Numerical studies of nonlinear axisymmetric waves on vortex filaments

Nonlinear Kelvin Waves;Vortex Breakdown;Vortex Core Dynamics;Vortex Filament;Weakly Nonlinear Stability

Mudkavi, Vidyadhar Yogeshwar ; Meiron, Daniel I.

University:California Institute of Technology

Department:Engineering and Applied Science

关键词: Nonlinear Kelvin Waves;    Vortex Breakdown;    Vortex Core Dynamics;    Vortex Filament;    Weakly Nonlinear Stability;   

Others  :  https://thesis.library.caltech.edu/2839/1/Mudkavi_vy_1991.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

The equations of Moore and Saffman (1971) are examined and are shown to contain the fast time scale equations governing the core waves on a straight vortex filament. The equations so derived are the same as those reported by Lundgren and Ashurst (1989) except for a correction term that allows for variation of the axial velocity structure within the vortex core. Numerical solutions of the Moore and Saffman equations are presented for various initial conditions consisting of wave-like perturbations on a cylindrical vortex, and they all show development of a jump in the core area. This has been advanced to be a mechanism for vortex breakdown by Lundgren and Ashurst. A comparison of the solutions of the Moore and Saffman equations with the solutions of the Navier-Stokes equations at high Reynolds number is presented for three different cases. In the first case a vortex with a very small perturbation is considered. The Moore and Saffman solution shows steepening of the initial wave resulting in the development of jump in the core area (shock). The Navier-Stokes solution shows bulging of the core. But, there is no indication of formation of a shock. In the second case a vortex with moderate perturbation is considered. The Moore and Saffman solution leads to a shock similar to the weak perturbation case. As before, the Navier-Stokes solution does not develop jump in the core area. However, development of a bubble of reversed flow is seen. In the third case, a jump in the core area in the solutions of the Navier-Stokes equations is seen for a strongly perturbed vortex. But the location and the sense of jump disagrees with jump that develops in the Moore-Saffman solution. Thus, the solutions of the Navier-Stokes equations and the Moore-Saffman equations show qualitative disagreement.Next, an extension of steady Kelvin waves for two different types of vorticity profiles is considered. In the first case, steady nonlinear waves are constructed via a perturbation method. In this case, the vorticity is nonzero inside the core and sharply drops to zero across the boundary. The shape of the core boundary is determined as part of the problem. The dependence of the Bernoulli function and the circulation function on the streamfunction are specified. This serves as the additional constraint necessary to determine the solution uniquely. The solutions are free of any vortex sheets. In the second case, nonlinear steady Kelvin waves on smooth vorticity distributions are constructed by means of a direct Newton method and a large order perturbation method. Instead of specifying the dependence of the Bernoulli function and the circulation function on the stream function as in the previous case, the solutions are restricted such that they have the same axial mean as the base flow. In both the approaches, regions of reversed flow are observed. This is the structure of bubble type of vortex breakdown.Next, an analysis of the weakly nonlinear stability of a columnar vortex is presented. It is shown that the amplitude, assumed to vary slowly in time and space, satisfies a cubic-nonlinear Schrodinger equation. Solutions are found to be unstable in the sense that the perturbations grow slowly in time. Solitary wave solutions are possible in this unstable case.