Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-13T07:06:50.873Z Has data issue: false hasContentIssue false
  • English
  • Français

Lagrangian drift near a wavy boundary in a viscous oscillating flow

Published online by Cambridge University Press:  10 July 2009

E. LARRIEU ,
E. J. HINCH  and
F. CHARRU
E. LARRIEU
Affiliation:
Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INPT/UPS 5502, 2, Avenue Camille Soula, 31400 Toulouse, France
E. J. HINCH
Affiliation:
CMS-DAMTP, Wilberforce Road, Cambridge CB3 0WA, UK
F. CHARRU*
Affiliation:
Institut de Mécanique des Fluides de Toulouse, UMR CNRS/INPT/UPS 5502, 2, Avenue Camille Soula, 31400 Toulouse, France
*
†Email address for correspondence: francois.charru@imft.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The formation of sand ripples in oscillating flows is thought to be due to a steady streaming current which near the bottom is towards the crests. We present quantitative observations of this mean flow over self-formed and artificial ripples, by observing the displacement of a coloured filament after a number of oscillations in the simple situation of viscous Couette flow. Confusingly, the filament moves in the ‘wrong’ direction, because it follows the Lagrangian mean flow. We calculate the Lagrangian mean flow. A complication is that the amplitudes of the oscillations in the experiments are not small. We compare the predictions with the experimental observations of displacements of the filament, showing good agreement.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 630 , 10 July 2009 , pp. 391 - 411
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Bagnold, R. 1946 Motion of waves in shallow water. interaction between waves and sand bottoms. Proc. R. Soc. Lond. A 187, 1–18.Google Scholar
Batchelor, G. K. 1967 An introduction to fluid dynamics. Cambridge University Press.Google Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.CrossRefGoogle Scholar
Blondeaux, P. 1990 Sand ripples under sea waves. Part 1. Ripple formation. J. Fluid Mech. 218, 1–17.CrossRefGoogle Scholar
Charru, F. 2006 Selection of the ripple length on a granular bed sheared by a liquid flow. Phys. Fluids 18, 121508.CrossRefGoogle Scholar
Charru, F. & Hinch, E. J. 2006 a Ripple formation on a particle bed sheared by a viscous liquid. Part 1. Steady flow. J. Fluid Mech. 550, 111–121.CrossRefGoogle Scholar
Charru, F. & Hinch, E. J. 2006 b Ripple formation on a particle bed sheared by a viscous liquid. Part 2. Oscillating flow. J. Fluid Mech. 550, 123–137.CrossRefGoogle Scholar
Charru, F., Larrieu, E., Dupont, J.-B. & Zenit, R. 2007 Motion of a particle near a rough wall in a viscous shear flow. J. Fluid Mech. 570, 431–453.CrossRefGoogle Scholar
Charru, F. & Mouilleron-Arnould, H. 2002 Instability of a bed of particles sheared by a viscous flow. J. Fluid Mech. 452, 303–323.CrossRefGoogle Scholar
Charru, F., Mouilleron-Arnould, H. & Eiff, O. 2004 Erosion and deposition of particles on a bed sheared by a viscous flow. J. Fluid Mech. 519, 55–80.CrossRefGoogle Scholar
Craik, A. D. D. 1982 The drift velocity of water waves. J. Fluid Mech. 116, 187–205.CrossRefGoogle Scholar
Du Toit, S. G., & Sleath, J. F. A. 1981 Velocity measurements close to rippled beds in oscillatory flow. J. Fluid Mech. 112, 71–96.CrossRefGoogle Scholar
Hara, T. & Mei, C. C. 1990 Oscillating flows over periodic ripples. J. Fluid Mech. 211, 183–209.CrossRefGoogle Scholar
