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Self-similar multi-layer exchange flow through a contraction

Published online by Cambridge University Press:  26 April 2006

Anders Engqvist
Anders Engqvist
Affiliation:
Department of Systems Ecology, Stockholm University, S-106 91 Stockholm, Sweden e-mail: ae@system.ecology.su.se
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Abstract

The multi-layer exchange equations for gravitationally driven flows between two basins with stable Boussinesq type of stratification in discrete layers, specified far upstream on either side of a connecting strait, result in a hydraulic control condition that must be satisfied at the narrowest part of the contraction, the control point. If one stagnant layer is present at the control point, the control condition that applies to all layers collectively may be separated into two such conditions that apply independently to two groups of layers going in opposite directions separated by the stagnant layer. Such bidirectional flow regimes exist if the structure of the prespecified density profiles permits each of the opposing groups to vertically reduce their thickness by the ratio 2/3 relative to their upstream thicknesses, leaving space for the stagnant layer to protrude through the contraction. Under these restrictions, the bidirectional flow is controlled by the fastest propagating wave mode and the stationary solution then relies on the superposition of two previously known unidirectional self-similar flow regimes that are completely decoupled. Techniques for their numerical computation are presented. The transition into loosely coupled and fully coupled flow is discussed. The decoupling principle also applies when several non-adjacent stagnant layers are simultaneously present at control in which case multiple groups of decoupled layers flow in alternating directions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 328 , 10 December 1996 , pp. 49 - 66
Copyright
© 1996 Cambridge University Press

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References

Aitken, A. C. 1958 Determinants and Matrices.. Oliver and Boyd.
Armi, L. 1986 The hydraulics of two flowing layers with different densities. J. Fluid Mech. 163, 27–58.Google Scholar
Armi, L. & Farmer, D. M. 1986 Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164, 27–51.Google Scholar
Armi, L. & Williams, R. 1993 The hydraulics of a stratified fluid flowing through a contraction. J. Fluid Mech. 251, 355–375.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, 1–22.Google Scholar
Baines, P. G. 1995 Topographic Effects in Stratified Flows.. Cambridge University Press.
Benjamin, T. B. 1981 Steady flows drawn from a stably stratified reservoir. J. Fluid Mech. 106, 245–260.Google Scholar
Benton, G. S. 1954 The occurrence of critical flow and hydraulic jumps in a multi-layered fluid system. J. Met. 11, 139–150.Google Scholar
Binnie, A. M. 1972 Research note: Hugoniot's method applied to stratified flow through a constriction. J. Mech. Engng Sci. 14, 72–73.Google Scholar
Craya, A. 1951 Critical regimes of flows with density stratification. Tellus 3, 28–42.Google Scholar
Dalziel, S. B. 1991 Two-layer hydraulics: a functional approach. J. Fluid Mech. 223, 135–163.Google Scholar
Dalziel, S. B. 1992 Maximal exchange in channels with nonrectangular cross sections. J. Phys. Oceanogr. 22, 1188–1206.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, 53–76.Google Scholar
Gill, A. E. 1977 The hydraulics of rotating-channel flow. J. Fluid Mech. 80, 641–671.Google Scholar
Gill, A. E. 1982 Atmosphere-Ocean Dynamics.. Academic.
Killworth, P. D. 1992 On hydraulic control in a stratified fluid. J. Fluid Mech. 237, 605–626.Google Scholar
Lee, J. D. & Su, C. H. 1977 A numerical method for stratified shear flows over a long obstacle. J. Geophys. Res. 82, 420–426.Google Scholar
Long, R. R. 1953 Some aspects of the flow of stratified fluids. I. Theoretical investigation. Tellus 5, 42–58.Google Scholar
Long, R. R. 1954 Some aspects of the flow of stratified fluids. II. Experiments with two-fluid system. Tellus 6, 97–115.Google Scholar
Long, R. R. 1955 Some aspects of the flow of stratified fluids. III. Continuous density gradients. Tellus 7, 341–357.Google Scholar
Long, R. R. 1977 Three-layer circulation in estuaries and harbors. Tellus 7, 415–421.Google Scholar
McClimans, T. A. 1990 Role of laboratory experiments and models in the study of sea strait processes. In The Physical Oceanography of Sea Straits (ed. L. J. Pratt), pp. 373–388. Kluwer.
Pratt, L. J. ed. 1990 The Physical Oceanography of Sea Straits. Kluwer.
Stigebrandt, A. 1990 On the response of the horizontal mean vertical density distribution in a fjord to low-frequency density fluctuations in the coastal water. Tellus 42A, 605–614.Google Scholar
Stommel, H. & Farmer, H. G. 1952 Abrupt change in width in two-layer open channel flow. J. Mar. Res. 11, 205–214.Google Scholar
Stommel, H. & Farmer, H. G. 1953 Control of salinity in an estuary by a transition. J. Mar. Res. 12, 13–20.Google Scholar
Su, C. H. 1976 Hydraulic jumps in an incompressible stratified fluid. J. Fluid Mech. 73, 33–47.Google Scholar
Wood, I. R. 1968 Selective withdrawal from a stably stratified fluid. J. Fluid Mech. 32, 209–223.Google Scholar
Wood, I. R. 1970 A lock exchange flow. J. Fluid Mech. 42, 671–687.Google Scholar