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Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

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Abstract

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional (2D) tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bangfuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.

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References

  1. Abramson, H. N. The dynamic behavior of liquids in moving containers. National Aeronautics and Space Administration Special Reports, SP-106, Washington, D. C. (1966)

  2. Miles, J. W. Nonlinear surface waves in closed basins. Journal of Fluid Mechanics, 75, 419–448 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Miles, J. W. Internally resonant surface waves in a circular cylinder. Journal of Fluid Mechanics, 149, 1–14 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Miles, J. W. Resonantly forced surface waves in a circular cylinder. Journal of Fluid Mechanics, 149, 15–31 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hutton, R. E. An investigation of resonant nonlinear nonplannar free surface oscillations of a fluid. National Aeronautics and Space Administration Technical Notes, TN-D-1870, Washington, D. C. (1963)

  6. Valtinsen, O. M. A nonlinear theory of sloshing in rectangular tanks. Journal of Ship Research, 18, 224–241 (1974)

    Google Scholar 

  7. Waterhouse, D. D. Resonant sloshing near a critical depth. Journal of Fluid Mechanics, 281, 313–318 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Valtinsen, O. M. Multidimensional modal analysis of nonlinear sloshing in a rectangular tank with finite water depth. Journal of Fluid Mechanics, 407, 201–234 (2000)

    Article  MathSciNet  Google Scholar 

  9. Whitham, G. B. Non-linear dispersion of water waves. Journal of Fluid Mechanics, 27, 399–412 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Balk, A. M. A Lagrangian for water waves. Physics of Fluids, 8, 416–420 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. La Rocca, M., Mele, P., and Armenio, V. Variational approach to the problem of sloshing in a moving container. Journal of Theoretical and Applied Fluid Mechanics, 1, 280–310 (1997)

    Google Scholar 

  12. Sirovich, L. New Perspectives in Turbulence, Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  13. Sriram, V., Sannasiraj, S. A., and Sundar, V. Numerical simulation of 2D sloshing waves due to horizontal and vertical random excitation. Applied Ocean Research, 28, 19–32 (2006)

    Article  Google Scholar 

  14. Frandsen, J. B. Numerical bridge deck studies using finite elements, part I: flutter. Journal of Fluids and Structures, 19, 171–191 (2004)

    Article  Google Scholar 

  15. Lin, P. A fixed-grid model for simulation of a moving body in free surface flows. Computers and Fluids, 36, 549–561 (2007)

    Article  MATH  Google Scholar 

  16. Löhner, R., Yang, C., and Oñate, E. On the simulation of flows with violent free surface motion. Computer Methods in Applied Mechanics and Engineering, 195, 5597–5620 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, Y. H., Hwang, W. S., and Ko, C. H. Numerical simulation of the three-dimensional sloshing problem by boundary element method. Journal of the Chinese Institute of Engineers, 23, 321–330 (2000)

    Article  Google Scholar 

  18. Chen, B. F. Nonlinear hydrodynamic pressures by earthquakes on dam face with arbitrary reservoir shapes. Journal of Hydraulic Research, 32, 401–413 (1994)

    Article  Google Scholar 

  19. Chen, B. F. and Nokes, R. Time-independent finite difference analysis of fully non-linear and viscous fluid sloshing in a rectangular tank. Journal of Computational Physics, 209, 47–81 (2005)

    Article  MATH  Google Scholar 

  20. Chen, B. F. Viscous fluid in a tank under coupled surge, heave and pitch motions. Journal of Waterway, Port, Coastal, and Ocean Engineering, 131, 239–256 (2005)

    Article  Google Scholar 

  21. Wu, C. H. and Chen, B. F. Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method. Ocean Engineering, 36, 500–510 (2009)

    Article  Google Scholar 

  22. Lapidus, L. and Pinder, G. F. Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley and Sons, New York (1982)

    MATH  Google Scholar 

  23. Hoffman, J. D. Numerical Methods for Engineers and Scientists, McGraw-Hill Inc., New York (1993)

    Google Scholar 

  24. Nakayama, T. and Washizu, K. The boundary element method applied to the analysis of two dimensional nonlinear sloshing problems. International Journal for Numerical Methods in Engineering, 17, 1631–1646 (1981)

    Article  MATH  Google Scholar 

  25. Wu, G. X. Second order resonance of sloshing in a tank. Ocean Engineering, 34, 2345–2349 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of System Science and Applied Mathematics, School of Science, Kunming University of Science and Technology, Kunming, 650083, P. R. China

    Zhi-qiang Luo  (罗志强)

  2. School of Engineering Sciences, University of Southampton, Southampton, SO17 1BJ, UK

    Zhi-min Chen  (阵志敏)

Authors
  1. Zhi-qiang Luo  (罗志强)
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  2. Zhi-min Chen  (阵志敏)
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Correspondence to Zhi-qiang Luo  (罗志强).

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Luo, Zq., Chen, Zm. Sloshing simulation of standing wave with time-independent finite difference method for Euler equations. Appl. Math. Mech.-Engl. Ed. 32, 1475–1488 (2011). https://doi.org/10.1007/s10483-011-1516-6

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  • DOI: https://doi.org/10.1007/s10483-011-1516-6

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Chinese Library Classification

2010 Mathematics Subject Classification

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