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

TL;DR

The paper develops Feynman-Kac-type stochastic representations for 2D incompressible viscous flows past a wall and leverages them to construct Monte-Carlo DNS via a random vortex method. It establishes well-posedness and robustness of regularised mean-field (McKean–Vlasov) dynamics, derives convergence guarantees for risk-free Monte-Carlo schemes (FMCRV/PMCRV), and provides explicit stability and consistency error bounds that yield convergence as $N\to\infty$, $h\to0$, and $δ\to0$, $R\to∞$. The wall case is treated by a reflected diffusion and boundary-vorticity mechanism, producing a wall-aware stochastic velocity representation with a wall Biot–Savart kernel $K_{D,δ}$. Numerical schemes and experiments in 2D demonstrate boundary-layer phenomena and the interaction between boundary and outer flows, validating the feasibility and accuracy of the Monte-Carlo approach for wall-bounded viscous flows.

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.

Figures (2)

• Figure 6.1: The boundary layer flow at different times $t$.
• Figure 6.2: The outer layer flow at different times $t$.

Theorems & Definitions (40)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.3
• Remark
• Corollary 3.1
• proof
• Remark
• Lemma 3.2