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

Madsen, Peter Hauge; Bingham, Harry B.; Liu, Hua

doi:10.1017/s0022112002008467

Cited by 302 publications

(286 citation statements)

References 43 publications

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“…Three-variable formulations, on the other hand, can have an infinite radius of convergence, as shown by Madsen and Agnon (2003). We focus here on the most accurate of these yet developed, that of Madsen, Bingham and Liu (2002). The Padé (4,4) version of this method is capable of accurately modelling nonlinear waves up to the point of breaking and out to relative water depths of approximately kd = 25, while accurate kinematics (the vertical variation of the flow) are obtained out to approximately kd = 10.…”

Section: Introductionmentioning

confidence: 99%

Nodal DG-FEM solution of high-order Boussinesq-type equations

et al. 2006

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Abstract. We present a discontinuous Galerkin finite element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one and two horizontal dimensions. The continuous equations are discretized using nodal polynomial basis functions of arbitrary order in space on each element of an unstructured computational domain. A fourth order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy and convergence of the model with both h (grid size) and p (order) refinement are verified for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar; and reflection of a steep solitary wave from a vertical wall. Test cases for two horizontal dimensions will be considered in a future paper.

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Section: Introductionmentioning

confidence: 99%

Nodal DG-FEM solution of high-order Boussinesq-type equations

et al. 2006

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“…Our main vehicle for establishing reference solutions for the transient wave problem is a numerical model, which solves the high-order Boussinesq formulation by Madsen, Fuhrman & Wang (2006), see also Madsen et al (2002Madsen et al ( , 2003. This method uses exact representations of the kinematic and dynamic free-surface conditions expressed in terms of surface velocities, and determines the vertical distribution of fluid velocity through a Padé-enhanced truncated series solution to the Laplace equation.…”

Section: Numerical Results and Comparisonsmentioning

confidence: 99%

“…The numerical solution procedure is based on finite-difference discretizations on an equidistant grid, and an explicit four-stage fourthorder Runge-Kutta scheme is used for the time integration. A detailed description of the scheme can be found in Madsen et al (2002) for one horizontal dimension. The examples presented in this section have been simulated using the following non-dimensional discretization: dx = 0.25-0.5 and dτ = 0.1-0.2 keeping the Courant number at Cr = dτ/dx = 0.4.…”

Section: Numerical Results and Comparisonsmentioning

confidence: 99%

“…As a step up in accuracy, various Boussinesq formulations are available, e.g. Peregrine (1967), Su & Gardner (1969), Madsen & Sørensen (1992), Nwogu (1993), and Madsen, Bingham & Liu (2002) and Madsen, Bingham & Schäffer (2003), and despite their very different levels of sophistication and accuracy with respect to dispersion and deep water capacities, they all simplify to the NSW equations in the dispersion-free shallow water limit.…”

Section: A New Characteristic Formulation Of the Unsteady Nsw Equationsmentioning

confidence: 99%

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Transient waves generated by a moving bottom obstacle: a new near-field solution

Madsen

Hansen

2012

J. Fluid Mech.

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We consider the classical problem of a single-layer homogeneous fluid at rest and a low, slowly varying, long and positive bottom obstacle, which is abruptly started from rest to move with a constant speed V. As a result a system of transient waves will develop, and we assume that locally in the region over the obstacle dispersion can be ignored while nonlinearity cannot. The relevant governing equations for the nearfield solution are therefore the nonlinear shallow water (NSW) equations. These are bidirectional and can be formulated in terms of a two-family system of characteristics. We analytically integrate and eliminate the backward-going family and achieve a versatile unidirectional single-family formulation, which covers subcritical, transcritical and supercritical conditions with relatively high accuracy. The formulation accounts for the temporal and spatial evolution of the bound waves in the vicinity of the obstacle as well as the development of the transient free waves generated at the onset of the motion. At some distance from the obstacle, dispersion starts to play a role and undular bores develop, but up to this point the new formulation agrees very well with numerical simulations based on a high-order Boussinesq formulation. Finally, we derive analytical asymptotic solutions to the new equations, providing estimates of the asymptotic surface levels in the vicinity of the obstacle as well as the crest levels of the leading non-dispersive free waves. These estimates can be used to predict the height and speed of the leading waves in the undular bores. The numerical and analytical solutions to the new single-family formulation of the NSW equations are compared to results based on the forced Korteweg-de Vries/Hopf equation and to numerical Boussinesq simulations.

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“…Later the linear model was extended to a non-linear model and effects of non-linearity were investigated (Imteaz & Imamura, 2001b). Madsen et al (2002) developed a model of multi-layered flow based on Boussinesq-type equations, which are suitable for shallow depth flow. Lynett and Liu (2004) developed another model of multi-layered flow using piecewise integration of Laplace equation for each individual layer and expanded the model for deep water.…”

Section: Wwwintechopencommentioning

confidence: 99%