The Linear and Nonlinear Shear
Instability of a Fluid Sheet
A theoretical and computational investigation of the inviscid Kelvin-Helmholtz instability of a two-dimensional fluid sheet is presented. Both linear and nonlinear analyses are performed. The study considers the temporal dilational (symmetric) and sinuous (antisymmetric) instability of a sheet of finite thickness, including the effect of surface tension and the density difference between the fluid in the sheet and the surrounding fluid. Previous linear-theory results are extended to include the complete range of density ratios and thickness-to-wavelength ratios. It is shown that all sinuous waves are stable when the dimensionless sheet thickness is less than a critical value that depends on the density ratio. At low density ratios, the growth rate of the sinuous waves is larger than that of the dilational waves, in agreement with previous results. At higher density ratios, it is shown that the dilational waves have a higher growth rate. The nonlinear calculations indicate the existence of sinuous oscillating modes when the density ratio is of the order of 1. Sinuous modes may result in ligaments interspaced by half a wavelength. Dilational modes grow monotonically and may result in ligaments interspaced by one wavelength.