Date of Award
2024
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
Sivaguru S. Ravindran
Committee Member
Chang-kwon Kang
Committee Member
John Bennewitz
Research Advisor
Sarma L. Rani
Subject(s)
Combustion chambers, Dynamics, Nonlinear theories, Theory of combustion, Physics
Abstract
The overall objective of this research is to analyze the nonlinear dynamics of combustors in one dimensional Cartesian coordinate. This research was carried out in four parts, with each successive part building on the previous one. The first part begins with investigation of the coupled nonlinear evolution of acoustic modes in a quasi 1-D duct with axially inhomogeneous mean velocity and mean thermodynamic properties. A novel modification of the Krylov--Bogoliubov method of averaging (KBMA) is developed to analytically solve the modal-amplitude equations that are linearly and nonlinearly coupled with both quadratic and cubic nonlinearities. Approximate analytical solutions based on the method of multiple scales (MMS) are also derived for the coupled modal-amplitude equations with cubic nonlinearities. Novel initial conditions are derived for the KBMA and MMS modal amplitude and phase equations that are essential to validate the analytical solutions using numerical solutions. The second part is an advancement of first part where we derive novel solutions based on the MMS to the modal amplitude equations constitute a multi-degree-of-freedom system of linearly and nonlinearly coupled ordinary differential equations with quadratic and cubic nonlinearities. Due to the presence of both quadratic and cubic terms, the perturbation expansion for the MMS solution necessary includes terms of three orders: $O(\epsilon)$, $O(\epsilon^2)$ and $O(\epsilon^3)$, where $\epsilon$ is the small parameter. The KBMA solutions are included in the second part to compare the MMS and KBMA approaches for the nonlinear equations with and without linear coupling. In the third part, a nonlinear coupled-oscillator model is developed that can predict the various stages of synchronization between the acoustic and heat-release oscillations in the lead-up to thermoacoustic instability in a laboratory scale dump combustor with a bluffbody-anchored flame. The coupled nonlinear equations governing the temporal evolution of pressure and heat-release-rate fluctuations are evolved along with an equation for the build-up of circulation at the dump plane, as well as an equation for vortex advection. The effects of vortex impingement on the flame are modeled as a localized, instantaneous source term in the flame oscillator equation. With the mean flow velocity as the control parameter, the nonlinear model is applied to investigate the acoustic–flame–vortex interactions. In the fourth step, we investigate the role of acoustic and combustion nonlinearities in triggering a stable cylindrical combustion chamber. Using the Galerkin method of weighted spatial integration, we derive the nonlinear equations governing the time-dependent amplitudes of the longitudinal modes of the combustion chamber. The modal amplitude equations include quadratic and cubic acoustic nonlinearities, as well as combustion nonlinearities. The latter, arising from unsteady combustion processes, are represented using three separate models that include Crocco’s pressure exponent–time lag model and two other nonlinear combustion response models.
Recommended Citation
Kathalagiri Vasantha kumar, Swarnalatha, "Nonlinear dynamics of acoustic fluctuations in one-dimensional ducts with and without unsteady combustion" (2024). Dissertations. 423.
https://louis.uah.edu/uah-dissertations/423