Numerical simulation of wave interaction with porous structures using an improved smoothed particle hydrodynamic method

A smoothed particle hydrodynamic (SPH) model is developed to simulate wave interaction with porous structures. The mean flow outside the porous structures is obtained by solving Reynolds Averaged Navier–Stokes (RANS) equations and the turbulence field is calculated by a large eddy simulation (LES) model. The porous flow is described by the spatially averaged Navier–Stokes type equations with the resistance effect of the porous media being represented by an empirical frictional source term. The interface boundaries between the porous flow and the outside flow are modeled by means of specifying a transition zone along the interface. The model is validated against other available numerical results and experimental data for wave damping over porous seabed with different levels of permeability. The validated model is then employed to investigate wave breaking over a submerged porous breakwater and good agreements between the SPH model results and the experimental data are obtained in terms of free surface displacement. In addition the predicted velocity, vorticity and pressure fields near the porous breakwater and in the breaking wave zone are also analyzed.     

Finite element analysis of flow control using porous media

A finite element model is established for simulating flow in and out a porous media. The extended Darcy equation inside the porous media and the Navier–Stokes equations in the fluid are coupled via the continuity condition at the interface between the two media. The model is firstly validated against the analytical and the numerical results available in literature. Then it is applied to simulate flow past a circular cylinder covered by a porous layer. The effect of the porous layer on the reduction of lift coefficient is investigated numerically. It is found that the lift reduction can be achieved by properly choosing the porous material. However, the amount of reduction greatly depends on the Reynolds number, the permeability and the Forchheimer coefficient.     

A coupled numerical model of wave interaction with porous medium

A new coupling model of wave interaction with porous medium is established in which the wave field solver is based on the two dimensional Reynolds Averaged Navier-Stokes (RANS) equations with a kε closure. Incident waves, which could be linear waves, cnoidal waves or solitary waves, are produced by a piston-type wave maker in the computational domain and the free surface is traced through the Piecewise Linear Interface Construction-Volume of Fluid (PLIC-VOF) method. Nonlinear Forchheimer equations are adopted to calculate the flow field within the porous media. By introducing a velocity–pressure correction equation, the wave field and the porous flow field are highly and efficiently coupled. The two fields are solved simultaneously and no boundary condition is needed at the interface of the internal porous flow and the external wave. The newly developed numerical model is used to simulate wave interaction with porous seabed and the numerical results agree well with the experimental data. The additional numerical tests are also conducted to study the effects of seabed thickness, porosity and permeability coefficient on wave damping and the pore water pressure responses.     

2-D numerical analysis of near-field flow at low-crested permeable breakwaters

This paper describes the capability of a numerical model named COrnell BReaking waves And Structures (COBRAS) [Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 359, 239–264; Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal and Ocean Engineering 125, 322–330, Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., Sakakiyama, T., 2000. A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions. Proc. Coastal Structures '99, 169–174] based on the Reynolds Averaged Navier–Stokes (RANS) equations to simulate the most relevant hydrodynamic near-field processes that take place in the interaction between waves and low-crested breakwaters. The model considers wave reflection, transmission, overtopping and breaking due to transient nonlinear waves including turbulence in the fluid domain and in the permeable regions for any kind of geometry and number of layers. Small-scale laboratory tests were conducted in order to validate the model, with different wave conditions and breakwater configurations. In the present study, regular waves of different heights and periods impinging on a wide-crested structure are considered. Three different water depths were tested in order to examine the influence of the structure freeboard. The experimental set-up includes a flow recirculation system aimed at preventing water piling-up at the lee of the breakwater due to overtopping. The applicability and validity of the model are examined by comparing the results of the numerical computations with experimental data. The model is proved to simulate with a high degree of agreement all the studied magnitudes, free surface displacement, pressure inside the porous structure and velocity field. The results obtained show that this model represents a substantial improvement in the numerical modelling of low-crested structures (LCS) since it includes many processes neglected previously by existing models. The information provided by the model can be useful to analyse structure functionality, structure stability, scour and many other hydrodynamic processes of interest.     

