Inertial waves and mean velocity profiles in a rotating pipe and a circular annulus with axial flow

TL;DR

This study derives and verifies inertial wave solutions in rotating pipes and annuli, revealing specific mean velocity profiles involving Bessel functions, supported by numerical simulations showing vortex clusters.

Contribution

It provides the first analytical solutions for inertial waves in rotating pipes and annuli, linking wave behavior to specific mean velocity profiles and validating with numerical data.

Findings

Mean velocity profiles follow Bessel functions of the first kind.

Excellent agreement between theory and direct numerical simulations.

Large-scale vortex clusters are observed in the bulk region.

Abstract

In this paper we solve the inviscid inertial wave solutions in a circular pipe or annulus rotating constantly about its axis with moderate angular speed. The solutions are constructed by the so-called helical wave functions. We reveal that the mean velocity profiles must satisfy certain conditions to accommodate the inertial waves at the bulk region away from boundary. These conditions require the axial and azimuthal components of the mean velocity take the shapes of the zeroth and first order Bessel functions of the first kind, respectively. The theory is then verified by data obtained from direct numerical simulations for both rotating pipe and circular annulus, and excellent agreement is found between theory and numerical results. Large scale vortex clusters are found in the bulk region where the mean velocity profiles match the theoretical predictions. The success of the theory in rotating pipe, circular annulus, and streamwise rotating channel suggests that such inertial waves are quite common in wall bounded flow with background rotation.