A 2DH Post-Boussinesq Model for Weakly Nonlinear Fully Dispersive Water Waves (original) (raw)

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.

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.

A Highly Nonlinear Boussinesq Wave Model of Improved Dispersion Characteristics

In the present study a modified Boussinesq-type model is derived to account for propagation of regular and irregular waves in two horizontal dimensions. An improvement of the linear and nonlinear characteristics of the model is obtained by optimizing the coefficients of each term in the momentum equation, expanding in this way its applicability in very deep waters and thus overcoming a dominant short-coming of most likewise models. The values of the coefficients were obtained by an inverse method in such a way to satisfy exactly the dispersion relation in terms of both first and second order analyse. The modified model was applied to simulate the propagation of regular and irregular waves in one horizontal dimension, in a variety of bottom profiles, such as constant depth, mild slope, submerged obstacles. The simulations are compared with experimental data and analytical results, indicating very good agreement in most cases.

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.

Mathematical and numerical modelling of dispersive water waves

2018

profesores con los que realicé dos estancias de investigación, y contribuyeron en los resultados obtenidos en esta tesis doctoral y en mi formación. Todo mi agradecimiento a mi familia por su incondicional apoyo. Gracias a todos. • the vertical dimension of the domain H is small compared with respect to the wavelength L, that is µ 1 = H L 1; where n is the Gauckler-Manning coefficient [185]. The system is completed with the corresponding initial conditions and, in the case of bounded domains, with the corresponding boundary conditions. In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water whereas the phase velocity of the system (0.0.5) is given by C 2 SW E = gH. The previous relation, which is also called a linear dispersion relation, reveals the dispersive character of the linear water wave theory and that SWE cannot take into account effects associated with dispersive waves. This also explains why the computed numerical simulation in Figure A.2 is shifted, since the speed propagation of the system (0.0.5), C SW E , is faster than the given by the linear theory, C Airy. H 2 ∂ xxt (uH) − 1 6 H 3 ∂ xxt u − τ b. (0.0.3) An asymptotic analysis in the limit kH → 0 shows that the dispersion relation is exact at order O(kH) 4 for the Peregrine system when compared with Airy theory, and therefore it makes a great model for the simulation of long-waves. Alternatively, the development of non-hydrostatic pressure models for coastal water waves has been the topic of many studies over the past 30 years. Non-hydrostatic models are capable of solving many relevant features of coastal water waves, such as dispersion, non-linearity, shoaling, refraction, diffraction, and run-up. The central hypothesis in the derivation consists in splitting the pressure into a hydrostatic and a non-hydrostatic part (see Casulli [52]). In this thesis, the non-hydrostatic pressure system derived by Sainte-Marie et al. in [21] was numerically approximated. This system can be derived after a standard depth-averaging process from the Euler equations and assuming a constant properties was proposed in [118] and numerically approximated. Concerning the nature of the dispersive PDE systems presented above, it is well known that the system (0.0.5) consists on a hyperbolic PDE system. However, the nature of the system (0.0.7), among others Boussinesq and non-hydrostatic systems, differs from a hyperbolic system, and responds instead to a mixed hyperbolic and elliptic problem. Some discussion and theoretical results about existence and uniqueness can be found in [167]. matrices. Sometimes their analytic expression is not available, making Roe schemes computationally expensive. Also, they do not satisfy in general an entropy inequality, as a consequence, an entropy-fix technique has to be added to capture the entropy solution in the presence of smooth transitions (see [154]). It is also well known that the use of incomplete Riemann solvers as Rusanov, Lax-Friedrichs, HLL, etc. allows one to reduce the CPU time required by a Roe solver which resolves all the characteristic fields (see, for instance, [106]). Although when combined with piecewise constant approximation Roe solvers give in general a better resolution of the discontinuities than incomplete Riemann solvers when combined with high order reconstructions the resolution may be indistinguishable. Therefore high order methods based on incomplete Riemann solvers may be more efficient than high order Roe methods. In this thesis, we use a class of computationally fast first order finite volume solvers: Polynomial Viscosity Matrix (PVM) methods introduced by Castro et al. in [40]. This class of incomplete simple Riemann solvers can be applied for general nonconservative hyperbolic systems, defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe linearisation, that overcome these difficulties. PVM schemes can be seen as the natural extension of the one proposed in [84] for balance laws, and, more generally, for nonconservative systems. Moreover, PVM schemes can be extended to high order by following the ideas developed in [25] and to two-dimensional systems following [41], based on a polynomial reconstruction of states. Another class of high order numerical methods considered in this thesis are the related to the numerical computation of stationary solutions: standard methods that solve correctly systems of conservation laws can fail in solving nonconservative systems when approaching equilibria or near to equilibria solutions. In the context of shallow water equations Bermúdez and Vázquez-Cendón introduced in [14] the condition called C-property: a a scheme is said to satisfy this condition if it solves correctly the steady-state [27] to solve the 2D shallow water system. In these references, authors intend to make easier the exploitation of CUDA-enabled platforms to accelerate PDE-based numerical simulations, by providing the suitable CUDA C programming foundations. For this purpose, the authors explain the adaptation to GPU of the finite volume numerical scheme to solve the 2D shallow water system. The CUDA implementation of a first order two-layer shallow water system solver is addressed in [75]. There also exists proposals to implement, using CUDA-enabled GPUs, high order schemes to simulate one-layer systems [47], [131] and to implement first-order schemes for one and two-layer systems on triangular meshes [28]. In [80], several finite volume numerical schemes to solve one-layer shallow water system on structured meshes are tuned to exploit the GPU execution model and efficiently implemented using CUDA C. The distributed implementation of numerical solvers for

