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Water entry of a flat elastic plate at high horizontal speed

  • M. Reinhard (a1), A. A. Korobkin (a1) and M. J. Cooker (a1)


The two-dimensional problem of an elastic-plate impact onto an undisturbed surface of water of infinite depth is analysed. The plate is forced to move with a constant horizontal velocity component which is much larger than the vertical velocity component of penetration. The small angle of attack of the plate and its vertical velocity vary in time, and are determined as part of the solution, together with the elastic deflection of the plate and the hydrodynamic loads within the potential flow theory. The boundary conditions on the free surface and on the wetted part of the plate are linearized and imposed on the initial equilibrium position of the liquid surface. The wetted part of the plate depends on the plate motion and its elastic deflection. To determine the length of the wetted part we assume that the spray jet in front of the advancing plate is negligible. A smooth separation of the free-surface flow from the trailing edge is imposed. The wake behind the moving body is included in the model. The plate deflection is governed by Euler’s beam equation, subject to free–free boundary conditions. Four different regimes of plate motion are distinguished depending on the impact conditions: (a) the plate becomes fully wetted; (b) the leading edge of the plate touches the water surface and traps an air cavity; (c) the free surface at the forward contact point starts to separate from the plate; (d) the plate exits the water. We could not detect any impact conditions which lead to steady planing of the free plate after the impact. It is shown that a large part of the total energy in the fluid–plate interaction leaves the main bulk of the liquid with the spray jet. It is demonstrated that the flexibility of the plate may increase the hydrodynamic loads acting on it. The impact loads can cause large bending stresses, which may exceed the yield stress of the plate material. The elastic vibrations of the plate are shown to have a significant effect on the fluid flow in the wake.


