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Stability of bedforms in laminar flows with free surface: from bars to ripples

Published online by Cambridge University Press:  23 December 2009

O. DEVAUCHELLE ,
L. MALVERTI ,
É. LAJEUNESSE ,
P.-Y. LAGRÉE ,
C. JOSSERAND  and
K.-D. NGUYEN THU-LAM
O. DEVAUCHELLE*
Affiliation:
Institut de Physique du Globe, 4 place Jussieu, 75252 Paris cedex 05, France Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
L. MALVERTI
Affiliation:
Institut de Physique du Globe, 4 place Jussieu, 75252 Paris cedex 05, France
É. LAJEUNESSE
Affiliation:
Institut de Physique du Globe, 4 place Jussieu, 75252 Paris cedex 05, France
P.-Y. LAGRÉE
Affiliation:
Institut Jean le Rond d'Alembert, CNRS, Université Pierre et Marie Curie, Boîte 161–162, Tour 55–65, 4 Place Jussieu, 75252 Paris Cedex 05, France
C. JOSSERAND
Affiliation:
Institut Jean le Rond d'Alembert, CNRS, Université Pierre et Marie Curie, Boîte 161–162, Tour 55–65, 4 Place Jussieu, 75252 Paris Cedex 05, France
K.-D. NGUYEN THU-LAM
Affiliation:
Institut Jean le Rond d'Alembert, CNRS, Université Pierre et Marie Curie, Boîte 161–162, Tour 55–65, 4 Place Jussieu, 75252 Paris Cedex 05, France
*
†Email address for correspondence: [email protected]
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Abstract

The present paper is devoted to the formation of sand patterns by laminar flows. It focuses on the rhomboid beach pattern, formed during the backswash. A recent bedload transport model, based on a moving-grains balance, is generalized in three dimensions for viscous flows. The water flow is modelled by the full incompressible Navier–Stokes equations with a free surface. A linear stability analysis then shows the simultaneous existence of two distinct instabilities, namely ripples and bars. The comparison of the bar instability characteristics with laboratory rhomboid patterns indicates that the latter could result from the nonlinear evolution of unstable bars. This result, together with the sensibility of the stability analysis with respect to the parameters of the transport law, suggests that the rhomboid pattern could help improving sediment transport models, so critical to geomorphologists.

