Transient Free Surface Green Function Research Articles

A Fully Coupled Time-Domain BEM-FEM Method for the Prediction of Symmetric Hydroelastic Responses of Ships with Forward Speed

This paper presents a direct time-domain method for the prediction of symmetric hydroelastic responses of ships progressing with forward speed in small amplitude waves. A transient time-domain free surface Green function is used for the idealisation of the seakeeping problem using an Earth fixed coordinate system. Free surface ship hydrodynamics are idealised in the time domain by a Green function, and forward speed effects are idealised by a space-state model. Modal actions are accounted for by Timoshenko beam structural dynamics. Flexible fluid structure interaction (FFSI) coupling is enabled by a body boundary condition, and a direct integration Newmark- β scheme is used to obtain symmetric dynamic responses. The method is validated against available published numerical and experimental results. A parametric study for different container ship hull forms confirms that (i) forward speed effects should be taken under consideration as far as practically possible and (ii) hull flexibility effects accounting for hull shear deformation and rotary inertia are more notable for slender hull forms.

Time domain TEBEM method of ship motion in waves with forward speed by using impulse response function formulation

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is proposed. The flow field is divided into inner and outer domain by an artificial matching surface. The inner domain includes ship-wetted surface, matching surface and free surface inside. The free-surface condition is formulated by perturbation about the double-body flow (DB model) or uniform incoming flow assumption (NK model). The outer domain contains matching surface, free surface outside and infinite far-field boundary, where only NK model is applied. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) by the impulse response function method for the inner and outer domain. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the BIEs. Matching conditions for the inner-domain and outer-domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. In this paper, several ships are selected to predict the hydrodynamic coefficients and displacement at different forward speed. The experiment measurements in physical tank is also used to validate the accuracy of numerical results. Good agreement of 6DOF ship motions in various wave headings can be obtained using TEBEM method.

Time-domain TEBEM method for mean drift force and moment of ships with forward speed under the oblique seas

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves is put forward. The flow field is divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain is surrounded by ship wetted surface and matching surface as well as part of the free surface. The free surface condition for the inner domain is formulated by perturbation about the double-body flow or uniform incoming flow assumption. The outer domain is surrounded by matching surface and the rest free surface as well as infinite far-field radiation boundary. The free surface condition for the outer domain is formulated by perturbation about uniform incoming flow. The simple Green function and transient free surface Green function are used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. Taylor Expansion Boundary Element Method (TEBEM) is utilized to solve the double-body flow and inner domain and outer domain unsteady flow BIE. Matching conditions for the inner domain flow and outer domain flow are enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on ship wetted surface is used to obtain the first-order and second-order wave forces (moments). The numerical predictions on the displacement, added resistance, sway mean drift force and yaw mean drift moment of the modified KVLCC2 ship at different forward speeds are investigated by the proposed TEBEM method. It is also compared with the other numerical results. The physical tank experiment results are also developed to validate the accuracy of numerical tank results. Compared with the experiment solutions, a good agreement can be obtained by TEBEM method.

Hydrodynamic Analysis and Motions of Ship with Forward Speed via a Three-Dimensional Time-Domain Panel Method

A new three-dimensional (3D) time-domain panel method is developed to solve the ship hydrodynamic problem and motions. For an advancing ship with a constant forward speed in regular waves, the ship’s hull can be discretized and processed into a number of quadrilateral panels. Based on Green’s theorem, an analytical expression for Froude–Krylov (F–K) forces evaluation on the quadrilateral panels is derived without accuracy loss. Within the linear potential theory, the transient free surface Green function (TFSGF) is applied to solve the boundary value problem. To improve the efficiency and numerical stability of TFSGF evaluation, a precise integration method with variable parameters setting for extended identity matrix is developed to compute the TFSGF in the computation domain. Then, radiation and diffraction forces can be evaluated by means of the impulse response function method. The Wigley I hull form is taken as a study case, and the computed hydrodynamic coefficients, wave exciting forces, and motions by the present method are compared with previous literature experimental data and prior published results. It manifests that the three-dimensional time-domain panel method proposed in this paper has good accuracy.

A Control strategy to improve the efficiency of point absorber wave energy converters in complex sea environments

A discrete control of latching is used to increase the bandwidth of the efficiency of the Wave Energy Converters (WEC) in regular and irregular seas. When latching control applied to WEC 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 vertical-cylinder as a point-absorber WEC is used for the numerical prediction of the different parameters. The absorbed maximum power from the sea is achieved with a three-dimensional panel method using Neumann-Kelvin approximation in which 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 calculated response amplitude operator, absorbed power, relative capture width, and efficiency of vertical-cylinder compared with analytical results.

