Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T14:56:07.683Z Has data issue: false hasContentIssue false
  • English
  • Français

Linearized no-slip boundary conditions at a rough surface

Published online by Cambridge University Press:  25 November 2013

Paolo Luchini
Paolo Luchini*
Affiliation:
DIIN, Università di Salerno, Italy
*
†Email address for correspondence: luchini@unisa.it
Get access Rights & Permissions [Opens in a new window]

Abstract

Linearized boundary conditions are a commonplace numerical tool in any flow problems where the solid wall is nominally flat but the effects of small waviness or roughness are being investigated. Typical examples are stability problems in the presence of undulated walls or interfaces, and receptivity problems in aerodynamic transition prediction or turbulent flow control. However, to pose such problems properly, solutions in two mathematical distinguished limits have to be considered: a shallow-roughness limit, where not only roughness height but also its aspect ratio becomes smaller and smaller, and a small-roughness limit, where the size of the roughness tends to zero but its aspect ratio need not. Here a connection between the two solutions is established through an analysis of their far-field behaviour. As a result, the effect of the surface in the small-roughness limit, obtained from a numerical solution of the Stokes problem, can be recast as an equivalent shallow-roughness linearized boundary condition corrected by a suitable protrusion coefficient (related to the protrusion height used years ago in the study of riblets) and a proximity coefficient, accounting for the interference between multiple protrusions in a periodic array. Numerically computed plots and interpolation formulas of such correction coefficients are provided.

JFM classification

Boundary Layers: Boundary Layers Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 737 , 25 December 2013 , pp. 349 - 367
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105–129.Google Scholar
Blake, J. R. 1971 A note on the image system for a Stokeslet in a no-slip boundary. Proc. Camb. Phil. Soc. 70, 303–310.Google Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183–204.CrossRefGoogle Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65–90.Google Scholar
Kamrin, K., Bazant, M. & Stone, H. A. 2010 Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor. J. Fluid Mech. 658, 409–437.Google Scholar
Luchini, P., Manzo, F. & Pozzi, A. 1991 Resistance of a grooved surface to parallel flow and cross-flow. J. Fluid Mech. 228, 87–109.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Sarkar, K. & Prosperetti, A. 1995 Effective boundary conditions for the Laplace equation with a rough boundary. Proc. R. Soc. Lond. A 451, 425–452.Google Scholar
Sarkar, K. & Prosperetti, A. 1996 Effective boundary conditions for Stokes flow over a rough surface. J. Fluid Mech. 316, 223–240.Google Scholar