Non-normality and nonlinearity in thermoacoustic instabilities

Analysis of thermoacoustic instabilities were dominated by modal (eigenvalue) analysis for many decades. Recent progress in nonmodal stability analysis allows us to study the problem from a different perspective, by quantitatively describing the short-term behavior of disturbances. The short-term evolution has a bearing on subcritical transition to instability, known popularly as triggering instability in thermoacoustic parlance. We provide a review of the recent developments in the context of triggering instability. A tutorial for nonmodal stability analysis is provided. The applicability of the tools from nonmodal stability analysis are demonstrated with the help of a simple model of a Rjike tube. The article closes with a brief description of how to characterize bifurcations in thermoacoustic systems

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

Investigation of the Effects of Oxidizer Temperature on the Stability of a Gas-Centered Swirl Coaxial Injector

Rocket engines achieve extraordinary high energy densities within the chamber in the form of high pressure turbulent combustion. Successful design of these engines requires sustained, stable operation of a combustor exposed to extreme thermal loads. Slight deviations in operating conditions can then incur consequences ranging from reduced performance up to catastrophic failure in the face of excess heat loading. Sustained periodic oscillations, termed combustion instabilities, are often encountered during development, as fluctuations produced by combustion noise couple with heat release modes by way of modulation of the feed system, injector hydrodynamics, chemical kinetics, and mixing and atomization process. Successful development of reliable, high performance rocket engines can be achieved either through a thorough understanding of both injector and combustor dynamics to mitigate these instabilities or through the laborious design/test iteration process. This document describes a two-fold work by the author. The first objective considers the acquisition of high-fidelity data sets of a single gas-centered swirl coaxial injector for use in the validation of computational models. Secondly, the stability of this injector was studied at two oxidizer inlet temperatures. Combustion stability was assessed through variation of the combustor geometry. Previous research shows that varying this geometry can either drive or dampen pressure oscillations. Testing was conducted on an experimental test bed equipped with modular sections to accommodate changing oxidizer post and chamber lengths. A single gas-centered swirl coaxial injector was used, with operating parameters based on the RD-180 injector element, such as equivalence and momentum flux ratios. Two oxidizer inlet temperatures were chosen. The first was oxygen combusted with gaseous hydrogen at lean conditions in a preburner to produce hot oxidizer near 700 K. The second was pure oxygen delivered at room temperature. Results from the test campaign revealed the system to be classically stable across all configurations and inlet conditions tested, with pressure perturbations less than 10% of the mean chamber pressure. Discriminating behavior was observed between the two oxidizer inlet temperatures. At elevated temperatures, peak-to-peak pressure oscillations observed throughout the system were small at less than 4% of the mean chamber pressure. There was no observed dependency of the amplitude on geometry. At ambient temperatures, the pressure oscillations ranged from 4% up to 7%. The increase in amplitudes were similar to that of the acoustic refection coefficient between the oxidizer and chamber gas, based on their acoustic specific impedance. An increase in the acoustic transmission coefficient was also observed, going from hot to ambient oxidizer. The increase in these two values would not necessarily lead to enhanced coupling between the chamber and resonance behavior in the post, but is expected to amplify pressure oscillations. At the ambient condition, clear variation in amplitudes were generated through manipulation of the geometry. The general trend matched previous experiments but was not followed by all tested configurations. It was determined that a methodology solely based on the effective resonator wavelength was not sufficient to predict the amplitude of pressure oscillations. Instead, a better predictor of amplitude was found based on the alignment of the system with postulated vortex generation from the injector face and impingement on the chamber walls. The time between local pressure oscillations and final impingement of the resulting vortices fell between one and two cycles of the fundamental longitudinal chamber mode, increasing linearly in strength as phase lag increased

Mathematical Analysis of Some Partial Differential Equations with Applications

In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2. In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature

