On the reattachment of a shock layer produced by an instantaneous energy release

Abstract

The behaviour of a strong shock wave, which is initiated by a point explosion and driven continuously outward by an inner contact surface (or a piston), is studied as a problem of multiple time scales for an infinite shock strength, $\dot{y}_{sh}/a_{\infty}\rightarrow \infty $, and a high shock-compression ratio, ρ_s/ρ_∞ ∼ 2γ/(γ − 1) ≡ ε⁻¹ [Gt ] 1. The asymptotic analyses are carried out for cases with planar and cylindrical symmetry in which the piston velocity is a step function of time. The solution shows that the transition from an explosion-controlled régime to that of a reattached shock layer is characterized by an oscillation with slowly-varying frequency and amplitude. In the interval of a scaled time 1 [Lt ] t [Lt ] ε⁻²/3(1+ν), the oscillation frequency is shown to be (1 + ν) (2π)^{−1t−½(1−ν)} and the amplitude varies as t^−¼(3+ν) matching the earlier results of Cheng et al. (1961). The approach to the large-time limit, ε^1/(1+ν)t → ∞ is found to involve an oscillation with a much reduced frequency, ¼π(1+ν)ε^−½t⁻¹, and with an amplitude decaying more rapidly like ε^−⅘t^−½(4+3ν); this terminal behaviour agrees with the fundamental mode of a shock/acoustic-wave interaction.