Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-19T01:12:43.788Z Has data issue: false hasContentIssue false
  • English
  • Français

The hydraulic jump in a viscous laminar flow

Published online by Cambridge University Press:  26 April 2006

F. J. Higuera
F. J. Higuera
Affiliation:
ETS Ingenieros Aeronáuticos, Pza Cardenal Cisneros, 3, 28040 Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

The hydraulic jump appearing in the viscous laminar flow of a thin liquid layer over a finite horizontal plate is studied using the boundary-layer approximation for the flow in and around the jump. The position and structure of the jump are determined by numerically solving the resulting problem with a boundary condition at the edge of the plate that expresses the matching of the layer with the shorter region where the liquid turns around and falls under the action of gravity. When the Froude number of the flow ahead of the jump is very large, the jump is much shorter than the horizontal extent of the layer, though still much longer than its depth. An asymptotic description of the inner structure of such a jump is given, building upon the analysis of Bowles & Smith for the short interaction region at the leading end of the jump. This structure consists of a fast moving separated flow in the upper part of the layer that progressively slows down by ingesting new fluid across its lower boundary, until the hydrostatically generated adverse pressure gradient makes it recirculate in the lower part of the layer. The effects of the surface tension and the cross-stream pressure variation owing to the curvature of the streamlines are taken into account in the jump and in the flow approaching the edge of the plate, showing that they can lead to quantitative and also qualitative changes of the jump structure, including a local breakdown of the boundary-layer approximation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 274 , 10 September 1994 , pp. 69 - 92
Copyright
© 1994 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

Batchelor, G. K. 1970 An Introduction to Fluid Dynamics, pp. 344–345. Cambridge University Press.
Bohr, T., Dimon, P. & Putkaradze, V. 1993 Shallow-water approach to the circular hydraulic jump. J. Fluid Mech. 254, 635–648.Google Scholar
Bouhadef, M. 1978 Etalement en couche mince d’un jet liquide cylindrique vertical sur un plan horizontal. Z. angew. Math. Phys. 29, 157–167.Google Scholar
Bowles, R. I. 1990 Applications of nonlinear viscous-inviscid interactions in liquid layer flows and transonic boundary layer transition. PhD thesis, University of London.
Bowles, R. I. & Smith, F. T. 1992 The standing hydraulic jump: theory, computations and comparisons with experiments. J. Fluid Mech. 242, 145–168.Google Scholar
Brown, S. N., Stewartson, K. & Williams, P. G. 1975 On expansive free interactions in boundary layers. Proc. R. Soc. Edinburgh 74A, 21.Google Scholar
Chaudhury, Z. H. 1964 Heat transfer in a radial liquid jet. J. Fluid Mech. 20, 501–511.Google Scholar
Craik, A. D. D., Latham, R. C., Fawkes, M. J. & Gribbon, P. W. F. 1981 The circular hydraulic jump. J. Fluid Mech. 112, 347–362.Google Scholar
Daniels, P. G. 1992 A singularity in thermal boundary layer flow on a horizontal surface. J. Fluid Mech. 242, 419–440.Google Scholar
Gajjar, J. S. B. & Smith, F. T. 1983 On hypersonic self-induced separation, hydraulic jumps and boundary layers with algebraic growth. Mathematika 30, 77–91.Google Scholar
Gañán, A. M. 1989 Análisis nodal de zonas líquidas axilsimétricas confinadas por tensión superficial. PhD thesis, University of Sevilla, Spain.
Glauert, M. B. 1956 The wall jet. J. Fluid Mech. 1, 625–643.Google Scholar
Higuera, F. J. 1993 Natural convection below a downward facing horizontal plate. Eur. J. Mech. B 12, 289–311.Google Scholar
Higuera, F. J. & Liñán, A. 1993 Choking conditions for nonuniform viscous flows. Phys. Fluids A 5, 768–770.Google Scholar
Ishigai, S., Nakanishi, S., Mizuno, M. & Imamura, T. 1977 Heat transfer of the impinging round water jet in the interference zone of film flow along the wall. Bull. JSME 20, 85–92.Google Scholar
Kurihara, M. 1946 Laminar flow in a horizontal liquid layer. Rep. of the Res. Inst. for Fluid Engng Kyusyu Imp. Univ. 3-1946, 11.
Larras, M. J. 1962 Ressaut circulaire sur fond parfaitement lisse. CR Acad. Sci. Paris 225, 837–839.Google Scholar
Liu, X. & Lienhard, V. 1993 The hydraulic jump in circular jet impingement and in other thin liquid films. Exps Fluids 15, 108–116.Google Scholar
Messiter, A. F. & Liñán, A. 1976 The vertical plate in laminar free convection: effects of leading and trailing edges and discontinuous temperature. Z. angew. Math. Phys. 27, 632–651.Google Scholar
Nakoryakov, V. E., Pokusaev, B. G. & Troyan, E. N. 1978 Impingement of an axisymmetric liquid jet on a barrier. Intl J. Heat Mass Transfer 21, 1175–1184.Google Scholar
Olsson, R. G. & Turkdogan, E. T. 1966 Radial spread of a liquid stream on a horizontal plate. Nature 211, 813–816.Google Scholar
Rahman, M. M., Hankey, W. L. & Faghri, A. 1991 Analysis of the fluid flow and heat transfer in a thin liquid film in the presence and absence of gravity. Intl J. Heat Mass Transfer 34, 103–114.Google Scholar
Smith, F. T. 1977 Upstream interaction in channel flow. J. Fluid Mech. 79, 631–655.Google Scholar
Smith, F. T. & Brotherton-Ratcliffe, R. V. 1990 Theoret. Comput. Fluid Dyn. 1, 21.
Smith, F. T. & Duck, P. W. 1977 Separation of jets and thermal boundary layers from a wall. Q. J. Mech. Appl. Maths 30, 143–156.Google Scholar
Tani, I. 1948 Flow separation in thin liquid layers. J. Phys. Soc. Japan 4, 212–215.Google Scholar
Watson, E. J. 1964 The radial spread of a liquid jet over a horizontal plate. J. Fluid Mech. 20, 481–499.Google Scholar