On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers

TL;DR

This paper investigates the asymptotic behavior of Galerkin approximations to the Navier-Stokes equations as the Grashof number becomes very large, introducing a new expansion method applicable in multiple settings.

Contribution

It introduces a novel asymptotic expansion technique for analyzing solutions of Galerkin approximations in the high Grashof number limit, applicable to various boundary conditions and force perturbations.

Findings

Convergent asymptotic expansions in G for certain solutions and forces

Methodology applies to both 2D and 3D Navier-Stokes equations

Results hold under no-slip and periodic boundary conditions

Abstract

We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions.