Volume-averaged Reynolds-averaged Navier-Stokes Equations Research Articles

Evaluating wave force and interface fluid velocity distribution at the vicinity of a breakwater by considering consolidation of porous media

Previous studies have successfully simulated the interaction between wave and porous media using volume-averaged Reynolds-averaged Navier-Stokes (VARANS) equations. However, the consolidation of porous media subject to wave loading affects the progressing wave, which is crucial to computation of wave force and wave-porous media interface fluid velocity. Research on this aspect is scarce. This study adopted the u -p approximation for Biot dynamic consolidation theory to quantify the consolidation of porous media. The continuity condition on the interfaces between the wave and porous seabed was considered to achieve their interaction. Coupling algorithms of wave and porous seabed were validated against experimental measurements in the literature. An engineering case of wave traveling over a breakwater was examined, and computation results of wave pressures and interface fluid velocities at the vicinity of a breakwater by VARANS equations and wave-porous seabed coupling algorithm of this study were compared to explore the influence of the consolidation effect on the progressing wave.

A numerical investigation of wave and current fields along bathymetry with porous media

A three-dimensional non-hydrostatic model based on Volume-Averaged, Reynolds-Averaged Navier–Stokes equations (VARANS) in the σ coordinate is presented. It is coupled with a non-linear k−ε turbulence model. The model's performance on wave interactions with porous structures is validated against laboratory data, including dam-break flows through a porous wall, solitary waves propagating over a submerged permeable obstacle, as well as periodic waves propagating over a submerged porous breakwater with wave breaking. Compared with models that use a volume-of-fluid approach, the high computational efficiency of the present model is demonstrated and stressed. Furthermore, the model is shown to be capable of simulating well wave transformation and wave-driven currents in a reef-lagoon-channel system, using the experimental data of Zheng et al. (2020). Next, the effects on wave-height, mean water level and wave-driven currents, caused by porous layers are studied. It is seen that the porous beds can reduce wave height about 43% on the reef flat, while wave setup is comparable (6% mean difference) between corresponding smooth and porous structures. Velocities of the wave-driven currents on the reef flat and the rip currents in the channel decrease by 16% and 21% with porous media. Moreover, results show only small discrepancies with varied porosities.

Application of a VARANS based resistance-type porosity model on simulating wave interactions with perforated caisson sitting on a rubble-mound foundation

Rubble-mound foundation can be regarded as a form of porous structure that is widely used in engineering practice. Porous medium flow must be considered when simulating wave interactions with perforated caisson sitting on a rubble-mound foundation. In this paper, volume-averaged/Reynolds averaged Navier-Stokes (VARANS) equations are adopted to describe the flow inside and outside the porous structure. Moreover, they are combined with the volume averaged k−ε model to simulate the turbulence effect. In addition, the three-step finite element method is applied to solve VARANS equations numerically, and the CLEAR-VOF method is applied to capture the free fluid surface. The resistance-type porosity numerical model is proposed and verified by several test cases and then used to simulate wave interactions with perforated caisson sitting on a rubble-mound foundation. The porosity model gives reliable results compared to the experiments with respect to the wave height, the pressure distribution, and the total vertical wave force. The model can be adopted to study the wave interaction with perforated caisson sitting on a rubble-mound foundation and used for the design of such caisson in engineering practice.

Characteristics of Breaking Wave Forces on Piles over a Permeable Seabed

Most offshore wind turbines are installed in shallow water and exposed to breaking waves. Previous numerical studies focusing on breaking wave forces generally ignored the seabed permeability. In this paper, a numerical model based on Volume-Averaged Reynolds Averaged Navier–Stokes equations (VARANS) is employed to reveal the process of a solitary wave interacting with a rigid pile over a permeable slope. Through applying the Forchheimer saturated drag equation, effects of seabed permeability on fluid motions are simulated. The reliability of the present model is verified by comparisons between experimentally obtained data and the numerical results. Further, 190 cases are simulated and the effects of different parameters on breaking wave forces on the pile are studied systematically. Results indicate that over a permeable seabed, the maximum breaking wave forces can occur not only when waves break just before the pile, but also when a “secondary wave wall” slams against the pile, after wave breaking. With the initial wave height increasing, breaking wave forces will increase, but the growth can decrease as the slope angle and permeability increase. For inclined piles around the wave breaking point, the maximum breaking wave force usually occurs with an inclination angle of α = −22.5° or 0°.

