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Generalized slip condition over rough surfaces

Published online by Cambridge University Press:  06 November 2018

Giuseppe A. Zampogna Open the ORCID record for Giuseppe A. Zampogna [Opens in a new window] ,
Jacques Magnaudet Open the ORCID record for Jacques Magnaudet [Opens in a new window]  and
Alessandro Bottaro Open the ORCID record for Alessandro Bottaro [Opens in a new window]
Giuseppe A. Zampogna
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, 31400 Toulouse, France
Jacques Magnaudet
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, 31400 Toulouse, France
Alessandro Bottaro*
Affiliation:
Dipartimento di Ingegneria Civile, Chimica e Ambientale, Università di Genova, Via Montallegro 1, 16145 Genova, Italy
*
†Email address for correspondence: [email protected]
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Abstract

A macroscopic boundary condition to be used when a fluid flows over a rough surface is derived. It provides the slip velocity $\boldsymbol{u}_{S}$ on an equivalent (smooth) surface in the form $\boldsymbol{u}_{S}=\unicode[STIX]{x1D716}{\mathcal{L}}\boldsymbol{ : }{\mathcal{E}}$, where the dimensionless parameter $\unicode[STIX]{x1D716}$ is a measure of the roughness amplitude, ${\mathcal{E}}$ denotes the strain-rate tensor associated with the outer flow in the vicinity of the surface and ${\mathcal{L}}$ is a third-order slip tensor arising from the microscopic geometry characterizing the rough surface. This boundary condition represents the tensorial generalization of the classical Navier slip condition. We derive this condition, in the limit of small microscopic Reynolds numbers, using a multi-scale technique that yields a closed system of equations, the solution of which allows the slip tensor to be univocally calculated, once the roughness geometry is specified. We validate this generalized slip condition by considering the flow about a rough sphere, the surface of which is covered with a hexagonal lattice of cylindrical protrusions. Comparisons with direct numerical simulations performed in both laminar and turbulent regimes allow us to assess the validity and limitations of this condition and of the mathematical model underlying the determination of the slip tensor ${\mathcal{L}}$.

JFM classification

Boundary Layers: Boundary Layers Flow Control: Drag reduction Flow Control: Flow Control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , pp. 407 - 436
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Copyright
© 2018 Cambridge University Press 

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