On the instability of pipe Poiseuille flow

Stability of the flow of incompressible viscous fluid in a circular pipe is studied numerically. A perturbation consisting of finite-amplitude two-dimensional and infinitesimal three-dimensional parts is imposed on the basic flow. The temporal evolution of the perturbation is analyzed by direct numerical calculation of the Navier–Stokes equations. The two-dimensional disturbances are independent of the streamwise coordinate and initially take the form of streamwise rolls. It is shown that the nonlinear development of two-dimensional perturbations results in substantial spanwise modulation of the streamwise velocity component manifesting itself as a formation of streaks and the occurrence of inflection points. The modulated mean flow is found to be highly unstable to the three-dimensional perturbations which are localized spatially near these points. An instability mechanism that includes the modulation of the flow by growing two-dimensional disturbances and the inflectional instability of the modulated flow to three-dimensional perturbations is proposed.