Numerical solution of the Navier-Stokes equations using the Path Tubes method

•We extend the Path Tubes method to the Navier-Stokes equations.•The method use an approach based on the mechanics of continuous media.•Were used values of Reynolds numbers typical of advection dominated flows.•The method proved to be accurate for the calculation of velocity fields.•The proposed methodology is able to work with coarse grids.

The present work addresses an extension of the Path Tubes method for solving the time-dependent Navier-Stokes equations for an incompressible Newtonian fluid. The approach used is a physically intuitive methodology whose formulation is based on the theoretical foundations of the mechanics of continuous media. This version of the Path Tubes method draws on a semi-Lagrangian time discretization that employs the Reynolds's transport theorem and a localization strategy. This time discretization can be seen as a transformation that acts on the Navier-Stokes equations, transforming this classical nonlinear model into linear partial differential equations of the (essentially) parabolic type.The result is 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. The Path Tubes method was implemented through parallel computing. For this, we use a computer equipped with shared-memory multiprocessors and the OpenMP software. After intensive numerical tests and using different values of Reynolds numbers typical of advection-dominated flows, the proposed method proved to be accurate and able to work with coarse grids.