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An analysis is presented of flame acceleration in a tube filled with flammable mixture, closed at one end and open to the atmosphere at its second end. Ignition takes place near the closed end; experiments then show that the flame accelerates, slows down and moves back, accelerates again, and may eventually reach considerable speeds. The one-dimensional analysis is based upon the assumption that the flame front propagates at a speed that is small compared to the speed of sound, and that depends upon the temperature and pressure perturbation due to acoustics that it encounters. A complete unsteady solution is constructed. The tube acoustics are set in motion by the expansion of the fluid due to ignition at the closed end. Subsequently, both spectrum and amplitude evolve because of the motion of the temperature interface, and because of forcing by the flame front, which itself depends upon the effect of acoustics on the state of the fluid that it encounters. Oscillations in the front position are stronger than in a previous study in which the flame propagation speed was taken to be constant. With the additional feedback now included, it is strong enough to result in strong flow reversal.

The structure of ZND waves under simple three step chain-branching kinetics is analyzed, assuming a slow initiation rate but arbitrary chain-branching activation energy. The analysis allows for a complete solution for the ZND wave in all cases, inside or outside the chain-branching explosion region, or close to the explosion limit. Results show that even when the von Neumann point is inside the explosion region, chain-branching effectively stops and the chain-branching radical concentration reaches a small near-steady value before all the reactant is consumed. Beyond that point, chemistry proceeds slowly, at a rate of the order of the initiation rate. For a von Neumann point relatively close to the limit, the reactant concentration is still quite significant when chain-branching stops, but diminishes for von Neumann points deeper inside the explosion region. The assumption that initiation is much slower than chain-branching is often quite accurate, in which case the length required for complete burn is orders of magnitude longer than the chain-branching length, so that as a practical matter, combustion never completes. In contrast, numerical simulation shows that under the same conditions, the cellular wave results in a more complete burn.

Flame acceleration in tubes is studied. A tube filled with flammable mixture is closed at one end and open to the atmosphere at its second end. When ignition takes place near the closed end, it is well-known from experiments that the flame may accelerate, oscillate and eventually reach considerable speeds. A one-dimensional analysis is presented, based upon the assumption that the flame front propagates at a speed that is small compared to the speed of sound. The analysis leads to a construction of the complete unsteady solution. Results from the analysis and from a numerical simulation are compared. They are similar enough to validate the analysis. The tube acoustics are set in motion by the expansion of the fluid due to ignition at the closed end. Subsequently, both spectrum and amplitude evolve because of the motion of the temperature interface, and because of forcing by the flame front, which the analysis precisely quantifies. Oscillations in the front position are strong enough to result in flow reversal. In addition, the induced periodic acoustic acceleration of the temperature and density interface will periodically make the flame front Rayleigh–Taylor unstable, which should result in the dramatic increase in the propagation speed seen in experiments.

We study multi-dimensional stability and perform high resolution two-dimensional numerical simulations of detonations with a four-step chain-branching reaction model. The reaction model is designed to approximate hydrogen chemistry. It consists of a chain-initiation step and a chain-branching step, both temperature-dependent with Arrhenius kinetics, followed by two pressure-dependent termination steps. Increasing the chain-branching activation energy shortens the ZND reaction length and leads to more unstable detonations, according to the stability analysis. Computations with four values of the chain-branching activation energy are performed both in narrow and wide channels. In the wider channel, all cases studied show distinct keystone-shaped regions, associated with substantial differences in reactivity across the shear layer hence of the time and distance until chain-branching takes place. As the chain-branching activation energy increases, cells take a shorter time to form, and the ratio cell length over width decreases. The cell size is dominated by the longer unstable wavelength even when high frequency modes are more unstable, but cells appear earlier in narrow channels than in wider ones. Initially, the cellular structure looks weaker, and the cell size is dominated by the shorter, more unstable wavelength, but eventually, it adapts to the longest unstable wavelength still consistent with the domain width.

Deflagration to detonation transition requires formation of a shock at least as strong as Chapman-Jouguet. Chemistry ahead of a flame, in the finite zone of reactive mixture behind the precursor shock, or, more generally, behind a source of an unburned mixture, always results in a hot spot appearing ahead of the flame. Diffusion-driven flames propagate at negligible speeds compared with the speed of sound and can thus be approximated as contact surfaces. Downstream, the hot spot triggers a weak detonation that slows down to the Chapman-Jouguet speed and quickly becomes a strong wave. Eventually, this wave reaches the contact surface: chemistry stops and a reflection moves upstream. If the source delivers fresh mixture at a velocity below a specific threshold, then the hot spot does not initially trigger a combustion wave on its upstream side, but only an inert shock. This shock, however, warms up the mixture, resulting in a much shorter ignition time. If ignition behind this second shock occurs before the reflection of the downstream wave on the contact surface reaches the location of this second ignition point, a second hot spot forms, on these much shorter time and length scales. This second hot spot triggers a second identical chain of events, which triggers a third one, and so forth, resulting in an embedded sequence of ignition and shock formation. Each of these appears slightly further downstream, on increasingly shorter scales, triggering the next events in the sequence. Eventually, a sequence of shock mergers results in a strong Chapman-Jouguet wave, completing transition to detonation. The condition for reflection to take longer than ignition yields a transition criterion. The analysis assumes high activation energy and a ratio of specific heat close to unity, and the transition criterion appears as a specific distinguished limit.

Slow chemistry in a finite zone of reactive mixture behind an inert shock results in the formation of a hot spot and eventually triggers ignition close to the surface where the shock originated, which, depending upon the specific scenario, could consist of a piston, a contact surface, or a flame. The current study assumed high activation energy and a ratio of the specific heats close to unity. The various scenarios can then be translated into different values of the acoustic reflection coefficient or of the ratio of the speeds of sound across the contact surface, with the piston being equivalent to a zero speed of sound downstream. Thus this study proceeds as a generalization of the work of Blythe and Crighton, who studied piston-induced ignition, for arbitrary downstream acoustic reflection coefficients. It is found that in all cases except for the piston, ignition occurs at a small distance from the contact surface. Initially, a shockless supersonic wave appears, moving toward the contact surface at a speed initially infinite but decreasing. For a ratio of the speeds of sounds below a critical value equal to the inverse of the Chapman-Jouguet (CJ) speed, this wave eventually reaches the contact surface at a speed still above the CJ value. But above this threshold, the speed reaches the CJ value, a shock appears, and the wave proceeds as a strong CJ wave. Finally, for higher temperatures behind the contact surface, a spatially uniform thermal, explosion eventually occurs in the largely unburned mixture ahead of the CJ wave before it reaches the contact surface. Thus, a variety of possible scenarios leading to detonation were uncovered. The last and strongest one is particularly applicable to flames, and it provides for a transition mechanism in the region immediately in front of the flame.