CFD/DEM coupled approach for the stability of caisson-type breakwater subjected to violent wave impact

Wave impacts on vertical caissons may cause breakwaters failures. This paper focuses on the analysis of the stability of breakwaters under violent wave impacts by using a triple-coupled Fluid-Porous-Solid model. The fluid model is described by the Volume-Averaged Reynolds-Averaged Navier-Stokes equations in which the nonlinear Forchheimer equations for the porous medium are added to the inertia terms. The solid model, based on the DDA method which is an implicit DEM method, has been used to analyze the movement and the stability of the caisson and armour units by taking into account the shapes of the armor units, as well as the contact between blocks. The developed model has been used for multiple purposes. Firstly, to estimate the variation of the maximum height of the impacting wave with the breakwater slope. A new formula has then been established for this purpose. Secondly, to analyze the influence of the porosity and of the thickness of the porous layer on the Turbulence Kinetic Energy (TKE) distribution around the breakwater structure. The results show that the higher the thickness, the lower the TKE intensity will be. Finally, the model has been used to analyze the stability of shaped armour units placed behind the caisson.

Development of An Integrated Numerical Model for Simulating Wave Interaction with Permeable Submerged Breakwaters Using Extended Navier–Stokes Equations

An integrated two-dimensional vertical (2DV) model was developed to investigate wave interactions with permeable submerged breakwaters. The integrated model is capable of predicting the flow field in both surface water and porous media on the basis of the extended volume-averaged Reynolds-averaged Navier–Stokes equations (VARANS). The impact of porous medium was considered by the inclusion of the additional terms of drag and inertia forces into conventional Navier–Stokes equations. Finite volume method (FVM) in an arbitrary Lagrangian–Eulerian (ALE) formulation was adopted for discretization of the governing equations. Projection method was utilized to solve the unsteady incompressible extended Navier–Stokes equations. The time-dependent volume and surface porosities were calculated at each time step using the fraction of a grid open to water and the total porosity of porous medium. The numerical model was first verified against analytical solutions of small amplitude progressive Stokes wave and solitary wave propagation in the absence of a bottom-mounted barrier. Comparisons showed pleasing agreements between the numerical predictions and analytical solutions. The model was then further validated by comparing the numerical model results with the experimental measurements of wave propagation over a permeable submerged breakwater reported in the literature. Good agreements were obtained for the free surface elevations at various spatial and temporal scales, velocity fields around and inside the obstacle, as well as the velocity profiles.

Investigation on A Resistance-Type Porosity Model and the Experimental Coefficients

The mathematical model based on the Volume-Averaged/Reynolds-Averaged Navier—Stokes (VARANS) equations has been adopted in recent years to generally simulate the interaction between waves and porous structures. However, it is still hard to determine the two experimental coefficients (α and β) included in VARANS equations. In the present study, VARANS equation is adopted to describe the flow inside and outside the porous structures uniformly, with applying the volume averaged k − e model to simulate the turbulence effect. A new calibration method is used to evaluate the accuracy of numerical simulation of wave motion on porous seabed with different coefficients, by taking the wave damping rate as an index. The calibration is achieved by completing a simulation matrix on two calibration cases. A region can be found in the parameter space to produce a lower error, which means that the simulation results are better consistent with the experimental results.

