| Abstract: | The article discusses the problem of the nonlinear oscillations of concentrated vortexes, For a vortical annulus and a spiral vortical filament, the article demonstrates the character of the change in the form and the frequency of the oscillations with an increase in the amplitude. In the case of standing waves, the solution is obtained in the form of a series with respect to the amplitude of the perturbations, with an accuracy up to terms of the third order of smallness, inclusive. For running waves, the solution is constructed using a method analogous to the Stewart method. The same problem is solved using direct numerical integration of the starting equations of motion. The values of the critical amplitudes which bring about the breakdown of a vortical annulus are obtained. It is shown that as a result of the nonlinearity of the equations in solution of the problem with the starting data higher harmonics can separate out intensively. For a spiral vortical filament, a study was made of the nonlinear interaction of the perturbations; it is shown, in particular, that a perturbation which is stable according to the linear theory may become unstable as a result of nonlinear interaction with neighboring perturbations along the length of the wave and of perturbations of the frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–49, May–June, 1973.The authors thank G. I. Petrov for his direction of the work, as well as V. Ya. Shkadov for his interest in the work. |
