The stability of nonlinear neutral modes in Hagen–Poiseuille flow

Abstract

The nonlinear stability of Hagen–Poiseuille flow through a pipe of circular crosssection subjected to non–symmetric disturbances is studied asymptotically at large Reynolds number. By introducing unsteady effects into the nonlinear critical layer, an evolution equation for the disturbance amplitude is derived and is found to possess an equilibrium solution first identified by Smith and Bodonyi. This solution is shown to provide a threshold amplitude, above which, on this scaling, perturbations experience finite–time blow–up, while, below the threshold, disturbances of all wavenumbers decay to zero.