Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

Meshless numerical simulation for fully nonlinear water waves

A meshless numerical model for nonlinear free surface water wave is presented in this paper. An approach of handling the moving free surface boundary is proposed. Using the fundamental solution of the Laplace equation as the radial basis functions and locating the source points outside the computational domain, the problem is solved by collocation of only a few boundary points. Present model is first applied to simulate the generation of periodic finite‐amplitude waves with high wave‐steepness and then is employed to simulate the modulation of monochromatic waves passing over a submerged obstacle. Good agreements are observed as compared with experimental data and other numerical models. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical simulations of wave interactions with vertical wave barriers using the SPH method

This paper focuses on the fluid boundary separation problem of the conventional dynamic solid boundary treatment (DSBT) and proposes a modified DSBT (MDSBT). Classic 2D free dam break flows and 3D dam break flows against a rectangular box are used to assess the performance of this MDSBT in free surface flow and violent fluid–structure interaction, respectively. Another test, water column oscillations in a U‐tube, is specially designed to reveal the applicability of dealing with two types of particular boundaries: the wet–dry solid boundary and the large‐curvature solid boundary. A comparison between the numerical results and the experimental data shows that the MDSBT is capable of eliminating the fluid boundary separation, improving the accuracy of the solid boundary pressure calculations and preventing the unphysical penetration of fluid particles. Using a 2D SPH numerical wave tank with MDSBT, the interactions between regular waves and a simplified vertical wave barrier are simulated. The numerical results reveal that the maximum horizontal force occurs at the endpoint of the vertical board, and with the enlargement of the relative submerged board length, the maximum moment grows linearly; furthermore, the relative average mass transportation under the breakwater initially increases to 11.14 per wave strike but is later reduced. The numerical simulation of a full‐scale 3D wave barrier with two vertical boards shows that the wave and structure interactions in the practical project are far more complicated than in the simplified 2D models. The SPH model using the MDSBT is capable of providing a reference for engineering designs. Copyright © 2014 John Wiley & Sons, Ltd.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

A lattice Boltzmann method for viscous free surface waves in two dimensions

We propose a new method based on the combination of the lattice Boltzmann equation (LBE) and the kinematic boundary condition (KBC) method to simulate viscous free surface wave in two dimensions. In our method, the flow field is modeled by LBE, whereas the free surface is explicitly tracked by the local height function, which is calculated by the KBC method. The free surface boundary condition (FSBC) for LBE is revised from previous researches. Interpolation‐supplemented lattice Boltzmann (ISLB) method is introduced, which enables our approach to be applied on arbitrary, nonuniform mesh grids. Five cases are simulated respectively to validate the LBE–KBC method: the stationary flow and the solitary waves simulated by the revised‐FSBC are more accurate than the one obtained by the former‐FSBC; numerical results of standing waves show that our method is compatible to the existing two‐dimensional finite‐volume scheme; cases of small amplitude Stokes wave and waves traveling over a submerged bar show good agreement on wave celerity, wavelength, wave amplitude and wave period between numerical results and corresponding analytical solutions and/or experiment data.Copyright © 2012 John Wiley & Sons, Ltd.     

A horizontally curvilinear non‐hydrostatic model for simulating nonlinear wave motion in curved boundaries

A horizontally curvilinear non‐hydrostatic free surface model that embeds the second‐order projection method, the so‐called θ scheme, in fractional time stepping is developed to simulate nonlinear wave motion in curved boundaries. The model solves the unsteady, Navier–Stokes equations in a three‐dimensional curvilinear domain by incorporating the kinematic free surface boundary condition with a top‐layer boundary condition, which has been developed to improve the numerical accuracy and efficiency of the non‐hydrostatic model in the standard staggered grid layout. The second‐order Adams–Bashforth scheme with the third‐order spatial upwind method is implemented in discretizing advection terms. Numerical accuracy in terms of nonlinear phase speed and amplitude is verified against the nonlinear Stokes wave theory over varying wave steepness in a two‐dimensional numerical wave tank. The model is then applied to investigate the nonlinear wave characteristics in the presence of dispersion caused by reflection and diffraction in a semicircular channel. The model results agree quantitatively with superimposed analytical solutions. Finally, the model is applied to simulate nonlinear wave run‐ups caused by wave‐body interaction around a bottom‐mounted cylinder. The numerical results exhibit good agreement with experimental data and the second‐order diffraction theory. Overall, it is shown that the developed model, with only three vertical layers, is capable of accurately simulating nonlinear waves interacting within curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure

A methodology for computing three‐dimensional interaction between waves and fixed bodies is developed based on a fully non‐linear potential flow theory. The associated boundary value problem is solved using a finite element method (FEM). A recovery technique has been implemented to improve the FEM solution. The velocity is calculated by a numerical differentiation technique. The corresponding algebraic equations are solved by the conjugate gradient method with a symmetric successive overrelaxation (SSOR) preconditioner. The radiation condition at a truncated boundary is imposed based on the combination of a damping zone and the Sommerfeld condition. This paper (Part 1) focuses on the technical procedure, while Part 2 [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 2. Numerical results and validation. International Journal for Numerical Methods in Fluids 2001] gives detailed numerical results, including validation, for the cases of steep waves interacting with one or two vertical cylinders. Copyright © 2001 John Wiley & Sons, Ltd.     

