Onsager Theory of Momentum Cascade in Wall-Bounded Turbulence

Onsager analyzed energy cascade exactly for individual flow realizations at infinite Reynolds number by spatial filtering. We study momentum cascade in wall-bounded turbulence by filtering out eddies of size h to the wall. We consider interior flows through channels and exterior flows past solid bodies, with surfaces either hydraulically rough or smooth. We show that skin friction τ_w and surface pressure p_w exist at the wall in the infinite-Re limit. But both space-time fields are obtained also by taking Re?∞ first and then matching to spatial flux of momentum as h,l?0. This deterministic “momentum cascade” does not generally occur in the traditional inertial sublayer, e.g. in a rough-wall pipe it occurs for h < roughness height. When the inviscid-limit velocity satisfies no-penetration at the wall, momentum flux vanishes and τ_w=0. Then only form drag remains for Re?∞ and wall pressure is obtained by taking the limit to the surface of the pressure of inviscid Euler solutions in the fluid interior. As an application, we show that Lighthill’s theory of vorticity generation holds even for Re?∞.Our analysis motivates novel LES methods for wall-bounded turbulence and applies also at finite Reynolds numbers.