This paper presents a numerical model for simulating wave interaction with porous structures. The model calculates the mean flow outside of porous structures based on the Reynolds averaged Navier-Stokes equations. The corresponding turbulence field is modeled by an improved k-e model. The flow in porous structures is described by the spatially averaged Navier-Stokes equations. The drag forces caused by the presence of a solid skeleton are modeled by the empirical linear and nonlinear frictional forms. The numerical model is first calibrated by simple experiments for flow passing through a porous dam with different porous media. Excellent agreements are obtained for the case using gravels with mean sizes of O (1cm) to O (10cm) as the materials for the porous dam. Reasonably good agreements are also obtained when small uniform glass beads with diameters of 3 mm are used. The calibrated numerical model is then employed to investigate the breaking wave overtopping a caisson breakwater, protected by a layer of armor units. Good agreements between numerical results and laboratory data are obtained in terms of both free surface displacement and overtopping rate. Different design scenarios are also studied numerically. The porous armor layer is effective in reducing the overtopping rate as well as in preventing the caisson breakwater from bottom scouring.

This paper presents a combined experimental and numerical effort to study solitary wave runup and rundown on beaches. Both nonbreaking and breaking solitary waves are investigated. A two-dimensional numerical model that solves both mean flow and turbulence is employed in this study. For the nonbreaking solitary wave on a steep slope, numerical results of the present model are verified by experimental data and numerical results obtained from the boundary integral equation method model, in terms of both velocity distribution and free surface profiles. The characteristics of flow patterns during runup and rundown phases are discussed. The vertical variations of the horizontal velocity component are large at some instances, implying that the shallow water approximation may be inaccurate even for the nonbreaking wave runup and rundown. For the breaking solitary wave on a mild slope, numerical results of the present model are compared with experimental data for free surface displacements. The present model is found to be more accurate than the depthaveraged equations models. Using this numerical model, the mean velocity field and turbulence distribution under the breaking wave are discussed.

A two-dimensional computer model is developed to simulate free surface flow interaction with a moving body. The model is based on the cut-cell technique in a fixed-grid system. In this model, a body is approximated by the partial cell treatment (PCT), in which an irregular body is represented by the volumetric fraction of solid in Cartesian cells. The body motion is tracked by Lagrangian method whereas the fluid motion around the body is solved by Eulerian method. The concept of "locally relative stationary (LRS)" is introduced in this study. In the LRS method, a source term is added locally to the conventional continuity equation on body surfaces to take account of body motions, which subsequently affects the computational results of fluid pressure and flow velocity around the body. The LRS method is incorporated into an earlier Reynolds averaged Navier–Stokes (RANS) equations model developed by Lin and Liu [A numerical study of breaking waves in the surf zone. J Fluid Mech 1998; 359: 239-64]. The new model is capable of simulating generic turbulent free surface flows and their interaction with a moving body or multiple moving bodies. A series of numerical experiments have been conducted to verify the accuracy of the model for simulation of moving body interaction with a free surface flow. These tests include the generation of a solitary wave with the prescribed wave paddle movements, water exit and water impact and entry of a horizontal circular cylinder, fluid sloshing in a horizontally excited tank, and the acceleration/deceleration of an elliptical cylinder near a water surface. Excellent agreements are obtained when numerical results are compared to available analytical, experimental, and other numerical results. The model is a simple-to-implement computational tool for simulating a moving body in turbulent free surface flows.

A three-dimensional numerical model is developed in this study to investigate the problem of wave–current–body interaction. The model solves the spatially averaged Navier–Stokes equations. Turbulence effects are modeled by a subgrid-scale (SGS) model using the concept of large eddy simulation (LES). The model is employed to study the wave–current interaction with a square cylinder that is mounted on the bottom and vertically pierces the free surface. The force analysis demonstrates that the presence of waves can reduce both the strength and frequency of vortex shedding induced by a uniform current due to the nonlinear wave–current interaction. The free surface elevation, strain rates of the mean flow, and eddy viscosity are found to closely correlate with the mechanism of vortex shedding. It is also shown that when the vortex shedding is neglected in the calculation such as by the potential flow approach, one may significantly underestimate the magnitude of in-line force. The energy spectral analysis reveals that there exist initiating, growing, and decaying regions for shedding vortices around the cylinder. In the vortex initiating region, both coherent and turbulent structures are nearly two-dimensional that become three-dimensional in the vortex growing region. The kinetic energy of both coherent and turbulent motions is dissipated in the vortex decaying region, within which the mean flow gradually returns back to two-dimensional.

