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Unsteady flow over a submerged source with low Froude number

Published online by Cambridge University Press:  04 August 2014

CHRISTOPHER J. LUSTRI  and
S. JONATHAN CHAPMAN
CHRISTOPHER J. LUSTRI
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia E-mail: [email protected]
S. JONATHAN CHAPMAN
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK
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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. Previous studies have considered linearized steady flow past a submerged source in infinite-depth fluids, in which exponential asymptotics were used to determine the behaviour of downstream longitudinal and transverse free-surface gravity waves. Here, unsteady flow past a submerged source in an infinite-depth fluid is investigated, with the free surface taken to be initially waveless. The source is taken to be weak, and the flow is linearized about the undisturbed solution. Exponential asymptotics are applied to determine the wave behaviour on the free surface in terms of the two-dimensional plan-view, in order to show how the free surface waves evolve over time and eventually tend to the steady solution.

Keywords

Free-surface potential flowsGravity wavesSingular perturbations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 5 , October 2014 , pp. 655 - 680
Copyright
Copyright © Cambridge University Press 2014 

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References

Abou-Dina, M. (2001) Nonlinear transient gravity waves due to an initial free-surface elevation over a topography. J. Comp. App. Math. 130 (1–2), 173–195.CrossRefGoogle Scholar
Abramowitz, M. & Stegun, I. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York.Google Scholar
Aoki, T., Koike, T. & Takei, Y. (2002) Vanishing of Stokes curves. In: Kawai, T. & Fujita, K. (editors), Microlocal Analysis and Complex Fourier Analysis, World Scientific, Singapore, pp. 1–22.Google Scholar
Chapman, S. J., King, J. R., Ockendon, J. R. & Adams, K. L. (1998) Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Proc. Roy. Soc. Lond. A 454 (1978), 2733–2755.Google Scholar
Chapman, S. J. & Mortimer, D. B. (2005) Exponential asymptotics and Stokes lines in a partial differential equation. Proc. Roy. Soc. Lond. A 461, 2385–2421.Google Scholar
Chapman, S. J. & Vanden-Broeck, J.-M. (2002) Exponential asymptotics and capillary waves. SIAM J. Appl. Math. 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
Cole, S. L. (1985) Transient waves produced by flow past a bump. Wave Mot. 7, 579–587.Google Scholar
Craik, A. D. D. (2004) The origins of water wave theory. Ann. Rev. Fluid Mech. 36 (1), 1–28.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 Press, New York.Google Scholar
Forbes, L. K., Hocking, G. C. & Stokes, T. E. (2008) On starting conditions for a submerged sink in a fluid. J. Eng. Math. 61, 55–68.Google Scholar
Gradshteyn, I. S. & Ryzhik, I. M. (1994) Table of Integrals, Series, and Products, Academic Press, New York.Google Scholar
Grimshaw, R. (2011) Exponential asymptotics and generalized solitary waves. In Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P. (editors), Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances, Vol. 523 of CISM Courses and Lectures, Springer, Vienna, pp. 71–120.Google Scholar
Grimshaw, R. & Joshi, N. (1995) Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation. SIAM J. Appl. Math. 55 (1), 124–135.Google Scholar
Havelock, T. H. (1917) Some cases of wave motion due to a submerged obstacle Proc. Roy. Soc. Lond. A 93 (654), 520–532.Google Scholar
Havelock, T. H. (1949) The wave resistance of a cylinder started from rest. Quart. J. Mech. App. Math. 2 (3), 325–334.Google Scholar
Howls, C. J., Langman, P. J. & Olde Daalhuis, A. B. (2004) On the higher-order Stokes phenomenon. Proc. Roy. Soc. Lond. A 460 (2121), 2285–2303.Google Scholar
