Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-28T08:34:35.953Z Has data issue: false hasContentIssue false

The standing hydraulic jump: theory, computations and comparisons with experiments

Published online by Cambridge University Press:  26 April 2006

R. I. Bowles
Affiliation:
Department of Mathematics, University College London. Gower Street. London WC1E 6BT, UK
F. T. Smith
Affiliation:
Department of Mathematics, University College London. Gower Street. London WC1E 6BT, UK

Abstract

In this theoretical and computational study of the flow of a liquid layer, under the influence of surface tension and gravity most notably, the nonlinear equations governing an interaction between viscous effects and the effects of surface tension, gravity and streamline curvature for the limit of large Reynolds numbers are derived. The aim is to make a comparison between the predictions of this theory and the experiments of Craik et al. on the axisymmetric hydraulic jump. Such a jump is commonly encountered in the everyday context of the initial filling of a kitchen sink, for example, and it is found in the present work that initially all the effects listed above can play a primary role in practice in the local jump phenomenon. As a first step here, the flow of the layer over a small obstacle is considered. It is seen that as surface tension becomes increasingly significant the upstream influence becomes more wave-like. Second, calculations and analysis of the nonlinear free interaction are presented and show wave-like behaviour upstream, followed downstream by a depth profile not unlike that in the typical hydraulic jump. The effects of gravity dominate those of surface tension downstream. Finally, comparisons are made with the experiments and show fair quantitative agreement, supporting the present proposition that these hydraulic jumps are caused by boundary-layer separation due to a viscous–inviscid interaction forced by downstream boundary conditions on, in this case, a fully developed, high-Froude-number liquid layer.

Type
Research Article
Copyright
© 1992 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

Benjamin, T. B. 1962 The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97112.Google Scholar
Bowles, R. I. 1990 Applications of nonlinear viscous-inviscid interactions in liquid layer flows & transonic boundary layer transition. Ph.D. thesis, University of London.
Brotherton-Ratcliffe, R. V. & Smith, F. T. 1986 Boundary layer effects in liquid layer flows. Ph.D. thesis, chapter 3, University of London.
Brown, S. N., Stewartson, K. & Williams, P. G. 1975 On expansive free interactions in boundary layers. Proc. R. Soc. Edinb. 74 A, 21.Google Scholar
Clarke, W. O. 1970 Tap-splash and the hydraulic jump. School Sci. Rev. 152, 67.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.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.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Larras, M. J. 1962 Ressaut circulaire sur fond parfaitement lisse. C. R. Acad. Sci. Paris 225, 837.Google Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.
Peregrine, D. H. 1974 Surface shear waves. J. Hydraul. Div. ASCE 100, 1215.Google Scholar
Rayleigh, Lord 1914 Proc. R. Lond. Soc. A 90, 324 and Scientific Papers, vol. 6, p. 250. Cambridge University Press.
Smith, F. T. 1976a Flow through constricted or dilated pipes and channels: Part 1. Q. J. Mech. Appl. Maths 29, 343.Google Scholar
Smith, F. T. 1976b Flow through constricted or dilated pipes and channels: Part 2. Q. J. Mech. Appl. Maths 29, 365.Google Scholar
Smith, F. T. 1977 Upstream interactions in channel flow. J. Fluid Mech. 79, 631.Google Scholar
Smith, F. T. 1978 Flow through symmetrically constricted tubes. J. Inst. Maths Applics. 21, 145.Google Scholar
Smith, F. T. 1982 On the high Reynolds number theory of laminar flows I.M.A. J. Appl. Maths 28, 207.Google Scholar
Smith, F. T. 1985 On large-scale eddy closure. J. Math. Phys. Sci. 19, 180.Google Scholar
Smith, F. T. 1988 Finite-time break-up can occur in any unsteady interactive boundary layer. Mathematika 35, 256.Google Scholar
Smith, F. T. & Duck, P. W. 1977 Separation of jets and thermal boundary layers from a wall. Q. J. Mech. Appl. Maths 30, 143.Google Scholar
Teles Da Silva, A. F. & Peregrine, D. H. 1988 Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281.Google Scholar
Watson, E. J. 1964 The radial spread of a liquid jet over a horizontal plane. J. Fluid Mech. 20, 481.Google Scholar