The role of the critical layer in the channel flow transition revisited

Abstract

The focus of this paper is on the development of a novel and simple linear modeling to interpret the (asymptotic) instability conditions for laminar channel (Poiseuille) flow, thereby giving insight on the major length and time scales of the subsequent mean turbulent regime. Although this is an old problem, recently there has been a renewed effort to understand how information on the wall turbulence scalings can be obtained by studying the flow linear dynamics, among other approaches by using the resolvent analysis. Here the classic asymptotic stability analysis is reformulated by means of a pseudo two-fluid model, a central inviscid flow motion, and a wall viscous one, forcing each other on their common interface streamline. The best agreement with the results of the celebrated Orr–Sommerfeld equation is obtained for the interface located around the elevation of the base flow average value, equal to 2/3 times the maximum velocity (frequency scale). It is argued that, in the early stage of perturbations growth, while the mechanism leading to instability (and that is related to the extraction of energy from the base flow via the Reynolds stress), is located near the wall in the critical layer region (already known result for inner length scale), the central region around the flow average velocity (i.e., the outer length scale is of order $${\mathcal {O}}(h)$$, with h being the channel half-height) is also deeply involved in the onset of the instability. The initial linear amplification of the wall viscous modes of the present analysis agrees with the major features of the wall functions driving the full turbulent profile, found by means of the resolvent analysis, in the subsequent nonlinear stage.