A nonlinear instability theory for a wave system inducing transition in spiral Poiseuille flow

Abstract

This paper is on the transition scenario of the class of spiral Poiseuille flows that results from the onset, propagation and evolution of disturbances according to mechanisms of Tollmien-Schlichting, and Taylor, acting simultaneously. The problem is approached from the fundamental point of view of following the growth of initially infinitesimally small disturbances into their nonlinear stage when the effect of Reynolds stresses makes itself felt. To this end a set of Generalised Nonlinear Orr–Sommerfeld, Squire and Continuity Equations is set up that enables accounting for effects of growth of initially infinitesimally small disturbances into nonlinearities through a rational iteration scheme. The present proposal closely follows the method put forth for this pupose in 1971 by Stuart and Stewartson in their seminal papers on the influence of nonlinear effects during transition in the bench-mark flows of the class of spiral Poiseuille flows; which are the plane-walled channel flow and the flow in the gap between concentric circular cylinders (Taylor instability).

The basic feature of the proposed method is the introduction of an Amplitude Parameter and of a slow/long- scale variable through which the effects of growing disturbances are accounted for within the framework of a rational iteration scheme. It is shown that the effect of amplified disturbances is capturable, as in the bench-mark flows, by a Ginzburg–Landau type differential equation for an Amplitude Function in terms of suitably defined slow/long-scale variables. However, the coefficients in this equation are numbers that depend upon the flow parameters of the spiral Poiseuille flow, which are a suitably defined Reynolds Number, the Swirl Number, and the geometric parameter of transverse curvature inherent in the flow geometry. The Ginzburg–Landau equation derived hints at the drastic changes in flow pattern that the spiral Poiseuille flow in transition may undergo, as its Swirl Number is taken from very small to very large values.