John, F. (1953) Two-dimensional potential flows with a free boundary. Comm. Pure Appl. Math. 6, 497–503.Google Scholar
Keller, J. B. & Ward, M. J. (1996) Asymptotics beyond all orders for a low Reynolds number flow. J. Eng. Math. 30 (1–2), 253–265.Google Scholar
Kelvin, B. W. T. (1887) On ship waves. Proc. Inst. Mech. Eng. 3, 409–434.Google Scholar
Liu, M. & Tao, M. (2001) Transient ship waves on an incompressible fluid of infinite depth. Phys. Fluids 13, 3610–3623.Google Scholar
Longuet-Higgins, M. S. (1980) A technique for time-dependent free-surface flows. Proc. Roy. Soc. Lond. A 371, 441–451.Google Scholar
Lu, D. (2009) Generation of free-surface gravity waves by an unsteady Stokeslet. Arch. App. Mech. 79, 311–322.Google Scholar
Lustri, C. J. & Chapman, S. J. (2013) Steady gravity waves due to a submerged source. J. Fluid Mech. 732, 660–686.Google Scholar
Lustri, C. J., McCue, S. W. & Binder, B. J. (2012) Free surface flow past topography: A beyond-all-orders approach. Euro. J. Appl. Math. 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. App. Math. 78 (4), 697–713.Google Scholar
Ockendon, J. R., Howison, S., Lacey, A. & Movchan, A. (1999) Applied Partial Differential Equations, Oxford University Press, New York.Google Scholar
Ockendon, J. R. & Wilmott, P. (1986) Matching and singularity distributions in inviscid flow. IMA J. App. Math. 37 (3), 199–211.Google Scholar
Ogilvie, T. F. (1968) Wave Resistance: The Low Speed Limit, Technical report, Michigan University, Ann Arbor, MI.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. App. Math. 55 (6), 1469–1483.Google Scholar
Peregrine, D. H. (1972) A line source beneath a free surface, Mathematics Research Center Technical Summary Report 1248, University of Wisconsin, Madison, WI.Google Scholar
Shen, M. (1969) Asymptotic theory of unsteady three-dimensional waves in a channel of arbitrary cross section. SIAM J. App. Math. 17 (2), 260–271.Google Scholar
Stokes, G. G. (1864) On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Cam. Phil. Soc. 10, 105.Google Scholar
Stokes, T., Hocking, G. & Forbes, L. (2003) Unsteady free-surface flow induced by a line sink. J. Eng. Math. 47, 137–160.Google Scholar
Trinh, P. H. (2011) Exponential asymptotics and Stokes line smoothing for generalized solitary waves. In: Steinrück, H., Pfeiffer, F., Rammerstorfer, F. G., Salençon, J., Schrefler, B. & Serafini, P. (editors), Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances, Vol. 523 of CISM Courses and Lectures, Springer, Vienna, pp. 121–126.Google Scholar
Trinh, P. H. & Chapman, S. J. (2010) Exponential Asymptotics and Free-surface Flows, PhD Thesis, University of Oxford.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
Tyvand, P. A. (1992) Nonlinear transient freesurface flow and dip formation due to a point sink. Phys. Fluids A 4, 671–676.Google Scholar
Tyvand, P. A. (1993) Unsteady free surface flow due to a line source. Phys. Fluids A 5, 1368–1375.Google Scholar
Tyvand, P. A. & Miloh, T. (1995a) Free-surface flow due to impulsive motion of a submerged circular cylinder. J. Fluid Mech. 286, 67–101.Google Scholar
Tyvand, P. A. & Miloh, T. (1995b) Free-surface flow generated by a small submerged circular cylinder starting from rest. J. Fluid Mech. 286, 103–116.Google Scholar
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. Lond. A 361 (1705), 207–224.Google Scholar
Wilmott, P. (1987) On the motion of a small two-dimensional body submerged beneath surface waves. J. Fluid Mech. 176, 465–481.Google Scholar
Xue, M. & Yue, D. K. P. (1998) ‘Nonlinear free-surface flow due to an impulsively started submerged point sink.’ J. Fluid Mech. 364, 325–347.Google Scholar
Zhu, S. & Zhang, Y. (1997) On nonlinear transient free-surface flows over a bottom obstruction. Phys. Fluids 9, 2598–2604.Google Scholar