Time-domain TEBEM method for wave added resistance of ships with forward speed

A numerical method for solving 3D unsteady potential flow problem of ship advancing in waves was put forward. The flow field was divided into an inner and an outer domain by introducing an artificial matching surface. The inner domain was surrounded by a ship-wetted surface and a matching surface as well as part of the free surface. The free-surface condition for the inner domain was formulated by perturbation about the double-body (DB) flow assumption. The outer domain was surrounded by a matching surface and the rest had a free surface as well as an infinite far-field radiation boundary. The free-surface condition for the outer domain was formulated by perturbation of the uniform incoming flow. The simple Green function and transient free-surface Green function were used to form the boundary integral equation (BIE) for the inner and outer domains, respectively. The Taylor expansion boundary element method (TEBEM) was adopted to solve the DB flow and inner-domain and outer-domain unsteady flow BIE. Matching conditions for the inner-domain flow and outer-domain flow were enforced by the continuity of velocity potential and normal velocity on the matching surface. Direct pressure integration on the ship-wetted surface was applied to obtain the first- and second-order wave forces. The numerical prediction on the displacement, acceleration and added resistance of the 14000-TEU container ship at different forward speeds were investigated by the proposed TEBEM method. Reynolds-Averaged Navier–Stokes (RANS) equations based on Computational Fluid Dynamics (CFD) method were adopted to compare with TEBEM method. The physical tank experiment results also validated the accuracy of the numerical tank results. Compared with the experimental solutions, TEBEM obtained good agreement with the RANS CFD method. TEBEM, however, was much more efficient and robust.

Prediction of Ship Motions Via a Three-Dimensional Time-Domain Method Following a Quad-Tree Adaptive Mesh Technique

Abstract A three-dimensional (3D) time-domain method is developed to predict ship motions in waves. To evaluate the Froude-Krylov (F-K) forces and hydrostatic forces under the instantaneous incident wave profile, an adaptive mesh technique based on a quad-tree subdivision is adopted to generate instantaneous wet meshes for ship. For quadrilateral panels under both mean free surface and instantaneous incident wave profiles, Froude-Krylov forces and hydrostatic forces are computed by analytical exact pressure integration expressions, allowing for considerably coarse meshes without loss of accuracy. And for quadrilateral panels interacting with the wave profile, F-K and hydrostatic forces are evaluated following a quad-tree subdivision. The transient free surface Green function (TFSGF) is essential to evaluate radiation and diffraction forces based on linear theory. To reduce the numerical error due to unclear partition, a precise integration method is applied to solve the TFSGF in the partition computation time domain. Computations are carried out for a Wigley hull form and S175 container ship, and the results show good agreement with both experimental results and published results.

Froude–Krylov nonlinear computations of three dimensional wave loads by a hybrid time domain boundary element method

Wave-induced loads on advancing ships are studied by using a three dimensional nonlinear hybrid high order boundary element method with the combination of the Rankine source Green function applied in the internal domain and the transient free surface Green function adopted in the external domain. A quadratic representation of both the boundary and the unknowns on the boundary is adopted. In the internal domain the forward speed problem is solved with the steady ship wave as the basis flow. In the external domain, an efficient algorithm combined with Chebyshev approximations and asymptotic expressions for the transient free surface Green function is adopted to the solution of radiation condition. The nonlinear Froude–Krylov and hydrostatic restoring forces resulted from large amplitude incident waves are taken into consideration over the instantaneous wetted hull surface. This paper presents an effective solution to the problems of excessive memory occupation and large time consuming in the long-duration simulation. A nonlinear numerical program is originally developed and the preliminary linear motions of Wigley-3 and S175 container hull at relatively high speed are analyzed. Furthermore, the nonlinear behavior of hydrodynamic forces, motions, and the wave loads of S175 hull are investigated in regular and irregular seas, especially the phenomenon of ‘super harmonics’ in severe regular waves are highlighted thoroughly. Comparisons with experiment results indicate the present method owns satisfactory numerical stability, accuracy, and efficiency for the practical application.

