Numerical solution of BVP for the incompressible Navier-Stokes equations at large Reynolds numbers

TL;DR

This paper develops finite element algorithms to compute stationary solutions of the incompressible Navier-Stokes equations at high Reynolds numbers, revealing non-uniqueness and stability issues in such flows.

Contribution

It introduces a computational approach using natural pressure-velocity variables and iterative linearizations to find multiple solutions at high Reynolds numbers.

Findings

Obtained multiple steady solutions at high Reynolds numbers.

Identified critical Reynolds number where solution non-uniqueness occurs.

Demonstrated algorithm effectiveness on cavity flow problem.

Abstract

The problems of numerical modeling of viscous incompressible fluid flows are widely considered in computational fluid dynamics. Stationary solutions of boundary value problems for the Navier-Stokes equations exist at large Reynolds numbers, but they are unstable and lead to transient or turbulent unsteady regimes. In addition, the solution of the boundary value problem at large values of Reynolds number may be non-unique. In this paper, we consider computational algorithms numerical algorithms for finding such stationary solutions. We use natural pressure-velocity variables under standard finite element approximation on triangular grids. Iterative methods with different linearizations of convective transport are used to test a two-dimensional problem of incompressible fluid flow in a square-section cavity with a movable top lid. The developed computational algorithm allowed us to obtain two solutions when the Reynolds number exceeds a critical value for flows in a cavity of semi-elliptical cross-section.