A spectral Petrov-Galerkin formulation for pipe flow II: Nonlinear transitional stages

Abstract

This work is devoted to the study of the nonlinear evolution of perturbations of Hagen-Poiseuille or pipe flow. We make use of a solenoidal spectral Petrov-Galerkin method for the spatial discretization of the Navier-Stokes equations for the perturbation field. For the time evolution, we use a semi-implicit time integration scheme. Special attention is given to the explicit treatment and efficient evaluation of the nonlinear terms. The hydrodynamic stability analysis is focused on the streak breakdown process by which two-dimensional streamwise-independent perturbations transiently modulate the basic flow, resulting in a profile which is linearly unstable with respect to three-dimensional perturbations. This mechanism is one possible route of transition to turbulence in subcritical shear flows.