Vortex methods in two-dimensional fluid dynamics
The authors discuss some recent results on vortex methods for the time evolution of planar incompressible fluids. Their analysis is macroscopic and is mostly based on a well-known and old idea: a two-dimensional incompressible ideal fluid behaves like a Hamiltonian system. The distribution of vorticity can be envisaged as formed by ''particles'', infinitesimal vortices, whose motion is described by Hamiltonian ordinary differential equations. Such dynamical systems are studied which yield solutions of the Euler equations. When the vorticity is concentrated on points, the evolution of the fluid reduces to a Hamiltonian system with a finite number of degrees of freedom, called the vortex model. Its relation to Euler flows under suitable limits is investigated here (sections 1-5). In the presence of viscosity a vortex dynamics can still be considered, the vortex motion being described by stochastic ordinary differential equations. This entails diffusion and delocalization of the vortices. The present point of view provides a good understanding of Navier-Stokes evolution in the absence of physical boundaries (sections 6-8). The physically most interesting case, in which obstacles occur in the fluid, is discussed in section 9, via the Chorin method. This case is more involved and has not been completely understood as yet.
time evolution of planar incompressible fluids
scientific article; zbMATH DE number 3868839