Incompressible SPH flow model for wave interactions with porous media

The paper presents an Incompressible Smoothed Particle Hydrodynamics (ISPH) method to simulate wave interactions with a porous medium. The SPH method is a mesh free particle modeling approach that is capable of tracking the large deformation of free surfaces in an easy and accurate manner. The ISPH method employs a strict incompressible hydrodynamic formulation to solve the fluid pressure and the numerical solution is obtained by using a two-step semi-implicit scheme. The ISPH flow model solves the unsteady 2D Navier–Stokes (NS) equations for the flows outside the porous media and the NS type model equations for the flows inside the porous media. The presence of porous media is considered by including additional friction forces into the equations. The developed ISPH model is first validated by the solitary and regular waves damping over a porous bed and the solitary wave interacting with a submerged porous breakwater. The convergence of the method and the sensitivity of relevant model parameters are discussed. Then the model is applied to the breaking wave interacting with a breakwater covered with a layer of porous materials. The computational results demonstrate that the ISPH flow model could provide a promising simulation tool in coastal hydrodynamic applications.     

Moving particle method for modeling wave interaction with porous structures

This paper presents a numerical model for simulating wave interaction with porous structures. Incompressible smoothed particle hydrodynamics in porous media (ISPHP) method is introduced in this study as a mesh free particle approach that is capable of efficiently tracking the large deformation of free surfaces in a Lagrangian coordinate system. The developed model solves two porous and pure fluid flows simultaneously by means of one equation that is equivalent to the unsteady 2D Navier–Stokes (NS) equations for the flows outside the porous media and the extended Forchheimer equation for the flows inside the porous media. Interface boundary between pure fluid and porous media is effectively modeled by the SPH integration technique. A two-step semi-implicit scheme is also used to solve the fluid pressure satisfying the fluid incompressibility criterion.The developed ISPHP model is then validated via different experimental and numerical data. Fluid flow pattern through porous dam with different porosities is studied and regular wave attenuation over porous seabed is investigated. As a practical case, wave running up and overtopping on a caisson breakwater protected by a porous armor layer are modeled. The results show good agreements between numerical and laboratory data in terms of free surface displacement, overtopping rate and pressure distribution. Based on this study, ISPHP model is an efficient method for simulating the coastal applications with porous structures.     

Numerical simulation of wave transformation and runup incorporating porous media wave absorber and turbulence models

A numerical wave tank is first established using the Navier–Stokes equations and the VOF method assuming laminar flow. The standard k–ε, realizable k–ε and RNG k–ε turbulent models are then incorporated to the numerical tank. An effective numerical method for wave absorption utilizing the energy-dissipating property of porous media is also included. To validate the accuracy of the proposed models, the propagation of a solitary wave, where analytical solution is available for comparison, is first simulated. This is followed by the simulation of irregular wave runup on a composite seawall, wave propagation over submerged bars and wave refraction and diffraction over an elliptic shoal, where experimental data are available for comparison. All computed results agree well with either the analytical solution or the experimental data.     

Numerical study on the characteristics of flow field and wave propagation near submerged breakwater on slope

In this study,characteristics of flow field and wave propagation near submerged breakwater on a sloping bed are investigated with numerical model. The governing equations of the vertical twodimensional model are Reynolds Averaged Navier Stokes equations. The Reynolds stress terms are closed by a nonlinear k ε turbulence transportation model. The free surface is traced through the PILC-VOF method. The proposed numerical model is verified with experimental results. The numerical result shows that the wave profile may become more asymmetrical when wave propa-gates over breakwater. When wave crest propagates over breakwater,the anticlockwise vortex may generate. On the contrary,when wave hollow propagates over breakwater,the clockwise vortex may generate. Meanwhile,the influenced zone of vortex created by wave crest is larger than that created by wave hollow. All the maximum values of the turbulent kinetic energy,turbulent dissi-pation and eddy viscosity occur on the top of breakwater. Both the turbulent dissipation and eddy viscosity increase as the turbulent kinetic energy increases. Wave energy may rapidly decrease near the breakwater because turbulent dissipation increases and energy in lower harmonics is transferred into higher harmonics.     

