Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-11T02:56:59.210Z Has data issue: false hasContentIssue false
  • English
  • Français

The phase lead of shear stress in shallow-water flow over a perturbed bottom

Published online by Cambridge University Press:  06 December 2010

PAOLO LUCHINI  and
FRANÇOIS CHARRU
PAOLO LUCHINI*
Affiliation:
Dipartimento di Ingegneria Meccanica, Università di Salerno, 84084 Fisciano (SA), Italy
FRANÇOIS CHARRU
Affiliation:
Institut de Mécanique des Fluides de Toulouse, CNRS–Université de Toulouse, 31400 Toulouse, France
*
†Email address for correspondence: luchini@unisa.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The analysis of flow over a slowly perturbed bottom (when perturbations have a typical length scale much larger than channel height) is often based on the shallow-water (or Saint-Venant) equations with the addition of a wall-friction term which is a local function of the mean velocity. By this choice, small sinusoidal disturbances of wall stress and mean velocity are bound to be in phase with each other. In contrast, studies of shorter-scale disturbances have long established that a phase lead develops between wall stress and mean velocity, with a crucial destabilizing effect on sediment transport along an erodible bed. The purpose of this paper is to calculate the wall-shear stress under large length-scale conditions and provide corrections to the Saint-Venant model.

JFM classification

Waves/Free-surface Flows: Channel flow Geophysical and Geological Flows: Sediment transport Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 665 , 25 December 2010 , pp. 516 - 539
Copyright
Copyright © Cambridge University Press 2010

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

Abrams, J. & Hanratty, T. J. 1985 Relaxation effects observed for turbulent flow over a wavy surface. J. Fluid Mech. 151, 443–455.Google Scholar
Aradian, A., Raphael, É. & de Gennes, P. G. 2002 Surface flows of granular materials: a short introduction to some recent models. C.R. Phys. 3, 187–196.Google Scholar
Balmforth, N. J. & Mandre, S. 2004 Dynamics of roll waves. J. Fluid Mech. 514, 1–33.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Belcher, S. E. & Hunt, J. C. R. 1998 Turbulent flow over hills and waves. Annu. Rev. Fluid Mech. 30, 507–538.CrossRefGoogle Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Brock, R. R. 1969 Development of roll-wave trains in open channels. J. Hydr. Div. ASCE 95-HY4, 1401–1427.Google Scholar
Chang, H.-C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Charru, F. 2006 Selection of the ripple length on a granular bed sheared by a liquid flow. Phys. Fluids 18, 121508.Google Scholar
Colombini, M. 2004 Revisiting the linear theory of sand dune formation. J. Fluid Mech. 502, 1–16.CrossRefGoogle Scholar
Engelund, F. 1970 Instability of erodible beds. J. Fluid Mech. 42, 225–244.Google Scholar
Engelund, F. & Fredsøe, J. 1982 Sediment ripples and dunes. Annu. Rev. Fluid Mech. 14, 13–37.CrossRefGoogle Scholar
Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 1–24.Google Scholar
Gradowczyk, M. H. 1968 Wave propagation and boundary instability in erodible-bed channels. J. Fluid Mech. 33, 93–112.Google Scholar
Hinze, J. O. 1975 Turbulence. McGraw Hill.Google Scholar
Hopfinger, E. J. 1983 Snow avalanche motion and related phenomena. Annu. Rev. Fluid Mech. 15, 47–76.Google Scholar
Jackson, P. S. & Hunt, J. C. R. 1975 Turbulent wind flow over a low hill. Q. J. R. Meteorol. Soc. 101, 929–955.Google Scholar
Jeffreys, H. 1925 The flow of water in an inclined channel of rectangular cross-section. Phil. Mag. (6) 49, 793–807.Google Scholar
Jimenez, J., Uhlmann, M., Pinelli, A. & Kawahara, G. 2001 Turbulent shear flow over active and passive porous surfaces. J. Fluid Mech. 442, 89–117.Google Scholar
Julien, P. Y. & Hartley, D. M. 1986 Formation of roll waves in laminar sheet flow. J. Hydraul. Res. 24, 5–17.Google Scholar
Lin, P. Y. & Hanratty, T. J. 1986 Prediction of the initiation of slugs with linear stability theory. Intl J. Multiphase Flow 12, 79–98.Google Scholar
Luchini, P. & Charru, F. 2010 Consistent section-averaged equations of quasi-one-dimensional laminar flow. J. Fluid Mech. 656, 337–341.CrossRefGoogle Scholar
Mellor, G. L. 1972 The large Reynolds number, asymptotic theory of turbulent boundary layers. Intl J. Engng Sci. 10, 851–873.Google Scholar
Needham, D. J. & Merkin, J. H. 1984 On roll waves down an open inclined channel. Proc. R. Soc. Lond. A 394, 259–278.Google Scholar
Nezu, I. & Rodi, W. 1986 Open-channel flow measurements with a laser Doppler anemometer. J. Hydr. Engng 112, 335–355.Google Scholar
Pedlosky, J. 2003 Waves in the Ocean and Atmosphere. Springer.Google Scholar
Richards, K. J. 1980 The formation of ripples and dunes on an erodible bed. J. Fluid Mech. 99, 597–618.Google Scholar
de Saint-Venant, A. J. C. B. 1871 Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154, 237–240.Google Scholar
Sumer, B. M. & Bakioglu, M. 1984 On the formation of ripples on an erodible bed. J. Fluid Mech. 144, 177–190.Google Scholar
Usha, R. & Uma, B. 1994 Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method. Phys. Fluids 16, 2679–2696.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Zanoun, E.-S., Durst, F. & Nagib, H. 2003 Evaluating the law of the wall in two-dimensional fully developed turbulent channel flows. Phys. Fluids 15, 3079–3089.Google Scholar