Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-23T21:59:05.709Z Has data issue: false hasContentIssue false
  • English
  • Français

The motion of a freely floating cylinder in the presence of a wall and the approximation of resonances

Published online by Cambridge University Press:  19 April 2016

P. McIver  and
R. Porter
P. McIver*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK
R. Porter
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A linear theory, based on wide-spacing and high-frequency approximations, is developed to describe resonant behaviour in two-dimensional water-wave problems involving a freely floating half-immersed cylinder in the presence of a vertical rigid wall. The theory is not able to describe the lowest-frequency resonance, but otherwise yields explicit approximations for the locations of resonances in the complex plane and for their corresponding residues. Two problems are investigated in detail: the time-domain motion following a vertical displacement of the cylinder from equilibrium, and the time-harmonic motion of a cylinder that is excited by an incident plane wave.

JFM classification

Waves/Free-surface Flows: Wave scattering Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 795 , 25 May 2016 , pp. 581 - 610
Check for updates
Copyright
© 2016 Cambridge University Press 

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

Ananthakrishnan, P. 2015 Viscosity and nonlinearity effects on the forces and waves generated by a floating twin hull under heave oscillation. Appl. Ocean Res. 51, 138–152.Google Scholar
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansion of Integrals. Dover.Google Scholar
Evans, D. V. & Porter, R. 2007 Wave-free motions of isolated bodies and the existence of trapped modes. J. Fluid Mech. 584, 225–234.Google Scholar
Faltinsen, O. M., Rognebakke, O. F. & Timokha, A. N. 2007 Two-dimensional resonant piston-like sloshing in a moonpool. J. Fluid Mech. 575, 359–397.CrossRefGoogle Scholar
Fitzgerald, C. J. & McIver, P. 2009 Approximation of near-resonant wave motion using a damped harmonic oscillator model. Appl. Ocean Res. 31, 171–178.Google Scholar
Fredriksen, A. G., Kristiansen, T. & Faltinsen, O. M. 2015 Wave-induced response of a floating two-dimensional body with a moonpool. Phil. Trans. R. Soc. Lond. A 373 (2033), 20140109.Google Scholar
Greenhow, M. 1986 High- and low-frequency asymptotic consequences of the Kramers–Kronig relations. J. Engng Maths 20, 293–306.CrossRefGoogle Scholar
Hazard, C. & Lenoir, M. 1993 Determination of scattering frequencies for an elastic floating body. SIAM J. Math. Anal. 24 (6), 1458–1514.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.Google Scholar
John, F. 1949 On the motion of floating bodies, I. Commun. Pure Appl. Maths 2, 13–57.Google Scholar
Lenoir, M., Vullierme-Ledard, M. & Hazard, C. 1992 Variational formulations for the determination of resonant states in scattering problems. SIAM J. Math. Anal. 23 (3), 579–608.Google Scholar
Leppington, F. G. 1973 On the radiation and scattering of short surface waves. Part 2. J. Fluid Mech. 59, 129–146.Google Scholar
Lewandowski, E. M. 2008 Multi-vessel seakeeping computations with linear potential theory. Ocean Engng 35, 1121–1131.CrossRefGoogle Scholar
Linton, C. M. & Evans, D. V. 1992 The radiation and scattering of surface waves by a vertical circular cylinder in a channel. Phil. Trans. R. Soc. Lond. A 338, 325–357.Google Scholar
Linton, C. M. & McIver, P. 2001 Handbook of Mathematical Techniques for Wave/Structure Interactions. Chapman & Hall/CRC.CrossRefGoogle Scholar
Martin, P. A. 1985 Multiple scattering of surface water waves and the null-field method. In Proceedings of the 15th Symposium on Naval Hydrodynamics held in Hamburg, Germany, 3–7 September 1984, pp. 119–132. National Academy Press.Google Scholar
Maskell, S. J. & Ursell, F. 1970 The transient motion of a floating body. J. Fluid Mech. 44, 303–313.CrossRefGoogle Scholar
McIver, M. & McIver, P. 2011 Water waves in the time domain. J. Engng Maths 70, 111–128.Google Scholar
McIver, P. 1994 Low-frequency asymptotics of hydrodynamic forces on fixed and floating structures. In Ocean Waves Engineering (ed. Rahman, M.), pp. 1–49. Computational Mechanics Publications.Google Scholar
McIver, P. 2005 Complex resonances in the water-wave problem for a floating structure. J. Fluid Mech. 536, 423–443.CrossRefGoogle Scholar
McIver, P. 2012 Linear water waves: the horizontal motion of a structure in the time domain. J. Fluid Mech. 709, 289–312.CrossRefGoogle Scholar
McIver, P. 2014 An extended wide-spacing approximation for two-dimensional water-wave problems in infinite depth. Q. J. Mech. Appl. Maths 67 (3), 445–468.Google Scholar
Mei, C. C., Stiassnie, M. & Yue, D. K.-P. 2005 Theory and Application of Ocean Surface Waves. World Scientific.Google Scholar
Meylan, M. H. 2014 The time-dependent motion of a floating elastic or rigid body in two dimensions. Appl. Ocean Res. 46, 54–61.CrossRefGoogle Scholar
Meylan, M. H. & Fitzgerald, C. 2014 The singularity expansion method and near trapping of linear water waves. J. Fluid Mech. 755, 230–250.CrossRefGoogle Scholar
Meylan, M. H. & Tomic, M. 2012 Complex resonances and the approximation of wave forcing for floating elastic bodies. Appl. Ocean Res. 36, 51–59.Google Scholar
Newman, J. N. 1977 The motions of a floating slender torus. J. Fluid Mech. 83, 721–735.CrossRefGoogle Scholar
Porter, R.2008 The solution to water wave scattering and radiation problems involving semi-immersed circular cylinders. http://www.maths.bris.ac.uk/∼marp/abstracts/semicyl_techrep.html.Google Scholar
Porter, R. & Evans, D. V. 2011 Estimation of wall effects on floating cylinders. J. Engng Maths 70, 191–204.Google Scholar
Porter, R. & Evans, D. V. 2009 Water-wave trapping by floating circular cylinders. J. Fluid Mech. 633, 311–325.Google Scholar
Robertson, D. N. 1986 An extension to the short-wave asymptotics of the transmission coefficient for the semi-submerged circular cylinder. J. Fluid Mech. 166, 325–340.CrossRefGoogle Scholar
Simon, M. 1985 The high-frequency radiation of water waves by oscillating bodies. Proc. R. Soc. Lond. A 401, 89–115.Google Scholar
Stakgold, I. 2000 Boundary Value Problems of Mathematical Physics. SIAM.Google Scholar
Takács, G. & Roha’-Ilkiv, B. 2012 Model Predictive Vibration Control. Springer.Google Scholar
Ursell, F. 1961 The transmission of surface waves under surface obstacles. J. Fluid Mech. 57, 638–668.Google Scholar
Ursell, F. 1964 The decay of the free motion of a floating body. J. Fluid Mech. 19, 305–319.Google Scholar
Yeung, R. W. & Seah, R. K. M. 2007 On Helmholtz and higher-order resonance of twin floating bodies. J. Engng Maths 58, 251–265.Google Scholar