Free-surface Green Function Research Articles

A precorrected-FFT higher-order boundary element method for wave–body problems

The boundary element method (BEM) has been widely applied in the field of wave interaction with offshore structures, but it is still not easy to use in resolving large-scale problems because of computing costs and computer storage being increased by O( N 2) for the traditional BEM. In this paper a precorrected Fast Fourier Transform (pFFT) higher-order boundary element method (HOBEM) is proposed for reducing the computational time and computer memory by O( N). By using a free-surface Green function for infinite water-depth, the disadvantage of the Fast Multipole Boundary Element Method (FMBEM)—i.e. unable to solve infinite deep-water wave problems—can be overcome. Numerical results from the problems of wave interaction with single- and multi-bodies show that the present method evidently has more advantages in saving memory and computing time, especially for large-scale problems, than the traditional HOBEM. In addition, the optimal variable of pFFT mesh is recommended to minimize time cost.

A free surface frequency domain green function with viscous dissipation and partial reflections from side walls

The free-surface Green function method is widely used in solving the radiation or diffraction problems caused by a ship or ocean structure oscillating on the waves. In the context of inviscid potential flow, hydrodynamic problems such as multi-body interaction and tank side wall effect cannot be properly dealt with based on the traditional free-surface frequency domain Green function method, in which the water viscosity is omitted and the energy dissipation effect is absent. In this paper, an open-sea Green function with viscous dissipation was presented within the theory of visco-potential flow. Then the tank Green function with a partial reflection from the side walls in wave tanks was formulated as a formal sum of open-sea Green functions representing the infinite images between two parallel side walls of the source in the tank. The new far-field characteristics of the tank Green function is vitally important for improving the validity of side-wall effects evaluation, which can be used in supervising the tank model tests.

Time Domain Prediction of Added Resistance of Ships

The prediction of the added resistance of the ships that can be computed from quadratic product of the first-order quantities is presented using the near-field method based on the direct pressure integration over floating body in time domain. The transient wave-body interaction of the first-order radiation and diffraction problems are solved as the impulsive velocity of the floating body by the use of a three dimensional panel method with Neumann-Kelvin method. These radiation and diffraction forces are the input for the solution of the equation of the motion that is solved by the use of the time marching scheme. The exact initial-boundary-value problem is linearized about a uniform flow, and recast as an integral equation using the transient free-surface Green function. A Wigley III hull form with forward speed is used for the numerical prediction of the different parameters. The calculated mean second-order added resistance and unsteady first-order impulse-response functions, hydrodynamics coefficients, exciting forces, and response amplitude operators are compared with experimental results.

Viscous and inviscid matching of three-dimensional free-surface flows utilizing shell functions

A methodology is presented for matching a solution to a three-dimensional free-surface viscous flow in an interior region to an inviscid free-surface flow in an outer region. The outer solution is solved in a general manner in terms of integrals in time and space of a time-dependent free-surface Green function. A cylindrical matching geometry and orthogonal basis functions are exploited to reduce the number of integrals required to characterize the general solution and to eliminate computational difficulties in evaluating singular and highly oscillatory integrals associated with the free-surface Green-function kernel. The resulting outer flow is matched to a solution of the Navier–Stokes equations in the interior region and the matching interface is demonstrated to be transparent to both incoming and outgoing free-surface waves.

3 차원 시간영역 근사비선형 2 차경계요소법에 의한 선체의 대진폭 운동 및 파랑하중 계산

A three-dimensional time-domain calculation method is of crucial importance in prediction of the motions and wave loads of a ship advancing in a severe irregular sea. The exact solution of the free surface wave-ship interaction problem is very complicated because of the essentially nonlinear boundary conditions. In this paper, an approximate body nonlinear approach based on the three-dimensional time-domain forward-speed free-surface Green function has been presented. The Froude-Krylov force and the hydrostatic restoring force are calculated over the instantaneous wetted surface of the ship while the forces due to the radiation and scattering potentials over the mean wetted surface. The time-domain radiation and scattering potentials have been obtained from a time invariant kernel of integral equations for the potentials which are discretized according to the second-order boundary element method (Hong and Hong 2008). The diffraction impulse-response functions of the Wigley seakeeping model advancing in transient head waves at various Froude numbers have been presented. A simulation of coupled heave-pitch motion of a long rectangular barge advancing in regular head waves of large amplitude has been carried out. Comparisons between the linear and the approximate body nonlinear numerical results of motions and wave loads of the barge at a nonzero Froude number have been made.

