Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-17T23:18:05.480Z Has data issue: false hasContentIssue false

The circular internal hydraulic jump

Published online by Cambridge University Press:  08 August 2008

S. A. THORPE*
Affiliation:
School of Ocean Sciences, Marine Science Laboratories, Bangor University, Menai Bridge, Anglesey LL59 5EY, UK
I. KAVCIC
Affiliation:
Department of Geophysics, Faculty of Science, University of Zagreb, Horvatovac bb, 10000 Zagreb, Croatia
*
Author to whom correspondence should be addressed at: ‘Bodfryn’, Glanrafon, Llangoed, Anglesey LL58 8PH, UK; oss413@sos.bangor.ac.uk

Abstract

Circular hydraulic jumps are familiar in single layers. Here we report the discovery of similar jumps in two-layer flows. A thin jet of fluid impinging vertically onto a rigid horizontal plane surface submerged in a deep layer of less-dense miscible fluid spreads radially, and a near-circular internal jump forms within a few centimetres from the point of impact with the plane surface. A jump is similarly formed as a jet of relatively less-dense fluid rises to the surface of a deep layer of fluid, but it appears less stable or permanent in form. Several experiments are made to examine the case of a downward jet onto a horizontal plate, the base of a square or circular container. The inlet Reynolds numbers, Re, of the jet range from 112 to 1790. Initially jumps have an undular, laminar form with typically 2–4 stationary waves on the interface between the dense and less-dense layers but, as the depth of the dense layer beyond the jump increases, the transitions become more abrupt and turbulent, resulting in mixing between the two layers. During the transition to a turbulent regime, single and sometimes moving multiple cusps are observed around the periphery of jumps. A semi-empirical model is devised that relates the parameters of the laboratory experiment, i.e. flow rate, inlet nozzle radius, kinematic viscosity and reduced gravity, to the layer depth beyond the jump and the radius at which an undular jump occurs. The experiments imply that surface tension is not an essential ingredient in the formation of circular hydraulic jumps and demonstrate that stationary jumps can exist in stratified shear flows which can be represented as two discrete layers. No stationary circular undular jumps are found, however, in the case of a downward jet of dense fluid when the overlying, less-dense, fluid is stratified, but a stationary turbulent transition is observed. This has implications for the existence of stationary jumps in continuously stratified geophysical flows: results based on two-layer models may be misleading. It is shown that the Froude number at which a transition of finite width occurs in a radially diverging flow may be less than unity.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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

Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Bell, T. H. 1974 Effects of shear on the properties of internal gravity waves. Deutsche Hydrograph Z. 27, 5762.CrossRefGoogle Scholar
Bohr, T., Dimon, P & Putkaradze, V. 1993 Shallow water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635648.CrossRefGoogle Scholar
Bowles, R. & Smith, F. 1992 The standing hydraulic jump: Theory, computations and comparisons with experiments. J. Fluid Mech. 242, 145168.CrossRefGoogle Scholar
Brown, G. L. & Roshko, A. 1974 On the density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775816.CrossRefGoogle Scholar
Bush, J. & Aristoff, J. 2003 The influence of surface tension on the circular hydraulic jump. J. Fluid Mech. 489, 229238.CrossRefGoogle Scholar
Bush, J. W. M., Aristoff, J. M. & Hosoi, A. E. 2006 An experimental investigation of the stability of the circular hydraulic jump. J. Fluid Mech. 558, 3352.CrossRefGoogle Scholar
Craik, A., Latham, R., Fawkes, M. & Gibbon, P. 1981 The circular hydraulic jump. J. Fluid Mech. 112, 347362.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press, Cambridge.Google Scholar
Duncan, J. H., Qiao, H. Philomin, V. & Wenz, A. 1999 Gentle spilling breakers: crest profile evolution. J. Fluid Mech., 379, 191222.CrossRefGoogle Scholar
Hassid, S., Regev, A. & Poreh, M. 2007 Turbulent energy dissipation in density jumps. J. Fluid Mech. 572, 112.CrossRefGoogle Scholar
Higuera, F. 1994 The hydraulic jump in a viscous laminar flow. J. Fluid Mech. 274, 6992.CrossRefGoogle Scholar
Holland, D. M., Rosales, R. R., Stefanica, D. & Tabak, E. G. 2002 Internal hydraulic jumps and mixing in two-layer flows. J. Fluid Mech. 470, 6383.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1992 Capillary rollers and bores. J. Fluid Mech. 240, 659679.CrossRefGoogle Scholar
Polzin, K., Speer, K. G., Toole, J. M. & Schmitt, R. W. 1996 Intense mixing of Antarctic Bottom Water in the equatorial Atlantic Ocean. Nature 380, 5456.CrossRefGoogle Scholar
Rayleigh, Lord 1914 On the theory of long waves and bores. Proc. R. Soc. Lond. A 90, 324327.Google Scholar
Scorer, R. 1972 Clouds of the World. Melbourne, Lothian Pub. Co.Google Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory, 2nd edn. Cambridge University Press.Google Scholar
Thorpe, S. A. 2007 Dissipation in hydraulic transitions in flows through abyssal channels. J. Mar. Res. 65, 147168.CrossRefGoogle Scholar
Thorpe, S. A. & Ozen, B. 2007 Are cascading flows stable? J. Fluid Mech. 589, 411432.CrossRefGoogle Scholar
Thurnherr, A. M., St Laurent, L. C., Speer, K. G., Toole, J. M. & Ledwell, J. R. 2005 Mixing associated with sills in a canyon on the mid-ocean ridge flank. J. Phys. Oceanogr. 35, 13701381.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Watson, E. M. 1964 The spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20, 481499.CrossRefGoogle Scholar
Wilkinson, D. L. & Wood, I. R. 1971 A rapidly varied flow phenomenon in a two-layer flow. J. Fluid Mech. 47, 241256.CrossRefGoogle Scholar