Free Surface Boundary Conditions Research Articles

Generalized finite difference method applied to solve seismic wave propagation problems. Examples of 3D simulations

The simulation of seismic wave propagation generally requires dealing with complex tridimensional geometries that are irregular in shape and have non‐uniform properties, features that make the application of the generalized finite difference method in this field interesting. This work continues the extensive developments by the research team focused on the simulation of seismic wave propagation in two‐dimensional domains. In this new contribution, the general formulation and the treatment of free surface boundary conditions are extended for the three‐dimensional case, and the results obtained from different examples are analyzed.

Elastic full-wave field simulation in 3D tunnel space with the variable staggered-grid finite-difference method in cylindrical coordinates

Tunnel advance detection technology is an important method for determining the structure of complex geological bodies in front of the tunnel face. Among the tunnel advance detection technologies, the seismic method is one of the most accurate methods with long detection distances. In seismic tunnel advance detection, the cylindrical configuration aggravates the complexity of the wave field in the 3D tunnel space and significantly influences the accuracy of the detection results. Thus, it is crucial to simulate an accurate seismic full-wave field of the tunnel space and to understand the propagation and wave-field characteristics of individual seismic waves for seismic tunnel advance detection. Usually, in 3D Cartesian coordinates, the tunnel wall is approximated with a staircase boundary, but it is not sufficiently accurate in shape and generates numerical dispersion in the simulation, especially in the presence of surface waves. Therefore, we developed a variable staggered-grid finite-difference method in cylindrical coordinates to simulate the elastic full-wave field in a 3D tunnel space. Properly setting free-surface boundary conditions solves the propagation of surface waves on the tunnel wall and face. The interference of the instability and discontinuity of the pole axis in the seismic wave field simulation was eliminated using our method. According to practical engineering situations, models containing three types of geological bodies in front of the tunnel face were designed. Their seismic wave propagation and wave field characteristics were analyzed. Compared with the simulation results in Cartesian coordinates, the results in cylindrical coordinates show that numerical dispersion is negligible and conclude that a higher signal-to-noise ratio and more accurate seismic wave field can be simulated with cylindrical coordinates in the 3D tunnel space. Our simulation method provides theoretical and practical guidance for analyzing and interpreting seismic wave fields in tunnel advance detection.

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.

Sigma mapping for drainage problems with a time-dependent water table

Regional groundwater flow forecasting is frequently accomplished using model equations based on Darcy's law, the continuity equation, and kinematic and dynamic boundary conditions for a time-dependent water table, to predict hydraulic heads and fluxes. For homogeneous and isotropic aquifers, the model reduces to Laplace's equation for the hydraulic head and a transient non-linear free surface boundary condition. These equations are solved in the literature and in computer packages by a variety of numerical methods, including finite difference, finite element, and boundary element methods. The solution of this model is not simple and subject to instability problems. Computations in a vertical plane using finite-difference methods are not easy due to the irregular mesh needed near a highly curved water table, frequently involving smoothing of the numerical results to ensure stability. A technique to avoid the complications related to mesh generation near a curved boundary and its tracking entails using mappings to produce transformed domains. However, this is not so frequent to compute unsteady seepages. In particular, the so-called sigma mapping is widely used to model irrotational water waves; it applies to any flow involving a time-dependent free surface boundary. This mapping transforms the domain into a rectangle, so that the free surface becomes static in the transformed plane. There is close similarity between the mathematical model describing irrotational water waves and that of unsteady seepage flow, e.g., both are governed by Laplace's equation involving relevant boundary conditions at the moving boundary. Both moving free surfaces are different in character, e.g., the seepage is dissipative whereas the water wave is dispersive. However, it is possible to transfer mathematical approximations between both fields of research. This transfer is exploited in this work and the sigma mapping is newly and successfully applied to seepage flows involving both a steady and unsteady water table either receiving, or not, a recharge from rainfall or an artificial source.