Comparative study on ship motions in waves based on two time domain boundary element methods

This paper presents a comparative study on the motions of a ship advancing in waves using two different three-dimensional time domain boundary element methods: a Transient Free surface Green's Function (TFGF) method and a Rankine Higher Order Boundary Element Method (HOBEM). Three models based on the HOBEM are also considered to identify the dominant factors affecting the accuracy of solutions: (i) the N–K model using Neumann–Kelvin (N–K) linearization, (ii) the N–K+DB model using the N–K linearization for the free surface boundary conditions and the double-body (DB) m-terms in the body boundary condition, (iii) the Rankine HOBEM based on DB linearization. The Wigley I and the Series 60 (CB = =0.7) are taken as study objects. The comparisons show that the Rankine HOBEM based on DB linearization is generally more accurate than the Rankine HOBEM with other two models and the TFGF method. The hydrodynamic coefficients are mainly affected by the m-terms, especially at low frequencies; while the influences of different linearizations of the free surface boundary conditions are negligible. Both the m-terms and the leading terms of steady velocity potential reserved in the free surface boundary conditions are important for motion responses, while the influence of the former is stronger than the latter.

Prediction of added resistance of a ship in waves at low speed

This paper presents predictions of the added resistance of a ship in waves at a low speed according to the IMO minimum propulsion power requirement by a hybrid Taylor expansion boundary element method (TEBEM). The flow domain is divided into two parts: the inner domain and the outer domain. The first-order TEBEM with a simple Green function is used for the solution in the inner domain and the zero order TEBEM with a transient free surface Green function is used for the solution in the outer domain. The TEBEM is applied in the numerical prediction of the motions and the added resistance in waves for three new designed commercial ships. The numerical results are compared with those obtained from the seakeeping model tests. It is shown that the prediction of the ship motions and the added resistance in waves are in good agreement with the experimental results. The comparison also indicates that the accuracy of the motion estimation is crucial for the prediction of the wave added resistance. In general, the TEBEM enjoys a satisfactory accuracy and efficiency to predict the added resistance in waves at a low speed according to the IMO minimum propulsion power requirement.

An efficient algorithm with new residual functions for the transient free-surface green function in infinite depth

Efficient evaluation of the transient free-surface Green function with sufficient accuracy is the key to employ time-domain panel method in solving hydrodynamic problems. In this paper, series and asymptotic expansions for the Green function and its two spatial derivatives are deduced in detail for the evaluation. New residual functions are derived straightforwardly from the resulted asymptotic formulas and it is concluded that the present residual functions decay rapidly than existing choices. The ultimate algorithm is the combination of Chebyshev approximations and asymptotic expressions, along with a special partitioning line introduced to speed up the evaluation. After comprehensive numerical validation, it is pointed out that the present algorithm owns at least 6 digits of accuracy and sufficient efficiency for practical application. Any efficient method for evaluating the Green function will use the asymptotic expansions. Furthermore, it should be pointed out that the present residual functions can also be conveniently applied to the scheme of interpolation in a precomputed table.

Fully coupled time domain solution for hydroelastic analysis of a floating body

A numerical method is developed to study the ship hydroelasticity problem in time domain. Boundary – Element Method (BEM) with three-dimensional transient free-surface Green function and Neumann – Kelvin approximation is used for the solution of the hydrodynamic part. The structural domain is modeled by finite element method (FEM) using 1D beam element in time domain. The coupling between BEM and FEM is made through body boundary condition in the hydrodynamic solution. Furthermore, direct integration scheme, i.e. Newmark –β method is used for obtaining the structural velocity and deformation. The efficiency and robustness of the proposed method are validated with available published results. Three different barge type structures and a large container ship are taken for the analysis. A good correspondence is observed between the proposed and reported numerical and experimental results. Also, it is observed that structural flexibility plays an important role while calculating structural deflection, shear force, bending moment etc. The developed numerical method seems to be robust, efficient and a very useful tool in predicting the hydrodynamic loads, structural deflections, shears force, bending moment etc. for flexible structure.

Time domain Rankine-Green panel method for offshore structures

To solve the numerical divergence problem of the direct time domain Green function method for the motion simulation of floating bodies with large flare, a time domain hybrid Rankine-Green boundary element method is proposed. In this numerical method, the fluid domain is decomposed by an imaginary control surface, at which the continuous condition should be satisfied. Then the Rankine Green function is adopted in the inner domain. The transient free surface Green function is applied in the outer domain, which is used to find the relationship between the velocity potential and its normal derivative for the inner domain. Besides, the velocity potential at the mean free surface between body surface and control surface is directly solved by the integration scheme. The wave exciting force is computed through the convolution integration with wave elevation, by introducing the impulse response function. Additionally, the nonlinear Froude-Krylov force and hydrostatic force, which is computed under the instantaneous incident wave free surface, are taken into account by the direct pressure integration scheme. The corresponding numerical computer code is developed and first used to compute the hydrodynamic coefficients of the hemisphere, as well as the time history of a ship with large flare; good agreement is obtained with the analytical solutions as well as the available numerical results. Then the hydrodynamic properties of a FPSO are studied. The hydrodynamic coefficients agree well with the results computed by the frequency method; the influence of the time interval and the truncated time is investigated in detail.

