##
**On the oblique water-entry problem of a rigid sphere.**
*(English)*
Zbl 0735.73060

Summary: The case of oblique water-entry of a rigid sphere into an ideal incompressible fluid is studied analytically in order to determine the hydrodynamical loads acting on the body. We consider the motion imparted to the fluid by an impulsively-started partially-submerged sphere under the large-impact approximation, in which the free surface is assumed flat and equipotential. Asymptotic small-time expressions are derived for both the vertical and horizontal time-dependent added masses and analytical expressions for the hydrodynamic forces are obtained by differentiating these added masses with respect to the instantaneous submergence depth. The resulting expressions are also compared with corresponding numerical solutions and with a known solution for a two-dimensional profile.

### MSC:

74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |

76B99 | Incompressible inviscid fluids |

45B05 | Fredholm integral equations |

45M05 | Asymptotics of solutions to integral equations |

### Keywords:

integral equation of first kind; small-time approximations; ideal incompressible fluid; hydrodynamical loads; impulsively-started partially-submerged sphere; large-impact approximation; asymptotic small- time expressions; vertical and horizontal time-dependent added masses; analytical expressions for the hydrodynamic forces; time-dependent impact problem### Citations:

Zbl 0677.76013
Full Text:
DOI

### References:

[1] | A.T. Chwang, Nonlinear hydrodynamic pressure on an accelerating plate. Phys. of Fluids 25 (1983) 383–387. · Zbl 0521.76025 |

[2] | I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York (1965). · Zbl 0918.65002 |

[3] | M.J. Greenhow, Water-entry and exit of a horizontal circular cylinder. Appl. Ocean Res. 10 (1988) 191–198. |

[4] | M.A. Grosenbaugh and R.W. Yeung, Nonlinear bow flows – An experimental and theoretical investigation. 17th Symp. on Naval Hydrodynamics, The Hague (1988) 195–214. |

[5] | E.W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, Cambridge (1931). · Zbl 0004.21001 |

[6] | A. Hulme, A ring-source integral-equation for the calculation of hydrodynamical forces exerted on floating bodies of revolution. J. Fluid Mech. 128 (1983) 387–412. · Zbl 0518.76016 |

[7] | S.W. Joo, W.W. Schultz and A.F. Messiter, Uniformly valid solution to the initial-value wavemaker problem. The Univ. of Michigan program in ship Hydrodynamics Rep. 89-01 (1989). |

[8] | A.A. Korobkin, Initial asymptotic behavior of the solution of the three-dimensional problem of the entry of a blunt body into an ideal liquid. Sov. Phys. Dokl. 30 (1985) 656–658. · Zbl 0637.76013 |

[9] | A.A. Korobkin, Inclined entry of a blunt profile into an ideal fluid. Fluid Dynamics 23 (1988) 443–447. · Zbl 0677.76013 |

[10] | A.A. Korobkin and V.V. Pukhnachov, Initial stage of water impact. Ann. Rev. Fluid Mech. 20 (1988) 159–185. |

[11] | H. Lamb, Hydrodynamics, Dover Publications, New York (1945). |

[12] | Y.L. Luke, Algorithm for the Computations of Mathematical Functions, Academic Press, New York (1977). · Zbl 0358.58005 |

[13] | P. McIver, Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201 (1989) 243–257. · Zbl 0667.76024 |

[14] | T. Miloh, Hydrodynamics of doformable contiguous spherical shapes in an incompressible invisoid fluid. J. Eng. Math. 11 (1977) 349–372. · Zbl 0368.76019 |

[15] | T. Miloh, Wave slam on a sphere penetrating a free-surface. J. Eng. Math. 15 (1981) 221–240. · Zbl 0479.76022 |

[16] | T. Miloh, Hamilton’s principle, Lagrange’s method and ship motion theory. J. Ship. Res. 28 (1984) 229–237. |

[17] | T. Miloh, On the initial stage slamming of a rigid sphere in a verical water entry, to appear in Appl. Ocean Research (1990). |

[18] | T. Miloh and L. Landweber, Generalization of the Kelvin-Kirchhoff equations for the motion of a body through a fluid. Phys. of Fluids 24 (1981) 6–9. · Zbl 0467.76029 |

[19] | M. Moghisi and P.T. Squire, An experimental investigation of the initial force of impact on a sphere striking a free-surface. J. Fluid Mech. 108 (1981) 133–146. |

[20] | P. Moon and D.E. Spencer, Field Handbook Springer, Berlin (1971). |

[21] | D.H. Peregrine, Flow due to a vertical plate moving in a channel. Unpublished note (1972). |

[22] | V.V. Pukhnachov and A.A. Korobkin, Initial asymptotic in problems of blunt body entrance into liquid. Proc. of 3rd International Conference on Numerical Ship Hydrodynamics, Paris (1981) 579–592. |

[23] | A.J. Roberts, Transient free-surface flows generated by a moving vertical plate. Quart. J. Mech. Appl. Math. 40 (1987) 129–158. · Zbl 0616.76020 |

[24] | L. Robin, Fonctions Spheriques de Legendre et Fonctions Spheroidales. Collection Technique et Scientifique du Centre National d’Etude des Telecommunications (1959). |

[25] | L. Sedov, The impact of a solid body floating on the surface of an incompressible fluid. TSAGI Report No. 187, Moscow (1934). |

[26] | I.H. Sneddon, The Use of Integral Transforms, McGraw-Hill, New York (1972). · Zbl 0237.44001 |

[27] | L. Trilling, The impact of a body on a water surface at an arbitrary angle, J. Appl. Phys. 21 (1950) 161–170. · Zbl 0035.42204 |

[28] | A.W. Trocsch and C.G. Kang, Hydrodynamic impact loads on three-dimensional bodies, Proc. 16th Symp, on Naval Hydrodynamics, Berkeley (1987) 537–538. |

[29] | P.A. Tyvand, On the interaction between a strong vortex pair and a free-surface, to appear in Physics of Fluids (1990). |

[30] | T. Vinje, On small time expansion of non linear free surface problems. Proc. 4th International Workshop on Water Waves and Floating Bodies, Oystese, Norway (1989) 245–247. |

[31] | Th. Von Karman, The impact on seaplane floats during landing. NACA TN 321 (1929). |

[32] | H. Wagner, Uber Stoss und Gleitvorgange an der Oberflache von Flussigkeiten ZAMM 12 (1932) 193–215. · JFM 58.0882.02 |

[33] | K.H. Wang and A. Chwang, Nonlinear free surface flow around an impulsively moving cylinder. J. Ship Res 33 (1989) 194–202. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.