Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T23:36:51.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Do waveless ships exist? Results for single-cornered hulls

Published online by Cambridge University Press:  06 October 2011

Philippe H. Trinh ,
S. Jonathan Chapman  and
Jean-Marc Vanden-Broeck
Philippe H. Trinh*
Affiliation:
PACM, Princeton University, Washington Road, Princeton, NJ 08544, USA
S. Jonathan Chapman
Affiliation:
OCIAM, Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, UK
Jean-Marc Vanden-Broeck
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
*
†Email address for correspondence: ptrinh@princeton.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider low-speed potential flow past a ship modelled as a semi-infinite two-dimensional body with constant draught. Is it possible to design the hull in such a way as to eliminate the waves produced downstream of the ship? In 1977, Vanden-Broeck & Tuck had conjectured that a single-cornered piecewise-linear hull will always generate a wake; in this paper, we show how recently developed tools in exponential asymptotics can be used to confirm this conjecture. In particular, we show how the formation of waves near a ship is a necessary consequence of singularities in the ship’s geometry (or its analytic continuation). Comprehensive numerical computations confirm the analytical predictions.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 685 , 25 October 2011 , pp. 413 - 439
Copyright
Copyright © Cambridge University Press 2011

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

1. Baba, E. 1976 Wave breaking resistance of ships. In Proc. Intl Seminar on Wave Resistance, Tokyo, pp. 75–92.Google Scholar
2. Boyd, J. P. 1991 A comparison of numerical and analytical methods for the reduced wave equation with multiple spatial scales. Appl. Numer. Maths 7, 453–479.CrossRefGoogle Scholar
3. Boyd, J. P. 1998 Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics. Kluwer.CrossRefGoogle Scholar
4. Boyd, J. P. 1999 The Devil’s invention: asymptotics, superasymptotics and hyperasymptotics. Acta Appl. 56, 1–98.CrossRefGoogle Scholar
5. Chapman, S. J. 1999 On the role of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension. Eur. J. Appl. Maths 10 (6), 513–534.CrossRefGoogle Scholar
6. Chapman, S. J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Maths 62 (6), 1872–1898.Google Scholar
7. Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.CrossRefGoogle Scholar
8. Combescot, R., Hakim, V., Dombre, T., Pomeau, Y. & Pumir, A. 1988 Analytic theory of the Saffman–Taylor fingers. Phys. Rev. A 37 (4), 1270–1283.CrossRefGoogle ScholarPubMed
9. Dagan, G. & Tulin, M. P. 1972 Two-dimensional free-surface gravity flow past blunt bodies. J. Fluid Mech. 51 (3), 529–543.CrossRefGoogle Scholar
10. Dias, F. & Vanden-Broeck, J.-M. 1993 Nonlinear bow flows with spray. J. Fluid Mech. 255, 91–102.CrossRefGoogle Scholar
11. Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic.Google Scholar
12. Farrow, D. E. & Tuck, E. O. 1995 Further studies of stern wavemaking. J. Austral. Math. Soc. B 36, 424–437.CrossRefGoogle Scholar
13. Forbes, L. K. 1983 Free-surface flow over a semi-circular obstruction, including the influence of gravity and surface tension. J. Fluid Mech. 127, 283–297.CrossRefGoogle Scholar
14. Gakhov, F. D. 1990 Boundary Value Problems. Dover.Google Scholar
15. King, A. C. & Bloor, M. I. G. 1990 Free-surface flow of a stream obstructed by an arbitrary bed topography. Q. J. Mech. Appl. Maths 43, 87–106.CrossRefGoogle Scholar
