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Nonlinear stern waves

  • J.-M. Vanden-Broeck (a1)

Steady two-dimensional potential flow past a semi-infinite flat-bottomed body is considered. This stern flow is assumed to separate tangentially from the body. Gravity waves of finite amplitude occur on the free surface. An exact relation between the amplitude of these waves and the Fronde number F is derived. It shows that these waves can exist only for F greater than the value F* = 2·23. This is slightly less than the value Fc = 2·26 at which breaking occurs. For F slightly larger than F*, the steepness is a multi-valued function of F, indicating the existence of more than one solution for these values of F. In addition, a numerical scheme based on an integro-differential equation formulation is derived to solve the problem in the fully nonlinear case. The shape of the free surface profile is computed for different values of F. As a check on the numerical results, they are shown to satisfy the exact relation between steepness and the Froude number.

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Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. Roy. Soc A 286, 183230.
Longuet-Higgins, M. S. 1975 Integral properties of periodic gravity waves of finite amplitude. Proc. Roy. Soc. A 342, 157174.
Longuet-Higgins, M. S. & Fox, M. J. H. 1978 Theory of the almost-highest wave. Part 2. Matching an analytic extension. J. Fluid Mech. 85, 769786.
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553578.
Vanden-Broeck, J.-M. & Tuck, E. O. 1977 Computation of near-bow or stern flows, using series expansion in the Froude number. In Proc. 2nd Int. Conf. Num. Ship Hydrodynamics, Berkeley.
Vanden-Broeck, J.-M., Schwartz, L. W. & Tuck, E. O. 1978 Divergent low Froude-number series expansion of non-linear free-surface flow problems. Proc. Roy. Soc. A 361, 207224.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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