Swirling turbulent pipe flows: Inertial region and velocity–vorticity correlations

Highlights: • Comparison of swirling flows due to Reynolds number effects. • Investigation into the mean-momentum equations. • Analysis of the velocity–vorticity correlations. Swirling pipe flows are studied here with an aim towards understanding the onset of the inertial region — where the turbulent-inertia term in the mean momentum equation is balanced by pressure gradient and viscous term is sub-dominant — as well as the clarifying the velocity–vorticity correlations that make up the turbulent inertia. To this end, we first manipulate the mean momentum equation in both axial and azimuthal directions and find some exact results in the inertial region, and carry out direct numerical simulations of swirling pipe flows at axial friction Reynolds numbers of 170 and 500. The swirl number considered in our simulations is $S\approx 0.3$, and we compare our results to non-swirling pipe flows at similar Reynolds numbers. We find that swirling produces a drag increase and an influence on the turbulence statistics similar to increasing the Reynolds number except for the streamwise turbulence intensity. An analysis on the axial and azimuthal mean momentum equations shows that swirling shifts the beginning of the inertial region wall-normal location closer to the wall. The turbulent inertia decomposition reveals that the near-wall region the velocity–vorticity correlations of the axial direction are similar to a 2D channel flow and interpreted as vorticity stretching/reorientation and dispersion, whereas in the new correlations in the azimuthal direction can also be given a similar physical meaning in the near-wall region. In the outer-region, however, the pipe axial correlations are different to the 2D-channel, and so are the azimuthal correlations. We find that the pipe has new a 'geometric' contribution in both axial and azimuthal directions that play an important role in contributing towards vorticity dispersion in the outer core region of a swirling pipe flow.