Vortices of large scale appearing in the 2D stationary Navier–Stokes equations at large Reynolds numbers

Abstract

We consider Kolmogorov’s problem for the two-dimensional (2D) Navier–Stokes equations. Stability of and bifurcation from the trivial solution are studied numerically. More specifically, we compute solutions with large Reynolds numbers with a family of prescribed external forces of increasing degree of oscillation. We find that, whatever the external force may be, a stable steady-state of simple geometric character exits for sufficiently large Reynolds numbers. We thus observe a kind of universal outlook of the solutions, which is independent of the external force. This observation is reinforced further by an asymptotic analysis of a simple equation called the Proudman–Johnson equation.