Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T05:03:44.631Z Has data issue: false hasContentIssue false
  • English
  • Français

Water entry of an expanding body with and without splash

Published online by Cambridge University Press:  15 January 2019

Y. A. Semenov Open the ORCID record for Y. A. Semenov [Opens in a new window]  and
G. X. Wu
Y. A. Semenov
Affiliation:
Department of Mechanical Engineering, University College London, London WC1E 6BT, UK
G. X. Wu*
Affiliation:
Department of Mechanical Engineering, University College London, London WC1E 6BT, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A self-similar flow generated by water entry of an expanding two-dimensional smooth and curved body is studied based on the incompressible velocity potential theory, with gravity and surface tension effects being ignored. At each expansion speed, mathematical solutions for detached flow with a splash jet and attached flow with jet leaning on the body surface are obtained, both of which have been found possible in experiment, corresponding to hydrophobic and hydrophilic bodies, respectively. The problem is solved using the integral hodograph method, which converts the governing Laplace equations into two integral equations along the half real and imaginary axes of a parameter plane. For the detached flow, the conditions for continuity and finite spatial derivative of the velocity at the point of flow departing form the body surface are imposed. It is found that the Brillouin–Villat criterion for flow detachment of steady flow is also met in this self-similar flow, which requires the curvatures of the free surface and the body surface to be the same at the detachment point. Solutions for the detached flow have been obtained in the whole range of the expansion speeds, from zero to infinity, relative to the water entry speed. For the attached flow there is a minimal expansion speed below which no solution cannot be obtained. Detailed results in terms of pressure distribution, free surface shape and streamlines and tip angle of the jet are presented. It is revealed that when solutions for both detached and attached flows exist, the pressure distributions on the cylinder surface are almost the same up to the point near the jet root. Beyond that point, the pressure relative to the ambient one drops to zero at the detachment point in the former, while it drops below zero in the jet attached on the body and then returns to zero at the contact point in the latter.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 924 - 950
Copyright
© 2019 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

