On the development of three-dimensional vortex breakdown in cylindrical regions

Three-dimensional and axisymmetric numerical simulations of the incompressible Navier-Stokes equations have been conducted to study the appearance and development of vortex breakdown in a family of columnar vortex flows in a straight pipe without wall friction. The numerical simulations show that the basic form of breakdown is axisymmetric, and a transition to helical breakdown modes is shown to be caused by a sufficiently large pocket of absolute instability inside the original axisymmetric “bubble” of recirculating flow. Depending on the values of the Reynolds and swirl parameters, two distinct unstable modes corresponding to azimuthal wave numbers n=+1 and n=+2 have been found to yield a helical or a double-helical breakdown mode, respectively. By means of a simple linear spatial stability analysis carried out in the sections of the pipe where the basic axisymmetric flow presents reverse flow, we have identified the frequencies and the dominant azimuthal wave numbers observed in the three-dimensional simulations.