Mode transition in Rotating Detonation Engines (RDEs) refers to an abrupt change in the number of detonation waves due to a change in inlet conditions such as the injected fuel reactivity and total pressure. Previous theories describing mode transition were based on 2D detonation tube models, and stipulate that the detonation wave (DW) height should be an integral multiple of the detonation cell width for stability. According to this mechanism, with changes in inlet conditions, the detonation cell width can change, and along with it the DW height resulting in mode transition. Through detailed numerical simulations in a 2D unrolled RDE geometry, an alternate mechanism for mode transition is proposed, along with a corresponding quantitative criterion that is validated using simulation data. We observed mode transition when the N_2 dilution of the injected fuel mixture was reduced, so that the more reactive, fresh mixture injected into the combustion chamber triggered a localized, micro-detonation to form. In the simulations, we observe the micro-detonation to eventually lead to a mode transition when τ_MD<τ_L, where τ_L is the time of revolution of the parent DW and τ_MD is the time required for a ‘micro-detonation’ to form. When this criteria was not satisfied, the parent DW consumed the fuel mixture in the hot spot, before a daughter wave could form. A relationship to predict the number of DWs following mode transition is also proposed and verified using simulation data.In the second part of this thesis, we describe detailed numerical simulations of a liquid fuel droplet impacted by a Mach 5 shock wave, considering the effects of chemical reactions and phase change due to evaporation. In our baseline case, a 5 μm, n-Dodecane fuel droplet is preheated to 460 K, and surrounded by preheated O_2 gas at 700 K. The fuel droplet undergoes significant deformation and morphological changes following shock impingement, as the droplet surface becomes unstable to the Kelvin-Helmholtz instability. The droplet core is also observed to eject a thin sheet near the equatorial plane, which is then stretched by the high-speed post-shock gas flow, affecting the late-time behavior. We find the observed dominant modes associated with the Kelvin-Helmholtz instability are in reasonable agreement with linear theory [1, 2], when the local conditions at the droplet surface are considered. The production of fuel vapors by the droplet impairs the growth of such surface instabilities, leading to reduced growth of the droplet surface area when compared with a non-evaporating droplet. Furthermore, an evaporation-induced Stefan flow is established which blows off the hot post-shock gasses surrounding the droplet, leading to droplet cooling. As the fuel vapors react, a diffusion flame is formed on the droplet-windward side, leading to intense droplet heating and enhanced vapor production in this region. In contrast, the leeward side of the droplet is occupied by pre-shock gasses entrapped in a low-pressure region formed by flow separation, resulting in lower temperatures and vapor production at that site. We investigated the effect of the Damkohler number on droplet evolution by varying the fuel reactivity, and found the flame thickness decreased with increasing reactivity in agreement with trends predicted by laminar diffusion flame theory. In addition, the greater consumption of fuel vapor in the region surrounding the droplet at higher reactivities, resulted in increased growth of the droplet surface area, and increased expansion rate of the ejected thin sheet structures. At the highest reactivity, secondary burning of fuel vapors was observed in the droplet wake, while the resulting flame eventually reattached to the droplet surface. Our results show significant spatial inhomogeneities are present in the droplet flowfield in all the cases investigated, which must be considered in the development of reduced order point-particle models for system-level simulations of detonation engines.