Explicit Representation for the Infinite-Depth Two-Dimensional Free-Surface Green's Function in Linear Water-Wave Theory

In this paper we derive an explicit representation for the two-dimensional free-surface Green's function in water of infinite depth, based on a finite combination of complex-valued exponential integrals and elementary functions. This representation can easily and accurately be evaluated in a numerical manner, and its main advantage over other representations lies in its simplicity and in the fact that it can be extended towards the complementary half-plane in a straightforward manner. It seems that this extension has not been studied rigorously until now, and it is required when boundary integral equations are extended in the same way. For the computation of the Green's function, the limiting absorption principle and a partial Fourier transform along the free surface are used. Some of its properties are also discussed, and an expression for its far field is developed, which allows us to state appropriately the involved radiation condition. This Green's function is then used to solve the two-dimensional infinite-depth water-wave problem by developing a corresponding boundary integral equation, whose solution is determined by means of the boundary element method. To validate the computations, a benchmark problem based on a half-circle is presented and solved numerically.

Time domain prediction of power absorption from ocean waves with latching control

A three-dimensional panel method using Neumann–Kelvin method is presented for the transient wave-body interaction problems in order to absorb maximum power from the sea. The exact initial boundary value problem is linearized about a uniform flow, and recast as an integral equation using the transient free-surface Green function. The hydrodynamics part of the solution including radiation and diffraction problem is solved as impulsive velocity problem. A discrete control of latching is used to increase the bandwidth of the efficiency of the wave energy converters (WEC). When latching control is applied to WEC in the case of off-resonance condition it increases the amplitude of the motion as well as absorbed power. It is assumed that the exciting force is known in the close future and that body is hold in position during the latching time. A heaving hemisphere as a point-absorber WEC is used for the numerical prediction of the different parameters. The calculated hydrodynamics coefficients, response amplitude operator, absorbed power, relative capture width of this device compared with analytical and other published results.

3 차원 시간영역 전진속도 자유표면 Green 함수와 2 차 경계요소법을 사용한 선체의 방사포텐셜 수치계산

The radiation potential of a ship advancing in waves is studied using the 3D time-domain forward-speed free-surface Green function and the Green integral equation. Numerical solutions are obtained by making use of the 2nd order BEM(Boundary Element Method) which make it possible to take account of the line integral along the waterline in a rigorous manner. The 6 degree of freedom motion memory functions of a hemisphere and the Wigley seakeeping model obtained by direct integration of the time-domain 3D potentials over the wetted surface are presented for various Froude numbers.

The Simulation of Ship Motions Using a B-Spline–Based Panel Method in Time Domain

In this paper, a B-spline-based higher-order method is developed for simulating three-dimensional ship motions with forward speed. The problem is formulated in time domain using a transient free surface Green function. The body geometry is defined by open uniform or nonuniform B-spline basis functions depending on the hull type, whereas the unknown field variables are described by open uniform B-spline basis functions. The collocation method is applied to discretize the integral equation and then solved for the unknown potentials and source strengths. Motion computations in head waves are carried out for three types of ship hulls: a mathematically defined Wigley hull, a typical containership (S175 hull), and a Series 60 hull. Results are obtained for regular and irregular waves and compared with available experimental and computational results. It is found that the results from the present method are in very good agreement with the published results, and in particular with experimental data. Long-duration simulations have also been carried out with an ordinary desktop PC (PIV with 512 MB RAM) to demonstrate the ability of the method to simulate motions over long periods without any visible deterioration using only modest computational resources.

Elementary water waves

New elementary farfield and nearfield time-harmonic water waves, observed from a Galilean reference frame, are given. These two related classes of waves are modifications of classical elementary plane waves that are obtained in a simple way, using only elementary fundamental considerations, by analyzing waves that slowly grow from rest at time T = −∞ and are bounded in the farfield. This approach circumvents the need for a radiation condition, which may indeed be regarded as (indirectly) satisfied ab initio by the elementary farfield and nearfield waves given here. The farfield waves are the product of classical elementary waves by a function, called radiation function, that is defined explicitly in terms of the dispersion function. Thus, this radiation function is valid not only for water waves, but more generally for a broad class of linear dispersive waves. For illustration and verification purposes, elementary stationary-phase considerations are used to determine the main properties (phase and group velocities, wave patterns, asymptote and cusp lines) of farfield waves, and two particular classes of water waves—steady ship waves and time-harmonic waves generated by an offshore structure—in uniform finite water depth are considered. The elementary nearfield waves can readily be used to construct free-surface Green functions or in a spectral representation of nearfield flows.