Model parameter design for modeling surface topography in VTI elastic finite-difference near-surface simulations

The near-surface anisotropy is one of the factors leading to an inaccurate velocity estimation in the shallow area, which has a strong impact on the static correction of land data and seismic imaging of the subsurface in depth. To investigate the anisotropy influence, the consideration of medium anisotropy in near-surface seismic modeling becomes necessary. In the oil and gas industry, finite-difference (FD) methods are commonly used for seismic wave modeling, thanks to their simplicity, accuracy, and high efficiency. However, for near-surface modeling, the implementation of free-surface boundary conditions and nonflat topography representation are the two main obstacles preventing the effectiveness of FD methods. To address these two issues in the vertical transversely isotropic anisotropic scenarios, we have developed a simple and efficient method for the discrete model design which can be easily applied to the conventional Cartesian-grid FD modeling. Our method involves: (1) the parameter-averaging method for implicitly implementing the stress-free condition by a modification of model anisotropy parameters near the (non) flat free-surface boundary and (2) an independent wavefield superposition with modeling results of different parameter configurations to accurately represent the rugged topography and significantly reduce the staircase diffractions caused by a staircase approximation of continuous surface in the Cartesian-grid discretization. For validation of this method, we conduct several numerical tests in 2D and 3D spaces. The accuracy is demonstrated by a comparison of the spectral-element solutions of SPECFEM in modeling the seismic wave propagation in the presence of an irregular free surface. From the aspect of computational efficiency, it is more promising in practical applications due to the use of wavefield superposition strategy in this method which does not require finer spatial sampling to eliminate the staircase diffractions.

Comparison of the Characteristics of Internal Wave Propagation under Two Surface-Boundary Conditions in Continuously Density-Stratified Oceans

Zhang, H.-s.; He, L.; Zhang, Z.-h., and Zhang, Y., 2023. Comparison of the characteristics of internal wave propagation under two surface boundary-conditions in continuously density-stratified oceans. Journal of Coastal Research, 39(2), 303–316. Charlotte (North Carolina), ISSN 0749-0208. Complete theoretical solutions were developed for the propagation of internal waves (IWs) with small amplitudes in a continuously density-stratified ocean under the free surface-boundary condition and rigid lid assumption. Additionally, a numerical model was established for the propagation of IWs under the free surface-boundary condition. In the proposed model, the Euler equations are employed as the governing equations and are discretized with an improved SIMPLE algorithm. In waters with a constant depth, a comparison between the numerical and theoretical results proved the reasonability and accuracy of the proposed method. Additionally, the numerical results under the free surface-boundary condition were compared with those under the rigid lid assumption to analyse the differences between the calculated velocities and density fields. Herein, the influences of characteristic parameters, including the period and amplitude of the incident waves and the coefficient of density and the mode number on the velocity fields under the two surface-boundary conditions, are discussed thoroughly. In the waters with nonuniform varied depths, numerical simulations on the IWs propagation were performed under the two surface-boundary conditions. Differences between the calculated physical fields of IWs in front of and behind the submerged dike are discussed. The detailed analyses showed that differences exist between the calculation results under the two surface-boundary conditions. In particular, they were obvious for the cases with high mode number and variable topography. The results are significant considering the propagation mechanism of IWs under different surface-boundary conditions in a continuously density-stratified ocean.

Alternative flow models, vector Green functions and boundary integral flow representations in ship and offshore hydrodynamics

Wave diffraction-radiation by large bodies such as offshore structures and ships advancing in regular waves or in calm water is widely analyzed via panel methods based on a boundary integral flow representation and a Green function that satisfies the relevant free-surface boundary condition. The boundary integral flow representations solved in existing panel methods express the flow potential in terms of distributions of sources and dipoles, related to a Green function and its gradient, over the body surface. Other notable features of existing boundary integral flow representations are that they can be ill-posed for a set of irregular frequencies, and that – for a ship advancing in calm water or in regular waves – they involve a troublesome line integral around the ship waterline. Several alternative boundary integral flow representations that are well suited for wave diffraction-radiation by offshore structures or by ships advancing in regular waves or in calm water are given in this study. These representations, based on an alternative linear flow model (given previously) and an alternative vector Green function (not previously considered), are weakly singular and do not involve a waterline integral. The influence coefficients related to these new flow representations consist of Rankine components associated with complementary types of weakly singular Rankine singularities, and Fourier components defined by a Fourier superposition of elementary waves. Both the Rankine and Fourier components of the influence coefficients can be evaluated simply and efficiently. In particular, the Fourier component can be evaluated via the Fourier–Kochin (FK) method, in which the dominant computational task is independent of the number of panels that approximate the body surface. Thus, the new boundary integral flow representations given in this study and the FK method expounded previously provide a practical new basis for the evaluation of three main classes of wave diffraction-radiation by ships and offshore structures via panel methods.

