Computation of Nonlinear Free-Surface Flows by a Meshless Numerical Method

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.

[1]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[2]  Nan-Jing Wu,et al.  Meshless numerical simulation for fully nonlinear water waves , 2006 .

[3]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[4]  Azzeddine Soulaïmani,et al.  A stabilized SPH method for inviscid shallow water flows , 2005 .

[5]  D. L. Young,et al.  Novel meshless method for solving the potential problems with arbitrary domain , 2005 .

[6]  D. Yue,et al.  COMPUTATION OF NONLINEAR FREE-SURFACE FLOWS , 1996 .

[7]  Qiang Zhu,et al.  Three-dimensional instability of standing waves , 2003, Journal of Fluid Mechanics.

[8]  Ole Secher Madsen,et al.  On the generation of long waves , 1971 .

[9]  Qingwei Ma,et al.  Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems , 2005 .

[10]  Serdar Beji,et al.  Experimental Verification of Numerical Model for Nonlinear Wave Evolutions , 1994 .

[11]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[12]  Qian Wang,et al.  Unstructured MEL modelling of nonlinear unsteady ship waves , 2005 .

[13]  Kazuo Nadaoka,et al.  Development of a numerical wave tank for analysis of nonlinear and irregular wave field , 1991 .

[14]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[15]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[16]  Y. Hon,et al.  A grid-free, nonlinear shallow-water model with moving boundary , 2004 .

[17]  Chongjiang Du,et al.  An element-free Galerkin method for simulation of stationary two-dimensional shallow water flows in rivers , 2000 .

[18]  Ching-Jer Huang,et al.  Generation and propagation of water waves in a two-dimensional numerical viscous wave flume , 2004 .

[19]  Dick K. P. Yue,et al.  A high-order spectral method for the study of nonlinear gravity waves , 1987, Journal of Fluid Mechanics.

[20]  John D. Fenton,et al.  A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions , 1982, Journal of Fluid Mechanics.

[21]  Stephan T. Grilli,et al.  A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom , 2001 .

[22]  Dick K. P. Yue,et al.  Numerical simulations of nonlinear axisymmetric flows with a free surface , 1987, Journal of Fluid Mechanics.

[23]  D. L. Young,et al.  Solutions of 2D and 3D Stokes laws using multiquadrics method , 2004 .

[24]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[25]  Yusong Cao,et al.  Three‐dimensional desingularized boundary integral methods for potential problems , 1991 .

[26]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[27]  J. Dold,et al.  The interaction between a solitary wave and a submerged semicircular cylinder , 1990, Journal of Fluid Mechanics.

[28]  Michael Isaacson.  Nonlinear-wave effects on fixed and floating bodies , 1982 .

[29]  E. O. Tuck,et al.  Nonlinear free-surface flow computations for submerged cylinders , 1995 .

[30]  Stephan T. Grilli,et al.  An efficient boundary element method for nonlinear water waves , 1989 .

[31]  Bruce J. West,et al.  A new numerical method for surface hydrodynamics , 1987 .

[32]  Chongjiang Du,et al.  Finite-point simulation of steady shallow water flows , 1999 .

[33]  John Moody,et al.  Fast Learning in Networks of Locally-Tuned Processing Units , 1989, Neural Computation.

[34]  Michael Selwyn Longuet-Higgins,et al.  The deformation of steep surface waves on water - I. A numerical method of computation , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.