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A spectral method for Faraday waves in rectangular tanks

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Abstract

A theoretical study of Faraday waves in an ideal fluid is presented. A novel spectral technique is used to solve the nonlinear boundary conditions, reducing the system to a set of nonlinear ordinary differential equations for a set of Fourier coefficients. A simple weakly nonlinear theory is derived from this solution and found to capture adequately the behaviour of the system. Results for resonance in the full nonlinear system are explored in various depth regimes. Time-periodic solutions about the main (subharmonic) resonance are also studied in both the full and weakly nonlinear theories, and their stability calculated using Floquet theory. These are found to undergo several bifurcations which give rise to chaos for appropriate parameter values. The system is also considered with an additional damping term in order to emulate some effects of viscosity. This is found to combine the two branches of the periodic solutions of a particular mode.

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References

  1. Faraday M (1831) On a peculiar class of acoustic resonances. Proc R Soc A 121: 299–340

    Google Scholar 

  2. Miles J, Henderson D (1990) Parametrically forced surface waves. Annu Rev Fluid Mech 22(1): 143–165

    Article  MathSciNet  ADS  Google Scholar 

  3. Edwards WS, Fauve S (1994) Patterns and quasi-patterns in the Faraday experiment. J Fluid Mech 278: 123–148

    Article  MathSciNet  ADS  Google Scholar 

  4. Benjamin TB, Ursell F (1954) The stability of the plane free surface of a liquid in vertical periodic motion. Proc R Soc A 225(1163): 505–515

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Tadjbakhsh I, Keller JB (1960) Standing surface waves of finite amplitude. J Fluid Mech 8(03): 442–451

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. Generalis SC, Nagata M (1995) Faraday resonance in a two-liquid layer system. Fluid Dyn Res 15(3): 145–165

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Miles J (1984) Nonlinear Faraday resonance. J Fluid Mech 146: 285–302

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Li Y (2004) Chaos in Miles’ equations. Chaos Solitons Fractals 22(4): 965–974

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Ibrahim RA (2005) Liquid sloshing dynamics: theory and applications. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  10. Frandsen JB (2004) Sloshing motions in excited tanks. J Comput Phys 196(1): 53–87

    Article  ADS  MATH  Google Scholar 

  11. Cariou A, Casella G (1999) Liquid sloshing in ship tanks: a comparative study of numerical simulation. Mar Struct 12(3): 183–198

    Article  Google Scholar 

  12. Bredmose H, Brocchini M, Peregrine D, Thais L (2003) Experimental investigation and numerical modelling of steep forced water waves. J Fluid Mech 490: 217–249

    Article  ADS  MATH  Google Scholar 

  13. Faltinsen OM, Rognebakke OF, Lukovsky IA, Timokha AN (2000) Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. J Fluid Mech 407: 201–234

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Faltinsen O, Timokha A (2001) An adaptive multimodal approach to nonlinear sloshing in a rectangular tank. J Fluid Mech 432(1): 167–200

    ADS  MATH  Google Scholar 

  15. Ferrant P, Touze DL (2001) Simulation of sloshing waves in a 3D tank based on a pseudo-spectral method. In: Proceedings of 16th international workshop on water waves and floating bodies, 2001, Hiroshima, Japan, pp 3–6

  16. Ikeda T (2007) Autoparametric resonances in elastic structures carrying two rectangular tanks partially filled with liquid. J Sound Vibration 302(4–5): 657–682

    Article  ADS  Google Scholar 

  17. Gavrilyuk I, Lukovsky I, Trotsenko Y, Timokha a (2006) Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 1. Linear fundamental solutions. J Eng Math 54(1): 71–88

    Article  MathSciNet  MATH  Google Scholar 

  18. Kalinichenko V, Sekerzh-Zen’kovich S (2007) Experimental investigation of Faraday waves of maximum height. Fluid Dyn 42(6): 959–965

    Article  ADS  Google Scholar 

  19. Penney WG, Price AT (1952) Part II. Finite periodic stationary gravity waves in a perfect liquid. Philos Trans R Soc Lond Ser A 244(882): 254–284

    Article  MathSciNet  ADS  Google Scholar 

  20. Taylor GI (1953) An experimental study of standing waves. Proc R Soc Lond Ser A 218(1132): 44–52

    Article  ADS  Google Scholar 

  21. Forbes LK, Chen MJ, Trenham CE (2007) Computing unstable periodic waves at the interface of two inviscid fluids in uniform vertical flow. J Comput Phys 221(1): 269–287

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Forbes LK (2010) Sloshing of an ideal fluid in a horizontally forced rectangular tank. J Eng Math 66(4): 395–412

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen MJ, Forbes LK (2011) Accurate methods for computing inviscid and viscous Kelvin–Helmholtz instability. J Comput Phys 230(4): 1499–1515

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Dias F, Dyachenko A, Zakharov V (2008) Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Phys Lett A 372(8): 1297–1302

    Article  ADS  MATH  Google Scholar 

  25. Hill DF (2002) The Faraday resonance of interfacial waves in weakly viscous fluids. Phys Fluids 14(1): 158–169

    Article  MathSciNet  ADS  Google Scholar 

  26. MacLachlan NW (1964) Theory and application of Mathieu functions. Dover, New York

    Google Scholar 

  27. Abramowitz M, Stegun I (1970) Handbook of mathematical functions. Dover, New York

    Google Scholar 

  28. Weideman J, Reddy S (2000) A MATLAB differentiation matrix suite. ACM Trans Math Softw 26(4): 465–519

    Article  MathSciNet  Google Scholar 

  29. Kalinichenko VA, Sekerzh-Zen’kovich SY (2010) Breakdown of parametric fluid oscillations. Fluid Dyn 45(1): 113–120

    Article  Google Scholar 

  30. Dormand JR, Prince PJ (1980) A family of embedded Runge-Kutta formulae. J Comput Appl Math 6(1): 19–26

    Article  MathSciNet  MATH  Google Scholar 

  31. von Winckel G (2004) Legendre-Gauss quadrature weights and nodes. MathWorks File Exchange. http://www.mathworks.com/matlabcentral/fileexchange/4540

  32. Seydel R (2009) Practical bifurcation and stability analysis. Springer, Berlin

    Google Scholar 

  33. Tao Shi W, Goodridge C, Lathrop D (1997) Breaking waves: bifurcations leading to a singular wave state. Phys Rev E 56(4): 4157–4161

    Article  ADS  Google Scholar 

  34. Horn DA, Imberger J, Ivey GN, Redekopp LG (2002) A weakly nonlinear model of long internal waves in closed basins. J Fluid Mech 467: 269–287

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Schultz WW, Vanden-Broeck J (1998) Highly nonlinear standing water waves with small capillary effect. J Fluid Mech 369: 253–272

    MathSciNet  ADS  MATH  Google Scholar 

  36. Newhouse S, Ruelle D, Takens F (1978) Occurrence of strange Axiom A attractors near quasi periodic flows on T m, m ≥ 3. Commun Math Phys 40: 35–40

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, TAS, 7004, Australia

    David E. Horsley & Lawrence K. Forbes

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  1. David E. Horsley
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  2. Lawrence K. Forbes
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Correspondence to David E. Horsley.

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Horsley, D.E., Forbes, L.K. A spectral method for Faraday waves in rectangular tanks. J Eng Math 79, 13–33 (2013). https://doi.org/10.1007/s10665-012-9562-0

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  • DOI: https://doi.org/10.1007/s10665-012-9562-0

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