In this work, the Prandtl-Batchelor theorem is extended to three-dimensional flows
slowly varying in one direction by using asymptotic techniques, and thus overcoming
the problem of having non-closed streamlines for recirculating three-dimensional
flows. The derived equations turned out to be an analogue of the quasi-cylindrical
equations used for describing behavior of streamwise vortices, rotating jets, vortex
breakdown phenomenon and some other problems. Hence, the derived equations
may be used for studying similar phenomena in non-axisymmetric cases. In order to
apply such a system of equations to particular problems, a computational code was
developed and validated by reproducing numerical results available in the literature.
This code was constructed in two parts, one part considered the parabolic system of
partial differential equations as decoupled from the Poisson equation and the second
part solved the nonlinear Poisson equation by using an iterative method. Finally,
these two algorithms were joined in order to solve the entire system. Once the code
was available, it was used to investigate possible non-axisymmetric effects on the
position of vortex breakdown phenomenon. The results of this study suggest that
non-axisymmetric effects precipitate the onset of vortex breakdown. From all this
work, two articles were written, one article was published in the Journal of Fluid
Mechanics (see Appendix D) and the second article will be submitted