Three-dimensional interaction of waves and porous coastal structures using OpenFOAM®. Part I: Formulation and validation

In this paper and its companion (Higuera et al., 2014--this issue), the latest advancements regarding Volume-averaged Reynolds-averaged Navier–Stokes (VARANS) are developed in OpenFOAM® and applied. A new solver, called IHFOAM, is programmed to overcome the limitations and errors in the original OpenFOAM® code, having a rigorous implementation of the equations. Turbulence modelling is also addressed for k-ϵ and k-ω SST models within the porous media. The numerical model is validated for a wide range of cases including a dam break and wave interaction with porous structures both in two and three dimensions. In the second part of this paper the model is applied to simulate wave interaction with a real structure, using an innovative hybrid (2D–3D) methodology.     

A coupled model of submerged vegetation under oscillatory flow using Navier–Stokes equations

This work presents a new model for wave and submerged vegetation which couples the flow motion with the plant deformation. The IH-2VOF model is extended to solve the Reynolds Average Navier–Stokes equations including the presence of a vegetation field by means of a drag force. Turbulence is modeled using a k–ε equation which takes into account the effect of vegetation by an approximation of dispersive fluxes using the drag force produce by the plant. The plant motion is solved accounting for inertia, damping, restoring, gravitational, Froude–Krylov and hydrodynamic mass forces. The resulting model is validated with small and large-scale experiments with a high degree of accuracy for both no swaying and swaying plants. Two new formulations of the drag coefficient are provided extending the range of applicability of existing formulae to lower Reynolds number.     

Parametric study of the wave-induced residual liquefaction around an embedded pipeline

Wave-induced liquefaction in a porous seabed around submarine pipeline may cause catastrophic consequences such as large horizontal displacements of pipelines on the seabed, sinking or floatation of buried pipelines. Most previous studies in relation to the wave and seabed interactions with embedded pipeline dealt with the wave-induced instaneous seabed response and possible resulting momentary liquefaction (where the soil is liquefied instantaneously during the passage of a wave trough), using theory of poro-elasticity. Studies for the interactions between a buried pipeline and a soil undergoing build-up of pore pressure and residual liquefaction have been comparatively rare. In this paper, this complicated process was investigated by using a new developed integrated numerical model with RANS (Reynolds averaged Navier–Stokes) equations used for governing the incompressible flow in the wave field and Biot consolidation equations used for linking the solid–pore fluid interactions in a porous seabed with embedded pipeline. Regarding the wave-induced residual soil response, a two-dimensional poro-elastoplastic solution with the new definition of the source term was developed, where the pre-consolidation analysis of seabed foundation under gravitational forces including the body forces of a pipeline was incorporated. The proposed numerical model was verified with laboratory experiment to demonstrate its accuracy and effectiveness. The numerical results indicate that residual liquefaction is more likely to occur in the vicinity of the pipeline compared to that in the far-field. The inclusion of body forces of a pipeline in the pre-consolidation analysis of seabed foundation significantly affects the potential for residual liquefaction in the vicinity of the pipeline, especially for a shallow-embedded case. Parametric studies reveal that the gradients of maximum liquefaction depth with various wave and soil characteristics become steeper as pipeline burial depth decreases.     

