Amplitude-dependent neutral modes in the Hagen-Poiseuille flow through a circular pipe

Abstract

An investigation is described for the nonlinear stability, at large Reynolds numbers R , of the Hagen-Poiseuille flow through a pipe of circular cross section when subjected to three-dimensional disturbances of typical relative size δ large enough to yield only a vanishingly small phase shift across the critical layer. A crucial size is δ = O ( R -⅓ ) since then this small phase shift is in tune with the small phase shift produced by the viscous wall layers. The critical layer, which is fully nonlinear and three-dimensional, and the wall layers, where the disturbance is greater than the basic flow and flow reversal occurs, are discussed in detail. Neutral solutions are then found to exist in the range c 01 < c 0 < 1 with N = 1, where c 0 is the non-dimensional wavespeed, c 01 = 0.284 and N is the azimuthal wavenumber; there is also evidence to suggest that no similar neutral solutions exist outside that range. The amplitude-dependence of the neutral modes follows and it is such that the cut-off value c 0 = c 01 + is approached as the amplitude shrinks, whereas centre modes with c 0 → 1 - are produced as the amplitude becomes relatively large.