Implementation of physical boundary conditions into computational domain in modelling of oscillatory bottom boundary layers

This paper discusses the importance of realistic implementation of the physical boundary conditions into computational domain for the simulation of the oscillatory turbulent boundary layer flow over smooth and rough flat beds. A mathematical model composed of the Reynolds averaged Navier–Stokes equation, turbulent kinetic energy (k) and dissipation rate of the turbulent kinetic energy (ε) has been developed. Control‐volume approach is used to discretize the governing equations to facilitate the numerical solution. Non‐slip condition is imposed on the bottom surface, and irrotational main flow properties are applied to the upper boundary. The turbulent kinetic energy is zero at the bottom, whereas the dissipation rate is approaching to a constant value, which is proportional to the kinematic viscosity times the second derivative of the turbulent kinetic energy. The output of the model is compared with the available experimental studies conducted in oscillatory tunnels and wave flume. It is observed that the irrotational flow assumption at the upper boundary is not realistic in case of water tunnels. Therefore, new upper boundary conditions are proposed for oscillatory tunnels. The data of wave flume show good agreement with the proposed numerical model. Additionally, several factors such as grid aspect ratio, staggered grid arrangement, time‐marching scheme and convergence criteria that are important to obtain a robust, realistic and stable code are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Analysis of efficient preconditioned defect correction methods for nonlinear water waves

Robust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

Unsteady RANS method for surface ship boundary layer and wake and wave field

Results are reported of an unsteady Reynolds‐averaged Navier–Stokes (RANS) method for simulation of the boundary layer and wake and wave field for a surface ship advancing in regular head waves, but restrained from body motions. Second‐order finite differences are used for both spatial and temporal discretization and a Poisson equation projection method is used for velocity–pressure coupling. The exact kinematic free‐surface boundary condition is solved for the free‐surface elevation using a body‐fitted/free‐surface conforming grid updated in each time step. The simulations are for the model problem of a Wigley hull advancing in calm water and in regular head waves. Verification and validation procedures are followed, which include careful consideration of both simulation and experimental uncertainties. The steady flow results are comparable to other steady RANS methods in predicting resistance, boundary layer and wake, and free‐surface effects. The unsteady flow results cover a wide range of Froude number, wavelength, and amplitude for which first harmonic amplitude and phase force and moment experimental data are available for validation along with frequency domain, linear potential flow results for comparisons. The present results, which include the effects of turbulent flow and non‐linear interactions, are in good agreement with the data and overall show better capability than the potential flow results. The physics of the unsteady boundary layer and wake and wave field response are explained with regard to frequency of encounter and seakeeping theory. The results of the present study suggest applicability for additional complexities such as practical ship geometry, ship motion, and maneuvering in arbitrary ambient waves. Copyright © 2001 John Wiley & Sons, Ltd.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

A fully non‐linear model for three‐dimensional overturning waves over an arbitrary bottom

An accurate three‐dimensional numerical model, applicable to strongly non‐linear waves, is proposed. The model solves fully non‐linear potential flow equations with a free surface using a higher‐order three‐dimensional boundary element method (BEM) and a mixed Eulerian–Lagrangian time updating, based on second‐order explicit Taylor series expansions with adaptive time steps. The model is applicable to non‐linear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. Arbitrary waves can be generated in the model, and reflective or absorbing boundary conditions specified on lateral boundaries. In the BEM, boundary geometry and field variables are represented by 16‐node cubic ‘sliding’ quadrilateral elements, providing local inter‐element continuity of the first and second derivatives. Accurate and efficient numerical integrations are developed for these elements. Discretized boundary conditions at intersections (corner/edges) between the free surface or the bottom and lateral boundaries are well‐posed in all cases of mixed boundary conditions. Higher‐order tangential derivatives, required for the time updating, are calculated in a local curvilinear co‐ordinate system, using 25‐node ‘sliding’ fourth‐order quadrilateral elements. Very high accuracy is achieved in the model for mass and energy conservation. No smoothing of the solution is required, but regridding to a higher resolution can be specified at any time over selected areas of the free surface. Applications are presented for the propagation of numerically exact solitary waves. Model properties of accuracy and convergence with a refined spatio‐temporal discretization are assessed by propagating such a wave over constant depth. The shoaling of solitary waves up to overturning is then calculated over a 1:15 plane slope, and results show good agreement with a two‐dimensional solution proposed earlier. Finally, three‐dimensional overturning waves are generated over a 1:15 sloping bottom having a ridge in the middle, thus focusing wave energy. The node regridding method is used to refine the discretization around the overturning wave. Convergence of the solution with grid size is also verified for this case. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical simulation of 3D free surface flows,with multiple incompressible immiscible phases. Applications to impulse waves

A numerical method for the solution to the density‐dependent incompressible Navier–Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume‐of‐fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density‐dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide‐generated impulse wave in a reservoir. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical modeling of wave interactions with coastal structures by a constrained interpolation profile/immersed boundary method

A high‐order difference method based multiphase model is proposed to simulate nonlinear interactions between water wave and submerged coastal structures. The model is based on the Navier–Stokes equations using a constrained interpolation profile (CIP) method for the flow solver, and employs an immersed boundary method (IBM) for the treatment of wave–structure interactions. A more accurate interface capturing scheme, the volume of fluid/weighed line interface calculation (VOF/WLIC) scheme, is adopted as the interface capturing method. A series of computations are performed to verify the application of the model for simulations of fluid interaction with various structures. These problems include flow over a fixed cylinder, water entry of a circular cylinder and solitary waves passing various submerged coastal structures. Computations are compared with the available analytical, experimental and other numerical results and good agreement is obtained. The results of this study demonstrate the accuracy and applications of the proposed model to simulate the nonlinear flow phenomena and capture the complex free surface flow. Copyright © 2015 John Wiley & Sons, Ltd.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.