Eulerian-Lagrangian Formulation for Compressible Navier-Stokes Equations

Abstract

The Eulerian-Lagrangian formulation of the (inviscid) Euler dynamics in terms of advected Weber-Clebsch potentials (Lamb, 1932), was extended by Constantin to cover the viscous Navier-Stokes dynamics (Constantin, 2001). Numerical studies (Ohkitani & Constantin, 2003), of this formulation of the Navier-Stokes equations concluded that the diffusive Lagrangian map becomes non-invertible under time evolution and requires resetting for its calculation. They proposed the observed sharp increase of the frequency of resettings as a new diagnostic of vortex reconnection. In previous work we were able (Cartes et al., 2007; 2009) to complement these results, using an approach that is based on a generalised set of equations of motion for the Weber-Clebsch potentials, that turned out to depend on a parameter τ, which has the unit of time for the Navier-Stokes case. Also to extend our formulation to magnetohydrodynamics, and thereby obtain a new diagnostic for magnetic reconnection. In this work we present a generalisation of the Weber-Clebsch variables in order to describe the compressible Navier-Stokes dynamics. Our main result is a good agreement between the dynamics for the velocity and density fields that come from the dynamics of Weber-Clebsch variables and direct numerical simulations of the compressible Navier-Stokes equations. We first present the inviscid Eulerian-Lagrangian theory, then Constantin’s extension to viscous fluids and derive our equations of motion for the Weber-Clebsch potentials that describe the compressible Navier-Stokes dynamics. Then, performing direct numerical simulations of the Taylor-Green vortex, we check that our formulation reproduces the compressible dynamics.