A case study of methods of series summation: Kelvin–Helmholtz instability of finite amplitude

We compute the singularities of the solution of the Birkhoff–Rott equation that governs the evolution of a planar periodic vortex sheet. Our approach uses the Taylor series obtained by Meiron et al. [J. Fluid Mech. 114 (1982) 283] for a flat sheet subject initially to a sinusoidal disturbance of amp...

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Bibliographic Details
Published in:Journal of computational physics 2003-05, Vol.187 (1), p.212-229
Main Authors: Khan, M.A.H., Tourigny, Y., Drazin, P.G.
Format: Article
Language:English
Subjects:
Birkhoff–Rott equation
Hermite–Padé approximation
Kelvin–Helmholtz instability
Series summation
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Summary:We compute the singularities of the solution of the Birkhoff–Rott equation that governs the evolution of a planar periodic vortex sheet. Our approach uses the Taylor series obtained by Meiron et al. [J. Fluid Mech. 114 (1982) 283] for a flat sheet subject initially to a sinusoidal disturbance of amplitude a. The series is then summed by using various generalisations of the Padé method. We find approximate values for the location and type of the principal singularity as a ranges from zero to infinity. Finally, the results are used as a basis to guide the choice of methods of summing series arising from problems in fluid mechanics.
ISSN:0021-9991
1090-2716
DOI:10.1016/S0021-9991(03)00096-2