UVM Theses and Dissertations
Format:
Print
Author:
Reardon, Michael S.
Dept./Program:
Mathematics
Year:
2012
Degree:
PhD
Abstract:
After a brief introduction, a model for solid propellant combustion in a three dimensional rectangular channel is re-introduced. The model was originally developed by Margolis and Armstrong in [28]. It incorporates a kinetic reaction zone separating a homogeneous solid fuel/oxidizer mixture from a gaseous products zone which is shown to collapse to a thin flame sheet in the asymptotic limit of a large modified nondimensional activation energy. The model equations derived via matched asymptotic expansions consist of a system of partial differential equations in the two separate regions on either side of the flame sheet linked by jump conditions at the interface. This system admits a basic solution where the flat flame sheet moves with constant velocity in the direction of the reagents.
The structure and stability properties of small perturbations in the temperature and front velocity about the basic solution is then investigated. A linear stability analysis shows that for certain physical parameter values a Hopf bifurcation occurs as the modified nondimensional activation energy is increased giving rise to persistent oscillatory solutions. Multiple scale expansions are utilized to describe changes in the amplitude and phase of perturbations to the basic solution in slow-time when the physical parameters are set in the weakly nonlinear regime. This analysis is performed in one and two dimensions and in the cases where a single mode or a pair of modes lose stability. The analytical studies result in a single ODE or a pair of coupled ODE's respectively of Stuart-Landau type which determine the amplitude of the linear mode(s).
The asymptotic model equations are then discretized using the Crank-Nicolson finite difference method and a direct numerical simulation of the asymptotic equations is carried out for certain parameter combinations. Lastly, the numerical results are compared with the analytical predictions and it is shown that for much of the physical parameter space the analytical and numerical solutions agree. In the physical parameter regimes where the analytical results predict supercritical Hopf bifurcations the slow-time amplitude functions are shown to approach orbits of finite radius in the complex plane where their limit cycle radius and rate of approach is dependent on problem parameters and possibly the initial conditions.
The structure and stability properties of small perturbations in the temperature and front velocity about the basic solution is then investigated. A linear stability analysis shows that for certain physical parameter values a Hopf bifurcation occurs as the modified nondimensional activation energy is increased giving rise to persistent oscillatory solutions. Multiple scale expansions are utilized to describe changes in the amplitude and phase of perturbations to the basic solution in slow-time when the physical parameters are set in the weakly nonlinear regime. This analysis is performed in one and two dimensions and in the cases where a single mode or a pair of modes lose stability. The analytical studies result in a single ODE or a pair of coupled ODE's respectively of Stuart-Landau type which determine the amplitude of the linear mode(s).
The asymptotic model equations are then discretized using the Crank-Nicolson finite difference method and a direct numerical simulation of the asymptotic equations is carried out for certain parameter combinations. Lastly, the numerical results are compared with the analytical predictions and it is shown that for much of the physical parameter space the analytical and numerical solutions agree. In the physical parameter regimes where the analytical results predict supercritical Hopf bifurcations the slow-time amplitude functions are shown to approach orbits of finite radius in the complex plane where their limit cycle radius and rate of approach is dependent on problem parameters and possibly the initial conditions.