Nonlinear evolution of two vortex sheets moving separately in uniform shear flows with opposite direction

Abstract

It has been considered that two close vortex sheets become unstable and evolve simultaneously when sufficiently strong uniform shears exist. However, Moore (Mathematika, 1976) suggested in his linear analysis that a vortex sheet evolves just as if the other vortex sheet were absent under certain conditions. In the current study, we investigate how the two vortex sheets evolve in the nonlinear region when they satisfy Moore's condition. We also consider density stratification, which is not included in Moore's analysis. Moore's estimate is only valid within linear theory; however, a motion suggested by Moore appears even in the nonlinear regime when Moore's condition is satisfied. We found that there is a case that a vortex sheet hardly deforms, even though the other sheet becomes unstable and largely deforms. We also show that there is a case that Moore's analysis is not effective even the condition is satisfied when a density instability exists in the system.

Keywords

• two vortex sheets
• multi-layer flow
• kelvin-helmholtz instability
• density stratification
• vortex method