TRANSIENT GROWTH IN CIRCULAR PIPE-FLOW .2. NONLINEAR DEVELOPMENT

The Navier-Stokes equations for circular pipe flow are integrated usin g direct numerical simulation for the case of transitional Reynolds nu mber. Previous work on linear disturbances (reported in Part I) is exp loited for the simulation of low to moderate amplitude disturbances wh ere it is found that the transient growth mechanism persists in the no nlinear development with the evolution attributable to the linear mech anism remaining of considerable significance. A hypothesis of Trefethe n 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 c ontinually perturbing the eigenmodes of the flow in such a way that ea ch mode is never permitted to relax to its least stable eigenstate (da mped in the subcritical case). In this way, the linear growth mechanis m can be regarded as an underpinning component of the general nonlinea r feedback insofar as it is the only part which can extract energy fro m the mean flow and thus yield a net increase in disturbance energy. T he physical aspects of the flow simulations are consistent with puff f ormation where, using a pair of helical waves as initial data, a sharp trailing front is formed naturally; axisymmetric ring vortices are ge nerated and the general flow characteristics are in broad agreement wi th experiment.