Weakly nonlinear analysis of short-wave elliptical instability

作者：Rodrigues Savio B^*; De Luca Jayme

来源：Physics of Fluids, 2009, 21(1): 014108.

DOI：10.1063/1.3068188

Two-dimensional inviscid flows with elliptical streamlines have linearly unstable three-dimensional modes close to the center of the ellipse. The linear instability equilibrates in the presence of enough viscosity, yielding a continuous set of neutrally stable plane waves known as Craik-Criminale waves (CCWs). For initial disturbance in a discrete superposition of plane waves our weakly nonlinear analysis yields a cubic ordinary differential equation for the slow-time evolution of the neutrally stable plane-wave amplitudes. In the subcritical region we find that all solutions eventually vanish, while in the supercritical region the amplitudes either saturate or have exponential growth; the existence of a finite-time blowup at cubic order being unlikely. The saturation to a steady state is possible only for special initial conditions, and for generic initial conditions the amplitudes saturate to a chaotic state. In contrast, a single CCW with exponential growth is still unstable, but it loses stability to cycles of growth and decay among the plane waves.

• 出版日期2009-1