A double-layer Boussinesq-type model for highly nonlinear and dispersive waves (original) (raw)
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Journal of Fluid Mechanics, 2002
A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed i...
Journal of Fluid Mechanics, 2003
The objective of this paper is to discuss and analyse the accuracy of various velocity formulations for water waves in the framework of Boussinesq theory. To simplify the discussion, we consider the linearized wave problem confined between the stillwater datum and a horizontal sea bottom. First, the problem is further simplified by ignoring boundary conditions at the surface. This reduces the problem to finding truncated series solutions to the Laplace equation with a kinematic condition at the sea bed. The convergence and accuracy of the resulting expressions is analysed in comparison with the target cosh-and sinh-functions from linear wave theory. First, we consider series expansions in terms of the horizontal velocity variable at an arbitrary z-level, which can be varied from the sea bottom to the still-water datum. Second, we consider the classical possibility of expanding in terms of the depth-averaged velocity. Third, we analyse the use of a horizontal pseudo-velocity determined by interpolation between velocities at two arbitrary z-levels. Fourth, we investigate three different formulations based on two expansion variables, being the horizontal and vertical velocity variables at an arbitrary z-level. This is shown to have a remarkable influence on the convergence and to improve accuracy considerably. Fifth, we derive and analyse a new formulation which doubles the power of the vertical coordinate without increasing the order of the horizontal derivatives. Finally, we involve the kinematic and dynamic boundary conditions at the free surface and discuss the linear dispersion relation and a spectral solution for steady nonlinear waves.
Fully Nonlinear Model for Water Wave Propagation from Deep to Shallow Waters
2011
A set of fully nonlinear Boussinessq-type equations (BTEs) with improved linear and nonlinear dispersive performance is presented. The equations are so that the highest order of the derivatives is three and they use the minimum number of unknowns: the free surface elevation and the horizontal velocity at a certain depth. The equations allow to reduce the errors both in linear frequency dispersion and shoaling below 0.30% for kh 5, and below 2.2% for kh 10, being k the wave number and h the water depth. The weakly nonlinear performance is also improved for kh 2. A simple fourth order explicit numerical scheme is presented so as to test the linear and nonlinear behavior of the model equations against analytical and experimental results.
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Numerical modeling of wave-ship interaction in shallow water over variable depth requires an accurate description of diffraction, refraction, reflection, and nonlinear wave-wave interaction. A computer program has been developed to solve time dependent Boussinesq-type hyperbolic long wave equations. The velocity at an arbitrary depth is expanded into an infinite series for the formulation of the extended Boussinesq equations. The numerical stability and dispersion characteristics are improved for increasing water depths. The partial differential equations are solved by using a fifth-order Adams-Bashforth-Moulton time marching multistep finite difference method. The results are compared with a second-order Crank Nicolson finite difference method and a Galerkin finite element method from previously published results. Results for linear and nonlinear waves are also compared with analytical and experimental data. The program will be integrated with a time-domain seakeeping program to si...
A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4
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A Boussinesq-type model is derived which is accurate to O(kh)4 and which retains the full representation of the fluid kinematics in nonlinear surface boundary condition terms, by not assuming weak nonlinearity. The model is derived for a horizontal bottom, and is based explicitly on a fourth-order polynomial representation of the vertical dependence of the velocity potential. In order to achieve a (4,4) Padé representation of the dispersion relationship, a new dependent variable is defined as a weighted average of the velocity potential at two distinct water depths. The representation of internal kinematics is greatly improved over existing O(kh)2 approximations, especially in the intermediate to deep water range. The model equations are first examined for their ability to represent weakly nonlinear wave evolution in intermediate depth. Using a Stokes-like expansion in powers of wave amplitude over water depth, we examine the bound second harmonics in a random sea as well as nonline...
