Analysis of the structure of vortex planar flows and their changes with time

A numerical approach is proposed for studying changes in the structures of vortex configurations of an ideal fluid with time. Numerical algorithms are based on the solution of the initial-boundary value problem for nonstationary Euler equations in terms of vorticity and stream function. For this purpose, the spectral messless vortex method is used. It is based on the approximation of the stream function by the Fourier series cut and the approximation of the vorticity field by the least squares method from its values in marker particles. To calculate the dynamics of particles, a Cauchy problem is solved. The scheme of the spectral-vortex method makes it possible to implement the algorithm for analyzing the “instantaneous structure” of the velocity field using the methods of the theory of dynamic systems. This includes the construction of an “instantaneous” vector flow field, its singular points, and saddle point separatrices. To study the dynamics of changes in structures with time, the field of the local Lyapunov exponents is calculated. The results of numerical modeling and the analysis of changes in the structure of vortex flows are presented on the basis of the proposed approaches for two types of boundary conditions. Under periodic boundary conditions, the vortex configuration consists of four vortex spots For the flow condition, the fluid flow in the channel with a given velocity at the boundary is considered. The calculations have shown the effectiveness of the proposed algorithms for a fine analysis of the emerging pattern of the velocity field of the vortex configuration.