Method of viscous dipole domains (VDD) for simulation of three-dimensional flows of viscous incompressible fluid. Physical mechanism of instability

Abstract

The viscous dipole domains method is a kind of the Lagrangian impulse-particle method using the diffusion velocity method and the particle strength exchange. We prefer to call particles as domains because a particle is associated with a concrete volume of fluid and moves with its velocity. When using diffusion velocity, we track a reference point that moves relative to the fluid. We assign this point a volume equal to the inverse density of points near it. This paper presents a detailed description of the method. The physical mechanism underlying the onset and evolution of numerical instability, which reflect the underlying physics of turbulent cascades, is revealed. A numerical stability criterion consistent with Kolmogorov scaling is derived. A new model for the diffusive motion of particles is proposed, ensuring a more uniform spatial distribution. The application of the developed method is demonstrated through numerical simulations of the evolution of vortex rings.