The flow motion of incompressible fluid can be described by Navier-Stokes equations with the continuity equation, which requires zero divergence of the velocity vector (i.e.,

In this paper, we develop an analytical technique in terms of series expansions to solve the mild-slope equation on an axisymmetric topography. This technique is applied to study the combined refraction and diffraction of plane monochromatic waves by a circular cylindrical island mounted on a paraboloidal shoal. By using the direct solution for the wave dispersion equation by Hunt [J. Waterw., Port, Coast., Ocean Div. Proc ASCE 4 (1979) 457], the mild-slope equation becomes explicit and it is then solved in terms of combined Fourier series and Taylor series. It is found that, to calculate the wave elevation along a perimeter with a specific radius, more terms in the Taylor series and angular modes in the Fourier series are needed for shorter waves. On the other hand, for the same incident wave, the outer the solutions are sought, more angular modes are needed to obtain the converged result of Fourier series. The comparison with the analytical solution based on the linear shallow-water equation by Homma [Geophys. Mag. 21 (1950) 199] is made for long wave incidence and excellent agreements are obtained. For long waves and waves in intermediate water depth, comparisons are made with other numerical results of the mild-slope equation and an equally good quality of agreement is achieved. D 2004 Elsevier B.V. All rights reserved.

A well-validated numerical model is employed to study solitary wave interaction with rectangular obstacles. The characteristics of wave transformation in terms of wave reflection, transmission, and dissipation (RTD) coefficients are investigated for various combination of obstacle length a and height b. Considering that a solitary wave will go through the fission process over a long obstacle or step, during which the wave profile continuously evolves that makes it difficult to define the transmission coefficient based on wave heights, we propose the integration of energy flux for the calculation of wave coefficients. A general integral energy equation is derived that serves as the basis of calculating RTD coefficients. This method is applied in this study for the obstacles with 0<b/h<1+2H/h and 0<a/h<l(where H is the wave height and h is the deep water depth), which cover the full range of structural type from a submerged obstacle to an emerged obstacle and from a thin plate to a step. For waves on steps, the present numerical results agree excellently with Lamb's [Lamb, H., 1932. Hydrodynamics, 6th Ed. Dover, New York] theory based on the long wave approximations and Seabra-Santos et al.'s [J. Fluid Mech. 176 (1987) 117] experimental data for both weakly nonlinear and fully nonlinear waves. The "edge-layer" theory developed by Sugimoto et al. [J. Phys. Soc. Jpn. 56 (1987) 1717], however, underestimates wave reflection significantly. For waves over obstacles, only the weakly nonlinear waves H/h=0.1 are considered. The RTD coefficients for different a/h and b/h are calculated and tabulated for the purpose of engineering application. The major differences between waves on a step and on a long obstacle are highlighted. The role of energy dissipation is explored and it is found that it can consume up to 60% of the total energy. The energy dissipation is mainly caused by vortex shedding and wave breaking that reduces wave transmission but has little impact on wave reflection.

In this paper, an analytical solution for linear long-wave reflection by an obstacle of general trapezoidal shape is explored. A closed-form expression in terms of first and second kinds of Bessel functions is obtained for the wave reflection coefficient, which depends on the relative lengths of the two slopes and top of the obstacle as well as the depth ratios in front of and behind the obstacle versus that above the obstacle. The analytical solution obtained in this study finds a few well-known analytical solutions to be its special cases, which include the wave reflection from a rectangular obstacle, an infinite step, and an infinite step behind a linear slope. The present analytical solution, however, covers a much wider range of problems. It is found that the periodicity of the wave reflection coefficient as the function of the relative length of the obstacle remains when two slopes are present but with a reduced magnitude. The phenomenon of zero wave reflection from the structure is special to a rectangular obstacle only, which disappears with the addition of a slope in front or at the rear. The new solution may be very useful in some engineering applications, for example, the design of a submerged breakwater of trapezoidal shape.

A three-dimensional numerical model has been developed to simulate stratied ows with free surfaces. The model is based on the Reynolds-averaged Navier-Stokes (RANS) equations with variable-uid density. The equations are solved in a transformed-coordinate system with the use of operatorsplitting method (Int. J. Numer. Meth. Fluids 2002; 38: 1045-1068). The numerical model is validated against the one-dimensional diusion problem and the two-dimensional density-gradient-ow. Excellent agreements are obtained between numerical results and analytical solutions. The model is then used to study transport phenomena of dumped sediments into a water body, which has been modelled as a strongly stratied-ow. For the two-dimensional problem, the numerical results compare well with experimental data in terms of mean particle falling velocity and spreading rate of the sediment cloud for both coarse and medium-size sediments. The model is also employed to study the dumping of sediments in a three-dimensional environment with the presence of free surface. It is found that during the descending process an annulus-like cloud is formed for ne sediments whereas a plate-like cloud for medium-size sediments. The model is proven to be a good tool to simulate strongly stratied free surface ows. Copyright.