Turbulence Appearance and Nonappearance in Thin Fluid Layers

Flows in fluid layers are ubiquitous in industry, geophysics, and astrophysics. Large-scale flows in thin layers can he considered two dimensional with bottom friction added. Here we find that the properties of such flows depend dramatically on the way they arc driven. We argue that a wall-driven (Couctte) flow cannot sustain turbulence, no matter how small the viscosity and friction. Direct numerical simulations (DNSs) up to the Reynolds number Re = 10(6) confirm that all perturbations die in a plane Couette flow. On the contrary, for sufficiently small viscosity and friction, perturbations destroy the pressure-driven laminar (Poiseuille) flow. What appears instead is a traveling wave in the form of a jet slithering between wall vortices. For 5 x 10(3) < Re < 3 x 10(4), the mean flow in most cases has remarkably simple structure: the jet is sinusoidal with a parabolic velocity profile, and vorticity is constant inside vortices, while the fluctuations are small. At higher Re, strong fluctuations appear, yet the mean traveling wave survives. Considering the momentum flux barrier in such a flow, we derive a new scaling law for the Re dependence of the friction factor and confirm it by DNS.