On the dynamics of the boundary vorticity for incompressible viscous flows

TL;DR

This paper derives a dynamical equation for boundary vorticity in incompressible viscous flows, revealing increased effective viscosity at the boundary and enabling boundary vorticity determination without detailed flow knowledge.

Contribution

It introduces a new dynamical equation for boundary vorticity and validates it through stochastic numerical simulations, enhancing understanding of boundary layer dynamics.

Findings

Boundary vorticity can be determined up to bounded errors over time.

Viscosity at the boundary effectively doubles, acting as if the fluid is more viscous.

The dynamical equation is validated via stochastic direct numerical simulations.

Abstract

The dynamical equation of the boundary vorticity has been obtained, which shows that the viscosity at a solid wall is doubled as if the fluid became more viscous at the boundary. For certain viscous flows the boundary vorticity can be determined via the dynamical equation up to bounded errors for all time, without the need of knowing the details of the main stream flows. We then validate the dynamical equation by carrying out stochastic direct numerical simulations (i.e. the random vortex method for wall-bounded incompressible viscous flows) by two different means of updating the boundary vorticity, one using mollifiers of the Biot-Savart singular integral kernel, another using the dynamical equations.