Hara, T., Mei, C. C. & Shum, K. T. 1992 Oscillating flows over periodic ripples of finite slope. Phys. Fluids A 4, 1373–1384.CrossRefGoogle Scholar
Kaneko, A. & Honji, H. 1979 Double structures of steady streaming in the oscillatory viscous flow over a wavy wall. J. Fluid Mech. 93 (4), 727–736.CrossRefGoogle Scholar
Lighthill, J. 1978 Acoustic streaming. J. Sound Vib. 61, 391–418.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Longuet-Higgins, M. S. 1981 Oscillating flow over steep sand ripples. J. Fluid Mech. 107, 1–35.CrossRefGoogle Scholar
Lyne, W. H. 1971 Unsteady viscous flow over a wavy wall. J. Fluid Mech. 50, 33–48.CrossRefGoogle Scholar
Marin, F. 2004 Eddy viscosity and Eulerian drift over rippled beds in waves. Coastal Engng 50, 139–159.CrossRefGoogle Scholar
Mei, C. C. & Yu, J. 1997 The instability of sand ripples under partially standing surface waves. Phys. Fluids 9, 1609–1620.CrossRefGoogle Scholar
Ourmières, Y. & Chaplin, J. R. 2004 Visualizations of the disturbed-laminar wave-induced flow above a rippled bed. Exp. Fluids 36, 908–918.CrossRefGoogle Scholar
Ourmières, Y. & Mouazé, D. 2007 Wave-induced boundary layer flows over a flat and rippled bed. J. Hydraul. Res. 45, 239–253.CrossRefGoogle Scholar
Rayleigh, L. 1883 On the circulation of air observed in Kundt's tubes and some allied acoustical problems. Phil. Trans. R. Soc. Lond. A 175, 1–21.Google Scholar
Riley, N. 1965 Oscillating viscous flows. Mathematika 12, 161–175.CrossRefGoogle Scholar
Riley, N. 2001 Steady streaming. Annu. Rev. Fluid Mech. 33, 43–65.CrossRefGoogle Scholar
Rousseaux, G., Yoshikawa, H., Stegner, A. & Wesfreid, J. E. 2004 Dynamics of transient eddy above rolling-grain ripples. Phys. Fluids 16, 1049–1058.CrossRefGoogle Scholar
Scandura, P. 2007 Steady streaming in a turbulent oscillating boundary layer. J. Fluid Mech. 571, 265–280.CrossRefGoogle Scholar
Schaflinger, U., Acrivos, A. & Stibi, H. 1995 An experimental study of viscous resuspension in a pressure-driven plane channel flow. Intl J. Multiphase Flow 21, 693–704.CrossRefGoogle Scholar
Schlichting, H. 1932 Berechnung ebener periodischer Grenzschichtstrmungen Phys. Z. 33, 327–335.Google Scholar
Shum, K. T. 1995 A numerical study of the wave-induced solute transport above a rippled bed. J. Fluid Mech. 299, 267–288.CrossRefGoogle Scholar
Sleath, J. F. A. 1974 Mass transport over a rough bed. J. Mar. Sci. 32, 13–24.Google Scholar
Sleath, J. F. A. 1976 On rolling-grain ripples. J. Hydraul. Res. 14, 69–81.CrossRefGoogle Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Camb. Trans. 8, 441–473.Google Scholar
Stuart, J. T. 1963 Unsteady boundary layers. In Laminary Boundary Layers (ed. Rosenhead, L.), pp. 349–408. Oxford University Press.Google Scholar
Unluata, U. & Mei, C. C. 1970 Mass transport in water waves J. Geophys. Res. 75, 7611–7618.CrossRefGoogle Scholar
Vittori, G. 1989 Non-linear viscous oscillatory flow over a wavy wall. J. Hydraul. Res. 27, 267–280.CrossRefGoogle Scholar
Vittori, G. & Blondeaux, P. 1990 Sand ripples under sea waves. Part 2. Finite-amplitude development. J. Fluid Mech. 218, 19–39.CrossRefGoogle Scholar
Wallner, J. & Schaflinger, U. 1998 Viscous resuspension of a sediment caused by oscillating stratified flows. Acta Mech. 127, 147–153.CrossRefGoogle Scholar
Zoueshtiagh, F., Thomas, P. J., Thomy, V. & Merlen, A. 2008 Micrometric granular ripple patterns in a capillary tube. Phys. Rev. Lett. 100, 054501.CrossRefGoogle Scholar