JFM classification

Geophysical and Geological Flows: Sediment transport
Type
Papers
Information
Journal of Fluid Mechanics , Volume 642 , 10 January 2010 , pp. 329 - 348
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Allen, J. R. L. 1982 Sedimentary Structures – Their Character and Physical Basis, Vol. II. Developments in Sedimentology, 30b 663, 395–405. Elsevier.Google Scholar
Andreotti, B., Claudin, P. & Douady, S. 2002 Selection of dune shapes and velocities. Part 2. A two-dimensional modelling. e Eur. Phys. J. B – Condens. Matter 28 (3), 341–352.Google Scholar
Bagnold, R. A. 1977 Bed load transport by natural rivers. Water Resour. Res. 13 (2), 303–312.CrossRefGoogle Scholar
Besio, G., Blondeaux, P. & Vittori, G. 2006 On the formation of sand waves and sand banks. J. Fluid Mech. 557, 1–27.CrossRefGoogle Scholar
Blondeaux, P. & Seminara, G. 1985 A unified bar-bend theory of river meanders. J. Fluid Mech. 157, 449–470.CrossRefGoogle Scholar
Callander, R. A. 1969 Instability and river channels. J. Fluid Mech. 36, 465–480.CrossRefGoogle Scholar
Chang, H. Y. & Simons, D. B. 1970 The bed configuration of straight sand beds when flow is nearly critical. J. Fluid Mech. 42, 491–495.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 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. & 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, 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
Coleman, S. E. & Eling, B. 2000 Sand wavelets in laminar open-channel flows. J. Hydraul. Res. 38 (5), 331–338.CrossRefGoogle Scholar
Colombini, M. 2004 Revisiting the linear theory of sand dune formation. J. Fluid Mech. 502, 1–16.CrossRefGoogle Scholar
Colombini, M. & Stocchino, A. 2005 Coupling or decoupling bed and flow dynamics: fast and slow sediment waves at high Froude numbers. Phys. Fluids 17, 036602.CrossRefGoogle Scholar
Defina, A. 2003 Numerical experiments on bar growth. Water Resour. Res. 39 (4), 1092.CrossRefGoogle Scholar
Devauchelle, O., Josserand, C., Lagrée, P. Y. & Zaleski, S. 2007 Morphodynamic modelling of erodible laminar channels. Phys. Rev. E 76 (5), 56318.CrossRefGoogle ScholarPubMed
Devauchelle, O., Malverti, L., Lajeunesse, É., Josserand, C., Lagrée, P. Y. & Métivier, F. 2009 Rhomboid beach pattern: a pocket sized bedform. J. Geophys. Res. (submitted).Google Scholar
Docoslis, A., Giese, R. F. & van Oss, C. J. 2000 Influence of the water–air interface on the apparent surface tension of aqueous solutions of hydrophilic solutes. Colloids Surf. B: Biointerfaces 19 (2), 147–162.CrossRefGoogle Scholar
Elbelrhiti, H., Claudin, P. & Andreotti, B. 2005 Field evidence for surface-wave-induced instability of sand dunes. Nature 437 (7059), 720–723.CrossRefGoogle ScholarPubMed
Exner, F. M. 1925 Über die Wechselwirkung zwischen Wasser und Geschiebe in Flüßen. Sitzenberichte Akad. Wiss. Wien 165 (3–4), 165–203.Google Scholar
Francalanci, S. & Solari, L. 2006 A particle tracking technique to study gravitational effects on bedload transport. In River, Coastal, and Estuarine Morphodynamics: RCEM 2005: Proceedings of the 4th IAHR Symposium on River Coastal and Estuarine Morphodynamics, 4–7 October 2005, Urbana, IL.CrossRefGoogle Scholar
Hall, P. 2006 Nonlinear evolution equations and the braiding of weakly transporting flows over gravel beds. Stud. Appl. Math. 117 (1), 27–69.CrossRefGoogle Scholar
Hersen, P. 2004 On the crescentic shape of barchan dunes. Eur. Phys. J. B – Condens. Matter Complex Syst. 37 (4), 507–514.CrossRefGoogle Scholar
Ikeda, H. 1983 Experiments on bedload transport, bed forms, and sedimentary structures using fine gravel in the 4-meter-wide flume. Environ. Res. Center Pap. 2, 1–78.Google Scholar
Ikeda, S., Parker, G. & Sawai, K. 1981 Bend theory of river meanders. Part 1. Linear development. J. Fluid Mech. 112, 363–377.CrossRefGoogle Scholar
Kapitza, P. L. 1948 Wave motion of a thin layer of a viscous liquidart I. Zh. Eksp. Teor. Fiz 18, 3.Google Scholar
Karcz, I. & Kersey, D. 1980 Experimental study of free-surface flow instability and bedforms in shallow flows. Sedimentary Geol. 27 (4), 263–300.CrossRefGoogle Scholar
Kennedy, J. F. 1963 The mechanics of dunes and antidunes in alluvial channels. J. Fluid Mech 16 (4), 521–544.CrossRefGoogle Scholar
Knaapen, M. A. F. & Hulscher, S. 2003 Use of a genetic algorithm to improve predictions of alternate bar dynamics. Water Resour. Res. 39 (9), 1231.CrossRefGoogle Scholar
Kouakou, K. K. J. & Lagrée, P. Y. 2005 Stability of an erodible bed in various shear flows. Eur. Phys. J. B – Condens. Matter 47 (1), 115–125.Google Scholar
Kuru, W. C., Leighton, D. T. & McCready, M. J. 1995 Formation of waves on a horizontal erodible bed of particles. Intl J. Multiph. Flow 21 (6), 1123–1140.CrossRefGoogle Scholar
Lagrée, P. Y. 2000 Erosion and sedimentation of a bump in fluvial flow. Comptes Rendus de l'Académie des Sciences Séries IIB Mechanics 328 (12), 869–874.CrossRefGoogle Scholar
Lagrée, P. Y. 2003 A triple deck model of ripple formation and evolution. Phys. Fluids 15, 2355.CrossRefGoogle Scholar
Lajeunesse, E., Malverti, L., Lancien, P., Armstrong, L., Métivier, F., Coleman, S., Smith, C., Davies, T., Cantelli, A. & Parker, G. 2009 Fluvial and subaqueous morphodynamics of laminar and near-laminar flows: a synthesis. Sedimentology (in press).CrossRefGoogle Scholar
Langlois, V. & Valance, A. 2005 Formation of two-dimensional sand ripples under laminar shear flow. Phys. Rev. Lett. 94 (24), 248001.CrossRefGoogle Scholar
Lin, S. P. & Chen, J. N. 1997 Elimination of three-dimensional waves in a film flow. Phys. Fluids 9, 3926.CrossRefGoogle Scholar
Morton, R. A. 1978 Large-scale rhomboid bed forms and sedimentary structures associated with hurricane washover. Sedimentology 25, 183–204.CrossRefGoogle Scholar
Parker, G., Seminara, G. & Solari, L. 2003 Bed load at low Shields stress on arbitrarily sloping beds: alternative entrainment formulation. Water Resour. Res. 39 (7), 1183.CrossRefGoogle Scholar
Reynolds, A. J. 1965 Waves on the erodible bed of an open channel. J. Fluid Mech. 22, 113–133.CrossRefGoogle Scholar
Sauermann, G., Kroy, K. & Herrmann, H. J. 2001 Continuum saltation model for sand dunes. Phys. Rev. E 64 (3), 31305.CrossRefGoogle ScholarPubMed
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183–222.CrossRefGoogle Scholar
Shields, A. 1936 Anwendung der Aehnlichkeitsmechanik und der Turbulenzforschung auf die Geschiebebewegung. Preussische Versuchsanstalt für Wasserbau und Schiffbau.Google Scholar
Singh, I. B. 1969 Primary sedimentary structures in Precambrian quartzites of Telemark, Southern Norway and their environmental significance. Nor. Geol. Tidsskr. 49, 1–31.Google Scholar
Stauffer, M. R., Hajnal, Z. & Gendzwill, D. J. 1976 Rhomboidal lattice structure: a common feature on sandy beaches. Can. J. Earth Sci. 13 (12), 1667–1677.CrossRefGoogle Scholar
Thompson, W. O. 1949 Lyons sandstone of Colorado Front Range. AAPG Bull. 33 (1), 52–72.Google Scholar
Valance, A. & Rioual, F. 1999 A nonlinear model for aeolian sand ripples. Eur. Phys. J. B – Condens. Matter 10 (3), 543–548.Google Scholar
Williamson, W. C. 1887 On some undescribed tracks of invertebrate animals from the Yoredale rocks, and on some inorganic phenomena, produced on tidal shores, simulating plant remains. Manchester Lit. Phil. Soc. Mem. Proc., Ser. 3 (10), 19–29.Google Scholar
Woodford, A. O. 1935 Rhomboid ripple mark. Am. J. Sci., 5th Ser. 29, 518–525.CrossRefGoogle Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar