Forced generation of solitary waves in a rotating fluid and their stability

[摘要] The primary objective of this graduate research is to study forced generation of solitary waves in a rotating fluid and their stability properties. For axisymmetric flow of a non-uniformly rotating fluid within a long cylindrical tube, an analysis is presented to predict the periodic generation of upstream-advancing vortex solitons by axisymmetric disturbance steadily moving with a transcritical velocity as a forcing agent. The phenomenon is simulated using the forced Korteweg-de Vries (fKdV) equation to model the amplitude function of the Stokes stream function for describing this family of rotating flows of an inviscid and incompressible fluid. The numerical results for the weakly nonlinear and weakly dispersive wave motion show that a sequence of well-defined axisymmetrical recirculating eddies is periodically produced and emitted to radiate upstream of the disturbance, soon becoming permanent in the form as a procession of vortex solitons, which we call vortons. Two primary flows, the Rankine vortex and the Burgers vortex, are adopted to exhibit in detail the process of producing the upstream vortons by the critical motion of a slender body moving along the central axis, with the Burgers vortex being found the more effective of the two in the generation of vortons. To investigate the evolution of free or forced waves within a tube of non-uniform radius, a new forced KdV equation is derived which models the variable geometry with variable coefficients. A set of section-mean conservation laws is derived specially for this class of rotational tube flows of an inviscid and incompressible fluid, in both differential and integral forms. A new aspect of stability theory is analyzed for possible instabilities of the axisymmetric solitary waves subject to non-axisymmetric disturbances. The present linear analysis based on the model equation involving the bending mode shows that the axisymmetric solitary wave is neutrally stable with respect to small bending mode disturbances. To study nonlinear interactions between the axisymmetric mode and bending mode, a new model is derived which consists of two coupled equations for disturbances of the two modes. The numerical results of the coupled equations show that the primary axisymmetric soliton appears to maintain its own entity, with some oscillations of its amplitude and an undular tail, inferring an interchange of energy between the two modes, when subject to small non-axisymmetric perturbations.

[发布机构] University:California Institute of Technology

[关键词] Engineering and Applied Science