Type Ia supernovae (SNe) are bright astrophysical explosions that form a remarkably homogeneous class of objects serving as the premier distance indicators for studying the expansion history of the Universe and the nature of dark energy. Despite the widespread acceptance of the surprising discovery of the acceleration of the expansion of the Universe and the existence of the mysterious dark energy driving it that followed from these studies, the progenitor systems of these explosions are unknown. Knowledge of the progenitor system is required to understand possible systematic effects due to properties of the parent stellar population or host galaxy. While several scenarios have been proposed, the most widely accepted one is the thermonuclear explosion of a near-Chandrasekhar-mass, carbon-oxygen white dwarf (WD). Under this scenario, the explosive burning begins near the center as a deflagration (subsonic burning) that transitions to a detonation (supersonic burning) some time later after the WD has expanded in response to the energy release. Turbulence, either pre-existing or generated by burning, serves to increase the surface area of the burning front, thus enhancing the fuel consumption rate. In addition, turbulence-flame interaction (TFI) may be responsible for deflagration-to-detonation transition (DDT). Simulations of this explosion scenario typically parameterize the DDT to occur when the flame reaches a particular density. I performed a suite of two-dimensional simulations with the compressible, hydrodynamics code FLASH to evaluate the influence of the DDT density on the average yield of radioactive Ni-56 that powers the SN light curve. In addition, I considered the compositional dependence of the DDT density to explore one way in which metallicity may influence the explosion outcome. My results have confirmed a new pathway to explain observed trends in the average peak brightness of SNe Ia with host galaxy metallicity. In a separate study, I address the basic physics of modeling flames and turbulent combustion. The disparate length scales in the SN necessitate use of a flame model to capture the effect of burning on unresolved scales. I implemented a method to measure the strength of unresolved turbulence, which is used to estimate the amount of wrinkling of the unresolved flame surface. In addition, the measure of turbulent strength may be used to improve the criterion by which DDT is initiated. These improvements will allow three-dimensional simulations of the early flame evolution in the presence of strong pre-existing turbulence. The research conducted for this dissertation has led to important insights into the explosion mechanism of SNe Ia. In addition, improvements to the model have allowed and will continue to allow simulations of unprecedented realism of the complex process of exploding WDs in a thermonuclear SN.