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Steady gravity waves due to a submerged source

Published online by Cambridge University Press:  12 September 2013

Christopher J. Lustri  and
S. Jonathan Chapman
Christopher J. Lustri*
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
S. Jonathan Chapman
Affiliation:
Oxford Center for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK
*
†Email address for correspondence: [email protected]
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Abstract

In the low-Froude-number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Steady linearized flow past a submerged source is considered, and exponential asymptotic methods are applied to determine the behaviour of the free-surface gravity waves. The free surface is found to contain longitudinal and transverse waves that switch on rapidly across curves known as Stokes lines on the free surface. The longitudinal waves are present everywhere downstream of the singularity, while the transverse waves are restricted to two downstream wedges. As the depth of the source approaches the surface, the familiar Kelvin-wedge wave behaviour is recovered.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 732 , 10 October 2013 , pp. 660 - 686
Copyright
©2013 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Akylas, T. R. 1987 Unsteady and nonlinear effects near the cusp lines of the Kelvin ship-wave pattern. J. Fluid Mech. 175, 333–342.CrossRefGoogle Scholar
Baba, E. 1976 Wave resistance of ships in low speed. Mitusubishi Tech. Bull. 109, 1–20.Google Scholar
Berry, M. V. 1988 Stokes phenomenon; smoothing a Victorian discontinuity. Pub. Math. de L’IHÉS 68, 211–221.Google Scholar
Berry, M. V. 1991 Asymptotics, superasymptotics, hyperasymptotics. In Asymptotics Beyond All Orders (ed. Segur, H., Tanveer, S. & Levine, H.), pp. 1–14. Plenum.Google Scholar
Berry, M. V. & Howls, C. J. 1990 Hyperasymptotics. Proc. R. Soc. Lond. A 430 (1880), 653–668.Google Scholar
Boyd, J. P. 1998 Weakly Non-local Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory. Mathematics and Its Applications, vol. 442, Kluwer.CrossRefGoogle Scholar
Boyd, J. P. 1999 The devils invention: Asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Maths 56 (1), 1–98.Google Scholar
Boyd, J. P. 2005 Hyperasymptotics and the linear boundary layer problem: why asymptotic series diverge. SIAM Rev. 47 (3), 553–575.Google Scholar
Chapman, S. J., King, J. R., Ockendon, J. R. & Adams, K. L. 1978 Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. R. Soc. Lond. A 454, 2733–2755.Google Scholar
Chapman, S. J., Lawry, J. M. H. & Tew, R. H. 1999 On the theory of complex rays. SIAM Rev. 41 (3), 417–509.Google Scholar
Chapman, S. J. & Mortimer, D. B. 2005 Exponential asymptotics and Stokes lines in a partial differential equation. Proc. R. Soc. Lond. A 461, 2385–2421.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2002 Exponential asymptotics and capillary waves. SIAM J. Appl. Maths 62 (6), 1872–1898.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. 2006 Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299–326.Google Scholar
Costin, O. & Kruskal, M. D. 1996 Optimal uniform estimates and rigourous asymptotics beyond all orders for a class of ordinary differential equations. Proc. R. Soc. Lond. A 452 (1948), 1057–1085.Google Scholar
Costin, O. & Kruskal, M. D. 1999 On optimal truncation of divergent series solutions of nonlinear differential systems. Proc. R. Soc. Lond. A 455 (1985), 1931–1956.Google Scholar
Dagan, G. & Tulin, M. P. 1972 Two-dimensional free-surface gravity flow past blunt bodies. J. Fluid Mech. 51 (3), 529–543.Google Scholar
Dingle, R. B. 1973 Asymptotic Expansions: Their Derivation and Interpretation. Academic.Google Scholar
Forbes, L. K. & Schwartz, L. W. 1982 Free surface flow over a semi-circular obstruction. J. Fluid Mech. 114, 299–314.Google Scholar
Grimshaw, R. 2011 Exponential asymptotics and generalized solitary waves. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P.), CISM Courses and Lectures, vol. 523, pp. 71–120. Springer.Google Scholar
Grimshaw, R. & Joshi, N. 1995 Weakly non-local solitary waves in a singularly perturbed Korteweg-de Vries equation. SIAM J. Appl. Maths 55 (1), 124–135.CrossRefGoogle Scholar
Havelock, T. H. 1917 Some cases of wave motion due to a submerged obstacle. Proc. R. Soc. Lond. A 93 (654), 520–532.Google Scholar
Havelock, T. H. 1919 Wave resistance: some cases of three-dimensional fluid motion. Proc. R. Soc. Lond. A 95 (670), 354–365.Google Scholar
Havelock, T. H. 1931 The wave resistance of a spheroid. Proc. R. Soc. Lond. A 131 (817), 275–285.Google Scholar
Havelock, T. H. 1932 The theory of wave resistance. Proc. R. Soc. Lond. A 138 (835), 339–348.Google Scholar
