Dynamic Free Surface Boundary Conditions Research Articles

Enhanced schemes for resolution of the continuity equation in projection-based SPH

To the best knowledge of the authors, the numerical resolution of the continuity equation, i.e., the simultaneous achievement of accurate and physically consistent velocity divergence and density fields, has not been investigated in the context of projection-based particle methods. In this study, based on the Velocity-divergence Error Mitigating (VEM)/Volume Conservation Shifting (VCS) schemes developed for the WCSPH method, which improve the numerical resolution of the continuity equation, VEM/VCS schemes are proposed for projection-based particle methods. Furthermore, two other novel schemes are introduced. First, a Dynamic VEM (DVEM) scheme is proposed, where VEM is dynamically activated only when the velocity divergence instantaneously diverge from the reference solution at individual particles. The second scheme is referred to as Enhanced VCS by reference density (EVCS), where precise imposition of dynamic free surface boundary condition is considered. Several benchmark tests with different features have been conducted, and it is shown that the proposed schemes enhance the numerical resolution of the continuity equation for incompressible flows, i.e., the divergence free velocity and the constant density conditions. Application of DVEM and EVCS has provided enhanced resolution of the continuity equation along with clear reductions in the numerical energy dissipation associated with the approximated additional numerical terms.

Linear Spectral Method for Simulating the Generation of Regular Waves by a Moving Bottom in a 3-dimensional Space

In this study, we introduce a linear spectral method capable of simulating wave generation and transformation caused by a moving bottom in a 3-dimensional space. The governing equations are linear dynamic free-surface boundary conditions and linear kinematic free-surface boundary conditions, which are solved in Fourier space. Solved velocity potential and free-surface displacement should satisfy continuity equation and kinematic bottom boundary condition. For numerical analysis, a 4th order Runge-Kutta method was utilized to analyze the time integral. The results obtained in Fourier space can be converted into velocity potential and free-surface displacement in a real space using inverse Fourier transform. Regular waves generated by various types of moving bottoms were simulated with the linear spectral method. Additionally, obliquely generated regular waves using specified bottom movements were simulated. The results obtained from the spectral method were compared to analytical solutions, showing good agreement between the two.

A regularized high-order diffusive smoothed particle hydrodynamics scheme without tensile instability

In the present work, we derive a novel high-order weakly compressible smoothed particle hydrodynamics scheme based on an accurate approximation of the pressure gradient and on the use of numerical Riemann fluxes. Specifically, a switch between non-conservative and conservative formulations of the pressure gradient is adopted close to the free surface, in order to fulfill the dynamic free-surface boundary condition and, at the same time, prevent the onset of the tensile instability in inner regions of the fluid domain. The numerical diffusion is obtained using Riemann solvers, with reconstruction/limitation of the left and right states derived from the Monotonic Upstream-centered Scheme for Conservation Laws technique. These allow for a high-order convergence rate of the diffusive terms that, for increasing spatial resolutions, results in a low numerical dissipation without tuning parameters. Regular particle distributions, which are crucial for the model accuracy, are obtained thanks to recent improvements in Particle Shifting Techniques. These are taken into account within the constitutive equations through a quasi-Lagrangian formalism. The energy balance of such a non-conservative formulation is derived, and an in-depth analysis of the term contributing to numerical dissipation is performed. The numerical investigation is carried out on several problems, illustrating the advantages of the present scheme with respect to conservative formulations. Since the proposed formulation does not intrinsically guarantee momenta conservation, the latter are monitored showing that the overall errors are generally small.

A Simplified Numerical Method for Simulating the Generation of Linear Waves by a Moving Bottom

In this study, simplified linear numerical method that can simulate wave generation and transformation by a moving bottom is introduced. Numerical analysis is conducted in wave number domain after continuity equation, linear dynamic and kinematic free surface boundary conditions and linear kinematic bottom boundary condition are Fourier transformed, and the results are expressed in space domain by an inverse Fourier transform. In the wavenumber domain, the dynamic free water surface boundary condition and the kinematic free water surface boundary condition are numerically calculated, and the velocity potential in the mean water level (<i>z</i> = 0) satisfies the continuity equation and the kinematic bottom boundary condition. Wave generation and transformation are investigated when the triangular and rectangular shape of bottoms move periodically. The results of the simplified numerical method are compared with the results of previous analytical solutions and agree well with them. Stability of numerical results according to the calculation time interval (△<i>t</i>) and the calculation wave number interval (△<i>k</i>) was also investigated. It was found that the numerical results were appropriate when △<i>t</i>≤<i>T</i>(period)/1000 and △<i>k</i>≤π/100.

