Premixed reactive mixtures of fuel and oxidizer often contain regions of higher temperature. These regions of raised temperature are generally called hot spots and have been shown to be the autoignition center in reactive mixtures. Linear temperature gradients and thermal stratification are often used to characterize their behavior. In this work, hot spots are modeled as a linear temperature gradient adjacent to a constant temperature plateau. This approach retains the simplicity of a linear temperature gradient, but captures the effects of a local temperature maximum of finite size. A one-step Arrhenius reaction for H₂-Air is used to model the reactive mixture.

A one dimensional model is considered first to characterize hot spot behavior based on the relation between how quickly the fuel reacts (excitation time) and the time it takes for fluid motion to be induced (acoustic time). Plateaus of three different initial sizes spanning two orders of magnitude are simulated. Each length corresponds to a different ratio of excitation time to acoustic time or acoustic timescale ratio. It is shown that ratios less than unity react at nearly isochoric conditions while ratios greater than unity react at nearly isobaric conditions. It is demonstrated that the gasdynamic response is characterized by the a priori prescribed hot spot acoustic timescale ratio. Based upon this ratio, it is shown that the plateau can have either a substantial or negligible impact on the reaction of a surrounding temperature gradient. This is explored further as the slope of the temperature gradient is varied. Plateaus with a particular acoustic timescale ratio are shown to facilitate detonation formation inside gradients that would otherwise not detonate.

This 1-D model is extended to two dimensions with symmetric and asymmetric plateau regions, modeled using both rectangular and elliptical geometries. Even with clear differences in behavior between one and two dimensions, the hot spot acoustic timescale ratio is shown to characterize the 2-D gasdynamic response. The relationship between one and two dimensions is explored using asymmetric plateau regions. It is shown that 1-D behavior is recovered over a finite time. Furthermore, the duration of this 1-D behavior is directly related to the asymmetry of the plateau.