Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-29T05:14:42.030Z Has data issue: false hasContentIssue false
  • English
  • Français

The unsteady expansion of a gas into a non-uniform near vacuum

Published online by Cambridge University Press:  12 April 2006

R. E. Grundy  and
R. Mclaughlin
R. E. Grundy
Affiliation:
Department of Applied Mathematics, University of St Andrews, Fife, Scotland
R. Mclaughlin
Affiliation:
Department of Applied Mathematics, University of St Andrews, Fife, Scotland
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper generalizes an earlier problem of Grundy (1972) by considering the expansion of a (uniform) initially contained gas into a low-density non-uniform ambient atmosphere of density ρ0r−k, where k > 0 and r is a non-dimensional radial co-ordinate. Regarding the flow as a perturbation of the perfect-vacuum expansion, we set up a boundary-value problem with boundary conditions on the contact front separating the two gases and on the strong shock which propagates into the ambient atmosphere. A large time solution to the problem can be developed by constructing an outer expansion valid near the contact front and an inner expansion valid near the shock. The matching process encounters two kinds of difficulty both of which imply that the large time solution is indeterminate from an asymptotic analysis alone.

The asymptotic analysis does show however that the shock velocity tends to a constant only for restricted values of k. For the remaining values the shock has a k-dependent power-law behaviour. The paper examines the location of the transition and determines the asymptotic power-law dependence of the shock velocity.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 81 , Issue 4 , 5 August 1977 , pp. 775 - 792
Copyright
© 1977 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Coutrant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Interscience.
Ellinwood, J. W. 1967 Asymptotic hypersonic-flow theory for blunted slender cones and wedges. J. Math. & Phys. 46, 281.Google Scholar
Freeman, N. C. 1965 In Research Frontiers in Fluid Dynamics (ed. R. J. Seeger & G. Temple), pp. 284–307. Interscience.
Grundy, R. E. 1968 Ph.D. thesis, University of London.
Grundy, R. E. 1969a Unsteady expansions into vacuum with spherical symmetry. J. Fluid Mech. 39, 529.Google Scholar
Grundy, R. E. 1969b Axially symmetric expansion of a monatomic gas from an orifice into a vacuum. Phys. Fluids 12, 2011.Google Scholar
Grundy, R. E. 1972 On the unsteady expansion of a gas into a near vacuum. J. Fluid Mech. 56, 97.Google Scholar
Mclaughlin, R. 1975 Ph.D. thesis, University of St Andrews.
Parker, E. N. 1963 Interplanetary Dynamical Processes. Interscience.
Sedov, L. I. 1959 Similarity and Dimensional Methods in Mechanics. Academic Press.
Stewartson, K. & Thompson, B. W. 1968 On one-dimensional unsteady flow at infinite Mach number. Proc. Roy. Soc. A 304, 255.Google Scholar
Stewartson, K. & Thompson, B. W. 1970 Eigenvalues for the blast wave. Phys. Fluids 13, 227.Google Scholar