Theory of unstable two-dimensional detonations: Genesis of the transverse waves

Abstract

The stability of two-dimensional piston-supported gaseous detonations is examined both numerically and analytically. Results presented are for a one-step, first-order, irreversible reaction obeying an Arrhenius rate expression. The numerical analysis has shown many interesting results. First, in channels whose widths are smaller than or of the order of the characteristic reaction length, transverse waves do not develop and only one-dimensional longitudinal oscillations are observed. Second, a critical channel width exists beyond which a transverse wave may develop. Third, as teh channel becomes wider and up till a second critical channel width, the transverse wave persists while its oscillation period becomes longer. Fourth, beyond that second critical channel width, new transverse waves form, whose wavelengths are equal to the wavelength corresponding to the second critical value. Moreover, at and beyond that critical channel width, the oscillation period attains a constant value, almost identical to that of the one-dimensional oscillation. These results demonstrate the existence of a most unstable wavelength which persists further on and governs the cell size in wide channels. An approximate linearized stability theory previously developed for a one-dimensional shock-reaction-zone complex has been extended to the two-dimensional case and the mechanism of oscillation is identified. It is found that the interactions between the irreversible temperature fluctuations (due to shock perturbation) and the finite reaction zone induce an oscillatory energy-source field which generates pressure waves. The latter, in turn, perturb the shock front and generate the temperature fluctuations, thereby completing the cycle. Stability limits as well as amplification (or attenuation) rates and oscillation periods are obtained for different transverse wave numbers at different degrees of overdrive. The amplification rates are found to decrease and become negative for larger wave numbers or for larger degrees of overdrive. In orther words, transverse waves do not develop in very narrow channels or in highly overdriven detonations because of attenuation. Moreover, it is also shown that for the most unstable wavelength (as observed in the numerical simulations), the oscillation period is almost equal to that of the one-dimensional oscillation.