Armand, J. L. & Cointe, R. 1987 Hydrodynamic impact analysis of a cylinder. J. Offshore Mech. Artic Engng 9, 237–243.Google Scholar
Bao, C. M., Wu, G. X. & Xu, G. D. 2016 Simulation of water entry of a two-dimension finite wedge with flow detachment. J. Fluids Struct. 65, 44–59.Google Scholar
Bao, C. M., Wu, G. X. & Xu, G. D. 2017 Simulation of freefall water entry of a finite wedge with flow detachment. Appl. Ocean Res. 65, 262–278.Google Scholar
Brillouin, M. 1911 Les surfaces de glissement de Helmholtz et la resistance des fluids. Ann. Chim. Phys. 23, 145–230.Google Scholar
Climent, H., Benitez, L., Rosich, F., Rueda, F. & Pentecote, N. 2006 Aircraft ditching numerical simulation. In 25th International Congress of the Aeronautical Sciences, Hamburg, Germany, 3–8 Sept. ICAS.Google Scholar
Dobrovol’skaya, Z. N. 1969 Some problems of similarity flow of fluid with a free surface. J. Fluid Mech. 36, 805–829.Google Scholar
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180–183.Google Scholar
Faltinsen, O. M. 2005 Hydrodynamics of High-speed Marine Vehicles. Cambridge University Press.Google Scholar
Greenhow, M. 1988 Water-entry and -exit of a horizontal circular cylinder. Appl. Ocean Res. 10 (4), 191–198.Google Scholar
Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic, 585 p.Google Scholar
Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2002 Deep- and shallow-water slamming at small and zero deadrise angles. J. Engng Maths. 42, 373–388.Google Scholar
Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2004 Oblique slamming, planing and skimming. J. Engng Maths 48, 321–337.Google Scholar
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215–230.Google Scholar
Joukovskii, N. E. 1890 Modification of Kirchhof’s method for determination of a fluid motion in two directions at a fixed velocity given on the unknown streamline. Math. Coll. 15, 121–278. (in Russian).Google Scholar
von Karman, T.1929 The impact of seaplane floats during landing. NACA Tech. Note 321.Google Scholar
Korobkin, A. A. 2004 Analytical models of water impact. Eur. J. Appl. Maths 15, 821–838.Google Scholar
Logvinovich, G. V. 1969 Hydrodynamics of Free-Boundary Flows. Naukova Dumka. Translated from Russian by the Israel Program for Scientific Translations, Jerusalem (1972).Google Scholar
Michell, J. H. 1890 On the theory of free stream lines. Phil. Trans. R. Soc. Lond. A 181, 389–431. doi:10.1098/rsta.1890.0006.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics, 5th edn. Dover.Google Scholar
Moore, M. R., Ockendon, H., Ockendon, J. R. & Oliver, J. M. 2015 Capillary and viscous perturbations to Helmholtz flows. J. Fluid Mech. 742, R1.Google Scholar
Sedov, L. I. 1993 Similarity and Dimensional Methods in Mechanics. CRC Press.Google Scholar
Semenov, Y. A. & Cummings, L. J. 2006 Free boundary Darcy flows with surface tension: analytical and numerical study. Eur. J. Appl. Maths 17, 607–631.Google Scholar
Semenov, Y. A. & Iafrati, A. 2006 On the nonlinear water entry problem of asymmetric wedges. J. Fluid Mech. 547, 231–256.Google Scholar
Semenov, Y. A. & Yoon, B-S. 2009 Onset of flow separation for the oblique water impact of a wedge. Phys. Fluids 21, 112103.Google Scholar
Semenov, Y. A., Wu, G. X. & Oliver, J. M. 2013 Splash jet generated by collision of two liquid wedges. J. Fluid Mech. 737, 132–145.Google Scholar
Semenov, Y. A. & Wu, G. X. 2013 The nonlinear problem of a gliding body with gravity. J. Fluid Mech. 727, 132–160.Google Scholar
Semenov, Y. A. & Wu, G. X. 2016 Water entry of an expanding wedge/plate with flow detachment. J. Fluid Mech. 797, 322–344.Google Scholar
Semenov, Y. A., Wu, G. X. & Korobkin, A. A. 2015 Impact of liquids with different densities. J. Fluid Mech. 766, 5–27.Google Scholar
Sun, H. & Faltinsen, O. M. 2007 The influence of gravity on the performance of planing vessels in calm water. J. Engng Maths 58, 91–107.Google Scholar
Sun, H. & Faltinsen, O. M. 2009 Water entry of a bow-flare ship section with roll angle. J. Mar. Sci. Technol. 14 (1), 69–79.Google Scholar
Tassin, A., Korobkin, A. A. & Cooker, M. J. 2014 On analytical models of vertical water entry of a symmetric body with separation and cavity initiation. Appl. Ocean Res. 48, 33–41.Google Scholar
Villat, H. 1914 Sur la validite des solutions de certains problemes d’hydrodynamique. J. Math. Pure Appl. 20, 231–290.Google Scholar
Vinje, T. & Brevig, P. 1981 Nonlinear two-dimensional ship motions. In Proceedings of 3rd International Conference on Numerical Ship Hydrodynamics, Paris, June, pp. 257–266. Bassin d’Essais des Carénes.Google Scholar
Wagner, H. 1932 Über Stoßund Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12, 192–215.Google Scholar
Wu, G. X & Sun, S. L. 2014 Similarity solution for oblique water entry of an expanding paraboloid. J. Fluid Mech. 745, 398–408.Google Scholar
Xu, G. D., Duan, W. Y. & Wu, G. X. 2008 Numerical simulation of oblique water entry of an asymmetrical wedge. Ocean Engng 35, 1597–1603.Google Scholar
Zhao, R. & Faltinsen, O. 1993 Water-entry of two-dimensional bodies. J. Fluid Mech. 246, 593–612.Google Scholar
Zhao, R., Faltinsen, O. & Aarsnes, J. 1997 Water entry of arbitrary two-dimensional sections with and without flow separation. In 21st Symposium on Naval Hydrodynamics, Trondheim, Norway.Google Scholar