Riblet Flow Model Based on an Extended FIK Identity

Abstract

International audienceLarge Eddy Simulations of zero-pressure-gradient turbulent boundary layers over riblets have been conducted. All along the controlled domain, riblets maintain a significant 11 % drag reduction with respect to the flat plate at the same R e τ (from 250 to 450). To compare the flows above riblets and a reference smooth wall, an appropriate vertical shift between the two surfaces is required. In the present study, the “vertical origin” is set using the identity of Fukagata, Iwamoto and Kasagi (FIK) in Phys. Fluids, vol. 14, 2002, L73. This identity, which provides a physically meaningful decomposition of the skin friction, has been extended to complex wall surfaces and constitutes the basis for the derivation of a new virtual origin. Using this FIK-based origin, it is shown that the complex interactions between the riblets and the near-wall turbulent structures can be taken into account by a simple shift of the two axes of the mean and turbulent velocity profiles. The appropriate upward shift Δu +, typical for drag reduction, is directly dependent on the skin friction on the riblets and on the reference smooth plate at the same R e τ

A theoretical decomposition of mean skin friction generation into physical phenomena across the boundary layer

“…Because the control is assumed to be inner scaled, A ∞ and B 1∞ are Reynolds-number invariant. At the same Re τ , U + ∞ becomes U + ∞ + U + , where U + = A ∞ + B 1∞ is Reynolds-number invariant, a prediction from the present decomposition coinciding with the observed behaviour of inner-scaled roughness or riblets (Jiménez 2004;García-Mayoral & Jiménez 2011;Spalart & McLean 2011;Aupoix, Pailhas & Houdeville 2012;Bannier, Garnier & Sagaut 2015). Note that strictly speaking, the present decomposition (2.8) does not apply to a non-smooth wall, but it is compatible with blowing/suction.…”

“…Applying the present decomposition (2.8) to drag reduction could suggest new strategies of flow control, in the same way as the FIK decomposition (1.7) has enabled significant achievements in this field (see e.g. Iwamoto et al (2005) in high-Reynolds-number channel flows, Kametani & Fukagata (2011), Kametani et al (2015 and Stroh et al (2015) in spatially-developing boundary layers, as well as Bannier et al (2015) with their extended FIK identity). Alternatively to the FIK strategy which focuses on the Reynolds shear-stress intensity, the present decomposition suggests that a flow control focusing on the turbulent kinetic-energy production level might be of interest.…”

“…One may also note that considering energy rather than momentum makes sense for applications such as flow control and friction drag reduction where the control energy consumption and the engine fuel consumption are involved, showing the interest of the present new decomposition. Besides, Bannier et al (2015) indicate that variants of the FIK identity may be obtained, for instance by integrating the momentum equation four times instead of three, leading to a different weighting of each zone of the boundary layer. This is reminiscent of the very mathematical nature of the derivation of the original FIK identity.…”

Friction factor decomposition for rough-wall flows: theoretical background and application to open-channel flows

“…Nikora and others proposed, for the first time, an explicit decomposition of the friction factor for hydraulically smooth-wall boundary layers, closed channels and pipes. Fukagata et al's (2002) approach has been subsequently expanded to cover three-dimensional flows over walls of complex geometry (Peet & Sagaut 2009;Bannier, Garnier & Sagaut 2015). Recently, two complementary decompositions have been suggested, also for smooth-wall flows, based on the mean energy balance (Renard & Deck 2016) and mean vorticity equation (Yoon et al 2016).…”

“…Friction factor decomposition and flow classification Equation (2.3) explicitly shows that the Darcy-Weisbach friction factor f can be split into at least five additive components, due to: (i) viscous stress; (ii) turbulent stress; (iii) dispersive stress; (iv) unsteadiness and spatial heterogeneity of double-averaged velocities and pressure; and (v) spatial heterogeneity of fluid stresses in the x-y plane. The proposed decomposition (2.3) can be viewed as a generalisation of the previous momentum-based decompositions (Fukagata et al 2002;Peet & Sagaut 2009;Bannier et al 2015), which are essentially focused on hydraulically smooth-wall flows, i.e. without accounting for important dynamic effects such as dispersive stresses and pressure drag that typically emerge in flows at a high roughness Reynolds number.…”