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Initial propagation of impulsively generated converging cylindrical and spherical shock waves

Published online by Cambridge University Press:  29 March 2006

Glen G. Bach  and
John H. Lee
Glen G. Bach
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal
John H. Lee
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal
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Abstract

An analytical description for the initial phases of collapse of a spherical or cylindrical shock wave in a perfect gas is given in the present paper. The shock wave is initiated by the instantaneous and uniform deposition of a finite quantity of energy per unit surface area at a finite radius R0. For the initial shock motion where xs = (Rs - R0)/R0 is small, analytical solutions are obtained by a power-series expansion of the dependent variables in xs. The classical self-similar solution for a strong planar blast wave is recovered as the present zero-order solution. Non-similar effects arising from both finite shock strengths and the presence of a characteristic length R0 are accounted for simultaneously in the present perturbation scheme. The analysis is carried out up to third order in xs. For very large values of the initiation energy where the shock wave remains strong throughout its collapse, it is found that the present perturbation solution can adequately describe a significant portion of the collapse processes. The solution indicates that the shock decays rather rapidly initially and later begins to accelerate as a result of the additional adiabatic compression of the shocked states due to flow-area convergence. However, for weak initiation where the energy released is small, the present perturbation solution is an asymptotic series and diverges very rapidly as the shock propagates away from the wall. The range of validity then is limited to very small values of xs.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 37 , Issue 3 , 10 July 1969 , pp. 513 - 528
Copyright
© 1969 Cambridge University Press

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References

Brode, H. 1955 J. Appl. Phys. 26, 766.
Butler, D. S. 1954 ARE Rep. no. 54/54.
Chester, W. 1954 Phil. Mag. 45, 1293.
Chisnell, R. F. 1955 Proc. Roy. Soc. A 232, 350.
Dennen R. S. & Wilson, L. N. 1962 Exploding Wirs, vol. 2 (edited by Chace & Moore). New York: Plenum Press.
Glass, I. I. 1965 UTIAS Rev. no. 25.
Guderley, G. 1942 Luftfahrtforschung, 19, 302.
Knystautas, R., Lee, B. H. K. & Lee, J. H. 1966 Paper presented at the 6th International Shock Tube Symposium, Freiburg, Germany. (To appear in the Phys. Fluids.)
Lee, B. H. K. 1966 Ph.D. Thesis. McGill University.
Lee, J. H. & Lee, B. H. K. 1965 Phys. Fluids, 8, 2148.
Payne, R. B. 1957 J. Fluid Mech. 2, 185.
Rae, W. J. & Kirchner, H. P. 1963 Cornell Aero. Lab. Rep. no. RM-1655-M-4, N63-16887.
Sakurai, A. 1954 J. Phys. Soc. Japan, 9, 256.
Stanyukovich, K. P. 1960 Unsteady Motion of Continuous Media (trans.). Oxford: Pergamon.
Welsh, R. L. 1966 University of California, Berkeley, Rep. no. AS-66-1.
Whitham, G. B. 1958 J. Fluid Mech. 4, 337.