Influence of small imperfections on the stability of plane Poiseuille flow and the limitation of Squire’s theorem

Abstract

In a recent paper [Luo and Wu, Phys. Fluids 16, 2852 (2004)], a theoretical model was proposed to illustrate the key role played by small imperfections in the subcritical transition of plane Poiseuille flow. In that model, the imperfections are represented by a steady but spatially periodic two-dimensional surface roughness, and a small oscillatory pressure gradient representing the inevitable background fluctuations. The present paper extends the previous work by considering an improved model, where the wall roughness is three-dimensional. A steady perturbation in the form of spatially periodic suction is also considered. In both cases, the resulting steady and unsteady components interact to produce a forcing that is in near resonance with a pair of oblique Tollmien-Schlichting (T-S) waves. The latter are generated as a result, and grow in proportion to the time during the initial stage. This excitation process and the subsequent nonlinear development of the T-S waves are studied by a weakly nonlinear analysis. A forced nonlinear (and nonlocal) amplitude equation is derived to provide a simple mathematical framework to link the critical Reynolds number for nonlinear instability to the level of imperfections. The theoretical result demonstrates that the onset critical Reynolds number is lower for three-dimensional than for two-dimensional disturbances if the imperfection exceeds a critical level, which turns out to be remarkably small. This offers a possible explanation as to why Squire’s theorem is applicable only when external perturbations are suppressed to an extremely low level. Further analysis of the nonlinear interactions indicates that the much reduced critical Reynolds number for three-dimensional external perturbations can be attributed to the fact that the excited pair of oblique T-S waves interact to generate extremely strong streaks, which then result in a very strong nonlinear destabilizing effect.