Abstract
This article can be considered as an extension of the paper of Fukagata et al. (Phys. Fluids 14:L73, 2002) who derived an analytical expression for the componential contributions into skin friction in a turbulent channel, pipe and plane boundary layer flows. In this paper, we extend theoretical analysis of Fukagata et al. limited to canonical cases with two-dimensional mean flow to a fully three-dimensional situation allowing complex wall shapes. We start our analysis by considering arbitrarily-shaped surfaces and then formulate a restriction on a surface shape for which the current analysis is valid. Theoretical formula for skin friction coefficient is thus given for streamwise and spanwise homogeneous surfaces of any shape, as well as some more complex configurations, including spanwise-periodic wavy patterns. Current theoretical analysis is validated using the results of Large Eddy Simulations of a turbulent flow over straight and wavy riblets with triangular and knife-blade cross-sections. Decomposition of skin friction into different componential contributions allows to analyze the influence of different dynamical effects on a drag modification by riblet-covered surfaces.
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Acknowledgements
Dr. Yves Charon (IFP, France) is gratefully acknowledged for many enlightening discussions. This project was supported by ANR as project ANR-PANH-READY.
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Appendix
Appendix
In this appendix, we spell out the formulas obtained for the components of friction after the double integration \(\int_{0}^{\eta^{\mathit{top}}}d\gamma\int _{0}^{\eta}d\gamma\) and summation over dζ→0 is applied to this equation. As before, γ denotes the local variable along the integration contour, and d γ — its differential. We will also be using the notations γ(η) and γ(ζ) to distinguish the integration along η or ζ lines, and d γ(η), d γ(ζ), referring to their differentials. The transformation of the four terms can be written as follows:
Integration by parts was used to transform multiple integrations over η to a single integration in the derivation of the formulas (9), (11) and (12). U b in (10) is the bulk velocity.
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Sagaut, P., Peet, Y. (2011). Theoretical Prediction of Turbulent Skin Friction on Geometrically Complex Surfaces. In: Stanislas, M., Jimenez, J., Marusic, I. (eds) Progress in Wall Turbulence: Understanding and Modeling. ERCOFTAC Series, vol 14. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9603-6_5
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DOI: https://doi.org/10.1007/978-90-481-9603-6_5
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