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

Adjoint characteristic decomposition of one-dimensional waves

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, J. G. et al, 2017, J. Computational Physics, vol. 341, 163-181). 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

On indirect noise in multicomponent nozzle flows

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.The author is supported by the Royal Academy of Engineering Research Fellowships Scheme

Global modes, receptivity, and sensitivity analysis of diffusion flames coupled with duct acoustics

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. 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 most 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 P\'eclet number have least influence. 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.This work is supported by the European Research Council through Project ALORS 2590620.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2014.32

Adjoint-based linear analysis in reduced-order thermo-acoustic models

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.L.M.’s PhD is supported by the European Research Council through Project ALORS 2590620. The Cambridge Philosophical Society (UK) is gratefully acknowledged for having partially covered travel costs for n3l conference, 2013.This is the final published version. It first appeared at http://multi-science.metapress.com/content/h751237233508647/?p=e2821b43d67e47229c5304862df2adce&pi=1

BENDING THE DOMING EFFECT IN STRUCTURE FROM MOTION RECONSTRUCTIONS THROUGH BUNDLE ADJUSTMENT

Structure from Motion techniques provides low-cost and flexible methods that can be adopted in arial surveying to collect topographic data with accurate results. Nevertheless, the so-called "doming effect", due to unfortunate acquisition conditions or unreliable modeling of radial distortion, has been recognized as a critical issue that disrupts the quality of the attained 3D reconstruction. In this paper we propose a novel method, that works effectively in the presence of a nearly flat soil, to tackle a posteriori the doming effect: an automatic ground detection method is used to capture the doming deformation flawing the reconstruction, which in turn is wrapped to the correct geometry by iteratively enforcing a planarity constraint through a Bundle Adjustment framework. Experiments on real word datasets demonstrate promising results

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

Electron drift velocity measurements in liquid krypton–methane mixtures

Abstract Electron drift velocities have been measured in liquid krypton, pure and mixed with methane at different concentrations (1–10% in volume) versus electric field strength, and a possible effect of methane on electron lifetime has been investigated. While no effect on lifetime could be detected, since lifetimes were in all cases longer than what measurable, a very large increase in drift velocity (up to a factor 6) has been measured

Quasi-BiHamiltonian Systems and Separability

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.Comment: 10 pages, AMS-LaTeX 1.1, to appear in J. Phys. A: Math. Gen. (May 1997

Compositional inhomogeneities as a source of indirect combustion noise

The generation of indirect combustion noise by compositional inhomogeneities is examined theoretically. For this, the compact nozzle theory of~\cite{MARBLE_CANDEL_JSV1977} 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.Financial support through NASA with award number NNX15AV04A and the Ford–Stanford Alliance project no. C2015-0590 is gratefully acknowledged.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2016.39