Relations for the skin-friction coefficient of canonical flows

It is shown that the relation discovered by Fukagata et al. (2002) for the skin-friction coefficient of free-stream boundary layers simplifies to the von Kármán momentum integral equation when the upper integration bound along the wall-normal direction is taken asymptotically large. If the upper bound is finite, the weighted contributions of the terms of the streamwise momentum equation depend spuriously on the bound itself. The family of infinite identities obtained by successive integrations also reduces to the von Kármán equation. For channel flows, we prove that only the original identity found by Fukagata et al. (2002) possesses a physical meaning as we show that the infinite family degenerates to the definition of skin-friction coefficient as the number of integrations grows asymptotically. The dependence on the number of repeated integrations is therefore also non-physical. By a twofold integration, we find an identity, valid for channel and pipe flows, that links the skin-friction coefficient with the integrated Reynolds stresses and the centreline mean velocity. We further use the identity found by Renard & Deck (2016) to decompose the momentum thickness as the sum of an energy thickness, a thickness related to the mean-flow wall-normal convection and a thickness linked to the mean-flow streamwise inhomogeneity. This decomposition is useful to interpret the skin-friction decomposition physically and for quantifying the role of the different momentum-equation terms on the friction drag.