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, ρs/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 ast−¼(3+ν)matching the earlier results of Chenget al.(1961). The approach to the large-time limit, ε1/(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.