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Water entry of a wedge with rolled-up vortex sheet

Published online by Cambridge University Press:  27 November 2017

Yuriy A. Semenov  and
G. X. Wu Open the ORCID record for G. X. Wu [Opens in a new window]
Yuriy 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: g.wu@ucl.ac.uk
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Abstract

The problem of asymmetric water entry of a wedge with the vortex sheet shed from its apex is considered within the framework of the ideal and incompressible fluid. The effects due to gravity and surface tension are ignored and the flow therefore can be treated as self-similar, as there is no length scale. The solution for the problem is sought through two mutually dependent parts using two different analytic approaches. The first one is due to water entry, which is obtained through the integral hodograph method for the complex velocity potential, in which the streamline on the body surface remains on the body surface after passing the apex, leading to a non-physical local singularity. The second one is due to a vortex sheet shed from the apex, and the shape of the sheet and the strength distribution of the vortex are obtained through the solution of the Birkhoff–Rott equation. The total circulation of the vortex sheet is obtained by imposing the Kutta condition at the apex, which removes the local singularity. These two solutions are nonlinearly coupled on the unknown free surface and the unknown vortex sheet. This poses a major challenge, which distinguishes the present formulation of the problem from the previous ones on water entry without a vortex sheet and ones on vortex shedding from a wedge apex without a moving free surface. Detailed results in terms of pressure distribution, vortex sheet, velocity and force coefficients are presented for wedges of different inner angles and heel angles, as well as the water-entry direction. It is shown that the vortex shedding from the tip of the wedge has a profound local effect, but only weakly affects the free-surface shape, overall pressure distribution and force coefficients.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 512 - 539
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Copyright
© 2017 Cambridge University Press 

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References

Armand, J. L. & Cointe, R. 1987 Hydrodynamic impact analysis of a cylinder. J. Offshore Mech. Artic Engng 9, 237–243.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor instability. In Proc. Sympos. Appl. Math., vol. XIII, pp. 55–76. American Mathematical Society.Google Scholar
Chekin, B. S. 1989 The entry of a wedge into incompressible fluid. Prikl. Mat. Mekh. 53, 300–307.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
Faltinsen, O. M. 2005 Hydrodynamics of High-speed Marine Vehicles. Cambridge University Press.Google Scholar
Gakhov, F. D. 1990 Boundary Value Problems. Dover.Google Scholar
Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic Press.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
Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2004 Oblique slamming, planing and skimming. J. Engng Maths 48, 321–337.Google Scholar
Judge, C., Troesch, A. & Perlin, M. 2004 Initial water impact of a wedge at vertical and oblique angles. J. Engng. Maths 48, 279–303.Google Scholar
Iafrati, A. 2000 Hydrodynamics of asymmetric wedges impacting the free surface. In European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS 2000. Barcelona, Spain, pp. 11–14.Google Scholar
Jones, M. A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.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. Sbornik. 15 (1), Moscow University, 121–278.Google Scholar
von Karman, T.1929 The impact of seaplane floats during landing. Washington, DC: NACA Tech. Note 321.Google Scholar
Korobkin, A. A. & Puknachov, V. V. 1988 Initial Stage of Water Impact. Annu. Rev. Fluid Mech. 20, 159–185.Google Scholar
Korobkin, A. A. 2004 Analytical models of water impact. Eur. J. Appl. Maths 15, 821–838.Google Scholar
Mei, X., Liu, Y. & Yue, D. K. P. 1999 On the water impact of general two-dimensional sections. Appl. Ocean Res. 21, 1–15.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
Moore, D. W. 1975 The rolling up of a semi-infinite vortex sheet. Proc. R. Soc. Lond. A 345, 417–430.Google Scholar
Oliver, J. M. 2007 Second-order Wagner theory for two-dimensional water-entry problems at small deadrise angles. J. Fluid Mech. 572, 59–85.CrossRefGoogle Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (03), 401–430.CrossRefGoogle Scholar
Riccardi, G. & Iafrati, A. 2004 Water impact of an asymmetric floating wedge. J. Engng Maths 49, 19–39.Google Scholar
Rott, N. 1956 Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1 (1), 111–128.Google Scholar
Saffman, P. G. 1993 Vortex Dynamics. Cambridge University Press.Google Scholar
Semenov, Y. A. & Iafrati, A. 2006 On the nonlinear water entry problem of asymmetric wedges. J. Fluid Mech. 547, 231–256.CrossRefGoogle 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. 2012 Asymmetric impact between liquid and solid wedges. Proc. R. Soc. Lond. A 469, 20120203.Google Scholar
Smith, J. H. B. 1968 Improved calculations of leading edge separation from slender delta wings. Proc. R. Soc. Lond. A 306, 87–90.Google Scholar
Xu., L. 2016 Numerical study of viscous starting flow past wedges. J. Fluid Mech. 801, 150–165.CrossRefGoogle 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.CrossRefGoogle Scholar
Xu, G. D., Duan, W. Y. & Wu, G. X. 2010 Simulation of water entry of a wedge through free fall in three degrees of freedom. Proc. R. Soc. Lond. A 466, 2219–2239.Google Scholar
Wagner, H. 1932 Über Stoßund Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12, 192–215.Google Scholar
Zhao, R., Faltinsen, O. M. & Aarsnes, J. 1996 Water entry of arbitrary two-dimensional sections with and without separation. In Procedings of the 21st Symposium on Naval Hydrodynamics, pp. 408–423. National Academy Press, Washington DC.Google Scholar