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

• Published in: Journal of computational physics Vol. 187; no. 1; pp. 212 - 229
• Main Authors: Khan, M.A.H., Tourigny, Y., Drazin, P.G.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2003
• Subjects: Birkhoff–Rott equation; Hermite–Padé approximation; Kelvin–Helmholtz instability; Series summation; Kelvin–Helmholtz instability; Series summation; Hermite–Padé approximation; Birkhoff–Rott equation
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/S0021-9991(03)00096-2

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.