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On the hydraulics of Boussinesq and non-Boussinesq two-layer flows

Published online by Cambridge University Press:  26 April 2006

Gregory A. Lawrence
Gregory A. Lawrence
Affiliation:
Department of Civil Engineering, University of British Columbia, Vancouver, BC, Canada, V6T 1W5
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Abstract

Exact expressions for the internal and external Froude numbers for two-layer flows are derived from the celerities of infinitesimal long internal and external waves, without recourse to the Boussinesq approximation. These expressions are functions of the relative density difference between the layers; the relative thickness of the layers; and the stability Froude number, which can be regarded as an inverse bulk Richardson number. A fourth Froude number, the composite Froude number, has been most often used in previous studies. However, the usefulness of the composite Froude number is shown to diminish as the stability Froude number increases. The potential confusion associated with having four Froude numbers of importance has been alleviated by deriving an equation interrelating them. This equation facilitates a comprehensive understanding of the hydraulics of two-layer flows.

It is demonstrated that in substantial portions of some flows (both Boussinesq and non-Boussinesq exchange flow through a contraction are presented as examples), the stability Froude number exceeds a critical value. In this case hydraulic analysis yields imaginary phase speeds corresponding to the instability of long internal waves. Various implications of this result are discussed.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 215 , June 1990 , pp. 457 - 480
Copyright
© 1990 Cambridge University Press

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References

Armi, L. 1975 The internal hydraulics of two flowing layers of different densities. PhD thesis, University of California, Berkeley.
Armi, L. 1986 The hydraulics of two flowing layers of different densities. J. Fluid Mech. 163, 27.Google Scholar
Armi, L. & Farmer, D. M. 1986 Maximal two-layer exchange flow through a contraction with barotropic net flow. J. Fluid Mech. 164, 27.Google Scholar
Armi, L. & Farmer, D. M. 1989 The flow of Mediterranean water through the Strait of Gibraltar. Prog. Oceanogr. 21, 1.Google Scholar
Artale, V. & Levi, D. 1988 Solitary waves in straits. In Nonlinear Evolutions (ed. J. J. Leon). World Scientific.
Baines, P. G. 1984 A unified description of two-layer flow over topography. J. Fluid Mech. 146, 127.Google Scholar
Baines, P. G. 1988 A general method for determining upstream effects in stratified flow of finite depth over long two-dimensional obstacles. J. Fluid Mech. 188, 1.Google Scholar
Benton, G. S. 1954 The occurrence of critical flow and hydraulic jumps in a multilayered fluid system. J. Met. 11, 139.Google Scholar
Corcos, G. M. & Sherman, F. S. 1976 Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech. 73, 241.Google Scholar
Denton, R. A. 1987 Locating and identifying hydraulic controls for layered flow through an obstruction. J. Hydraul. Res. 25, 281.Google Scholar
Farmer, D. M. & Armi, L. 1986 Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic flow. J. Fluid Mech. 164, 53.Google Scholar
Farmer, D. M. & Armi, L. 1989 The flow of Atlantic water through the Strait of Gibraltar. Prog. Oceanogr. 21, 1.Google Scholar
Gill, A. E. 1982 Atmosphere—Ocean Dynamics. Academic.
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39.Google Scholar
Henderson, F. M. 1966 Open Channel Flow. Macmillan.
Houghton, D. D. & Isaacson, E. 1970 Mountain winds. Stud. Num. Anal. 2, 21.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509.Google Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135.Google Scholar
Korn, G. A. & Korn, T. M. 1968 Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. McGraw-Hill.
Lawrence, G. A. 1985 The hydraulics and mixing of two-layer flow over an obstacle. PhD thesis, University of California, Berkeley.
Lawrence, G. A. 1987 Steady flow over an obstacle. J. Hydraul. Engng 113, 981.Google Scholar
Long, R. R. 1954 Some aspects of the flow of stratified fluids. II. Experiments with a two-fluid system. Tellus 6, 97.Google Scholar
Long, R. R. 1956 Long waves in a two-fluid system. J. Met. 13, 70.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496.Google Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. Blackie & Sons.
Schijf, J. B. & Schönfeld, J. C. 1953 Theoretical considerations on the motion of salt and fresh water. In Proc. of the Minn. Intl Hyd. Conv. Joint meeting of the IAHR and Hyd. Div. ASCE, p. 321.
Scotti, R. S. & Corcos, G. M. 1972 An experiment on the stability of small disturbances in a stratified free shear flow. J. Fluid Mech. 52, 499.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441.Google Scholar
Stommel, H. & Farmer, H. G. 1953 Control of salinity in an estuary by a transition. J. Mar. Res. 12, 13.Google Scholar
Thorpe, S. A. 1973 Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731.Google Scholar
Thorpe, S. A. 1987 Transitional phenomena of turbulence in stratified fluids. J. Geophys. Res. 92, 5231.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Wood, I. R. 1968 Selective withdrawal from a stably stratified fluid. J. Fluid Mech. 32, 209.Google Scholar
Wood, I. R. 1970 A lock exchange flow. J. Fluid Mech. 42, 671.Google Scholar