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Nonlinear interactions between a free-surface flow with surface tension and a submerged cylinder

Published online by Cambridge University Press:  07 April 2010

R. M. MOREIRA  and
D. H. PEREGRINE
R. M. MOREIRA*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
D. H. PEREGRINE
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Present address: Computational Fluid Dynamics Laboratory, Fluminense Federal University, Rua Passo da Pátria 156, bl.D, sl.563A, Niterói, RJ 24210-240, Brazil. Email address for correspondence: roger@vm.uff.br.
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Abstract

A submerged cylinder in a uniform stream flow is approximated by a horizontal doublet, following Lamb's classical method. A linear steady solution including surface tension effects is derived, showing that under certain conditions small-scale ripples are formed ahead of the cylinder, while a train of ‘gravity-like’ waves appear downstream. Surface tension effects and a dipole are included in the fully nonlinear unsteady non-periodic boundary-integral solver described by Tanaka et al. (J. Fluid Mech., vol. 185, 1987, pp. 235–248). Nonlinear effects are modelled by considering a flat free surface or the linear stationary solution as an initial condition for the fully nonlinear irrotational flow programme. Long-run computations show that these unsteady flows approach a steady solution for some parameters after waves have radiated away. In other cases the flow does not approach a steady solution. Interesting features at the free surface such as the appearance of ‘parasitic capillaries’ near the crest of gravity waves and the formation of capillary–gravity waves upstream of the cylinder are found.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 485 - 507
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

‡

Professor Peregrine passed away before this paper was completed. This manuscript was prepared for publication by the first author.

References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Ceniceros, H. D. & Hou, T. Y. 1999 Dynamic generation of capillary waves. Phys. Fluids 11, 1042–1050.CrossRefGoogle Scholar
Chen, B. & Saffman, P. G. 1980 Steady gravity–capillary waves on deep water. Part II. Numerical results for finite amplitude. Stud. Appl. Math. 62, 95–111.CrossRefGoogle Scholar
Crapper, G. D. 1970 Non-linear capillary waves generated by steep gravity waves. J. Fluid Mech. 40, 149–159.CrossRefGoogle Scholar
Dagan, G. 1971 Free-surface gravity flow past a submerged cylinder. J. Fluid Mech. 49, 179–192.CrossRefGoogle Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31, 301–346.CrossRefGoogle Scholar
Dold, J. W. 1992 An efficient surface-integral algorithm applied to unsteady gravity waves. J. Comput. Phys. 103, 90–115.CrossRefGoogle Scholar
Dommermuth, D. 2000 The initialization of nonlinear waves using an adjustment scheme. Wave Mot. 32, 307–317.CrossRefGoogle Scholar
Forbes, L. K. 1983 Free-surface flow over a semicircular obstruction, including the influence of gravity and surface tension. J. Fluid Mech. 127, 283–297.CrossRefGoogle Scholar
Grandison, S. & Vanden-Broeck, J.-M. 2006 Truncation approximations for gravity–capillary free-surface flows. J. Engng Math. 54, 89–97.CrossRefGoogle Scholar
Havelock, T. H. 1926 The method of images in some problems of surface waves. Proc. R. Soc. Lond. A 115, 268–280.Google Scholar
Havelock, T. H. 1936 The forces on a circular cylinder submerged in a uniform stream. Proc. R. Soc. Lond. A 157, 526–534.Google Scholar
Hogan, S. J. 1979 Some effects of surface tension on steep water waves. J. Fluid Mech. 91, 167–180.CrossRefGoogle Scholar
Hogan, S. J. 1980 Some effects of surface tension on steep water waves. Part 2. J. Fluid Mech. 96, 417–445.CrossRefGoogle Scholar
Hogan, S. J. 1981 Some effects of surface tension on steep water waves. Part 3. J. Fluid Mech. 110, 381–410.CrossRefGoogle Scholar
Hogan, S. J. 1983 Energy flux in capillary–gravity waves. Phys. Fluids 26, 1206–1209.CrossRefGoogle Scholar
Jervis, M. T. 1996 Some effects of surface tension on water waves and water waves at a wall. PhD thesis, University of Bristol, Bristol.Google Scholar
Lamb, H. 1913 On some cases of wave-motion on deep water. Ann. Mat. 3, xxi, 237.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Longuet-Higgins, M. S. 1963 Generation of capillary waves by steep gravity waves. J. Fluid Mech. 16, 138–159.CrossRefGoogle Scholar
Longuet-Higgins, M. S. & Cokelet, E. D. 1976 The deformation of steep surface waves on water. Part I. A numerical method of computation. Proc. R. Soc. Lond. A 350, 1–26.Google Scholar
Maleewong, M., Grimshaw, R. & Asavanant, J. 2005 Free surface flow under gravity and surface tension due to an applied pressure distribution. Part II. Bond number less than one-third. Eur. J. Mech. B 24, 502–521.CrossRefGoogle Scholar
Milewski, P. A. & Vanden-Broeck, J.-M. 1999 Time dependent gravity–capillary flows past an obstacle. Wave Mot. 29, 63–69.CrossRefGoogle Scholar
Moreira, R. M. 2001 Nonlinear interactions between water waves, free surface flows and singularities. PhD thesis, University of Bristol, Bristol.CrossRefGoogle Scholar
Perlin, M. & Schultz, W. W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241–274.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1986 Numerical Recipes – the Art of Scientific Computing. Cambridge University Press.Google Scholar
Rottman, J. W. & Olfe, D. B. 1979 Numerical calculations of steady gravity–capillary waves using an integro-differential formulation. J. Fluid Mech. 94, 777–793.CrossRefGoogle Scholar
Ruvinsky, K. D., Feldstein, F. I. & Freidman, G. I. 1991 Numerical simulation of the quasi-stationary stage of ripple excitation by steep gravity–capillary waves. J. Fluid Mech. 230, 339–353.CrossRefGoogle Scholar
Ruvinsky, K. D. & Freidman, G. I. 1981 The generation of capillary–gravity waves by steep gravity waves. Izv. Atmos. Ocean. Phys. 17, 548–553.Google Scholar
Schwartz, L. W. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary–gravity waves. J. Fluid Mech. 95, 119–139.CrossRefGoogle Scholar
Scullen, D. & Tuck, E. O. 1995 Nonlinear free-surface flow computations for a submerged cylinder. J. Ship Res. 39, 185–193.Google Scholar
Tanaka, M., Dold, J. W., Lewy, M. & Peregrine, D. H. 1987 Instability and breaking of a solitary wave. J. Fluid Mech. 185, 235–248.CrossRefGoogle Scholar
Tuck, E. O. 1965 The effect of nonlinearity at the free surface on flow past a submerged cylinder. J. Fluid Mech. 22, 401–414.CrossRefGoogle Scholar
Tyvand, P. A. & Miloh, T. 1995 a Free surface flow due to impulsive motion of a submerged circular cylinder. J. Fluid Mech. 286, 67–101.CrossRefGoogle Scholar
Tyvand, P. A. & Miloh, T. 1995 b Free surface flow generated by a small submerged circular cylinder starting from rest. J. Fluid Mech. 286, 103–116.CrossRefGoogle Scholar
Vanden-Broeck, J.-M. 2002 Wilton ripples generated by a moving pressure distribution. J. Fluid Mech. 451, 193–201.CrossRefGoogle Scholar
Whitham, G. B. 1999 Linear and Nonlinear Waves. John Wiley.CrossRefGoogle Scholar
Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688–700.CrossRefGoogle Scholar