Realization of Incompressible Navier-Stokes Flow as Superposition of Transport Processes for Clebsch Potentials
arXiv:2010.16048 · doi:10.1063/5.0035339
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.
16 pages