A point explosion in a cold exponential atmosphere

Abstract

The problem considered is that of a strong shock propagating from a point energy source into a cold atmosphere whose density varies exponentially with altitude. An explicit analytic solution is obtained by taking the flow field as ‘locally radial’ and using an integral method with an energy constraint. A scaling law is given which eliminates the parametric dependence of the solution on the explosion energy, scale height, and atmospheric density at the point of the explosion. The scaling law also transforms the infinity of solutions for various polar angles into two distinct solutions which show that all motions of the ascending portion of the shock may be scaled from the vertically upward behaviour and all motions of the descending portion of the shock may be scaled from the vertically downward behaviour. The limit in the lateral direction of both of the fundamental solutions corresponds to the case of the uniform density atmosphere. The results for the uniform density atmosphere show remarkable agreement with the exact Taylor—Sedov results. Comparison with finite difference calculations of Troutman & Davis for the vertically upward and downward directions shows excellent agreement with respect to the prediction of shock propagation velocity, position, and the flow variables behind the shock. A scaling law for the time, shock velocity, and pressure for different values of the adiabatic exponent γ is proposed which correlates the results of the present analysis for different values of γ over the entire range of shock positions where the analysis applies. The solution shows that, contrary to the result obtained by Kompaneets, there is no theoretical limit as to how far downward a strong shock may propagate. The far field behaviour of the shock wave in the upward and downward directions is found to be of the same form as the self-similar asymptotic solutions obtained by Raizer for a plane shock. It is shown by relaxing the energy constraint in the vertically downward direction that the asymptotic result obtained agrees closely with that obtained by Raizer. The energy constraint, however, is the appropriate one for all but the far field behaviour. The far field limit of the present solution in the upward direction is found to compare favourably with the approximate asymptotic calculations of Hayes for an ascending curved shock. The empirical concept of ‘modified Sachs scaling’ for calculating the overpressure is considered and shown within this model to have a justification in the downward direction but a limited range of applicability in the upward direction.