Arbitrary Bottom Research Articles

Calculation of the Initial Elevation of the Water Surface at the Source of a Tsunami in a Basin with Arbitrary Bottom Topography

Calculation of the Initial Elevation of the Water Surface at the Source of a Tsunami in a Basin with Arbitrary Bottom Topography

Perturbation analysis of nonlinear partial reflected wave on a sloping bottom

In this paper, we use the perturbation method to develop a new mathematical derivation to describe nonlinear partial standing wave over uniformly sloping bottoms. In the Lagrangian coordinate system, the particle trajectories are obtained as a function of the nonlinear order parameter ɛ and the bottom slope α to the second order of the perturbation. The setups and mean return flow for an arbitrary bottom in the Lagrangian framework are also found as part of the solutions. The direct influences of reflection coefficient, wave steepness and bottom slope on the surface profiles of partially reflected waves are also derived. This nonlinear analytical solution is verified by reduction to the classical Stokes solution of progressive waves in both the deep water and constant depth limit. This solution allows to describe the successive deformation of partial reflected wave profiles and water particle trajectories prior to wave breaking to be described. The dynamic properties including mass transport, and Lagrangian mean level and radiation stress for nonlinear partially reflected waves on various sloping bottoms are investigated.

Nonlinear time‐domain wave‐structure interaction: A parallel fast integral equation approach

AbstractWe report on the development and validation of a new numerical wave tank (NWT) solving fully nonlinear potential flow (FNPF) equations, as a more efficient variation of Grilli et al.'s NWT [Grilli et al., A fully nonlinear model for three‐dimensional overturning waves over arbitrary bottom. Int J Numer Methods Fluids. 2001; 35:829‐867], which was successful at modeling many wave phenomena, including landslide‐generated tsunamis, rogue waves, and the initiation of wave breaking over slopes. This earlier NWT combined a three dimensional MII (mid‐interval interpolation) boundary element method (BEM) to an explicit mixed Eulerian–Lagrangian time integration. The latter was based on second‐order Taylor series expansions for the mesh geometry and Dirichlet free surface boundary condition for the potential, requiring high‐order derivatives to be computed in space and time. Here, to be able to solve large scale wave‐structure interaction problems for surface‐piercing bodies of complex geometry, of interest for ocean engineering and naval hydrodynamics applications, the NWT is reformulated to use cubic B‐spline meshes and the BEM solution is accelerated with a parallelized fast multipole method (FMM) based on ExaFMM, one of the fastest open source FMM to date. The NWT accuracy, convergence, and scaling are first assessed for simple cases, by comparing results with those of the earlier MII‐NWT as a function of mesh size and other model parameters. The relevance of the new NWT for solving the targeted applications is then demonstrated for surface piercing fixed cylinders, for which we show that results agree well with theoretical and experimental data for wave elevation and hydrodynamic forces.

Reduction of wave impact on seashore as well as seawall by floating structure and bottom topography

The three-dimensional problem involving diffraction of water wave by a finite floating rigid dock over an arbitrary bottom is studied for two cases (i) in the absence of wall (ii) in the presence of wall. The problem is handled for its solution with the aid of step method. Here both asymmetric and symmetric arbitrary bottom profile is approximated using successive steps. Step approximation helps to apply the matched eigenfunction expansion method, in result, system of algebraic equations are obtained which are solved to determine the hydrodynamic quantities, namely, force experienced by rigid floating dock as well as rigid seawall, free surface elevation, transmission & reflection coefficients associated with transmission & reflected waves respectively. The effects of various structural and system parameters are examined on these hydrodynamics quantities. The appropriate values of length and thickness of dock, water depth and angle of incidence provide the salient information to marine and coastal engineers to design the offshore structures and creation of parabolic trench on the bottom. The present results are compared with known results for special case of bottom topography. The energy balance relation is derived and checked.

