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A point explosion in a cold exponential atmosphere

Published online by Cambridge University Press:  28 March 2006

Dallas D. Laumbach  and
Ronald F. Probstein
Dallas D. Laumbach
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
Ronald F. Probstein
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 35 , Issue 1 , 16 January 1969 , pp. 53 - 75
Copyright
© 1969 Cambridge University Press

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References

Andriankin, E. I., Kogan, A. M., Kompaneets, A. S. & Krainov, V. P. 1962 The propagation of a strong explosion in a nonhomogeneous atmosphere. Zh. Prikl. Mekh. Tekh. Fiz. no. 6, 3–7.Google Scholar
Chernyi, G. G. 1957 The problem of a point explosion Dokl. AN SSSR 112, 213–16.Google Scholar
Chernyi, G. G. 1959 Introduction to Hypersonic Flow. English translation (trans, and ed. R. F. Probstein), 1961. New York: Academic Press.
Chernyi, G. G. 1960 Application of integral relationships in problems of propagation of strong shock waves. Prikl. Mat. Mekh. 24, 121–5; J. Appl. Math. Mech. 24, 159–65.Google Scholar
Chernyi, G. G. 1961 Integral methods for the calculation of gas flows with strong shock waves. Prikl. Mat. Mekh. 25, 101–7; J. Appl. Math. Mech. 25, 138–47.Google Scholar
Grover, R. & Hardy, J. W. 1966 The propagation of shocks in exponentially decreasing atmospheres Astrophys. J. 143, 48–60.Google Scholar
Hayes, W. D. 1968a Self-similar strong shocks in an exponential medium J. Fluid Mech. 32, 305–15.Google Scholar
Hayes, W. D. 1968b The propagation upward of the shock wave from a strong explosion in the atmosphere J. Fluid Mech. 32, 317–31.Google Scholar
Hayes, W. D. & Probstein, R. F. 1966 Hypersonic Flow Theory. Vol. I. Inviscid Flows, 2nd ed. New York: Academic Press.
Karlikov, V. P. 1955 Solution of the linearized axisymmetrie problem of a point explosion in a medium with variable density Dokl. AN SSSR, 101, 1009–12.Google Scholar
Kompaneets, A. S. 1960 A point explosion in an inhomogeneous atmosphere. Dokl. AN SSSR, 130, 1001–3; Soviet Phys. Doklady, 5, 46–8.Google Scholar
Lutzky, M. & Lehto, D. L. 1968 Shock propagation in spherically symmetric exponential atmospheres Phys. Fluids, 11, 1466–1472.Google Scholar
Raizer, Yu. P., 1963 Motion produced in an inhomogeneous atmosphere by a plane shock of short duration. Dokl. AN SSSR, 153, 551–4; Soviet Phys. Doklady. 8, 1056–8 (1964).Google Scholar
Raizer, Yu. P. 1964 The propagation of a shock wave in a nonhomogeneous atmosphere in the direction of decreasing density. Zh. Prikl. Mekh. Tekh. Fiz. no. 4, 49–56.Google Scholar
Sedov, L. I. 1946 Propagation of strong blast waves Prikl. Mat. Mekh. 10, 241–50.Google Scholar
Sedov, L. I. 1957 Similarity and Dimensional Methods in Mechanics, 4th edition, English transl. (ed. M. Holt). 1959. New York: Academic Press.
Taylor, G. I. 1950 The formation of a blast wave by a very intense explosion. Proc. R. Soc A 201, 159–86.Google Scholar
Troutman, W. W. & Davis, C. W. 1965 The two-dimensional behavior of shocks in the atmosphere. Air Force Weapons Lab., Rept. AFWL-TR-65-151.Google Scholar
Zel'’ovich, Ya. B. & Raizer, YU. P. 1966 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, 2nd ed. English translation (ed. W. D. Hayes and R. F. Probstein), vol. II, 1967. New York: Academic Press.