Weakly nonlinear analysis on the Kelvin-Helmholtz instability

¹  Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics Beijing 100088, China
²  School of Mechanics and Civil Engineering, China University of Mining and Technology Beijing 100083, China
³  Department of Physics, Zhejiang University - Hangzhou 310027, China
⁴  Center for Applied Physics and Technology, Peking University - Beijing 100871, China
⁵  Institute for Fusion Theory and Simulation, Zhejiang University - Hangzhou 310027, China

Corresponding author: ye_wenhua@iapcm.ac.cn

Received: 7 February 2009
Accepted: 16 March 2009

Abstract

A weakly nonlinear model is proposed for the Kelvin-Helmholtz instability in two-dimensional incompressible fluids. The second- and third-harmonic generation effects of single-mode perturbation, as well as the nonlinear correction to the exponential growth of the fundamental modulation are analyzed. An important resonance in the mode-coupling process is found. The nonlinear saturation time depends on the initial perturbation amplitude and the density ratio of the two fluids, but the nonlinear saturation amplitude depends only on the initial perturbation amplitude. The weakly nonlinear result is supported by numerical simulation. The practical system of boundary layer containing thermal conductivity is analyzed. Their nonlinear saturation amplitude can be predicted by our weakly nonlinear model.