Hydroelastic analysis of cantilever plate in time domain

Numerical solutions for the hydroelastic problems of bodies are studied directly in the time domain using Neumann–Kelvin formulation. In the hydrodynamic part of problem, the exact initial boundary value problem is linearized using the free stream as a basis flow, replaced by the boundary integral equation applying Green theorem over the transient free surface Green function. The resultant boundary integral equation is discretized using quadrilateral elements over which the value of the potential is assumed to be constant and solved using the trapezoidal rule to integrate the memory or convolution part in time. In the structure part of the problem, the finite element method is used to solve the hydroelastic problem. The Mindlin plate as a bending element, which includes transverse shear effect and rotary inertia effect are used. The present numerical results show acceptable agreement with experimental, analytical, and other published numerical results.

A higher-order-coupled boundary element and finite element method for the wave forcing of a floating elastic plate

We present a higher-order method to calculate the motion of a floating, shallow draft, elastic plate of arbitrary geometry subject to linear wave forcing at a single frequency. The solution is found by coupling the boundary element and finite element methods. We use the same nodes, basis functions, and maintain the same order in both methods. Two equations are derived that relate the displacement of the plate and the velocity potential under the plate. The first equation is derived from the elastic plate equation. The discrete version of this equation is very similar to the standard finite element method elastic plate equation except that the potential of the water is included in a consistent manner. The second equation is based on the boundary integral equation which relates the displacement of the plate and the potential using the free-surface Green function. The discrete version of this equation, which is consistent with the order of the basis functions, includes a Green matrix that is analogous to the mass and stiffness matrices of the classical finite element method for an elastic plate. The two equations are solved simultaneously to give the potential and displacement. Results are presented showing that the method agrees with previous results and its performance is analysed.

The eigenfunction expansion of the infinite depth free surface Green function in three dimensions

A new representation of the infinite depth free surface Green function in three dimensions is derived. This representation is in the eigenfunction expansion of an outgoing wave centred at the source point of the Green function. Such a representation allows the calculation of the scattered potential in terms of the eigenfunctions of an outgoing wave. Furthermore this new representation of the Green function is found to compare favourably with existing representations in terms of its naive numerical evaluation. We also show that the eigenfunction representation can be retained in a new coordinate system which is not centred at the source point of the Green function.

3D free‐surface flow computation using a RANSE/Fourier–Kochin coupling

AbstractA coupling method for numerical calculations of steady free‐surface flows around a body is presented. The fluid domain in the neighbourhood of the hull is divided into two overlapping zones. Viscous effects are taken in account near the hull using Reynolds‐averaged Navier–Stokes equations (RANSE), whereas potential flow provides the flow away from the hull. In the internal domain, RANSE are solved by a fully coupled velocity, pressure and free‐surface elevation method. In the external domain, potential‐flow theory with linearized free‐surface condition is used to provide boundary conditions to the RANSE solver. The Fourier–Kochin method based on the Fourier–Kochin formulation, which defines the velocity field in a potential‐flow region in terms of the velocity distribution at a boundary surface, is used for that purpose. Moreover, the free‐surface Green function satisfying this linearized free‐surface condition is used. Calculations have been successfully performed for steady ship‐waves past a serie 60 and then have demonstrated abilities of the present coupling algorithm. Copyright © 2003 John Wiley & Sons, Ltd.

Spectral Shell and Perfectly Transparent Open-Boundary Condition for Unsteady Wave-Body Interactions

This work, presented in the OMAE-2002 Special Symposium in honor of John V. Wehausen, is dedicated to him, an admirable teacher, colleague, and scholar. A general open boundary condition for time-dependent wave-body interaction problems is developed. The technique is based on domain decomposition and a use of the unsteady free-surface Green function in an outer region. The boundary condition results in a shell condition which may be applied to a variety of interior solution methods. When the interior solution method is a boundary-integral method, an implicit matching can be done. When a volume-discretization method is used in the interior region, a newly developed explicit matching procedure is performed. Demonstrations of the transparent properties of this shell are made for several types of unsteady problems.

