Vortex breakdown in a three-dimensional swirling flow
Résumé
Time-dependent swirling flows inside an enclosed cylindrical rotor-stator cavity with aspect ratio H/R = 4, larger than the ones usually considered in the literature, are studied. Within a certain range of governing parameters, vortex breakdown phenomena can arise along the axis. Very recent papers exhibiting some particular three-dimensional effects have stimulated new interest in this topic. The study is carried out by a numerical resolution of the three-dimensional Navier-Stokes equations, based on high-order spectral approximations in order to ensure very high accuracy of the solutions. The first transition to an oscillatory regime occurs through an axisymmetric bifurcation (a supercritical Hopf bifurcation) at Re = 3500. The oscillatory regime is caused by an axisymmetric mode of centrifugal instability of the vertical boundary layer and the vortex breakdown is axisymmetric, being composed of two stationary bubbles. For Reynolds numbers up to Re = 3500, different three-dimensional solutions are identified. At Re = 4000, the flow supports the k = 5 mode of centrifugal instability. By increasing the rotation speed to Re = 4500, the vortex breakdown evolves to an S-shaped type after a long computational time. The structure is asymmetric and gyrates around the axis inducing a new time-dependent regime. At Re = 5500, the structure of the vortex breakdown is more complex: the upper part of the structure takes a spiral form. The maximum rotation speed is reached at Re = 10000 and the flow behaviour is now chaotic. The upper structure of the breakdown can be related to the spiral-type. Asymmetric flow separation on the container wall in the form of spiral arms of different angles is also prominent.