Onsager Theory of Turbulence, the Josephson–Anderson Relation, and the D’Alembert Paradox

The Josephson–Anderson relation, valid for the incompressible Navier–Stokes solutions which describe flow around a solid body, equates the power dissipated by drag instantaneously to the flux of vorticity across the flow lines of the potential Euler solution considered by d’Alembert. Its derivation involves a decomposition of the velocity field into this background potential-flow field and a solenoidal field corresponding to the rotational wake behind the body, with the flux term describing a transfer from the interaction energy between the two fields and into kinetic energy of the rotational flow. We establish the validity of the Josephson–Anderson relation for the weak solutions of the Euler equations obtained in the zero-viscosity limit, with one transfer term due to inviscid vorticity flux and the other due to a viscous skin-friction anomaly. Furthermore, we establish weak forms of the local balance equations for both interaction and rotational energies. We define nonlinear spatial fluxes of these energies and show that the asymptotic flux of interaction energy to the wall equals the anomalous skin-friction term in the Josephson–Anderson relation. However, when the Euler solution satisfies a condition of vanishing normal velocity at the wall, then the anomalous term vanishes. In this case, we can show also that the asymptotic flux of rotational energy to the wall must vanish and we obtain in the rotational wake the Onsager–Duchon–Robert relation between viscous dissipation anomaly and inertial dissipation due to scale-cascade. In this way we establish a precise connection between the Josephson–Anderson relation and the Onsager theory of turbulence, and we provide a novel resolution of the d’Alembert paradox.