Application and Verification of Deepwater Green Function for Water Waves

An efficient method for the numerical evaluation of the free-surface Green function for deep-water application was presented by Telste & Noblesse (1986) and again by Ponizy et al (1994). A FORTRAN code was include in their 1986 paper. The numerical method makes use of known mathematical functions. Numerical values of some of these mathematical functions were depicted, but no verification on the accuracy of the Green function routine in an application was given. The purpose of this paper is to compare their numerical values in the near-field and far-field regions with other similar computation for the infinite-depth Green function. The results of the infinite-depth Green function are also compared with the results from the finite-depth Green function that is more time consuming. Based on the accuracy of the various methods, the regions of application of the efficient deepwater Green function formulation and the effect of the water depth on the Green function are discussed. Any regions of inaccurate results are noted. Forces on submerged offshore structures and motions of the floating structures are determined using the lower-order panel method and these different Green function routines. The results on the motions of a semisubmersible in various water depths are compared and the accuracy of these routines in various regions is shown. The presented results will help the hydrodynamicist and designer to evaluate the suitable formulation and its regions of application for the accurate analysis and design of offshore structures.

A variational equation for the wave forcing of floating thin plates

A variational equation is presented for floating thin plates subject to wave forcing. This equation is derived from the thin plate equation of motion by including the wave forcing using the free-surface Green function. This variational equation combines the standard method for solving the motion of a thin plate (a variational equation) with the standard method for solving the wave forcing of a floating body (the free-surface Green function method). Solutions of the variational equation are presented for some simple thin plate geometries using polynomial basis functions. The variational equation is extended to the case of plates of variable properties and to multiple plates and further solutions are presented.

The Forward Speed Diffraction Problem

This paper examines the theory and computational methods behind predicting the linear unsteady motion of a ship with steady forward speed in waves. The focus is on the wave exciting force impulse-response function as computed via the transient free-surface Green function. The linear equation of motion for a ship in waves was first written in a rational form, using the concept of the impulse-response function, by Cummins (1962). Some years later King et al (1988) added the corresponding wave exciting force in its appropriate convolution form. We extend this work by clarifying the definition of the impulsive incident wave in following seas, and show it to be easily computable. Continuing truncated calculations towards infinite time becomes especially important in following waves, and the method suggested by Bingham et al (1994) is employed here. A novel filtering scheme is also introduced to prevent short wave contamination of the solution. These developments allow calculations in following waves to be presented for the first time using this approach. The integral equation formulation of the linear seakeeping problem is reviewed in some detail, and the relevant equations derived. Transient Haskind relations for bodies with forward speed are also derived although, like their frequency-domain counterparts, these are only approximate. Computed, first-order exciting forces and response-amplitude operators for real ship geometries, in head and following seas, are presented that demonstrate the usefulness of the transient approach for the diffraction problem.

Numerical Evaluation of Free-Surface Green Functions

A very simple and efficient method for computing the nonoscillatory near-field terms in the expressions for the Green functions, and their gradients, for wave diffraction/radiation by an offshore structure and steady ship waves in deep water is presented. The Green functions are decomposed into three terms corresponding to simple (Rankine) singularities, wave fields, and nonoscillatory near-field (local) flow components. The method which is presented for approximating the latter nonoscillatory near-field components is based on the use of a coordinate-transformation and a function-transformation. The coordinate-transformation maps the unbounded domain of definition of the Green function into a finite domain (unit square or cube) of transformed coordinates. The function-transformation expresses the near-field components, which are singular at the origin, in terms of functions that are regular everywhere. Proper coordinate and function transformations reduce the problem of approximating singular functions in unbounded domains into that of approximating smoothly varying functions within finite domains. The latter task can be accomplished in a number of ways, including the use of linear table interpolation presented in the study.

On the free-surface Green's function for channel problems

New expressions are derived for the free-surface Green's function appropriate to channel problems. Both finite and infinite depth are considered. The infinite series contained in these new representations converge much more rapidly than the series that are obtained if a simple set of images is used to account for the channel walls, but the individual terms in the series are more complicated.