Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-23T20:30:54.804Z Has data issue: false hasContentIssue false
  • English
  • Français

On a self-similar solution for the decay of turbulent bursts

Published online by Cambridge University Press:  16 July 2009

S. P. Hastings  and
L. A. Peletier
S. P. Hastings
Affiliation:
Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15205, USA
L. A. Peletier
Affiliation:
Department of Mathematics and Computer Science, Leiden University, Postbus 9512, 2300 RA Leiden, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 4 , December 1992 , pp. 319 - 341
Copyright
Copyright © Cambridge University Press 1992

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

[A]Aronson, D. G. 1980 Density-dependent interaction - diffusion systems. In Dynamics and modelling of reactive systems (eds. Stewart, W. E., Harmon Ray, W. & Conley, C. C.). Academic Press, pp. 161–176.CrossRefGoogle Scholar
[B]Barenblatt, G. I. 1983 Selfsimilar turbulence propagation from an instantaneous plane source. In Nonlinear dynamics and turbulence (eds. Barenblatt, G. I., loos, G. & Joseph, D. D.). Pitman, pp. 48–60.Google Scholar
[BER]Barenblatt, G. I., Entov, V. M. & Ryzhik, V. M. 1990 Theory of fluid flow through natural rocks. Kluwer, p. 182.Google Scholar
[BGL]Barenblatt, G. I., Galerkina, N. L. & Luneva, M. V. 1987 Evolution of turbulent bursts. Inzhenerno-Fizicheskii Zh. (J. Engg. Phys.), 53, 733–740 (in Russian).Google Scholar
[CG]Chen, L.-Y. & Goldenfeld, N. 1992 Renormalisation group theory for the propagation of a turbulent burst. Phys. Rev. A., 45, 5572–5577.CrossRefGoogle Scholar
[DS]Diaz, J. I. & Saa, J. 1992 Uniqueness of very singular self-similar solution of the parabolic p Laplacian operator with absorption. Preprint.CrossRefGoogle Scholar
[GP]Gilding, B. H. & Peletier, L. A. 1976 On a class of similarity solutions of the porous media equation. J. Math. Anal. Appl. 55, 351–368.CrossRefGoogle Scholar
[KPV1]Kamin, S., Peletier, L. A. & Vazquez, J. L. 1991 On the Barenblatt equation of elastoplastic filtration. Indianna University Math. J., 40, 1333–1362.CrossRefGoogle Scholar
[KPV2]Kamin, S., Peletier, L. A. & Vazquez, J. L. 1989 Classification of singular solutions of a nonlinear diffusion equation. Duke Math. J. 58, 601–615.CrossRefGoogle Scholar
[KVa]Kamin, S. & Vazquez, J. L. 1991 The propagation of turbulent bursts. IMA preprint Series no. 843, 08.Google Scholar
[KVe]Kamin, S. & Veron, L. 1988 Existence and uniqueness of the very singular solution for the porous media equation with absorption. J. d'Anal. Math. 51, 245–258.CrossRefGoogle Scholar
[K]Kolmogorov, A. N. 1942 Equations of turbulent motion of an incompressible fluid. Izvestia Akademii Nauk SSSR, Serija Fizicheskaya 6, 56–58.Google Scholar
[MY]Monin, A. S. & Yaglom, A. M. 1971 Statistical fluid mechanics: mechanics of turbulence, Vol. I. MIT Press.Google Scholar
[P]Peletier, L. A. 1981 The porous media equation. In Applications of nonlinear analysis in the physical sciences (eds. Amann, H., Bazley, N. & Kirchgässner, K.). Pitman, 229–241.Google Scholar
[PT]Peletier, L. A. & Terman, D. 1986 A very singular solution of the porous media equation with absorption. J. Diff. Equ. 65, 396–410.CrossRefGoogle Scholar