Spinning detonations and stability considerations

It is curious that the basis for the discovery of transverse structures in detonations existed more than 50 years ago. Peculiar detonations that appeared to propagate along helical paths in tubes with round cross sections 

If it is accepted that a spinning mode of propagation exists, then an acoustic theory may be applied to predict the spin frequency, the slope of the helical path on the tube wall, and other characteristics of spinning detonations [75]-[76]. For a planar Chapman-Jouguet wave in an infinitely long tube and in a frame of reference in which the burnt gas is at rest, the circumferential velocity of a traveling tangential acoustic wave at the wall 

The experimental prevalence of multiheaded spin has prompted categorical statements to the effect that all self-sustaining detonations exhibit significant three-dimensional structure [25]. The belief that no Chapman-Jouguet detonations are planar rests on the conclusion that all such planar waves are unstable to certain nonplanar disturbances. This conclusion is difficult to substantiate in general because of the complexity of the needed 

It should be understood that since the stability predictions involve reaction-rate properties, planar Chapman-Jouguet detonations are stable for suitable rate functions. For example, if the rate of heat release decreases monotonically with an increasing extent of reaction behind the shock, then the mechanism for the instability is absent. The failure to find Chapman-Jouguet detonations without transverse structures reflects the inability to encounter real chemical systems with reaction-rate properties suitable for stability. 

To obtain a rough physical understanding of the mechanism of the instability, attention may be focused first on a planar detonation subjected to a one-dimensional, time-dependent perturbation. Since the instability depends on the wave structure, a model for the steady detonation structure is needed to proceed with a stability analysis. As the simplest structure model, assume that properties remain constant at their Neumann-spike values for an induction distance after which all of the heat of combustion is released instantaneously. If v is the gas velocity with respect to the shock at the Neumann condition, then may be expressed approximately in terms of the explosion time given by equation (B-57) as Z = vt. From normal-shock relations for an ideal gas with constant specific heats in the strong-shock limit, the Neumann-state conditions are expressible by v/vq = po/p — 

The discussion given here has been qualitative and does not constitute a correct stability analysis. Even for the simplified model adopted, consideration should be given to jump conditions for interactions at the shock and 

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