Realization of Incompressible Navier-Stokes Flow as Superposition of Transport Processes for Clebsch Potentials

TL;DR

This paper reformulates incompressible Navier-Stokes equations using Clebsch potentials, revealing new transport process representations, entropy measures, and flow classifications that extend beyond traditional two-dimensional flow analysis.

Contribution

It introduces a novel Clebsch potential-based formulation of Navier-Stokes equations, including a new entropy functional and flow classification beyond 2D flows.

Findings

Navier-Stokes can be expressed as transport equations for Clebsch potentials.

A Shannon-type entropy functional with non-negative growth rate is defined.

A broader class of flows with non-positive entropy growth and vanishing vortex stretching is identified.

Abstract

In ideal fluids, Clebsch potentials occur as paired canonical variables associated with the Hamiltonian description of the Euler equations. This paper explores the properties of the incompressible Navier-Stokes equations when the velocity field is expressed through a complete set of paired Clebsch potentials. First, it is shown that the incompressible Navier-Stokes equations can be cast as a system of transport (convection-diffusion) equations where each Clebsch potential plays the role of a generalized distribution function. The diffusion operator associated with each Clebsch potential departs from the standard Laplacian due to a term depending on the Lie-bracket of the corresponding Clebsch pair. It is further shown that the Clebsch potentials can be used to define a Shannon-type entropy measure, i.e. a functional, different from energy and enstrophy, whose growth rate is non-negative. As a consequence, the flow must vanish at equilibrium. This functional can be interpreted as a measure of the topological complexity of the velocity field. In addition, the Clebsch parametrization enables the identification of a class of flows, larger than the class of two dimensional flows, possessing the property that the vortex stretching term identically vanishes and the growth rate of entrophy is non-positive.