Influence of small imperfections on the stability of plane Poiseuille flow: A theoretical model and direct numerical simulation

Abstract

It is well known that the stability of plane Poiseuille flow is extremely sensitive to small imperfections that are inevitably present. In this paper, a simple model is proposed, in which the imperfections are represented by a steady but spatially periodic surface roughness and a small oscillatory pressure gradient. A steady perturbation in the form of spatially periodic suction is also considered. For both cases, the resulting steady and unsteady components interact to produce a forcing that is in resonance with the Tollmien–Schlichting (TS) wave. The latter is excited as a result, and grows in proportion to time during the initial stage. The subsequent nonlinear development of the TS wave is shown to be governed by a forced (nonlocal) amplitude equation, which provides a simple framework to link the subcritical nonlinear instability explicitly to external forcing. The validity of the amplitude equation was checked against the direct numerical simulation (DNS), carried out in the case of wall suction. The results indicate that for a fixed level of external disturbance, there exist two distinctive regimes of response: A large-amplitude regime for the Reynolds numbers above a critical value, and a small-amplitude regime below it. The large-amplitude response regime in DNS was found to correspond to the occurrence of a finite-time singularity in the solution to the amplitude equation. This observation allows the critical Reynolds number for the large-amplitude response regime to be defined as one that divides the singular and bounded solutions of the amplitude equation. Estimate based on this equation shows that presence of small imperfections may reduce the critical Reynolds number for subcritical instability to values well below 5772.22.