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Acheson, D. J. 1990 Elementary Fluid Dynamics. Clarendon.
Baker, C. T. 1977 The Numerical Treatment of Integral Equations. Clarendon.
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes, and Cavities. Academic.
Cointe, R., Fontaine, E., Molin, B. & Scolan, Y.-M. 2004 On energy arguments applied to the hydrodynamic impact force. J. Engng Maths 48, 305319.
Cooker, M. J. 1996 Sudden changes in a potential flow with a free surface due to impact. Q. J. Mech. Appl. Maths 49 (4), 581591.
Donnell, L. H. 1976 Beams, Plates and Shells. McGraw-Hill.
Faltinsen, O. M. 2005 Hydrodynamics of High-Speed Marine Vehicles. Cambridge University Press.
Faltinsen, O. M., Kvålsvold, J. & Aarsnes, J. 1997 Wave impact on a horizontal elastic plate. J. Mar. Sci. Technol. 2, 87100.
Faltinsen, O. M. & Semenov, Y. A. 2008 Nonlinear problem of flat-plate entry into an incompressible liquid. J. Fluid Mech. 611, 151173.
Gakhov, F. D. 1966 Boundary Value Problems. Pergamon.
Hewitt, I. J., Balmforth, N. J. & McElwaine, J. N. 2011 Continual skipping on water. J. Fluid Mech. 669, 328353.
Hicks, P. D. & Purvis, R. 2010 Air cushioning and bubble entrapment in three-dimensional droplet impacts. J. Fluid Mech. 649, 135163.
Hicks, P. D. & Smith, F. T. 2011 Skimming impacts and rebounds on shallow liquid layers. Proc. R. Soc.Lond. A 467, 653674.
Howison, S. D., Morgan, J. D. & Ockendon, J. R. 1997 A class of codimension-two free boundary problems. SIAM Rev. 39 (2), 221253.
Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2004 Oblique slamming, planing and skimming. J. Engng Maths 48, 321337.
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215230.
Iafrati, A. & Korobkin, A. A. 2008 Hydrodynamic loads during early stage of flat plate impact onto water surface. Phys. Fluids 20, 082104.
Khabakhpasheva, T. I. & Korobkin, A. A. 2013 Oblique impact of a smooth body on a thin layer of inviscid liquid. Proc. R. Soc.Lond. A 469 (2151).
King, F. W. 2009 Hilbert Transforms, Vol. 1. Cambridge University Press.
Korobkin, A. A. 1994 Blunt-body penetration into a slightly compressible liquid. In Proceedings of 20th Symposium on Naval Hydrodynamics, Santa Barbara, pp. 179–186. Office of Naval Research.
Korobkin, A. A. 1995 Wave impact on the bow end of a catamaran wet deck. J. Ship Res. 39 (4), 321327.
Korobkin, A. A. 1998 Wave impact on the centre of an Euler beam. J. Appl. Mech. Tech. Phys. 39, 770781.
Korobkin, A. A. 2004 Analytical models of water impact. Eur. J. Appl. Maths 15, 821838.
Korobkin, A. A. 2007 Second-order Wagner theory of wave impact. J. Engng Maths 58, 121139.
Korobkin, A. A. & Khabakhpasheva, T. I. 2006 Regular wave impact onto an elastic plate. J. Engng Maths 55, 127150.
Meyerhoff, W. K. 1965a Die Berechnung hydroelastischer Stöße. Schiffstechnik 12 60, 1830.
Meyerhoff, W. K. 1965b Die Berechnung hydroelastischer Stöße. Schiffstechnik 12 61, 4964.
Moore, M. R., Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2012a A note on oblique water entry. J. Engng Maths doi:10.1007/s10665-012-9570-0.
Moore, M. R., Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2012b Three-dimensional oblique water-entry problems at small deadrise angles. J. Fluid Mech. 711, 259280.
Newman, J. N. 1977 Marine Hydrodynamics. MIT.
Oliver, J. M. 2002 Water entry and related problems. D.Phil thesis, University of Oxford.
Oliver, J. M. 2007 Second-order Wagner theory for two-dimensional water-entry problems at small deadrise angles. J. Fluid Mech. 572, 5985.
Reinhard, M., Korobkin, A. & Cooker, M. J. 2012a The bounce of a blunt body from a water surface at high horizontal speed. In 27th International Workshop on Water Waves and Floating Bodies, Copenhagen, pp. 153–156. Technical University of Denmark.
Reinhard, M., Korobkin, A. A. & Cooker, M. J. 2012b Elastic plate impact into water at high horizontal speed with early water detachment. In 6th International Conference on Hydroelasticity in Marine Technology 2012, Tokyo, pp. 1–10. University of Tokyo Press.
Rosselini, L., Hersen, F., Clanet, C. & Bocquet, L. 2005 Skipping stones. J. Fluid Mech. 543, 137146.
Scolan, Y.-M. & Korobkin, A. A. 2003 Energy distribution from vertical impact of a three-dimensional solid body onto the flat free surface of an ideal fluid. J. Fluids Struct. 17 (2), 275286.
Scolan, Y.-M. & Korobkin, A. A. 2012 Hydrodynamic impact (Wagner) problem and Galin’s theorem. In 27th International Workshop on Water Waves and Floating Bodies, Copenhagen. Technical University of Denmark.
Sedov, L. I. 1940 On the theory of unsteady planing and the motion of a wing with vortex separation. NACA Tech Rep. 942.
Semenov, Y. A. & Yoon, B. S. 2009 Onset of flow separation for the oblique water impact of a wedge. Phys. Fluids 21, 112103.
Smiley, R. F. 1951 An experimental study of water-pressure distributions during landings and planing of a heavily loaded rectangular flat-plate model. NACA Tech Rep. 2453.
Ulstein, T. 1995 Nonlinear effects of a flexible stern seal bag on cobblestone oscillations of an SES. PhD thesis, Norwegian Institute of Technology, Trondheim.
Ulstein, T. & Faltinsen, O. M. 1996 Two-dimensional unsteady planing. J. Ship Res. 40 (3), 200210.
Vorus, W. S. 1996 A flat cylinder theory for vessel impact and steady planing resistance. J. Ship Res. 40 (2), 89106.
Wagner, H. 1932 Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12, 193215.
Zhao, R. & Faltinsen, O. M. 1993 Water entry of two-dimensional bodies. J. Fluid Mech. 246, 593612.
Zhao, R., Faltinsen, O. M. & Aarsnes, J. 1996 Water entry of arbitrary two-dimensional sections with and without flow separation. In Proceedings of 21st Symposium on Naval Hydrodynamics, Trondheim, pp. 408–423, US National Research Council.
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Water entry of a flat elastic plate at high horizontal speed

  • M. Reinhard (a1), A. A. Korobkin (a1) and M. J. Cooker (a1)


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