A FULLY NONLINEAR BOUSSINESQ MODEL FOR WATER WAVE PROPAGATION

Hong-sheng Zhang, Hua-wei Zhou, Guang-wen Hong, Jian-min Yang

Abstract


A set of high-order fully nonlinear Boussinesq-type equations is derived from the Laplace equation and the nonlinear boundary conditions. The derived equations include the dissipation terms and fully satisfy the sea bed boundary condition. The equations with the linear dispersion accurate up to [2,2] padé approximation is qualitatively and quantitatively studied in details. A numerical model for wave propagation is developed with the use of iterative Crank-Nicolson scheme, and the two-dimensional fourth-order filter formula is also derived. With two test cases numerically simulated, the modeled results of the fully nonlinear version of the numerical model are compared to those of the weakly nonlinear version.

Keywords


Boussinesq type equations; fully nonlinear; numerical model; dissipation terms; comparison

References


Abbott, M.B., McCowan, A.D., Warren, I.R., 1984. Accuracy of short-wave numerical models, Journal of Hydraulic Engineering 110(10), 1287-1301. http://dx.doi.org/10.1061/(ASCE)0733-9429(1984)110:10(1287)

Agnon, Y., Madsen, P.A. Schäffer, H.A., 1999. A new approach to high-order Boussinesq models, Journal of. Fluid Mechanics 399, 319-333. http://dx.doi.org/10.1017/S0022112099006394

Berkhoff J. C. W., Booij, N., Radder, A. C., 1982. Verification of numerical wave propagation models for simple harmonic linear water waves, Coastal Eng., 6: 255-279. http://dx.doi.org/10.1016/0378-3839(82)90022-9

Chen, Y., Liu,P. L-F.,1995. Modified Boussinesq equations and associated parabolic models for water wave propagation, Journal of Fluid Mechanics 288, 351-381. http://dx.doi.org/10.1017/S0022112095001170

Gobbi, M. F., Kirby, J. T., Wei, G., 2000. A Fully Nonlinear Boussinesq model for Surface Waves, Part 2.Extension to O(kh)4. Journal of Fluid Mechanics 405, 181-210. http://dx.doi.org/10.1017/S0022112099007247

Hong Guangwen, 1997. High-Order Models of Nonlinear and Dispersive Wave in Water of Varying Depth with Arbitray Sloping Bottom, China Ocean Engineering, Vol.11, No.3, pp.243-260.

Ito, Y., Tanimoto, K., 1972. A method numerical analysis of wave propagation application to wave diffraction and refraction[C], Proc.13th Conf.Coastal Engineering, 503-522.

Madsen, P. A., Murray, R., Sørensen, O. R., 1991. A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engineering, 15, 371-388. http://dx.doi.org/10.1016/0378-3839(91)90017-B

Madsen, P.A., Bingham, H.B., Liu, H., 2002. A new Boussinesq method for fully nonlinear waves from shallow to deep water, Journal of Fluid Mechanics 462, 1-30. http://dx.doi.org/10.1017/S0022112002008467

Nwogu, O., 1993. Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, 119(6), 618-638. http://dx.doi.org/10.1061/(ASCE)0733-950X(1993)119:6(618)

Peregrine, D.H., 1967. Long waves on a beach slope, Journal of Fluid Mechanics, 27(4), 815-827. http://dx.doi.org/10.1017/S0022112067002605

Shapiro, R.1970. Smoothing filtering and boundary effects, Geophys. Space Phys., 8(2): 359-387. http://dx.doi.org/10.1029/RG008i002p00359

Shi, F. Y., Dalrymple, R. A., Kirby, J. T., et al., 2001. A fully nonlinear Boussinesq model in generalized curvilinear coordinates, Coastal Engineering, 42,337-358. http://dx.doi.org/10.1016/S0378-3839(00)00067-3

Wei, G., Kirby, J.T., 1995. Time-dependent numerical code for extended Boussinesq equations,Journal of Waterway, Port, Coastal, and Ocean Engineering 121(5), 251-261. http://dx.doi.org/10.1061/(ASCE)0733-950X(1995)121:5(251)

Wei,G., Kirby, J.T., Grilli, S.T., Subramanya, R., 1995. A fully nonlinear Boussinesq model for surface waves: Part 1. Highly nonlinear unsteady waves, J. Fluid Mechanics 294,71-92. http://dx.doi.org/10.1017/S0022112095002813

Wilmott, C.J., 1991. On the validation of models, Phys. Geogr. 2, 219-232.

Zhang, H.S., 2000. Numerical simulation of nonlinear wave propagation, PhD Dissertation, Hohai University, Nanjing, China(in Chinese with an English abstract).

Zhang, H.S., Hong, G.W., Ding, P.X., 2001. Numerical modeling of standing waves with threedimensional non-linear wave propagation model, China Ocean Engineering 15(4), 521-530.

Zhang, H. S., Zhu. L. S. and You, Y. X., 2005. A numerical model for wave propagation in curvilinear coordinates, Coastal Engineering, 52, 513-533. http://dx.doi.org/10.1016/j.coastaleng.2005.02.004

Zhang,H.S., Wang, W.Y., Feng,W.J., et al. 2010. A numerical model for nonlinear wave propagation on non-uniform current, China Ocean Engineering, Vol.24(1), 17-31.

Zhang, H. S., Yu, X.W., Yang, J.M., et al. 2009. Verification and application of the improved numerical model for nonlinear wave propagation[J], Chinese Journal of Hydrodynamics, 24(3):674-678(in Chinese).

Zou, Z.L., 1999. Higher order Boussinesq equations, Ocean Engineering 26, 767-792. http://dx.doi.org/10.1016/S0029-8018(98)00019-5


Full Text: PDF

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.