Fast relaxation of a viscous vortex in an external flow

Date: 14 April 2025 - 17 April 2025

Abstract:

The aim of these lectures is to present a rigorous mathematical analysis of a classical model for the interaction of viscous vortices in two-dimensional incompressible flows. The object of our study is a concentrated vortex that evolves according to the Navier-Stokes equations and is further advected by a smooth, divergence-free external field. The latter is typically generated by other vortical structures that may be present in different regions of the fluid domain.

We first study the idealized situation where the initial vorticity is a point vortex. Using a power series expansion in a small parameter related to the size of the vortex core, we compute an approximate solution of the system that accurately describes, in the regime of high Reynolds numbers, the motion of the vortex center and the deformation of the streamlines under the shear stress of the external flow.

We then consider the more realistic case where the initial vorticity is a sharply concentrated Gaussian vortex. Such data can be described as ill-prepared, in the sense that the evolution exhibits a transient regime near initial time where the solution quickly relaxes to the approximation computed in the previous step. This phenomenon, which is due to enhanced dissipation in the vortex core, occurs on a time scale that depends on the Reynolds number and is much shorter than the diffusive time scale.

The material presented in these lectures is based on joint work with Martin Donati (Grenoble).