The Linear and Nonlinear Shear
Instability of a Fluid Sheet
Abstract
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.