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

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