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

Efficient Calculation of Hydrodynamic Loads on Offshore Wind Substructures Including Slamming Forces

Abstract Estimation of the hydrodynamic loads based on strip theory using the Morrison equation provides an inexpensive method for load estimation for the offshore industry. The advantage of this approach is that it requires only the undisturbed wave kinematics along with inertia and viscous force coefficients. Over the recent years, the development in numerical wave tank simulations makes it possible to simulate nonlinear 3-h sea states, with computational times in the order of real time. This presents the possibility to calculate loads using wave spectrum input in numerical simulations with reasonable computational time and effort. In the current paper, the open-source fully nonlinear potential flow model REEF3D::FNPF is employed for calculating the nonlinear wave kinematics. Here, the Laplace equation for the velocity potential is solved on a σ-coordinate mesh with the nonlinear free surface boundary conditions to close the system. A technique to calculate the total acceleration on the σ-coordinate grid is introduced which makes it possible to apply strip theory in a moving grid framework. With the combination of strip theory and 3-h wave simulations, a unique possibility to estimate the hydrodynamic loads in real time for all discrete positions in space within the domain of the numerical wave tank is presented in this paper. The numerical results for inline forces on an offshore wind mono-pile substructure are compared with measurements, and the new approach shows good agreement.

Elastic anisotropic finite-difference full-wave modeling and imaging of 2D tilted transversely isotropic (TTI) media

Imaging below the tilted transversely isotropic (TTI) media pose serious problems and distortions of the subsurface geological targets of interest for hydrocarbon exploration, which is very common in fold-thrust belts or active tectonic areas like the Rocky Mountain foothills of the Canada or foothills of the Himalaya. To model and image below the TTI media having symmetry axis orthogonal to dipping anisotropic layers, a 2D TTI thrust sheet is considered for elastic anisotropic finite-difference full-wave modeling and generation of synthetic seismic data using the robust staggered-grid scheme of wave propagation. Free-surface boundary conditions at the top and absorbing boundary conditions for the other three sides of the model have been imposed to reduce undesirable edge effects and suppress dispersions. The synthetic seismic data generated for the 2D TTI thrust model are migrated using the anisotropic Kirchhoff pre-stack time (PSTM) and depth migration (PSDM) technique followed by the migration velocity analysis (MVA) algorithm. The traveltime computations are made using robust eikonal equations with fourth-order elastic anisotropic finite-difference solutions to handle large tilts (ν) along with anisotropic ray-tracing to image below the TTI thrust sheet. The MVA algorithm adopted also estimates the anisotropic parameters ε and δ accurately within the acceptable error limit through successive iterations of inversion and parameter update, which results in sufficient flattening of the reflection events in the common image gathers (CIGs). This is a powerful and stable methodology to handle large tilt (ν = 60°) of the steeply dipping reflectors in complex geological structure like the TTI thrust model without much distortion of CIGs for which the symmetry axis of each TTI block is set orthogonal to the bottom of the reflectors having different tilts.

A generalized weak-scatterer approximation for nonlinear wave–structure interaction in marine hydrodynamics

In this paper, a generalized weak-scatterer (GWS) approximation is proposed for solving nonlinear wave–structure interaction problems. In contrast to the original weak-scatterer (OWS) theory, where the approximated free surface boundary conditions (FSBCs) are Taylor-expanded in the vertical direction from the incident wave surface, we apply Taylor series expansion in an arbitrary direction which, in particular, is tangential to the boundary of the floating structure close to the waterline. This leads to generalized kinematic and dynamic FSBCs for the radiated and scattered waves, along with corresponding expressions of the wave loads. Accordingly, an Arbitrary Lagrangian–Eulerian (ALE) approach is adopted to track the free-surface properties. The new GWS method is more consistent than the OWS model in that the wave markers do not separate from the body surface at the waterline for structures with flare. An Immersed-Boundary Adaptive Harmonic Polynomial Cell (IB-AHPC) method is implemented to solve the corresponding boundary value problems (BVPs) for both the velocity potential and the Lagrangian acceleration potential at each time step. The new formulation introduces additional convective terms in the FSBCs, making them similar to the seakeeping problems for ships with forward speed, and this requires special treatment to avoid instability in the time-domain simulations. Based on a matrix-based eigenvalue stability analysis, we illustrate that stable solutions can be achieved by introducing an upwind-biased scheme to discretize the convective terms in the kinematic FSBC. The proposed model is verified by three wave diffraction problems in regular waves, including a submerged circular cylinder, a rounded-corner rectangular ship section, and a trapezoidal section with a large flare angle.

