Onsager approach to wall bounded turbulence

Turbulent drag laws observed with both smooth and rough walls imply divergences of velocity gradients at the wall and/or in the flow interior in the infinite-Re limit. The fluid motions cannot then be understood as strong solutions of the PDE’s. To obtain a valid dynamical description, regularization of the divergences is necessary. Coarse-graining the Navier-Stokes equation filters out eddies of size <ℓ and a windowing function is needed to remove eddies of distance <h to the wall, thus introducing two arbitrary regularization length-scales. The regularized equations for resolved eddies correspond to the weak formulation of Navier-Stokes equation and contain, in addition to the usual turbulent stress, an inertial drag force modeling momentum exchange with unresolved near-wall eddies. By an Onsager-type argument, we derive for smooth-wall channel flow an upper bound on skin friction by a Blasius -1/4 law. Our result is a deterministic version of Prandtl’s relation with the 1/7 power-law profile of the mean streamwise velocity, with also an assumption on scaling of the Reynolds stress with wall distance. Our approach of filtering small-scale motions and windowing out near-wall eddies offers a rigorous framework for large-eddy simulation modeling in the presence of solid walls.