Solitary solution method for incompressible Navier-Stokes PDE

TL;DR

This paper introduces a novel solitary solution method for incompressible Navier-Stokes PDEs that leverages space contraction to obtain compact solutions, enabling validation across various geometries and meshes.

Contribution

The method systematically isolates nonlinear responses in one spatial dimension using space contraction, allowing for flexible validation of numerical schemes for Navier-Stokes equations.

Findings

Successfully validated numerical schemes for Euler and Navier-Stokes PDEs.

Enhanced solution compactness through numerical integration of contracting domains.

Applicable to arbitrary unstructured meshes and complex geometries.

Abstract

The method exploits the contraction of space to systematically obtain compact solitary solutions. The latter is provided for the incompressible Euler and Navier-Stokes PDE. The nonlinear response of momentum advection is moved into a term for contracting space. Then the linear continuity PDE is solved by means of arbitrarily selected closure functions. The contracting space is then split into two variables. The compactness of some solutions is enhanced by numerically integrating the contracting domain while retaining a solution for the nonlinear PDE. The validation of numerical schemes is demonstrated for the Euler and Navier-Stokes PDE. As the nonlinear response is isolated in only one spatial dimension, the method permits to validate arbitrary unstructured meshes and domain geometries by introducing the spatial dimension n+1.