Date of Award

2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mechanical and Aerospace Engineering

Committee Chair

Sarma L. Rani

Committee Member

Kader Frendi

Committee Member

Robert Frederick

Committee Member

Babak Shotorban

Committee Member

Sivaguru Ravindran

Subject(s)

Combustion--Research., Combustion chambers--Mathematical models.

Abstract

The primary objective of this dissertation is to develop and investigate various analytical methods to predict thermoacoustic instabilities in premixed combustion systems. The analytical models derived as part of this study are of four main types: (1) Acoustically consistent, linear modal analysis method to predict the longitudinal and transverse combustion instabilities in a dump combustor; (2) Novel level set method for deriving flame surface-area response to incident acoustic fluctuations; (3) Novel approximate analytical solutions to acoustic waves in inhomogeneous media; and (4) Investigation of the limit-cycle behavior of nonlinear acoustic wave equation with combustion source term. The linear modal analysis method that was developed in this study, has a number of novel and distinguishing features when compared to prior works on combustion instability. (i) Combustion instabilities are a thermoacoustic phenomenon, i.e.~they are manifested as self-excited acoustic oscillations that are sustained by a feedback loop between the acoustic perturbations and the flame heat-release fluctuations. Therefore, first and foremost, an instability model must be able to predict the natural acoustic modes of the combustor in the absence of combustion. Our model satisfies this criterion by successfully predicting the acoustic modes of ducts with multiple discontinuities in cross-sectional area. Such a consistency testing of an instability model had not been performed previously. (ii) New acoustically consistent matching conditions with distinct forms for the purely axial and non-axial modes were developed and applied at the zonal interfaces of a duct, whereas prior studies employed the conventional mass, momentum, and energy balances at the interfaces. For the purely axial modes, acoustic mass velocity and total pressure are mathced across the interface while for non-axial modes, the continuity of acoustic velocity and pressure fluctuations is applied. The new matching conditions are essential to accurately predict the duct acoustic modes. (iii) Effects of edge conditions on the linear modal analysis of ducts with area discontinuities are analyzed in great detail. Edge conditions are constraints that need to be satisfied in addition to the matching conditions at an area discontinuity. (iv) Through a novel approach, the effects of the fluctuating heat-release source term in the acoustic wave equation are directly incorporated into the modified axial wavenumbers in the combustion region. This approach obviates the need for applying a separate matching condition across the flame. (v) The analytical model presented in this work, is the first to account for realistic mean flame shapes in combustion instability analysis, whereas prior models assumed that combustion occurred in a cross-sectional plane of zero thickness. A new $G$-equation level-set method was developed that describes flame surface-area response to acoustic oscillations incident on the flame. This method presents a different paradigm when compared to an approach that has existed for at least two decades. In this method, we directly solve for the level set fluctuations $G'$ in terms of velocity fluctuations, and relate the flame surface-area oscillations to $G'$, whereas in the conventional $f$-approach, the level-set $G$ is expressed as $G(x,y,t) = x - f(y,t)$ and a solution for $f$ is sought. In the absence of turbulent flame-speed fluctuations, the response functions from the present $G$-equation approach are in good agreement with those from the conventional $f$-equation approach. However, when turbulent flame-speed fluctuations are included, the two approaches differ, principally in the flame response to axial velocity fluctuations. This $G$-equation approach is more generalized since the effects of flame-speed fluctuations are reflected in both the axial and tranverse velocity response functions; whereas in the $f$-equation approach, this inclusion predominantly affects the axial velocity fluctuations. As part of this work, an analytical Wentzel-Kramers-Brillouin (WKB)-type approximation for a 2-D acoustic duct with axial gradients in temperature, axial mean flow and duct cross-sectional area has been developed. Standard WKB method uses two principal approximations: (i) the amplitude of the wave varies slowly compared to its frequency and (ii) the mean properties vary slowly in space. The modified WKB method developed in this study relaxes the latter assumption of slowly varying mean properties. Using this novel WKB-type solutions along with the acoustically consistent modal analysis framework that was developed earlier, we are able to compute acoustic resonant frequencies as well as predict lonigtudinal unstable modes of a duct with discontinuities, which has not been done before. In this study, the effects of nonlinear acoustic and combustion source terms on the limit-cycle behavior of thermoacoustic instabilities in a Rijke tube has also been investigated. First, the relevant parameters that dictate the linear instbility viz., convective time-lag and mean heat-release rate are identified. And in the linearly unstable regime, the limit-cycle amplitudes for the first two longitudinal modes of a duct are computed. It was found that at higher mean heat release rates, combustion source term nonlinearities are more important to achieve limit-cycle amplitudes than purely acoustic nonlinearities.

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