A generalisation of the theory of geometrical shock dynamics

Abstract

Abstract. The study of the propagation of a shock down a tube of slowly varying cross sectional area has proved to be most valuable in understanding the dynamics of shocks. A particular culmination of this work has been the theory of geometrical shock dynamics due to Whitham (1957Whitham ( , 1959. In this theory the motion of a shock may be approximately computed independently of a determination of the flow field behind the shock. In this paper the propagation of a shock down such a tube is reconsidered. It is found that the motion of the shock is described by an infinite sequence of ordinary differential equations. Each equation is coupled to all of its predecessors but only to its immediate successor, a feature which allows the system to be closed by truncation. Of particular relevance is the demonstration that truncation at the first equation in the sequence yields the A-M relation that is the basis for Whitham's highly successful theory. Truncation at the second equation yields the next level of approximation. The equations so obtained are investigated with analytic solutions being found in the strong shock limit for the propagation of cylindrical and spherical shock waves. Implementation of the theory in the numerical scheme of geometrical shock dynamics allows the computation of shock motion in more general geometries. In particular, investigation of shock diffraction by convex corners of large angular deviation successfully yields the observed inflection point in the shock shape near the wall. The theory developed allows account to be taken of non-uniform flow conditions behind the shock. This feature is of particular interest in consideration of underwater blast waves in which case the flow behind the shock decays approximately exponentially. Application of the ideas developed here provides an excellent description of this phenomenon.

A fast-marching like algorithm for geometrical shock dynamics

“…The shock adjusts itself to changes in the geometry only [29]. As explained by Best [8], Whitham considered the motion of a shock into a uniform gas at rest, down a tube of slowly varying cross sectional area, A, and under some physically grounded hypothesis, he obtained an expression relating the local shock Mach number, M, to A, now known as the A-M relation [31] (see also (9) and (4)). The GSD model reads…”

“…In practice, GSD has proven to be fairly accurate for diffraction around a corner, non-regular Mach reflection [29], or accelerating shocks and shown only little deviation for expanding decelerating flows [5]. Whitham's model has been extended to take into account unsteady flow behind the shock [8,9,10], non-uniform gases properties [20], and has been applied, among others, to imploding shock waves [11,1], atmospheric propagation [7], detonation in explosives [2,3,6], supersonic engine unstart [27] and astrophysics [14].…”

Shock wave focusing using geometrical shock dynamics

“…͑2.5͒ was assumed to be independent of entropy. On the other hand, Holl 3 and Best 20 use the Rankine Hugoniot relation for the conservation of energy to write an over-determined system which they use to derive A(M ) for water. Cates 18 has shown that this approach can lead to unnecessary error when calculating the Mach number from pressure.…”

Untitled

“…The truncated system, which is closed, can be regarded as a good approximation of the infinite hierarchy of the system governing shock propagation [22]. It is worth mentioning that an infinite sequence of ordinary differential equations, which hold on the shock front, was also derived by Best [31] in an attempt to describe shock motion in one dimension. However, his approach, unlike the one presented here, is based on CCW approximation and admits the fact that the application of a C + characteristic equation at the shock is somewhat ad-hoc.…”