Reynolds stress modelling of flow in compound channels with vegetated floodplains

Flow in compound channel with vegetated floodplain is complex and efficient modelling should include the effects of vegetation on velocity, secondary flow and shear stress. In the present study, the Volume-averaged Reynolds-averaged Navier–Stokes equations, in conjunction with a Reynolds Stress turbulence model, are solved numerically for a non-symmetrical compound channel of a trapezoidal main channel and a vegetated floodplain. The numerical results agree well with available experimental data, while the model is capable to reproduce the evolution of vortices. The cross-sectional flow characteristics reveal the momentum exchange mechanism between main channel and floodplain due to increased shear stresses and turbulence anisotropy near the vegetation interface. Also, the recently improved analytical Shiono and Knight method [1991. Turbulent open-channel flows with variable depth across the channel. J Fluid Mech. 222:617–646] is applied for the determination of the depth-averaged velocity and shear stress, together with simple Manning calculations.

Numerical simulation of turbulent exchange flow in aquatic canopies

ABSTRACTIn the present study, lock-exchange flows through rigid emergent vegetation are investigated numerically. The volume-averaged Reynolds-averaged Navier–Stokes equations and the transport equations of the renormalization group k-ε turbulence model, which include additional terms due to vegetation, are solved numerically. The drag coefficient CD is estimated as a function of the local volume-averaged velocity, which varies in space and time. The dynamics of gravity currents in regions covered by vegetation (fully and partly) is examined. The solid volume fraction of vegetation ϕ ranges from 0.03 to 0.35. The numerical model is validated with available experimental data for sparse vegetation. The characteristics of gravity currents in dense vegetation, where there is no sufficient information, are investigated. Ιt is found that the toe velocity in the open area is influenced by the adjacent vegetation for ϕ > 0.104. The currents become drag dominated from the beginning of the exchange flow for ϕ > 0.141.

2-D integrated numerical modeling for the potential of solitary wave-induced residual liquefaction over a sloping porous seabed

A 2-D integrated numerical model is developed for liquefaction due to the build-up of pore pressure in porous sloping seabed subject to solitary wave loading. In the integrated 2-D model, the propagation of a solitary wave over a porous sloping seabed is governed using the volume-averaged Reynolds averaged Navier–Stokes equations, in which discontinuity of the flow (i.e., wave breaking due to shoaling, hydraulic jump during the wave drawdown phase) can be captured with $$k-\epsilon $$ model, while Biot’s consolidation equations are used for linking the solid–pore fluid interaction. Regarding the wave-induced residual soil response, a new 2-D pore pressure build-up model is developed with the new definition of the source term where the phase-resolved oscillatory shear stress is involved. The initial consolidation state of the sloping seabed foundation is considered under hydrostatic load using theory of poro-elasticity. The numerical results indicate that compared to a 1:6 slope, the wave-breaking process is more likely to occur in the case of a mild 1:15 slope due to wave shoaling and that a mild 1:15 slope experiences a longer duration of the wave run-up and drawdown compared to that in a steep 1:6 slope. Furthermore, the results suggest that the potential for liquefaction first occurs near the intersection between the initial shore line and the bed boundary. Then it will be extended both laterally and vertically to the neighboring points. The depth of the liquefaction zone increases and the width of the liquefaction zone decreases as the bed slope increases. Parametric studies indicate that the build-up of residual pore pressure can accumulate to a large value in the case of soil with lower permeability and lower relative density under larger wave loading.

Non-hydrostatic modeling of wave interactions with porous structures

This paper presents a three-dimensional non-hydrostatic wave model NHWAVE for simulating wave interactions with porous structures. The model calculates the porous media flow based on well-balanced volume-averaged Reynolds-averaged Navier–Stokes equations (VARANS) in σ coordinate. The turbulence field within the porous structures is simulated by an improved k−ϵ model. To account for the temporally varying porosity at the grid cell center introduced by the variation of free surface elevation, the porosity is updated at each time step in the computation. The model is calibrated and validated using a wide range of laboratory measurements, involving dam-break flow through porous media, 3D solitary wave interactions with a porous structure, 2D solitary wave interactions with a submerged permeable obstacle as well as periodic wave breaking over a submerged porous breakwater with steep slopes. The model is shown to be capable of well simulating wave reflection, diffraction, wave breaking and wave transmission through porous structures, as well as the turbulent flow field around the permeable structures. It is also demonstrated in the paper that the current non-hydrostatic model is computationally much more efficient than the existing porous flow models based on VOF approach. The non-hydrostatic wave model NHWAVE can be a useful tool for studying wave–structure interactions.

