Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-21T12:21:42.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Centrifugal instability of an oscillatory flow over periodic ripples

Published online by Cambridge University Press:  26 April 2006

Tetsu Hara  and
Chiang C. Mei
Tetsu Hara
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Chiang C. Mei
Affiliation:
Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

An oscillating flow over a sandy beach can initiate and enhance the formation of bed ripples, with crests perpendicular to the direction of the ambient oscillation. Under certain circumstances, bridges may develop to span adjacent ripple crests, resulting in a brick pattern. It has been suggested that the onset of this transition is due to a three-dimensional centrifugal instability of an otherwise two-dimensional flow over periodic long-crested ripples. Here we analyse theoretically such an instability by assuming that the ripples are rigid and smooth. Two complementary cases are studied. We first consider a weak ambient oscillation over ripples of finite slope in Case (i). The three-dimensional disturbance is found to be localized in a small region either along the crests or along the troughs. In Case (ii) we analyse finite oscillations over ripples of mild slope. The region influenced by the instability is now comparable with a ripple wavelength and the unstable disturbance along adjacent ripples may interact with each other. Four types of harmonic and subharmonic instabilities are found. The associated steady streaming close to the ripple surface shows various tendencies of possible sand accumulations, some of which appear to be qualitatively relevant to the initiation of brick-patterned ripples.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 217 , August 1990 , pp. 1 - 32
Copyright
© 1990 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Bagnold, R. A. 1946 Motion of waves in shallow water; interaction between waves and sand bottoms. Proc. R. Soc. Lond. A 187, 1–15.Google Scholar
Batchelor, G. K. 1967 An introduction to Fluid Mechanics. Cambridge University Press.
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Davis, S. H. 1976 The stability of time-periodic flows. Ann. Rev. Fluid Mech. 8, 57–74.Google Scholar
Hall, P. 1984 On the stability of the unsteady boundary layer on a cylinder oscillating transversely in a viscous fluid. J. Fluid Mech. 146, 347–367.Google Scholar
Hara, T. 1990 Nonlinear dynamics near a wavy boundary on the sea surface or the sea bottom. Ph.D. thesis, MIT.
Hara, T. & Mei, C. C. 1990 Oscillating flows over periodic ripples. J. Fluid Mech. 211, 183–209.Google Scholar
Honji, H. 1981 Streaked flow around an oscillating circular cylinder. J. Fluid Mech. 107, 509–520.Google Scholar
Kaneko, A. & Honji, H. 1979 Double structures of steady streaming in the oscillatory viscous flow over a wavy wall. J. Fluid Mech. 93, 727–736.Google Scholar
Kerczek, C. von & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layer. J. Fluid Mech. 62, 753–773.Google Scholar
Li, H. 1954 Stability of oscillatory laminar flow along a wall. Beach Erosion Board, US Army Corps Engng, Wasington DC, Tech. Memo, 47.
Lyne, W. H. 1971 Unsteady viscous flow over a wavy wall. J. Fluid Mech. 50, 33–48.Google Scholar
Matsunaga, N. & Honji, H. 1980 Formation mechanism of brick-pattern ripples. Rep. Res. Inst. Appl. Mech., Kyushu Univ., vol. 28, no. 88, pp. 27–38.
Sarpkaya, T. 1986 Force on a circular cylinder in viscous oscillatory flow at low Keulegan—Carpenter numbers. J. Fluid Mech. 165, 61–71.Google Scholar
Sergeev, S. I. 1966 Fluid oscillations in pipes at moderate Reynolds numbers. Mekh. Zhidk. Gaza 1, no. 1, 168–170. (Transl. In Fluid Dyn. 1, no. 1, 121–122.)Google Scholar
Sleath, J. F. A. 1984 Sea Bed Mechanics. Wiley-Interscience.
Sleath, J. F. A. & Ellis, A. C. 1978 Ripple geometry in oscillatory flow. University of Cambridge. Dept. of Engng Rep. A/Hydraul./TR2.
Thompson, C. 1987 Stability of a Stokes boundary layer. J. Acoust. Soc. Am. 81, 861–873.Google Scholar
Vittori, G. 1989 Non-linear viscous oscillatory flow over a small amplitude wavy wall. J. Hydraul. Res. 27, 267–280.Google Scholar
Yih, C. S. 1968 Instability of unsteady flows or configurations. Part 1. Instability of a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 31, 737–751.Google Scholar