Large Eddy Simulations of gaseous flames in gas turbine combustion chambers

Recent developments in numerical schemes, turbulent combustion models and the regular increase of computing power allow Large Eddy Simulation (LES) to be applied to real industrial burners. In this paper, two types of LES in complex geometry combustors and of specific interest for aeronautical gas turbine burners are reviewed: (1) laboratory-scale combustors, without compressor or turbine, in which advanced measurements are possible and (2) combustion chambers of existing engines operated in realistic operating conditions. Laboratory-scale burners are designed to assess modeling and funda- mental flow aspects in controlled configurations. They are necessary to gauge LES strategies and identify potential limitations. In specific circumstances, they even offer near model-free or DNS-like LES computations. LES in real engines illustrate the potential of the approach in the context of industrial burners but are more difficult to validate due to the limited set of available measurements. Usual approaches for turbulence and combustion sub-grid models including chemistry modeling are first recalled. Limiting cases and range of validity of the models are specifically recalled before a discussion on the numerical breakthrough which have allowed LES to be applied to these complex cases. Specific issues linked to real gas turbine chambers are discussed: multi-perforation, complex acoustic impedances at inlet and outlet, annular chambers.. Examples are provided for mean flow predictions (velocity, temperature and species) as well as unsteady mechanisms (quenching, ignition, combustion instabil- ities). Finally, potential perspectives are proposed to further improve the use of LES for real gas turbine combustor designs

Bibliographies of the Jet Propulsion Laboratory, Numbers 39-1 through 39-5. Cumulative Bibliographic Index No. 41-1, Jan. 1938 - Jun. 1964

Cumulative index of Jet Propulsion Laboratory bibliographie

Nonlinear thermoacoustic oscillations of a ducted laminar premixed flame

This thesis was awarded the Leonardo da Vinci prize in 2013 for outstanding doctoral research in fluid mechanics, turbulence, combustion and acoustics (a Europe wide award). This thesis was also awarded the silver award of the ERCOFTAC (European research community on flow, turbulence and combustion) 10th Osborne Reynolds Award in 2013.Finding limit cycles and their stability is one of the central problems of nonlinear thermoacoustics. However, a limit cycle is not the only type of self-excited oscillation in a nonlinear system. Nonlinear systems can have quasi-periodic and chaotic oscillations. This thesis examines the different types of oscillation in a numerical model of a ducted premixed flame, the bifurcations that lead to these oscillations and the influence of external forcing on these oscillations. Criteria for the existence and stability of limit cycles in single mode thermoacoustic systems are derived analytically. These criteria, along with the flame describing function, are used to find the types of bifurcation and minimum triggering amplitudes. The choice of model for the velocity perturbation field around the flame is shown to have a strong influence on the types of bifurcation in the system. Therefore, a reduced order model of the velocity perturbation field in a forced laminar premixed flame is obtained from Direct Numerical Simulation. It is shown that the model currently used in the literature precludes subcritical bifurcations and multi-stability. The self-excited thermoacoustic system is simulated in the time domain with many modes in the acoustics and analysed using methods from nonlinear dynamical systems theory. The transitions to the periodic, quasiperiodic and chaotic oscillations are via sub/supercritical Hopf, Neimark-Sacker and period-doubling bifurcations. Routes to chaos are established in this system. It is shown that the single mode system, which gives the same results as a describing function approach, fails to capture the period-$2$, period-$k$, quasi-periodic and chaotic oscillations or the bifurcations and multi-stability seen in the multi-modal case, and underpredicts the amplitude. Instantaneous flame images reveal that the wrinkles on the flame surface and pinch off of flame pockets are regular for periodic oscillations, while they are irregular and have multiple time and length scales for quasi-periodic and chaotic oscillations. Cusp formation, their destruction by flame propagation normal to itself, and pinch-off and rapid burning of pockets of reactants are shown to be responsible for generating a heat release rate that is a highly nonlinear function of the velocity perturbations. It is also shown that for a given acoustic model of the duct, many discretization modes are required to capture the rich dynamics and nonlinear feedback between heat release and acoustics seen in experiments. The influence of external harmonic forcing on self-excited periodic, quasi-periodic and chaotic oscillations are examined. The transition to lock-in, the forcing amplitude required for lock-in and the system response at lock-in are characterized. At certain frequencies, even low-amplitude forcing is sufficient to suppress period-$1$ oscillations to amplitudes that are 90$\%$ lower than that of the unforced state. Therefore, open-loop forcing can be an effective strategy for the suppression of thermoacoustic oscillations. This thesis shows that a ducted premixed flame behaves similarly to low-dimensional chaotic systems and that methods from nonlinear dynamical systems theory are superior to the describing function approach in the frequency domain and time domain analysis currently used in nonlinear thermoacoustics.University Gas Turbine Partnership, Rolls Royce Plc., EPSRC, Dorothy Hodgkin Postgraduate Award, Trinity college fellowships, Ford foundation fellowship, Rouse Ball and Eddington fellowships, American Society of Mechanical Engineers, International Gas Turbine Institute, The Combustion Institute