Boussinesq Model for Weakly Nonlinear Fully Dispersive Water Waves
Journal of Waterway, Port, Coastal, and Ocean Engineering, 2009
In the present work a new post-Boussinesq type dispersive wave propagation model is proposed. It is developed for fully dispersive and weakly nonlinear irregular waves. The momentum equations include only one frequency dispersion term, expressed through convolution integrals, which are estimated using appropriate impulse functions. The model is applied to simulate the propagation of regular and irregular waves using a simple explicit scheme of finite differences. The results of the simulations are compared with experimental data. The comparisons show that the method is capable of simulating weakly nonlinear dispersive wave propagation over finite water depth, as well as breaking wave-induced currents, in a satisfactory way.
Higher-order Boussinesq equations for two-way propagation of shallow water waves
Standard perturbation methods are applied to Euler's equations of motion governing the capillary-gravity shallow water waves to derive a general higher-order Boussinesq equation involving the small-amplitude parameter, α = a/ h 0 , and long-wavelength parameter, β = (h 0 /l) 2 , where a and l are the actual amplitude and wavelength of the surface wave, and h 0 is the height of the undisturbed water surface from the flat bottom topography. This equation is also characterized by the surface tension parameter, namely the Bond number τ = Γ /ρgh 2 0 , where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity. The general Boussinesq equation involving the above three parameters is used to recover the classical model equations of Boussinesq type under appropriate scaling in two specific cases: (1) | 1 3 − τ | | β, and (2) | 1 3 − τ | = O(β). Case 1 leads to the classical (ill-posed and well-posed) fourth-order Boussinesq equations whose dispersive terms vanish at τ = 1 3. Case 2 leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [P. Daripa, W. Hua, A numerical method for solving an illposed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput. 101 (1999) 159–207] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. The relationship between the sixth-order Boussinesq equation and fifth-order KdV equation is also established in the limiting cases of the two small parameters α and β.
A 2DH Post-Boussinesq Model for Weakly Nonlinear Fully Dispersive Water Waves
Coastal Engineering 2008 - Proceedings of the 31st International Conference, 2009
In the present work a new post-Boussinesq type dispersive wave propagation model is proposed. It is developed for fully dispersive and weakly nonlinear irregular waves. The momentum equations include only one frequency dispersion term, expressed through convolution integrals, which are estimated using appropriate impulse functions. The model is applied to simulate the propagation of regular and irregular waves using a simple explicit scheme of finite differences. The results of the simulations are compared with experimental data. The comparisons show that the method is capable of simulating weakly nonlinear dispersive wave propagation over finite water depth, as well as breaking wave-induced currents, in a satisfactory way.
Computational models for weakly dispersive nonlinear water waves
Computer Methods in Applied Mechanics and Engineering, 1998
Numerical methods for the two and three dimensional Boussinesq equations governing weakly non-linear and dispersive water waves are presented and investigated. Convenient handling of grids adapted to the geometry or bottom topography is enabled by nite element discretization in space. Staggered nite di erence schemes are used for the temporal discretization, resulting in only two linear systems to be solved during each time step. E cient iterative solution of linear systems is discussed. By introducing correction terms in the equations, a fourth-order, two-level temporal scheme can be obtained. Combined with (bi-) quadratic nite elements the truncation errors of this scheme can be made of the same order as the neglected perturbation terms in the analytical model, provided that the element size is of the same order as the characteristic depth. We present analysis of the proposed schemes in terms of numerical dispersion relations. Veri cation of the schemes and their implementations is performed for standing waves in a closed basin with constant depth. More challenging applications cover plane incoming waves on a curved beach and earthquake induced waves over a shallow seamount. In the latter example we demonstrate a signi cantly increased computational e ciency when using higher order schemes and bathymetry adapted nite element grids.
Nonlinear progressive waves in water of finite depth — an analytic approximation
Coastal Engineering, 2007
An analytical solution using homotopy analysis method is developed to describe the non-8 linear progressive waves in water of finite depth. The velocity potential of the wave is ex-9 pressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by 10 continuous mapping. Unlike the perturbation method, the present approach is not dependent 11 on small parameters. Thus solutions are possible for steep waves. Furthermore, a significant 12 improvement of the convergence rate and region is achieved by applying Homotopy-Padé 13 Approximants. The calculated wave characteristics of the present solution agree well with 14 previous numerical and experimental results.