Transient growth in circular pipe flow. II. Nonlinear development

Abstract

The Navier–Stokes equations for circular pipe flow are integrated using direct numerical simulation for the case of transitional Reynolds number. Previous work on linear disturbances (reported in Part I) is exploited for the simulation of low to moderate amplitude disturbances where it is found that the transient growth mechanism persists in the nonlinear development with the evolution attributable to the linear mechanism remaining of considerable significance. A hypothesis of Trefethen et al. [Science 261, 578 (1993)] concerning the role of nonlinearity in the transition process and ultimately in turbulence is elucidated and given support. It is suggested that nonlinearity is essential in continually perturbing the eigenmodes of the flow in such a way that each mode is never permitted to relax to its least stable eigenstate (damped in the subcritical case). In this way, the linear growth mechanism can be regarded as an underpinning component of the general nonlinear feedback insofar as it is the only part which can extract energy from the mean flow and thus yield a net increase in disturbance energy. The physical aspects of the flow simulations are consistent with puff formation where, using a pair of helical waves as initial data, a sharp trailing front is formed naturally; axisymmetric ring vortices are generated and the general flow characteristics are in broad agreement with experiment.