Fourier solution of two-dimensional Navier Stokes equation with periodic boundary conditions and incompressible flow

TL;DR

This paper presents a Fourier-based approximate solution to the 2D Navier-Stokes equations with periodic boundaries, analyzing how the solution's accuracy depends on the size of the basis vector space.

Contribution

It introduces a finite Fourier basis approach for solving the 2D Navier-Stokes equations and explores extrapolation to the infinite basis case.

Findings

Approximate solutions vary with basis size.

Extrapolation suggests potential failure for certain initial conditions.

Full basis solution may not exist for all initial velocities.

Abstract

An approximate solution to the two dimensional Navier Stokes equation with periodic boundary conditions is obtained by representing the x any y components of fluid velocity with complex Fourier basis vectors. The chosen space of basis vectors is finite to allow for numerical calculations, but of variable size. Comparisons of the resulting approximate solutions as they vary with the size of the chosen vector space allow for extrapolation to an infinite basis vector space. Results suggest that such a solution, with the full basis vector space and which would give the exact solution, would fail for certain initial velocity configurations when initial velocity and time t exceed certain limits.