| The subject of this dissertation is to develop reliable analytical and numerical methods for the study of the nonlinear stability of a class of slender, incompressible axisymmetric swirling flows. We desire to understanding early nonlinear evolution and possibly gain insight into phenomena such as solitary waves observed on vortex filaments and strongly nonlinear phenomena like vortex breakdown.; We use an extension of the method used by Leibovich & Ma [1982] for the development of the equations, which differs in some significant aspects from the formulation of the aforementioned authors. We find, in agreement with Leibovich & Ma [1982], that the complex envelope amplitude of weakly nonlinear asymmetric waves is governed by the cubically nonlinear Schrödinger equation (NLS), however our form for coefficients differs from those of Leibovich & Ma [1982]. Most significantly, our formulation includes an axisymmetric disturbance component at second order (a full order lower than Leibovich & Ma) and thus permits energy exchange between asymmetric and axisymmetric disturbance components. The resulting equations also explicitly demonstrate the possibility of singular points where the group (not phase) velocity of linear disturbance equals the local axial flow velocity (group-velocity critical layer in our terminology) and where the NLS coefficients blow up, thus providing a wavenumber selection mechanism for the weakly nonlinear evolution. After implementation of numerical algorithm, our goal is to investigate the effects of weak nonlinearities on the stability of axisymmetric columnar flows. The analysis is applied to several model vortical flows, namely the Q-vortex and the Batchelor [1964] trailing line vortex. |