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.

Numerical simulation of three-dimensional breaking waves on a gravel slope using a two-phase flow Navier–Stokes model

Wave breaking is mainly a three-dimensional flow problem characterized by wave energy dissipation due to turbulence. The understanding of the wave breaking mechanism on a beach is essential in studying coastal processes. The complexity of the wave-induced turbulence flow is also increased by the presence of a two-phase flow, which introduces buoyancy effects. In this work a set of numerical experiments is carried out on wave breaking on a gravel slope. The influence of a one-phase and two-phase flow and the permeability effect of the beach are investigated numerically by means of a Navier–Stokes model known as IH-3VOF, which considers the volume-averaged Reynolds-averaged Navier–Stokes (VARANS) equations (del Jesus, 2011 [3]) to characterize the flow within the porous media. The accuracy of the VARANS equations is demonstrated by means of comparisons with laboratory data. The results are found to be within a 2% error in terms of wave height prior to the broken wave, and up to a 10% error after then, and in the order of 0.20 s in the time domain for the worst case. A further analysis of wave evolution on a permeable beach with alongshore variation of porosity is studied. Three-dimensional wave breaking and post-breaking wave transformations alongshore are analysed according to porosity values.

An integrated model for the wave-induced seabed response around marine structures: Model verifications and applications

In this study, an integrated model (PORO-WSSI II) for wave–seabed–structure interactions (WSSI) is developed by combining (i) the Volume-Averaged Reynolds-Averaged Navier–Stokes (VARANS) equations for wave motions in a fluid domain and the porous media flows in porous structures, and (ii) the dynamic Biot's equations for a porous elastic seabed. The effects of the porous flow in seabed and marine structures, and the fluid exchange at the interface between the fluid domain and solid domain on wave characteristics are considered. The finite difference two-step projection method and the forward time difference method are adopted to solve the VARANS equations. The volume of fluid (VOF) method is applied to track water free-surface. The finite element method and the Generalized Newmark method are respectively adopted for the space discretization and time discretization of the dynamic Biot's equations. A one-way integrating method is developed to integrate the VARANS equations with the dynamic Biot's equations. Several experimental data available in the literature are used to validate the integrated model. An overall agreement between the numerical results and the experiment data indicates that the integrated model developed for the WSSI problem is highly reliable. The integrated model is then applied to investigate the dynamic response of a large-scale composite breakwater on a seabed, and the mechanism of WSSI. Numerical results indicate that there are intensive fluid exchanges between the water body and the seabed and strong seepage forces in the seabed under the ocean wave loading. The excessive upward seepage force leads to the liquefaction of the seabed in the region under wave trough. There is a liquefaction zone in the seabed close to the bottom corner of the rubble mound, which may lead to foundation instability of the composite breakwater. The parametric study indicates that the wave characteristics have a significant impact on the liquefaction properties (depth, width and area).

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.

A numerical model for wave motions and turbulence flows in front of a composite breakwater

A mathematical model based on the Volume-Averaged/Reynolds Averaged Navier-Stokes (VARANS) equations is developed to describe surface wave motions in the vicinity of a coastal structure, which could be either a rigid solid structure or a permeable structure or a combination of both. In the VARANS equations, the volume-averaged Reynolds stress is modeled by adopting the nonlinear eddy viscosity assumption. The model equations for the volume-averaged turbulent kinetic energy and its dissipation rate are derived by taking the volume-average of the standard k− ϵ equations. Because of the volume-averaging process, the effects of the small-scale turbulence in porous media are introduced. The performance of the model is checked by comparing numerical solutions with the experimental data related to a composite breakwater reported by Sakakiyama and Liu [Coast. Eng. 121 (2001) 117].