Havelock, T. H. 1949 The wave resistance of a cylinder started from rest. Q. J. Mech. Appl. Maths 2 (3), 325–334.Google Scholar
Howls, C. J., Langman, P. J. & Olde Daalhuis, A. B. 2004 On the higher-order Stokes phenomenon. Proc. R. Soc. Lond. A 460 (2121), 2285–2303.Google Scholar
Jones, D. S. 1997 Introduction to Asymptotics: a Treatment Using Nonstandard Analysis, p. 160. World Scientific.Google Scholar
Keller, J. B. 1979 The ray theory of ship waves and the class of streamlined ships. J. Fluid Mech. 91 (3), 465–488.CrossRefGoogle Scholar
Keller, J. B. & Ward, M. J. 1996 Asymptotics beyond all orders for a low Reynolds number flow. J. Engng Maths 30 (1–2), 253–265.Google Scholar
Kelvin, Lord 1887 On ship waves. Proc. Inst. Mech. Engrs 3, 409–434.Google Scholar
Lamb, H. 1879 Hydrodynamics. Cambridge University Press.Google Scholar
Liu, M. & Tao, M. 2001 Transient ship waves on an incompressible fluid of infinite depth. Phys. Fluids 13, 3610–3623.Google Scholar
Lunde, J. K. 1951 On the linearized theory of wave resistance for displacement ships in steady and accelerated motion. Trans.-Soc. Nav. Archit. Mar. Eng 59, 25–76.Google Scholar
Lustri, C. J., McCue, S. W. & Binder, B. J. 2012 Free surface flow past topography: a beyond-all-orders approach. Eur. J. Appl. Maths 23 (4), 441–467.Google Scholar
Lustri, C. J., McCue, S. W. & Chapman, S. J. 2013 Exponential asymptotics of free surface flow due to a line source. IMA J. Appl. Maths 78 (4), 697–713.Google Scholar
Maruo, H. & Suzuki, K. 1977 Wave resistance of a ship of finite beam predicted by the low speed theory. J. Soc. Nav. Arch. Japan 142, 1–7.Google Scholar
Michell, J. H. 1898 The wave-resistance of a ship. Phil. Mag. 45 (5), 106–123.Google Scholar
Noblesse, F. 1981 Alternative integral representations for the Green function of the theory of ship wave resistance. J. Engng Maths 15 (4), 241–265.Google Scholar
Noblesse, F. 1982 The Green function in the theory of radiation and diffraction of regular water waves by a body. J. Engng Maths 16, 137–169.Google Scholar
Ockendon, J. R., Howison, S., Lacey, A. & Movchan, A. 1999 Applied Partial Differential Equations. Oxford University Press.Google Scholar
Ockendon, J. R. & Wilmott, P. 1986 Matching and singularity distributions in inviscid flow. IMA J. Appl. Maths 37 (3), 199–211.Google Scholar
Ogilvie, T. F. 1968 Wave resistance: the low speed limit. In Tech. Rep. Ann Arbor, MI, Michigan University.Google Scholar
Olde Daalhuis, A. B. 1998 Hyperasymptotic solutions of higher-order linear differential equations with a singularity of rank one. Proc. R. Soc. Lond. A 454 (1968), 1–29.Google Scholar
Olde Daalhuis, A. B., Chapman, S. J., King, J. R., Ockendon, J. R. & Tew, R. H. 1995 Stokes phenomenon and matched asymptotic expansions. SIAM J. Appl. Maths 55 (6), 1469–1483.CrossRefGoogle Scholar
Paris, R. B. & Wood, A. D. 1995 Stokes phenomenon demystified. IMA Bull. 31, 21–28.Google Scholar
Peregrine, D. H. 1972 A line source beneath a free surface. In Mathematics Research Center Technical Summary Report 1248. University of Wisconsin.Google Scholar
Peters, A. S. 1949 A new treatment of the ship wave problem. Commun. Pure Appl. Maths 2 (2–3), 123–148.Google Scholar
Segur, H., Tanveer, S. & Levine, H. (Eds) 1991 Asymptotics Beyond All Orders. Plenum.Google Scholar
Stokes, G. G. 1864 On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Camb. Phil. Soc. 10, 106–128.Google Scholar
Trinh, P. H. 2011 Exponential asymptotics and Stokes line smoothing for generalized solitary waves. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P.), CISM Courses and Lectures, vol. 523, pp. 121–126. Springer.Google Scholar
Trinh, P. H. & Chapman, S. J. 2013a New gravity-capillary waves at low speeds. Part 1. Linear geometries. J. Fluid Mech. 724, 367–391.Google Scholar
Trinh, P. H. & Chapman, S. J. 2013b New gravity-capillary waves at low speeds. Part 2. Nonlinear geometries. J. Fluid Mech. 724, 392–424.Google Scholar
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. 2011 Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413–439.Google Scholar
Ursell, F. 1960 On Kelvin’s ship-wave pattern. J. Fluid Mech. 8, 418–431.CrossRefGoogle Scholar
Vanden-Broeck, J.-M., Schwartz, L. W. & Tuck, E. O. 1978 Divergent low-Froude number series expansion of nonlinear free-surface flow problems. Proc. R. Soc. Lond. A 361 (1705), 207–224.Google Scholar
Ward, M. J. & Kropinski, M.-C. 2011 Asymptotic methods for pde problems in fluid mechanics and related systems with strong localized perturbations in two-dimensional domains. In Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances (ed. Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P.), CISM Courses and Lectures, vol. 523, pp. 23–70. Springer.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Springer.CrossRefGoogle Scholar