Mass Transport in the Stokes Edge Wave for Constant Arbitrary Bottom Slope in a Rotating Ocean

Abstract The Lagrangian mass transport in the Stokes surface edge wave is obtained from the vertically integrated equations of momentum and mass in a viscous rotating ocean, correct to the second order in wave steepness. The analysis is valid for bottom slope angles β in the interval 0 < β ≤ π/2. Vertically averaged drift currents are obtained by dividing the fluxes by the local depth. The Lagrangian mean current is composed of a Stokes drift (inherent in the waves) plus a mean Eulerian drift current. The latter arises as a balance between the radiation stresses, the Coriolis force, and bottom friction. Analytical solutions for the mean Eulerian current are obtained in the form of exponential integrals. The relative importance of the Stokes drift to the Eulerian current in their contribution to the Lagrangian drift velocity is investigated in detail. For the given wavelength, the Eulerian current dominates for medium and large values of β, while for moderate and small β, the Stokes drift yields the main contribution to the Lagrangian drift. Because most natural beaches are characterized by moderate or small slopes, one may only calculate the Stokes drift in order to assess the mean drift of pollution and suspended material in the Stokes edge wave. The main future application of the results for large β appears to be for comparison with laboratory experiments in rotating tanks.

1022 底面地形の違いによる孤立波の砕波形状に関する実験的数値的研究(OS-5 可変形性境界を含む流れ(2))

1022 底面地形の違いによる孤立波の砕波形状に関する実験的数値的研究(OS-5 可変形性境界を含む流れ(2))

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 © 2001 John Wiley & Sons, Ltd.

Spectrum-Based Prediction Model for Random Wave Transformation over Arbitrary Bottom Topography

This paper proposes a one-dimensional and a horizontally two-dimensional random wave transformation models based on a spectral wave equation with a probabilistic bore-type energy dissipation term, and examines the validity of the wave models by comparing the predictions with the observations. The present spectrum-based prediction models with a probabilistic dissipation term axe rather heuristic but robust to use. Since the present random wave transformation models consist of spectral and probabilistic models, the wave model can be called as a hybrid random wave transformation model. The model solves the spatial evolution of complex Fourier amplitudes from which energy spectra are calculated. In addition, water surface elevations are obtained from the calculated complex amplitudes by using the inverse Fast Fourier Transform. Furthermore, representative wave heights and periods are obtained from the water surface elevations.We apply the one-dimensional wave model to predict transformations of double peak spectral waves, small and prototype scales' experimental waves and field waves over uniform slope and arbitrary bathymetry. In addition, the horizontally two-dimensional wave model is applied to predict transformations over an elliptic shoal, a conic shoal and field bathymetry, and also employed to investigate the mach-reflection. The comparison between the various observations by laboratory and field experiments and the predictions by the wave models shows good agreements.

A NUMERICAL MODEL FOR REFRACTION OF LINEAR AND CNOIDAL WAVES

A numerical model for refraction of linear and cnoidal waves over an arbitrary bottom is presented. The model, which is based on the ray theory of refraction (11), utilizes linear theory in deep and intermediate water and cnoidal theory in shallow water. The model permits one to determine nearshore wave properties using a nonlinear wave theory. Nearshore wave properties such as wave height, celerity, and wave angle determined using the linear-cnoidal refraction model may vary considerably from those determined using a linear refraction model. In general, cnoidal waves refract less, travel faster, and shoal higher than linear waves. This result can have a considerable effect on coastal engineering design, particularly in the areas of coastal structure design and longshore transport computations.

AN ENERGY REFERENCE LINE FOR DISSIPATIVE WATER WAVES ON A CURRENT

The energy principle known from steady hydraulics (i.e. the concept of an energy or total head line and a horizontal energy reference line) has been successfully extended to a dissipative current wave motion over an arbitrary bottom. The shear stress is assumed confined to a thin bottom layer, and the current is vertically uniform. However, it is allowed to have vorticity about vertical axes. The waves are described by a second order Stokian approach. The important new energy height equation states that the sum of a total current wave (energy) head and a current wave dissipation head is constant along a streamline. The total head is a sum of four terms: The mean/water surface height above datum, plus a current velocity head, plus a mean wave velocity head, and minus an interaction term head. The current-wave dissipation head is a sum of two terms: A current dissipation head minus a wave dissipation head. The usefulness of this new energy height equation, when calculating current depth refraction, is demon...

