Stephan T. Grilli

A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves

Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

A fully non-linear model for three-dimensional overturning waves over an arbitrary bottom

An accurate three-dimensional numerical model, applicable to strongly non-linear waves, is proposed. The model solves fully non-linear potential flow equations with a free surface using a higher-order three-dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second-order explicit Taylor series expansions with adaptive time steps. The model is applicable to non-linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16-node cubic ‘sliding’ quadrilateral elements, providing local inter-element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well-posed in all cases of mixed boundary conditions. Higher-order tangential derivatives, required for the time updating, are calculated in a local curvilinear co-ordinate system, using 25-node ‘sliding’ fourth-order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio-temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two-dimensional solution proposed earlier. Finally, three-dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright

Modeling of waves generated by a moving submerged body. Applications to underwater landslides

In this work, a boundary element model of an underwater landslide is developed. Careful gridding and time stepping techniques are demonstrated that produce highly accurate simulation results based on conservation of volume. A sensitivity analysis shows how to optimize simulation accuracy. The general techniques demonstrated herein are applicable to arbitrary moving submerged bodies.

An efficient boundary element method for nonlinear water waves

Abstract The paper presents a computational model for highly nonlinear 2-D water waves in which a high order Boundary Element Method is coupled with a high order explicit time stepping technique for the temporal evolution of the waves. The choice of the numerical procedures is justified from a review of the literature. Problems of the wave generation and absorption are investigated. The present method operates in the physical space and applications to four different wave problems are presented and discussed (space periodic wave propagation and breaking, solitary wave propagation, run-up and radiation, transient wave generation). Emphasis in the paper is given to describing the numerical methods used in the computation.

Development of a 3D numerical wave tank for modeling tsunami generation by underwater landslides

A three-dimensional (3D) numerical wave tank (NWT) solving fully nonlinear potential flow theory, with a higher-order boundary element method (BEM), is modified to simulate tsunami generation by underwater landslides. New features are added to the NWT to model underwater landslide geometry and motion and specify corresponding boundary conditions in the BEM model. In particular, a new snake absorbing piston boundary condition is implemented to remove reflection from the onshore and offshore boundaries of the NWT. Model results are favorably compared to recent laboratory experiments. Sensitivity analyses of numerical results to the width and length of the discretization are conducted, to determine optimal numerical parameters. The effect of landslide width on tsunami generated is estimated. Results show that the two-dimensional approximation is applicable when the ratio of landslide width over landslide length is greater than 2. Numerical accuracy is examined and found to be excellent in all cases.

Numerical Simulation of the 2011 Tohoku Tsunami Based on a New Transient FEM Co-seismic Source: Comparison to Far- and Near-Field Observations

In this work, we simulate the 2011 M9 Tohoku-Oki tsunami using new coseismic tsunami sources based on inverting onshore and offshore geodetic data, using 3D Finite Element Models (FEM). Such FEMs simulate elastic dislocations along the plate boundary interface separating the stiff subducting Pacific Plate from the relatively weak forearc and volcanic arc of the overriding Eurasian plate. Due in part to the simulated weak forearc materials, such sources produce significant shallow slip (several tens of meters) along the updip portion of the rupture near the trench. To assess the accuracy of the new approach, we compare observations and numerical simulations of the tsunamis far- and near-field coastal impact for: (i) one of the standard seismic inversion sources (UCSB; Shaoet al.2011); and (ii) the new FEM sources. Specifically, results of numerical simulations for both sources, performed using the fully nonlinear and dispersive Boussinesq wave model FUNWAVE-TVD, are compared to DART buoy, GPS tide gauge, and inundation/runup measurements. We use a series of nested model grids with varying resolution (down to 250 m nearshore) and size, and assess effects on model results of the latter and of model physics (such as when including dispersion or not). We also assess the effects of triggering the tsunami sources in the propagation model: (i) either at once as a hot start, or with the spatiotemporal sequence derived from seismic inversion; and (ii) as a specified surface elevation or as a more realistic time and space-varying bottom boundary condition (in the latter case, we compute the initial tsunami generation up to 300 s using the non-hydrostatic model NHWAVE). Although additional refinements are expected in the near future, results based on the current FEM sources better explain long wave near-field observations at DART and GPS buoys near Japan, and measured tsunami inundation, while they simulate observations at distant DART buoys as well or better than the UCSB source. None of the sources, however, are able to explain the largest runup and inundation measured between 39.5° and 40.25°N, which could be due to insufficient model resolution in this region (Sanriku/Ria) of complex bathymetry/topography, and/or to additional tsunami generation mechanisms not represented in the coseismic sources (e.g., splay faults, submarine mass failure). This will be the object of future work.