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.

Numerical Investigation on Higher-Order Harmonic Waves Induced by a Submerged Inclined Plate

In this paper, a two-dimensional time-domain numerical flume has been established to model and investigate nonlinear interactions between nonlinear surface waves and a submerged inclined thin plate. The model solves the Laplace equation and the fully nonlinear free surface boundary conditions within the framework of potential flow theory based on the high-order boundary element method. The mixed Euler–Lagrangian method is applied to update the water surface at each time step, and the fourth-order Runge–Kutta method for time stepping. A so-called four-point method was employed to separate the second-order harmonic free and bounded wave that has the same wave frequency but different wave celerity in front of and behind the submerged plate. It is found that the amplitude of the second-order harmonic free wave increases with the inclination angle of the submerged plate, and the level/amount of the increase is larger for a larger wave steepness. In addition, the amplitudes of both the second-order reflected and transmitted waves are found to increase with the wave steepness, and their empirical relationships are derived for potential use in practical engineering.

Comparison of wave diffraction forces on a surface-piercing body for various free-surface grid update schemes

This study analyzed the difference in wave diffraction force of a surface piercing body according to the grid update scheme of free surface nodes in the time integration process. A two-dimensional fully nonlinear numerical wave tank based on potential flow theory and boundary element method was used for numerical analysis. The MEL (mixed Eulerian–Lagrangian) method was used for free surface node treatment. This method was divided into a semi-Lagrangian approach and a full Lagrangian approach (material node approach).The Runge–Kutta 4th-order integration method was used to integrate the free surface boundary condition temporally. In the internal sub-time step of the time integration process, the frozen coefficient scheme and the full update scheme were used according to the node update scheme, respectively. The vertical diffraction forces of each frequency component for node update schemes were compared. The difference in the 2nd-order vertical forces according to wave frequency, wave steepness, time step interval, and node spacing was investigated. A more precise time-domain numerical analysis method was proposed for high-order external forces in the wave-floating body interaction.

Hydroelastic potential flow solver suited for nonlinear wave dynamics in ice-covered waters

The aim of this paper is to introduce a fully nonlinear numerical finite element solver for the simulation of nonlinear wave processes in the presence of a solid ice sheet. In this study, solid ice cover referred to the size of the ice sheet and denoted that the length of the ice sheet was many times larger than the longest relevant wavelength. The complexity of the ice sheet characteristics was assumed to be homogeneous, isotropic and in the linear elastic plate regime so that the deformation process could be modeled by the Kirchhoff–Love plate ansatz. The method presented was verified and validated for different ice dimensions and wave scenarios. At the beginning, the implementation of the flexural rigidity to the free surface boundary condition was verified by comparing the analytical wave-ice dispersion relation to simulation results with small amplitude regular waves and varying ice dimensions. Afterwards, the general applicability was validated by means of wave-ice experiments. The experiments were performed in the ice tank at Hamburg Ship Model Basin HSVA comprising regular waves and transient wave groups.

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.

Numerical and experimental study on free-surface wave generation by a submerged moving body in a towing tank

Free surface waves were generated by a submerged moving body in a towing tank using a numerical and a physical towing tank. Unlike a circulating water channel, a towing tank allows easy to control of acceleration and deacceleration range of the body motion. Numerical and experimental analysis can make the same moving conditions of the submerged body. Using a Numerical Towing Tank (NTT), free surface elevation was calculated according to body velocities, submerged depths and body volumes. The measured experimental data agreed well with the results of a fully nonlinear simulation and the data was also compared with the Russell comprehensive error method. The nonlinearity of the free surface waves increased as the submerged depth of the body became shallower and the body velocity became faster. The increase in wave height according to the increase of the body velocity or the increase in the wave height according to the decrease in the submerged depth of the body generally showed a proportional relationship. Overall, the nonlinear free surface boundary condition needs to be applied when calculating the free surface wave due to the motion of a submerged body. The ratio of wave height difference according to body displacement was similar to the ratio (percentage) of increase in body displacement to NACA shape displacement as body velocity becomes faster.