Surface Contours and Flow Pattern of a Perfectly Plastic Three-Dimensional Ice Sheet With Arbitrary Bottom and Edge Topography

AbstractThe differential equation determining the elevations of a perfectly plastic three-dimensional steady-state ice sheet is set up. Analytical solutions of the equation are obtained in two simple cases, viz. (1) an ice sheet on a horizontal base with an arbitrary curve as edge and (2) an ice sheet ona plane but sloping bed, with an edge composed of straight-line segments. The solutions are discussed in particular with reference to the development of ice divides and ice streams.

Surface Contours and Flow Pattern of a Perfectly Plastic Three-Dimensional Ice Sheet With Arbitrary Bottom and Edge Topography

Abstract The differential equation determining the elevations of a perfectly plastic three-dimensional steady-state ice sheet is set up. Analytical solutions of the equation are obtained in two simple cases, viz. (1) an ice sheet on a horizontal base with an arbitrary curve as edge and (2) an ice sheet ona plane but sloping bed, with an edge composed of straight-line segments. The solutions are discussed in particular with reference to the development of ice divides and ice streams.

Computation of Wave Heights Due to Refraction and Friction

The equations for the wave orthogonals and the wave heights are presented for depth refraction of regular, long-crested, small amplitude surface gravity waves over an arbitrary bottom, using time as the independent variable. The effect of turbulent bottom friction on the wave height is included. The computer outputs are in the form of automatically plotted wave orthogonals with the wave heights written at discrete points along these paths. Systematic tests are presented for water areas with straight and parallel bottom contours. The errors inherently connected with the introduction of a grid plus a set of formulas for the differentiation and interpolation are examined for a horizontal bottom with an infinite long sinusoidal threshold. For one plane sloping bottom, information is given for the influence of the initial curvature of the wave front, bottom roughness, and initial wave height.

Acoustic ray tracing on the general-purpose electronic-analog computer

A simulation of the trajectory of underwater acoustic rays is implemented on a general-purpose electronic-analog computer. Equations defining the phenomena are sum marized, assumptions are listed, and a preferred form of the equations is developed to provide maximum attain able accuracy of solution. Included are considerations of the velocity of sound as an arbitrary function of both depth and range, and the effects of reflections from both the surface and an arbitrary bottom. A complete computer diagram is presented and special circuits are described. Typical results of depth versus range are shown for vary ing launch depths and launch angles. A method of ob taining the difference in time required for rays leaving a common emitter at various angles to reach a common receiver is also derived. By use of perturbations, Taylor series expansions, and the method of steepest descents to insert the constraint cos θ/c = constant, the overall error in this highly sensitive computation has been reduced to 0.01 per cent or better.

Acoustic Ray Tracing on the General-Purpose Electronic-Analog Computer

A simulation of the trajectory of underwater acoustic rays is implemented on a general-purpose electronic-analog computer. Equations defining the phenomena are summarized, assumptions are listed, and a preferred form of the equations is developed to provide maximum attainable accuracy of solution. Included are considerations of the velocity of sound as an arbitrary function of both depth and range, and the effects of reflections from both the surface and an arbitrary bottom. A complete computer diagram is presented and special circuits are described. Typical results of depth versus range are shown for varying launch depths and launch angles. A method of obtaining the difference in time required for rays leaving a common emitter at various angles to reach a common receiver is also derived. By use of perturbations, Taylor series expansions, and the method of steepest descents to insert the constraint cos ?/c = constant, the overall error in this highly sensitive computation has been reduced to 0.01 per cent or better.

Underwater Acoustic Ray Analysis

A simulation of the trajectory of underwater acoustic rays was implemented on a general purpose electronic analog computer. Equations defining the phenomena are summarized, assumptions are listed and included, and a preferred form of the equations is developed to provide maximum attainable accuracy of solution. Included are considerations of the velocity of sound as an arbitrary function of both depth and range and the effects of reflections from the surface and an arbitrary bottom. Complete computer diagram is presented and special circuits are described. Typical results of depth versus range are shown for varying launch depths and launch angles. A method of obtaining the difference in time required for rays leaving a common emitter at various angles to reach a common receiver is also covered. Finally, provision for computing spreading loss as a function of range is described.