Control of Wave Energy Converters for Maximum Power Absorption with Time Domain Analysis

A discrete control of latching is used to increase the bandwidth of the efficiency of the Wave Energy Converters (WEC) in regular and irregular seas. When latching control applied to WEC 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 vertical-cylinder as a point-absorber WEC is used for the numerical prediction of the different parameters. The absorbed maximum power from the sea is achieved with a three-dimensional panel method using Neumann-Kelvin approximation in which 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 calculated response amplitude operator, absorbed power, relative capture width, and efficiency of vertical-cylinder compared with analytical results.

A note on the relative efficiency of methods for computing the transient free-surface Green function

A number of papers have appeared recently on computing the time-domain, free-surface Green function. Two papers in particular, Chuang et al. (2007) and Li et al. (2015) considered the method developed by Clement (1998) who showed that this Green function is the solution to a fourth-order Ordinary Differential Equation (ODE). This ODE has been suggested as a means for speeding up the calculation of the Green function coefficients compared to the standard algorithms developed for example by Newman (1992). Clement solved the ODE using the classical fourth-order, four-step Runge–Kutta scheme (RK44) with a fixed time step size. The two papers mentioned above proposed alternative numerical methods which are claimed to be more efficient. In this note we consider the relative efficiency of these four methods on a representative test case, and conclude that the standard method is the most efficient. Of the ODE-based methods, the method of Chuang et al. (2007) is found to be slightly more efficient than the RK44 method, while the method of Li et al. (2015) is at least an order of magnitude less efficient. It is also pointed out that ODE methods have yet to be extended to include finite water depth.

Time domain prediction of seakeeping behaviour of catamarans

The numerical predictions of the seakeeping characteristics of catamarans with and without forward speed are presented using a direct time domain approximation. Boundary-Integral Equation Method (BIEM) with three-dimensional transient free surface Green function and Neumman–Kelvin linearization is used for the solution of the hydrodynamic interactions in the gap between catamaran hulls and solved as impulsive velocity potential. The numerical results have shown that the response of the fluid between multi-hulls can affect seakeeping performance and influence the hydrodynamic forces and the motion amplitude operators of the floating multi-bodies. It is also shown numerically if the critical reduced frequency is smaller than the ratio of vessel length to hull separation, the wave interactions between hulls are very significant due to trapped waves in the gap of twin-hull. A hemi-spheroid catamaran, Wigley catamaran, and DUT high speed catamaran hull forms are used for the numerical analyses and the comparisons of the present ITU-WAVE numerical results for added-mass, damping coefficients, exciting force, and response amplitude operators show satisfactory agreement with existing numerical and experimental results.

A precise computation method of transient free surface Green function

The transient free surface Green function (TFSGF) is of great importance in the prediction of unsteady ship motions with forward speed. In this numerical approach, the wave part of the TFSGF and its spatial derivatives are obtained by solving the fourth-order ordinary differential equations (ODEs) based on the semi-analytical Precise Integration Method (PIM). Theoretical derivations show that in fact the horizontal and vertical derivatives can be expressed by TFSGF, which means only one ODE needs to be solved. The stability and accuracy of this method is demonstrated by the comparison with other method as well as the analytical solutions. Additionally, the proposed method is applied to solve the radiation problem of a floating hemisphere at zero speed using the time domain Rankine–Green method. The numerical hydrodynamic coefficients show good agreement with the analytical solutions.

Time domain prediction of hydroelasticity of floating bodies

The numerical predictions of the hydroelasticity of floating bodies with and without forward speed are presented using a direct time domain approximation. Boundary-Integral Equation Method (BIEM) with three-dimensional transient free surface Green function and Neumman–Kelvin approximation is used for the solution of the hydrodynamic part and solved as impulsive velocity potential whilst Euler–Bernoulli beam approach is used for the structural analysis with analytically defined modeshapes. The hydrodynamic and structural parts are then fully coupled through modal analysis for the solution of the hydroelastic problem. A stiff structure is then studied assuming that contributions of rigid body modes are much bigger than elastic modes. A rectangular barge with zero speed and Wigley hull form with forward speed are used for the numerical analyses and the comparisons of the present Istanbul Technical University (ITU)-WAVE numerical results for response amplitude operator, bending moment, shear force, etc. show satisfactory agreement with existing experimental results.

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.

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.