The Josephson-Anderson relation for wall drag in classical turbulent channel flows

Turbulence in pressure-driven channels and pipes is often viewed as a “cascade” of momentum to the wall, but can be viewed also as an “inverse cascade” of vorticity away from the wall. Huggins exploited this idea to derive a “Josephson-Anderson” (JA) relation for incompressible channel flow, recently extended also to flows around solid bodies. These exact relations calculate drag by motion of vorticity relative to the background potential flow originating from the Kelvin minimum energy theorem. Since such Helmholtz-like decompositions into potential and rotational fields are non-unique, a good choice can make computations easier while keeping the two fields orthogonal. We modify Huggins’ JA relation to be more suitable for turbulence. We apply the relation, with spectral accuracy, to periodic channel data obtained from the Johns Hopkins Turbulence Database. Calculations of the spectral contents of vorticity flux reveal that wall-attached eddies are majorly responsible for vorticity flux within the inertial sublayer. Filtering out small scales allows us to identify structures providing more than 50% of the total drag generated in the log layer while occupying only 8% of the volume.