Studies Of Higher-order Boussinesq Equations

Boussinesq equations are studied through three aspects:theoretical models, the numerical schemes and applicability of numerical models.1?About Boussinesq theroies:(1)Based on the Boussinesq equations for rapidly varying depth, an enhanced Boussinesq model is derived by introducing additional terms with four parameters. The linear dispersion is accurate to Pade(4,4)expansion of linear Stokes's dispersion, the shoaling property is appilcalbe for water depth kh?6 and nonlinear property is accurate to kh?1.05 within 5%error.(2)The equations with two parameters, equations with six parameters and the enhanced Boussinesq equations are further enhanced, which enables them applicable to rapidly varying depth. Analytical solutions to these models and well-known Boussinesq models are derived for Bragg reflection case and the results are discussed.(3) The wave current interaction is anylazed when current is of order O(?1/2),and a Boussinesq model with moderate current is derived.(4)Another parameter is introduced to Beji and Nadaoka's equations, and the shoaling property of the new extended Boussinesq model is more accurate for water depth kh< 3.2?About numerical schemes in numerical modelsBased on the higher-order Boussinesq equations, numerical models are established under the unstaggered grids and the numerical models are solved by predictor-corrector scheme of finite difference method. During the solving process, three different schemes are used in time-marching:Crank-Nicolson scheme, Frog-loop scheme and a composite fourth order Adams-Bashforth-Moulton scheme. The different difference schemes are used in spatial derivatives. Source function and eddy visocity term are added to these numerical models for internal wave generation and simulating wave breaking, respectively. Comparions among numerical results upon wave propagation over a submerged breakwater are done, the effect of the different schemes are discussed.3?About the applicability of the numerical models(1) Numerical models are used to simulate nonbreaking waves propagation over a submerged breakwater, through the comparions between numerical results and experimental data, the effects nonlinear property and dispersion property are dicussed. Numerical simulations are also presented for breaking waves propagation over a submerged breakwater, through the comparisons between numerical results and experimental data, the four models with eddy viscosity terms are validated.(2)Numerical simulations based on improved enhanced Boussinesq model for rapidly varying topography, improved 6-parameter model and improved 2-parameter model are carried out to study Bragg reflection over sandbars and the numerical results are compared with the experimental data, the well agreement shows that the present three models can be applicable to varying topographies.(3)Boussinesq model with moderate current,2-parameter model and 6-parameter model are employed to numerically simulate wave-current interaction, and the applicability of these model is investigated considering different wave nonlinearity.(4)2-D models, i.e.,2-parameter model, Boussinesq model for rapidly topography and 6-parameter model are used to simulate wave propagation over three typical topographies and are validated through comparing numerical results against experimental data.