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An asymptotic derivation of the linear stability of the square-wave detonation using the Newtonian limit

Mark Short

1996
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Proceedings of the Royal Society A
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The two-dimensional linear stability of a detonation wave characterized by a one-step irreversible Arrhenius reaction is examined by a tw o-param eter asym ptotic approach. The first is the limit of high activation energy in which the underlying steady detona tion structure tends to the classical square-wave profile. The second is due to Blythe &; Crighton ( P r o c. R. Soc. Lond. A 426, 189-209 (1989)) and assumes the Newto limit in which the ratio of the isotropic sound speed to the isotherm
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... ed to the isotherm al sound speed is close to unity. It is found th a t under two possible choices of distinguished limit between the two param eters, analytical forms for the p erturbation variables in the induction zone and equilibrium zones of the perturbed detonation can be found in normal mode form. A dispersion relation describing the growth rate of the p ertu rb a tions is then obtained in one case through a com patibility condition on the structure of the perturbation eigenfunctions at the flame front, and in the other through a matching of the perturbation variables across the flame front into the equilibrium zone, where an acoustic radiation condition is imposed. I n tr o d u c tio n For sufficiently high activation energies, a steady detonation wave characterized by an irreversible one-step Arrhenius reaction exhibits the form of a square-wave profile. This consists of a chemically frozen shock wave followed by an induction period of vanishingly small reaction term inated by a th in flame zone (or fire). In the limit of high activation energy, the complete structure of the steady square-wave detonation can be derived analytically using formal m atched asym ptotic techniques (Kassoy & Clarke 1985) . Zaidel (1961) has considered the stability of the steady square-wave detonation structure. He assumed th a t there is no heat generation w ithin the induction zone and so the evolution of induction-zone perturbations could be described by the equa tions of non-reactive acoustics. By solving similar equations in the equilibrium zone, Zaidel obtained a stability spectrum which, according to the further investigations of Erpenbeck (1963) , consisted of an infinite number of unstable eigenvalues. How ever, for perturbations of a frequency commensurate w ith the length of the induction zone, the reaction of the fluid w ithin the induction zone, however small, cannot be neglected and the assumptions made by Zaidel cannot be justified. Buckmaster & Neves (1988) later presented a correct asym ptotic formulation of the full detonation stability problem in the limit of high activation energy for one dimensional disturbances w ith wavelengths on the order of the induction-zone length.

doi:10.1098/rspa.1996.0117
fatcat:smfcndbcvfhbtnz5rhathla5ry