Optimal asymptotic behavior of the vorticity of a viscous flow past a two-dimensional body

TL;DR

This paper derives a new asymptotic expansion for the vorticity in a 2D viscous flow past a body, revealing behavior outside the wake that differs from previous models and depends on flow data.

Contribution

It introduces an improved asymptotic expansion for vorticity that applies outside the wake, unlike previous results, highlighting unique 2D flow characteristics.

Findings

Vorticity decays algebraically inside the wake

Vorticity decays exponentially outside the wake

Asymptotic term depends on flow data, not just linearization

Abstract

The asymptotic behavior of the vorticity for the steady incompressible Navier-Stokes equations in a two-dimensional exterior domain is described in the case where the velocity at infinity $\boldsymbol{u}_{\infty}$ is nonzero. It is well known that the asymptotic behavior of the velocity field is given by the fundamental solution of the Oseen system which is the linearization of the Navier-Stokes equation around $\boldsymbol{u}_{\infty}$. The vorticity has the property of decaying algebraically inside a parabolic region called the wake and exponentially outside. The previously proven asymptotic expansions of the vorticity are relevant only inside the wake because everywhere else the remainder is larger than the asymptotic term. Here we present an asymptotic expansion that removes this weakness. Surprisingly, the found asymptotic term is not given by the Oseen linearization and has a power of decay that depends on the data. This strange behavior is specific to the two dimensional problem and is not present in three dimensions.