Application of the Path Tubes Method to the Navier-Stokes Equations

Abstract

This work deals with an extension of the Path Tubes method for the solution of the time-dependent Navier-Stokes equations for an incompressible Newtonian fluid. Departing from a physically intuitive methodology based on the theoretical basis of the mechanics of continuous media, a robust numerical technique is obtained. This version of the Path Tubes method draws on a semi-Lagrangian time-discretization that employs the Reynolds’ transport theorem, and a localization approach, to establish an implicit semi-Lagrangian algorithm that allows the use of classical schemes for spatial discretization, such as central-difference formulas, without the need to use upwind techniques, or high-order corrections for time derivatives. Some of the extensive numerical tests are shown herein, in particular for Reynolds’ numbers typical of advection dominated flows. The tests show the method is accurate, even for coarse grids.