16. Kostyukov, A. A. 1968 Theory of Ship Waves and Wave Resistance. Effective Communications Inc, (English translation).Google Scholar
17. Kotik, J. & Newman, D. J. 1964 A sequence of submerged dipole distributions whose wave resistance tends to zero. J. Math. Mech. 13, 693–700.Google Scholar
18. Madurasinghe, M. A. D. 1988 Splashless ship bows with stagnant attachment. J. Ship Res. 32 (3), 194–202.CrossRefGoogle Scholar
19. Madurasinghe, M. A. D. & Tuck, E. O. 1986 Ship bows with continuous and splashless flow attachment. J. Austral. Math. Soc. B 27, 442–452.CrossRefGoogle Scholar
20. Noble, B. & Beighton, S. 1980 Error estimates for three methods of evaluating Cauchy principal value integrals. J. Inst. Maths Applics 26, 431–446.CrossRefGoogle Scholar
21. Ogilvie, T. F. 1968 Wave resistance: The low speed limit. Tech Rep. Michigan University, Ann Arbor.Google Scholar
22. Olver, S. 2011 Computing the Hilbert transform and its inverse. Maths Comput. 80, 1745–1767.CrossRefGoogle Scholar
23. Trinh, P. H. 2010 Exponential asymptotics and free-surface flows. PhD thesis, University of Oxford.Google Scholar
24. Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2010 The existence and non-existence of waveless ships. In Proc. 25th Intl Workshop on Water Waves and Floating Bodies, Harbin, China, Harbin Engineering University.Google Scholar
25. Tuck, E. O. 1991a Ship-hydrodynamic free-surface problems without waves. J. Ship Res. 35 (4), 277–287.CrossRefGoogle Scholar
26. Tuck, E. O. 1991b Waveless solutions of wave equations. In Proc. 6th International Workshop on Water Waves and Floating Bodies. MIT.Google Scholar
27. Tuck, E. O. 1994 The planing splash. In Proc. 9th Intl Workshop on Water Waves and Floating Bodies (ed. M. Ohkusu). Kyushu University, Japan.Google Scholar
28. Tuck, E. O. & Roberts, A. J. 1997 Bow-like free surfaces under gravity. Phil. Trans. R. Soc. Lond. A 355, 655–677.Google Scholar
29. Tuck, E. O. & Vanden-Broeck, J.-M. 1984 Splashless bow flows in two-dimensions. In Proc. 15th Symposium Naval Hydrodynamics. National Academy Press.Google Scholar
30. Tulin, M. P. 2005 Reminiscences and reflections: ship waves, 1950–2000. J. Ship Res. 49 (4), 238–246.CrossRefGoogle Scholar
31. Vanden-Broeck, J.-M. 1980 Nonlinear stern waves. J. Fluid Mech. 96 (3), 603–611.CrossRefGoogle Scholar
32. Vanden-Broeck, J.-M. 1985 Nonlinear free-surface flows past two-dimensional bodies. In Advances in Nonlinear Waves, vol. 2, pp. 31–42. Pitman Publishing Inc.Google Scholar
33. Vanden-Broeck, J.-M. 2010 Gravity–Capillary Free-Surface Flows. Cambridge University Press.CrossRefGoogle Scholar
34. Vanden-Broeck, J.-M. & Tuck, E. O. 1977 Computation of near-bow or stern flows using series expansion in the Froude number. In 2nd Intl Conf. on Numerical Ship Hydrodynamics, Berkeley, California, University of California, Berkeley.Google Scholar
35. Vanden-Broeck, J.-M. & Tuck, E. O. 1994 Flow near the intersection of a free surface with a vertical wall. SIAM J. Appl. Maths 54 (1), 1–13.CrossRefGoogle Scholar
36. Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves, Encyclopedia of Physics , IX. Springer.Google Scholar
37. Xie, X. & Tanveer, S. 2002 Analyticity and nonexistence of classical steady Hele–Shaw fingers. Commun. Pure Appl. Maths 56 (3), 353–402.Google Scholar
38. Yeung, R. W. 1991 Nonlinear bow and stern waves – inviscid and viscous solutions. In Mathematical Approaches in Hydrodynamics, pp. 349–369. SIAM.Google Scholar