivity analysis). We illustrate the structural sensitivity by calculating the effect of a second hot wire with a small heat-release parameter. In a single calculation, this shows how the second hot wire changes the growth rate and frequency of the small oscillations, as a function of its position in the tube. We then examine the components of the structural sensitivity in order to determine the passive control mechanism that has the strongest influence on the growth rate. We find that a force applied to the acoustic momentum equation in the opposite direction to the instantaneous velocity is the most stabilizing feedback mechanism. We also find that its effect is maximized when it is placed at the downstream end of the tube. This feedback mechanism could be supplied, for example, by an adiabatic mesh. We illustrate the base-state sensitivity by calculating the effects of small variations in the damping factor, the heat-release time-delay coefficient, the heat-release parameter, and the hot-wire location. The successful application of sensitivity analysis to thermo-acoustics opens up new possibilities for the passive control of thermo-acoustic oscillations by providing gradient information that can be combined with constrained optimization algorithms in order to reduce linear growth rates. © Cambridge University Press 2013.
Magri L, Balasubramanian K, Sujith RI, et al., 2013, Non-normality in combustion-acoustic interaction in diffusion flames: A critical revision, Journal of Fluid Mechanics, Vol: 733, Pages: 681-683, ISSN: 0022-1120
© 2013 Cambridge University Press. Perturbations in a non-normal system can grow transiently even if the system is linearly stable. If this transient growth is sufficiently large, it can trigger self-sustained oscillations from small initial disturbances. This has important practical consequences for combustion-acoustic oscillations, which are a persistent problem in rocket and aircraft engines. Balasubramanian & Sujith (J. Fluid Mech., vol. 594, 2008, pp. 29-57) modelled an infinite-rate chemistry diffusion flame in an acoustic duct and found that the transient growth in this system can amplify the initial energy by a factor, Gmax, of the order of 105 to 107. However, recent investigations by L. Magri and M. P. Juniper have brought to light certain errors in that paper. When the errors are corrected, Gmax is found to be of the order of 1 to 10, revealing that non-normality is not as influential as it was thought to be.
Magri L, Juniper MP, 2013, A theoretical approach for passive control of thermoacoustic oscillations: Application to ducted flames, Journal of Engineering for Gas Turbines and Power, Vol: 135, ISSN: 0742-4795
In this paper, we develop a linear technique that predicts how the stability of a thermoacoustic system changes due to the action of a generic passive feedback device or a generic change in the base state. From this, one can calculate the passive device or base state change that most stabilizes the system. This theoretical framework, based on adjoint equations, is applied to two types of Rijke tube. The first contains an electrically heated hot wire, and the second contains a diffusion flame. Both heat sources are assumed to be compact, so that the acoustic and heat release models can be decoupled. We find that the most effective passive control device is an adiabatic mesh placed at the downstream end of the Rijke tube. We also investigate the effects of a second hot wire and a local variation of the cross-sectional area but find that both affect the frequency more than the growth rate. This application of adjoint sensitivity analysis opens up new possibilities for the passive control of thermoacoustic oscillations. For example, the influence of base state changes can be combined with other constraints, such as that the total heat release rate remains constant, in order to show how an unstable thermoacoustic system should be changed in order to make it stable. Copyright © 2013 by ASME.
Magri L, Galvanetto U, 2012, Example of a non-smooth Hopf bifurcation in an aero-elastic system, Mechanics Research Communications, Vol: 40, Pages: 26-33, ISSN: 0093-6413
We investigate a typical aerofoil section under dynamic stall conditions, the structural model is linear and the aerodynamic loading is represented by the Leishman-Beddoes semi-empirical dynamic stall model. The loads given by this model are non-linear and non-smooth, therefore we have integrated the equation of motion using a Runge-Kutta-Fehlberg algorithm equipped with event detection. The main focus of the paper is on the interaction between the Hopf bifurcation typical of aero-elastic systems, which causes flutter oscillations, and the discontinuous definition of the stall model. The paper shows how the non-smooth definition of the dynamic stall model can generate a non-smooth Hopf bifurcation. The mechanisms for the appearance of limit cycle attractors are described by using standard tools of the theory of dynamical systems such as phase plots and bifurcation diagrams. © 2012 Elsevier Ltd © 2012 Elsevier Ltd. All rights reserved.
Doan NAK, Polifke W, Magri L, Physics-Informed Echo State Networks for Chaotic Systems Forecasting, Lecture Notes in Computer Science, Vol: 11539, ISSN: 0302-9743
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.
Traverso T, Magri L, Data Assimilation in a Nonlinear Time-Delayed Dynamical System with Lagrangian Optimization, Lecture Notes in Computer Science, Vol: 11539, ISSN: 0302-9743
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.
Galvanetto U, Magri L, Use of the theory of dynamical systems for transient problems: application to the switching-on problem, AIMETA - Theoretical and applied mechanics conference
This paper addresses the problem of dynamical systems with parameters varying in time (transient systems). A method to predict their behaviour is proposed. This class of transient systems can be seen as the evolution of an ordinary steady system into another ordinary steady system, for both of which the classical theory of dynamical systems holds. The evolution from a steady system to the other is driven by a transient force, which is regarded as a map between the two steady systems. We apply our method to a system which is subjected to a transient excitation, that is neither constant nor periodic, to simulate the effect of switching-on procedures.
