The temporal evolution of neutral modes in the impulsively started flow through a circular pipe and their connection to the nonlinear stability of Hagen–Poiseuille flow

Abstract

The linear stability of the impulsively started flow through a pipe of circular crosssection is studied at high Reynolds number R. A crucial non-dimensional time of O(R 7/9 ) is identified at which the disturbance acquires internal flow characteristics. It is shown that even if the disturbance amplitude at this time is as small as O(R −22/27 ) the subsequent evolution of the perturbation is nonlinear, although it can still be followed analytically using a multiple-scales approach. The amplitude and wave speed of the nonlinear disturbance are calculated as functions of time and we show that as t → ∞, the disturbance evolves into the long-wave limit of the neutral mode structure found by Smith & Bodonyi in the fully developed Hagen-Poiseuille flow, into which our basic flow ultimately evolves. It is proposed that the critical amplitude found here forms a stability boundary between the decay of linear disturbances and 'bypass' transition, in which the fully developed state is never attained.

The emergence of localized vortex–wave interaction states in plane Couette flow

“…For example, the linear stability of spatially-developing pipe and channel flows is governed by modes with an O(R) wavelength, comparable to the lengthscale over which the basic flow is developing (Smith and Bodonyi 1980). The corresponding neutral modes in temporally-developing pipe flow exhibit the same behaviour (Walton 2002). A further situation where the linearized version of (2.12) is relevant is the stability of the axial flow in a concentric annulus, where the equations describe the travelling-wave instability close to the cut-off in radius ratio or sliding velocity (Gittler 1993, Walton 2004.…”

The stability of developing pipe flow at high Reynolds number and the existence of nonlinear neutral centre modes

“…Moreover, these asymptotic structures remain essentially intact as the disturbance size is increased, and so this high Reynolds number approach allows us to incorporate the effects of nonlinearity in a self-consistent manner. In Walton (2002) the mode structure on the upper branch of the weakly nonlinear neutral curve was analyzed at large Reynolds number for nonsymmetric disturbances proportional to exp(iN θ). There it was found that for N = 1 the asymptotic structure develops into the travelling-wave structure for HPF proposed by Smith & Bodonyi (1982), with the time dependence of the basic flow having a purely parametric effect on the nonlinear modes.…”

“…It is shown in Walton (2002) that, at least as far as nonsymmetric modes are concerned, curvature effects first enter the upper branch stability structure at viscous times of O(R −2/9 ) after the fluid is set into motion. This occurs because at this time, the radial velocity in the wall layer has grown to a sufficient size to balance the curvatureinduced phase shift from the critical layer.…”

“…This phase shift is then reduced to zero by viscous processes within the wall layer provided the disturbance amplitude A is dependent in a specific way on the wavenumbers α and N. One of the features of this work was the fact that very little numerical computation was necessary in order to determine A(α, N ) and a key result was that this travelling wave structure only exists if N = 1. Motivated by this work, Walton (2002) showed, by considering impulsively-started pipe flow, that the Smith & Bodonyi neutral criteria arise from linear upper-branch stability properties as the disturbance size is increased. This study also showed that modes with N > 1 can be supported up to a finite value of the time parameter t say, with only the N = 1 mode surviving once the basic flow becomes fully-developed.…”

“…Our original motivation for the study we describe here was to investigate whether a similar analysis to Walton (2002) could be performed for N = 0. If so, it would be interesting to know whether such modes disappear at a finite time after the fluid has been set into motion or whether they can survive as nonlinear disturbances to HPF.…”