Integral relations for the skin-friction coefficient of canonical flows

TL;DR

This paper derives simplified integral relations for the skin-friction coefficient in canonical flows, clarifying the influence of Reynolds stresses and introducing new identities and thickness decompositions to quantify drag contributions.

Contribution

It presents new simplified integral identities for skin-friction coefficients and introduces a novel two-fold integration identity for channel and pipe flows, extending previous work.

Findings

Reynolds stresses do not influence wall-shear in free-stream boundary layers.

Identities for skin-friction simplify to the von Karman equation with large upper bounds.

Decomposition of skin-friction into contributions from different integral thicknesses.

Abstract

We show that the Fukagata et al. (2002)'s identity for free-stream boundary layers simplifies to the von Karman momentum integral equation relating the skin-friction coefficient and the momentum thickness when the upper bound in the integrals used to obtain the identity is taken to be asymptotically large. If a finite upper bound is used, the terms of the identity depend spuriously on the bound itself. Differently from channel and pipe flows, the impact of the Reynolds stresses on the wall-shear stress cannot be quantified in the case of free-stream boundary layers because the Reynolds stresses disappear from the identity. The infinite number of alternative identities obtained by performing additional integrations on the streamwise momentum equation also all simplify to the von Karman equation. Analogous identities are found for channel flows, where the relative influence of the physical terms on the wall-shear stress depends on the number of successive integrations, demonstrating that the laminar and turbulent contributions to the skin-friction coefficient is only distinguished in the original identity discovered by Fukagata et al. (2002). In the limit of large number of integrations, these identities degenerate to the definition of skin-friction coefficient and a novel two-fold integration identity is found for channel and pipe flows. In addition, we decompose the skin-friction coefficient as the sum of the change of integral thicknesses with the streamwise direction, following the study of Renard & Deck (2016). We utilize an energy thickness and an inertia thickness, which is composed of a thickness related to the mean-flow wall-normal convection and a thickness linked to the streamwise inhomogeneity of the mean streamwise velocity. The contributions of the different terms of the streamwise momentum equation on the friction drag is thus quantified by these integral thicknesses.