Collapse, Symmetry Breaking, and Hysteresis in Swirling Flows

The paper reviews striking features of swirling flows—collapse, swirl generation, vortex breakdown, hysteresis, and axisymmetry breaking—and the mechanisms involved with the help of conical similarity solutions of the Navier-Stokes equations. The strong accumulation of axial and angular momenta, observed in tornadoes and flows over delta wings, corresponds to collapse, i.e. the singularity development in these solutions. Bifurcation of swirl explains the threshold character of swirl development in capillary and electrovortex flows. Analytical solutions for fold catastrophes and hysteresis reveal why there are so few stable states and why the jump transitions between the states occur—features typical of tornadoes, of flows over delta wings, and in vortex devices. Finally, the divergent instability explains such effects as the splitting of a tornado and the development of spiral branches in tree and near-wall swirling flows.