Propagation of solitary waves over a submerged permeable breakwater

We study the interactions between a non-breaking solitary wave and a submerged permeable breakwater experimentally and numerically. The particle image velocimetry (PIV) technique is employed to measure instantaneous free surface displacements and velocity fields in the vicinity of a porous dike. The porous medium, consisting of uniform glass spheres, is mounted on the seafloor. Due to the limited size of each field of view (FOV) for high spatial resolution purposes, four FOVs are set in order to form a continuous flow field around the structure. Quantitative mean properties are obtained by ensemble averaging 30 repeated instantaneous measurements. The Reynolds decomposition method is then adopted to separate the velocity fluctuations for each trial to estimate the turbulent kinetic energy. In addition, a highly accurate two-dimensional model with the volume of fluid interface tracking technique is used to simulate an idealized volume-averaged porous medium. The model is based on the Volume-Averaged Reynolds Averaged Navier–Stokes equations coupled with the non-linear k–ε turbulence closure solver. Comparisons are performed between measurements and numerical results for the time histories of the free surface elevation recorded by wave gauges and the spatial distributions of free surface displacement with the corresponding velocity and turbulent kinetic energy around the permeable object imaged by the PIV system. Fairly good agreements are obtained. It is found that the measured and modeled turbulent intensities on the weather side are much larger than those on the lee side of the object, and that the magnitude of the turbulent intensity increases with increasing wave height of a solitary wave at a constant water depth. The verified numerical model is then used to estimate the energy reflection, transmission and dissipation using the energy integral method by varying the aspect ratio and the grain size of the permeable obstacle.     

Breaking solitary wave evolution over a porous underwater step

Solitary wave evolution over a shelf including porous damping is investigated using Volume-Averaged Reynolds Averaged Navier–Stokes equations. Porous media induced damping is determined based on empirical formulations for relevant parameters, and numerical results are compared with experimental information available in the literature. The aim of this work is to investigate the effect of wave damping on soliton disintegration and evolution along the step for both breaking and non-breaking solitary waves. The influence of several parameters such as geometrical configuration (step height and still water level), porous media properties (porosity and nominal diameter) or solitary wave characteristics (wave height) is analyzed. Numerical simulations show the porous bed induced wave damping is able to modify wave evolution along the step. Step height is observed as a relevant parameter to influence wave evolution. Depth ratio upstream and downstream of the edge appears to be the more relevant parameter in the transmission and reflection coefficients than porosity or the ratio of wave height–water depth. Porous step also modifies the fission and the solitary wave disintegration process although the number of solitons is observed to be the same in both porous and impermeable steps. In the absence of breaking, porous bed triggers a faster fission of the incident wave into a second and a third soliton, and the leading and the second soliton reduces their amplitude while propagating. This decrement is observed to increase with porosity. Moreover, the second soliton is released before on an impermeable step. Breaking process is observed to dominate over the wave dissipation at the porous bottom. Fission is first produced on a porous bed revealing a clear influence of the bottom characteristics on the soliton generation. The amplitude of the second and third solitons is very similar in both impermeable and porous steps but they evolved differently due to the effect of bed damping.     

Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part I. Solitary waves

Interactions between a solitary wave and a submerged rectangular obstacle are investigated both experimentally and numerically. The Particle Image Velocimetry (PIV) technique is used to measure the velocity field in the vicinity of the obstacle. The generation and evolution of vortices due to flow separation at the corners of the obstacle are recorded and analyzed. It is found that although the size of the vortex at the weatherside of the obstacle is smaller than that at the leeside, the turbulence intensity is, however, stronger. A numerical model, based on the Reynolds Averaged Navier–Stokes (RANS) equations with a k–ϵ turbulence model, is first verified with the measurements. Overall, the agreement between the numerical results and laboratory velocity measurements is good. Using the RANS model, a series of additional numerical experiments with different wave heights and different heights of the rectangular obstacle are then performed to test the importance of the energy dissipation due to the generation of vortices. The corresponding wave transmission coefficient, the wave reflection coefficient and the energy dissipation coefficient are calculated and compared with solutions based on the potential flow theory. As the height of the obstacle increases to D/h=0.7, the energy dissipation inside the vortices can reach nearly 15% of the incoming wave energy.     