Corner problems and global accuracy in the boundary element solution of nonlinear wave flows

Abstract The numerical model for nonlinear wave propagation in the physical space, developed by Grilli, et al. 12,13 , uses a higher-order BEM for solving Laplaces equation, and a higher-order Taylor expansion for integrating in time the two nonlinear free surface boundary conditions. The corners of the fluid domain were modelled by double-nodes with imposition of potential continuity. Nonlinear wave generation, propagation and runup on slopes were successfully studied with this model. In some applications, however, the solution was found to be somewhat inaccurate in the corners and this sometimes led to wave instability after propagation in time. In this paper, global and local accuracy of the model are improved by using a more stable free surface representation based on quasi-spline elements and an improved corner solution combining the enforcement of compatibility relationships in the double-nodes with an adaptive integration which provides almost arbitrary accuracy in the BEM numerical integrations. These improvements of the model are systematically checked on simple examples with analytical solutions. Effects of accuracy of the numerical integrations, convergence with refined discretization, domain aspect ratio in relation with horizontal and vertical grid steps, are separately assessed. Global accuracy of the computations with the new corner solution is studied by solving nonlinear water wave flows in a two-dimensional numerical wavetank. The optimum relationship between space and time discretization in the model is derived from these computations and expressed as an optimum Courant number of ∼0.5. Applications with both exact constant shape waves (solitary waves) and overturning waves generated by a piston wavemaker are presented in detail.

Numerical modeling of wave breaking induced by fixed or moving boundaries

In this paper, several numerical aspects of an existing model for fully nonlinear waves are improved and validated to study wave breaking due to shoaling over a gentle plane slope and wave breaking induced by a moving lateral boundary.The model is based on fully nonlinear potential flow theory and combines a higher-order Boundary Element Method (BEM) for solving Laplaces equation at a given time and Lagrangian Taylor expansions for the time updating of the free surface position and potential. An improved numerical treatment of the boundary conditions at the intersection between moving lateral boundaries and the free surface (corner) is implemented and tested in the model, and the free surface interpolation method is also improved to better model highly curved regions of the free surface that occur in breaking waves. Finally, a node regridding technique is introduced to improve the resolution of the solution close to moving boundaries and in breaker jets.Examples are presented for solitary wave propagation, shoaling, and breaking over a 1:35 slope and for wave breaking induced by a moving vertical boundary. Using the new methods, both resolution and extent of computations are significantly improved compared to the earlier model, for similar computational efforts. In all cases computations can be carried out up to impact of the breaker jets on the free surface.

Numerical study of three-dimensional overturning waves in shallow water

Simulations in a three-dimensional numerical wave tank are performed to investigate the shoaling and breaking of solitary waves over a sloping ridge. The numerical model solves fully nonlinear potential flow equations with a high-order boundary-element method combined with an explicit time-integration method, expressed in a mixed Eulerian–Lagrangian formulation. Analyses of shoaling and breaking-wave profiles and kinematics (both on the free surface and within the flow) are carried out. It is observed that the transverse modulation of the ridge topography induces threedimensional effects on the time evolution, shape and kinematics of breaking waves. Comparisons of two- and three-dimensional results in the middle cross-section of the ridge, however, show remarkable similarities, especially for the shape and dynamics of the plunging jet.

Dual-reciprocity BEM based on global interpolation functions

Abstract Several global shape functions are introduced to interpolate the body-force term in the dual-reciprocity boundary-element method. These global-interpolation functions, which include polynomial, trigonometric, and hyperbolic series, can be used in place of the locally based radial shape function. For the several examples presented, the global functions have demonstrated superior convergence properties.