Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-vvt5l Total loading time: 0.534 Render date: 2022-06-30T09:13:13.985Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Gravity currents and related phenomena

Published online by Cambridge University Press:  28 March 2006

T. Brooke Benjamin
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, La Jolla

Abstract

This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.

In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.

Type
Research Article
Copyright
© 1968 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

Bata, G. & Bogich, K. 1953 Some observations on density currents in the laboratory and in the field. Proc. Minn. Internat. Hydraulics Conv., Univ. of Minnesota, p. 387.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. Roy. Soc. A 224, 448.Google Scholar
Berson, F. A. 1958 Some measurements on undercutting cold air Quart. J. Roy. Meteorol. Soc. 84, 1.Google Scholar
Binnie, A. M. 1952 The flow of water under a sluice-gate Quart. J. Mech. appl. Math. 5, 395.Google Scholar
Braucher, E. P. 1950 Initial characteristics of density current flow. S.M. thesis, Massachusetts Inst. of Technology.
Clarke, R. H. 1961 Mesostructure of dry cold fronts over featureless terrain J. Meteorology, 18, 715.Google Scholar
De, S. C. 1955 Contributions to the theory of Stokes waves Proc. Camb. Phil. Soc. 51, 713.Google Scholar
Goldstein, S. (Ed.) 1938 Modern Developments in Fluid Dynamics. Oxford University Press. (Dover edition, 1965.)
Ippen, A. T. & Harleman, D. R. F. 1952 Steady-state characteristics of subsurface flow Proc. NBS Symp. on Gravity Waves, Nat. Bur. Stand. Circ. 521, 79.Google Scholar
Kármán, T. VON 1940 The engineer grapples with nonlinear problems Bull. Am. Math. Soc. 46, 615.Google Scholar
Keulegan, G. H. 1949 Interfacial instability and mixing in stratified flows J. Res. Nat. Bur. Stand. 43, 487.Google Scholar
Keulegan, G. H. 1957 An experimental study of the motion of saline water from locks into fresh water channels. Nat. Bur. Stand. Rept. 5168.Google Scholar
Keulegan, G. H. 1958 The motion of saline fronts in still water. Nat. Bur. Stand. Rept. 5831.Google Scholar
Keunen, P. H. 1950 Turbidity currents of high density. 18th Internat. Geol. Congr., London, Repts. Pt. 8, p. 44.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press. (Dover edition, 1945.)
Michell, A. G. M. 1893 The highest waves in water. Phil. Mag. (v), 36, 430.Google Scholar
Middleton, G. V. 1966 Experiments on density and turbidity currents. 1. Motion of the head Canad. J. Earth Sci. 3, 523.Google Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. New York: Hafner.
Prandtl, L. & Tietjens, O. G. 1934 Applied Hydro- and Aerodynamics. New York: McGraw-Hill. (Dover edition, 1957.)
Schijf, J. B. & Schönfeld, J. C. 1953 Theoretical considerations on the motion of salt and fresh water. Proc. Minn. Internat. Hydraulics Conv., Univ. of Minnesota, p. 321.Google Scholar
Southwell, R. V. & Vaisey, G. 1946 Relaxation methods applied to engineering problems. XII: Fluid motions characterized by ‘free’ streamlines. Phil. Trans. Roy. Soc. A 240, 117.Google Scholar
Wood, I. R. 1966 Studies in unsteady self preserving turbulent flows. Univ. of New South Wales, Water Research Lab., Rept. no. 81.Google Scholar
Yih, C.-S. 1947 A study of the characteristics of gravity waves at a liquid interface. M.S. thesis, State Univ. of Iowa.
Yih, C.-S. 1965 Dynamics of Nonhomogeneous Fluids. New York: Macmillan.
Zukoski, E. E. 1966 Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes J. Fluid Mech. 25, 821.Google Scholar
949
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Gravity currents and related phenomena
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Gravity currents and related phenomena
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Gravity currents and related phenomena
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *