Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-28T18:42:44.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Impact of a falling jet

Published online by Cambridge University Press:  06 July 2010

PAUL CHRISTODOULIDES  and
FRÉDÉRIC DIAS
PAUL CHRISTODOULIDES
Affiliation:
Faculty of Engineering and Technology, Cyprus University of Technology, Limassol, Cyprus
FRÉDÉRIC DIAS*
Affiliation:
CMLA, ENS Cachan and CNRS, UniverSud, 61 avenue du President Wilson, F-94235 Cachan cedex, France School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland
*
†Email address for correspondence: dias@cmla.ens-cachan.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Given the complexity of the problem of the impact of a mass of liquid on a solid structure, various simplified models have been introduced in order to obtain some insight on particular aspects of the problem. Here the steady flow of a jet falling from a vertical pipe, hitting a horizontal plate and flowing sideways is considered. Depending on the elevation H of the pipe relative to the horizontal plate and the Froude number F, the flow can either leave the pipe tangentially or detach from the edge of the pipe. When the flow leaves tangentially, it can either be diverted immediately by the plate or experience squeezing before being diverted. First, the problem is reformulated using conformal mappings. The resulting problem is then solved by a collocation Galerkin method; a particular form is assumed for the solution, and certain coefficients in that representation are then found numerically by satisfying Bernoulli's equation on the free surfaces at certain discrete points. The resulting equations are solved by Newton's method, yielding various configurations of the solution based on the values of F and H. The pressure exerted on the plate is computed and discussed. For a fixed value of F, the maximum pressure along the plate goes through a minimum as H increases from small to large values. Results are presented for the three possible configurations: (i) tangential departure from the pipe and no squeezing, (ii) tangential departure from the pipe followed by squeezing of the liquid and (iii) detachment of the liquid from the pipe (with subsequent squeezing).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 657 , 25 August 2010 , pp. 22 - 35
Copyright
Copyright © Cambridge University Press 2010

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

Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google Scholar
Birkhoff, G. & Carter, D. 1957 Rising plane bubbles. J. Math. Mech. 6 (4), 769–779.Google Scholar
Braeunig, J.-P., Brosset, L., Dias, F. & Ghidaglia, J.-M. 2009 Phenomenological study of liquid impacts through 2D compressible two-fluid numerical simulations. In Proceedings of the Nineteenth International Offshore and Polar Engineering Conference, Osaka, Japan, 21–26 June, 2009, vol. III, pp. 21–28.Google Scholar
Braeunig, J.-P., Brosset, L., Dias, F., Ghidaglia, J.-M. & Maillard, S. 2010 On the scaling problem for impact pressure caused by sloshing. Journal of Ship Research (submitted).Google Scholar
Christodoulides, P. & Dias, F. 2009 Impact of a rising stream on a horizontal plate of finite extent. J. Fluid Mech. 621, 243–258.Google Scholar
Christodoulides, P., Dias, F., Ghidaglia, J.-M. & Kjerland, M. 2010 On the effect of compressibility on the impact of a falling jet. In Proceedings of the Twentieth International Offshore and Polar Engineering Conference, Beijing, China, 20–25 June, 2010, vol. III, pp. 53–61.Google Scholar
Couët, B. & Strumolo, G. S. 1987 The effects of surface tension and tube inclination on a two-dimensional rising bubble. J. Fluid Mech. 184, 1–14.CrossRefGoogle Scholar
Dias, F. & Tuck, E. O. 1993 A steady breaking wave. Phys. Fluids A 5, 277–279.Google Scholar
Garabedian, P. R. 1957 On steady-state bubbles generated by Taylor instability. Proc. R. Soc. Lond. A 241, 423–431.Google Scholar
Hocking, G. C. 1985 Cusp-like free-surface flows due to a submerged source or sink in the presence of a flat or sloping bottom. J. Aust. Math. Soc. B 26, 470–476.Google Scholar
Hocking, G. C. 1992 Flow from a vertical slot into a layer of finite depth. Appl. Math. Modelling 16, 300–306.Google Scholar
Inogamov, N. A. & Oparin, A. M. 2003 Bubble motion in inclined pipes. JETP 97, 1168–1185.CrossRefGoogle Scholar
Korobkin, A. 1995 Impact of two bodies one of which is covered by a thin layer of liquid. J. Fluid Mech. 300, 43–58.Google Scholar
Lee, J. & Vanden-Broeck, J.-M. 1993 Two-dimensional jets falling from funnels and nozzles. Phys. Fluids A 5, 2454–2460.Google Scholar
Lee, J. & Vanden-Broeck, J.-M. 1998 Bubbles rising in an inclined two-dimensional tube and jets falling along a wall. J. Aust. Math. Soc. B 39, 332–349.Google Scholar
Merino, F. 1996 Fluid flow through a vertical slot under gravity. Appl. Math. Modelling 20, 934–939.Google Scholar
Michallet, H., Mathis, C., Maissa, P. & Dias, F. 2001 Flow filling a curved pipe. Trans. ASME J. Fluids Engng 123, 686–691.Google Scholar
Milne-Thomson, L. M. 1996 Theoretical Hydrodynamics, 5th edn. Dover.Google Scholar
Modi, V. 1985 Comment on ‘Bubbles rising in a tube and jets falling from a nozzle.’ Phys. Fluids 28 (11), 3432–3433.Google Scholar
Peregrine, D. H. & Kalliadasis, S. 1996 Filling flows, cliff erosion and cleaning flows. J. Fluid Mech. 310, 365–374.CrossRefGoogle Scholar
Peregrine, D. H. & Thais, L. 1996 The effect of entrained air in violent water wave impacts. J. Fluid Mech. 325, 377–397.Google Scholar
Tuck, E. O. & Dixon, A. 1989 Surf-skimmer planing hydrodynamics. J. Fluid Mech. 205, 581–592.Google Scholar
Tuck, E. O. & Vanden-Broeck, J.-M. 1984 A cusp-like free-surface flow due to a submerged source or sink. J. Aust. Math. Soc. B 25, 443–450.Google Scholar
Vanden-Broeck, J.-M. 1984 Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids 27 (5), 1090–1093.Google Scholar
Vanden-Broeck, J.-M. 1986 Pointed bubbles rising in a two-dimensional tube. Phys. Fluids 29 (5), 1343–1344.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 1991 Axisymmetric jet falling from a vertical nozzle and bubble rising in a tube of circular cross section. Phys. Fluids A 3 (2), 258–262.Google Scholar