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Propagation of spherical and cylindrical N-waves

  • P. L. Sachdev (a1) (a2) and R. Seebass (a1)

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.

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References

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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).
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Taylor, T. D., Ndefo, E. & Masson, B. S. 1972 J. Omp. Phy 9, 99.
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Propagation of spherical and cylindrical N-waves

  • P. L. Sachdev (a1) (a2) and R. Seebass (a1)

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