A Fully Dispersive Weakly Nonlinear Model for Water Waves

Proceedings of the Royal Society of London a Mathematical Physical and Engineering Sciences, 1997

A fully dispersive weakly nonlinear water wave model is developed via a new approach named the multiterm-coupling technique, in which the velocity field is represented by a few vertical-dependence functions having different wave-numbers. This expression of velocity, which is approximately irrotational for variable depth, is used to satisfy the continuity and momentum equations. The Galerkin method is invoked to obtain a solvable set of coupled equations for the horizontal velocity components and shown to provide an optimum combination of the prescribed depth-dependence functions to represent a random wave-field with diversely varying wave-numbers. The new wave equations are valid for arbitrary ratios of depth to wavelength and therefore it is possible to recover all the well-known linear and weakly nonlinear wave models as special cases. Numerical simulations are carried out to demonstrate that a wide spectrum of waves, such as random deep water waves and solitary waves over constant depth as well as nonlinear random waves over variable depth, is well reproduced at affordable computational cost.

A Fully-Dispersive Nonlinear Wave Model and its Numerical Solutions

Coastal Engineering 1994, 1995

A set of fully-dispersive nonlinear wave equations is derived by introducing a velocity expression with a few vertical-dependence functions and then applying the Galerkin method, which provides an optimum combination of the verticaldependence functions to express an arbitrary velocity field under wave motion. The obtained equations can describe nonlinear non-breaking waves under general conditions, such as nonlinear random waves with a wide-banded spectrum at an arbitrary depth including very shallow and far deep water depths. The single component forms of the new wave equations, one of which is referred to here as "time-dependent nonlinear mild-slope equation", are shown to produce various existing wave equations such as Boussinesq and mild-slope equations as their degenerate forms. Numerical examples with comparison to experimental data are given to demonstrate the validity of the present wave equations and their high performance in expressing not only wave profiles but also velocity fields.

A Fourier–Boussinesq method for nonlinear water waves

European Journal of Mechanics - B/Fluids, 2005

A Boussinesq method is derived that is fully dispersive, in the sense that the error of the approximation is small for all 0 kh < ∞ (k the magnitude of the wave number and h the water depth). This is made possible by introducing the generalized (2D) Hilbert transform, which is evaluated using the fast Fourier transform. Variable depth terms are derived both in mild-slope form, and in augmented mild-slope form including all terms that are linear in derivatives of h. A spectral solution is used to solve for highly nonlinear steady waves using the new equations, showing that the fully dispersive behavior carries over to nonlinear waves. A finite-difference-FFT implementation of the method is also described and applied to more general problems including Bragg resonant reflection from a rippled bottom, waves passing over a submerged bar, and nonlinear shoaling of a spectrum of waves from deep to shallow water.  2004 Elsevier SAS. All rights reserved.

A new Boussinesq method for fully nonlinear waves from shallow to deep water

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...

A double-layer Boussinesq-type model for highly nonlinear and dispersive waves

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2009

We derive and analyse, in the framework of the mild-slope approximation, a new double-layer Boussinesq-type model that is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential, thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. ( Agnon et al. 1999 J. Fluid Mech. 399 , 319–333) and enhanced by Madsen et al. ( Madsen et al. 2003 Proc. R. Soc. Lond. A 459 , 1075–1104), which consists of constructing infinite-series Taylor solutions to the Laplace equation, to truncate them at a finite order and to use Padé approximants, and the double-layer approach of Lynett & Liu ( Lynett & Liu 2004 a Proc. R. Soc. Lond. A 460 , 2637–2669), which allows lowering the order of derivatives. We formulate the model in terms of a static Dirichlet–Neumann operator translated from the free surface to the still-water level, and we derive...