Newton's Method to Determine Fourier Coefficients and Wave Properties for Deep Water Waves

Since Chappelear developed a Fourier approximation method, considerable research efforts have been made. On the other hand, Fourier approximations are unsuitable for deep water waves. The purpose of this study is to provide a Fourier approximation suitable even for deep water waves and a numerical method to determine the Fourier coefficients and the wave properties. In addition, the convergence of the solution was tested in terms of its order. This paper presents a velocity potential satisfying the Laplace equation and the bottom boundary condition (BBC) with a truncated Fourier series. Two wave profiles were derived by applying the potential to the kinematic free surface boundary condition (KFSBC) and the dynamic free surface boundary condition (DFSBC). A set of nonlinear equations was represented to determine the Fourier coefficients, which were derived so that the two profiles are identical at specified phases. The set of equations was solved using Newton’s method. This study proved that there is a limit to the series order, i.e., the maximum series order is N=12, and that there is a height limitation of this method which is slightly lower than the Michell theory. The reason why the other Fourier approximations are not suitable for deep water waves is discussed.

An Analytical Study of Regular Waves Generated by Bottom Wave Makers in a 3-Dimensional Wave Basin

Analytical solutions for regular waves generated by bottom wave makers in a 3-dimensional wave basin were derived in this study. Bottom wave makers which have triangular, rectangular and combination of two shapes were adopted. The 3-dimensional velocity potential was derived based on the linear wave theory with the bottom moving boundary condition, kinematic and dynamic free surface boundary conditions in a wave basin. Then, analytical solutions of 3-dimensional particle velocities and free surface displacement were derived from the velocity potential. The solutions showed physically valid results for regular waves generated by bottom wave makers in a wave basin. The analytical solution for obliquely propagating wave generation from bottom wave maker which works like a snake was also derived. Numerical results of the solution agree well with theoretically predicted results.

Application of normalized SPH method to inviscid fluid flows with large deformation of free surface

A new normalized smoothed particle hydrodynamics (SPH) method has been presented recently and applied to transient solid mechanics problems of small deformation. The method can overcome boundary deficiency of the conventional SPH method without troublesome inversion of corrective matrix and provide convenience for treating boundary conditions. In this paper, a formulation of the normalized SPH method is derived for modeling inviscid fluid flows with large deformation of free surface. The treatment of dynamic free surface boundary conditions in the present method is addressed and theoretically compared with that in the conventional SPH method. A method based on kernel estimation is proposed for calculation of boundary summation and compared with the method based on α-shape. The normalized SPH method is applied to modeling inviscid free surface flows. Numerical results demonstrate that the present method is capable of modeling large deformation of free surface and can produce higher accuracy than the conventional SPH method, especially at boundary regions. The newly developed kernel estimation method for the calculation of boundary summation is more efficient than the α-shape-based method, and makes the computational efficiency of the normalized SPH method close to the conventional SPH method. To simulate violent free surface flows, a particle shifting technique is introduced in the present method. With this technique, the method is applied to simulating extremely large deformation, fragmentation and reconnection of free surface.

Development of Analytical Solutions on Velocities of Regular Waves Generated by Bottom Wave Makers in a Flume

Analytical solutions for two-dimensional velocities of regular waves generated by bottom wave makers in a flume were derived in this study. Triangular and rectangular bottom wave makers were adopted. The velocity potential was derived based on the linear wave theory with the bottom moving boundary condition, kinematic and dynamic free surface boundary conditions. Then, analytical solutions of two-dimensional particle velocities were derived from the velocity potential. The velocity potential and two-dimensional particle velocities which were derived as complex integral equations were numerically calculated. The solutions showed physically valid results as velocities of regular waves generated by bottom wave makers in a flume.

