Hostname: page-component-7c8c6479df-27gpq Total loading time: 0 Render date: 2024-03-27T11:41:02.453Z Has data issue: false hasContentIssue false

Propagation of spherical and cylindrical N-waves

Published online by Cambridge University Press:  29 March 2006

P. L. Sachdev
Affiliation:
Graduate School of Aerospace Engineering, Cornell University, Ithaca, N.Y. Present address: Department of Applied Mathematics, Indian Institute of Science, Bangalore-560012.
R. Seebass
Affiliation:
Graduate School of Aerospace Engineering, Cornell University, Ithaca, N.Y.

Abstract

An implicit predictor–corrector difference scheme is employed to study the propagation of spherical and cylindrical N-waves governed by the modified Burgers equation \[ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x}+\frac{\nu u}{2t}=\frac{\delta}{2}\frac{\partial^2u}{\partial x^2}, \] where ν = 0, 1 or 2 for plane, cylindrical and spherical symmetry respectively. The numerical scheme is first tested by computing the plane solution and comparing it with theexact analyticsolution obtained by Lighthill (1956) through the Hopf-Cole transformation.

Our numerical solutions for the non-planar N-waves show that variation of the ‘lobe’ Reynolds number, which may be used as a measure of the importance of viscous diffusion, can be accurately determined by the analysis which is strictly valid only for large Reynolds numbers. This is true even when shock wave is well diffused end the ‘lobe’ Reynolds number is as small as ½.

Type
Research Article
Copyright
© 1973 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

Batedun, H. 1915 Mon. Weather Rev. 43, 163.
Burqers, J. M. 1948 Advances in Appl. Mech. 1, 171.
Cole, J. D. 1951 Quart. Appl. Math. 9, 225.
Douglas, J. & Jones, B. V. 1963 J. SOC. Id. Appl. Math. 11. 195.
Hopf, E. 1960 Commun. Pure Appl. Math. 3, 201.
Lighthill, M. J. 1956 Surveys in Mechanics (ed. G. K. Batchelor & R. Davis), pp. 250351. Cambridge University Press.
Leibovich, S. & Seebass, A. R. 1972 Nonlinear Waves, chap. 4. Cornell University Press (to appear).
Rusnov, V. V. 1961 Zh. Nyck. Math. 1, 267.
Taylor, T. D., Ndefo, E. & Masson, B. S. 1972 J. Omp. Phy 9, 99.