Structure of a Turbulent Puff in Pipe Flow

Abstract

The flow in a circular pipe of radius a driven by a constant and uniform axial force is investigated over a range of Reynolds numbers, including the critical value for laminar–turbulent transition. The Navier–Stokes equation for an incompressible viscous fluid is solved numerically by a spectral method as the initial value problem with the no-slip boundary condition on the pipe wall and periodic boundary conditions with period 16πa in the axial direction. The initial condition is given by the Hagen–Poiseuille flow (corresponding to the external force) superimposed with perturbations of finite amplitude. The long-term behaviour of the flow is qualitatively different depending on the Reynolds number Re=Ua⁄ν, where U is the centerline velocity of the above Hagen–Poiseuille flow and ν is the kinematic viscosity of the fluid. For intermediate Reynolds numbers (3300≤Re≤4000) we find a locally turbulent region, called a “puff”, which is advected downstream with velocity close to the mean axial velocity. The velocity fluctuations in the puff change quasi-periodically in time. The upstream boundary, called the “trailing edge” of the puff, sharply divides the laminar and turbulent regions. The mean shape of the trailing edge is determined by the least-squares method using the cross-axial velocity fluctuations in the puff. This will be useful as a reference when the flow structure and/or generation mechanism of a turbulent puff are analysed.