Right Type Departmental Bulletin Paper

Abstract. We study the convergence of the two-dimensional stationary Kolmogorov flows as the Reynolds number increases to infinity. Since the flows to be considered are stationary solutions of the Navier-Stokes equations, they are smooth whatever the Reynolds number may be. However, in the limit of infinite Reynolds number, they can, at least theoretically, converge to a non-smooth function. Through numerical experiments, we will show that, under a certain condition, some smooth solutions of the Navier-Stokes equations converge to a non-smooth solution of the Euler equations and develop internal layers. Therefore the Navier-Stokes flows are “ nearly singular ” for large Reynolds numbers. In view of this nearly singular solutions, we propose a possible scenario of turbulence, which is of intermediate nature between Leray’s scenario and Ruelle-Takens’s one. \S 1. Introduction. We study how stationary solutions to the Navier-Stokes equations depends on the Reynolds number as the number tends to infinity. Specifically, we consider incompressible viscous flows in $2D $ flat tori. The equations of motion are as follows: $u \frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu\triangle u-\frac{\partial p}{\partial x}+\gamma\sin(\pi y/b) $