Computation of Nonlinear Free-Surface Flows by a Meshless Numerical Method
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 134, Issue 2
Abstract
A meshless numerical model for nonlinear free surface water waves is presented in this paper. Using the fundamental solution of the Laplace equation as the radial basis functions and locating the source points outside the computational domain, the problem is solved by collocation of boundary points. The present model is first applied to simulate the generation of periodic finite-amplitude waves with high wave steepness and then is employed to simulate the modulation of monochromatic waves passing over a submerged obstacle. Very good agreements are observed when comparing the present results with an analytical solution, experimental data, and other numerical results.
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References
Ata, R., and Soulaimani, A. (2005). “A stabilized SPH method for inviscid shallow water flows.” Int. J. Numer. Methods Fluids, 47, 139–159.
Beylkin, G., Coifman, R., and Rokhlin, V. (1991). “Fast wavelet transforms and numerical algorithms I.” Commun. Pure Appl. Math., 44, 141–183.
Cao, Y. S., Schultz, W. W., and Beck, R. F. (1991). “Three-dimensional desingularized boundary integral methods for potential problems.” Int. J. Numer. Methods Fluids, 12, 785–803.
Cooker, M. J., Peregrine, D. H., and Skovggard, O. (1990). “The interaction between a solitary wave and a submerged semicircular cylinder.” J. Fluid Mech., 215, 1–22.
Dommermuth, D. G., and Yue, D. K. P. (1987a). “A high-order spectral method for the study of nonlinear gravity waves.” J. Fluid Mech., 184, 267–288.
Dommermuth, D. G., and Yue, D. K. P. (1987b). “Numerical simulations of nonlinear axisymmetric flows with a free surface.” J. Fluid Mech., 178, 195–219.
Dong, C. M., and Huang, C. J. (2004). “Generation and propagation of water waves in a two-dimensional numerical viscous wave flume.” J. Waterway, Port, Coastal, Ocean Eng., 130(3), 143–153.
Du, C. J. (1999). “Finite-point simulation of steady shollow water flows.” J. Hydraul. Eng., 125(6), 621–630.
Du, C. J. (2000). “An element-free Galerkin method for simulation of stationary two-dimensional shallow water flows in rivers.” Comput. Methods Appl. Mech. Eng., 182, 89–107.
Fenton, J. D., and Rienecker, M. M. (1982). “A Fourier method for solving nonlinear water-wave problems—Application to solitary-wave interactions.” J. Fluid Mech., 120, 267–281.
Franke, C. (1982). “Scattered data interpolation: Test of some methods.” Math. Comput., 38, 181–200.
Golberg, M. A., and Chen, C. S. (1998). “The method of fundamental solutions for potential, Helmholtz and diffusion problems.” Boundary integral methods—Numerical and mathematical aspects, Computational Mechanics Publications, Southhampton, UK; Billercia, MA: WIT Press, 103–176.
Grilli, S. T., Guyenne, P., and Dias, F. (2001). “A fully nonlinear model for three-dimensional overturning waves over an arbitrary bottom.” Int. J. Numer. Methods Fluids, 35, 829–867.
Grilli, S. T., Skourup, J., and Svendsen, I. A. (1989). “An efficient boundary element method for nonlinear water waves.” Eng. Anal. Boundary Elem., 6, 97–107.
Hackbusch, W., and Nowak, Z. P. (1989). “On the fast matrix multiplication in the boundary element method by panel clustering.” Numerische Mathematik, 54, 463–491.
Hardy, R. L. (1971). “Multiquadric equations of topography and other irregular surfaces.” J. Geophys. Res., 76, 1905–1915.
Issacson, M. (1982). “Nonlinear wave effects on fixed and floating bodies.” J. Fluid Mech., 120, 267–281.
Kansa, J. E. (1990). “Multiquadrics—A scattered data approximation scheme with applications to computational fluid dynamics—II. Solutions to parabolic, hyperbolic and elliptic partial differential equations.” Comput. Math. Appl., 19, 127–145.
Longuet-Higgins, H. S., and Cokelet, E. D. (1976). “The deformation of steep waves on water. I. A numerical method of computation.” Proc. R. Soc. London, A350, 1–26.
Ma, Q. (2005). “Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems.” J. Comput. Phys., 205, 611–625.
Madsen, O. S. (1971). “On the generation of long waves.” J. Geophys. Res., 76, 8672–8683.
Moody, J., and Darken, C. (1989). “Fast-learning in networks of local tuned processing units.” Neural Comput., 1, 281–294.
Ohyama, T., Beji, S., Nadaoka, K., and Battjes, J. A. (1994). “Experimental verification of numerical model for nonlinear wave evolutions.” J. Waterway, Port, Coastal, Ocean Eng., 120, 637–644.
Ohyama, T., and Nadaoka, K. (1991). “Development of a numerical wave tank for analysis of nonlinear and irregular wave field.” Fluid Dyn. Res., 8, 231–251.
Rokhlin, V. (1985). “Rapid solution of integral equations of classical potential theory.” J. Comput. Phys., 60, 187–207.
Scullen, D., and Tuck, E. O. (1995). “Nonlinear free-surface flow computations for submerged cylinders.” J. Ship Res., 39, 185–193.
Tsai, W. T., and Yue, D. K. P. (1996) “Computation of nonlinear free-surface flows.” Annu. Rev. Fluid Mech., 28, 249–278.
Wang, Q. X. (2005). “Unstructured MEL modeling of nonlinear unsteady ship waves.” J. Comput. Phys., 210, 368–385.
West, B. J., Brueckner, K., Janda, R. S., Milder, D., and Milton, R. (1987). “A new numerical method for surface hydrodynamics.” J. Geophys. Res., [Oceans], 92, 11803–11824.
Wu, N. J., Tsay, T. K., and Young, D. L. (2006). “Meshless simulation for fully nonlinear water waves.” Int. J. Numer. Methods Fluids, 50, 219–234.
Young, D. L., Chen, K. H., and Lee, C. W. (2005). “Novel meshless method for solving the potential problems with arbitrary domain.” J. Comput. Phys., 209, 290–321.
Young, D. L., Jane, S. J., Lin, C. Y., Chiu, C. L., and Chen, K. C. (2004). “Solution of 2D and 3D Stokes law using multiquadratics method.” Eng. Anal. Boundary Elem., 28, 1233–1243.
Zhou, X., Hon, Y. C., and Cheung, K. F. (2004). “A grid-free, nonlinear shallow-water model with moving boundary.” Eng. Anal. Boundary Elem., 28, 967–973.
Zhu, Q., Liu, Y. M., and Yue, D. K. P. (2003). “Three-dimensional instability of standing waves.” J. Fluid Mech., 496, 213–242.
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© 2008 ASCE.
History
Received: Jan 31, 2006
Accepted: Mar 3, 2007
Published online: Mar 1, 2008
Published in print: Mar 2008
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