On the enhancement of boundary layer skin friction by turbulence: an angular momentum approach

Abstract

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.

Integral relations for the skin-friction coefficient of canonical flows

“…(2002). During the final revision stages of the present work, we became aware that was also discovered by Elnahhas & Johnson (2022) for channel flows.…”

“…These comments are also valid for the original identities by Fukagata et al (2002). During the final revision stages of the present work, we became aware that (3.21) was also discovered by Elnahhas & Johnson (2022) for channel flows.…”

“…The interesting study by Elnahhas & Johnson (2022) is particularly worth mentioning because their identity expresses the skin-friction coefficient of free-stream boundary layers as the sum of the Blasius friction coefficient and an integrated contribution of the Reynolds stresses, thereby distinguishing the contribution of the laminar flow and the nonlinear fluctuations in transitional or turbulent boundary layers.…”

Integral relations for the skin-friction coefficient of canonical flows

“…For channel flows, only the original identity found by Fukagata et al (2002) possesses a physical meaning and we have proved that the infinite family degenerates to the definition of skin-friction coefficient as the number of integrations grows asymptotically. By a two-fold integration, we have found an identity, valid for channel and pipe flows, that links the skin-friction coefficient with the integrated Reynolds stresses and the centerline mean velocity (the identity for channel flows was also discovered by Elnahhas & Johnson (2022)).…”

“…It is noted, however, that the RD decomposition (3.22) and identities emerging from it, such as (3.37), do not distinguish the laminar and the turbulent contributions to the skin-friction coefficient for any flow, while the FIK identity achieves this task for confined flows and the identity discovered by Elnahhas & Johnson (2022) does so for free-stream boundary layers. As demonstrated by Renard & Deck (2016), the difference in the skin-friction coefficient between a laminar and a turbulent boundary layer at the same Reynolds number based on ∆ (for which E is identical) is dominated by P.…”