Near identity transformations for the Navier-Stokes equations

TL;DR

This paper introduces a framework using near identity transformations to describe the Navier-Stokes equations, linking diffusive particle path transformations with active velocity and vorticity, and analyzing their properties and reset conditions.

Contribution

It presents a novel formulation of Navier-Stokes equations through near identity maps, connecting diffusive transformations with physical quantities and providing reset criteria based on enstrophy.

Findings

Active velocity derived from diffusive path transformation and virtual velocity.

Active vorticity obtained via diffusive path transformation and virtual vorticity.

Lower bounds on reset times related to maximum enstrophy.

Abstract

The Navier-Stokes equations and their various approximations can be described in terms of near identity maps, that are diffusive particle path transformations of physical space. The active velocity is obtained from the diffusive path transformation and a virtual velocity using the Weber formula. The active vorticity is obtained from the diffusive path transformation and a virtual vorticity using a Cauchy formula. The virtual velocity and the virtual vorticity obey diffusive equations, which reduce to passive advection formally, if the viscosity is zero. Apart from being proportional to the viscosity, the coefficients of these diffusion equations involve second derivatives of the near identity transformation and are related to the Christoffel coefficients. If and when the near-identity transformation departs excessively from the identity, one resets the calculation. Lower bounds on the minimum time between two successive resettings are given in terms of the maximum enstrophy.