On the spatial evolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at Re = 3000. Part 1. Subcritical disturbance

Abstract

We have performed a numerical study on the transition of a cylindrical pipe flow under the influence of a localized disturbance in the form of periodic suction and blowing (PSB) applied at the pipe wall. We focus here on the so-called receptivity problem where the spatial evolution of this disturbance is studied as it travels downstream through the pipe. The study is carried out by means of two techniques: an eigenmode expansion solution (EES) and a full nonlinear direct numerical simulation (DNS). The EES is based on an analytical expansion in terms of the eigenfunctions of the linear operator which follows from the equations of motion expressed in a cylindrical coordinate system. The DNS is formulated in terms of a spectral element method.We restrict ourselves to a so-called subcritical disturbance, i.e. the flow does not undergo transition. For very small amplitudes of the PSB disturbance the results of the EES and DNS techniques agree excellently. For larger amplitudes nonlinear interactions come into play which are neglected in the EES method. Nevertheless, the results of both methods are consistent with the following transition scenario. The PSB excites a flow perturbation that has the same angular wavenumber and frequency as the imposed disturbance itself. This perturbation is called the fundamental mode. By nonlinear self-interaction of this fundamental mode higher-order harmonics, both in the angular wavenumber and frequency, are generated. It is found that the harmonic with angular wavenumber 2, i.e. twice the wavenumber of the fundamental mode, and with zero frequency grows strongly by a linear process known as transient growth. As a result the (perturbed) pipe flow downstream of the disturbance region develops extended regions of low velocity, known as low-speed streaks. At large disturbance amplitudes these low-speed streaks show the development of high wavenumber oscillations and it is expected that at even higher disturbance amplitudes these oscillations become unstable and turbulent flow will set in.Our result agrees (at least qualitatively) with the transition scenario in a plane Poiseuille flow as discussed by Reddy et al. (1998) and Elofson & Alfredson (1998).