Random vortex dynamics and Monte-Carlo simulations for wall-bounded viscous flows

TL;DR

This paper develops a stochastic integral framework for wall-bounded viscous flows, deriving exact random vortex dynamics and proposing convergent numerical schemes, demonstrated through simulations of boundary layer flows.

Contribution

It introduces a novel stochastic integral representation for viscous flows and establishes convergence of new numerical schemes for random vortex dynamics.

Findings

Exact random vortex dynamics derived for wall-bounded flows

Numerical schemes with proven convergence for vortex simulations

Demonstrated flow motion in boundary layer through simulations

Abstract

Functional integral representations for solutions of the motion equations for wall-bounded incompressible viscous flows, expressed (implicitly) in terms of distributions of solutions to stochastic differential equations of McKean-Vlasov type, are established by using a perturbation technique. These representations are used to obtain exact random vortex dynamics for wall-bounded viscous flows. Numerical schemes therefore are proposed and the convergence of the numerical schemes for random vortex dynamics with an additional force term is established. Several numerical experiments are carried out for demonstrating the motion of a viscous flow within a thin layer next to the fluid boundary.