An analytical solution for blast waves

Abstract

An approximate analytical method that is valid for the very low shock Mach number regime of a blast wave is described in the present paper. The method is based on the assumption of a power-law density profile behind the blast wave. The exponent of the power-law density profile is determined for each local shock Mach number from the mass integral. Solution of the differential equations of continuity and momentum then yield the particle velocity and the pressure distributions. The dependence of the shock decay coefficient on the shock velocity is determined from the energy integral. Comparison with the exact numerical solution of Goldstine and von Neumann and other existing analytical solutions indicates that the present solution is surprisingly accurate for the very low shock strength regime as compared to existing analytical solutions. HE theory of point blast waves is a very fundamental one in gasdynamics and has been applied to a variety of problems in hypersonic aerodynamics/'2 astrophysics,3-4 and hypervelocity impact.5'6 The validity of the classical selfsimilar solution of von Neumann,7 Taylor,8 and Sedov9 is confined to the early time regime when the shock wave is very strong (i.e., 1/MS2 —>• 0). For intermediate times when the shock strength is finite, small departures from the classical self-similar solution due to counterpressu re effects is accounted for in the perturbation solution of Sakurai10 and the quasi-similar solution of Oshima. 11 For the asymptotic motion at late times when l/Ms2 -*-1, solutions have been obtained by Whitham 12 and Sedov.9 Apart from exact numerical solutions,13 there appears to be a lack of an accurate analytical solution that provides an adequate description of the blast motion for the entire regime. In the present paper we describe an approximate analytical method that gave a surprisingly accurate solution for the entire propagation regime of a blast wave. The present method is due to Rae 6 who laid down most of the essential steps of the analyses but did not carry out the work to completion. Following Rae, the analysis was further developed by Lee. 14 The essence of the method is to assume a power-law density profile behind the blast wave, the exponent of which is determined from the mass integral. This then enables the particle velocity profile to be obtained from the differential equation of mass conservation. With the form for the density and particle velocity profiles known, the momentum equation can be integrated directly to determine the pressure profile. Substituting these profiles into the energy integral then yields a first-order differential equation for the dependence of the decay coefficient on the shock Mach number. The integration of this equation with the given boundary conditions then completes the solution of the problem. It should be noted that the assumption of a powerlaw density profile was first proposed by Porzel, 15 and it was Rae6 who pointed out the use of the energy integral for the solution of the problem. At the completion of the present work, it was found that a similar attempt has been made by Sakurai 16 to obtain an analytical solution valid for the entire propagation regime of the blast wave. In Sakurai's work, a linear velocity profile