Blonigan P, Fernandez P, Murman SM, et al., Towards a Chaotic Adjoint for LES, Center for Turbulence Research Proceedings of the Summer Program
Adjoint-based sensitivity analysis methods are powerful tools for engineers who use ow simulations for design. However, the conventional adjoint method breaks down for scale-resolving simulations like large-eddy simulation (LES) or direct numerical simula- tion (DNS), which exhibit the chaotic dynamics inherent in turbulent ows. Sensitivity analysis based on least-squares shadowing (LSS) avoids the issues encountered by con- ventional methods, but has a high computational cost. The following report outlines a new, more computationally e cient formulation of LSS, non-intrusive LSS, and estimates its cost for several canonical ows using Lyapunov analysis.
Juniper M, Magri L, Bauerheim M, et al., Sensitivity analysis of thermo-acoustic eigenproblems with adjoint methods, Center for Turbulence Research Proceedings of the Summer Program
This paper outlines two new applications of adjoint methods in the study of thermoacoustic instability. The first is to calculate gradients for the active subspace method, which is used in uncertainty quantification. The second is to calculate gradients in a nonlinear thermo-acoustic Helmholtz solver. Two methods are presented. The first, which uses the discrete adjoint approach, is specifically for nonlinear Helmholtz eigenvalue problems that are solved iteratively. The second, which uses a hybrid adjoint approach, is more general and can be applied to both problems.
Magri L, See YC, Ihme M, et al., Adjoint sensitivity analysis of thermo-acoustic/hydrodynamic instabilities in turbulent combustion chambers, Center for Turbulence Research Proceedings of the Summer Program
In this paper, we de ne a mathematically consistent set of thermo-acoustic equations via asymptotic multiple scale methods in the low-Mach number limit. The nal thermoacoustic equations consist of reacting low-Mach number (LMN) equations for hydrodynamic phenomena and acoustic (AC) equations. The two sets of equations are two-way coupled. The coupling terms depend on which multiple scales are used. We derive and discuss the coupling terms for three distinct limits: double-time-double-space (2T-2S); double-time-single-space (2T-1S); and single-time-double-space (1T-2S). We linearize the thermo-acoustic equations around the mean ow, which is obtained by time averaging Large-Eddy simulations. We show that only 1T-2S provides a two-way coupled linearized system. In the other limits, the coupling from the AC to the LMN is of higher order. We perform global direct and adjoint analysis to identify unstable modes and passivefeedback mechanisms to stabilize/lower the frequency of the oscillation. Preliminary results are shown for a dual-swirl gas turbine combustor and a simpli ed dump combustor.
Qadri U, Magri L, Ihme M, et al., Optimal ignition placement in diffusion flames by nonlinear adjoint looping, Center for Turbulence Research Proceedings of the Summer Program
We derive and implement a gradient-based optimization framework to identify the best place to ignite a mixture of fuel and oxidizer. We have used the time-averaged in- crease in thermal energy as the cost function that indicates the success of ignition. We use direct numerical simulation (DNS) of the low Mach number Navier–Stokes equations with single-step finite-rate reaction chemistry to model spark ignition of an axisymmetric methane jet in air.We model the spark as a two-dimensional Gaussian function. We solve the adjoint of these equations to obtain the sensitivity of the cost function with respect to the position of the spark. We find that the optimal locations track the stoichiometric mixture fraction surface in the flow. At low Reynolds numbers, the optimization prefers locations downstream that allow upstream flame propagation.
Nair V, Nastac G, Magri L, et al., Inspecting Lagrangian coherent structures in turbulent combustion simulations, Center for Turbulence Research Proceedings of the Summer Program
In this study, fnite time Lyapunov exponent (FTLE) fields from turbulent reacting flows are evaluated. The unsteady turbulent velocity field is obtained from large-eddy simulations, and reacting (DLR Flame A) and non-reacting unconfined jet configurations are considered. Backward FTLE analysis on the velocity field is performed, and the anal- ysis shows that the ridges approximate the flow structure associated with the organized fuid structure. To examine the merit of this method in identifying coherent structures, we compare the results with the well-known Q-criterion.
Labahn J, Nastac G, Magri L, et al., Determining dynamic content of turbulent flow LES using the Lyapunov exponent, Center for Turbulence Research Annual Research Briefs
Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) have been em- ployed for computing turbulent flows . While DNS resolves all turbulent scales involved in the dynamics with no physical modeling, LES represents the energy contained in the large scales, and effects of the smaller scales are taken into account either explicitly through a subgrid scale model or implicitly through the numerical dissipation of the nu- merical method. Besides numerical algorithms, two factors determine the quality of LES: the physical model or dissipation of the subgrid scales (SGS), which are filtered out in the governing equations, and the filter width, which describes the numerical resolution of the resolved scales.
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