Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances

Abstract

A theoretical study is presented of the spatial stability of flow in a circular pipe to small but finite axisymmetric disturbances. The disturbance is represented by a Fourier series with respect to time, and the truncated system of equations for the components up to the second-harmonic wave is derived under a rational assumption concerning the magnitudes of the Fourier components. The solution provides a relation between the damping rate and the amplitude of disturbance. Numerical calculations are carried out for Reynolds numbers R between 500 and 4000 and βR [les ] 5000, β being the non-dimensional frequency. The results indicate that the flow is stable to finite disturbances as well as to infinitesimal disturbances for all values of R and βR concerned.