Analytical Solution for Bichromatic Waves on Linearly Varying Currents

It is well-known that the bound long waves play an important role on beach morphology. The existence of the current will modify the intensity of the waves. To examine the more realistic problem where the current is non-uniform, a third-order analytical solution for bichromatic waves on currents with constant vorticity is derived using a perturbation method. Earlier derivations of theories for interactions between waves and currents have been limited to cases of bichromatic waves on depth-uniform currents and cases of monochromatic waves on linear shear currents. Unlike the derivations of monochromatic waves, the moving-frame method cannot be used in the case of bichromatic waves because there are multiple waves with different celerities, and the flow can in no way be treated as steady-state. Moreover, for shear currents where the flows are rotational, the velocity potential cannot be simply defined. For these reasons, it is difficult to carry out mathematical operations when the dynamic free surface boundary condition is applied. However, with the consideration of the wave part of the fluid motion remaining irrotational, some of the terms in the expanded boundary conditions can be ignored; thus, the derivations can be further processed. As a result, the third-order explicit expressions for the stream function, the velocity potential, and the surface elevation can be solved. The nonlinear dispersion relation is also derived to account for interacting wave components with different frequencies and amplitudes which can be used to solved for wavenumbers of both wave trains. The obtained solutions are verified by reducing to those of previous results for monochromatic waves and uniform currents. Furthermore, the influence of constant vorticity on the wave kinematics is illustrated. A comparison between following/opposing currents is also carried out. Finally, the effects of the shear current on the strength of the bound long waves are also illustrated.

A Fourier Series Approximation for Deep-water Waves

Dean (1965) proposed the use of the root mean square error (RMSE) in the dynamic free surface boundary condition (DFSBC) and kinematic free-surface boundary condition (KFSBC) as an error evaluation criterion for wave theories. There are well known wave theories with RMSE more than 1%, such as Airy theory, Stokes theory, Dean’s stream function theory, Fenton’s theory, and trochodial theory for deep-water waves. However, none of them can be applied for deep-water breaking waves. The purpose of this study is to provide a closed-form solution for deep-water waves with RMSE less than 1% even for breaking waves. This study is based on a previous study (Shin, 2016), and all flow fields were simplified for deep-water waves. For a closed-form solution, all Fourier series coefficients and all related parameters are presented with Newton’s polynomials, which were determined by curve fitting data (Shin, 2016). For verification, a wave in Miche’s limit was calculated, and, the profiles, velocities, and the accelerations were compared with those of 5th-order Stokes theory. The results give greater velocities and acceleration than 5th-order Stokes theory, and the wavelength depends on the wave height. The results satisfy the Laplace equation, bottom boundary condition (BBC), and KFSBC, while Stokes theory satisfies only the Laplace equation and BBC. RMSE in DFSBC less than 7.25×10-2% was obtained. The series order of the proposed method is three, but the series order of 5th-order Stokes theory is five. Nevertheless, this study provides less RMSE than 5th-order Stokes theory. As a result, the method is suitable for offshore structural design.

Enhancement of δ-SPH for ocean engineering applications through incorporation of a background mesh scheme

The improved Smoothed Particle Hydrodynamics (SPH) method with numerical diffusive terms, which is referred to as δ-SPH, has been widely applied into the studies of engineering problems, especially in the fields of ocean and coastal engineering. Nonetheless, there are still some drawbacks associated with δ-SPH model as in case of other particle methods. These drawbacks correspond to presence of unphysical pressure noise as well as imprecise satisfaction of dynamic free-surface boundary condition in numerical simulations. Through derivation and implementation of a so-called Background Mesh (BM) scheme and enforcing the spatial continuity of calculated source term of pressure equation (Equation of State; EOS), we present enhancements for δ-SPH in providing a more accurate pressure field as well as a more precise reproduction of free-surface. The enhancing effects of the newly proposed δ-SPH+BM are portrayed through several ocean engineering-related test cases including dam break and violent sloshing flows in both two and three dimensions. The incorporation of BM scheme is shown to result in enhanced calculation of pressure field as well as improved satisfaction of dynamic free-surface boundary condition as a consequence of enhanced density and pressure fields.

