Role of chemical kinetics on detonation stability and cell sizes

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The detonation cell width is a crucial parameter in many applications, including safety, because it is relatively easy to obtain experimentally and because many practical features and properties are found to scale with the cell width. The cell width is ultimately a function of the chemical kinetics characterizing the reactive mixture, but there is no clear way to determine it directly knowing the kinetic scheme. Indeed kinetic schemes, even simple ones, include many scales, some of which are also very temperature-sensitive, so that the range of scales determined by the kinetics is very large. We hypothesize that the cell width is related to the most unstable transverse stability modes. The motivation for this is that both width and modes are transverse scales, and that there are no other obvious transverse scales that can be defined. We propose initially to test this by comparing cell sizes obtained by numerical simulation with stability results for three step chain-branching schemes in which, as is often the case for realistic chain-branching kinetics including hydrogen-air, the steady planar wave length is effectively determined by initiation, and the reaction zone itself is much thinner. In that situation, stability analysis becomes difficult and our experience is that attempts to resolve both zones may not be doable because of the scale disparity: wavelengths resolving the reaction zone, where instability is known to initiate, are then very short when compared with the initiation length. Yet, such a situation is quite typical and it needs to be addressed, motivating the proposed work. Furthermore, it is also well-known, mainly from experiments, that the cell width is invariably much larger than the steady planar wave thickness. Thus, if there is indeed a relationship between cell sizes and transverse stability, then if the wavelength of unstable transverse modes is much larger than the steady planar wave thickness, which is itself much larger than the reaction zone hence longitudinal mode wavelengths, the analysis will need to be based upon a multiple scale model, an exercise which is expected to be quite challenging. Assuming results validate the current hypothesis, next, the stability analysis will be extended to arbitrary complex schemes also featuring a structure in which initiation is much longer than the zone of main reaction. Stability analysis consists in an eigenvalue problem for a perturbation to the steady planar solution, which is determined first. Thus stiffness issues associated with kinetic steps with high activation energy can be dealt with by locally using the Jacobian of the kinetics matrix along the planar steady wave, which can be obtained and diagonalized once and for all when computing the steady planar solution, perhaps using a symbolic package. Then these results are already available and simply used in the iterative search for stability modes.