Flows in cylindrical coordinates with two walls

Abstract

The interaction of axial-symmetric vortices with solid walls necessitates cylindrical coordinates for a more accurate representation. Verzicco & Orlandi (1996a) have shown that a Cartesian coordinate system introduces a perturbation of wave number n = 4 of an amplitude is the greater the closer are the boundaries. Orlandi & Verzicco (1993a, 1993b) thus used cylindrical coordinates to study the three dimensional interaction of a perturbed vortex ring with a solid wall. They reproduced the experiment by Cerra et al. (1972) and explained features difficult to understand by flow visualizations. The analysis of the rate of strain tensor in the principal axis permitted to understand why the secondary ring, generated at the wall, is more unstable than the primary ring. The simulations were performed by a numerical method similar to that described in Chapt.10. A non-uniform coordinate, in the radial direction, clusters the grid points in the regions where the vorticity evolves and locates the external boundary as far as possible. The grid in the direction normal to the wall was uniform and few modifications were made to adapt the code, with periodicity in z, to flows in the presence of walls normal to z. The modifications consisted of an expansion in cos harmonics in the z direction to evaluate the ”pressure”, and in a few simple changes to impose the no-slip conditions at the walls. The code, in a certain sense, was inefficient, in fact it, was not possible to have sufficient resolution to reproduce the thin vorticity layers forming at the wall when the Reynolds number is high. The study was limited to the minimum Reynolds number at which the three dimensional perturbations are not dumped by viscous effects.