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Unsteady flow over a submerged source with low Froude number

  • CHRISTOPHER J. LUSTRI (a1) and S. JONATHAN CHAPMAN (a2)
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.

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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), 173195.
Abramowitz, M. & Stegun, I. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York.
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. 122.
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), 27332755.
Chapman, S. J. & Mortimer, D. B. (2005) Exponential asymptotics and Stokes lines in a partial differential equation. Proc. Roy. Soc. Lond. A 461, 23852421.
Chapman, S. J. & Vanden-Broeck, J.-M. (2002) Exponential asymptotics and capillary waves. SIAM J. Appl. Math. 62 (6), 18721898.
Chapman, S. J. & Vanden-Broeck, J.-M. (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299326.
Cole, S. L. (1985) Transient waves produced by flow past a bump. Wave Mot. 7, 579587.
Craik, A. D. D. (2004) The origins of water wave theory. Ann. Rev. Fluid Mech. 36 (1), 128.
Dagan, G. & Tulin, M. P. (1972) Two-dimensional free-surface gravity flow past blunt bodies. J. Fluid Mech. 51 (3), 529543.
Dingle, R. B. (1973) Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York.
Forbes, L. K., Hocking, G. C. & Stokes, T. E. (2008) On starting conditions for a submerged sink in a fluid. J. Eng. Math. 61, 5568.
Gradshteyn, I. S. & Ryzhik, I. M. (1994) Table of Integrals, Series, and Products, Academic Press, New York.
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. 71120.
Grimshaw, R. & Joshi, N. (1995) Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation. SIAM J. Appl. Math. 55 (1), 124135.
Havelock, T. H. (1917) Some cases of wave motion due to a submerged obstacle Proc. Roy. Soc. Lond. A 93 (654), 520532.
Havelock, T. H. (1949) The wave resistance of a cylinder started from rest. Quart. J. Mech. App. Math. 2 (3), 325334.
Howls, C. J., Langman, P. J. & Olde Daalhuis, A. B. (2004) On the higher-order Stokes phenomenon. Proc. Roy. Soc. Lond. A 460 (2121), 22852303.
John, F. (1953) Two-dimensional potential flows with a free boundary. Comm. Pure Appl. Math. 6, 497503.
Keller, J. B. & Ward, M. J. (1996) Asymptotics beyond all orders for a low Reynolds number flow. J. Eng. Math. 30 (1–2), 253265.
Kelvin, B. W. T. (1887) On ship waves. Proc. Inst. Mech. Eng. 3, 409434.
Liu, M. & Tao, M. (2001) Transient ship waves on an incompressible fluid of infinite depth. Phys. Fluids 13, 36103623.
Longuet-Higgins, M. S. (1980) A technique for time-dependent free-surface flows. Proc. Roy. Soc. Lond. A 371, 441451.
Lu, D. (2009) Generation of free-surface gravity waves by an unsteady Stokeslet. Arch. App. Mech. 79, 311322.
Lustri, C. J. & Chapman, S. J. (2013) Steady gravity waves due to a submerged source. J. Fluid Mech. 732, 660686.
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), 441467.
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), 697713.
Ockendon, J. R., Howison, S., Lacey, A. & Movchan, A. (1999) Applied Partial Differential Equations, Oxford University Press, New York.
Ockendon, J. R. & Wilmott, P. (1986) Matching and singularity distributions in inviscid flow. IMA J. App. Math. 37 (3), 199211.
Ogilvie, T. F. (1968) Wave Resistance: The Low Speed Limit, Technical report, Michigan University, Ann Arbor, MI.
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), 14691483.
Peregrine, D. H. (1972) A line source beneath a free surface, Mathematics Research Center Technical Summary Report 1248, University of Wisconsin, Madison, WI.
Shen, M. (1969) Asymptotic theory of unsteady three-dimensional waves in a channel of arbitrary cross section. SIAM J. App. Math. 17 (2), 260271.
Stokes, G. G. (1864) On the discontinuity of arbitrary constants which appear in divergent developments. Trans. Cam. Phil. Soc. 10, 105.
Stokes, T., Hocking, G. & Forbes, L. (2003) Unsteady free-surface flow induced by a line sink. J. Eng. Math. 47, 137160.
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. 121126.
Trinh, P. H. & Chapman, S. J. (2010) Exponential Asymptotics and Free-surface Flows, PhD Thesis, University of Oxford.
Trinh, P. H. & Chapman, S. J. (2013a) New gravity-capillary waves at low speeds. Part 1. Linear geometries. J. Fluid Mech. 724, 367391.
Trinh, P. H. & Chapman, S. J. (2013b) New gravity-capillary waves at low speeds. Part 2. Nonlinear geometries. J. Fluid Mech. 724, 392424.
Trinh, P. H., Chapman, S. J. & Vanden-Broeck, J.-M. (2011) Do waveless ships exist? Results for single-cornered hulls. J. Fluid Mech. 685, 413439.
Tyvand, P. A. (1992) Nonlinear transient freesurface flow and dip formation due to a point sink. Phys. Fluids A 4, 671676.
Tyvand, P. A. (1993) Unsteady free surface flow due to a line source. Phys. Fluids A 5, 13681375.
Tyvand, P. A. & Miloh, T. (1995a) Free-surface flow due to impulsive motion of a submerged circular cylinder. J. Fluid Mech. 286, 67101.
Tyvand, P. A. & Miloh, T. (1995b) Free-surface flow generated by a small submerged circular cylinder starting from rest. J. Fluid Mech. 286, 103116.
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), 207224.
Wilmott, P. (1987) On the motion of a small two-dimensional body submerged beneath surface waves. J. Fluid Mech. 176, 465481.
Xue, M. & Yue, D. K. P. (1998) ‘Nonlinear free-surface flow due to an impulsively started submerged point sink.’ J. Fluid Mech. 364, 325347.
Zhu, S. & Zhang, Y. (1997) On nonlinear transient free-surface flows over a bottom obstruction. Phys. Fluids 9, 25982604.
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
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