A Finite Volume Based Fully Nonlinear Potential Flow Model for Water Wave Problems

A new Fully Nonlinear Potential Flow (FNPF) numerical model has been developed for the simulation of nonlinear water wave problems. At each time step, the mixed boundary value problem for the flow field is spatially discretised by Finite Volume Method (FVM) and the kinematic and dynamic free surface boundary conditions are defined in a semi-Eulerian-Lagrangian form, which are used to update the wave elevation and velocity potential on the free surface. In the numerical model, waves are generated through a relaxation zone and absorbed by an artificial damping zone at the inlet and outlet of the numerical wave tank (NWT), respectively. Instead of a five-point smoothing technique, a more versatile fourth-order technique is developed to eliminate the possible saw-tooth instability at the free surface. Test cases with increasing complexities, such as wave generation and absorption, 2- and 3-Dimensional wave shoaling, and wave-cylinder interaction are simulated to assess its accuracy, convergence, and robustness. For all the cases considered, satisfactory agreements of free surface elevation and wave-induced forces against the experimental measurements and other existing numerical results are achieved. The developed numerical model fully utilises the existing functionalities in OpenFOAM and has the potential to provide an effective alternative to other FNPF based models for constructing a hybrid numerical wave tank model through its coupling with the multiphase flow models in OpenFOAM.

On the initial values of kinematic and dynamic free-surface boundary conditions for water wave problems

The kinematic and dynamic boundary conditions on the free surface of a fluid should be posed for water wave problems. In the framework of potential theory for an inviscid and incompressible fluid with an irrotational motion, the combined boundary condition, which involves the velocity potential only, is often used by eliminating the elevation terms mathematically. Such a combination is correct for the solutions in the frequency domain, and is not feasible for an initial-boundary-value problem in the time domain since it leads to a totally different physical formulation. The correct initial conditions for pure gravity waves and hydroelastic waves are presented.

Dynamic analysis of wave action on an OWC wave energy converter under the influence of viscosity

Oscillating water column (OWC) device is one of the most promising wave energy converters (WECs). Besides the energy conversion efficiency, the survivability should also be considered for a design purpose in the process of wave energy exploitation. In the present study, by introducing the artificial viscous terms into the dynamic free surface boundary condition and Bernoulli equation, a fully nonlinear numerical model based on higher-order boundary element method (HOBEM) is adopted to model the wave dynamics of an OWC device. The viscosity effects on the wave force (i.e., ΔF) is investigated by comparing the predicted wave force by the numerical model with and without these viscous terms. The effects of the chamber geometry parameters, such as front wall draft, chamber width and opening ratio (i.e., air orifice width), on ΔF are investigated. The results indicate that the viscosity effect on the wave force on the seaside surface of the front wall is larger than that on its shoreside surface. The viscosity effect on the total horizontal wave force on the front wall increases with the increase of front wall draft in some extent. The influence of the viscosity on the horizontal wave force increases with opening ratio decreasing due to the increasing air pressure inside the chamber.

On pressure forcing techniques for numerical modelling of unidirectional nonlinear water waves

A method of incorporating pressure forcing into a nonlinear potential flow wave model is presented. A semi-analytical pseudo-spectral method is used to calculate dynamic response of a water body exposed to evolving local pressure distribution. Surface slope coherent and slope proportional pressure functions are directly applied through a pressure term appearing in the dynamic free-surface boundary condition of a formulated initial boundary-value problem. First, a monochromatic pressure distribution is used to generate steady regular waves of permanent form. The pressure-induced wave motion exhibits stable harmonic structure for deepwater, transitional water and shallow water waves. In the next step, a more complex pressure system is used to initiate multi-component wave propagation. It is demonstrated that the proposed method provides well-posed initial conditions for studying various water wave scenarios within a framework of nonlinear potential flow solutions.

A viscous damping model for piston mode resonance

A viscous damping model is proposed based on a simplified equation of fluid motion in a moonpool or the narrow gap formed by two fixed boxes. The model takes into account the damping induced by both flow separation and wall friction through two damping coefficients, namely, the local and friction loss coefficients. The local loss coefficient is determined through specifically designed physical model tests in this work, and the friction loss coefficient is estimated through an empirical formula found in the literature. The viscous damping model is implemented in the dynamic free-surface boundary condition in the gap of a modified potential flow model. The modified potential flow model is then applied to simulate the wave-induced fluid responses in a narrow gap formed by two fixed boxes and in a moonpool for which experimental data are available. The modified potential flow model with the proposed viscous damping model works well in capturing both the resonant amplitude and frequency under a wide range of damping conditions.

