A three-dimensional numerical model has been developed to simulate stratified flows with free surfaces. The model is based on the Reynolds-averaged Navier-Stokes (RANS) equations with variable fluid density. The equations are solved in a transformed σ-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 diffusion problem and the two-dimensional density-gradient flow. 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 stratified flow. 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 fine sediments whereas a plate-like cloud for medium-size sediments. The model is proven to be a good tool to simulate strongly stratified free surface flows. Copyright

A three-dimensional numerical model based on the full Navier-Stokes equations (NSE) inσ-coordinate is developed in this study. The σ-coordinate transformation is first introduced to map the irregular physical domain with the wavy free surface and uneven bottom to the regular computational domain with the shape of a rectangular prism. Using the chain rule of partial differentiation, a new set of governing equations is derived in the σ-coordinate from the original NSE defined in the Cartesian coordinate. The operator splitting method (Li and Yu, Int. J. Num. Meth. Fluids 1996; 23: 485-501), which splits the solution procedure into the advection, diffusion, and propagation steps, is used to solve the modified NSE. The model is first tested for mass and energy conservation as well as mesh convergence by using an example of water sloshing in a confined tank. Excellent agreements between numerical results and analytical solutions are obtained. The model is then used to simulate two-and three-dimensional solitary waves propagating in constant depth. Very good agreements between numerical results and analytical solutions are obtained for both free surface displacements and velocities. Finally, a more realistic case of periodic wave train passing through a submerged breakwater is simulated. Comparisons between numerical results and experimental data are promising. The model is proven to be an accurate tool for consequent studies of wave-structure interaction. Copyright

A 3D multiple-layer r-coordinate model has been developed to simulate surface wave interaction with various types of structures including submerged structures, immersed structures, and floating structures. This model is the extension of the earlier model [Lin P, Li CW. A r-coordinate three-dimensional numerical model for surface wave propagation. Int J Numer Methods Fluid 2002; 38 (11): 1045-68] that solves Navier-Stokes equations in the transformed r-coordinate, which is especially efficient for simulation of wave propagation over varying topography. By introducing the layered r-coordinates, the present model overcomes the difficulty encountered by the earlier model in calculating waves past a depth discontinuity, e.g., a submerged rectangular breakwater. Furthermore, with the employment of 3-layer r-coordinate the present model is able to simulate flow interaction with an immersed body or a floating body. The new model is validated against an established Volume-Of-Fluid (VOF) model [Lin P, Liu PL-F. A numerical study of breaking waves in the surf zone. J Fluid Mech 1998; 359: 239-64] for the 2D solitary wave interaction with a submerged, immersed, or floating rectangular obstacle. For the solitary wave interaction with a submerged breakwater, the numerical results are also compared to the experimental data by Zhuang and Lee [A viscous rotational model for wave overtopping over marine structure. In Proc 25th Int Conf Coast Eng, ASCE, 1996. p. 2178-91] and very good agreements have been obtained for velocities in the vortex behind the structure. Finally, the present model is used to simulate 3D wave interaction with a Very Large Floating Structure (VLFS) above a submerged shoal. It is proved that the model is an accurate and efficient numerical tool to investigate different wave-structure interactions problems.

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.