Mathematical Modeling and Numerical Approximations of Combustion Instability Frequencies and Growth Rates

This dissertation presents a mathematical model and numerical simulations to determine the resonant frequencies and their associated growth rates for longitudinal modes in a combustion system similar to that found in a rocket engine. The mathematical model, which is applicable to a two-duct system with a thin flame between the two ducts, each of which having constant area and properties, considers the case of axial mean velocity and uses a vibrating wall at the inlet to select the frequency so that all modes may be found. The model is applied to the acoustics equations describing pressure and velocity fluctuations, derived from the linearized Euler equations. The combustion process is modeled with the n-τ flame transfer function. A novel PDE-constrained optimization formulation is proposed to determine the resonant frequencies, which are sought as the maximizers of a pseudo-energy functional of the pressure field. To compute the PDE-constrained optimization problem, a combination of the secant and bisection methods is used as the iterative solver. Within each iterative step, the PDE system is solved with a numerical method. Both a finite volume method and a discontinuous Galerkin method are used to solve the one-dimensional acoustics equations to gauge the performance of the proposed mathematical approach and to validate it using analytical results. In addition, the discontinuous Galerkin method is extended to solve the two-dimensional acoustics equations. After the resonant frequencies are determined, the growth rates are obtained for each resonant frequency by using the least squares method to fit the pressure profile to a geometric sum that describes the acoustics of the system. Computer simulations are provided to show efficiency and accuracy of the proposed mathematical and numerical approaches.