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Nonlinear Modelling of Sloshing Waves and Solitary Waves in Shallow Basins

Authors: Mohammad R. Jalali, Mohammad M. Jalali


The earliest theories of sloshing waves and solitary waves based on potential theory idealisations and irrotational flow have been extended to be applicable to more realistic domains. To this end, the computational fluid dynamics (CFD) methods are widely used. Three-dimensional CFD methods such as Navier-Stokes solvers with volume of fluid treatment of the free surface and Navier-Stokes solvers with mappings of the free surface inherently impose high computational expense; therefore, considerable effort has gone into developing depth-averaged approaches. Examples of such approaches include Green–Naghdi (GN) equations. In Cartesian system, GN velocity profile depends on horizontal directions, x-direction and y-direction. The effect of vertical direction (z-direction) is also taken into consideration by applying weighting function in approximation. GN theory considers the effect of vertical acceleration and the consequent non-hydrostatic pressure. Moreover, in GN theory, the flow is rotational. The present study illustrates the application of GN equations to propagation of sloshing waves and solitary waves. For this purpose, GN equations solver is verified for the benchmark tests of Gaussian hump sloshing and solitary wave propagation in shallow basins. Analysis of the free surface sloshing of even harmonic components of an initial Gaussian hump demonstrates that the GN model gives predictions in satisfactory agreement with the linear analytical solutions. Discrepancies between the GN predictions and the linear analytical solutions arise from the effect of wave nonlinearities arising from the wave amplitude itself and wave-wave interactions. Numerically predicted solitary wave propagation indicates that the GN model produces simulations in good agreement with the analytical solution of the linearised wave theory. Comparison between the GN model numerical prediction and the result from perturbation analysis confirms that nonlinear interaction between solitary wave and a solid wall is satisfactorilly modelled. Moreover, solitary wave propagation at an angle to the x-axis and the interaction of solitary waves with each other are conducted to validate the developed model.

Keywords: Solitary Waves, nonlinearity, Green–Naghdi equations, Even harmonic components of sloshing waves

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[1] R. A. Ibrahim, Liquid sloshing dynamics theory and applications. Cambridge, Cambridge University Press, 2005, Introduction.
[2] M. R. Jalali, One-Dimensional and Two-Dimensional Green-Naghdi Equation Solvers for Shallow Flow over Uniform and Non-Uniform Beds. Edinburgh, The University of the Edinburgh, 2016, pp. 20–67.
[3] R. H. Stewart, Introduction to Physical Oceanography. Texas, Texas A & M University, 2008, pp. 273–280.
[4] A. E. Green and P. M. Naghdi. ‘‘Directed Fluid Sheets’’, Proc. of the Royal Society of London. Series A, Mathematical and Physical Sciences J., vol. 347, no. 1651, pp. 447–473, 1976.
[5] W. C. Webster and J. J. Shields, ‘‘Applications of high-level Green-Naghdi theory to fluid flow problems’’, W. G. Price, P. Temarel, and A. J. Keane, Eds, In Dynamics of Marine Vehicles and Structures in Waves, Elsevier Science, Amsterdam, 1991.
[6] Z. Demirbilek and W. C. Webster, Application of the Green-Naghdi theory of fluid sheets to shallow-water wave problems, Report 1, Model Development. US Army Engineers Waterways Experiment Station, Coastal Engineering Research Center, DC, Technical Report CERC–92–11, 1992.
[7] R. C. Ertekin, (1984) Soliton Generation by Moving Disturbances in Shallow Water: Theory, Computation and Experiment. California, University of California, 1984.
[8] J. J. Shields and W. C. Webster. ‘‘On direct methods in water-wave theory’’. Fluid Mechanics J., vol. 197: pp. 171–199, 1988.
[9] J. W. Kim and R. C. Ertekin, ‘‘A numerical study of nonlinear wave interaction in regular and irregular seas: irrotational Green-Naghdi model’’. Marine Structures J., vol. 13, no. 45, pp. 331–347, 2000.
[10] O. Le Métayer, S. Gavrilyuk, and S. Hank. ‘‘A numerical scheme for the Green-Naghdi model’’, Computational Physics J., vol. 229, no. 6, pp. 2034–2045, 2010.
[11] M. Hayatdavoodi and R. C. Ertekin. ‘‘Wave forces on a submerged horizontal plate. Part II: Solitary and cnoidal waves’’. Fluids and Structures J., vol. 54: pp. 580–596, April 2015.
[12] R. C. Ertekin, M. Hayatdavoodi, and J. W. Kim, ‘‘On some solitary and cnoidal wave diffraction solutions of the Green-Naghdi equations’’. Applied Ocean Research J., vol. 47: pp. 125–137, August 2014.
[13] B. B. Zhao, R. C. Ertekin, W. Y. Duan and M. Hayatdavoodi, (2014) ‘‘On the steady solitary-wave solution of the Green-Naghdi equations of different levels’’. Wave Motion J., vol. 51, no. 8, pp. 1382–1395, 2014.
[14] M. R. M. Haniffah, Wave Evolution on Gentle Slopes-Statistical Analysis and Green-Naghdi Modelling. Oxford, University of Oxford, 2013, pp. 73–92.
[15] M. R. Jalali and A. G. L Borthwick, ‘‘One-dimensional and two-dimensional Green–Naghdi equations for sloshing in shallow basins’’, Proc. Of the Inst. of Civil Engineers-Engineering and Computational Mechanics, vol. 170, no. 2: pp. 49–70, 2017.
[16] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes the art of scientific computing. Solution of linear algebraic equations. Cambridge, Cambridge University Press, 2007.
[17] T. Johannessen and C. Swan. ‘‘A laboratory study of the focusing of transient and directionally spread surface water waves’’. Proc. of the Royal Society of London. Series A, Mathematical, Physical and Engineering Sciences, vol. 457, no. 2008, pp. 971–1006, 2001.
[18] A. C. Hunt, P. H. Taylor, A. G. L. Borthwick, and P.K. Stansby. ‘‘Phase inversion and the identification of harmonic structure in coastal engineering experiments’’. in Pro. 29th Int. Conf. on Coastal Engineering, Lisbon, 2004, pp. 1047–1059.