A NONLINEAR AND DISPERSIVE 3D MODEL FOR COASTAL WAVES USING RADIAL BASIS FUNCTIONS

Accurate wave propagation models are required for the design of coastal structures and the evaluation of coastal risks. Nonlinear and dispersive effects are particularly important in the nearshore environment. Two-dimensional cross-shore (2DV) wave models can be used as a preliminary step in coastal studies, but 3D models are needed to capture fully the effects of alongshore bathymetric variations, variable wave incidence, the presence of coastal or harbor structures, etc. Yates and Benoit (2015) developed a numerical model based on fully nonlinear potential flow theory. By assuming non-overturning waves, the kinematic and dynamic free surface boundary conditions are expressed as evolution equations of the free surface elevation and velocity potential, following Zakharov (1968). At each time step, the free surface vertical velocity is estimated by solving the Laplace equation for the velocity potential in the domain. Following Tian and Sato (2008), a spectral approach is used to expand the velocity potential in the vertical as a linear combination of Chebyshev polynomials. The accuracy of the 2DV model was validated with several non-breaking experimental test cases (Benoit et al., 2014; Raoult et al., 2016). Here the model is extended to 3D using scattered nodes (for flexibility) to discretize the horizontal domain. Spatial derivatives are estimated at each node using a linear combination of the function values at neighboring points using Radial Basis Functions (RBF) (Wright and Fornberg, 2006). The accuracy of the method depends on the number of neighboring nodes (Nsten) and the chosen RBF type (e.g. multiquadric, Gaussian, polyharmonic spline (PHS), thin plate spline, etc.), with associated shape factor C for some of them.

Nonlinear and viscous effects on the hydrodynamic performance of a fixed OWC wave energy converter

The hydrodynamic performance of a fixed Oscillating Water Column (OWC) device is experimentally and numerically investigated. Based on the time-domain higher-order boundary element method (HOBEM), by introducing an artificial viscosity term in the dynamic free surface boundary condition, a fully nonlinear numerical wave model is used to simulate the hydrodynamic performance of an OWC device. A set of comprehensive experiments for regular waves is carried out to validate the numerical results as well as to investigate the nonlinear effects on the hydrodynamic performance of OWC. The mechanism of the nonlinear phenomenon is investigated based on the analysis of the experimental and numerical results. The influence of the wave nonlinearity and the viscosity on the hydrodynamic efficiency is quantified by comparing the linear and nonlinear numerical results. It was found that the hydrodynamic efficiency increases with the nonlinearity and viscosity when the incident wave amplitude is small. When the incident wave amplitude is large, the hydrodynamic efficiency is reduced by the weakened transmission of the second-order harmonic wave component due to the strong wave nonlinearity. However, when the wave amplitude is between these two regimes, the wave is weakly nonlinear, the efficiency decreases with the wave amplitude due to the combined effect of the nonlinearity and viscosity.

Iterative Rankine HOBEM analysis of hull-form effects in forward-speed diffraction problem

A time-domain numerical algorithm based on the higher-order boundary element method and the iterative time-marching scheme is proposed for seakeeping analysis. The ship waves generated by a hull advancing at a constant forward speed in incident waves and the resultant diffraction forces acting on the hull are computed to investigate the hull-form effects on the hydrodynamic forces. A rectangular computational domain travelling at ship's speed is considered. An artificial damping beach for satisfying the radiation condition is installed at the outer portion of the free surface except the downstream side. An iterative time-marching scheme is employed for updating both kinematic and dynamic free-surface boundary conditions for numerical accuracy and stability. The boundary integral equation is solved by distributing higher-order boundary elements over the wetted body surface and the free surface. The hull-form effects on the naval hydrodynamics are investigated by comparing three different Wigley models. Finally, the corresponding unsteady wave patterns and the wave profiles around the hulls are illustrated and discussed.

A Continuous Desingularized Source Distribution Method Describing Wave–Body Interactions of a Large Amplitude Oscillatory Body

A Rankine source method with a continuous desingularized free surface source panel distribution is developed to solve numerically a wave–body interaction problem with nonlinear boundary conditions. A body undergoes forced oscillatory motion in a free water surface and the variation of wetted body surface is captured by a regridding process. Free surface sources are placed in continuous panels, rather than points in isolation, over the calm water surface, with free surface collocation points placed on the calm water surface. Nonlinear kinematic and dynamic free surface boundary conditions along the collocation points on the calm water surface are solved in a time domain simulation based on a Lagrange time dependent formulation. Compared with isolated desingularized source points distribution methods, a significantly reduced number of free surface collocation points with sparse distribution are utilized in the present numerical computation. The numerical scheme of study is shown to be computationally efficient and the accuracy of numerical solutions is compared with traditional numerical methods as well as measurements.