Josephson-Anderson Relation as Diagnostic of Turbulent Drag Reduction by Polymers*

The detailed Josephson-Anderson relation, which equates instantaneously the volume-integrated vorticity flux and the work by pressure-drop, has been the key to drag-reduction in superconductors and superfluids. We employ a classical version of this relation to investigate the dynamics of polymer drag-reduced channel flows, particularly in the High-extent Drag Reduction (HDR) regime which is known to exhibit strong space-time intermittency. We show that high drag is not created instantaneously by near-wall coherent vortex structures as assumed in prior works. The vorticity flux in turbulent channel flow is described by competing wall normal fluxes of spanwise vorticity, namely a drag-increasing "down-gradient" flux away from the wall and a drag-decreasing "up-gradient" flux towards the wall. The coherent vortex structures in HDR flows can produce a net ``up-gradient'' flux of vorticity toward the wall, which instead reduces instantaneous drag. Increase of wall-vorticity and skin friction due to this up-gradient flux occurs after an apparent lag of several advection times, increasing with Weissenberg number. This increasing lag appears due to polymer damping of up-gradient nonlinear vorticity transport arising from large-scale eddies in the logarithmic layer. The relatively greater polymer damping of down-gradient transport due to small-scale eddies results in lower net vorticity flux and hence lower drag. The Josephson-Anderson relation thus provides an exact tool to diagnose the mechanism of polymer drag-reduction in terms of vorticity dynamics and it explains also prior puzzling observations on transient drag-reduction, as for centerline-release experiments in pipe flow.