Journal of Theoretical
and Applied Mechanics

42, 1, pp. 3-20, Warsaw 2004

Random vortex method for three dimensional flows. Part I: Mathematical background

Andrzej Styczek, Piotr Duszyński, Marta Poćwierz, Jacek Szumbarski
The paper presents a mathematical formulation of the Lagrangian method suitable for numerical simulation of 3D viscous incompressible flows. The vorticity field is approximated by a large ensemble of vortex particles which move with the fluid (advection) and perform random walks (diffusion). The charges of the particles change with time due to the stretching term in the governing equation. The construction of the vortex particles ensures that the approximated vorticity field is strictly divergence-free at any time instant. The boundary condition at the surface of an immersed body is satisfied by the creation of new vortex particles near the surface. Various properties of induced velocity and vorticity fields are also discussed.
Keywords: vortex methods; vortex stretching; Fokker-Planck-Kolmogorow equation; Itô stochastic differential equations