The decay of a swirling perturbation of pipe Poiseuille flow

Abstract

Secondary flows consisting of two pairs of vortices arise when two fluid streams meet at a confluence, such as in the airways of the human lung during expiration or at the vertebrobasilar junction in the circulatory system, where the left and right vertebral arteries converge. In this paper the decay of these secondary flows is studied by considering a four-vortex perturbation from Poiseuille flow in a straight, three-dimensional pipe. A polynomial eigenvalue problem is formulated and the exact solution for the zero Reynolds numberRis derived analytically. This solution is then extended by perturbation analysis to produce an approximation to the eigenvalues forR≪ 1. The problem is also solved numerically for 0 ≤R≤ 2,000 by a spectral method, and the stability of the computed eigenvalues is analysed using pseudospectra. For all Reynolds numbers, the decay rate of the swirling perturbation is found to be governed by complex eigenvalues, with the secondary flows decaying more slowly asRincreases. A comparison with results from an existing computational study of merging flows shows that the two models give rise to similar secondary flow decay rates.