Scaling laws in turbulence.

Abstract

Following the idea that dissipation in turbulence at high Reynolds number is dominated by singular events in space-time and described by solutions of the inviscid Euler equations, we draw the conclusion that in such flows, scaling laws should depend only on quantities appearing in the Euler equations. This excludes viscosity or a turbulent length as scaling parameters and constrains drastically possible analytical pictures of this limit. We focus on the drag law deduced by Newton for a projectile moving quickly in a fluid at rest. Inspired by this Newton's drag force law (proportional to the square of the speed of the moving object in the limit of large Reynolds numbers), which is well verified in experiments when the location of the detachment of the boundary layer is defined, we propose an explicit relationship between the Reynolds stress in the turbulent wake and quantities depending on the velocity field (averaged in time but depending on space). This model takes the form of an integrodifferential equation for the velocity which is eventually solved for a Poiseuille flow in a circular pipe.