Robert A. Dalrymple

Waves over Soft Muds: A Two-Layer Fluid Model

Abstract The problem of water waves propagating over a mud bottom, characterized as a laminar viscous fluid, is treated in several ways. First, two complete models are present, each valid for different lower (mud) layer depths, and second, a boundary layer model is presented as an appendix for the case where the lower layer is thick with respect to the boundary layer. These models are compared to the shallow water model and experimental results of Gade (1957, 1958) and agree well. The results show that extremely high wave attenuation rates are possible when the thickness of the lower layer is the same order as the internal boundary layer thickness and when the lower layer is thick.

State-of-the-art of classical SPH for free-surface flows

Smoothed Particle Hydrodynamics (SPH) is the most widely established mesh-free method which has been used in several fields as astrophysics, solids mechanics and fluid dynamics. In the particular case of computational fluid dynamics, the model is beginning to reach a maturity that allows carrying out detailed quantitative comparisons with laboratory experiments. Here the state-of-the-art of the classical SPH formulation for free-surface flow problems is described in detail. This is demonstrated using dam-break simulations in 2-D and 3-D. The foundations of the method will be presented using different derivations based on the method of interpolants and on the moving least-squares approach. Different methods to improve the classic SPH approach such as the use of density filters and the corrections of the kernel function and its gradient are examined and tested on some laboratory cases

Boussinesq modeling of a rip current system

In this study, we use a time domain numerical model based on the fully nonlinear extended Boussinesq equations [Wei et al., 1995] to investigate surface wave transformation and breaking-induced nearshore circulation. The energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum equations, with the viscosity strongly localized on the front face of the breaking waves. Wave run-up on the beach is simulated using a moving shoreline technique. We employ quasi fourth-order finite difference schemes to solve the governing equations. Satisfactory agreement is found between the numerical results and the laboratory measurements of Haller et al. [1997], including wave height, mean water level, and longshore and cross-shore velocity components. The model results reveal the temporal and spatial variability of the wave-induced nearshore circulation, and the instability of the rip current in agreement with the physical experiment. Insights into the vorticity associated with the rip current and wave diffraction by underlying vortices are obtained.

REFLECTION AND TRANSMISSION FROM POROUS STRUCTURES UNDER OBLIQUE WAVE ATTACK

The linear theory for water waves impinging obliquely on a vertically sided porous structure is examined. For normal wave incidence, the reflection and transmission from a porous breakwater has been studied many times using eigenfunction expansions in the water region in front of the structure, within the porous medium, and behind the structure in the down-wave water region. For oblique wave incidence, the reflection and transmission coefficients are significantly altered and they are calculated here. Using a plane-wave assumption, which involves neglecting the evanescent eigenmodes that exist near the structure boundaries (to satisfy matching conditions), the problem can be reduced from a matrix problem to one which is analytic. The plane-wave approximation provides an adequate solution for the case where the damping within the structure is not too great. An important parameter in this problem is Γ 2 = ω 2 h ( s - i f )/ g , where ω is the wave angular frequency, h the constant water depth, g the acceleration due to gravity, and s and f are parameters describing the porous medium. As the friction in the porous medium, f , becomes non-zero, the eigenfunctions differ from those in the fluid regions, largely owing to the change in the modal wavenumbers, which depend on Γ 2 . For an infinite number of values of ΓF 2 , there are no eigenfunction expansions in the porous medium, owing to the coalescence of two of the wavenumbers. These cases are shown to result in a non-separable mathematical problem and the appropriate wave modes are determined. As the two wavenumbers approach the critical value of Γ 2 , it is shown that the wave modes can swap their identity.

A parabolic equation for the combined refraction-diffraction of Stokes waves by mildly varying topography

A parabolic equation governing the leading-order amplitude for a forward-scattered Stokes wave is derived using a multiple-scale perturbation method, and the connection between the linearized version and a previously derived approximation of the linear mild slope equation is investigated. Two examples are studied numerically for the situation where linear refraction theory leads to caustics, and the nonlinear model is shown to predict the development of wave-jump conditions and significant reductions in amplitude in the vicinity of caustics.

SPHysics - development of a free-surface fluid solver - Part 1: Theory and formulations

A free-surface fluid solver called SPHysics is presented. Part 1 provides a description of the governing equations based on Smoothed Particle Hydrodynamics (SPH) theory. The paper describes the formulations implemented in the code including the classical SPH formulation along with enhancements like density filtering, arbitrary Lagrange-Euler (ALE) schemes and the incorporation of Riemann solvers for particle-particle interactions. Second-order time stepping schemes are presented along with the boundary conditions employed which can handle floating objects to study fluid-structure interaction. In addition, the model implementation is briefly described. This information will be used in Part 2, where the efficiency of the code is discussed, along with several study cases.

SPH on GPU with CUDA

A Smoothed Particle Hydrodynamics (SPH) method for free surface flows has been implemented on a graphical processing unit (GPU) using the Compute Unified Device Architecture (CUDA) developed by Nvidia, resulting in tremendous speed-ups. The entire SPH code, with its three main components: neighbor list construction, force computation, and integration of the equation of motion, is computed on the GPU, fully exploiting its computational power. The simulation speed achieved is one to two orders of magnitude faster than the equivalent CPU code. Example applications are shown for paddle-generated waves in a basin and a dam-break wave impact on a structure. GPU implementation of SPH permits high resolution SPH modeling in hours and days rather than weeks and months on inexpensive and readily available hardware.

Propagation of obliquely incident water waves over a trench

The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries ; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission ; however, reflection is not total owing to the finiteness of the obstacle. Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution. 1. Introduction The problem of the diffraction of incident waves by a finite obstacle in an otherwise infinite and uniform domain remains of general interest in linear wave theory. Several geometries of interest can be schematized by domains divided into separate regions by vertical geometrical boundaries, with the fluid depth being constant in each subdomain. Representative two-dimensional problems, with the boundary geometry uniform in the direction normal to the plane of interest, include those of elevated rectangular sills, fixed or floating rectangular obstacles at the water surface, and submerged trenches. The approach to the solution of problems of this type has typically been to construct solutions for each constant-depth subdomain in terms of eigenfunction expansions of the velocity potential ; the solutions are then matched at the vertical boundaries, resulting in sets of linear integral equations which must be truncated to a finite number of terms and solved numerically. One of the earliest solutions of this type was given by Takano (1960), who studied the cases of normal wave incidence on an elevated sill and fixed obstacle at the surface. In this study, we employ a modification of Takano’s method, discussed in

An approximate model for nonlinear dispersion in monochromatic wave propagation models

3. Newman (19653) also employed an integral-equation approach to study reflection and transmission of waves normally incident on a single step between finite- and infinite-depth regions. A variational approach, developed by Schwinger to study discontinuitiesin waveguides (see Schwinger & Saxon 1968) has been used by Miles (1967), to study Newman’s single-step problem, and by Mei & Black (1969), who studied the symmetric elevated sill and a floating rectangular cylinder. Lassiter (1 972), using the variational approach, studied waves normally incident on a rectangular trench where the water depths before and after the trench are constant but not necessarily equal, referred to here as the asymmetric case. Lee &

A fully nonlinear Boussinesq model in generalized curvilinear coordinates

Abstract A method is proposed for smoothly matching an approximate, shallow-water dispersion relation to an analytically obtained relation for intermediate and deep water. The method provides a correct limit for increasing water depth in the case of weakly non-linear waves, and provides a smooth prediction of wave parameters for the entire range of water depth. The model is applied to a parabolic equation form of the combined refraction-diffraction model, and numerical results are presented in comparison to published data.