Calculation of three-dimensional flows of an incompressible fluid based on a dipole representation of vorticity

A study was conducted to calculate three-dimensional flows of an incompressible fluid based on a dipole representation of vorticity. new approach to the solution of time-dependent three-dimensional Navier-Stokes equations of a viscous incompressible fluid in the Lagrangian coordinates was suggested. The method was based on the introduction of a new vector function D called the density dipoles through which the velocity and vorticity in the flow field were expressed. The equations for the evolution of the density of dipoles corresponding to varying hydrodynamic fields were derived and the boundary conditions for it in the form of integrated equation were formulated. A corresponding numerical method or the method of dipole domains (MDD) was constructed with a fundamental distinction from the known three-dimensional vortex methods that used the unclosed vortex elements.