Using RANS to simulate vortex generation and dissipation around impermeable submerged double breakwaters

In this paper, a Reynolds Averaged Navier–Stokes (RANS) model was developed to simulate the vortex generation and dissipation caused by progressive waves passing over impermeable submerged double breakwaters. The dynamics of the turbulence are described by introducing a k–ɛ model with Boussinesq closure. The Height Function (HF) is implemented to define the free-surface configuration. The governing equations are discretized by means of a finite volume method based on a staggered grid system with variable width and height. The feasibility of the numerical model was verified through a series of comparisons of numerical results with the existing analytical solutions and the experimental data. The good agreements demonstrate the satisfactory performance of the developed numerical model. The flow separation mechanism both near the upstream and the downstream edges of the obstacles demonstrates the physical and expected nature of development of the flow. The present model provides an accurate and efficient tool for the simulation of flow field and wave transformation near coastal structures without breaking.     

Surf zone characterization for integration with remote sensing data

A numerical model that solves the unsteady, incompressible, Reynolds averaged, Navier–Stokes equations has been utilized to simulate 57 cases of monochromatic, breaking waves over a sloping bed. The Volume of Fluid technique is used to track the complex, discontinuous free surface and the Renormalized Group turbulence model is used for closure. The model is validated by comparing predictions with Particle Image Velocimetry data and other empirical results. The model results are used to determine a relationship between the incipient wave breaking height and the maximum orbital velocity as well as a relationship between surf zone width and breaker type. Such expressions may be useful for remote sensing methods like Synthetic Aperture Radar to derive breaker height and classification from image data.     

Wave deformation and vortex generation in water waves propagating over a submerged dike

The Navier–Stokes equations and the exact free surface boundary conditions are solved to simulate wave deformation and vortex generation in water waves propagating over a submerged dike. Incident waves are generated by a piston-type wavemaker set up in the computational domain. Numerical results are compared with experimental data in order to confirm the validity of the numerical model. The fast Fourier transform and a wave resolution technique are applied to decompose the transformed waves and the higher harmonics. Effects of different parameters on wave transformation and vortex generation are studied systematically. These parameters include the Ursell number, the Keulegan–Carpenter number, the water depth ratio, the Reynolds number, the length aspect ratio of the dike, and the type of dike.     

Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part II: Cnoidal waves

Vortex generation and evolution due to flow separation around a submerged rectangular obstacle under incoming cnoidal waves is investigated both experimentally and numerically. The Particle Image Velocimetry (PIV) technique is used in the measurement. Based on the PIV data, a characteristic velocity, phrased in terms of incoming wave height, phase speed, dimension of the obstacle, and a local Reynolds number are proposed to describe the intensity of vortex. The numerical model, which solves the two dimensional Reynolds Averaged Navier Stokes (RANS) equations, is used to further study the effects of wave period on the vortex intensity. Measurements for the mean and turbulent velocity fields further indicate that the time history of the intensity of fluid turbulence is closely related to that of the vortex intensity.     

Propagation of a solitary wave over rigid porous beds

The unsteady two-dimensional Navier–Stokes equations and Navier–Stokes type model equations for porous flows were solved numerically to simulate the propagation of a solitary wave over porous beds. The free surface boundary conditions and the interfacial boundary conditions between the water region and the porous bed are in complete form. The incoming waves were generated using a piston type wavemaker set up in the computational domain. Accuracy of the numerical model was verified by comparing the numerical results with the theoretical solutions. The main characteristics of the flow fields in both the water region and the porous bed were discussed by specifying the velocity fields. Behaviors of boundary layer flows in both fluid and porous bed regions were also revealed. Effects of different parameters on the wave height attenuation were studied and discussed. The results of this numerical model indicate that for the investigated incident wave as the ratio of the porous bed depth to the fluid depth exceeds 10, any further increase of the porous bed depth has no effect on wave height attenuation.